Index theory in nonlinear analysis 9811372861, 9789811372865

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Chungen Liu

Index theory in nonlinear analysis

Index theory in nonlinear analysis

Chungen Liu

Index theory in nonlinear analysis

123

Chungen Liu School of Mathematical and Information Science Guangzhou University Guangzhou, Guangdong, China

ISBN 978-981-13-7286-5 ISBN 978-981-13-7287-2 (eBook) https://doi.org/10.1007/978-981-13-7287-2 Mathematics Subject Classification: 37J45, 34C25, 53D12, 58J20 © Springer Nature Singapore Pte Ltd. 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Foreword

This book contains three aspects of index theories with applications. From them, we choose one aspect to demonstrate how to use the L-index theory to study the multiplicity of brake orbits on a symmetric convex hypersurface. We begin from a famous Seifert conjecture about the number of brake orbits on a compact hypersurface in R2n .

1 Seifert Conjecture Let us recall the famous conjecture proposed by H. Seifert in his pioneer work [268] in 1948 concerning the multiplicity of brake orbits of certain Hamiltonian systems in R2n . We assume H ∈ C 2 (R2n , R) possesses the following form: H (p, q) =

1 A(q)p · p + V (q), 2

(1)

where p, q ∈ Rn , A(q) is a positive definite n×n symmetric matrix for any q ∈ Rn , A is C 2 , V ∈ C 2 (Rn , R) is the potential energy. The solution of the following problem for Hamiltonian system x˙ = J H  (x), x = (p, q), τ p(0) = p( ) = 0. 2

(2) (3)

is called a brake orbit. Moreover, if h is the total energy of a brake orbit (q, p), ¯ i.e., H (p(t), q(t)) = h, then V (q(0))  = V (q(τ  )) = h and q(t) ∈  ≡ {q ∈ 0 −I Rn |V (q) ≤ h} for all t ∈ R, where J = is the standard symplectic matrix I 0 and I is the n × n identity matrix. v

vi

Foreword

In [268] of 1948, H. Seifert studied the existence of brake orbit for system (2)–(3) with the Hamiltonian function H in the form of (1) and proved that the set Jb () of brake orbits on the energy surface  = H −1 (h) is not empty, i.e., Jb () = ∅, ¯ is bounded and homeomorphic to the provided V  = 0 on ∂, V is analytic, and  n n unit ball B1 (0) in R . The set of geometrically distinct brake orbits on the energy surface  is denoted by J˜b (). Then, in the same paper, he proposed the following conjecture which is still open for n ≥ 2 now: #

J˜b () ≥ n under the same conditions.

We note that for the Hamiltonian function, 1 2  2 2 |p| + aj qj , 2 n

H (p, q) =

q, p ∈ Rn ,

j =1

/ Q for all i = j and q = (q1 , q2 , . . . , qn ). There are exactly n where ai /aj ∈ geometrically distinct brake orbits on the energy hypersurface  = H −1 (h).

2 The Generalized Seifert Conjecture in Linear Symplectic Space 2n For the standard symplectic  space  (R , ω0 ) with ω0 (x, y) = J x, y , an involution −I 0 matrix defined by N = is clearly anti-symplectic, i.e., N J = −J N . The 0 I fixed point set of N and −N are the Lagrangian subspaces L0 = {0} × Rn and L1 = Rn × {0} of (R2n , ω0 ), respectively. In general, we suppose H ∈ C 2 (R2n \{0}, R)∩C 1 (R2n , R) satisfies the following reversible condition:

H (Nx) = H (x),

∀ x ∈ R2n .

(4)

We consider the following fixed energy problem of nonlinear Hamiltonian system with Lagrangian boundary conditions x(t) ˙ = J H  (x(t)), H (x(t)) = h, x(0) ∈ L0 , x(τ/2) ∈ L0 .

(5) (6) (7)

Here, we require that h is a regular value of the function H . It is clear that a solution (τ, x) of (5)–(7) is a characteristic chord on the contact submanifold  := H −1 (h) = {y ∈ R2n | H (y) = h} of (R2n , ω0 ) and satisfies

Foreword

vii

x(−t) = N x(t),

(8)

x(τ + t) = x(t).

(9)

In general, this kind of τ -periodic characteristic (τ, x) is called a brake orbit on the hypersurface . We note that the problem (2)–(3) with the Hamiltonian function H defined in (1) is a special case of the problem (5)–(7). We denote by Jb (, H ) the set of all brake orbits on . Two brake orbits (τi , xi ) ∈ Jb (, H ), i = 1, 2 are equivalent if the two brake orbits are geometrically the same, i.e., x1 (R) = x2 (R). We denote by [(τ, x)] the equivalence class of (τ, x) ∈ Jb (, H ) in this equivalence relation and by J˜b (, H ) the set of [(τ, x)] for all (τ, x) ∈ Jb (, H ). In fact, J˜b (, H ) is the set of geometrically distinct brake orbits on , which is independent on the choice of H . So from now on, we simply denote it by J˜b (), and in the notation [(τ, x)], we always assume x has minimal period τ . We also denote by J˜ () the set of all geometrically distinct closed characteristics on . The number of elements in a set S is denoted by # S. It is well-known that # J˜b () (and also # J˜ ()) is only depending on , that is to say, for simplicity, we take h = 1; if H and G are two C 2 functions satisfying (4) and H := H −1 (1) = G := G−1 (1), then # Jb (H ) =# Jb (G ). So we can consider the brake orbit problem in a more general setting. Let  be a C 2 compact hypersurface in R2n bounding a compact set C with nonempty interior. Suppose  has nonvanishing Gaussian curvature and satisfies the reversible condition N( − x0 ) =  − x0 := {x − x0 |x ∈ } for some x0 ∈ C. Without loss of generality, we may assume x0 = 0. We denote the set of all such hypersurfaces in R2n by Hb (2n). For x ∈ , let n (x) be the unit outward normal vector at x ∈ . Note that, here, by the reversible condition, there holds n (N x) = N n (x). We consider the dynamics problem of finding τ > 0 and a C 1 smooth curve x : [0, τ ] → R2n such that x(t) ∈ ,

x(t) ˙ = J n (x(t)), x(−t) = N x(t),

x(τ + t) = x(t),

(10) for all t ∈ R.

(11)

A solution (τ, x) of the problem (10)–(11) determines a brake orbit on . Now, the generalized Seifert conjecture can be represented as Generalized Seifert conjecture: For any  ∈ Hb (2n), there holds #

J˜b () ≥ n.

We can take the above estimate as a result on the number of Lagrangian intersection (more precisely the Legendre intersection since  ∩ L0 is a Legendre submanifold of the contact manifold ) of “reversible” Hamiltonian map ϕb : #

{ ∩ L0 ∩ ϕb (L0 )} ≥ n.

The famous Arnold conjecture has relation with the Lagrangian boundary problem, which says that the number of Lagrangian intersection of a Hamiltonian map on a

viii

Foreword

closed symplectic manifold M can be estimated from below by the Betti number of M in the nondegenerate case and by the cuplength of M (cf. [8]). In this direction, one can refer to [100, 137], and [191]. We denote by Hbc (2n) = { ∈ Hb (2n)|  is strictly convex }, Hbs,c (2n) = { ∈ Hbc (2n)| −  = }. For the multiplicity of the brake orbits on a symmetrical compact convex hypersurface, we have the following result: Theorem A For any  ∈ Hbs,c (2n), there holds #

J˜b () ≥ n.

3 L-Index Theory Historically, as far as the author knows, the classification and an index theory for linear Hamiltonian systems with periodic boundary condition began with the work of H. Amann and E. Zehnder in their paper [5] of 1980. They established the corresponding index theory for linear Hamiltonian systems with constant coefficients. In their celebrated paper [57] of 1984, C. Conley and E. Zehnder defined an index theory for any nondegenerate path in symplectic group  Sp(2n) =  0 −I n {M ∈ GL(R2n )| M T J M = J } with n ≥ 2, where J = . This index In 0 was further defined for the nondegenerate paths in Sp(2) by Y. Long and E. Zehnder in [231] of 1990. The index theory for degenerate linear Hamiltonian systems was defined by Y. Long in [224] and C. Viterbo in [280] of 1990 independently. Then, in the paper[225] of 1997, this index theory was further extended to any symplectic paths, and an axiomatic characterization of the index theory was given by Y. Long. We now call this index the Maslov-type index. For a symplectic path γ starting from the identify matrix, we denote its Maslov-type index by (i1 (γ ), ν1 (γ )) ∈ Z × {0, 1, · · · , 2n}. In [226] of 1999, Y. Long established a generalized index function theory (iω (γ ), νω (γ )) parameterized by ω in the unit circle U of the complex plane C for every symplectic path γ starting from the identity matrix. The Maslov-type index theory for symplectic paths is essentially a classification of the periodic linear Hamiltonian systems. So it is natural to use this index theory to study the periodic solutions of a Hamiltonian system.

Foreword

ix

We denote the set of symplectic paths defined in the interval [0, τ ] by Pτ (2n) = {γ ∈ C([0, τ ], Sp(2n))|γ (0) = I2n }. In [232] of 2006, Y. Long, D. Zhang, and C. Zhu studied the multiple solutions of the brake orbit problem on a convex hypersurface; in this paper, they introduced two indices (μ1 (γ ), ν1 (γ )) and (μ2 (γ ), ν2 (γ )) for symplectic path γ ∈ Pτ (2n). After that, in [189] of 2007, C. Liu introduced an index theory associated with a Lagrangian subspace for symplectic paths. For a symplectic path γ ∈ Pτ (2n), and a Lagrangian subspace L, by definition, the L-index is assigned to a pair of integers (iL (γ ), νL (γ )) ∈ Z × {0, 1, · · · , n}. This index theory is suitable for studying the Lagrangian boundary value problems (L-solution, for short) related to nonlinear Hamiltonian systems. In [192], C. Liu applied this index theory to study the L-solutions of some asymptotically linear Hamiltonian systems. The indices μ1 (γ ) and μ2 (γ ) are essentially special cases of the L-index iL (γ ) for Lagrangian subspaces L0 = {0} × Rn and L1 = Rn × {0}, respectively, up to a constant n. The key ingredients in the study of the multiplicity of brake orbits on a symmetric convex hypersurface are the index iteration formulas (Bott-type formulas) and the (L0 , L1 )-estimate. For simplicity, we suppose γ ∈ P(2n), i.e., we take τ = 1. For j ∈ N, we define the j -times iteration path γ j : [0, j ] → Sp(2n) of γ by γ 1 (t) = γ (t), t ∈ [0, 1],  γ 2 (t) =

γ (t), t ∈ [0, 1], Nγ (2 − t)γ (1)−1 Nγ (1), t ∈ [1, 2],

and in general, for k ∈ N, we define γ (2) = Nγ (1)−1 N γ (1) and ⎧ ⎪ γ (t), t ∈ [0, 1], ⎪ ⎪ ⎪ ⎪ ⎨ Nγ (2 − t)γ (1)−1 Nγ (1), t ∈ [1, 2], γ 2k−1 (t) = · · · · · · ⎪ ⎪ ⎪ Nγ (2k − 2 − t)Nγ (2)k−1 , t ∈ [2k − 3, 2k − 2], ⎪ ⎪ ⎩ γ (t − 2k + 2)γ (2)k−1 , t ∈ [2k − 2, 2k − 1], ⎧ ⎪ γ (t), t ∈ [0, 1], ⎪ ⎪ ⎪ ⎪ ⎨ Nγ (2 − t)γ (1)−1 Nγ (1), t ∈ [1, 2], 2k γ (t) = · · · · · · ⎪ ⎪ ⎪ γ (t − 2k + 2)γ (2)k−1 , t ∈ [2k − 2, 2k − 1], ⎪ ⎪ ⎩ Nγ (2k − t)Nγ (2)k , t ∈ [2k − 1, 2k].

(12)

(13)

x

Foreword

For γ ∈ Pτ (2n), we define γ k (τ t) = γ˜ k (t) with γ˜ (t) = γ (τ t).

(14)

Proposition B (mixed (L0 , L1 )-concavity estimate) For γ ∈ Pτ (2n), let P = γ (τ ). If iL0 (γ ) ≥ 0, iL1 (γ ) ≥ 0, i1 (γ ) ≥ n, γ 2 (t) = γ (t − τ )γ (τ ) for all t ∈ [τ, 2τ ], then iL1 (γ ) + SP+2 (1) − νL0 (γ ) ≥ 0.

(15)

+ (1) is the splitting number of the symplectic matrix M (see [226] and where SM [223]).

Proposition C (Bott-type formulas) Suppose γ ∈ Pτ (2n), for the iteration symplectic paths γ k defined in (12)–(14) above, when k is odd, there hold iL0 (γ k ) = iL0 (γ 1 ) +

k−1 2

νL0 (γ k ) = νL0 (γ 1 ) +

i=1 iωk2i (γ



2 ),

k−1 2

2 i=1 νω2i (γ ),

(16)

k

when k is even, there hold L0 (γ 1 ) + iL0 (γ k ) = iL0 (γ 1 ) + i√ −1

νL0 (γ k ) =

k2 −1

2 i=1 iωk2i (γ ),

k2 −1 L0 νL0 (γ 1 ) + ν√ (γ 1 ) + i=1 νω2i (γ 2 ), −1 k

(17)

√

where ωk = eπ −1/k and (iω (γ ), νω (γ )) is the ω index pair of the symplectic L0 L0 path γ introduced in [226] and the index pair (i√ (γ 1 ), ν√ (γ 1 )) was defined in −1 −1 [209]. For an L0 -solution (x, τ ) of a nonlinear Hamiltonian system x(t) ˙ = J H  (x(t)), x(0) ∈ L0 , x(τ ) ∈ L0 , the linearized system at x of the above nonlinear Hamiltonian system is a linear Hamiltonian system z˙ (t) = J B(t)z(t), B(t) = H  (x(t)). Its fundamental solution γx : [0, τ ] → Sp(2n) is a symplectic path starting from the identity matrix, so we have the L0 -index (iL0 (x), νL0 (x)) defined by (iL0 (x), νL0 (x)) = (iL0 (γx ), νL0 (γx )).

Foreword

xi

The k-times iteration (k ∈ N) of x denoted by (x k , kτ ) is also an L0 -solution of the above nonlinear Hamiltonian system; we write (iL0 (x k ), νL0 (x k )) as (iL0 (x, k), νL0 (x, k)). The formulas (16)–(17) help us to understand the indices of the iteration sequence x k . We will, in this book, establish some important properties of {(iL0 (x, k), νL0 (x, k))}k∈N and get a proof of Theorem A.

4 Index Information from Variational Method For  ∈ Hbs,c (2n), let j :  → [0, +∞) be the gauge function of  defined by j (0) = 0,

and

j (x) = inf{λ > 0 |

x ∈ C}, λ

∀x ∈ R2n \ {0},

where C is the domain enclosed by . Define H (x) == (j (x))2 , ∀x ∈ R2n .

(18)

Then, H ∈ C 2 (R2n \{0}, R) ∩ C 1,1 (R2n , R). We consider the following fixed energy problem: x(t) ˙ = J H (x(t)),

(19)

H (x(t)) = 1,

(20)

x(−t) = N x(t), x(τ + t) = x(t),

(21) ∀ t ∈ R.

(22)

Denote by Jb () the set of all solutions (τ, x) of problem (19)–(22) and by J˜b () the set of all geometrically distinct solutions of (19)–(22). For S 1 = R/Z, as in [232], we define the Hilbert space E by    1,2 1 2n  E = x ∈ W (S , R ) x(−t) = N x(t),



1

for all t ∈ R and

x(t)dt = 0 .

0

The inner product on E is given by (x, y) = 0

1

x(t), ˙ y(t) dt. ˙

(23)

xii

Foreword

The C 1,1 Hilbert manifold M ⊂ E associated with  is defined by   M = x ∈ E  

1 0



H∗ (−J x(t))dt ˙

1

= 1 and

J x(t), ˙ x(t) dt < 0 ,

(24)

0

where H∗ is the Fenchel conjugate function of the function H defined by H∗ (y) = max{(x · y − H (x))| x ∈ R2n }.

(25)

Let Z2 = {−id, id} be the usual Z2 group. We define the Z2 -action on E by −id(x) = −x,

id(x) = x,

∀x ∈ E.

Since H∗ is even, M is symmetric to 0, i.e., Z2 invariant. M is a paracompact Z2 -space. We define (x) =

1 2



1

J x(t), ˙ x(t) dt,

(26)

0

then is a Z2 invariant function and ∈ C ∞ (E, R). We denote by  the restriction of to M . Proposition D If # J˜b () < +∞, there is a sequence {ck }k∈N , such that − ∞ < c1 < c2 < · · · < ck < ck+1 < · · · < 0, ck → 0

as k → +∞.

(27) (28)

For any k ∈ N, there exists a brake orbit (τ, x) ∈ Jb () with τ being the minimal period of x and m ∈ N satisfying mτ = (−ck )−1 such that for z(x)(t) = (mτ )−1 x(mτ t) −

1 (mτ )2



mτ

x(s)ds,

t ∈ S1,

(29)

0

z(x) ∈ M is a critical point of  with  (z(x)) = ck and iL0 (x, m) ≤ k − 1 ≤ iL0 (x, m) + νL0 (x, m) − 1.

(30)

The above proposition says that for each critical value ck , k ∈ N, we have a brake orbit xk on  and an integer number mk such that its mk -times iteration with the index interval [iL0 (xk , mk ), iL0 (xk , mk ) + νL0 (xk , mk ) − 1] covers the positive integer number k − 1. This means that every positive integer number can be covered by this kind of index interval of an L0 -solution. So the indices of the L0 -solutions (xk , mk ) are widespread like the set N.

Foreword

xiii

5 Index Jumping Formulas In the following, we write (iL0 (γ , k), νL0 (γ , k)) = (iL0 (γ k ), νL0 (γ k )) for any symplectic path γ ∈ Pτ (2n) and k ∈ N, where γ k is defined by (12) and (13). We have the following jumping formulas for the L0 -index. For a symplectic path γ , its L0 -mean index i¯L0 (γ ) is defined by iL (γ , k) i¯L0 (γ ) = lim 0 . k→∞ k Proposition E Let γj ∈ Pτj (2n) for j = 1, · · · , q. Let Mj = γj2 (2τj ) = Nγj (τj )−1 N γj (τj ), for j = 1, · · · , q. Suppose i¯L0 (γj ) > 0,

j = 1, · · · , q.

Then there exist infinitely many (R, m1 , m2 , · · · , mq ) ∈ Nq+1 such that (i) (ii) (iii) (iv) (v)

νL0 (γj , 2mj ± 1) = νL0 (γj ), + iL0 (γj , 2mj − 1) + νL0 (γj , 2mj − 1) = R − (iL1 (γj ) + n + SM (1) − νL0 (γj )), j iL0 (γj , 2mj + 1) = R + iL0 (γj ), ν1 (γj2 , 2mj ± 1) = ν1 (γj2 ), + (1) − ν1 (γj2 )), i1 (γj2 , 2mj − 1) + ν1 (γj2 , 2mj − 1) = 2R − (i(γj2 ) + 2SM j

(vi) i1 (γj2 , 2mj + 1) = 2R + i1 (γj2 ), 2nj

where we have set i1 (γj2 , nj ) = i1 (γj

2nj

), ν1 (γj2 , nj ) = ν1 (γj

) for nj ∈ N.

Proposition F For any (τ, x) ∈ Jb (, 2) and m ∈ N, there hold iL0 (x, m + 1) − iL0 (x, m) ≥ 1,

(31)

iL0 (x, m + 1) + νL0 (x, m + 1) − 1 ≥ iL0 (x, m + 1) > iL0 (x, m) + νL0 (x, m) − 1. (32)

From Propositions D, E, and F, we can get that there are some L0 -solutions on  with their indices contributing densely as N, but in some sense, the jumping steps of the iterated indices of a L0 -solution are wide. So there should be enough geometrically distinct brake orbits (at least n) on . This is the main idea of the proof of Theorem A.

Preface

This monograph presents fundamental methods and topics in the relative Morse index theories and their efficient applications to some boundary value problems for Hamiltonian systems and partial differential equations. The Morse theory belongs to the categories of differential topology and nonlinear analysis. The finitedimensional Morse theory deals with the topological properties of functions defined on finite-dimensional manifolds. The infinite-dimensional Morse theory concerns the topological properties of functionals defined on infinite-dimensional topological spaces such as Banach manifolds or, specially, Hilbert manifolds. One of the differences between the finite-dimensional Morse theory and the infinite-dimensional one is that in the latter case, the Morse index may be infinite. When the Morse index of a critical point of the functional is finite, one can fully understand the topological properties of the functional. But in many nonlinear variational problems, the Morse index at a critical point of the functional is infinite, namely, the functional is strongly indefinite. For instance, the functionals, with respect to first-order Hamiltonian systems, are strongly indefinite. When the Morse index of a critical point of a functional is infinite, one should define the so-called relative Morse index to understand the topological properties of the functional. The Conley-Zehnder index theory or, more generally, the Maslov-type index theory for periodic linear Hamiltonian systems is such a relative Morse index theory. For a linear Hamiltonian system, its fundamental solution is a symplectic path starting from the identity. In 1984, C. Conley and E. Zehnder in their celebrated paper [57] introduced the socalled Conley-Zehnder index theory for the nondegenerate symplectic paths in the real symplectic matrix group Sp(2n) with n ≥ 2. In 1990, Y. Long and E. Zehnder in [231] generalized this index theory to the nondegenerate case with n = 1. Then, Y. Long in [224] and C. Viterbo in [280] further extended this index theory in 1990 independently to degenerate symplectic paths which are fundamental solutions of linear Hamiltonian systems. In [225], this index theory was further extended to continuous symplectic paths together with an axiomatic characterization for this index theory. In the literature after that, one usually calls this index the Maslovtype index. The book [223], written by Y. Long published in 2002, summarized the Maslov-type index theory for symplectic paths which are suitable for studying xv

xvi

Preface

periodic solutions of nonlinear Hamiltonian systems. The Maslov-type index theory for symplectic paths is essentially a classification of the periodic linear Hamiltonian systems. Besides the problem of periodic solutions of nonlinear Hamiltonian systems, the important problems are the canonical boundary value problems for nonlinear Hamiltonian systems, such as the Lagrangian submanifold boundary value problems (L-boundary value problems for short), which solve the Hamiltonian systems for solutions x satisfying x(0) ∈ L and x(τ ) ∈ L for Lagrangian submanifolds L and L , and the boundary value problems related with symplectic matrix P (the P -boundary value problems for short) which solve the Hamiltonian systems for solutions x satisfying x(τ ) = P x(0). These two kinds of boundary value problems were considered by A. Weinstein in [292–294]. As a sequel of the book [223], this book introduces some new index theories which are suitable for investigating some nonperiodic problems such as L-boundary value problems or P boundary value problems of nonlinear Hamiltonian systems. The main point of the topics in this book is to systematically deal with the nonperiodic solution problems (open-string problems) or the symmetric periodic solution problems of Hamiltonian systems. The aims of this book are: (1) to give an introduction to the index theory for symplectic paths with Lagrangian boundary conditions (L-index theory for short) and its iteration theory, which form a basis of Morse theoretical study about Hamiltonian systems with Lagrangian boundary conditions. The brake orbit problem is such a Lagrangian boundary problem. (2) to give an introduction to the index theory for symplectic paths with P boundary conditions (P -index theory for short) and its iteration theory, which form a basis of Morse theoretical study about Hamiltonian systems with P boundary conditions. A periodic solution of some delay differential equations (systems) can be viewed as a P -boundary solution of Hamiltonian systems. (3) to give an introduction to the index theory of abstract operator equations, where the essential spectrum of the linear self-adjoint operator possesses a gap on the real axis, and give applications of this theory to periodic and boundary value problems for some wave equations. (4) to establish the relations among various index theories. (5) to introduce some index theoretical methods in the study of nonlinear variational problems. This book is divided into 11 chapters, with the first two chapters as its foundation parts covering some materials of linear symplectic spaces and the Maslov-type index theory published in [223]. The main body of this book contains two parts. The first part are Chaps. 3, 4, 5, 6, and 7, covering the theoretical material-relative Morse index theory, P -index theory, L-index theory, Maslov-type index theory for paths of Lagrangian pair, and relations among these index theories. The second part are Chaps. 8, 9, 10, and 11, which contain some applications of these index theories to the minimal periodic problems and subharmonic solution problems of Hamiltonian systems with symmetry, multiplicity of brake orbits, existence, and multiplicity for periodic-boundary value problems of some wave equations.

Preface

xvii

The first three chapters are foundation of this book. If one wants to skip some chapters, we suggest that from Chap. 4, one can skip directly to Chap. 8; from Chap. 5, one can skip to Chaps. 9 and 10; and from Chap. 3, one can skip directly to Chap. 11. I am deeply indebted to Professor Yiming Long, who introduced me to this interesting area, for his constant encouragement and help. I have benefited a lot from his brilliant ideas and insights. I would also like to thank the National Natural Science Foundation of China, the 973 Program of the Science and Technology Ministry of China, and the Research Fund for the Doctoral Program of Higher Education of the Education Ministry of China for their support. I am grateful to Professors Long Chen and Yifeng Yu of the Mathematical Department of the University of California, Irvine, for their hospitality and help when I visited UCI from June to September 2016. During that period, this book was prepared. Nankai University, Tianjin, China Guangzhou University, Guangzhou, China July 2017

Chungen Liu

Contents

1

Linear Algebraic Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Linear Symplectic Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Symplectic Matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Lagrangian Subspaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Linear Hamiltonian Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Eigenvalues of Symplectic Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 12 17 19

2

A Brief Introduction to Index Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Maslov Type Index i1 (γ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 ω-Index iω (γ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23 24 28

3

Relative Morse Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Relative Index via Galerkin Approximation Sequences . . . . . . . . . . . . 3.2 Relative Morse Index via Orthogonal Projections . . . . . . . . . . . . . . . . . . 3.3 Morse Index via Dual Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 The Definition of Index Pair in Case 1 and 2. . . . . . . . . . . . . . 3.3.2 The Definition of Index Pair in Case 3 . . . . . . . . . . . . . . . . . . . . 3.4 Saddle Point Reduction for the General Cases. . . . . . . . . . . . . . . . . . . . . .

35 35 40 41 42 48 49

4

The P -Index Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 P -Index Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Relative Index via Saddle Point Reduction Method . . . . . . . . . . . . . . . . 4.3 Galerkin Approximation for the (P , ω)-Boundary Problem of Hamiltonian Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 (P , ω)-Index Theory from Analytical Point of View . . . . . . . . . . . . . . . 4.5 Bott-Type Formula for the Maslov Type P -Index . . . . . . . . . . . . . . . . . . 4.6 Iteration Theory for P -Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1 Splitting Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 Abstract Precise Iteration Formulas . . . . . . . . . . . . . . . . . . . . . . . 4.6.3 Iteration Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55 55 66 69 76 79 87 87 89 91

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Contents

The L-Index Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Definition of L-Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 The Properties of the L-Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 The Relations of iL (γ ) and i1 (γ ) . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 L-Index for General Symplectic Paths . . . . . . . . . . . . . . . . . . . . 5.2 The (L, L )-Index Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Understanding the Index i P (γ ) in View of the Lagrangian  Index iLL (γ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 The Relation with the Morse Index in Calculus Variations . . . . . . . . 5.5 Saddle Point Reduction Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Galerkin Approximation Formulas for L-Index . . . . . . . . . . . . . . . . . . . . 5.7 Dual L-Index Theory for Linear Hamiltonian Systems . . . . . . . . . . . . 5.8 The (L, ω)-Index Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9 The Bott Formulas of L-Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10 Iteration Inequalities of L-Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10.1 Precise Iteration Index Formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10.2 Iteration Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

121 122 127 134 140 144 147 156 156 158

6

Maslov Type Index for Lagrangian Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Lagrangian Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Maslov Type Index for a Pair of Lagrangian Paths . . . . . . . . . . . . . . . . . 6.3 Hörmander Index Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

161 161 164 171

7

Revisit of Maslov Type Index for Symplectic Paths . . . . . . . . . . . . . . . . . . . . 7.1 Maslov Type Index for Symplectic Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 The ω-Index Function for P -Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 The Concavity of Symplectic Paths and (ε, L0 , L1 )-Signature . . . . 7.4 The Mixed (L0 , L1 )-Concavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

177 177 179 180 211

8

Applications of P -Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 The Existence of P -Solution of Nonlinear Hamiltonian Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 The Existence of Periodic Solutions for Delay Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 M-Boundary Problem of a Hamiltonian System . . . . . . . . . 8.2.2 Delay Differential Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.3 Poisson Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.4 First Order Delay Hamiltonian Systems . . . . . . . . . . . . . . . . . . 8.2.5 Second Order Delay Hamiltonian Systems. . . . . . . . . . . . . . . . 8.2.6 Background and Related Works . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.7 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 The Minimal Period Problem for P -Symmetric Solutions . . . . . . . . .

219

5

95 95 101 107 111 117

219 222 222 224 226 228 230 231 231 236

Contents

9

10

11

Applications of L-Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 The Existence of L-Solutions of Nonlinear Hamiltonian Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 The Minimal Period Problem for Brake Solutions . . . . . . . . . . . . . . . . . 9.3 Brake Subharmonic Solutions of First Order Hamiltonian Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xxi

253 253 261 270

Multiplicity of Brake Orbits on a Fixed Energy Surface . . . . . . . . . . . . . . . 10.1 Brake Orbits of Nonlinear Hamiltonian Systems . . . . . . . . . . . . . . . . . . . 10.1.1 Seifert Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.2 Some Related Results Since 1948 . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.3 Some Consequences of Theorem 1.2 and Further Arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Proofs of Theorems 1.2 and 1.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

275 275 277 278

The Existence and Multiplicity of Solutions of Wave Equations . . . . . . 11.1 Variational Setting and Critical Point Theories . . . . . . . . . . . . . . . . . . . . . 11.1.1 Critical Point Theorems in Case 1 and Case 2 . . . . . . . . . . . . 11.1.2 Critical Point Theorems in Case 3 . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Applications: The Existence and Multiplicity of Solutions for Wave Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.1 One Dimensional Wave Equations. . . . . . . . . . . . . . . . . . . . . . . . . 11.2.2 n-Dimensional Wave Equations . . . . . . . . . . . . . . . . . . . . . . . . . . .

293 293 293 302

279 280

304 304 314

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

Chapter 1

Linear Algebraic Aspects

1.1 Linear Symplectic Spaces In this book, we define by N, Z, R and C the sets of all natural, integral, real and complex numbers respectively. For a matrix M, we denote its transpose by M T . For any n ∈ N and any field K, denote by K n the linear space formed by all the column vectors of the form x = (x1 , · · · , xn )T with xi ∈ K. We usually treat x ∈ K n as an n × 1 matrix with no explain. Let L(K n ) denote the group of all n × n matrices with entries in the field K, and Ls (K n ) the subset of L(K n ) consists of symmetric matrices. Any linear map T : K n → K n corresponds to a matrix T ∈ L(K n ) in the usual way. We will not distinguish these two objectors. Let V be an m-dimensional vector space over R with its dual space V ∗ , and let ω˜ : V × V → R be a bilinear map. Definition 1.1 The map ω˜ ∗ : V → V ∗ is the linear map defined by ω˜ ∗ (v)(u) = ω(v, ˜ u). The kernel of ω˜ ∗ is the subspace U of V Definition 1.2 A skew-symmetric bilinear map ω˜ is symplectic (or nondegenerate) if ω˜ ∗ is bijective, i.e., U = {0}. The map ω˜ is then called a linear symplectic structure (or symplectic form) on V , and (V , ω) ˜ is called a symplectic vector space. The following are immediate properties of a linear symplectic structure ω: ˜ ω(u, ˜ v) = −ω(v, ˜ u), ∀ u, v ∈ V . ω(u, ˜ v) = 0 for some u ∈ V and any v ∈ V implies u = 0. The map ω˜ ∗ : V → V ∗ is a bijection. By algebraic arguments, from the skewsymmetric of ω˜ and dim U = 0, we have that dim V = 2n is even. Let (V , ω) ˜ be a symplectic vector space. u, v ∈ V are symplectic orthogonal and denoted by u⊥ω˜ v, if ω(u, ˜ v) = 0. For subspaces A, B of V , we define A⊥ω˜ B if

© Springer Nature Singapore Pte Ltd. 2019 C. Liu, Index theory in nonlinear analysis, https://doi.org/10.1007/978-981-13-7287-2_1

1

2

1 Linear Algebraic Aspects

u⊥ω˜ v for any u ∈ A and v ∈ B. We note that u⊥ω˜ u for any u ∈ V . For any linear subspace E of V , we define ˜ v) = 0, ∀ v ∈ E}. E ⊥ω˜ = {u ∈ V | ω(u, Then E ⊥ω˜ is a linear subspace of V , and there holds (E ⊥ω˜ )⊥ω˜ = E, dim E + dim E ⊥ω˜ = 2n. Definition 1.3 A linear subspace E of the symplectic vector space (V , ω) ˜ is isotropic, if E ⊂ E ⊥ω˜ , coisotropic, if E ⊃ E ⊥ω˜ , Lagrangian, if E = E ⊥ω˜ , symplectic, if E ∩ E ⊥ω˜ = {0}. From the definition, we see that a linear subspace L of (V , ω) ˜ is a Lagrangian subspace iff dim L = 12 dim V = n and ω| ˜ L = 0 that is ω(u, ˜ v) = 0 for any u, v ∈ L. A subspace E ⊂ V is a symplectic subspace iff (E, ω| ˜ E ) is a symplectic vector space. Lemma 1.4 A dimension 2n symplectic vector space (V , ω) ˜ has a basis e1 , · · · , en , f1 , . . . , fn satisfying ω(e ˜ i , fj ) = δij and ω(e ˜ i , ej ) = 0 = ω(f ˜ i , fj ). Proof Fix a non-zero vector e1 ∈ V . By the non-degeneracy of ω, ˜ there exists a vector f1 ∈ V satisfying ω(e ˜ 1 , f1 ) = 1. So the subspace E = span{e1 , f1 } is a dimension 2 symplectic subspace of (V , ω). ˜ If dim V = 2, the lemma is proved. If dim V = 2n > 2, then E ⊥ω˜ is a 2(n − 1) dimensional symplectic subspace of (V , ω), ˜ by an induction argument on the dimension the proof is complete.   The basis e1 , · · · , en , f1 , . . . , fn in the above lemma is called a symplectic basis of (V , ω). ˜ With a symplectic basis e1 , · · · , en , f1 , . . . , fn , we can write n  (xi ei + yi fi ), xi , yi ∈ R. So the symplectic vector space (V , ω) ˜ u = i=1

is symplectically isomorphic to the standard linear symplectic space (R2n , ω˜ 0 ) n  with ω˜ 0 (z1 , z2 ) = (−xi2 yi1 + xi1 yi2 ) = (J z1 , z2 ) = (z2 )T · J · z1 for zk = i=1

(x1k , · · · , xnk , y1k , · · · , ynk )T , k = 1, 2. Here the standard symplectic matrix J is   n  0 −In defined by J = . In other word, ω˜ 0 = dxi ∧ dyi . For zi = (xi , yi ) ∈ In 0 i=1 Rn × Rn , i = 1, 2, there holds ω˜ 0 (z1 , z2 ) = x1 , y2 − x2 , y1 , where ·, · denotes the standard inner product in Rn . A Lagrangian subspace L ⊂ (R2n , ω˜ 0 ) is a dimensional n subspace with ω˜ 0 (z1 , z2 ) = 0 for all z1 , z2 ∈ L.

1.2 Symplectic Matrices

3

Definition 1.5 Let (Vk , ω˜ k ), k = 1, 2 be two dimension 2n symplectic vector spaces. A linear invertible map T : V1 → V2 is called a linear symplectic map if ω˜ 2 (T u, T v) = ω˜ 1 (u, v), ∀ u, v ∈ V1 ,

(1.1)

that is to say T ∗ ω˜ 2 = ω˜ 1 . In this case, the two symplectic vector spaces are called symplectic isomorphic. The linear map T is a symplectic isomorphism from (V1 , ω˜ 1 ) to (V2 , ω˜ 2 ). So in this sense, the map T : (V , ω) ˜ → (R2n , ω˜ 0 ), T (u) = (x1 , · · · , xn , y1 , · · · , yn ) as stated above is a symplectic map. Corollary 1.6 Any two symplectic vector spaces with the same dimension are symplectic isomorphic. Proof Since the isomorphic relation is an equivalent relation, and every dimension 2n symplectic vector space (V , ω) ˜ is symplectic isomorphic to the standard symplectic space (R2n , ω˜ 0 ), we prove the corollary.   We note that the standard orthonormal basis of R2n is a symplectic basis of the standard symplectic space (R2n , ω˜ 0 ). In general, for linear symplectic space (R2n , ω), ˜ there exists a skew-symmetric non-degenerate matrix J such that ω(u, ˜ v) = v T · J −1 · u, namely, ω˜ =

2n 

(1.2)

a ij dxi ∧ dxj , here J −1 = (a ij ). Then the 2n × 2n matrix T as

i,j =1

˜ to (R2n , ω˜ 0 ) should satisfy T T J T = J −1 . a symplectic map from (R2n , ω)

1.2 Symplectic Matrices Definition 2.1 A 2n × 2n real matrix M ∈ L(R2n ) is called a symplectic matrix if it corresponds to a linear symplectic map M : (R2n , ω˜ 0 ) → (R2n , ω˜ 0 ) in the standard orthonormal basis. So M is a symplectic matrix if and only if M T J M = J . We denote by Sp(2n) the set of symplectic matrices, i.e., Sp(2n) = {M ∈ L(R2n )|M T J M = J }. In general for the linear symplectic space (R2n , ω) ˜ with the corresponding skewsymmetric non-degenerate matrix J defined in (1.2), the matrix M is called J symplectic if M T J −1 M = J −1 .

4

1 Linear Algebraic Aspects

From definition, we know that any symplectic matrix is non-singular. We remind that in the following proposition T : (R2n , ω) ˜ → (R2n , ω˜ 0 ) is a symplectic map, and the corresponding matrix with respect to the standard basis is still denoted by T . Proposition 2.2 M is a J -symplectic matrix if and only if M  ≡ T MT −1 is a symplectic matrix. Proof By direct computation (T MT −1 )T J T MT −1 = (T T )−1 M T T T J T MT −1 = (T T )−1 M T J −1 MT T = (T T )−1 J −1 T −1 = J.   The above proposition says that the following diagram is symplectic. T

(R2n⏐, ω) ˜ −−−−→ (R2n ,⏐ω˜ 0 ) ⏐ ⏐ M⏐ M ⏐    T

(R2n , ω) ˜ −−−−→ (R2n , ω˜ 0 ). Lemma 2.3 Suppose A, B ∈ Sp(2n). Then A−1 , AT ∈ Sp(2n) and AB ∈ Sp(2n). Moreover, I2n and J ∈ Sp(2n). Proof From AT J A = J , we get (A−1 )T J A−1 = J , so A−1 ∈ Sp(2n). Taking inverse in the second equality, we have AJ AT = J , so AT ∈ Sp(2n). If further B T J B = J , then B T AT J AB = B T J B = J , so AB ∈ Sp(2n). That I2n and J ∈ Sp(2n) are obvious.   So Sp(2n) is a group, which is called symplectic group. From Proposition 2.2, we see that the set of all 2n × 2n J -symplectic matrices is also a group which is denoted by SpJ (2n). We call it the J -symplectic group. We have T SpJ (2n)T −1 = Sp(2n). Lemma 2.4 Any symplectic matrix M ∈ Sp(2n) has the square block form 

AB CD



with both AT C and B T D being symmetric, and AT D − C T B = In . The proof of this lemma follows directly from definition.

1.2 Symplectic Matrices

5



 √ AB Lemma 2.5 For any symplectic matrix M = , we set (A −1 + CD √ √ B)(C −1 + D)−1 := P + Q −1 with P and Q being real n × n matrices. Then we have that P and Q are symmetric matrices and Q is positively definite. Proof We have √ √ √ (C T −1 + D T )(A −1 + B) = D T B − C T A + −1(C T B + D T A), √ √ √ (AT −1 + B T )(C −1 + D) = B T D − AT C + −1(B T C + AT D). By the results in Lemma 2.4, we obtain √ √ (P + Q −1)T = P + Q −1. √ We now prove Q > 0. It is suffice to prove Re((A−iB)(D+iC)−1 ) > 0. i = −1. For any x ∈ Rn , x = 0, set y = (D + Ci)−1 x = a + bi, a, b ∈ Rn . Then there holds x = (D + Ci)(a + bi) = Da − Cb + i(Ca + Db) ∈ Rn . It implies that Ca + Db = 0.

(1.3)

By definition we have Re (x T (A − Bi)(D + Ci)−1 x) = Re((a T + bT i)(D T + C T i)(A − Bi)(a + bi)) = a T (D T A + C T B)a − 2a T (C T A − D T B)b − bT (D T A + C T B)b. (1.4) From (1.3), we have a T C T Ba = −bT D T Ba, bT D T Ab = −a T C T Ab, and by using Lemma 2.4 and (1.4), we obtain Re(x T (A − Bi)(D + Ci)−1 x) = |a|2 + |b|2 + bT (B T C − C T B)b.

(1.5)

We note that the matrix B T C − C T B is anti-symmetric, so we have bT (B T C − C T B)b = 0 and so we have Re(x T (A − Bi)(D + Ci)−1 x) = |a|2 + |b|2 > 0.   The following lemma says that any symplectic matrix has its polar decomposition.

6

1 Linear Algebraic Aspects

Lemma 2.6 ([227]) Any symplectic matrix M ∈ Sp(2n) can be represented in the form M = P U,

(1.6)

where P is a symmetric symplectic positive definite matrix, U is a symplectic orthogonal matrix, and they are uniquely determined by M. Proof In fact any non-singular square matrix has unique polar decomposition (1.6) with √ A being symmetric positive definite and U being orthogonal matrix. Let P = MM T . Then P is symmetric positive definite. Let U = P −1 M. From U U T = P −1 MM T P −1 = P −1 P 2 P −1 = I2n , we obtain that U is orthogonal. Suppose we have two polar decompositions M = P1 U1 = P2 U2 , then U1T P1 = U2T P2 . Thus P12 = P1 U1 U1T P1 = A2 U2 U2T P2 = P22 . Since P1 and P2 are symmetric positive definite, it implies that P1 = P2 , so U1 = U2 . Next we need to prove that P and U are symplectic. From M T J M = J , we have M = J −1 [(P U )T ]−1 J = J −1 P −1 J J −1 (U T )−1 J. It is easy to see that we have another polar decomposition M = P  U  with P  = J −1 P −1 J and U  = J −1 (U T )−1 J . So we have P = J −1 P −1 J and U = J −1 (U T )−1 J . Thus both P and U are symplectic.   It is well known that every positive definite symmetric matrix P can be uniquely represented by P = exp(Q), where Q is a real symmetric matrix (the logarithm of P ), and exp(Q) =

∞  1 i Q. i! i=0

Lemma 2.7 ([223, 227]) A symmetric positive definite matrix P ∈ L(R2n ) is symplectic, if and only if it has the form P = exp(Q),

1.2 Symplectic Matrices

 where Q =

7

 A B , with A and B being symmetric n × n matrices. B −A

Proof Since P is symmetric, Q is symmetric. Necessity Since P = exp(Q) is symplectic, exp(Q) = J −1 exp(−Q)J = exp(−J −1 QJ ). Since −J −1 QJ is real symmetric, by the uniqueness of the above representation we obtain Q = −J −1 QJ.

(1.7)



 AB in (1.7) yields D = −A and C = B. By QT = Q, we have CD B T = B and AT = A.   A B Sufficiency If Q = with AT = A, B T = B, then Q solves the Eq. (1.7), B −A and so P is symplectic.   Setting Q =

From the viewpoint of Lie group, Sp(2n) is a Lie group, its Lie algebra is sp(2n) = {M ∈ L(R2n )|M T J +J M = 0}. It is valuable to note that the exponential map exp : sp(2n) → Sp(2n) is not surjective, i.e., exp(sp(2n)) is only a proper subgroup of Sp(2n). The above result shows that the subset of symplectic symmetric positive definite matrices lies in the range exp(sp(2n)) of the exponential map. Lemma 2.8 ([31], Theorem IV.2.2) For any compact connected Lie group G with its Lie algebra g, the exponential map is surjective, i.e., there holds G = exp(g). Corollary 2.9 We have Osp(2n) ≡ Sp(2n) ∩ O(2n) = exp(sp(2n) ∩ o(2n)), where sp(2n) ∩ o(2n) = {N ∈ L(R2n ) | N T J + J N = 0, N T + N = 0}.

(1.8)

Proof Since the Lie subgroup Sp(2n) ∩ O(2n) is connected and compact, it is a direct consequence of Lemma 2.8.   As a consequence, we have the following result.

8

1 Linear Algebraic Aspects

Proposition 2.10 For any M ∈ Sp(2n), there exist matrices A and B satisfying AT = A, J A + AJ = 0, B T = −B, J B − BJ = 0 such that M = exp(A) exp(B).

(1.9)

Proof From Lemma 2.6, we have the polar decomposition M = PU

(1.10)

with P being symplectic symmetric positive definite and U being symplectic orthogonal. So the result follows directly from Lemma 2.7 and Corollary 2.9   Example For t ∈ R, the matrix tJ ∈ sp(2n) ∩ o(2n), and exp(tJ ) = =

∞ k  t k=0 ∞ 

k!

m=0

Jk ∞

 (−1)m−1 t 2m−1 (−1)m t 2m I2n + J (2m)! (2m − 1)!

= (cos t)I2n + (sin t)J

(1.11)

m=1

is an orthogonal symplectic matrix. Lemma 2.11 ([223, 227]) An orthogonal matrix U ∈ L(R2n ) is symplectic, if and only if it has the form  U=

A B −B A



where AT B is symmetric, and AT A + B T B = In , or equivalently A ± unitary matrices. Proof An orthogonal matrix U ∈ L(R2n ) is symplectic, if and only if U T (I2n +  We set U =

√ √ −1J )U = I2n + −1J.

 AB . It implies that CD √ √ −1C T )(A − −1C) = In , √ √ √ (AT + −1C T )(B − −1D) = − −1In , √ √ (B T + −1D T )(B − −1D) = In , √ √ √ (B T + −1D T )(A − −1C) = −1In .

(AT +

(1.12) √

−1B are

1.2 Symplectic Matrices

9

√ √ From the first two equalities, we obtain A − −1C = D + √−1B, so D = A and C = −B. Together with the first equality, we see that A√+ −1B are unitary √   matrix. Thus A − −1B is also unitary matrix. That is A ± −1B ∈ U (n). The above lemma establishes actually an isomorphism between the symplectic orthogonal group and the unitary group Sp(2n) ∩ O(2n)  U (n). Note that in the sense of the topological groups, this is also a homeomorphism. Corollary 2.12 ([227]) For any M ∈ Sp(2n), there holds det M = 1.

(1.13)

Proof By the polar decomposition (1.6), we have det M = det P det U. From the M T J M = J and det J = 0, we have (det M)2 = 1. Since det P > 0, we have det P = 1. Taking a complex matrix  T =

√  −1In In √ , In − −1In

we have T UT √

−1

 =

A+

√  −1B 0 √ . 0 A − −1B

√ −1B) = | det(A + −1B)|2 = 1.   AB Lemma 2.13 For any symplectic matrix M = , there hold CD

So det U = det(A +

−1B) det(A −

√

 

(A −

√ √ −1C)(A + −1C)−1 ∈ U (n)

(1.14)

(D −

√ √ −1B)(D + −1B)−1 ∈ U (n).

(1.15)

and

Proof We only prove (1.14). Since M is non-singular, we get A ± singular. By Lemma 2.4, we obtain (AT +

√

−1C T )(A −

√

−1C is non-

√ √ √ −1C) = (AT − −1C T )(A + −1C).

10

1 Linear Algebraic Aspects

So there holds √ √ √ √ (AT − −1C T )−1 (AT + −1C T )(A − −1C)(A + −1C)−1 √ √ √ √ = (AT − −1C T )−1 (AT − −1C T )(A + −1C)(A + −1C)−1 = In .   In the same way, we get (A +

√ √ −1C)(A − −1C)−1 ∈ U (n)

(1.16)

(D +

√ √ −1B)(D − −1B)−1 ∈ U (n).

(1.17)

and

Lemma 2.14 Let α1 , · · · , α2n denote the columns of a symplectic matrix M respectively. Then L1 = span{α1 , · · · , αn } and L2 = span{αn+1 , · · · , α2n } are Lagrangian subspaces of (R2n , ω˜ 0 ). For any Lagrangian subspace L of (R2n , ω˜ 0 ), there is a symplectic orthogonal matrix U such that L = U (L0 ) with L0 = {0} × Rn ⊂ R2n . The matrix U is determined uniquely up to an orthogonal matrix On ∈ O(n) in the sense that for all such matrices U  , there is a matrix On ∈ O(n) such that U = U



On 0 0 On

 .

Proof It is clear that M(L) is Lagrangian subspace of (R2n , ω˜ 0 ) for any Lagrangian 2n subspace L of  (R ,ω˜ 0 ) and any M ∈ Sp(2n). AB Let M = , then CD 

B D



 =

AB CD

   0 . · In

The Lagrangian subspace L0 = {0} × Rn is the subspace span{en+1 , · · · , e2n } of R2n , where e1 , · · · , e2n is the standard basis of R2n ). So L2 = M(L0 ) is a Lagrangian subspace of (R2n , ω˜ 0 ). We can prove that L1 is a Lagrangian subspace of (R2n , ω˜ 0 ) in a similar way. Suppose L is a Lagrangian subspace of (R2n , ω˜ 0 ) with orthonormal basis βn+1 , · · · , β2n . We extend it to an orthogonal basis of R2n as β1 , · · · , β2n such that

1.2 Symplectic Matrices

11

ω˜ 0 (βi , βn+j ) = δij , i, j = 1, · · · , n. Then the matrix U with the columns β1 , · · · , β2n is an orthogonal matrix and satisfies L = U (L0 ). 

 AC , then by the orthogonality and by the fact that L is BD Lagrangian subspace of (R2n , ω˜ 0 ), there hold

In fact, let U =



C T DT

       C  0 −In C = 0. = In , C T D T In 0 D D

√ They are exactly the conditions in Lemma 2.11, i.e., D ± −1C ∈ U (n). Since U is orthogonal symplectic, then D = A and C = −B. In other word, βi = J −1 βn+i , i = 1, · · · , n. The orthonormal basis βn+1 , · · · , β2n of L determine the orthogonal symplectic matrix U . For another orthonormal basis αn+1 , · · · , α2n of L, there is an orthogonal   matrix On ∈ O(n) such that αn+i = On (βn+i ), i = 1, · · · , n. Suppose −B  is the matrix with its ith-column being αn+i , then A 

−B  A



 =

−BOn AOn

 .

i.e., 

U =



A −B  B  A



 =

A −B B A



On 0 0 On

 .  

A part of Lemma 2.14 was proved in [238]. We denote by (n) the set of all Lagrangian subspaces of (R2n , ω˜ 0 ). From Lemmas 2.11 and 2.14, we obtain that (n)  U (n)/O(n).

(1.18)

 Ai Bi with Given any two 2ki × 2ki matrices of square block form, Mi = Ci D i i = 1, 2, we define an operation -sum (symplectic direct sum) of M1 and M2 to be the 2(k1 + k2 ) × 2(k1 + k2 ) symplectic matrix given by (cf. [223, 226]) 

12

1 Linear Algebraic Aspects

⎛

A1 ⎜ 0 M1  M2 = ⎜ ⎝ C1 0

0 A2 0 C2

B1 0 D1 0

⎞ 0 B2 ⎟ ⎟. 0 ⎠ D2

(1.19)

We denote by M k the k-fold -sum M  · · ·  M.

1.3 Lagrangian Subspaces For a 2n dimensional real linear symplectic space (V , ω), ˜ we have a complex structure J : V → V satisfying ω(J ˜ v, J w) = ω(v, ˜ w) for any v, w ∈ V , ω(v, ˜ J w) = v, w is an inner product of V and J 2 = −I2n . Associated to this situation, there is a Hermitian inner product (·, ·) on V defined by (v, w) = −ω(J ˜ v, w) −

√ −1ω(v, ˜ w).

√ We understand (V , J ) as a complex vector space by identifying (a + b −1)v with (a + bJ )v for all a, b ∈ R and v ∈ V . Let Lag(V ) denote the set of Lagrangian subspaces in (V , ω). ˜ For the standard symplectic space (R2n , ω˜ 0 ), we denote the set of its Lagrangian subspaces by (n), i.e., (n) = Lag(R2n ). Let L ∈ Lag(V ) be a Lagrangian subspace of (V , ω) ˜ . By definition dim L = n and ω(v, ˜ w) = 0 for all v, w ∈ L. Suppose {ej }j =1,··· ,n is a basis of L with ei , ej = δij , then by direct computation, we have (ei , ej ) = δij . So {ej }j =1,··· ,n is also an orthogonal basis of the complex vector space (V , J ) with the Hermitian inner product (·, ·). Thus any Lagrangian subspace L of (V , ω) ˜ is the real span L =

n 

R{ej }

j =1

of some orthogonal basis {ej }j =1,··· ,n of the complex vector space (V , J ), and as a complex vector space we have V =

n  j =1

C{ej } =

n 

R{ej } ⊕ R{J ej }.

j =1

We denote the unitary group of the complex vector space (V , (·, ·)) by U˜ (n), that ˜ ˜ is to say for every linear mapping U˜ ∈ U˜ (n), there holds n (U v, U w) = (v, w) for all v, w ∈ V . Based on the Lagrangian subspace L = j =1 R{ej }, there is a natural mapping φ : U˜ (n) → Lag(V )

(1.20)

1.3 Lagrangian Subspaces

13

n sending A ∈ U˜ (n) to the real span φ(A) = j =1 R{Aej }. This mapping is obviously surjective and in fact it give rise to a bijection  ˜ φ(L) : U˜ (n)/O(n) −→ Lag(V )

which depends only on L(not on the choice of basis {ej }j =1,··· ,n and the choice of   

O 0 ˜ ˜ the complex structure J ), where O(n) = O := : O ∈ O(n) . In order to 0 O see this clearly, by Lemma 1.4, we choose a symplectic basis such that (V , ω) ˜ can 2n , ω ˜ ) with the complex structure be viewed as the standard symplectic space (R 0   0 −In , which is the standard symplectic matrix. Suppose U˜ ∈ U˜ (n). J = In 0 Since V is a real 2n dimensional space, we can assume U˜ in the form   AB ˜ U= . CD Then by direct computation, from (U˜ v, U˜ w) ≡ (v, w), we have AT A + C T C = In , B T B + D T D = In , AT B + C T D = 0

(1.21)

AT D − C T B = In , AT C = C T A, B T D = D T B.

(1.22)

and

The conditions in (1.21) means that U˜ is an orthogonal matrix. The conditions √ in (1.22) means that U˜ is a symplectic matrix. So A = D, B = −C and A ± −1B are unitary matrices from Lemma 2.11. Therefore, we have U˜ (n) = Sp(2n) ∩ O(2n). of L, then e1 , · · · , en , J e1 , · · · , J en is a If {ej }j =1,··· ,n is a orthogonal basis √ orthogonal basis of V . Taking J e as −1ej , we get v = (z1 , · · · , zn )T with j √ zj = xj + −1yj for a vector v˜ ∈ V with v˜ = (x1 , · · · , xn , y1 , · · · , yn )T = n  √ (xi ei + yi J ei ). In this sense, we see that an unitary matrix U = A − −1B ∈ i=1

U (n) acts on a complex vector v = (z1 , · · · ,  zn )T is the same thing as that A B an orthogonal symplectic matrix U˜ = acts on the real vector v˜ = −B A (x1 , · · · , xn , y1 , · · · , yn )T . Thus we can identify U and U˜ in this sense. In fact, the mapping

14

1 Linear Algebraic Aspects

ψ : U (n) → Osp(2n), ψ(U ) = U˜ is a group isomorphism, where we have denoted the orthogonal symplectic subgroup of Sp(2n) by Osp(2n). I.e. Osp(2n) = Sp(2n) ∩ O(2n) ∼ = U (n). It sends an orthogonal matrix O(n)  C : L → L to a 2n × 2n orthogonal symplectic matrix   C 0 ψ(C) = C˜ = . It implies that 0 C ∼ ˜ U˜ (n)/O(n) = U (n)/O(n). Lemma 3.1 ([32]) Let L1 and L2 be two Lagrangian subspaces in V . Then (1) eJ θ L2 is a Lagrangian subspace of V for all θ ; (2) There exists an  ∈ (0, π ) such that L1 ∩ eJ θ L2 = {0} for all 0 < |θ | < . Proof We only need to prove the statement (2). Let a = dimR (L1 ∩ L2 ) and let {ej }j =1,··· ,n be an orthogonal basis of L1 with the first a of it form a basis of L1 ∩L2 . Let U and W be the complex subspaces of V generated respectively by {ej }j =1,··· ,a and {ej }j =a+1,··· ,n , i.e. U=

a  j =1

C{ej },

W =

n 

C{ej }.

(1.23)

j =a+1

Then we have V = U ⊕ W with L1 ∩ L2 ⊂ U.

(1.24)

By dimension counting, the real span of {ej }j =1,··· ,a is L1 ∩ L2 and the real span of {ej }j =a+1,··· ,n is L1 ∩ W . Thus we have L1 = (L1 ∩ L2 ) ⊕ (L1 ∩ W ). In a similar manner, we choose an orthogonal basis {ej }j =1,··· ,n of L2 such that ej = ej , j = 1, · · · , a. Let W  be the complex subspace of V generated by  {ej }j =a+1,··· ,n , i.e., W  = nj=a+1 C{ej }. Then we have the following orthogonal sum decomposition: V = U ⊕ W  , L2 = (L1 ∩ L2 ) ⊕ (L2 ∩ W  ). Since the inner product ·, · is nondegenerate. U has a unique orthogonal complement, so we have W = W  . Thus we have two orthogonal decompositions: L1 = (L1 ∩ L2 ) ⊕ (L1 ∩ W ), L2 = (L1 ∩ L2 ) ⊕ (L2 ∩ W ).

(1.25)

Since {ej }j =1,··· ,a is a real basis for L1 ∩ L2 and a complex basis for U , the multiplication by eJ θ , 0 < |θ | < π , on {ej }j =1,··· ,a gives us an R-linear independent one {eJ θ ej }j =1,··· ,a . In other words, we have

1.3 Lagrangian Subspaces

15

(L1 ∩ L2 ) ∩ eJ θ (L1 ∩ L2 ) = {0}.

(1.26)

However, we have (L1 ∩ W ) ∩ (L2 ∩ W ) = (L1 ∩ L2 ) ∩ W ⊂ U ∩ W = {0}, so the subspaces L1 ∩W and L2 ∩W are transverse to each other. Since transversality is an open condition, there exists  > 0 such that (L1 ∩ W ) ∩ eJ θ (L2 ∩ W ) = {0} for all |θ | < .

(1.27)

Combining (1.25), (1.26) and (1.27), we have, for 0 < |θ | < , L1 ∩ eJ θ L2 = (L1 ∩ L2 ) ∩ eJ θ (L1 ∩ L2 ) ⊕ (L1 ∩ W ) ∩ eJ θ (L2 ∩ W ) = {0}.   The following results are due to Arnold [9], Hörmander [143] (see also [76]). Theorem 3.2 The set of Lagrangian subspaces Lag(V ) is a connected regular algebraic subvariety of the Grassmann variety GV ,n of n-dimensional linear 0 subspaces of V , dim Lag(V ) = n(n+1) 2 . For any L ∈ Lag(V ),  (V , L) =   {L ∈ Lag(V )| L ∩ L = {0}} is (algebraically) diffeomorphic to the vector space Symm(L ) of all symmetric bilinear forms on L , for any L ∈ 0 (V , L). Moreover, 0 (V , L) is open and dense in Lag(V ). Finally the tangent space TL Lag(V ) is canonically isomorphic to Symm(L ) for every L ∈ Lag(V ). Proof Let L, L ∈ Lag(V ), L ∩ L = {0}. Then each n-dimensional subspace M that is transversal to L is of the form {x + Ax| x ∈ L } for some linear map A : L → L. Then Q(M)(x, y) = ω(Ax, ˜ y), x, y ∈ L

(1.28)

defines a bilinear form on L , which is symmetric if and only if M is Lagrangian. In fact, if M ∈ Lag(V ), there holds ω(x ˜ + Ax, y + Ay) = 0, ∀ x, y ∈ L . It implies ω(Ax, ˜ y) = ω(Ay, ˜ x), so Q(M) is symmetric. The reverse is obvious. So Q defines a bijective mapping: 0 (V , L) → Symm(L ), the inverse being an algebraic embedding: Symm(L ) → GV ,n . So the Q’s form a set of local coordinates of Lag(V ); one can prove that 2n of the 0 (V , N ) cover Lag(V ). This proves the first part of the theorem.

16

1 Linear Algebraic Aspects

Clearly 0 (V , L) is open in Lag(V ). For its density we choose, for any L ∈ ˜ is an open neighborhood Lag(V ), an L˜ ∈ 0 (V , L) ∩ 0 (V , L ). Then 0 (V , L) ˜ corresponds to the nonsingular of L , identified with Symm(L), in which 0 (V , L) elements of Symm(L), which is dense. Clearly Q depends on the choice of L , L, but for fixed L its differential at L does not depend on L. In fact, let L ∈ 0 (V , L ), then L = {z + Bz| ∈ L} for some linear map B : L → L . If Lˆ is close enough to L , Lˆ still is transversal both ˜ y ∈ L }, for some to L and L and we can write Lˆ = {x + Ax| x ∈ L } = {y + Ay|    ˜ Ay ˜ = z + Bz for linear maps A : L → L, A˜ : L → L . Then x + Ax = y + Ay, ˜ − BAx). Taking some z ∈ L, so z = Ax, y = x − Bz, x + Ax = (x − BAx) + A(x symplectic product with u ∈ L : ˜ u) − ω( ˜ ω(Ax, ˜ u) = ω( ˜ Ax, ˜ ABAx, u).

(1.29)

The second term in the right hand side vanishes of second order when Lˆ → L , so ˜ have the same differential at Lˆ = L . Q and Q   Theorem 3.3 For any k ≥ 1, L ∈ Lag(V ), the set k (V , L) = {L ∈ Lag(V )| dim(L ∩ L ) = k}

(1.30)

 is a regular part of its closure k (V , L) = l≥k l (V , L), which in turn is a in Lag(V ). If L ∈ connected algebraic subvariety of Lag(V ) of codimension k(k+1) 2 k k  (V , L), then the tangent space TL  (V , L) corresponds to the Q ∈ Symm(L ) such that Q(x, y) = 0 for all x, y ∈ L ∩ L . Proof In view of Theorem 3.2 also 0 (V , L) ∩ 0 (V , L ) is dense in Lag(V ) so it contains at leat some N . Let Q be the coordinization: 0 (V , N ) → Symm(L ) described in the proof of Theorem 3.2. Then, if L ∈ k (V , L), we have L˜ ∈ ˜ = k. The reason is that k (V , L) ∩ 0 (V , N ) if and only if dim ker(Q(L) − Q(L)) for L˜ = {x + Bx| x ∈ L } and L = {y + Ay| y ∈ L }, where B : L → N and A : L → N , x + Bx ∈ L ∩ L˜ ⇔ (A − B)x = 0. On a suitable basis of L we can write     00 T U ˜ Q(L) = , , Q(L) = 0S UT W where S is a nonsingular (n − k) × (n − k) matrix. If W is sufficiently small, then S − W is still nonsingular and rank(Q(L) − ˜ = n − k if and only if T = −U D, U T = (S − W )D for some D. So the Q(L)) equation L˜ ∈ k (V , L) reads T = −U (S −W )−1 U T if L˜ is close to L . This proves and also that that k (V , L) is a smooth algebraic manifold of codimension k(k+1)  2  T U TL k (V , L) corresponds to the symmetric matrices such that T = 0. UT W

1.4 Linear Hamiltonian Systems

17

To prove that k (V , L) is connected we remark that L → L ∩ L defines a ⊥ fibration: k (V , L) → GL,k , with fiber over L0 equal to Lag(L0 ω˜ /L0 ). To explain ⊥ ⊥ this, let L0 ⊂ L, dim L0 = k. Then L0 ⊂ L = L⊥ω˜ ⊂ L0 ω˜ . Furthermore L0 ω˜ /L0 is a symplectic vector space with symplectic form ω˜ modL0 defined by ⊥

˜ 1 , e2 ), e1 , e2 ∈ L0 ω˜ . ω˜ modL0 (e1 + L0 , e2 + L0 ) = ω(e

(1.31) ⊥

If L ∈ Lag(V ), L ∩ L = L0 , then also L0 ⊂ L = (L )⊥ω˜ ⊂ L0 ω˜ and it is easily verified that L → L /L0 is an isomorphism between the fiber over L0 and ⊥ ⊥ Lag(L0 ω˜ /L0 ). Because both GL,k and Lag(L0 ω˜ /L0 ) are connected it follows that   k (V , L) is connected.

1.4 Linear Hamiltonian Systems A linear Hamiltonian system can be written in the following way z˙ (t) = J B(t)z(t),

(1.32)

where B(t) is a continuous 2n × 2n symmetric matrix function. Its fundamental solution γ (t) is a differentiable 2n × 2n matrix function satisfying 

γ˙ (t) = J B(t)γ (t), γ (0) = I2n .

For the reason of simplicity, we also call γ (t) the fundamental solution of B(t). Example For the constant coefficient Hamiltonian system z˙ (t) = J Bz(t), the fundamental solution is γ (t) = exp(tJ B). Lemma 4.1 The linear Hamiltonian system (1.32) always possesses fundamental solution γ (t). Furthermore, every solution x(t) of (1.32) satisfying initial condition z(0) = ξ ∈ R2n can be represented as z(t) = γ (t)ξ.

(1.33)

Proof It is a well known result. For reader’s convenience, we give a proof here. Suppose {e1 , · · · , e2n } is the standard basis of R2n . Consider the initial value problem

18

1 Linear Algebraic Aspects



x(t) ˙ = J B(t)x(t), x(0) = ei .

(1.34)

We know that the problem (1.34) has unique solution αi (t). We write ξ = (a1 , · · · , a2n )T . Then the solution of the initial value problem 

z˙ (t) = J B(t)z(t), Z(0) = ξ

can be written as z(t) =

2n 

ak αk (t) = γ (t)ξ,

k=1

where the matrix γ (t) is composed by taking αi (t) as its ith-column. So γ (t) is the fundamental solution of (1.32).   Lemma 4.2 The fundamental solution γ (t) of (1.32) is a symplectic path, i.e., γ (t) ∈ Sp(2n) for any t ∈ R. On the other hand, any C 1 -symplectic path γ (t) with γ (0) = I2n is the fundamental solution of some linear Hamiltonian system, i.e., J γ˙ (t)γ (t)−1 is symmetric for any t ∈ R. Proof By direct computation d T T T dt (γ (t) J γ (t)) = γ˙ (t) J γ (t) + γ (t) J γ˙ (t) T T T = γ (t) B(t)J J γ (t) + γ (t) J J B(t)γ (t) = 0,

we have γ (t)T J γ (t) = constant. Since γ (0)T J γ (0) = J , we prove that γ (t)T J γ (t) ≡ J, t ∈ R.

(1.35)

If γ (t) is a C 1 symplectic path satisfying γ (0) = I2n , take B(t) = −J γ˙ (t)γ (t)−1 . We should prove B T (t) = B(t). We have B T (t) = γ T (t)−1 γ˙ (t)T J. From (1.35) there holds γ T (t)−1 = −J γ (t)J . By differentiating (1.35), there holds γ˙ (t)T J = −γ T (t)J γ˙ (t)γ −1 (t). Since γ (t)T is symplectic B T (t) = J γ (t)J γ T (t)J γ˙ (t)γ −1 (t) = −J γ˙ (t)γ −1 (t) = B(t).  

1.5 Eigenvalues of Symplectic Matrices

19

Lemma 4.3 For any matrix M ∈ Sp(2n), there is a C 0 -symmetric positive definite matrix function B(t) such that M = γ (1), where γ (t) is the fundamental solution of B(t). We call the property stated in Lemma 4.3 the positive definite path connectivity of Sp(2n). Proof From Proposition 2.10, for M ∈ Sp(2n), there exist matrices A and B satisfying AT = A, J A + AJ = 0, B T = −B, J B − BJ = 0 such that M = exp(A) exp(B).

(1.36)

Set γ1 (t) = exp(tA) exp(tB), then γ1 (0) = I2n and γ1 (1) = M. From Lemma 4.2, we have B1 (t) := −J γ˙1 (t)γ1 (t)−1 is symmetric for any t ∈ R. So γ1 (t) is the fundamental solution of B1 (t). If B1 (t) > 0, the proof is done. In general, we set γ (t) = exp(2kπ tJ )γ1 (t), k ∈ Z. Then γ (0) = I2n and γ (1) = M. B(t) = −J γ˙ (t)γ (t)−1 = 2kπ + exp(2kπ tJ )B1 (t) exp(−2kπ tJ ). is symmetric positive definite for large integer k.

 

1.5 Eigenvalues of Symplectic Matrices Denote by U = {z ∈ C| |z| = 1} the unit circle on the complex plane C. For any m × m matrix M, we denote by σ (M) the spectrum of M, i.e., σ (M) = {λ ∈ C| det(M − λIm ) = 0}. Suppose λ ∈ σ (M). The geometric multiplicity μg (λ) of λ is defined to be dimC kerC (M − λIm ). Denote the complex root vector space of M belonging to λ by Eλ (M) = ∪k≥1 kerC (M − λIm )k ⊂ Cm . The algebraic multiplicity μa (λ) of λ ∈ σ (M) is defined to be dimC Eλ (M). It is clear that μg (λ) ≤ μa (λ). If μg (λ) = μa (λ), λ ∈ σ (M) is called semi-simple, in this case, each irreducible invariant subspace of Eλ (M) is one dimensional. Fix M ∈ Sp(2n). Its characteristic polynomial is defined by fM (λ) = det(M − λI2n ), λ ∈ C.

20

1 Linear Algebraic Aspects

This is a polynomial of degree 2n in λ with real coefficients. Since J −1 M T J M = I2n and det M = 1, there holds fM (λ) = det(M − λJ −1 M T J M) = det(I2n − λM T ). Thus fM (λ) = λ2n fM (λ−1 ). Therefore fM (λ) can be written in the symmetric form fM (λ) = λn

n 

ak (λk + λ−k ),

k=0

where ak ∈ R, k = 0, 1, · · · , n and an = 1. The following result is very clear now. Lemma 5.1 Suppose M ∈ Sp(2n). If λ ∈ σ (M), then also λ¯ , λ−1 and λ¯ −1 belong to σ (M), and each of them possesses the same geometric and algebraic multiplicities as λ if it is different from λ. Particularly, if 1 or −1 ∈ σ (M), then its algebraic multiplicity is even. Definition 5.2 ([223, 226]) We define the homotopic set of M ∈ Sp(2n) by (M) = {N ∈ Sp(2n) | σ (N) ∩ U = σ (M) ∩ U and dimC kerC (N − λI ) = dimC kerC (M − λI ), ∀λ ∈ σ (M) ∩ U}. The path connected component of (M) which contains M is denoted by 0 (M), and is called the homotopy component of M in Sp(2n). Definition 5.3 We say that a symplectic matrix M ∈ Sp(2n) is elliptic if σ (M) ⊂ U, and M is hyperbolic if σ (M) ∩ U = ∅. M is autonomous hyperbolic if σ (M) ∩ U = {1} and the algebraic multiplicity of 1 ∈ σ (M) is 2. The total algebraic multiplicities of eigenvalues λ ∈ σ (M) belonging to U is called the elliptic hight of M and denote it by e(M). The linear Hamiltonian system (1.32) in the interval [0, τ ] is elliptic (resp. hyperbolic, or autonomous hyperbolic) if M = γ (τ ) is elliptic (resp. hyperbolic, or autonomous hyperbolic), where γ (t) the fundamental solution of (1.32). A linear system x˙ = M(t)x is called positively (resp. negatively) stable if all its real solutions remain bounded for all t > 0 (resp. t < 0). It is called stable if it is both positively and negatively stable, that is, its real solutions are bounded for all times t ∈ R. If the coefficient matrix B(t) in (1.32) is τ -periodic and the linear system is not elliptic in [0, τ ], then the linear Hamiltonian system must be unstable. For this topic, the readers can refer the monograph [78] for further arguments.

1.5 Eigenvalues of Symplectic Matrices

21

Notes and Comments As far as our knowledge, there are many monographs and lectures taking linear symplectic spaces, symplectic group and linear Hamiltonian systems as preliminary material. For example, the monographs [78] of I. Ekeland, [223, 227] of Y. Long, [270] of A. C. da Silva and [238] of D. McDuff and D. Salamon. In Ekeland’s book [78], there is an introduction to the Floquet theory of linear Hamiltonian systems and some arguments of stability of linear Hamiltonian systems. In Long’s books [223, 227], some basic properties about the symplectic matrices were introduced, and the normal forms for eigenvalues on the unit circle of symplectic matrices were developed.

Chapter 2

A Brief Introduction to Index Functions

For n ∈ N, we recall that the symplectic group is defined as Sp(2n) ≡ Sp(2n, R) = {M ∈ L(R2n ) | M T J M = J }, 

 0 −In , In is the identity matrix on Rn , and L(R2n ) is the space of In 0 2n × 2n real matrices. Without confusion, we shall omit the subindex of the identity matrices. For τ > 0, suppose H ∈ C 2 (Sτ × R2n , R), where Sτ ≡ R/(τ Z). Let x be a τ -periodic solution of the nonlinear Hamiltonian system where



x(t) ˙ = J H (t, x(t)), x ∈ R2n .

(2.1)

Then the fundamental solution of the linearized Hamiltonian system of (2.1) at x y(t) ˙ = J B(t)y, y ∈ R2n ,

(2.2)



with B(t) = H (t, x(t)), is a path γx ∈ C([0, +∞), Sp(2n)) with γx (0) = I . In order to study the properties of periodic solutions of (2.1), C. Conley and E. Zehnder in 1984 established an index theory in their celebrated work [57] for nondegenerate paths in Sp(2n) started from the identity matrix with n ≥ 2. This index theory was extended to the nondegenerate case with n = 1 by Y. Long and E. Zehnder in [231]. The index theory for the degenerate Hamiltonian systems was established by Y. Long in [224] and C. Viterbo in [280] via different methods. Then in [225], Y. Long further established the index theory for the general degenerate symplectic paths. In this book we call it the Maslov-type index for periodic boundary condition to distinguished it from the Maslov type index theories with some different boundary conditions introduced in the sequel of this book. In [226], Y. Long introduced the ω-index theory parametrized by all ω on the unit circle in the complex plane, and used it to establish the Bott-type formula and the iteration © Springer Nature Singapore Pte Ltd. 2019 C. Liu, Index theory in nonlinear analysis, https://doi.org/10.1007/978-981-13-7287-2_2

23

24

2 A Brief Introduction to Index Functions

theory of the Maslov-type index for periodic boundary condition. We define the set of symplectic paths by P(2n) ≡ Pτ (2n) = {γ ∈ C([0, τ ], Sp(2n)) | γ (0) = I }. For any γ ∈ Pτ (2n), ω ∈ U, as in [226] the ω-nullity of γ is defined by νω (γ ) = dimC kerC (γ (τ ) − ωI ). If νω (γ ) > 0, we say that the symplectic path γ is ω-degenerate, otherwise it is called ω-non-degenerate. The ω-index theory assigns a pair of integers (iω (γ ), νω (γ )) ∈ Z × {0, 1, . . . 2n} to each γ ∈ Pτ (2n). When ω = 1, it coincides with the Maslov-type index theory for periodic boundary condition. If γ (t) is the fundamental solution of the linear system (2.2), we denote the index pair of γ also by (iω (B), νω (B)). It is a classification of the linear Hamiltonian systems. If x(t) is a τ -periodic solution of the nonlinear Hamiltonian system (2.1), we denote (iω (x), νω (x)) = (iω (B), νω (B)) in this case. We refer to the monographs [223] and [227] for more details and applications of this index theory. In Chaps. 4 and 5 below, we will introduce some new development of the Maslov type index theories which are suitable to be used in the studies of Hamiltonian systems with non-periodic boundary conditions such as P -boundary condition (P -index) and Lagrangian boundary condition (L-index).

2.1 Maslov Type Index i1 (γ ) We now recall the definition of the Maslov-type index for periodic boundary condition. For simplicity we take τ = 1 now. For a symplectic path γ ∈ P1 (2n), suppose its polar decomposition is γ (t) = U (t)P (t) with U (t) ∈ O(2n) ∩ Sp(2n) and P (t) the symmetrical positive definite symplectic matrix for t ∈ [0, 1]. We recall that a symplectic matrix M is non-degenerate if det(M − I ) = 0. γ is non-degenerate if det(γ (1) − I ) = 0. We denote the set of nondegenerate symplectic matrices by Sp(2n)∗ and the nondegenerate symplectic paths by P1 (2n)∗ . Sp(2n)0 = Sp(2n) \ Sp(2n)∗ , P1 (2n)0 = P1 (2n) \ P1 (2n)∗ the degenerate set and the degenerate path set. Sp(2n)∗ is 2-path connected. M + and M − belong to different path connected components of Sp(2n)∗ , where M + = D(2)n , M − = D(−2)  D(2)(n−1) , D(a) = diag{a, a −1 }, and M k is the k-folds symplectic sum defined in (1.19). We choose a symplectic path β : [0, 1] → Sp(2n)∗ with β(0) = γ (1) and β(1) ∈ {M + , M − }. By joining γ with β, we get a path γ˜ = β ∗ γ ∈ P1 (2n)∗ . Its polar decomposition is γ˜ (t) =

2.1 Maslov Type Index i1 (γ )

25

  ˜ ˜ √ A(t) B(t) ˜ ˜ ˜ ˜ ˜ − −1B(t) ∈ U (n). U (t)P (t). Writing U (t) = , we have A(t) ˜ ˜ −B(t) A(t) √ √ ˜ ˜ ˜ − −1B(t)) = e −1θ(t) , then (θ˜ (1) − θ˜ (0))/π := k ∈ Z, and Suppose det(A(t) there holds  k is odd, if β(1) = M − , (2.3) k is even, if β(1) = M + . Definition 1.1 ([57, 231]) For a non-degenerate path γ ∈ P1 (2n)∗ , we define i1 (γ ) = k, ν1 (γ ) = 0.

(2.4)

If γ ∈ P1 (2n)0 is a degenerate path, then one can perturb the path near the end point γ (τ ) slightly to get a non-degenerate path γs for s ∈ [−1, 1] such that ⎧ ⎪ γ0 = γ , ⎪ ⎪ ⎪ ⎪ ⎨ γs (t) = γ (t), ∀ 0 ≤ t ≤ 1 − , 0 <  0, ⎪ ⎪ ⎩ i (γ ) − i (γ  ) = ν (γ ), s > 0, s  < 0. 1 s 1 s 1 The perturbed path γs can be chosen as γs (t) = γ (t)esρ (t)J with the continuous function ρ defined as ⎧ 0 ≤ t ≤ 1 − , ⎨ 0, ρ (t) = (2 − )(t − 1 + ), 1 −  < t < 1 − 2 , ⎩ t, 1 − 2 ≤ t ≤ 1. Definition 1.2 ([223, 224]) For a degenerate path γ ∈ P1 (2n)0 , we define i1 (γ ) = i1 (γ−s ), s ∈ (0, 1],

ν1 (γ ) = dimC kerC (γ (1) − I ).

For the properties of the index pair (i1 (γ ), ν1 (γ )), we refer the monographs [223] and [227] for detail arguments. We now prove some new results about this index theory. Theorem 1.3 For any  γ =

A B −B A

√

 ∈ P1 (2n) ∩ O(2n) (orthogonal path),

suppose λj (t) =√e −1θj (t) , j = 1, · · · , n are eigenvalues of the unitary matrix function A(t) − −1B(t) with θj ∈ C([0, 1], R). There holds

26

2 A Brief Introduction to Index Functions

i1 (γ ) = 2

n 

 E

j =1

θj (1) − θj (0) 2π

where E(a) = max{k ∈ Z| k < a}. Proof It is easy to see that 1 ∈ σ (A(t) − Step 1 Suppose γ ∈ part we have 

√



 =

+ n,

(2.5)

−1B(t)) if and only if 1 ∈ σ (γ (1)).

P1 (2n)∗ , then θj (1)−θj (0)

θ˜j (1) − θ˜j (0) 2π



= 2kπ, k ∈ Z. So for the integer

   θj (1) − θj (0) θj (1) − θj (0) =E , 2π 2π

(2.6)

and θ˜j (1) − θ˜j (0) ∈ 2Z + 1, π where [a] = max{k ∈ Z|k ≤ a} is the Gaussian function. So we have θ˜j (1) − θ˜j (0) = 2π



 θ˜j (1) − θ˜j (0) 1 + . 2π 2

Thus we get     θ˜j (1) − θ˜j (0) θ˜j (1) − θ˜j (0) θj (1) − θj (0) =2 + 1. +1=2 π 2π 2π

(2.7)

Taking sum from j = 1 to j = n, we have i1 (γ ) =

n ˜  θj (1) − θ˜j (0) j =1

π



n    θj (1) − θj (0) = +1 . 2 2π

(2.8)

j =1

And (2.8) implies (2.5). Step 2 Suppose γ ∈ P1 (2n)0 , there are some θj with   θj (1) − θj (0) θj (1) − θj (0) = 2π 2π and others with

(2.9)

2.1 Maslov Type Index i1 (γ )

27

  θj (1) − θj (0) θj (1) − θj (0) > . 2π 2π

(2.10)

In the two cases (2.9) and (2.10), by some suitable rotations and a similar argument as in the above step, we get   θ˜−s,j (1) − θ˜−s,j (0) θj (1) − θj (0) 1 =E + . 2π 2π 2

(2.11)

Thus we have i1 (γ ) = i1 (γ−s ) = 2

n  j =1

 E

θj (1) − θj (0) 2π

 + n.  

Remark 1.4 By the same reason as in Step 2 above, we have −n ≤ i1 (γ ) −

θγ (1) − θγ (0) < n, π

√ √ where θγ ∈ C([0, 1], R) is defined by det(A(t) − −1B(t)) = e −1θγ (t) , i.e., n  θγ (t) = θj (t). The left hand side equality holds only if γ (1) = I2n . j =1

Theorem 1.5 For a symmetric positive definite path γ ∈ Pτ (2n), there holds 1 i1 (γ ) = − ν1 (γ ) ∈ [−n, 0]. 2

(2.12)

Proof We note that if γ ∈ Pτ (2n)∗ , by definition, we can connect γ (1) with M + by symmetric positive definite path. So there holds i1 (γ ) = 0. If γ ∈ Pτ (2n)0 , the multiplicity of the eigenvalue 1 ∈ σ (γ (1)) is even. I.e., ν1 (γ ) = 2k ∈ 2Z. By k-times rotations, we get γs (t) = Rh1 (stθ0 ) · · · Rhk (stθ0 )γ (t). Thus by definition again, we get 1 i1 (γ ) = −k = − ν1 (γ ) ≥ −n. 2  

28

2 A Brief Introduction to Index Functions

2.2 ω-Index iω (γ ) The ω-index theory for continuous symplectic paths starting from the identity matrix I2n was first established in [226]. In this subsection we give a brief introduction of this ω-index theory without proofs and refer to [223] and [226] for the details. For any ω ∈ U, where U is the unite circle in complex plane, and M ∈ Sp(2n), as in [226] we define Dω (M) = (−1)n−1 ω−n det(M − ωI ). One can easily see that Dω (M) = Dω¯ (M) for all ω ∈ U, M ∈ Sp(2n) and D ∈ C ∞ (U × Sp(2n), R). For ω ∈ U, we set Sp(2n)± ω = {M ∈ Sp(2n)| ± Dω (M) < 0}, − Sp(2n)∗ω = Sp(2n)+ ω ∪ Sp(2n)ω ,

Sp(2n)0ω = Sp(2n) \ Sp(2n)∗ω ,

and ∗ (2n) = {γ ∈ Pτ (2n)|γ (τ ) ∈ Sp(2n)∗ω }. Pτ,ω

Definition 2.1 For any τ > 0 and γ ∈ Pτ (2n), we define νω (γ ) = dimC kerC (γ (τ ) − ωI ),

∀ω ∈ U.

Definition 2.2 For τ > 0 and ω ∈ U, given two paths γ0 and γ1 ∈ Pτ (2n), if there exists a map δ ∈ C([0, 1] × [0, τ ], Sp(2n)) such that δ(0, ·) = γ0 (·), δ(1, ·) = γ1 (·), δ(s, 0) = I and νω (δ(s, ·)) is constant for 0 ≤ s ≤ 1, the paths γ0 and γ1 are ωhomotopic on [0, τ ] along δ(·, τ ) and we write γ0 ∼ω γ1 . If γ0 ∼ω γ1 for all ω ∈ U, then γ0 and γ1 are homotopic on [0, τ ] along δ(·, τ ) and we write γ0 ∼ γ1 . As well known, every M ∈ Sp(2n)  has itsunique polar decomposition M = AU , √ u1 −u2 with u = u1 + −1u2 ∈ L(Cn ) where A = (MM T )1/2 , U = u2 u1 being a√unitary matrix. For γ ∈ Pτ (2n), we have the corresponding u(t) = u1 (t)+ −1u2 (t) √ ∈ U (n). So there exists a continuous real function (t) satisfying detu(t) = exp( −1(t)). We define τ (γ ) = (τ ) − (0) ∈ R which depends only on γ . ∗ (2n), we can connect γ (τ ) to M − or M + by a path β within For any γ ∈ Pτ,ω n n ∗ Sp(2n)ω and get a product path β ∗ γ defined by β ∗ γ (t) = γ (2t) if 0 ≤ t ≤ τ/2, β ∗ γ (t) = β(2t − τ ) if τ/2 ≤ t ≤ τ . Then k≡

1 τ (β ∗ γ ) ∈ Z. π

2.2 ω-Index iω (γ )

29

This integer k is independent of the special choice of the path β. As in [226], we define iω (γ ) = k ∈ Z. 0 (2n) = P (2n)  P ∗ (2n), we define For γ ∈ Pτ,ω τ τ,ω

iω (γ ) = inf{ iω (β)| β ∈ Pτ∗ (2n) and β is sufficiently C 0 -close to γ }. Theorem 2.3 ([226]) For any γ ∈ Pτ (2n) and ω ∈ U, the above definition yields (iω (γ ), νω (γ )) ∈ Z × {0, 1, · · · , 2n}, which is called the ω-index of γ . Note that the Maslov-type index coincides with the 1-index for any γ ∈ Pτ (2n): iω (γ ) = i1 (γ ),

νω (γ ) = ν1 (γ ), ω = 1.

This ω-index theory generalizes also corresponding Bott functions (ω) and N (ω) for closed geodesics defined in [28] and Ekeland index functions for convex Hamiltonian systems defined in [78]. For any γ ∈ Pτ (2n), we define the iteration path γ˜ ∈ C([0, +∞), Sp(2n)) of γ by γ˜ (t) = γ (t − j τ )γ (τ )j ,

for j τ ≤ t ≤ (j + 1)τ and j ∈ {0} ∪ N,

and denote by γ m = γ˜ |[0,mτ ] for m ∈ N. Theorem 2.4 ([226]) For any γ ∈ Pτ (2n) and k ∈ N, there hold i1 (γ k ) =



iω (γ ), ν1 (γ k ) =

ωk =1

iω0 (γ k ) =





νω (γ ),

(2.13)

ωk =1

iω (γ ), νω0 (γ k ) =

ωk =ω0

1 i1 (γ k ) = i¯1 (γ ) ≡ lim k→∞ k 2π

0



νω (γ ),

(2.14)

iexp(√−1θ) (γ ) dθ ∈ R.

(2.15)

ωk =ω0 2π

i¯1 (γ ) is called the Maslov-type mean index per period τ of γ ∈ Pτ (2n). Equation (2.13) is called the Bott-type formula for the index pair (i1 , ν1 ). Note that there holds i¯1 (γ k ) = k i¯1 (γ ),

∀γ ∈ Pτ (2n), k ∈ N.

(2.16)

30

2 A Brief Introduction to Index Functions

Theorem 2.5 (Homotopy invariance, [226]) For any two paths γ0 and γ1 ∈ Pτ (2n), if γ0 ∼ω γ1 on [0, τ ], there hold iω (γ0 ) = iω (γ1 ),

νω (γ0 ) = νω (γ1 ).

Theorem 2.6 (Symplectic additivity, [226]) For any γj ∈ Pτ (2nj ) with nj ∈ N, j = 0, 1, there holds iω (γ0  γ1 ) = iω (γ0 ) + iω (γ1 ). Lemma 2.7 ([223]) For a symplectic path γ ∈ P(2n) with γ (1) = M1  M2 , Mj ∈ Sp(2nj ), j = 1, 2, n1 + n2 = n, there exists two symplectic paths γj ∈ P(2nj ) such that γ ∼ γ1  γ2 and γj (1) = Mj . As proved in [226], the homotopic invariance, symplectic additivity, and the values on elements in Pτ (2) ∪ Pτ (4) uniquely determine the ω-index theory. For any τ > 0, γ ∈ Pτ (2n), the ω-index pair (iω (γ ), νω (γ )) is determined by the homotopic class of γ in Pτ (2n). In particular, iω (γ ) is completely determined by the homotopic component 0 (γ (τ )) up to an additive constant, and νω (γ ) is completely determined by 0 (γ (τ )). We define the basic normal forms of eigenvalues in U as:   λb N1 (λ, b) = , λ = ±1, b = ±1, 0, 0λ 

 cos θ − sin θ R(θ ) = , θ ∈ (0, π ) ∪ (π, 2π ), sin θ cos θ   R(θ ) B , θ ∈ (0, π ) ∪ (π, 2π ), N2 (ω, B) = 0 R(θ )   b b B = 11 12 , bij ∈ R, b12 = b21 . b21 b22 A basic normal form M is trivial, if for sufficiently small α > 0, MR((t −1)α)n possesses no eigenvalue on U for t ∈ [0, 1), and is non-trivial otherwise. Note that the matrix N1 (λ, b) is non-trivial if λb = −1 and trivial if λb = −1. For the Maslov-type index function iω (γ ), we have the following result which was proved in [197] (see also [196] and [223]). Proposition 2.8 ([197]) 1o For any γ ∈ P(2n) and ω ∈ U \ {1}, there always holds i1 (γ ) + ν1 (γ ) − n ≤ iω (γ ) ≤ i1 (γ ) + n − νω (γ ).

(2.17)

2.2 ω-Index iω (γ )

31

2o The left equality in (2.17) holds for some ω ∈ U+ \{1} (or U− \{1}) if and only if there holds I2p N1 (1, −1)q K ∈ 0 (γ (τ )) for some non-negative integers p and q satisfying 0 ≤ p + q ≤ n and K ∈ Sp(2(n − p − q)) with σ (K) ⊂ U \ {1} satisfying that all eigenvalues of K located within the arc between 1 and ω including ω in U+ (or U− ) possess total multiplicity n − p − q. If ω = −1, all eigenvalues of K are in U \ R and those in U+ \ R (or U− \ R) are all Krein negative (or positive) definite. If ω = −1, it holds that −I2s N1 (−1, 1)t H ∈ 0 (K) for some non-negative integers s and t satisfying 0 ≤ s + t ≤ n − p − q, and some H ∈ Sp(2(n − p − q − s − t)) satisfying σ (H ) ⊂ U \ R and that all elements in σ (H ) ∩ U+ (or σ (H ) ∩ U− ) are all Krein-negative (or Kreinpositive) definite. 3o The left equality in (2.17) holds for all ω ∈ U \ {1} if and only if I2p  N1 (1, −1)(n−p) ∈ 0 (γ (τ )) for some integer p ∈ [0, n]. Specifically in this case, all the eigenvalues of γ (τ ) equal to 1 and ντ (γ ) = n + p ≥ n. 4o The right equality in (2.17) holds for some ω ∈ U+ \{1} (or U− \{1}) if and only if there holds I2p  N1 (1, 1)r  K ∈ 0 (γ (τ )) for some non-negative integers p and r satisfying 0 ≤ p + r ≤ n and K ∈ Sp(2(n − p − r)) with σ (K) ⊂ U \ {1} satisfying the condition that all eigenvalues of K located within the closed arc between 1 and ω in U+ \ {1} (or U− \ {1}) possess total multiplicity n − p − r. If ω = −1, all eigenvalues in σ (K)∩U+ (or σ (K)∩U− ) are all Krein positive (or negative) definite; if ω = −1, there holds (−I2s )N1 (−1, 1)t H ∈ 0 (K) for some non-negative integers s and t satisfying 0 ≤ s + t ≤ n − p − r, and some H ∈ Sp(2(n − p − r − s − t)) satisfying σ (H ) ⊂ U \ R and that all elements in σ (H ) ∩ U+ (or σ (H ) ∩ U− ) are all Krein positive (or negative) definite. o 5 The right equality in (2.17) holds for all ω ∈ U \ {1} if and only if I2p  N1 (1, 1)(n−p) ∈ 0 (γ (τ )) for some integer p ∈ [0, n]. Specifically in this case, all the eigenvalues of γ (τ ) must be 1, and there holds ντ (γ ) = n + p ≥ n. 6o Both equalities in (2.17) hold for all ω ∈ U \ {1} if and only if γ (τ ) = I2n . The following results are useful iteration inequalities for the Maslov-type index i1 (γ ). Proposition 2.9 ([197]) 1o For any γ ∈ P(2n) and m ∈ N, there always holds mi¯1 (γ ) − n ≤ i1 (γ m ) ≤ mi¯1 (γ ) + n − ν1 (γ m ).

(2.18)

2o The right equality in (2.18) holds for all m ∈ N if and only if I2p  N1 (1, −1)(n−p) ∈ 0 (γ (τ )) for some integer number p ∈ [0, n]. Especially in this case, all the eigenvalues of γ (τ ) are equal to 1 and ν1 (γ ) = n + p ≥ n. 3o The left equality in (2.18) holds for all m ∈ N if and only if I2p  N1 (1, 1)(n−p) ∈ 0 (γ (τ )) for some integer number p ∈ [0, n]. Especially in this case, all the eigenvalues of γ (τ ) equal to 1 and ντ (γ ) = n + p ≥ n. 4o Both equalities in (2.18) hold for all m ∈ N if and only if γ (τ ) = I2n .

32

2 A Brief Introduction to Index Functions

Proposition 2.10 ([197]) 1o For any γ ∈ P(2n) and m ∈ N, there always holds m(i1 (γ )+ν1 (γ )−n)+n−ν1 (γ ) ≤ i1 (γ m ) ≤ m(i1 (γ )+n)−n−(ν1 (γ m )−ν1 (γ )). (2.19) 2o The left equality of (2.19) holds for some m > 1 if and only if I2p N1 (1, −1)q  K ∈ 0 (γ (τ )) for some non-negative integers p and q satisfying p + q ≤ n and some K ∈ Sp(2(n − p − q)) satisfying σ (K) ⊂ U \ {1}. If m = 2 and r = n − p − q > 0, then N1 (−1, −1)r ∈ 0 (K). If m ≥ 3 and r = n − p − q > 0, then R(θ1 )  · · ·  R(θr ) ∈ 0 (K) for some θj ∈ (0, π ) satisfying the condition mθ that 0 < 2πj ≤ 1 with 1 ≤ j ≤ r. In this case, all eigenvalues of K on U+ \ {1} √ − {1}) are located on the open arc between 1 and exp(2π −1/m) (and (on U \ √ exp(−2π −1/m)) in U+ (in U− ) and are all Krein negative (or Krein positive) definite. 3o The right equality of (2.19) holds for some m > 1 if and only if I2p N1 (1, 1)r  K ∈ 0 (γ (τ )) for some non-negative integers p and r satisfying p + r ≤ n and some K ∈ Sp(2(n − p − r)) with σ (K) = {−1} satisfying the following conditions: If m > 2, we must have n = p + r. If m = 2 and n − p − r > 0, there holds N1 (−1, 1)t  N1 (−1, −1)s ∈ 0 (K) for some non-negative integers s and t satisfying s + t = n − p − r. 4o The two equalities of (2.19) hold for some m = m1 and m = m2 ≥ 2 respectively if and only if γ (τ ) = I2p N1 (−1, −1)(n−p) for some non-negative integer p ≤ n. Here p < n happens only when m1 = m2 = 2. Proposition 2.11 ([233]) For any γ ∈ P(2n), and any m1 , m2 ∈ N, there hold ν1 (γ m1 ) + ν1 (γ m2 ) − ν1 (γ (m1 ,m2 ) ) −

e(γ (τ )) 2

≤ i1 (γ m1 +m2 ) − i1 (γ m1 ) − i1 (γ m2 ) ≤ ν1 (γ (m1 ,m2 ) ) − ν1 (γ m1 +m2 ) +

e(γ (τ )) , 2

where (m1 , m2 ) is the greatest common divisor of m1 and m2 . Proposition 2.12 ([233]) For any γ ∈ P(2n), and any m ∈ N, there hold ν1 (γ m ) −

e(γ (τ )) e(γ (τ )) ≤ i1 (γ m+1 ) − i1 (γ m ) − i1 (γ ) ≤ ν1 (γ ) − ν1 (γ m+1 ) + . 2 2 (2.20)

We remind that the upper and√lower semi-circle U± in U is defined by U+ = {e −1θ | θ ∈ [0, π ]} and U− = {e −1θ | θ ∈ [π, 2π ]}. √

2.2 ω-Index iω (γ )

33

Definition 2.13 ([226]) For any M ∈ Sp(2n) and ω ∈ U, choose τ > 0 and γ ∈ Pτ (2n) with γ (τ ) = M, and define ± (ω) = lim iexp ± √−1ω (γ ) − iω (γ ). SM →0+

These two integers are independent of the choice of τ > 0 and the path γ . They are called the splitting numbers of M at ω. + − + − It is easy to see that SM (1) = SM (1) and SM (−1) √ = SM (−1). When ω ∈ / σ (M), ± −1θ 0 , θ0 ∈ (0, π ], there holds we have SM (ω) = 0. So from definition, for ω0 = e + (1) + iω0 (γ ) = i1 (γ ) + SM



(S + (e

√

−1θ

) − S − (e

√

−1θ

− )) − SM (e

√

−1θ0

).

θ∈(0,θ0 )

(2.21) From the definition and the Bott type formula (2.14), we know that ± SM k (ω0 ) =



± SM (ω).

(2.22)

ωk =ω0

As a special case, we have ± ± ± SM 2 (1) = S (1) + S (−1).

(2.23)

We recall that the Krein form is defined by Gx, y

∀ x, y ∈ C2n ,

√ where G = −1J and ·, · is the standard Hermitian inner product in C2n . This quadratic form is symmetric in C2n . We suppose M ∈ Sp(2n). It is well known that if λ, μ ∈ σ (M), and λμ¯ = 1, then the generalized eigen-subspaces Eλ (M) and Eμ (M) are G-orthogonal. When |λ| = 1, then the restriction of G to Eλ (M) is non-degenerate. I.e., for x ∈ Eλ (M), Gx, y = 0 for any y ∈ Eλ (M) implies x = 0. Definition 2.14 For M ∈ Sp(2n), if λ ∈ σ (M) with |λ| = 1. Denote the total multiplicities of positive and negative eigenvalues of G|Eλ (M) by p and q respectively. The integer pair (p, q) is called the Krein type number of λ. If q = 0, λ is Krein positive. If p = 0, λ is Krein negative. Lemma 2.15 ([226]) For any M ∈ Sp(2n) and ω ∈ U, denote by (p, q) the Krein ± type number of ω ∈ σ (M). The splitting numbers SM (ω) are constant on 0 (M), and satisfy

34

2 A Brief Introduction to Index Functions ± 0 ≤ SM (ω) ≤ dimC kerC (M − ωI ), + − 0 ≤ SM (ω) ≤ p, 0 ≤ SM (ω) ≤ q, + − SM (ω) = SM (ω). ¯

We note that (2.17) can be deduced from formula (2.21) and Lemma 2.15. Lemma 2.16 ([223]) For the splitting numbers, we have the following list. + − (SM (1), SM (1)) = (1, 1), for M = N1 (1, b) with b = 0 or 1. + − (SM (1), SM (1)) = (0, 0), for M = N1 (1, −1). + − (SM (−1), SM (−1)) = (1, 1), for M = N1 (−1, b) with b = 0 or −1. + − (SM (−1), S M (−1))√= (0, 0), for M = N1 (−1, 1). √ + − −1θ (e√−1θ )) = (0, 1), for M = R(θ ) with θ ∈ (0, π ) ∪ (π, 2π ). (SM (e√ ), SM + − (SM (e√ −1θ ), SM (e −1θ )) = (1, 1), for M = N2 (ω, b) being non-trivial with ω = e √−1θ , θ ∈ R.√ + − (7) (S√M (e −1θ ), SM (e −1θ )) = (0, 0), for M = N2 (ω, B) being trivial with ω = e −1θ ,√θ ∈ R. √ + − (8) (SM (e −1θ ), SM (e −1θ )) = (0, 0), for any ω ∈ U and M ∈ Sp(2n) satisfying ω ∈ σ (M).

(1) (2) (3) (4) (5) (6)

Chapter 3

Relative Morse Index

3.1 Relative Index via Galerkin Approximation Sequences Let E be a separable Hilbert space, and Q = A − B : E → E be a bounded self-adjoint linear operators with B : E → E being a compact self-adjoint operator. Suppose that N = ker Q and dim N < +∞. Q|N ⊥ is invertible. P : E → N is the orthogonal projection. We denote 0 < d ≤ 14 #(Q|N ⊥ )−1 #−1 . Suppose  = {Pk |k = 1, 2, · · · } is the Galerkin approximation sequence of A with (1) Ek := Pk E is finite dimensional for all k ∈ N, (2) Pk → I strongly as k → +∞ (3) Pk A = APk . For a self-adjoint operator T , we denote by M ∗ (T ) the eigenspaces of T with eigenvalues belonging to (0, +∞), {0} and (−∞, 0) with ∗ = +, 0 and ∗ = −, respectively. We denote by m∗ (T ) = dim M ∗ (T ). Similarly, we denote by Md∗ (T ) the d-eigenspaces of T with eigenvalues belonging to (d, +∞), (−d, d) and (−∞, −d) with ∗ = +, 0 and ∗ = −, respectively. We denote by m∗d (T ) = dim Md∗ (T ). For any adjoint operator L, we denote L = (L|I mL )−1 . Lemma 1.1 There exists m0 ∈ N such that for all m ≥ m0 , there hold m− (Pm (Q + P )Pm ) = m− d (Pm (Q + P )Pm )

(3.1)

m− (Pm (Q + P )Pm ) = m− d (Pm QPm ).

(3.2)

and

Proof The proof of (3.1) is essential the same as that of Theorem 2.1 of [95], we note that dim ker(Q + P ) = 0.

© Springer Nature Singapore Pte Ltd. 2019 C. Liu, Index theory in nonlinear analysis, https://doi.org/10.1007/978-981-13-7287-2_3

35

36

3 Relative Morse Index

By considering the operators Q + sP and Q − sP for small s > 0, for example s < min{1, d/2}, there exists m1 ∈ N such that − m− d (Pm QPm ) ≤ m (Pm (Q + sP )Pm ), ∀ m ≥ m1

(3.3)

and − 0 m− d (Pm QPm ) ≥ m (Pm (Q − sP )Pm ) − md (Pm QPm ), ∀ m ≥ m1 .

(3.4)

In fact, the claim (3.3) follows from Pm (Q + sP )Pm = Pm QPm + sPm P Pm and for x ∈ Md− (Pm QPm ), d (Pm (Q + sP )Pm x, x) ≤ −d#x#2 + s#x#2 ≤ − #x#2 . 2 The claim (3.4) follows from that for x ∈ M − (Pm (Q − sP )Pm ), (Pm QPm x, x) ≤ s(Pm P Pm x, x) < d#x#2 . By the Floquet theory, for m ≥ m1 we have m0d (Pm QPm ) = dim N = dim I m(Pm P Pm ), and by I m(Pm P Pm ) ⊆ Md0 (Pm QPm ) we have I m(Pm P Pm ) = Md0 (Pm QPm ). It is easy to see that Md0 (Pm QPm ) ⊆ Md+ (Pm (Q + sP )Pm ). By using Pm (Q − sP )Pm = Pm (Q + sP )Pm − 2sPm P Pm we have m− (Pm (Q−sP )Pm ) ≥ m− (Pm (Q+sP )Pm )+m0d (Pm QPm ), ∀ m ≥ m1 . Now (3.2) follows from (3.3), (3.4), and (3.5). M − (Q

(3.5)  

M − (Q)

Since + P) = and the two operators Q + P and Q have the same negative spectrum, moreover, Pm (Q + P )Pm → Q + P and Pm QPm → Q strongly, one can prove (3.2) by the spectrum decomposition theory. Lemma 1.2 Let B be a linear self-adjoint compact operator as above. Then m0d (Pm (A − B)Pm ) eventually becomes a constant independent of m and for large m, there holds m0d (Pm (A − B)Pm ) = m0 (A − B).

(3.6)

3.1 Relative Index via Galerkin Approximation Sequences

37

Proof It is easy to show that there is a constant m1 > 0 such that for m ≥ m1 dim Pm ker(A − B) = dim ker(A − B). Since B is compact, there is m2 ≥ m1 such that for m ≥ m2 #(I − Pm )B# ≤ 2d. Take m ≥ m2 , let Em = Pm ker(A − B) we have



Ym , then Ym ⊆ Im(A − B). For y ∈ Ym

y = (A − B) (A − B)y = (A − B) (Pm (A − B)Pm y + (I − Pm )By). It implies #Pm (A − B)Pm y# ≥ 2d#y#, ∀y ∈ Ym . Thus we have m0d (Pm (A − B)Pm ) ≤ m0 (A − B).

(3.7)

On the other hand, for x ∈ Pm ker(A − B), there exists y ∈ ker(A − B), such that x = Pm y. Since Pm → I strongly, there exists m3 ≥ m2 such that for m ≥ m3 #I − Pm #
0 and m∗ > 0 such that

38

3 Relative Morse Index

m0d (Pm (A − B(s))Pm ) = m0 (A − B(s)), m > m∗ , where B(s) = (1 − s)B1 + sB2 , s ∈ [0, 1]. Theorem 1.3 For any two operators B1 and B2 ∈ Ls (E) with B1 < B2 , there is m∗ > 0 such that  − (P (A − B )P ) − m (P (A − B )P ) = m0 (A − B(s)), m > m∗ . m− m 2 m m 1 m d d s∈[0,1)

(3.10) − 0 0 Proof Denote by m− d (s) = md (Pm (A − B(s))Pm ), md (s) = md (Pm (A − 0 0 B(s))Pm ) = m (A − B(s)). If m (A − B(s0 )) = 0, then there is a neighborhood B(s0 , δ) of s0 such that for s ∈ B(s0 , δ), there hold m0d (Pm (A − B(s))Pm ) = 0 m0 (A − B(s)) = 0. Thus m− d (s) is constant in B(s0 , δ). If m (A − B(s0 )) = 0, − − we claim that md (s0 + 0) − md (s0 ) = ν(s0 ). In fact, on one side hand, by the continuity of the eigenvalue of continuous operator function, we have m− (s0 + 0) − m− (s0 ) ≤ ν(s0 ). On the other side hand, since (A − B(s0 )) > (A − B(s)), for s0 < s, we see that m0 (A − B(s)) = 0 for s > s0 but s − s0 small enough. So m0d (s) = m0d (Pm (A − B(s))Pm ) = 0. Since Pm (A − B(s0 ))Pm ≥ Pm (A − B(s))Pm , − 0 0 so we have m− d (s0 + 0) + md (s0 + 0) ≥ md (s0 ) + md (s0 ). Thus the claim is true by using Lemma 1.1. Therefore we have the equality (3.10).  

Lemma 1.4 Let B be a linear self-adjoint compact operator. Then the difference of the d-Morse indices − m− d (Pm (A − B)Pm ) − md (Pm APm )

(3.11)

eventually becomes a constant independent of m, where d > 0 is determined by the operators A and A − B. A similar result was proved in [38]. Proof We can choose B0 < 0 and B0 < B, so we have − m− d (Pm (A − B)Pm ) − md (Pm APm ) − = (m− d (Pm (A − B)Pm ) − md (Pm (A − B0 )Pm )) − −(m− d (Pm APm ) − md (Pm (A − B0 )Pm )).

  Definition 1.5 For the self-adjoint bounded Fredholm operator A with a Galerkin approximation sequence  and the self-adjoint compact operator B on Hilbert space E, we define the relative Morse index by − I (A, A − B) = m− d (Pm (A − B)Pm ) − md (Pm APm ),

m ≥ m∗ ,

(3.12)

3.1 Relative Index via Galerkin Approximation Sequences

39

where m∗ > 0 is a constant large enough such that the difference in (3.12) becomes a constant independent of m ≥ m∗ . By Lemma 1.4 we have the following Remark 1.6 Let E˜ be another separable Hilbert space, A˜ be a linear self-adjoint ˜ Fredholm operator on E˜ and B be a compact linear self-adjoint operator on E. There holds ˜ (A ⊕ A) ˜ − (B ⊕ B)) ˜ = I (A, A − B) + I (A, ˜ A˜ − B), ˜ I (A ⊕ A, ˜ ⊕y) = Ax ⊕ Ay ˜ and (B ⊕ B)(x ˜ ⊕y) = Bx ⊕ By ˜ for x ⊕y ∈ E⊕ E. ˜ where (A⊕ A)(x The spectral flow for a parameter family of linear self-adjoint Fredholm operators was introduced by Atiyah, Patodi and Singer in [11]. The following result shows that the relative index in Definition 1.5 is a spectral flow. Lemma 1.7 For the operators A and B in Definition 1.5, there holds I (A, A − B) = −sf{A − sB, 0 ≤ s ≤ 1},

(3.13)

where sf(A − sB, 0 ≤ s ≤ 1) is the spectral flow of the operator family A − sB, s ∈ [0, 1] (cf. [316]). Proof For simplicity, we set Isf (A, A − B) = −sf{A − sB, 0 ≤ s ≤ 1} which is exact the relative Morse index defined in [316]. By the Galerkin approximation formula in Theorem 3.1 of [316], Isf (A, A − B) = Isf (Pm APm , Pm (A − B)Pm )

(3.14)

if ker(A) = ker(A − B) = 0. By (2.17) of [316], we have Isf (Pm APm , Pm (A − B)Pm ) = m− (Pm (A − B)Pm ) − m− (Pm APm ) − = m− d (Pm (A − B)Pm ) − md (Pm APm )

= I (A, A − B)

(3.15)

for d > 0 small enough. Hence (3.13) holds in the nondegenerate case. In general, if ker(A) = 0 or ker(A − B) = 0, we can choose d > 0 small enough such that ker(A + dId) = ker(A − B + dId) = 0, here Id : E → E is the identity operator. By (2.14) of [316] we have Isf (A, A − B) = Isf (A, A + dId) + Isf (A + dId, A − B + dId) +Isf (A − B + dId, A − B) = Isf (A + dId, A − B + dId) = I (A + d · Id, A − B + d · Id)

40

3 Relative Morse Index

= m− (Pm (A − B + dId)Pm ) − m− (Pm (A + dId)Pm ) − = m− d (Pm (A − B)Pm ) − md (Pm APm ) = I (A, A − B).

(3.16)

In the second equality of (3.16) we note that Isf (A, A + dId) = Isf (A − B + dId, A − B) = 0 for d > 0 small enough since the spectrum of A is discrete and B is a compact operator, in the third and the forth equalities of (3.16) we have applied (3.15).   A similar way to define the relative index of two operators was shown in [38]. A different way to study the relative index theory was presented in [91].

3.2 Relative Morse Index via Orthogonal Projections In general, let E be a separable Hilbert space, for any self-adjoint operator A on E, there is a unique A-invariant orthogonal splitting E = E+ (A) ⊕ E− (A) ⊕ E0 (A),

(3.17)

where E0 (A) is the null space of A, A is positive definite on E+ (A) and negative definite on E− (A). We denote by PA the orthogonal projection from E to E− (A). For any bounded self-adjoint Fredholm operator F and a compact self-adjoint operator T on E, PF − PF −T is compact (cf. Lemma 2.7 of [316]), where PF : E → E− (F) and PF −T : E → E− (F − T ) are the respective projections. Then by Fredholm operator theory, PF |E− (F −T ) : E− (F − T ) → E− (F) is a Fredholm operator. Here and in the sequel, we denote by ind (·) the Fredholm index of a Fredholm operator. Definition 2.1 For any bounded self-adjoint Fredholm operator F and a compact self-adjoint operator T on E, the relative Morse index pair (μF (T ), υF (T )) is defined by μF (T ) = ind(PF |E− (F −T ) ).

(3.18)

υF (T ) = dim E0 (F − T ).

(3.19)

and

Let {Fθ |θ ∈ [0, 1]} be a continuous path of self-adjoint Fredholm operators on the Hilbert space E. It is well known that the concept of spectral flow Sf (Fθ ) was first introduced by Atiyah, Patodi and Singer in [11], and then extensively studied in [32, 98, 259, 260, 316]. The following proposition displays the relationship between spectral flow and the relative Morse index defined above.

3.3 Morse Index via Dual Methods

41

Proposition 2.2 (cf. [40]) Suppose that, for each θ ∈ [0, 1], Fθ − F0 is a compact operator on E, then ind(PF0 |E− (F1 ) ) = −sf (Fθ ). Thus, from Definition 2.1, μF0 (T ) = −sf (Fθ , 0 ≤ θ ≤ 1), where Fθ = F − θ T , T is a compact operator. Moreover, if σ (T ) ⊂ [0, ∞) and 0∈ / σP (T ), from the definition of Spectral flow, we have μF0 (T ) = −sf (Fθ , 0 ≤ θ ≤ 1)  υF (θ T ) = θ∈[0,1)

=



dim E0 (F − θ T ).

(3.20)

θ∈[0,1)

3.3 Morse Index via Dual Methods Let H be an infinitely dimensional separable Hilbert space with inner product (·, ·)H and norm # · #H . Denote by O(H ) the set of all linear self-adjoint operators on H . For A ∈ O(H ), we denote by σ (A) the spectrum of A and σe (A) the essential spectrum of A. We define three subsets of O(H ) as follows Oe− (μ) = {A ∈ O(H )| σe (A) ∩ (−∞, μ) = ∅ and σ (A) ∩ (−∞, μ) = ∅}, Oe+ (μ) = {A ∈ O(H )| σe (A) ∩ (μ, +∞) = ∅ and σ (A) ∩ (μ, +∞) = ∅}, Oe0 (a, b) = {A ∈ O(H )| σe (A) ∩ (a, b) = ∅ and σ (A) ∩ (a, b) = ∅}. We note that if μ = +∞ and A ∈ Oe− (μ), then σe (A) = ∅. If σe (A) = ∅ and A ∈ Oe− (μ) for some μ, then −∞ < μ < +∞ is a real number. Setting λ− = inf(σe (A)), we have −∞ < λ− < +∞ is real number and A ∈ Oe− (λ− ). Similarly, if μ = −∞ and A ∈ Oe+ (μ), then σe (A) = ∅. If σe (A) = ∅ and A ∈ Oe+ (μ) for some μ, then −∞ < μ < +∞ is a real number. Setting λ+ = sup(σe (A)), we have −∞ < λ+ < +∞ is real number and A ∈ Oe+ (λ+ ). If the operator A is fixed and A ∈ Oe− (μ) or A ∈ Oe+ (μ), we always write it in A ∈ Oe− (λ− ) or A ∈ Oe+ (λ+ ) with λ∓ in the above sense. We remind that inf ∅ = +∞ and sup ∅ = −∞. We have a remark that when A ∈ O(H ) with σe (A) ∩ (−∞, μ) = ∅ and σ (A) ∩ (−∞, μ) = ∅, the index iA− (B) in case 1 below is still well defined for

42

3 Relative Morse Index

− − B ∈ L− s (H, λ ), but all the indices are the same, i.e., iA (B) is a constant function − − − on Ls (H, λ ) and so the index iA (B) take no further information. We also have a similar remark for other two cases. Let A ∈ O(H ) satisfying σ (A) \ σe (A) = ∅. Now, we consider the following cases:

Case 1. A ∈ Oe− (λ− ). Case 2. A ∈ Oe+ (λ+ ). Case 3. A ∈ Oe0 (λa , λb ), −∞ < λa < λb < +∞ and λa , λb ∈ / σ (A). Denote Ls (H ) the set of all linear bounded self-adjoint operators on H . − + + Corresponding to Case 1, Case 2 and Case 3, we define L− s (H, λ ), Ls (H, λ ) 0 and Ls (H, λa , λb ) three subsets of Ls (H ) respectively by − − L− s (H, λ ) = {B ∈ Ls (H ), B < λ · I },

(3.21)

+ + L+ s (H, λ ) = {B ∈ Ls (H ), B > λ · I },

(3.22)

L0s (H, λa , λb ) = {B ∈ Ls (H ), λa · I < B < λb · I

(3.23)

and

where I is the identity map on H , the inequality B < λ− · I means that there exists δ > 0 such that (λ− − δ) · I − B is positive define, B > λ+ · I and λa · I < − + + B < λb · I have similar meanings. It is easy to see L− s (H, λ ), Ls (H, λ ) and 0 Ls (H, λa , λb ) are open and convex subsets of Ls (H ). We will define the index pair (iA∓ (B), νA∓ (B)) and (iA0 (B), νA0 (B)) in three cases.

3.3.1 The Definition of Index Pair in Case 1 and 2 In this subsection, we will give the definition of our index pair in Case 1 and Case 2. Let’s begin with some useful lemmas. Lemma 3.1 ([284]) If A ∈ Oe− (λ− ) (or A ∈ Oe+ (λ+ ) ), for any B ∈ − + + L− s (H, λ ) (or B ∈ Ls (H, λ )), dim ker(A − B) < ∞. Further more, if 0 ∈ σ (A − B), then 0 is an isolated point spectrum. − Proof We only consider the situation of A ∈ Oe− (λ− ) and B ∈ L− s (H, λ ), the proof for other situations is similar. Since B is a bounded self-adjoint operator on H , we can choose a number k ∈ R satisfying

k∈ / σ (A), k · I < B.

(3.24)

Then we have A − k · I is invertible and 0 < B − k · I < (λ− − k) · I.

(3.25)

3.3 Morse Index via Dual Methods

43

That is to say B − k · I is a positive definite self-adjoint operator on H and #B − k · I # < λ− − k.

(3.26)

If σ (A) ∩ (−λ− + 2k, λ− ) = ∅, that is σ (A − k · I ) ∩ (−λ− + k, λ− − k) = ∅. From (3.26) it is easy to see 0 ∈ / σ (A − B). Now, we assume σ (A) ∩ (−λ− + 2k, λ− ) = ∅.

(3.27)

For simplicity, we denote Ak := A−k ·I and Bk := B −k ·I , then A−B = Ak −Bk , specially ker(A−B) = ker(Ak −Bk ) and σ (A−B) = σ (Ak −Bk ). We need the idea of Lyapunov-Schmidt reduction which is also the idea of saddle point reduction. Let E(z) be the spectral measure of Ak . From (3.26) and (3.27), there exists δ > 0 small enough such that #Bk # < λ− − k − δ,

(3.28)

and σ (Ak ) ∩ (−λ− + k + δ, λ− − k − δ) = ∅. Denote P0 :=

λ− −k−δ −(λ− −k−δ)

1dE(z),

and P1 := I − P0 . Then H has the decomposition H = H0 ⊕ H1 with H0 = P0 H and H1 = P1 H . From our assumption there are only finite eigenvalues of Ak in set (−λ− + k + δ, λ− − k − δ), each of them has finite dimensional eigenspace. So H0 is a finite dimensional space. Denote E = D(|Ak |1/2 ), since k ∈ / σ (A) and 0 ∈ / σ (Ak ), thus E is a Hilbert space with the inner product (·, ·)E and corresponding norm # · #E defined by (x, y)E := (|Ak |1/2 x, |Ak |1/2 y)H , ∀x, y ∈ E, #x#2E := (x, x)E , ∀x ∈ E. We also have the following decomposition E = E0 ⊕ E1 ,

(3.29)

44

3 Relative Morse Index

with E0 = E ∩ H0 and E1 = E ∩ H1 . The operator Ak and Bk will define two bounded self-adjoint operators A˜ k and B˜ k on E by (A˜ k x, y)E := (Ak x, y)H , ∀x, y ∈ E, and (B˜ k x, y)E := (Bk x, y)H , ∀x, y ∈ E. It is easy to see A˜ k = |Ak |−1 Ak and B˜ k = |Ak |−1 Bk . Thus ker(Ak − Bk ) = ker(A˜ k − B˜ k ). Further more, we can write A˜ k and B˜ k in the block form    ˜ A˜ k,1 0 ˜ k = Bk,11 , B A˜ k = ˜ B˜ k,21 0 Ak,2

 B˜ k,12 , B˜ k,22

(3.30)

with respect to the decomposition (3.29). For any u ∈ E, u = x + y with x ∈ E0 and y ∈ E1 , the equation A˜ k u = B˜ k u can be rewritten as       x x A˜ k,1 0 B˜ k,11 B˜ k,12 = ˜ . ˜ ˜ Bk,21 Bk,22 0 Ak,2 y y That is 

A˜ k,1 x = B˜ k,11 x + B˜ k,12 y, A˜ k,2 y = B˜ k,21 x + B˜ k,22 y.

From (3.28), the definitions of P0 , P1 , A˜ k,2 and B˜ k,22 , we have A˜ k,2 is invertible on ˜ E1 and #A˜ −1 k,2 Bk,22 #E < 1. Thus we have y = (A˜ k,2 − B˜ k,22 )−1 B˜ k,21 x, and A˜ k u = B˜ k u ⇐⇒ A˜ k,1 x = [B˜ k,11 + B˜ k,12 (A˜ k,2 − B˜ k,22 )−1 B˜ k,21 ]x. It’s easy to see dim ker(A − B) ≤ dim(E0 ) < ∞ and we have proved the first part. In order to prove the rest part of the lemma, by arguing indirectly, assume 0 is

3.3 Morse Index via Dual Methods

45

− not an isolated point spectrum of A − B for some fixed B ∈ L− s (H, λ ). That is to say Hε := Qε H is an infinite dimensional space for any small ε > 0, where Qε is ˆ a projection map corresponding to the spectral measure E(z) of A − B defined by

Qε =

ε/2 −ε/2

ˆ 1d E(z).

Now we can choose ε > 0 small enough, such that B < (λ− − ε) · I and define a self-adjoint operator Cε =

ε/2

−ε/2

ˆ zd E(z).

− Then we have B + Cε < (λ− − 12 ε) · I , that is to say B + Cε ∈ L− s (H, λ ). On the other hand we have ker(A − B − Cε ) = Hε , which is contradict to the fact dim ker(A − B − Cε ) < ∞. Thus we have proved the lemma.   − Now for any B ∈ L− s (H, λ ) and k ∈ R satisfying (3.24), consider the bounded − self-adjoint operator TB,k on H defined by − − − TB,k := Bk−1 − A−1 k , ∀B ∈ Ls (H, λ ).

(3.31)

Firstly, the invertible map Bk−1 establishes the one-to-one correspondence between − − ) and ker(A − B), so we have dim ker(TB,k ) = dim ker(A − B). Secondly, ker(TB,k from (3.25) and the definition of H1 , we have − (TB,k y, y)H > c(y, y)H , ∀y ∈ H1 ,

for some fixed c > 0. Since H has the decomposition H = H0 ⊕ H1 and − has only finite dimensional negative definite subspace, that is dim H0 < ∞, TB,k − ) has only finite points with finite dimensional eigenvalue to say (−∞, 0) ∩ σ (TB,k space. Summed up, we have the following lemma. − Lemma 3.2 ([284]) Suppose A ∈ Oe− (λ− ). For any B ∈ L− s (H, λ ) and k ∈ R satisfying (3.24), there is an orthogonal decomposition of H

H = HT−− ⊕ HT0 − ⊕ HT+− , B,k

B,k

B,k

− such that TB,k is negative definite, zero and positive definite on HT−− , HT0 − and B,k

HT+− respectively. Further more B,k

dim HT−− < ∞, dim HT0 − = dim ker(A − B). B,k

B,k

B,k

46

3 Relative Morse Index

− Thus if A ∈ Oe− (λ− ), for any B ∈ L− s (H, λ ) and k ∈ R satisfying (3.24) we − − denote the Morse index pair of TB,k by (iA,k (B), νA− (B)), that is − iA,k (B) := dim HT−− , νA− (B) := dim HT0 − .

(3.32)

B,k

B,k

− (B) depends on the choose of k. But we will show that Of cause the index iA,k − − − iA,k (B1 ) − iA,k (B2 ) will not depend on k for any fixed B1 , B2 ∈ L− s (H, λ ), it only depends on B1 , B2 and A. For this purpose, we need the following lemma. − Lemma 3.3 ([284]) Suppose A ∈ Oe− (λ− ). For any B1 , B2 ∈ L− s (H, λ ) satisfying B1 < B2 , we have − − iA,k (B2 ) − iA,k (B1 ) =



ν(A − (1 − s)B1 − sB2 )

s∈[0,1)

for any k ∈ R \ σ (A) and k · I < B1 , B2 . Where ν(P ) = dim ker P is the nullity of the linear operator P . − Proof Denote i(s) := iA,k (B(s)) and ν(s) = ν(A − B(s)), where B(s) := (1 − s)B1 + sB2 . Since B1 < B2 , we have B(s1 ) < B(s2 ), for any 0 ≤ s1 < s2 ≤ 1, so we have

Bk−1 (s1 ) > Bk−1 (s2 ) > 0, 0 ≤ s1 < s2 ≤ 1, and T (s1 ) > T (s2 ), 0 ≤ s1 < s2 ≤ 1, where T (s) := Bk−1 (s) − A−1 k and the map T (s) : [0, 1] → Ls (H ) is continuous. − Firstly, from the definition of iA,k (·), it’s easy to see i(s) is left continuous and − 0 ≤ i(s1 ) ≤ i(s2 ) ≤ iA,k (B2 ), ∀ 0 ≤ s1 < s2 ≤ 1.

Further more, for s0 ∈ [0, 1], if ν(s0 ) = 0 then i(s) is continuous at s0 . If ν(s) = 0, we have i(s + 0) − i(s) = ν(s). In fact, by the continuity of the eigenvalue of continuous operator function, we have i(s + 0) − i(s) ≤ ν(s). On the other side, since T (s1 ) > T (s2 ), for s1 < s2 , we see that i(s +0)−i(s) ≥ ν(s). From the above − properties of i(s) and the fact that i(s) ∈ [0, iA,k (B2 )] ∩ Z, thus there are only finite number of s ∈ [0, 1] such that ν(s) = 0 and − − iA,k (B2 ) − iA,k (B1 ) =



ν(A − (1 − s)B1 − sB2 ).

s∈[0,1)

Thus we have proved the lemma.

 

3.3 Morse Index via Dual Methods

47

Remark 3.4 With the same methods, if A ∈ Oe+ (λ+ ), we can define the index pair + for B ∈ L+ s (H, λ ), with any k ∈ R satisfying k∈ / σ (A), k · I > B.

(3.33)

In this case, we redefine the operators Ak := k · I − A and Bk := k · I − B. Consider the bounded self-adjoint operator + + + := Bk−1 − A−1 TB,k k , ∀B ∈ Ls (H, λ ). + It is easy to see TB,k has finite dimensional negative definite space and + + = dim ker(A − B). Similarly, define iA,k (B) the dimension of the dim ker TB,k + + negative definite space of TB,k and νA (B) = dim ker(A − B).

And we also have the following lemma. + Lemma 3.5 ([284]) If A ∈ Oe+ (λ+ ), for any B1 , B2 ∈ L+ s (H, λ ) satisfying B2 < B1 , we have  + + (B2 ) − iA,k (B1 ) = ν(A − (1 − s)B1 − sB2 ) iA,k s∈[0,1)

for any k ∈ R \ σ (A) and k · I > B1 , B2 . The proof is similar to the proof of Lemma 3.3, so we omit it here. Let B − ∈ − + + + L− s (H, λ ) and B ∈ Ls (H, λ ) be fixed. ∓ Definition 3.6 ([284]) If A ∈ Oe∓ (λ∓ ), for any B ∈ L∓ s (H, λ ), define the index ∓ ∓ pair (iA (B), νA (B)) by ∓ ∓ (B) − iA,k (B ∓ ), iA∓ (B) := iA,k

νA∓ (B) := dim ker(A − B), with some k ∈ R \ σ (A) satisfying ∓k · I > ∓B, and ∓k · I > ∓B ∓ . We note that νA− (B) = νA+ (B) when they all make sense, so in sequel, we write it in νA (B) for simplicity. The definition is well defined, we will prove that it only depends on the choice of − ˜ B ∓ . We only consider the case of B ∈ L− s (H, λ ), by Lemma 3.3, for any k ∈ R − − ˜ ˜ satisfying B, B < k · I and k < λ , − − − ˜ − − ˜ − (B) − iA,k (B − ) = (iA,k (k · I ) − iA,k (B − )) − (iA,k (k · I ) − iA,k (B)) iA,k   ν(A − (1 − s)B − − s k˜ · I ) − ν(A − (1 − s)B − s k˜ · I ), = s∈[0,1)

s∈[0,1)

48

3 Relative Morse Index

where the right hand side does not depend on the choice of k and we have proved that the definition of iA∓ (B) is well defined. In this definition, for the fixed operators B ∓ , we have iA∓ (B ∓ ) = 0. For any other choice of the operators B ∓ , the corresponding index is different up to a constant. Remark 3.7 A. It is worth to note that we don’t care about the choice of B ∓ though they have affects on the definition of iA∓ (B). What we care about is the difference between ∓ iA∓ (B1 ) and iA∓ (B2 ) with B1 , B2 ∈ L∓ s (H, λ ). From the definition, we have ∓ ∓ ∓ ∓ iA (B2 ) − iA (B1 ) = iA,k (B2 ) − iA,k (B1 ), that is to say iA∓ (B2 ) − iA∓ (B1 ) is the difference between the Morse indices of TB∓2 ,k and TB∓1 ,k . Thus for any B ∈ ∓ ∓ L∓ s (H, λ ), we call iA (B) the relative Morse index of B. B. In the definition of our index iA∓ (B), the number k, in fact the operator k · I , can ˆ For example, in Definition 3.6, be replaced by bounded self-adjoint operator B. − ), we can choose a bounded self-adjoint operator B ˆ (H, λ for any B ∈ L− s ˆ ˆ satisfying 0 ∈ / σ (A − B) and B < B. C. If the self-adjoint operator A on H has no essential spectrum that is to say A has compact resolvent (i.e., A is Fredholm operator), in this case we define λ− = − +∞ then L− s (H, λ ) = Ls (H ) and for any B ∈ Ls (H ) we can also define the − index pair (iA (B), νA− (B)). Definition 3.8 Suppose A is a Fredholm operator on H . For any bounded selfadjoint operator B ∈ Ls (H ), we define the index pair (iA (B), νA (B)) by (iA (B), νA (B)) = (iA− (B), νA− (B)). By Lemma 3.3 it’s easy to see the index pair (iA (B), νA (B)) will coincide with the indexes defined in [69, 282] and [283] up to a constant, so the index theory introduced in this subsection can be regarded as a generalization of the ones in [69, 282] and [283].

3.3.2 The Definition of Index Pair in Case 3 Now, as a supplement to Case 1 and Case 2, we are in the position of defining the index pair in Case 3. Let A ∈ Oe0 (λa , λb ) for some λa , λb ∈ R, from the definition of Oe0 (λa , λb ), we can assume that λa , λb ∈ / σ (A). Firstly, we have the same result of Lemma 3.1 for any B ∈ L0s (H, λa , λb ). We give a brief proof here. Choose the b in Lemma 3.1. If k ∈ / σ (A), the rest part of the proof will be number k = λa +λ 2 same as done before. Otherwise, k ∈ σ (A), since there is no essential spectrum in (λa , λb ), k is an eigenvalue of A with finitely dimensional eigenspace. Let Pk the projection map on the eigenspace of k, so Pk is a self-adjoint operator with finite rank. Then k ∈ / σ (A+Pk ), the results in Lemma 3.1 will keep valid for (A+Pk )−B.

3.4 Saddle Point Reduction for the General Cases

49

Since Pk is a finite rank self-adjoint operator, the results in Lemma 3.1 will keep valid for (A + Pk ) − (B + Pk ). Secondly, compared to the Case 1 and Case 2, instead of using dual variational method to define the index here, we define the index pair for B ∈ L0s (H, λa , λb ) by b the Morse index of saddle point reduction. Recall k = λa +λ 2 , and Pk defined above, let  A − k · I, k∈ / σ (A), Ak := A − k · I + Pk , k ∈ σ (A), and  Bk :=

B − k · I, k∈ / σ (A), B − k · I + Pk , k ∈ σ (A).

Thus, we can also define the Hilbert space E, the bounded self-adjoint operator A˜ k and B˜ k on E, and we also have the decomposition as (3.29) and the block form (3.30) of A˜ k and B˜ k , the only difference is the definition of P0 . Let E(z) be the spectral measure of Ak , redefine the projection map P0 by P0 =

(λb −λa )/2 −(λb −λa )/2

1dE(z).

With the above discussion, we have the following definition. Definition 3.9 ([284]) Assume A ∈ Oe0 (λa , λb ). For any B ∈ L0s (H, λa , λb ), define the index pair (iA0 (B), νA0 (B)) by iA0 (B) := m− (A˜ k,1 − [B˜ k,11 + B˜ k,12 (A˜ k,2 − B˜ k,22 )−1 B˜ k,21 ]), νA0 (B) := dim ker(A − B), where A˜ k,1 − [B˜ k,11 + B˜ k,12 (A˜ k,2 − B˜ k,22 )−1 B˜ k,21 ] is a self-adjoint operator on finite dimensional space E0 , and m− (P ) is the Morse index of the quadratic form (P z, z)H defined by the self-adjoint operator P .

3.4 Saddle Point Reduction for the General Cases Let H be a Hilbert space with inner product (·, ·)H and norm # · #H , A ∈ O(H ) be a self-adjoint linear operator with compact resolvent and dense domain D(A). B ∈ Ls (H ) is a bounded self-adjoint linear operator on H with its operator norm / σ (A). Denote by N = ker(A) the kernel of A and P0 = H → N #B#H < c, ±c ∈ the projection. We set A˜ = A + P0 . Denote by Eλ the spectral resolution of the ˜ we define the projections on H by self-adjoint operator A,

50

3 Relative Morse Index

P=

c

−c

dEλ , P

+

=

+∞

dEλ , P

−

c

=

−c

−∞

dEλ .

The Hilbert space H possesses an orthogonal decomposition H = H + ⊕ H − ⊕ X, where H ± = P ± H , and X = PH is a finite dimensional space. We consider the quadratic functional f (z) =

1 ((A − B)z, z)H , z ∈ D(A) ⊂ H. 2

Theorem 4.1 ([284]) There exist a function a ∈ C 2 (X, R) and a linear map u : X → H satisfying the following conditions: 1o The map u has the form u(x) = w(x) + x with Pw(x) = 0. 2o The function a satisfies a(x) = f (u(x)) =

1 1 ((A − B)u(x), u(x))H = ((A − B  )x, x)H , 2 2

where B  : X → X is defined in (3.37) below. 3o x ∈ X is a critical point of a, if and only if z = u(x) is a critical point of f , i.e. z = u(x) ∈ ker(A − B). ˜ 1/2 ), since 0 ∈ ˜ thus E is a Hilbert space with the Proof Denote E = D(|A| / σ (A), inner product (·, ·)E and corresponding norm # · #E defined by ˜ 1/2 x, |A| ˜ 1/2 y)H , ∀x, y ∈ E, (x, y)E := (|A| #x#2E := (x, x)E , ∀x ∈ E. We also have the following decomposition E = E0 ⊕ E1 ,

(3.34)

with E0 = E ∩ X and E1 = E ∩ (H + ∪ H − ). The operators A and B will define two bounded self-adjoint operator A¯ and B¯ on E by ¯ y)E := (Ax, y)H , ∀x, y ∈ E, (Ax, and ¯ y)E := (Bx, y)H , ∀x, y ∈ E. (Bx,

3.4 Saddle Point Reduction for the General Cases

51

˜ −1 A and B¯ = |A| ˜ −1 B. Thus It is easy to see A¯ = |A| ¯ = ker(A − B). ker(A¯ − B) Further more, we can write A¯ and B¯ in the block form     B¯ 11 B¯ 12 A¯ 1 0 ¯ ¯ , B= ¯ ¯ , A= 0 A¯ 2 B21 B22

(3.35)

with respect to the decomposition (3.34). For any u ∈ E, u = x + y with x ∈ E0 and y ∈ E1 , the equation ¯ = Bu ¯ Au can be rewritten as       x x A¯ 1 0 B¯ 11 B¯ 12 = . B¯ 21 B¯ 22 0 A¯ 2 y y That is 

A¯ 1 x = B¯ 11 x + B¯ 12 y A¯ 2 y = B¯ 21 x + B¯ 22 y.

From (3.28), the definitions of P, P ± , A¯ 2 and B¯ 22 , we have A¯ 2 is invertible on E1 and #A¯ −1 B¯ 22 #E < 1. Thus we have 2

y = w(x) = (A¯ 2 − B¯ 22 )−1 B¯ 21 x, and ¯ − Bu ¯ = 0 ⇐⇒ A¯ 1 x − [B¯ 11 + B¯ 12 (A¯ 2 − B¯ 22 )−1 B¯ 21 ]x = 0. Au

(3.36)

So u(x) = x + w(x) = x + (A¯ 2 − B¯ 22 )−1 B¯ 21 x satisfies all the required properties with B  defined as B  = B¯ 11 + B¯ 12 (A¯ 2 − B¯ 22 )−1 B¯ 21 .

(3.37)  

We note that in the nonlinear case, the same result is true. But in the proof one should use the contraction mapping principle and the implicit function theorem. We refer the papers [4, 5, 33, 34] and [223] for general settings.

52

3 Relative Morse Index

Definition 4.2 ([284]) For any B ∈ Ls (H ) with #B#H < c, we define ¯ E0 − B  ), νA (B) = dim ker(A − B). μcA (B) = m− (A| Theorem 4.3 ([284]) For any two operators B1 , B2 ∈ Ls (H ) with #Bi #H < c, i = 1, 2 and B1 < B2 , there holds  νA ((1 − s)B1 + sB2 ). (3.38) μcA (B2 ) − μcA (B1 ) = s∈[0,1)

Proof We set Bs = (1 − s)B1 + sB2 , i(s) = μcA ((1 − s)B1 + sB2 ), ν(s) = νAc ((1 − s)B1 + sB2 ) and as (x) =

1 ¯ ((A1 − Bs )x, x)E , 2

where Bs = ((1 − s)B¯ 1 + s B¯ 2 )11 + ((1 − s)B¯ 1 + s B¯ 2 )12 (A¯ 2 − ((1 − s)B¯ 1 + s B¯ 2 )22 )−1 ((1 − s)B¯ 1 + s B¯ 2 )21 . We denote b(s) = A¯ 1 − Bs . For any s0 ∈ [0, 1], if ν(s0 ) = 0, that is to say b(s0 ) has zero nullity subspace of E0 , so from the continuous dependent of the quadratic function as on s, there exists a neighbourhood U (s0 ) of s0 in [0, 1], such that i(s) = i(s0 ) and ν(s) = ν(s0 ) = 0, ∀s ∈ U (s0 ).

(3.39)

If ν(s0 ) = 0, we have the following decomposition E0 = E0− ⊕ E00 ⊕ E0+ , such that b(s0 ) is negative definite, zero and positive definite on E0− , E00 and E0+ respectively. For any x0 ∈ ker b(s0 ) with #x0 # = 1, that is b(s0 )x0 = 0. Define a smooth function a(s) : [0, 1] → R by a(s) := (b(s)x0 , x0 )E , and we have a(s0 ) = 0. From the definition of b(s) and denote ξ(s) := (A¯ 2 − B¯ 22 (s))−1 B¯ 21 (s) for simplicity, we have ¯ a(s) = ((A¯ − B(s))(x 0 + ξ(s)x0 ), (x0 + ξ(s)x0 ))E , and ¯ 0 ))(x0 + ξ(s0 )x0 ) = 0. (A¯ − B(s So, we have

3.4 Saddle Point Reduction for the General Cases

53

a  (s0 ) = −(B¯  (s0 )(x0 + ξ(s0 )x0 ), (x0 + ξ(s0 )x0 )E = −((B2 − B1 )(x0 + ξ(s0 )x0 ), (x0 + ξ(s0 )x0 )H . Since B1 < B2 and x0 = 0, we have a  (s0 ) < 0. Summing up, there exists δ > 0, such that a(s) < 0 for any s ∈ (s0 , s0 + δ). So from the continuous of b(s), there exists δ¯ ≤ δ, such that ¯ (b(s)x, x)E < 0, ∀x ∈ E0− ⊕ E00 , s ∈ (s0 , s0 + δ), ¯ s0 ) (b(s)x, x)E > 0, ∀x ∈ E00 , s ∈ (s0 − δ, and ¯ (b(s)x, x)E > 0, ∀x ∈ E0+ , s ∈ (s0 , s0 + δ). That is to say ¯ i(s) = i(s0 ) + ν(s0 ) and ν(s) = 0, ∀s ∈ (s0 , s0 + δ)

(3.40)

¯ s0 ). i(s) = i(s0 ) and ν(s) = 0, ∀s ∈ (s0 − δ,

(3.41)

and

So from (3.39), (3.40) and (3.41), we have μcA (B2 ) − μcA (B1 ) =



νA ((1 − s)B1 + sB2 ).

s∈[0,1)

Thus we have proved the lemma.

 

From Theorem 4.3, we know that μcA (B2 ) − μcA (B1 ) is independent of c for any B1 and B2 ∈ Ls (H ) with c > max{#B1 #H , #B2 #H }. In fact, for any such two operators, we can choose an operator B0 ∈ Ls (H ) such that B0 < Bi , i = 1, 2 and #B0 #H < c. Then we have μcA (B2 ) − μcA (B1 ) = (μcA (B2 ) − μcA (B0 )) − (μcA (B1 ) − μcA (B0 )), which is independent of c. Definition 4.4 ([284]) For any B ∈ Ls (H ), we define μA (B) = μcA (B) − μcA (0), c > #B#H . So the index pair (μA (B), νA (B)) is well defined.

(3.42)

54

3 Relative Morse Index

From Theorems 1.3 and 4.3, we have the following result. Theorem 4.5 ([284]) Suppose both the indices I (A, A − B) and μA (B) are well defined for the operator pair (A, B). Then we have I (A, A − B) = μA (B).

(3.43)

Proof Firstly, we claim that for the positively definite operator B > 0, (3.43) is true. In fact, since I (A, A − 0) = μA (0) = 0, there holds I (A, A − B) =



m0 (A − sB) =

s∈[0,1)



νA (sB) = μA (B).

s∈[0,1)

In general, we choose a positively definite operator B0 such that B < B0 , so we have  m0 (A−(1−s)B −sB0 ) = μA (B0 )−μA (B). I (A, A−B0 )−I (A, A−B) = s∈[0,1)

Therefore from I (A, A − B0 ) = μA (B0 ), we have the desired equality (3.43).

 

Chapter 4

The P -Index Theory

4.1 P -Index Theory If the Hamiltonian function H ∈ C 2 (R×R2n , R) satisfying H (t +τ, P x) = H (t, x) with P ∈ Sp(2n), it is natural to consider the following nonlinear Hamiltonian system with P -boundary condition 



x(t) ˙ = J H (t, x(t)), x(τ ) = P x(0).

(4.1)

So if x is a solution of the problem Eq. (4.1), the matrix B(t) = H  (t, x(t)) should satisfy the following condition B(t + τ ) = (P −1 )T B(t)P −1 . The fundamental solution of the linear system z˙ (t) = J B(t)z(t)

(4.2)

in the interval [0, τ ] is a symplectic path γ (t), i.e., γ ∈ Pτ (2n). Definition 1.1 ([202]) For any γ ∈ Pτ (2n), P ∈ Sp(2n), ω ∈ U, the (P , ω)-nullity νωP (γ ) is defined by νωP (γ ) = dimC kerC (γ (τ ) − ωP ).

(4.3)

We note that for the fundamental solution γ ∈ Pτ (2n) of the linear Hamiltonian system (4.2), from Lemma 4.1 of Chap. 1, the (P , 1)-nullity ν1P (γ ) is just the dimensional of the real solution space of the linear Hamiltonian system (4.2) satisfying the boundary condition x(τ ) = P x(0). And the (P , ω)-nullity νωP (γ ) © Springer Nature Singapore Pte Ltd. 2019 C. Liu, Index theory in nonlinear analysis, https://doi.org/10.1007/978-981-13-7287-2_4

55

56

4 The P -Index Theory

is just the complex dimensional of the complex solution space of the linear Hamiltonian system (4.2) satisfying the boundary condition x(τ ) = ωP x(0). In the following, we define the (P , ω)-index part iωP (γ ) parameterized by all ω on the unit circle for any symplectic path γ starting from the identity, and study some of its basic properties. We fix n ∈ N, τ > 0 and firstly introduce some notations and definitions. For P , M ∈ Sp(2n), ω ∈ U, we define Dω,P (M) = (−1)n−1 ω−n det (M − ωP ), Sp(2n)± ω,P = {M ∈ Sp(2n) | ±Dω,P (M) < 0}, − Sp(2n)∗ω,P = Sp(2n)+ ω,P ∪ Sp(2n)ω,P ,

Sp(2n)0ω,P = Sp(2n) \ Sp(2n)∗ω,P . For τ > 0, we define four subsets of Pτ (2n) by 0 ∗ P Pτ,ω (2n)

= {γ ∈ Pτ (2n) | γ (0) ∈ Sp(2n)0ω,P , γ (τ ) ∈ Sp(2n)∗ω,P },

0 0 P Pτ,ω (2n)

= {γ ∈ Pτ (2n) | γ (0) ∈ Sp(2n)0ω,P , γ (τ ) ∈ Sp(2n)0ω,P },

∗ ∗ P Pτ,ω (2n)

= {γ ∈ Pτ (2n) | γ (0) ∈ Sp(2n)∗ω,P , γ (τ ) ∈ Sp(2n)∗ω,P },

∗ 0 P Pτ,ω (2n)

= {γ ∈ Pτ (2n) | γ (0) ∈ Sp(2n)∗ω,P , γ (τ ) ∈ Sp(2n)0ω,P }. 0 P Pτ,ω (2n)

0 0 = 0P Pτ,ω (2n) ∪ ∗P Pτ,ω (2n),

∗ P Pτ,ω (2n)

∗ ∗ = 0P Pτ,ω (2n) ∪ ∗P Pτ,ω (2n),

0 P Pτ,ω (2n)

∗ 0 = 0P Pτ,ω (2n) ∪ 0P Pτ,ω (2n),

∗ P Pτ,ω (2n)

∗ 0 = ∗P Pτ,ω (2n) ∪ ∗P Pτ,ω (2n).

We denote Sp(2n)∗ω = Sp(2n)∗ω,I , Sp(2n)0ω = Sp(2n)0ω,I .

4.1 P -Index Theory

57

For any two paths γ1 : [0, τ ] → Sp(2n) and γ2 : [0, τ ] → Sp(2n) with γ1 (τ ) = γ2 (0), we define their joint path by γ2 ∗ γ1 (t) =

γ1 (2t),

0 ≤ t ≤ τ/2,

γ2 (2t − τ ), τ/2 ≤ t ≤ τ.



 cos θ − sin θ . sin θ cos θ Let D(a) = diag(a, a −1 ) for a ∈ R \ {0}, and define For θ ∈ R define R(θ ) =

Mn+ = D(2)n , Mn− = D(−2)  D(2)(n−1) . − − We have Mn+ ∈ Sp(2n)+ ω and Mn ∈ Sp(2n)ω . From Lemma 2.6 of Chap. 1, every M ∈ Sp(2n) has its unique polar decomposition M = AU , where A = (MM T )1/2 is a symmetric symplectic positive definite matrix, U is  a symplectic orthogonal matrix. Therefore U can be written  √ u1 −u2 , where u = u1 + −1u2 is a unitary matrix. So for every as U = u2 u1 path γ in C([0, τ ], Sp(2n)) we can associate uniquely a path u(t) in the unitary group on Cn to it.√Let  : [0, τ ] → R be any continuous real function satisfying det u(t) = exp( −1(t)). We define the rotation number of γ on [0, τ ] by τ (γ ) = (τ ) − (0) which depends only on the symplectic path γ but not on the choice of the function .

Definition 1.2 For τ > 0 and ω ∈ U, given two paths γ0 , γ1 ∈ C([0, τ ], Sp(2n)), if there exists a map δ ∈ C([0, 1] × [0, τ ], Sp(2n)) such that δ(0, ·) = γ0 (·), δ(1, ·) = γ1 (·), and both dimC kerC (δ(s, 0) − ωI ) and dimC kerC (δ(s, 1) − ωI ) are constant for s ∈ [0, 1], then γ0 and γ1 are ω-homotopic on [0, τ ] and we write γ0 ∼ ω γ1 . This homotopy possesses fixed end points if δ(s, 0) = γ0 (0), δ(s, τ ) = γ0 (τ ) for all s ∈ [0, 1]. The following lemma studies the relation between the above homotopy and the homotopy which fixes the end points. It generalizes the Lemma 5.2.2 in [223] but with almost the same proof. Lemma 1.3 If γ0 , γ1 ∈ C([0, τ ], Sp(2n)) possess common end points γ0 (0) = γ1 (0), γ0 (τ ) = γ1 (τ ). Suppose γ0 ∼ ω γ1 on [0, τ ] via a homotopy δ ∈ C([0, 1] × [0, τ ], Sp(2n)) such that δ(·, 0), δ(·, τ ) are contractible in Sp(2n). Then the homotopy δ can be modified to fix the end points all the time, i.e. δ(s, 0) = γ0 (0), δ(s, τ ) = γ0 (τ ) for all s ∈ [0, 1]. Remark 1.4 It is known that Spωk (2n) = {M ∈ Sp(2n)| dimC kerC (M − ωI ) = k} is contractible in Sp(2n) for 0 ≤ k ≤ 2n(see for example p14 of [227] for a proof for the case ω = 1 and k = 0), so the closed curves δ(·, 0) ⊂ Spωk (2n) and δ(·, τ ) ⊂ Spωl (2n) for some integers k and l must be contractible in Sp(2n).

58

4 The P -Index Theory

Lemma 1.5 ([223]) If γ0 , γ1 ∈ C([0, τ ], Sp(2n)) possess common end points γ0 (0) = γ1 (0), γ0 (τ ) = γ1 (τ ), then τ (γ0 ) = τ (γ1 ) if and only if γ0 ∼ γ1 on [0, τ ] with fixed points. By [57] and [231], Sp(2n)∗ω contains precisely two connected components ∗ ∗ ± Sp(2n)± ω which contains Mn respectively. Thus for any γ ∈ P Pτ,ω (2n), we can connect P −1 γ (0) to Mn+ or Mn− by β0 in Sp(2n)∗ω , connect P −1 γ (τ ) to Mn+ or Mn− by β1 in Sp(2n)∗ω , so we get a joint path β1 ∗ P −1 γ ∗ β0− . Here we remind that the path β0− is the backward path of β0 , i.e., β0− (t) = β0 (τ − t). By the definition of ∗ Mn± , k ≡ τ (β1 ∗P −1 γ ∗β0− )/π is an integer, in this case we say γ ∈ ∗P Pτ,ω,k (2n). ∗ Since Sp(2n)ω is simply connected in Sp(2n), this integer k is independent of the choice of the paths β0 , β1 . This integer also satisfies k∈

2Z,

if β0 (τ ) = β1 (τ ),

2Z + 1,

if β0 (τ ) = β1 (τ ).

Definition 1.6 ([186, 202]) The (P , ω)-index iωP (γ ) of a path γ ∈ Pτ (2n) is defined as ∗ (1) We define iωP (γ ) = k, if γ ∈ ∗P Pτ,ω,k (2n). ∗ 0 (2) If γ ∈ P Pτ,ω (2n), then ∗ iωP (γ ) = inf{iωP (β) | β ∈ P Pτ,ω (2n)

and β is sufficiently C 0 close to γ in Pτ (2n)}. ∗ (2n), then (3) If γ ∈ 0P Pτ,ω

iωP (γ ) = max{ iωP (β) | β ∈ C([0, τ ], Sp(2n)), β(0) ∈ Sp(2n)∗ω,P and β is sufficiently C 0 close to γ }. Where by saying that β is sufficiently C 0 -close to γ in Pτ (2n) we mean that β is close enough to γ in the topology of Pτ (2n) so that there exists a homotopy ∗ (2n) δ : [0, 1] → Pτ (2n) with the properties δ(0) = β, δ(1) = γ , and δ(s) ∈ ∗P Pτ,ω for all 0 ≤ s < 1. In another way, we define the Maslov type (P , ω)-index in the following way by using the ω-index function. Definition 1.7 ([202]) For any τ > 0, ω ∈ U, P ∈ Sp(2n) and γ ∈ Pτ (2n), we define the Maslov type (P , ω)-index iωP (γ ) = iω (P −1 γ ∗ ξ ) − iω (ξ ), where ξ ∈ Pτ (2n) such that ξ(τ ) = P −1 γ (0) = P −1 .

(4.4)

4.1 P -Index Theory

59

Remark 1.8 The above definition depends only on γ but not on ξ , and therefore it is well defined. Definitions 1.1, 1.6, and 1.7 give a pair of integers (iωP (γ ), νωP (γ )) ∈ Z × {0, 1, . . . 2n} for any γ ∈ Pτ (2n) and P ∈ Sp(2n). We call iωP (γ ) the (P , ω)index of γ and νωP (γ ) the (P , ω)-nullity of γ . When P = I , the Maslov type P -index in [186] is the (P , 1)-index. We show it in Theorem 1.9 below. When ω = 1, we denote (i P (γ ), ν P (γ )) = (iωP (γ ), νωP (γ )). We also denote P i (γ ) = i P (B) when γ is the fundamental solution of (4.2). From the definition, iωP (I ) = 0. When P = I , ω = 1, if we use Definition 1.7 to define iωP (γ ), now we take ξ ≡ I , then i I (γ ) = i1 (P −1 γ ∗ ξ ) − i1 (ξ ) = i1 (γ ) − (−n) = i1 (γ ) + n. Thus i I (γ ) and i1 (γ ) are different up to n, where i1 (γ ) is the Maslov-type index for periodic boundary condition introduced in Chap. 2 (cf. [223–225]). Theorem 1.9 ([202]) The two definitions of index iωP (γ ) in Definitions 1.6 and 1.7 are consistent. Proof ∗ (2n), we choose ξ ∈ P (2n) such that ξ(τ ) = P −1 , and we can (1) If γ ∈ ∗P Pτ,ω τ −1 connect P γ (0) to Mn+ or Mn− by a path β0 : [0, τ ] → Sp(2n)∗ω , connect P −1 γ (τ ) to Mn+ or Mn− by a path β1 : [0, τ ] → Sp(2n)∗ω . Then we have

iω (P −1 γ ∗ ξ ) − iω (ξ ) =τ (β1 ∗ P −1 γ ∗ ξ )/π − τ (β0 ∗ ξ )/π =τ (β1 ∗ P −1 γ )/π + τ (ξ )/π − (τ (β0 ) + τ (ξ ))/π =τ (β1 ∗ P −1 γ )/π − τ (β0 )/π =τ (β1 ∗ P −1 γ ∗ β0− )/π. 0 (2n), by the Definition 6.2.13 of the Maslov-type index in [223], (2) If γ ∈ ∗P Pτ,ω we have

iω (P −1 γ ∗ ξ ) ∗ = inf{iω (P −1 β ∗ ξ ) | β ∈ P Pτ,ω (2n),

β is sufficiently C 0 -close to γ in Pτ (2n)} ∗ = inf{iωP (β) | β ∈ P Pτ,ω (2n),

β is sufficiently C 0 -close to γ in Pτ (2n)} + iω (ξ ).

60

4 The P -Index Theory

We get the conclusion by putting the second term on the left. (3) If γ ∈ 0P Pτ,ω (2n), first we choose ξ in Pτ (2n) such that ξ(τ ) = P −1 .  Then we choose ξ ∈ Pτ (2n) is sufficiently C 0 -close to ξ in Pτ (2n) such that  ∗ ξ (τ ) ∈ Sp(2n)ω,P ; further, we choose β is sufficiently C 0 -close to γ such that  P −1 β(0) = ξ (τ ), β(τ ) = γ (τ ).  So we have P −1 γ ∗ ξ ∼ P −1 β ∗ ξ on [0, τ ] with fixed points and obtain iωP (γ ) = iω (P −1 γ ∗ ξ ) − iω (ξ ) 

= iω (P −1 β ∗ ξ ) − iω (ξ ) 

= iωP (β) + iω (ξ ) − iω (ξ ).



By the definition of Maslov-type index, iω (ξ ) − iω (ξ ) ≥ 0, the equality can be  achieved when ξ = ξ−s , β = s γ , ∀s ∈ (0, 1]. We will explain it in sequel.   Definition 1.10 For ω ∈ U, given two paths γ0 and γ1 in Pτ (2n), if there exists a map δ ∈ C([0, 1] × [0, τ ], Sp(2n)) such that δ(0, ·) = γ0 (·), δ(1, ·) = γ1 (·), δ(s, 0) = I and dim ker(δ(s, τ ) − ωP ) are constant for s ∈ [0, 1], then we say they are (P , ω)-homotopic, and write γ0 ∼ Pω γ1 . ∗ (2n) for j = 0, 1. Proposition 1.11 For ω ∈ U, P ∈ Sp(2n), suppose γj ∈ P Pτ,ω Then iωP (γ0 ) = iωP (γ1 ) if and only if γ0 ∼ Pω γ1 .

Proof Sufficiency. It is the same as the following Theorem 1.14. ∗ (2n) Necessity. We choose ξ ∈ Pτ (2n) such that ξ(τ ) = P −1 , γj ∈ P Pτ,ω together with iωP (γ0 ) = iωP (γ1 ) imply that there exists a map δ ∈ C([0, 1] × [0, τ ], Sp(2n)) such that P −1 γ0 ∗ ξ ∼ ω P −1 γ1 ∗ ξ . We can modify δ such that δ(s, t) = ξ(2t), 0 ≤ s ≤ 1, 0 ≤ t ≤ τ2 . So we have P −1 γ0 ∼ω P −1 γ1 with the   initial point fixed. It implies γ0 ∼ Pω γ1 . Modifying the result in Corollary 6.2.10 of [223], we get the following result. Proposition 1.12 If γ0 , γ1 ∈ Pτ (2n) with the same end points possess the same Maslov (P , ω)-index if and only if they can be continuously deformed to each other with the end points fixed. Proposition 1.13 ([186, 202]) For ω ∈ U, suppose Pj ∈ Sp(2nj ), γj ∈ Pτ (2nj ) for j = 0, 1. Then iωP (γ0  γ1 ) = iωP0 (γ0 ) + iωP1 (γ1 ),

(4.5)

where P = P0  P1 and γ0  γ1 (t) = γ0 (t)  γ1 (t) for all t ∈ [0, τ ]. Proof We choose ξj ∈ Pτ (2nj ) such that ξj (τ ) = Pj−1 and set ξ(t) = ξ0  ξ1 (t) ∈ Pτ (2n0 + 2n1 ).

4.1 P -Index Theory

61

Suppose W is the product matrices ofthe elementary  transformations of changing γ0 (t) 0 W −1 . Note that W = W −1 , the rows satisfying γ0 (t)  γ1 (t) = W 0 γ1 (t) W2 = I. Then we obtain iωP (γ0  γ1 ) = iω (P −1 (γ0  γ1 ) ∗ ξ ) − iω (ξ )   −1   P0 γ0 0 −1 W ∗ ξ − iω (ξ ) = iω W 0 P1−1 γ1 = iω ((P0−1 γ0 ∗ ξ0 )  (P1−1 γ1 ∗ ξ1 )) − iω (ξ0  ξ1 ) = iω (P0−1 γ0 ∗ ξ0 ) + iω (P1−1 γ1 ∗ ξ1 ) − iω (ξ0 ) − iω (ξ1 ) = iωP0 (γ0 ) + iωP1 (γ1 ).   Theorem 1.14 ([186, 202]) Let ω ∈ U, P ∈ Sp(2n). For any two paths γ0 and γ1 ∈ Pτ (2n), if γ0 ∼ Pω γ1 on [0, τ ], then iωP (γ0 ) = iωP (γ1 ), νωP (γ0 ) = νωP (γ1 ).

(4.6)

Proof By Definition 1.10 and γ0 ∼ Pω γ1 , we obtain νωP (γ0 ) = νωP (γ1 ). To prove the first equality in (4.6), we choose a path ξ ∈ Pτ (2n) so that ξ(τ ) = P −1 . By γ0 ∼ Pω γ1 , we obtain P −1 γ0 ∗ ξ ∼ ω P −1 γ1 ∗ ξ . Then according to Theorem 6.2.3 in [223], we have iω (P −1 γ0 ∗ ξ ) = iω (P −1 γ1 ∗ ξ ). This proves the first equality of (4.6).   Next we study the degenerate paths, and get the most important property of rotational perturbation paths. Fix ω ∈ U, P ∈ Sp(2n) and P = I , γ ∈ 0P Pτ,ω (2n). Using Theorem 7.5 in [226], there exists W1 ∈ Sp(2n) such that (7.8) in [226] holds for M = ξ(τ ). As discussed in [226], we define Q(M, s1 , . . . , sp+2q ) ≡ MW1−1 Rm1 (s1 θ0 ) · · · Rmp+2q (sp+2q θ0 )W1 .

(4.7)

For any (s, t) ∈ [−1, 1] × [0, τ ], we define the rotational perturbation paths of ξ ξs (t) = ξ(t)W1−1 Rm1 (sρ(t)θ0 ) · · · Rmp+2q (sρ(t)θ0 )W1 . Where si , mi , θ0 , ρ(t) are all defined as in [226].   For t0 ∈ (0, τ ), Let κ ∈ C 2 ([0, τ ], [0, 1]) such that κ(t) = 0 for t0 ≤ t ≤ τ ,  κ(t) ˙ ≤ 0 for 0 ≤ t ≤ τ , κ(0) = 1 and κ(0) ˙ = 0. Whenever t0 ∈ (0, τ ) is sufficiently  −1 close to 0, there holds P γ ([0, t0 ]) ⊂ B (P −1 ). For any (s, t) ∈ [−1, 1] × [0, τ ], we define the left rotational perturbation paths of γ

62

4 The P -Index Theory s γ (t)

= γ (t)W1−1 Rm1 (sκ(t)θ0 ) · · · Rmp+2q (sκ(t)θ0 )W1 .

(4.8)

Corresponding, we define s (P −1 γ ) = P −1 s γ . Note that ξs (τ ) = s (P −1 γ )(0), ∗ s γ (0) ∈ Sp(2n)ω,P . 0 (2n), we using Theorem 7.5 in [226], there exists Similarly, when γ ∈ P Pτ,ω  W2 ∈ Sp(2n) such that (7.8) in [226] holds for M = γ (τ ). As discussed in [226], we define 





Q (M , s1 , . . . , sp +2q  ) ≡ M W2−1 Rm (s1 θ0 ) · · · Rm

  p +2q

1

(sp +2q  θ0 )W2 .

(4.9)

We define the right rotational perturbation paths of γ γs (t) = γ (t)W2−1 Rm (sρ(t)θ0 ) · · · Rm 1

  p +2q

(sρ(t)θ0 )W2 .

(4.10)

∗ (2n). Corresponding, we define (P −1 γ )s = P −1 γs . Note that γs ∈ P Pτ,ω

Theorem 1.15 ([202]) For ω ∈ U, P ∈ Sp(2n) and 0 < s ≤ 1. If γ ∈ 0P Pτ,ω (2n), the left perturbation paths of γ defined by (4.8) satisfy iωP (s γ ) − iωP (−s γ ) = − dim ker(P −1 − ωI ). If γ ∈ satisfy

0 P Pτ,ω (2n),

(4.11)

the right rotational perturbation paths of γ defined by (4.10) iωP (γs ) − iωP (γ−s ) = νωP (γ ).

(4.12)

Proof If γ ∈ 0P Pτ,ω (2n), for any (s, t) ∈ [−1, 1] × [0, τ ], we define ζs (t) = Q(P −1 , st/τ, . . . , st/τ ) for s ∈ [−1, 1]. Note that ξs (τ ) = ζs (τ ). Fix s ∈ (0, 1]. Then by Lemma 1.5 we obtain: • • • •

ζ−s ∗ ξ ∼ ξ−s on [0, τ ] with fixed points; ζs ∗ ξ ∼ ξs on [0, τ ] with fixed points; − P −1 γ ∗ ζ−s ∼ −s (P −1 γ ) on [0, τ ] with fixed points; −1 − P γ ∗ ζs ∼ s (P −1 γ ) on [0, τ ] with fixed points;

where ζs− (t) = ζs (τ − t), 0 ≤ t ≤ τ is different from ζs only with two opposite orientations. Naturally, we get −s (P −1 γ ) ∗ ξ−s ∼ s (P −1 γ ) ∗ ξs on [0, τ ] with fixed points, then iω (−s (P −1 γ ) ∗ ξ−s ) = iω (s (P −1 γ ) ∗ ξs ). Therefore (4.11) follows from Definition 1.7 and Theorem 5.4.1 in [223]. Equation (4.12) is directly obtained by Definition 1.7 and Theorem 5.4.1 in [223].  

4.1 P -Index Theory

63

0 (2n). Theorem 1.16 ([202]) For ω ∈ U and τ > 0. Let P ∈ Sp(2n), γ ∈ P Pτ,ω ∗ (2n) which are sufficiently C 0 -close to γ , there Then for any paths α and β ∈ P Pτ,ω holds

| iωP (β) − iωP (α) |≤ νωP (γ ).

(4.13)

Proof We choose ξ ∈ Pτ (2n) such that ξ(τ ) = P −1 . By Theorem 6.1.8 in [223], we obtain iω ((P −1 γ )−s ∗ ξ ) ≤ iω ((P −1 α) ∗ ξ ) ≤ iω ((P −1 γ )s ∗ ξ ). By Definition 2.6, we get iωP (γ−s ) ≤ iωP (α) ≤ iωP (γs ) = iωP (γ−s ) + νωP (γ ).

(4.14)  

The proof is complete by (4.14). (R2n , ω) ˜

As in Sect. 1.2, the general symplectic space has its symplectic transform group as the J -symplectic matrix group SpJ (2n). We denote by Pτ,J (2n) = {γ ∈ C 0 ([0, τ ], SpJ (2n))|γ (0) = I } the set of J -symplectic paths. From Proposition 2.2 of Chap. 1, we have T Pτ,J (2n)T −1 = Pτ (2n), ˜ → (R2n , ω˜ 0 ) defined in Sect. 1.1 where the symplectic transform T : (R2n , ω) T corresponds to a matrix T satisfying T J T = J −1 . Definition 1.17 For ω ∈ U, P ∈ SpJ (2n), γ ∈ Pτ,J (2n), the (P , ω)-index of γ is defined as iωJ ,P (γ ) = iωP0 (γ0 ), νωJ ,P (γ ) = νωP0 (γ0 ),

(4.15)

where P0 = T P T −1 , and γ0 (t) = T γ (t)T −1 . Remark 1.18 I. We note that there holds νωJ ,P (γ ) = dimC kerC (γ (τ ) − ωP ).

(4.16)

II. The Definition 1.17 is natural in the following sense. Consider the linear Hamiltonian system 

x(t) ˙ = J B(t)x(t), x(τ ) = P x(0).

(4.17)

64

4 The P -Index Theory

If we take x(t) = T −1 x0 (t), the linear Hamiltonian system (4.17) becomes 

x˙0 (t) = J B0 (t)x0 (t), x0 (τ ) = P0 x0 (0),

(4.18)

where B0 (t) = −(T −1 )T B(t)T −1 . Suppose γ (t) and γ0 (t) be the fundamental solutions of the linear Hamiltonian systems (4.17) and (4.18), respectively. Then there holds γ0 (t) = T γ (t)T −1 .

(4.19)

For a general continuous symplectic path ρ : [a, b] → Sp(2n), we define its Maslov type P -index as follow. Definition 1.19 iˆP (ρ) = i P (γb ) − i P (γa ),

(4.20)

where γa ∈ P(2n) is a symplectic path ended at ρ(a) and γb = ρ ∗ γa ∈ P(2n) is a symplectic path ended at ρ(b) which is the composite of γa with ρ. We remind that for the constant path γ = I there holds i P (I ) = 0, so iˆP (γ ) = i P (γ ) for γ ∈ P(2n). Lemma 1.20 The index iˆP (ρ) is well defined, i.e., it is independent of the choice of γa . Proof For two symplectic paths γa , γa ∈ P(2n), we denote γb , γb ∈ P(2n) correspondingly as in Definition 1.19. We should prove i P (γb ) − i P (γb ) = i P (γa ) − i P (γa ).

(4.21)

Taking ξ as in Definition 1.17, then it should be proved that i1 (P −1 γb ∗ ξ ) − i1 (P −1 γb ∗ ξ ) = i1 (P −1 γa ∗ ξ ) − i1 (P −1 γa ∗ ξ ).

(4.22)

Using the pole decomposition, we suppose P −1 γb ∗ ξ = Pb Ub , P −1 γb ∗ ξ = Pb Ub , P −1 γa ∗ ξ = Pa Ua , P −1 γa ∗ ξ = Pa Ua , and det Ub = e

√

−1b

, det Ub = e

√

−1b

, det Ua = e

√

−1a

, det Ua = e

√

−1a

.

By definition, the functions ’s have the path additivity, we prove the equality (4.22) in the non-degenerate cases. For the degenerate cases, since the two paths have the

4.1 P -Index Theory

65

same end points, we can prove the equality (4.22) by taking non-degenerate paths close to the degenerate ones and taking the infimum in every terms.   Theorem 1.21 The index iˆP has the following properties (1) (Affine Scale Invariance). For k > 0, l ≥ 0, we have the affine map ϕ : [a, b] → [ka + l, kb + l] defined by ϕ(t) = kt + l. For a given continuous path ρ : [ka + l, kb + l] → Sp(2n), there holds iˆP (ρ) = iˆP (ρ ◦ ϕ).

(4.23)

(2) (Homotopy Invariance rel. End Points). If δ : [0, 1] × [a, b] → Sp(2n) is a continuous map with δ(0, t) = ρ1 (t), δ(1, t) = ρ2 (t), δ(s, a) = ρ1 (a) = ρ2 (a) and δ(s, b) = ρ1 (b) = ρ2 (b) for s ∈ [0, 1], then iˆP (ρ1 ) = iˆP (ρ2 ).

(4.24)

(3) (Path Additivity). If a < b < c, and ρ[a,c] : [a, c] → Sp(2n) is concatenate path of ρ[a,b] and ρ[b,c] , the, there holds iˆP (ρ[a,c] ) = iˆP (ρ[a,b] ) + iˆP (ρ[b,c] ).

(4.25)

(4) (Symplectic Additivity). Let Pk ∈ Sp(2nk ), ρk : [a, b] → Sp(2nk ), k = 1, 2, P = P1  P2 , ρ = ρ1  ρ2 . Then we have iˆP (ρ) = iˆP1 (ρ1 ) + iˆP2 (ρ2 ).

(4.26)

(5) (Symplectic Invariance). For any M ∈ Sp(2n), there holds iˆMP (Mρ) = iˆP (ρ), iˆP M (ρM) = iˆP (ρ).

(4.27)

(6) (Normalization).  For P = I and ρ : [−ε, ε] → Sp(2) with ε > 0 small and 11 ρ(t) = eJ t , we have 01 (i) iˆI (ρ) = 1; (ii) iˆI (ρ[−ε,0] ) = 0; (iii) iˆI (ρ[0,ε] ) = 1.

(4.28)

Proof We prove the statement (6) only. The remainders are direct consequence of the definition. From definition, iˆI (ρ) = i1 (γε ) − i1 (γ−ε ).

66

4 The P -Index Theory

But it is easy to see i1 (γ−ε ) = 0 and i1 (γε ) = 1. This proves statement (i). The other two statements are similar.   We will see that the six properties in Theorem 1.21 character the index iˆP uniquely. In the next chapter, we will define an index μP (γ ) as the Maslov type index of a pair of Lagrangian paths and prove that this index also satisfies the six properties of Theorem 1.21. So we have iˆP (γ ) = i P (γ ) = μP (γ ) for every γ ∈ P(2n). Notes and comments The Maslov P -index theory for a symplectic path was first studied in [68] and [186] independently for any symplectic matrix P with different treatment. In fact, in the case of P is a special orthogonal matrix, the (P , ω)-index theory and its iteration theory were studied in [71].

4.2 Relative Index via Saddle Point Reduction Method Let B ∈ C([0, τ ], Ls (2n)) be a continuous symmetric 2n×2n matrix value function, we denote by (iω (B), νω (B)) = (iω (γB ), νω (γB )) the ω-index of γB , where γB is the fundamental solution of the linear Hamiltonian system x(t) ˙ = J B(t)x(t). Lemma 2.1 Suppose B0 , B1 ∈ C(R, Ls (2n)) such that B0 < B1 and Bi (t + τ ) = Bi (t), i = 0, 1, there holds iω (B1 ) − iω (B0 ) =



νω ((1 − s)B0 + sB1 ).

(4.29)

s∈[0,1)

Proof By the saddle point reduction formula of ω-index (cf. [223]), there holds m− (Bi ) = dω + iω (Bi ), i = 0, 1. Therefore, by using the boundary condition to define the operators A, Bi , there holds iω (B1 ) − iω (B0 ) = m− (B1 ) − m− (B0 ) = μcA (B1 ) − μcA (B0 ). The remainder is the same as the proof of Theorem III.4.3, we remind in this case, the nullity in the formula (3.38) should be ω-nullity.   We now denoted by γP (t) the symplectic path with γP (0) = I and γP (τ ) = P . The specific choice is as follows. For P ∈ Sp(2n), we know that there exists a unique polar decomposition P = AU , where A = exp (M1 ), M1 satisfying

4.2 Relative Index via Saddle Point Reduction Method

M1T J + J M1 = 0 and M1T = M1 ;

67

(4.30)

U is a symplectic orthogonal matrix. Sp(2n) ∩ O(2n) is a compact connected Lie group and its Lie algebra is sp(2n)∩o(2n) constituted by the matrices M2 satisfying M2T J + J M2 = 0 and M2T + M2 = 0.

(4.31)

Then there exists a matrix M2 ∈ sp(2n) ∩ o(2n) such that U = exp (M2 ). So P takes the form P = exp (M1 ) exp (M2 ) (cf. Proposition I.2.10). We set P (t) = exp (tM1 /τ ) exp (tM2 /τ ). Note that P (t) ∈ Pτ (2n) and P (τ ) = P. In order to get the relationship between the Morse index and the Maslov (P , ω)index, we should demand that γP (0)T J γ˙P (0) = γP (τ )T J γ˙P (τ ),

(4.32)

then the corresponding B(t) satisfies B(0) = B(τ ) while using Theorem 3.2 (one needs the matrix function B˜ γP (t) in (4.125) below be τ -periodic). So when P = exp (M) with M T J + J M = 0, we take γP (t) = exp (tM/τ ). It can be easily verified that γP (t) satisfies (4.32). When P has the general form P = exp (M1 ) exp (M2 ), M1 and M2 satisfy (4.30) and (4.31) respectively, (4.32) does not always hold, we should do some modification on P (t). We let

!(t) = P

⎧ ⎪ ⎪ ⎨I,

0 ≤ t ≤ τ/3,

P (3t − τ ), ⎪ ⎪ ⎩P ,

τ/3 ≤ t ≤ 2τ/3, 2τ/3 ≤ t ≤ τ.

!(t) ∈ Pτ (2n) and P !(τ ) = P . We get P ! (t) ∈ C 1 ([0, τ ], Sp(2n)) by Note that P !(t) near t = τ/3 and t = 2τ/3 in the space of taking small perturbation of P symplectic paths, the end points will be fixed in the process of the perturbation. At ˙ (0) = ! (t) ∼ P ! !(t) on [0, τ ] with fixed end points and P the moment, we have P  ˙ ! (τ ) = 0. We set γP (t) = P ! (t), then γP (t) satisfies (4.32). Obviously, P (t) ∼ P !(t) on [0, τ ] with fixed end points, thus γP (t) ∼ P (t) on [0, τ ] with fixed end P points and iωP (γP (t)) = iωP (P (t)) by Proposition 1.12. Lemma 2.2 ([194]) Suppose B (P −1 )T B(t)P −1 , there holds

∈

C(R, Ls (2n)) satisfying B(t + τ )

˜ νωP (B) = νω (B),

=

(4.33)

˜ where B(t) = γP (t)T J γ˙P (t) + γP (t)T B(t)γP (t). Proof It is easy to check that the fundamental solution of the linear Hamiltonian ˜ system x(t) ˙ = J B(t)x(t) is the following symplectic path γ2 (t) = γP (t)−1 γ (t). −1 But γ2 (τ ) = γP (τ ) γ (τ ) = P −1 γ (τ ). Thus by definition, there holds

68

4 The P -Index Theory

˜ νωP (B) = dim ker(γ (τ ) − P ) = dim ker(P −1 γ (τ ) − I ) = νω (B).   Theorem 2.3 ([186, 202]) Let γ , γP , and γ2 ∈ Pτ (2n) be defined as above, there holds iωP (γ ) − iωP (γP ) =

iω (γ2 ),

ω = 1,

iω (γ2 ) + n,

ω = 1.

(4.34)

Thus the number iω (γ2 ) + iωP (γP ) depends only on P but not on the choice of M1 and M2 , where M1 and M2 are defined by P = exp (M1 ) exp (M2 ) as above. Proof If P = I , then γP ≡ I and γ2 = γ , by the definition of Maslov ω-index in [223] and Definition 1.7, we have iωI (γ ) − iωI (γP ) = iω (γ2 ) − iω (I ) =

iω (γ2 ),

ω = 1,

iω (γ2 ) + n,

ω = 1.

If P = I , when ω = 1, it is just the result of Theorem 4.2 in [186]. Next we prove the case of ω = 1. We define three symplectic paths γ3 , γ5 ∈ Pτ (2n) and γ4 ∈ C([0, τ ], Sp(2n)) by γ3 (t) = γP (t)−1 , γ4 (t) = P −1 γ (t), γ5 = γ4 ∗ γ3 . Then we have γ5 ∼ω γ2 and the homotopy is defined by δ(s, t) =

0 ≤ t ≤ τ/2,

γ3 (2st), γP ((2 − 2s)t + (2s

− 1)τ )−1 γ (2t

− τ ), τ/2 ≤ t ≤ τ.

Hence we have iω (γ2 ) = iω (γ5 ).

(4.35)

By the definition of (P , ω)-index, we get iω (γ5 ) = iωP (γ ) + iω (γ3 ). We note that P −1 P (t) ∼ω P (τ − t)−1 on [0, τ ] with fixed points

(4.36)

4.3 Galerkin Approximation for the (P , ω)-Boundary Problem of Hamiltonian. . .

69

by the homotopy map ! δ (s, t) = e−M2 estM2 /τ e(t−τ )M1 /τ e(1−s)tM2 /τ . By the choice of γP (t), we know that γP (t) ∼ P (t) on [0, τ ] with fixed end points. Thus P −1 γP (t) ∼ω γ3 (τ − t) on [0, τ ] with fixed points.

(4.37)

Therefore by definition we have iω (γ3 ) = iω (P −1 γP (τ − t)) = −iωP (γP (t)) + iω (P −1 γP (t) ∗ P −1 γP (τ − t))

(4.38)

= −iωP (γP (t)).  

We complete the proof from (4.35), (4.36), and (4.38).

Lemma 2.4 ([194]) Suppose B0 , B1 ∈ C(R, Ls (2n)) satisfying B(t + τ ) = (P −1 )T B(t)P −1 and B0 < B1 , there holds 

i P (B1 ) − i P (B0 ) =

ν P ((1 − s)B0 + sB1 ).

(4.39)

s∈[0,1)

Proof From Theorem 2.3, there holds i P (B1 ) − i P (B0 ) = i1 (B˜ 1 ) − i1 (B˜ 0 ) =



ν1 ((1 − s)B˜ 0 + s B˜ 1 ).

s∈[0,1)

We see that (1 − s)B˜ 0 (t) + s B˜ 1 (t) = γP (t)T J γ˙P (t) + γP (t)T [(1 − s)B0 (t) + sB1 (t)]γP (t) = B˜ s (t) with Bs (t) = (1 − s)B0 (t) + sB1 (t). Now from Lemma 2.2, we get the result (4.39).   Theorem 2.5 ([194]) Suppose B (P −1 )T B(t)P −1 , there holds

∈

C(R, Ls (2n)) satisfying B(t + τ )

I (A, A − B) = μA (B) = i P (B).

=

(4.40)

Proof We only need to prove the case P = I . From definition, we have i P (0) = 0. So by using Lemma 2.4 and the same computations as in the proof of Theorem 4.5, we have the formulas (4.40).  

4.3 Galerkin Approximation for the (P , ω)-Boundary Problem of Hamiltonian Systems √

Let τ > 0, ω = eθ −1 ∈ U. Define W ω = Wτω to be the subspace of L2 ([0, τ ], C2n ) composed by all y with the form

70

4 The P -Index Theory

y(t) =



√

e

−1(θ+2kπ )t/τ

ak , ak ∈ C2n ,

k∈Z

satisfying #y#2 = #y#21/2,2 =

 ((θ + 2kπ )2 + 1)τ |ak |2 < ∞. k∈Z

This space is a Hilbert space with the norm # · # and the inner product ·, · . Define Wkω = Wkω,+ Wkω,− with Wkω,± = e

√

−1(θ+2kπ )t/τ

(J ±

√ −1I )R2n .

(4.41)

Then Wω =



Wkω .

(4.42)

k∈Z

For P ∈ Sp(2n), define the (P , ω)-space by ω = {x ∈ L2 ([0, τ ], C2n ) | x(t) = γP (t)ξ(t), ξ(t) ∈ W ω }. WγωP = Wτ,γ P

(4.43)

By (4.42) and (4.43), correspondingly we have the space splitting WγωP =



WγωP ,k ,

k∈Z

where WγωP ,k = {x ∈ L2 ([0, τ ], C2n ) | x(t) = γP (t)ξ(t), ξ(t) ∈ Wkω }. And we

similarly define Wγω,± the subspaces of WγωP ,k with WγωP ,k = Wγω,+ ⊕ Wγω,− . P ,k P ,k P ,k We define the norm # · #1 and the inner product ·, · 1 in the space WγωP such that it is a Hilbert space. For zi (t) = γP (t)ξi (t), i = 1, 2, define #z1 #1 = #ξ #, z1 , z2 1 = ξ1 , ξ2 . We define the operator AωP : WγωP → WγωP by AωP x, y 1

= 0

τ

(−J x(t), ˙ y(t))dt, ∀x, y ∈ WγωP ,

(4.44)

where (·, ·) is the standard Hermitian product in C2n . Then A is bounded linear and self-adjoint with finite dimensional kernel N = ker AωP . The range of AωP is closed and the restriction AωP |N ⊥ is invertible. Define the functional on WγωP by

4.3 Galerkin Approximation for the (P , ω)-Boundary Problem of Hamiltonian. . .

1 f (x) = AωP x, x 1 − 2



τ 0

H (t, x)dt, ∀x ∈ WγωP .

71

(4.45)

Define the compact self-adjoint operator BPω : WγωP → WγωP by BPω z1 , z2 1

τ

= 0

(B(t)z1 (t), z2 (t))dt, ∀z1 , z2 ∈ WγωP , B(t) = H  (t, x(t)).

! ω : Wγω → Wγω by We define an operator M P P P !Pω x, y 1 = M



τ 0

(−J γ˙P (t)γP (t)−1 x(t), y(t))dt, ∀x, y ∈ WγωP .

(4.46)

(4.47)

!=M ! ω . Then we have For simplicity, we set A = AωP , B = BPω and M P ! y 1 = (A − M)x,



τ

(−J ξ˙ (t), η(t))dt,

(4.48)

0

where x(t) = γP (t)ξ(t), y(t) = γP (t)η(t), ξ , η ∈ W ω . By direct computation, we get ! = {λ± }, λ± = ± σ (A − M) k k

θ + 2kπ , ((θ + 2kπ )2 + 1)τ

(4.49)

! each eigenvalue λ± k of A − M has multiplicity 2n and the corresponding eigenspace ω,± is WγP ,k .  Let Zm,γP = |k|≤m WγωP ,k and Pm : WγωP → Zm,γP the projection map. It is readily seen that  = {Pm | m = 1, 2, · · · } be the Galerkin approximation sequence ! with respect to A − M: (1) Zm,γP = Pm WγωP is finite dimensional for all m ∈ N, (2) Pm → I strongly as m → +∞, ! := Pm (A − M) ! − (A − M)P ! m = 0. (3) [Pm , A − M] Define the operator Aω : W ω → W ω by

τ

A ξ, η = ω

(−J ξ˙ (t), η(t))dt, ∀ξ, η ∈ W ω .

(4.50)

0

and the compact self-adjoint operator B ω : W ω → W ω by

τ

B ω ξ, η =

(B(t)ξ(t), η(t))dt

(4.51)

0

for a symmetric matrix function B(t) ∈ C(Sτ , L(R2n )). By the Floquet theory we have

72

4 The P -Index Theory

νω (B) = dim ker(Aω − B ω ).

(4.52)

 ω ω = ω ω → Z ω the projection map. Then the Let Zm m |k|≤m Wk and Pm : W ω ω sequence  = {Pm | m = 1, 2, · · · } be the Galerkin approximation sequence of Aω : ω = P ω W ω is finite dimensional for all m ∈ N, (1) Zm m (2) Pmω → I strongly as m → +∞, (3) Pmω Aω = Aω Pmω , ∀ m ≥ 1.

For a self-adjoint operator S, we denote by M ∗ (S) the eigenspaces of S with eigenvalues belonging to (0, +∞), {0} and (−∞, 0) with ∗ = +, 0 and ∗ = −, respectively. We denote by m∗ (S) = dim M ∗ (S). Similarly, for any d > 0, we denote by Md∗ (S) the d-eigenspaces of S with eigenvalues belonging to [d, +∞), (−d, d) and (−∞, −d] with ∗ = +, 0 and ∗ = −, respectively. We also denote by m∗d (S) = dim Md∗ (S). We denote S  = (S|I mS )−1 , Pm SPm = (Pm SPm )|Zm,γP : ω → Zω . Zm,γP → Zm,γP and Pmω SPmω = (Pmω SPmω )|Zmω : Zm m Lemma 3.1 For any symmetric matrix function B(t) ∈ C(Sτ , L(R2n )), there exists an m1 > 0 such that for m ≥ m1 , there holds dim ker(Pmω (Aω − B ω )Pmω ) ≤ dim ker(Aω − B ω ),

(4.53)

where B ω is defined by (4.112). Proof We follow the idea of [95]. There is an m0 > 0 such that for m ≥ m0 , dim Pmω ker(Aω − B ω ) = dim ker(Aω − B ω ).

(4.54)

For otherwise, there exists ξj ∈ ker(Aω − B ω ) ∩ (I − Pmωj )W ω such that #ξj # = 1. Note that Aω ξj = (I − Pmωj )B ω ξj . Then we have #Aω ξj # ≥ #(Aω ) #−1 > 0, and #(I − Pmωj )B ω ξj # ≤ #(I − Pmωj )B ω # → 0 as j → +∞, a contradiction. Thus (4.54) holds. ω = X ⊕ Y . Then we have Take m ≥ m0 , let Xm = Pmω ker(Aω − B ω ) and Zm m m Ym ⊂ I m(Aω − B ω ). Let d = 14 #(Aω − B ω ) #−1 . Since B ω is compact, we have

4.3 Galerkin Approximation for the (P , ω)-Boundary Problem of Hamiltonian. . .

73

#(I − Pmω )B ω # → 0 as m → +∞. Hence there is an m1 ≥ m0 such that for m ≥ m1 , #(I − Pmω )B ω # ≤ 2d.

(4.55)

For m ≥ m1 , ∀η ∈ Ym , we have η = (Aω − B ω ) (Aω − B ω )η = (Aω − B ω ) (Pmω (Aω − B ω )Pmω η + (Pmω − I )B ω η). This implies that #η# ≤

1 #P ω (Aω − B ω )Pmω η#. 2d m

(4.56)

Thus by (4.54) and (4.56) we have dim ker(Pmω (Aω − B ω )Pmω ) ≤ dim Xm = dim ker(Aω − B ω ).   Theorem 3.2 ([202]) For any symmetric matrix function B(t) ∈ C(Sτ with the Maslov ω-index (iω (B), νω (B)) and any constant 0 < d ≤ 14 #(Aω − B ω ) #−1 , there exists an m∗ > 0 such that for m ≥ m∗ we have

, L(R2n ))

1 ω dim Zm − iω (B) − νω (B), 2 1 ω ω ω ω ω dim Zm m− + iω (B), d (Pm (A − B )Pm ) = 2 ω ω ω ω m+ d (Pm (A − B )Pm ) =

m0d (Pmω (Aω − B ω )Pmω ) = νω (B). Proof We follow the idea of [94]. We distinguish two cases. Case 1.

νω (B) = 0. By (4.52) and Lemma 6.5, for m ≥ m1 we obtain that dim M 0 (Pmω (Aω − B ω )Pmω ) = dim ker(Aω − B ω ) = 0.

Since B ω is compact, there exists m2 ≥ m1 such that for m ≥ m2 , #(I − Pmω )B ω # ≤

1 #(Aω − B ω )−1 #−1 . 2

Then Pmω (Aω − B ω )Pmω = (Aω − B ω )Pmω + (I − Pmω )B ω Pmω implies that

(4.57)

74

4 The P -Index Theory

#Pmω (Aω − B ω )Pmω ξ # ≥

1 ω #(Aω − B ω ) #−1 #ξ #. ∀ξ ∈ Zm . 2

Thus Md∗ (Pmω (Aω − B ω )Pmω ) = M ∗ (Pmω (Aω − B ω )Pmω ) for ∗ = +, −, 0. As proved in [226], there is m∗ > 0 such that for m ≥ m∗ the relation (4.57) holds. Case 2. νω (B) > 0. By Lemma 3.1 there exists m3 > 0 such that for m ≥ m3 , m0d (Pmω (Aω − B ω )Pmω ) ≤ νω (B).

(4.58)

For otherwise, there exists η ∈ Md0 (Pmω (Aω − B ω )Pmω ) ∩ Ym , # η #= 1, where ω Zm = Pmω ker(Aω − B ω ) ⊕ Ym , dim Pmω ker(Aω − B ω ) = νω (B).

Then #(Pmω (Aω − B ω )Pmω )η# ≤ d#η# contradicts to #η# ≤ B ω )Pmω #.

1 ω ω 2d #Pm (A

−

Let γ be the fundamental solution of (1.2) and γs be as described in [226]. For s ∈ [−1, 1], we define Bs (t) = −J γ˙s (t)γs−1 (t), t ∈ [0, τ ]. Let Bsω be the compact operator defined by (4.112) corresponding to Bs (t). By the results in [226] we have M 0 (Aω − Bsω ) = 0 for s = 0, #Bsω − B ω # → 0 as s → 0. iω (γs ) − iω (γ−s ) = νω (B), iω (γ−s ) = iω (B), ∀s ∈ (0, 1].

(4.59)

ω # ≤ 1 d. By case 1, (4.58), (4.59) and the Choose 0 < s0 ≤ 1 such that #B ω − B±s 2 0 fact that ω ω Pmω (Aω − B±s )Pmω = Pmω (Aω − B ω )Pmω + Pmω (B ω − B±s )Pmω , 0 0

there exists m∗ ≥ m3 such that for m ≥ m∗ , ω ω ω ω + ω ω ω ω m+ d (Pm (A − B )Pm ) ≤ m (Pm (A − Bs0 )Pm ) =

1 ω dim Zm − iω (B) − νω (B), 2

ω ω ω ω + ω ω ω ω 0 ω ω ω ω m+ d (Pm (A − B )Pm ) ≥ m (Pm (A − B−s0 )Pm ) − md (Pm (A − B )Pm )

≥

1 ω dim Zm − iω (B) − νω (B). 2

4.3 Galerkin Approximation for the (P , ω)-Boundary Problem of Hamiltonian. . .

75

Hence, m0d (Pmω (Aω − B ω )Pmω ) = νω (B) and ω ω ω ω m+ d (Pm (A − B )Pm ) =

1 ω dim Zm − iω (B) − νω (B). 2

Similarly, we have ω ω ω ω − ω ω ω ω m− d (Pm (A − B )Pm ) ≤ m (Pm (A − B−s0 )Pm ) =

1 ω dim Zm + iω (B), 2

ω ω ω ω − ω ω ω ω 0 ω ω ω ω m− d (Pm (A − B )Pm ) ≥ m (Pm (A − Bs0 )Pm ) − md (Pm (A − B )Pm )

≥

1 ω dim Zm + iω (B), 2

ω ω ω ω Hence m− d (Pm (A − B )Pm ) =

1 2

ω + i (B) and the proof is complete. dim Zm ω

 

Theorem 3.3 ([202]) For 0 < d ≤ 14 #(AωP −BPω ) #−1 , there exists an m∗ > 0 such that for m ≥ m∗ we have ω ω m+ d (Pm (AP − BP )Pm ) =

1 2 1 2

dim Zm,γP + iωP (γP ) − iωP (γ ) − νωP (γ ),

ω ω m− d (Pm (AP − BP )Pm ) =

1 2 1 2

dim Zm,γP − iωP (γP ) + iωP (γ ),

ω = 1,

dim Zm,γP − iωP (γP ) + iωP (γ ) − n,

ω = 1.

dim Zm,γP

+ iωP (γP ) − iωP (γ ) − νωP (γ ) + n,

m0d (Pm (AωP − BPω )Pm ) = νωP (γ ).

ω = 1, ω = 1.

(4.60)

Proof Let x(t) = γP (t)ξ(t), ξ ∈ W ω . Then we have (A − B)x, x 1 τ = [(−J x(t), ˙ x(t)) − (B(t)x(t), x(t))]dt

0



0

τ

=

τ

= 0

[(−J ξ˙ (t), ξ(t)) − (γP (t)T J γ˙P (t)ξ(t), ξ(t)) − (γP (t)T B(t)γP (t)ξ(t), ξ(t))]dt !γP (t)ξ(t), ξ(t))]dt, [(−J ξ˙ (t), ξ(t)) − (B

!γP (t) = γP (t)T J γ˙P (t) + we remind that we have set A = AωP , B = BPω and B T !γP (t) is γP (t) B(t)γP (t). By the definition of γP (t) and B(t), we know that B ! ! symmetric and BγP (0) = BγP (τ ). Consider the following linear Hamiltonian systems !γP (t)z(t), z(t) ∈ R2n . z˙ (t) = J B

(4.61)

76

4 The P -Index Theory

Suppose ! γ (t) is the fundamental solution of (4.125). Then by direct computation, we obtain γ (t) = γP (t)−1 γ (t) = γ2 (t). ! By Theorem 3.2, there exists an m∗ > 0 such that for m ≥ m∗ such that 1 + ω ω ω ω !ω !γP ) − νω (B !γP ), m+ − iω (B dim Zm d (Pm (A − B)Pm ) = md (Pm (A − BγP )Pm ) = 2 1 − ω ω ω ω !ω !γP ), + iω (B m− dim Zm d (Pm (A − B)Pm ) = md (Pm (A − BγP )Pm ) = 2 !γω )Pmω ) = νω (B !γP ), m0d (Pm (A − B)Pm ) = m0d (Pmω (Aω − B P (4.62) !γω be the compact operator defined by (4.112) corresponding to B !γP (t). where B P ω , #(Aω − B !γω ) #−1 = #(A − B) #−1 . Hence (4.60) Note that dim Zm,γP = dim Zm P !γP ) = νω (γ2 ) = νωP (γ ). follows from (4.34), (4.130) and νω (B  

4.4 (P , ω)-Index Theory from Analytical Point of View Let Qω = Aω − B ω : W ω → W ω . We note that Aω is a bounded self-adjoint operator, B ω is a compact self-adjoint operator. Suppose N ω := ker Qω , then N ω is a finite dimensional subspace of W ω and Qω |N ω ⊥ is invertible. We denote by P ω : W ω → N ω the orthogonal projection. Set d = 14 #(Qω |N ω ⊥ )−1 #−1 . Recall that  ω = {Pmω | m = 1, 2, · · · } is the Galerkin approximation sequence of Aω . From Lemma III.1.4, we have the following result. Lemma 4.1 The difference between the d-Morse indices − ω ω ω ω ω ω ω m− d (Pm (A − B )Pm ) − md (Pm A Pm )

(4.63)

and m0d (Pmω (Aω − B ω )Pmω ) eventually becomes a constant independent of m, and for large m there holds m0d (Pmω (Aω − B ω )Pmω ) = m0 (Aω − B ω ).

(4.64)

Following Definition III.1.5, we have the following definition for relative ω-index. Definition 4.2 We define the relative index by − ω ω ω ω ω ω ω ∗ I (Aω , Aω −B ω ) = m− d (Pm (A −B )Pm )−md (Pm A Pm ), m ≥ m ,

(4.65)

where m∗ > 0 is a constant large enough such that the difference in (4.63) becomes a constant independent of m ≥ m∗ .

4.4 (P , ω)-Index Theory from Analytical Point of View

77

Remark 4.3 Following Lemma III.1.7, up to a sign, the relative index I (Aω , Aω − B ω ) is exactly the spectral flow of the operator family Aω − sB ω . Precisely, there holds I (Aω , Aω − B ω ) = −sf (Aω − sB ω ). !γω is !γω : W ω → W ω the compact self-adjoint operator. The operator B Let B P P defined by !γω ξ, η = B P



τ 0

!γP (t)ξ(t), η(t))dt, (B

(4.66)

where !γP (t) = γP (t)T J γ˙P (t) + γP (t)T B(t)γP (t) B

(4.67)

and B(t) ∈ C([0, τ ], L(R2n )) is a symmetric matrix function. Define the operator B¯ γωP : W ω → W ω by B¯ γωP ξ, η =



τ 0

(B¯ γP (t)ξ(t), η(t))dt,

(4.68)

where B¯ γP (t) = γP (t)T J γ˙P (t).

(4.69)

!γP (t) and B¯ γP (t) Recall that γP ∈ Pτ (2n) satisfies γP (τ ) = P . So we have that B !γP (0) = B !γP (τ ), B¯ γP (0) = B¯ γP (τ ). are both symmetric matrix functions and B Therefore we can give the following definition γ

γ

Definition 4.4 ([202]) We defined the index pair (jωP (B), nωP (B)) of B by !γω ) − I (Aω , Aω − B¯ γω ), jωγP (B) = I (Aω , Aω − B P P !γω ). nγωP (B) = m0 (A − B) = m0 (Aω − B P

(4.70)

Theorem 4.5 For ω ∈ U and P ∈ Sp(2n), we have jωγP (B) = iωP (γ ),

(4.71)

nγωP (B) = νωP (γ ),

(4.72)

where γ (t) is the fundamental solution of z˙ (t) = J B(t)z(t). Thus the two definitions of the (P , ω)-index are consistent.

78

4 The P -Index Theory

Proof Recalling the notations in Theorem 2.3, by (4.70) and (4.65), we have !γω ) − I (Aω , Aω − B¯ γω ) jωγP (B) = I (Aω , Aω − B P P − ω ω ω ω ω ω !ω = m− d (Pm (A − BγP )Pm ) − md (Pm A Pm )   − ω ω ω ω ω ω ¯ω − m− d (Pm (A − BγP )Pm ) − md (Pm A Pm ) − ω ω ω ω ω ω ¯ω !ω = m− d (Pm (A − BγP )Pm ) − md (Pm (A − BγP )Pm )

= iω (γ2 ) − iω (γ3 ), !γP (t)z(t) where γ2 (t) = γP (t)−1 γ (t) is the fundamental solution of z˙ (t) = J B −1 ¯ and γ3 (t) = γP (t) is the fundamental solution of z˙ (t) = J BγP (t)z(t). The last equality is achieved by Theorem 3.2. When P = I , then γ2 = γ and γ3 ≡ I , so iω (γ2 ) − iω (γ3 ) = iω (γ ) − iω (I ) = iωI (γ ).

(4.73)

When P = I , in the Definition 1.7, we take ξ(t) = γ3 (t) = γP (t)−1 , then by (4.35) and (4.36), we obtain iω (γ2 ) − iω (γ3 ) = iωP (γ ).

(4.74)

!γP ) = νωP (γ ), and by Floquet theory, (4.57) and (4.64) we get the Note that νω (B second equality.   Definition 4.6 If z(t) is a solution of the nonlinear system z˙ (t) = J H  (t, z(t)), x(t) ∈ R2n and Bz (t) = H  (t, z(t)), we define jωγP (z) = jωγP (Bz ), νωγP (z) = νωγP (Bz ). Remark 4.7 We note that the relation (4.32) and the homotopy in (4.37) are the only key ingredients in the arguments of Sects. 4.3 and 4.4, so ∀ γ0 ∈ Pτ (2n) satisfies γ0 (τ ) = P and γ0 (0)T J γ˙0 (0) = γ0 (τ )T J γ˙0 (τ ), P −1 γ0 (t) ∼ω γ0 (τ − t)−1 on [0, τ ] with fixed points,

(4.75) (4.76)

we can obtain all results in these two sections if replacing γP by γ0 . Thus the index γ γ pair (jω0 (B), nω0 (B)) is not dependent on the choice of γ0 in this sense and we γ γ P denote (jω (B), nPω (B)) := (jω0 (B), nω0 (B)).

4.5 Bott-Type Formula for the Maslov Type P -Index

79

4.5 Bott-Type Formula for the Maslov Type P -Index In this section, we establish the Bott-type iteration formula for the Maslov type P -index theory. For P ∈ Sp(2n), we consider the following P -boundary value problem 

x(t) ˙ = J H  (t, x(t)), x(τ ) = P x(0),

(4.77)

where H ∈ C 2 (R × R2n , R) satisfying H (t + τ, P x) = H (t, x), H  (t, x) is the gradient of H with respect to the variable x. Clearly we have P T H  (t + τ, P x)P = H  (t, x). Let x(t) be a solution of (4.77). Linearizing the Hamiltonian system (4.77) at the solution x we get a linear Hamiltonian system y˙ = J B(t)y,

(4.78)

where B(t) = H  (t, x(t)) satisfies B(t + τ ) = H  (t + τ, x(t + τ )) = H  (t + τ, P x(t)) = (P −1 )T H  (t, x(t))P −1 . (4.79) It implies P T B(t + τ )P = B(t) (we say B satisfying (P , τ ) periodic condition). We can extend the definition of B(t) from [0, τ ] to [0, +∞) in the obvious way. Recall that Pˆ τ (2n) = {γ ∈ C 1 ([0, τ ], Sp(2n)) | γ (0) = I, γ˙ (τ ) = P γ˙ (0)P −1 γ (τ )}. Note that the set Pˆ τ (2n) formed by the fundamental solutions of all the linear Hamiltonian systems (4.78) with continuous symmetric and (P , τ )-periodic coefficients and Pˆ τ (2n) is dense in Pτ (2n). Suppose that the continuous symplectic path γ : [0, τ ] → Sp(2n) is the fundamental solution of (4.78). For k ∈ N, we define Bk (t) = B(t)|[0, kτ ]

(4.80)

and define the k-times iteration path γ k : [0, kτ ] → Sp(2n) of γ by

γ k (t) =

⎧ ⎪ γ (t), ⎪ ⎪ ⎪ ⎪ ⎪ P γ (t − τ )P −1 γ (τ ), ⎪ ⎪ ⎪ ⎨P 2 γ (t − 2τ )(P −1 γ (τ ))2 , ⎪ P 3 γ (t − 3τ )(P −1 γ (τ ))3 , ⎪ ⎪ ⎪ ⎪ ⎪ ······ ⎪ ⎪ ⎪ ⎩ k−1 P γ (t − (k − 1)τ )(P −1 γ (τ ))k−1 ,

t ∈ [0, τ ], t ∈ [τ, 2τ ], t ∈ [2τ, 3τ ], t ∈ [3τ, 4τ ], t ∈ [(k − 1)τ, kτ ].

(4.81)

80

4 The P -Index Theory

Note that γ k (t) is the fundamental solution of y(t) ˙ = J Bk (t)y(t). We suppose P k = exp (M¯ 1 ) exp (M¯ 2 ), where M¯ 1 , M¯ 2 satisfy (4.30) and (4.31) respectively. When the interval get changed from [0, τ ] to [0, kτ ], we have Pk (t) = exp (t M¯ 1 /kτ ) exp (t M¯ 2 /kτ ) naturally and Pk (t) ∈ Pτ (2n). Note that Pk (kτ ) = P k . We obtain γP k (t) by the same argument of the choice of γP (t). We extend γP (t) from [0, τ ] to [0, +∞) by γP (t + τ ) = P · γP (t). And we define γ¯P k (t) = γP (t)|[0,kτ ] . Note that γ¯P (t) = γP (t) and γ¯P k (kτ ) = P k . In order to simplify the calculation of the iteration of Malsov P -index, we want to replace γP k (t) by γ¯P k (t). In view of Remark 4.7, (4.75) holds with γ¯P k (t) on [0, kτ ], so we only need the following lemma. Lemma 5.1 For the iterated path γ¯P k (t), we have the following ω-homotopy P −k γ¯P k (t) ∼ω γ¯P k (kτ − t)−1 on [0, kτ ] with fixed points.

(4.82)

Proof As consequences of definitions, we get ⎧ ⎪ P −k γP (t), ⎪ ⎪ ⎪ ⎪ −k+1 γ (t − τ ), ⎪ P ⎪P ⎪ ⎪ ⎨P −k+2 γ (t − 2τ ), P P −k γ¯P k (t) = ⎪ ······ ⎪ ⎪ ⎪ ⎪ ⎪ P −2 γP (t − (k − 2)τ ), ⎪ ⎪ ⎪ ⎩ −1 P γP (t − (k − 1)τ ),

t ∈ [0, τ ], t ∈ [τ, 2τ ], t ∈ [2τ, 3τ ],

(4.83)

t ∈ [(k − 2)τ, (k − 1)τ ], t ∈ [(k − 1)τ, kτ ].

⎧ ⎪ γP (τ − t)−1 P −(k−1) , ⎪ ⎪ ⎪ ⎪ ⎪ γP (2τ − t)−1 P −(k−2) , ⎪ ⎪ ⎪ ⎨γ (3τ − t)−1 P −(k−3) ,

t ∈ [0, τ ], t ∈ [τ, 2τ ],

t ∈ [2τ, 3τ ], P ⎪ ······ ⎪ ⎪ ⎪ ⎪ ⎪ γP ((k − 1)τ − t)−1 P −1 , t ∈ [(k − 2)τ, (k − 1)τ ], ⎪ ⎪ ⎪ ⎩ γP (kτ − t)−1 , t ∈ [(k − 1)τ, kτ ]. (4.84) Note that P −k γ¯P k (t) and γ¯P k (kτ − t)−1 have the same values at t = 0, τ, 2τ, · · · , (k − 1)τ, kτ , which are P −k , P −k+1 , · · · , P −1 , I , respectively. We only prove here that P −k γ¯P k (t) and γ¯P k (kτ − t)−1 are homotopic on [0, τ ] with fixed end points given by δ1 (s, t), the rest is similar, suppose P −k γ¯P k (t) and γ¯P k (kτ − t)−1 are homotopic on [j τ, (j + 1)τ ] with fixed end points given by δj (s, t), j = 1, 2, k − 1, so we complete the proof by the homotopy map δk−1 ∗ δk−2 ∗ · · · ∗ δ1 . If we have constructed a homotopic map δ(s, t) such that P −k γ¯P k (t) ∼ω γ¯P k (kτ − t)−1 on [0, τ ] along δ(·, τ ), according to Lemma 1.3 and Remark 1.4, the homotopy δ can be modified to fix the end points all the time. γ¯P k (kτ − t)−1 =

4.5 Bott-Type Formula for the Maslov Type P -Index

81

In the following, we use Corollary 6.2.2, Corollary 6.2.5 and Corollary 6.2.10 in [223] repeatedly (or apply Theorem II.2.5 and Theorem II.2.6) to give a direct proof of P −k γ¯P k (t) ∼ω γ¯P k (kτ − t)−1 on [0, τ ] with fixed points. P −k γP (t) ∼ω γP (τ − t)−1 P −(k−1) on [0, τ ] with fixed end points ⇐⇒ γP (t)P k−1 ∼ω P k γP (τ − t)−1 on [0, τ ] with fixed end points ⇐⇒ iω (γP (t)P k−1 ) = iω (P k γP (τ − t)−1 ) = iω (γP (τ − t)−1 P k ) ⇐⇒ γP (t)P k−1 ∼ω γP (τ − t)−1 P k on [0, τ ] with fixed end points ⇐⇒ γP (t) ∼ω γP (τ − t)−1 P on [0, τ ] with fixed end points ⇐⇒ iω (γP (t)) = iω (γP (τ − t)−1 P ). The first and the forth implications “ ⇐⇒ ” follow from the definition of homotopy of two paths, the second and third “ ⇐⇒ ” follow from the results of Corollary 6.2.5 and Corollary 6.2.10 in [223], and the last “ ⇐⇒ ” follows from Corollary 6.2.2 in [223]. Recall that P (t) = exp (tM1 /τ ) exp (tM2 /τ ), ∀t ∈ [0, τ ]. Then we obtain   iω (P (τ − t)−1 P ) = iω exp (M2 )P (τ − t)−1 P exp (−M2 )   = iω exp (tM2 /τ ) exp (tM1 /τ )   = iω exp (−tM2 /τ ) exp (tM2 /τ ) exp (tM1 /τ ) exp (tM2 /τ ) = iω (P (t)). Thus P −k P (t) ∼ω P (τ − t)−1 P −(k−1) on [0, τ ] with fixed end points. Note that γP (t) ∼ P (t) on [0, τ ] with fixed end points, thus P −k γP (t) ∼ω γP (τ − t)−1 P −(k−1) on [0, τ ] with fixed end points.   For any τ > 0, γ ∈ Pˆ τ (2n) and ω ∈ U, we define γ¯

k

k

k

γ¯

k

jωP (γ k ) = jωP (γ k ), nPω (γ k ) = nωP (γ k ). k

(4.85)

k

When ω = 1, we denote jωP (γ k ) and nPω (γ k ) by jP k (γ k ), nP k (γ k ) respectively. We define the Hilbert space Wk1 = W 1/2,2 (Skτ , C2n ), where Skτ = R/(kτ Z), and the Hilbert space k

WkP = {x ∈ L2 ([0, kτ ], C2n ) | x(t) = γ¯P k (t)ξ(t), ξ(t) ∈ Wk1 }.

(4.86)

82

4 The P -Index Theory

Recall that WγωP = {x ∈ L2 ([0, τ ], C2n ) | x(t) = γP (t)ξ(t), ξ(t) ∈ W ω }. If k

ωk = 1, we can identify WγωP with the subspace {x ∈ WkP | x(t + τ ) = ωP x(t)} Pk

of Wk . k We define two self-adjoint operators and a quadratic form on WkP by Ak x, y 1 =

kτ



kτ

(−J x(t), ˙ y(t))dt, Bk x, y 1 =

(Bk (t)x(t), y(t))dt,

0

0

Qk (x, y) = (Ak − Bk )x, y 1 .

(4.87) (4.88)

We also define a quadratic form on WγωP by Q(x, y) = (A − B)x, y 1 ,

(4.89)

where the operators A and B are defined by (4.44) and (4.46). Lemma 5.2 ([202]) Let k ≥ 1, P ∈ Sp(2n) be given. For ωk = 1, then WγωP are orthogonal subspace of Wγ1¯ k for Qk , and Wγ1¯ k splits into a direct sum P

P

Wγ1¯

Pk

Proof Any x(t) ∈ Wγ1¯

Pk

x(t) = γ¯P k (t)

=



WγωP .

(4.90)

ωk =1

can be written as:  p∈Z

"

√ # 2pπ t −1 exp ap , ap ∈ C2n . kτ

(4.91)

For q = 0, 1, . . . , k − 1, we denote by C(q) the set of all p such that p − q ∈ kZ. We may write x(t) =

k−1 

xq (t), with xq (t) = γ¯P k (t)

q=0

 p∈C(q)

"

√ # 2pπ t −1 exp ap . kτ

(4.92)

We then check that 

xq (t + τ ) = P · γ¯P k (t)

p∈C(q)

"

√ √ # 2pπ t −1 + 2pπ τ −1 exp ap kτ

(4.93)

= ωq P xq (t), $ √ % ω where ωq = exp 2π qk −1 . So we have xq ∈ WγPq . When q runs from 0 to k − 1, then ω runs through the k-th roots of unity. Hence we get the splitting (4.90).

4.5 Bott-Type Formula for the Maslov Type P -Index

83

Letting x ∈ WγωP and y ∈ WγλP , where ω and λ are k-th roots of unity, there holds

j +1

[(−J x(t), ˙ y(t)) − (B(t)x(t), y(t))] dt

j

= ωλ¯



(4.94)

j j −1

[(−J x(t), ˙ y(t)) − (B(t)x(t), y(t))] dt.

Then we get

kτ

Qk (x, y) =

[(−J x(t), ˙ y(t)) − (Bk (t)x(t), y(t))] dt

0

τ k−1  j ¯ (ωλ) [(−J x(t), ˙ y(t)) − (B(t)x(t), y(t))] dt = 0

j =0

=

k−1  ¯ j Q(x, y). (ωλ) j =0

When ω = λ, k−1 k−1   (ωλ¯ )j = (ωω) ¯ j = k. j =0

j =0

When ω = λ, k−1  ¯ k 1 − (ωλ) ¯ j = (ωλ) = 0. 1 − ωλ¯ j =0

It implies WγωP and WγλP are orthogonal for Qk , ∀ω = λ and Qk (x, y) = kQ(x, y), x, y ∈ WγωP .

(4.95)  

Based on the results of the previous sections, following the idea of [28], we now prove the following result which is called the Bott-type formula for the (P , ω)index. Theorem 5.3 ([202]) For any τ > 0, ω0 ∈ U, γ ∈ Pτ (2n) and k ∈ N, we have k

iωP0 (γ k ) =

 ωk =ω0

k

iωP (γ ), νωP0 (γ k ) =

 ωk =ω0

νωP (γ ).

(4.96)

84

4 The P -Index Theory

Proof We first assume that ω0 = 1. By definition, there hold nP k (γ k ) = m0 (Ak − Bk ), nPω (γ ) = m0 (A − B). By (4.95) and Lemma 5.2, we get nP k (γ k ) =



nPω (γ ).

ωk =1

Let x(t), y(t) ∈ Wγ1¯ k , we set x(t) = γ¯P k (t)ξ(t), y(t) = γ¯P k (t)η(t), ξ , η ∈ Wk1 . P Then we have Ak x, y 1 =

kτ

[(−J x(t), ˙ y(t))

0

=

kτ

0

=

kτ

[(−J ξ˙ (t), η(t)) − (γ¯P k (t)T J γ˙¯P k (t)ξ(t), η(t)) [(−J ξ˙ (t), η(t)) − (B¯ k (t)ξ(t), η(t))]dt,

0

where B¯ k (t) = γ¯P k (t)T J γ˙¯P k (t) ⎧ ⎪ γP (t)T J γ˙P (t), ⎪ ⎪ ⎪ ⎨γ (t − τ )T J γ˙ (t − τ ), P P = ⎪ ······ ⎪ ⎪ ⎪ ⎩ γP (t − (k − 1)τ )T J γ˙P (t − (k − 1)τ ),

t ∈ [0, τ ], t ∈ [τ, 2τ ],

(4.97)

t ∈ [(k − 1)τ, kτ ].

We can easily see that B¯ k (t) is symmetric by the definition of γ¯P k (t) and τ -periodic. (Ak − Bk )x, y 1 kτ = [(−J x(t), ˙ y(t)) − (Bk (t)x(t), y(t))]dt

0 kτ

= 0



[(−J ξ˙ (t), η(t)) − (γ¯P k (t)T J γ˙¯P k (t)ξ(t), η(t))

− (γ¯P k (t)T Bk (t)γ¯P k (t)ξ(t), η(t))]dt kτ

= 0

&k (t)ξ(t), η(t))]dt, [(−J ξ˙ (t), η(t)) − (B

4.5 Bott-Type Formula for the Maslov Type P -Index

85

where !k (t) = γ¯P k (t)T J γ˙¯P k (t) + γ¯P k (t)T Bk (t)γ¯P k (t) B ⎧ T T ⎪ t ∈ [0, τ ], ⎪ ⎪γP (t) J γ˙P (t) + γP (t) B(t)γP (t), ⎪ ⎪ T T ⎪ ⎪ ⎨γP (t − τ ) J γ˙P (t − τ ) + γP (t − τ ) B(t − τ )γP (t − τ ), t ∈ [τ, 2τ ], = ······ ⎪ ⎪ ⎪ ⎪ γP (t − (k − 1)τ )T J γ˙P (t − (k − 1)τ )+ ⎪ ⎪ ⎪ ⎩γ (t − (k − 1)τ )T B(t − (k − 1)τ )γ (t − (k − 1)τ ), t ∈ [(k − 1)τ, kτ ]. P P (4.98)

It is also symmetric and τ -periodic. We define A1k : Wk1 → Wk1 by A1k ξ, η =

kτ

(−J ξ˙ (t), η(t))dt.

(4.99)

!k (t)ξ(t), η(t))dt, (B

(4.100)

0

!1 : W 1 → W 1 by Define B k k k !k1 ξ, η = B



kτ

0

!k (t) is defined by (4.98). where B Define B¯ k1 : Wk1 → Wk1 by B¯ k1 ξ, η =



kτ

(B¯ k (t)ξ(t), η(t))dt,

0

where B¯ k (t) is defined by (4.97). By definition, there hold !k1 ) − I (A1k , A1k − B¯ k1 ), jP k (γ k ) = I (A1k , A1k − B !γω ) − I (Aω , Aω − B¯ γω ), jωP (γ ) = I (Aω , Aω − B P P

(4.101)

!γω and B¯ γω is defined by (4.66) and (4.68) respectively. where the operator B P P As a special case of Lemma 5.2, we have Wk1 =

 ωk =1

W ω.

(4.102)

86

4 The P -Index Theory

We can treat W ω as W ω = {y ∈ Wk1 | y(τ ) = ωy(0)}, which is the subspace of Wk1 . Taking ξ ∈ W ω and η ∈ W λ , where ω and λ are k-th roots of unity, by (4.97) and (4.98) there hold (A1k

!k1 )ξ, η = −B



kτ

(−J ξ˙ (t), η(t))dt −



0

kτ

!k (t)ξ(t), η(t))dt (B

0

τ k−1  j !γP (t)ξ(t), η(t))] dt ¯ = (ωλ) [(−J ξ˙ (t), η(t)) − (B 0

j =0

=

k−1  !γω )ξ, η , ¯ j (Aω − B (ωλ) P j =0

(4.103)

(A1k − B¯ k1 )ξ, η =



kτ

(−J ξ˙ (t), η(t))dt −

0



kτ

(B¯ k (t)ξ(t), η(t))dt

0

τ k−1  j ¯ = (ωλ) [(−J ξ˙ (t), η(t)) − (B¯ γP (t)ξ(t), η(t))] dt j =0

=

0

k−1  (ωλ¯ )j (Aω − B¯ γωP )ξ, η , j =0

(4.104) !γP (t), B¯ γω and B¯ γP (t) satisfy (4.66), (4.67), (4.68), and (4.69) !γω , B where B P P respectively. By (4.103) and (4.104), we have !k1 ) = I (A1k , A1k −B



!γω ), I (A1k , A1k −B¯ k1 ) = I (Aω , Aω −B P

ωk =1

 ωk =1

I (Aω , Aω −B¯ γωP ). (4.105)

So by (4.101), we get jP k (γ k ) =

 ωk =1

jωP (γ ), nP k (γ k ) =

 ωk =1

We complete the proof by Theorem 4.5 and (4.106).

nPω (γ ).

(4.106)

4.6 Iteration Theory for P -Index

87 k

ω0 For the general case, we should replace the space WkP by Wkτ,γ defined k $ √ % √P (2π q+θ0 ) −1 with ω0 = exp(θ0 −1). It is in (4.43) and ωq by ωq, ω0 = exp k k easy to see ωq, ω0 = ω0 . For the general case of γ ∈ Pτ (2n). We can choose β ∈ Pˆ τ (2n) such that β(τ ) = γ (τ ) and β ∼ γ with fixed end points. This homotopy can be extended to [0, 1] × [0, kτ ] for any k ∈ N. Thus we obtain β k ∼ γ k , ∀k ∈ N. By Proposition 1.12, for all k ∈ N and ω ∈ U we obtain k

k

k

k

iωP (γ k ) = iωP (β k ), νωP (γ k ) = νωP (β k ).

(4.107)

Together with (4.96) for β, we obtain it for γ . This completes the proof of Theorem 5.3.  

4.6 Iteration Theory for P -Index 4.6.1 Splitting Numbers Definition 6.1 For any τ > 0, P ∈ Sp(2n), γ ∈ Pτ (2n), m ∈ N, and ω ∈ U, we define m

m

m

m

iωP (γ , m) = iωP (γPm ), νωP (γ , m) = νωP (γPm ).

(4.108)

Where the iteration path γPm ∈ Pmτ (2n) is defined by (4.81). m

m

If the subindex ω = 1, we simply write (i(γ , m), ν(γ , m)), (i P (γ , m), ν P (γ , m)) etc, and omit the subindex 1 when there is no confusion. Fix τ > 0, P ∈ Sp(2n) and a path γ ∈ Pτ (2n), next we study the properties of the index function iωP (γ ) of γ at ω as a function ω ∈ U, and use the short notations when γ is a fixed path i P (ω) = iωP (γ ), ν P (ω) = νωP (γ ).

(4.109)

¯ ν P (ω) = ν P (ω). ¯ i P (ω) = i P (ω),

(4.110)

By definition we have

Lemma 6.2 ([203]) i P (·) is locally constant on U \ (σ (P −1 γ (τ )) ∪ σ (P −1 )), and thus is constant on each connected component of U \ (σ (P −1 γ (τ )) ∪ σ (P −1 )). It holds that ν P (ω) = 0, ∀ω ∈ U \ (σ (P −1 γ (τ )) ∪ σ (P −1 )).

(4.111)

88

4 The P -Index Theory

Proof Note that U ∩ (σ (P −1 γ (τ )) ∪ σ (P −1 )) contains at most 4n points. For any ω0 ∈ U \ (σ (P −1 γ (τ )) ∪ σ (P −1 )), let N (ω0 ) be an open connected neighborhood of ω0 in U \ (σ (P −1 γ (τ )) ∪ σ (P −1 )). By Definition 1.1 we get ν P (ω) = 0 for ∗ (2n) for all ω ∈ N (ω ). As all ω ∈ N (ω0 ). Then (4.111) holds and γ ∈ ∗P Pτ,ω 0 in Definition 1.7, we can connect P −1 to Mn+ or Mn− by β0 in Sp(2n)∗ω0 , connect P −1 γ (τ ) to Mn+ or Mn− by β1 in Sp(2n)∗ω0 . By the compactness of the image of β0 , β1 and the openness of Sp(2n)∗ω0 in Sp(2n), we can further require N (ω0 ) to be so small that both β0 and β1 are completely located within Sp(2n)∗ω for all ω ∈ N (ω0 ). We choose a path ξ ∈ Pτ (2n) so that ξ(τ ) = P −1 . Then by definition, this implies iω (P −1 γ ∗ ξ ) = iω0 (P −1 γ ∗ ξ ), iω (ξ ) = iω0 (ξ ) for all ω ∈ N (ω0 ). Thus by Definition 1.7, we have i P (ω) = i P (ω0 ) for all ω ∈ N (ω0 ).   Corollary 6.3 ([203]) The discontinuity points of i P (·) and ν P (·) are contained in U ∩ σ (P −1 γ (τ )) ∩ σ (P −1 ). Analogy to the definition of splitting numbers about Maslov-type index, we give the following definition. Definition 6.4 ([203]) For any P ∈ Sp(2n), M ∈ Sp(2n) and ω ∈ U, choosing τ > 0 and γ ∈ Pτ (2n) with γ (τ ) = M, we define ± P SM (ω)

P √ = lim iexp(±ε (γ ) − iωP (γ ). −1)ω ε→0+

(4.112)

± ± (ω) = SP±−1 M (ω) − SP±−1 (ω). Thus P SM (ω) is Lemma 6.5 ([203]) We have P SM well defined in the sense that it is independent of the path γ ∈ Pτ (2n) with γ (τ ) = M appearing in (4.112).

Proof Choose ξ ∈ Pτ (2n) such that ξ(τ ) = P −1 , then by definition we obtain ± P SM (ω)

P √ = lim iexp(±ε (γ ) − iωP (γ ) −1)ω ε→0+

= lim (iexp(±ε√−1)ω (P −1 γ ∗ ξ ) ε→0+

− iexp(±ε√−1)ω (ξ )) − iω (P −1 γ ∗ ξ ) + iω (ξ ) = lim (iexp(±ε√−1)ω (P −1 γ ∗ ξ ) − iω (P −1 γ ∗ ξ )) ε→0+

− lim (iexp(±ε√−1)ω (ξ ) − iω (ξ )) ε→0+

=

SP±−1 M (ω) − SP±−1 (ω).

Since SP±−1 M (ω) and SP±−1 (ω) are independent of the choice of the path γ , so we complete the proof.  

4.6 Iteration Theory for P -Index

89

Lemma 6.6 ([203]) For P , M ∈ Sp(2n) and ω ∈ U, there hold ± P SM (ω)

if ω ∈ U \ (σ (P −1 γ (τ )) ∪ σ (P −1 )),

= 0,

+ P SM (ω)

− = P SM (ω). ¯

(4.113) (4.114)

For Pi , Mi ∈ Sp(2ni ) with i = 0 and 1, P = P1  P2 , there holds ± P SM0 M1 (ω)

± ± = P1 SM (ω) + P2 SM (ω), ∀ω ∈ U. 0 1

(4.115)

Proof This is a direct consequence of Definition 6.4, Lemma 6.5 and the symplectic additivity of Maslov P -index.  

4.6.2 Abstract Precise Iteration Formulas Definition 6.7 ([203]) We define the P -mean index of γ ∈ Pτ (2n) by m

i¯P (γ ) ≡

i P (γ , m) lim . m→+∞ m

(4.116)

The following result tall us that the P -mean index i¯P (γ ) is well defined and always finite. Proposition 6.8 ([203]) For any τ > 0 and γ ∈ Pτ (2n) there hold ¯ −1 γ ∗ ξ ) − i(ξ ¯ )= 1 i¯P (γ ) = i(P 2π



2π

0

i P√−1θ (γ ) dθ, e

(4.117)

¯ where i(ρ) is the mean index of ρ for the Maslov-type index in the periodic case, m

1 ν P (γ , m) ν¯ (γ ) ≡ lim = m→+∞ m 2π



2π

P

0

νe√−1θ (P −1 γ ∗ ξ ) dθ = 0.

(4.118)

Especially, i¯P (γ ) is always a finite real number. Proof This is a direct consequence of Theorem 5.3 and (2.15).   For P , M ∈ Sp(2n), we define CP (M) =

 0 0, then R(θ1 )  ·  R(θu ) ∈ 0 (K) for some θj ∈ (0, π ) satisfying the condition that 0 < mθj /2π ≤ 1 with 1 ≤ j ≤ u. In this case, all eigenvalues of√K on U+ \ {1} (or U− √ \ {1}) are located on the open arc between 1 and exp(2π −1/m) (or exp(−2π −1/m)) in U+ (in U− ) and are all Krein-negative (or Krein-positive) definite. (3) The right equality of (4.130) holds for some m ≥ 2 if and only if I2p  N1 (1, 1)r  K ∈ 0 (P −1 γ (τ )) for some non-negative integers p and r satisfying 0 ≤ p + r ≤ n and some K ∈ Sp(2(n − p − r)) with σ (K) ⊂ U \ {1} satisfying the following conditions: If m = 2 and n − p − r > 0, it holds that −I2s  N1 (−1, 1)t ∈ 0 (K) for some non-negative integers s1 and t satisfying s + t = n − p − r. If m ≥ 3, we must have n = p + r. (4) The two equalities of (4.130) hold for some m1 and m2 ≥ 2 respectively if and only if P −1 γ (τ ) = I2p  N1 (−1, −1)n−p for some non-negative integers p ≤ n. Here p < n happens only when m1 = m2 = 2. Proof By Theorem 5.3 with ω0 = 1, summing (4.125) over all m-th roots of unity, we obtain (m − 1)(i P (γ , 1) + ν P (γ , 1) − n) + (m − 1)i1 (ξ )  m iω (ξ ) + i P (γ , 1) ≤ i P (γ , m) − ωm =1,ω=1

≤ (m − 1)(i P (γ , 1) + n) + (m − 1)i1 (ξ )   νωP (γ ) − iω (ξ ) + i P (γ , 1). − ωm =1,ω=1

ωm =1,ω=1

This yields (4.130). The equality conditions follow from Proposition II.2.10.

 

94

4 The P -Index Theory

Proposition 6.14 ([203]) For any path γ ∈ Pτ (2n), P ∈ Sp(2n), set M = γ (τ ) and extend γ to [0, ∞) by (4.81). Then for any m1 and m2 ∈ N we have νP

m1

(γ , m1 ) + ν P

m2

(γ , m2 ) − ν P

+ ν(ξ, m1 + m2 ) − ≤ iP

(m1 +m2 )

≤ νP

(m1 ,m2 )

(γ , (m1 , m2 )) − ν(ξ, (m1 , m2 ))

e(P −1 M) e(P −1 ) − 2 2

(γ , m1 + m2 ) − i P

(γ , (m1 , m2 )) − ν P

+ ν(ξ, (m1 , m2 )) +

(m1 ,m2 )

m1

(γ , m1 ) − i P

(m1 +m2 )

e(P −1 M) 2

m2

(γ , m2 )

(γ , m1 + m2 ) − ν(ξ, m1 ) − ν(ξ, m2 )

+

e(P −1 ) . 2

(4.131)

where (m1 , m2 ) is the greatest common divisor of m1 and m2 . Proof Proposition 6.14 is a direct result of (4.123) and Proposition II.2.11.

 

The following result is a direct corollary of Proposition 6.14. Corollary 6.15 ([203]) For any path γ ∈ Pτ (2n), P ∈ Sp(2n), set M = γ (τ ) and extend γ to [0, ∞) by (4.81). Then for any m ∈ N we have m

ν P (γ , m) − ν(ξ, 1) + ν(ξ, m + 1) − ≤ iP

(m+1)

e(P −1 M) e(P −1 ) − 2 2

m

(γ , m + 1) − i P (γ , m) − i P (γ , 1)

≤ ν P (γ , 1) − ν P

(m+1)

(γ , m + 1) − ν(ξ, m) +

(4.132) e(P −1 M) e(P −1 ) + . 2 2

Proposition 6.16 ([203]) For any path γ ∈ Pτ (2n), P ∈ Sp(2n), set M = γ (τ ) and extend γ to [0, ∞) by (4.81). Suppose that there exist A1 , A2 ∈ Sp(2n) and B1 , B2 ∈ Sp(2n − 2) such that −1 = A−1 P −1 M = A−1 1 (N1 (1, 1)  B1 )A1 , P 2 (N1 (1, 1)  B2 )A2 .

(4.133)

Then for any m ∈ N we have m

ν P (γ , m) − ν(ξ, 1) + ν(ξ, m + 1) − ≤ iP

(m+1)

e(P −1 M) e(P −1 ) − +1 2 2

m

(γ , m + 1) − i P (γ , m) − i P (γ , 1)

≤ ν P (γ , 1) − ν P

(m+1)

(γ , m + 1) − ν(ξ, m) +

e(P −1 M) e(P −1 ) + − 1. 2 2 (4.134)

Proof It is a direct result of Theorem 5.3 and Proposition II.2.12.

 

Chapter 5

The L-Index Theory

5.1 Definition of L-Index In this subsection for L ∈ (n) we will define an index pair for any symplectic path γ ∈ Pτ (2n) with L-boundary condition(L-index for short). Comparing with the Maslov-type index theory for periodic boundary condition which is suitable to be used in the study of periodic solution of a Hamiltonian systems, the L-index theory is suitable to be used in the study of Hamiltonian systems with L-boundary condition. Firstly, we consider a special case. Suppose L = L0 = {0} ⊕ Rn ⊂ R2n which is a Lagrangian subspace of the linear symplectic space (R2n , ω0 ). For a symplectic path γ (t), we write it in the following form:  γ (t) =

 S(t) V (t) , T (t) U (t)

(5.1)

where S(t), T (t), V (t),  U (t) are n × n matrices. The n vectors coming from the V (t) column of the matrix are linear independent and they span a Lagrangian U (t) subspace of (R2n , ω0 ). Particularly, at t = 0, this Lagrangian subspace is L0 = {0} ⊕ Rn . For L0 = {0} ⊕ Rn , we define the following two subspaces of Sp(2n) by Sp(2n)∗L0 = {M ∈ Sp(2n)| detVM = 0}, Sp(2n)0L0 = {M ∈ Sp(2n)| detVM = 0}, 

 SM VM for M = . We note that in [232] the singular set Sp(2n)0L0 and an index T M UM theory with this singular set was defined via an analytic method. © Springer Nature Singapore Pte Ltd. 2019 C. Liu, Index theory in nonlinear analysis, https://doi.org/10.1007/978-981-13-7287-2_5

95

96

5 The L-Index Theory

Since the space Sp(2n) is path connected, and the n × n non-degenerate matrix space has two path connected components, one with detVM > 0, and another with detVM < 0, the space Sp(2n)∗L0 has two path connected components as well. We denote by Sp(2n)± L0 = {M ∈ Sp(2n)| ± detVM > 0}, − 0 then we have Sp(2n)∗L0 = Sp(2n)+ L0 ∪ Sp(2n)L0 . We call Sp(2n)L0 the L0 ∗ degenerate subspace of Sp(2n) and Sp(2n)L0 the L0 -non-degenerate subspace of Sp(2n). We denote the corresponding symplectic path spaces by

Pτ (2n)∗L0 = {γ ∈ Pτ (2n)| γ (τ ) ∈ Sp(2n)∗L0 } and Pτ (2n)0L0 = {γ ∈ Pτ (2n)| γ (τ ) ∈ Sp(2n)0L0 }. If γ ∈ Pτ (2n)0L0 , we call it the L0 -degenerate symplectic path, otherwise, if γ ∈ Pτ (2n)∗L0 , we call it the L0 -nondegenerate symplectic path. For simplicity, we fix τ = 1 now. Definition 1.1 We define the L0 -nullity of any symplectic path γ ∈ P1 (2n) by νL0 (γ ) ≡ dim kerL0 (γ (1)) := dim ker V (1) = n − rankV (1)

(5.2)

with the n × n matrix function V (t) defined in (5.1).   √ V (t) We note that rank = n, so the complex matrix U (t) ± −1V (t) is U (t) invertible. We define a complex matrix function by Q(t) = Qγ (t) = [U (t) −

√

−1V (t)][U (t) +

√ −1V (t)]−1 .

(5.3)

From Lemma I.2.13 we know that the matrix Q(t) ∈ U (n) is unitary matrix for any t ∈ [0, 1]. We denote by  M+ =

0 In −In 0



 , M− =

0 Dn −Dn 0

 , Dn = diag(−1, 1, · · · , 1).

It is clear that M± ∈ Sp(2n)± L0 . For a path γ ∈ P1 (2n)∗L0 , we first adjoin it with a simple symplectic path starting from J = −M+ , i.e., we define a symplectic path by  γ˜ (t) =

I cos (1−2t)π + J sin (1−2t)π , t ∈ [0, 1/2]; 2 2 γ (2t − 1), t ∈ [1/2, 1],

(5.4)

5.1 Definition of L-Index

97

then we choose a symplectic path β(t) in Sp(2n)∗L0 starting from γ (1) and ending − at M+ or M− according to γ (1) ∈ Sp(2n)+ L0 or γ (1) ∈ Sp(2n)L0 , respectively. We now define a joint path by  γ¯ (t) = β ∗ γ˜ :=

γ˜ (2t), t ∈ [0, 1/2], β(2t − 1), t ∈ [1/2, 1].

(5.5)

By the definition, we see that the symplectic path γ¯ starting from −M+ and ending at either M+ or M− . As above, we define ¯ Q(t) = [U¯ (t) −  for γ¯ (t) =

√

−1V¯ (t)][U¯ (t) +

√ −1V¯ (t)]−1 .

(5.6)

 ¯ S(t) V¯ (t) ¯ : [0, 1] → R . We can choose a continuous function  T¯ (t) U¯ (t)

such that ¯ detQ(t) = e2

√

¯ −1(t)

.

(5.7)

¯ ¯ − (0)) ∈ Z and it does By the above arguments, we see that the number π1 ((1) ¯ not depend on the choice of the function (t). We note that there is a positive continuous function ρ : [0, 1] → (0, +∞) such that det(U¯ (t) −

√ √ ¯ −1V¯ (t)) = ρ(t)e −1(t) .

Definition 1.2 ([189]) For a symplectic path γ ∈ P1 (2n)∗L0 , we define the L0 -index of γ by iL0 (γ ) =

1 ¯ ¯ ((1) − (0)). π

(5.8)

We now describe the index of L0 -nondegenerate symplectic path γ ∈ P1 (2n)∗L0 by another way. In (5.3) we know that Qγ (t) ∈ U (n) for any t ∈ [0, 1] (here the subscript γ in Qγ is to indicate the dependence of√γ ). By the non-degenerate condition, we have detV (1) = 0. Suppose λj (t) = e2 −1θj (t) are the eigenvalues of Qγ (t) for j = 1, · · · , n. The existence of the continuous functions θj (t) comes from that by the continuity of the matrix value function Qγ (t), we know that the eigenvalues λj (t) are all continuous functions on the interval [0, 1], so by take the logarithm we get the continuity θj : [0, 1] → R/(2π Z), moreover, by a locally continuous extension method, one can get continuous functions θj : [0, 1] → R with θj (0) = 2kπ for some fix k ∈ Z, for example k = 0. More precisely, for any t0 ∈ [0, 1], there is a small δ0 > 0 such that in N(t0 ) := (t0 − δ0 , t0 + δ0 ) ∩ [0, 1] the functions θj0 : N(t0 ) → R are well defined(not unique, the difference of any two components may be 2kπ for some k ∈ Z). For another choice t1 ∈ [0, 1] with N(t0 ) ∩ N (t1 ) = ∅ then by a suitable choice of the component of the functions, one

98

5 The L-Index Theory

can get well defined continuous functions θj1 : N(t1 ) → R such that θj0 (t) = θj1 (t) for t ∈ N (t0 ) ∩ N (t1 ). Now one can start from t = 0 and extend step by step to define the continuous functions θj : [0, 1] → R by using the compactness of [0, 1]. Proposition 1.3 ([189]) For γ ∈ P1 (2n)∗L0 , with the above notations, there holds iL0 (γ ) =

 n   θj (1) − θj (0) j =1

π

,

(5.9)

where [a] = max{k ∈ Z| k ≤ a}. Proof We take a symplectic path β : [a, b] → Sp(2n)∗L0 . Suppose β(t) =   Aβ (t) Cβ (t) with detCβ (t) = 0. Consider the determinant of the unitary matrix Bβ (t) Dβ (t) √ √ detQβ (t) := det(Dβ (t) − −1Cβ (t))(Dβ (t) + −1Cβ (t))−1 √ √ = det(Dβ (t)Cβ−1 (t) − −1I )(Dβ (t)Cβ−1 (t) + −1I )−1 . It is easy to see that Dβ (t)Cβ−1 (t) is a symmetric matrix function. So the eigenvalue of the matrix Qβ (t) can be written as √ √ aj2 (t) − 1 2aj (t) −1 −1 − 2 ∈ S 1 , aj (t) ∈ R. ζβ,j (t) = = 2 √ a (t) + 1 a (t) + 1 aj (t) + −1 j j aj (t) −

√

It is clear that if we write ζβ,j (t) = e2 −1ϑβ,j (t) , there holds ϑβ,j (t) ∈ / π Z. By the  ϑ (b)−ϑ (a)  continuity, we have  β,j π β,j  < 1. We now just take β as in (5.5), and get a joint path β ∗ γ . Then we have 

   ϑγ ,j (1) − ϑγ ,j (0) ϑβ∗γ ,j (1) − ϑβ∗γ ,j (0) = . π π

But there holds   ϑβ∗γ ,j (1) − ϑβ∗γ ,j (0) ϑβ∗γ ,j (1) − ϑβ∗γ ,j (0) 1 = + . π π 2 As in the definition of γ˜ , we have join a rotation path ξ(t) = I cos t + J sin t, t : π 2 → 0. By direct computation, we have ϑξ,j (0) − ϑξ,j ( π2 ) 1 =− . π 2 Thus as in (5.5), we have

5.1 Definition of L-Index

99

    ϑγ ,j (1) − ϑγ ,j (0) ϑγ¯ ,j (1) − ϑγ¯ ,j (0) θj (1) − θj (0) = = . π π π Taking summation from j = 1 to n, we get (5.9).

 

Remark 1.4 For the non-degenerate case, if we choose ϑγ ,j (0) = ϑβ∗γ ,j (0) = 0 and ϑγ ,j (1) = ϑβ,j (0), then by the above computations, we have −

ϑβ,j (1) − ϑβ,j (0) 1 1 < < 2 π 2

and thus β (1) − β (0)  ϑβ,j (1) − ϑβ,j (0) n n < = < . 2 π π 2 n

−

j =1

Note that iL0 (γ ) = −

n γ (1) − γ (0) β (1) − β (0) + + , 2 π π

we get −n < iL0 (γ ) −

γ (1) − γ (0) < 0. π

By the arguments in Theorem 1.21 below, for any γ ∈ P1 (2n), −n ≤ iL0 (γ ) −

γ (1) − γ (0) < 0. π

The left side equality holds only for γ (1) = I2n . For a L0 -degenerate symplectic path γ ∈ Pτ (2n)0L0 , its L0 -index is defined by the infimum of the indices of the nearby nondegenerate symplectic paths. Definition 1.5 ([189]) For a symplectic path γ ∈ Pτ (2n)0L0 , we define the L0 -index of γ by iL0 (γ ) = inf{iL0 (γ˜ )| γ˜ ∈ Pτ (2n)∗L0 , and γ˜ is sufficiently close to γ }.

(5.10)

In the general situation, let L be any linear Lagrangian subspace of R2n . Now we are ready to define the index for any symplectic path γ ∈ Pτ (2n) with L-boundary condition. We know from Lemma I.2.14 that (n) = U (n)/O(n), this means that for any linear subspace L ∈ (n), there is an orthogonal symplectic matrix P =   √ A −B with A ± −1B ∈ U (n), such that P L0 = L. P is uniquely determined B A

100

5 The L-Index Theory

by L up to an orthogonal matrix C ∈ O(n). It means that for any other choice  P satisfying  above conditions, there exists a matrix C ∈ O(n) such that P = C0 P . We define the conjugated symplectic path γc ∈ Pτ (2n) of γ by γc (t) = 0 C P −1 γ (t)P . Definition 1.6 ([189]) We define the L-nullity of any symplectic path γ ∈ Pτ (2n) by νL (γ ) ≡ dim kerL (γ (1)) := dim ker Vc (1) = n − rankVc (1).

(5.11)

The n × n matrix function Vc (t) is defined in (5.1) with the symplectic path γ replaced by γc , i.e.,  γc (t) =

 Sc (t) Vc (t) . Tc (t) Uc (t)

(5.12)

The L-nullity νL (γ ) is well defined. In fact, for any C ∈ O(n), we have 

C −1 0 0 C −1



Sc (t) Vc (t) Tc (t) Uc (t)



C0 0 C



 =

C −1 Sc (t)C C −1 Vc (t)C C −1 Tc (t)C C −1 Uc (t)C

 ,

(5.13)

and dim ker Vc (1) = dim ker C −1 Vc (1)C. For any L ∈ (n), we define the following two subspaces of Sp(2n) by Sp(2n)∗L = {M ∈ Sp(2n)| detVc = 0}, Sp(2n)0L = {M ∈ Sp(2n)| detVc = 0}, where Vc is defined by Mc = P

−1

 MP =

Sc Vc T c Uc

 .

The space Sp(2n)∗L has two path connected components as well. We denote the two components by Sp(2n)± L = {M ∈ Sp(2n)| ± detVc > 0}, − 0 then we have Sp(2n)∗L = Sp(2n)+ L ∪ Sp(2n)L . We call Sp(2n)L the L-degenerate ∗ subspace of Sp(2n) and Sp(2n)L the L-non-degenerate subspace of Sp(2n). We denote the corresponding symplectic path spaces by

Pτ (2n)∗L = {γ ∈ Pτ (2n)| γ (1) ∈ Sp(2n)∗L }

5.1 Definition of L-Index

101

and Pτ (2n)0L = {γ ∈ Pτ (2n)| γ (1) ∈ Sp(2n)0L }. If γ ∈ Pτ (2n)0L , we call it the L-degenerate symplectic path, otherwise, if γ ∈ Pτ (2n)∗L , we call it the L-nondegenerate symplectic path. Definition 1.7 ([189]) For a symplectic path γ ∈ Pτ (2n), we define the L-index of γ by iL (γ ) = iL0 (γc ).

(5.14)

By the arguments (5.12) and (5.13), we see that the L-index iL (γ ) is well defined. In fact, for any other choice P  , the conjugated symplectic path associated it is γc (t)



Sc (t) Vc (t) Tc (t) Uc (t)





C −1 0 = = 0 C −1   −1 C Sc (t)C C −1 Vc (t)C . = C −1 Tc (t)C C −1 Uc (t)C





C0 γc (t) 0 C



The associated unitary matrix defined in (2.3) becomes √ −1Vc (t)][Uc (t) + −1Vc (t)]−1 √ √ = C −1 [Uc (t) − −1Vc (t)][Uc (t) + −1Vc (t)]−1 C.

Qc (t) = [Uc (t) −

√

Thus for any symplectic path γ ∈ Pτ (2n), the L-index pair (iL (γ ), νL (γ )) ∈ Z × {0, 1, · · · , n} is well defined.

5.1.1

The Properties of the L-Indices

We only consider the properties of the L-indices with L = L0 in this section. All of the properties are valid for the general cases. Definition 1.8 ([189]) For two symplectic paths γ0 , γ1 ∈ Pτ (2n), we say that they are L0 -homotopic and denoted by γ0 ∼L0 γ1 , if there is a map δ : [0, 1] → Pτ (2n) such that δ(j ) = γj for j = 0, 1, and νL0 (δ(s)) is constant for s ∈ [0, 1]. Theorem 1.9 ([189]) If γ0 , γ1 ∈ Pτ (2n)∗L0 , then iL0 (γ0 ) = iL0 (γ1 ) if and only if γ0 ∼L0 γ1 .

102

5 The L-Index Theory

Proof Connecting γj (1) to M+ or M− by a path βj : [0, 1] → Sp(2n)∗L0 for j = 0, 1, we have γj ∼L0 βj ∗ γj along βj and by definition iL0 (γj ) = iL0 (βj ∗ γj ) where the joint path β ∗ γ is defined in (5.5). Thus γ0 ∼L0 γ1 if and only if β0 ∗   γ0 ∼L0 β1 ∗ γ1 . By Definition 1.2, it is equivalent to iL0 (γ0 ) = iL0 (γ1 ). For any C ∈ O(n), from the definition, we have iL0 (γ ) = iL0 (γ C ), νL0 (γ ) = νL0 (γ C ), ∀ γ ∈ Pτ (2n),

(5.15)

where the symplectic path γ C is defined by  γ (t) = C

C 0 0 C





C −1 0 γ (t) 0 C −1

 .

Lemma 1.10 ([189]) If γ ∈ Pτ (2n)0L0 , and νL0 (γ ) = k, there is matrix C ∈ O(n) and θ0 > 0 sufficient small satisfying νL0 (γs1 ) = k − 1 for s ∈ [−1, 1] and s = 0, where     −1 C0 C 0 1 = Rh (stθ0 )γ C (t), γs (t) := Rh (stθ0 ) (5.16) γ (t) 0 C −1 0 C and Rh (θ ) is a rotation matrix defined in (5.18) below. Furthermore, if k = 1, then γs (1) and γ−s (1) belong to different connected component of Sp(2n)∗L0 , i.e., there holds 1 detVs1 (1)detV−s (1) < 0.

(5.17)



 S(t) V (t) Proof Suppose γ (t) = and dim ker V (1) = k, 0 < k ≤ n. There T (t) U (t) exists an orthogonal matrix C ∈ O(n) such that CV (1)C −1 =



 A0 , A ∈ L(Rn−k , Rn−k ), detA = 0, B0

and with the same block form decomposition, we have CU (1)C −1 =



U11 U12 U21 U22

 .

5.1 Definition of L-Index

103

T A+U T B = 0. Since γ ∈ Pτ (2n), we have U T (1)V (1) = V T (1)U (1), and then U12 22 By the rank k condition, we have U22 = 0 (otherwise, by detA = 0, we have U12 = 0). Suppose U22 = (ui,j )k×k and ui,j = 0 for some 1 ≤ i, j ≤ k. Now we construct the symplectic matrices (rotation matrix) Rm (θ ) = (αj,l ) by

⎧ αm,m = αn+m,n+m = cos θ, ⎪ ⎪ ⎨ αn+m,m = −αm,n+m = sin θ, ⎪ α = 1, for k = m, n + m, ⎪ ⎩ j,j αj,l = 0, for any other cases.

(5.18)

Thus we have  Rn−k+i (θ )

C 0 0 C



S(1) V (1) T (1) U (1)



C −1 0 0 C −1



 =

Sθ Vθ T θ Uθ



with  Vθ =

 0 Vn−k , ∗ −U22 (θ )

(5.19)

and U22 (θ ) is a matrix function induced from U22 by multiplying its i-th row by sin θ and by multiplying other rows by 0. It is clear that dim ker Vθ = dim ker V (1) − 1 for |θ | = 0 small. Thus (5.16) follows by choosing h = n − k + i. If k = 1, then U22 (θ ) = c sin θ for some constant c = 0. Equation (5.17) follows from (5.19).   From Lemma 1.10, by induction, we have j +1

j

νL0 (γs ) − νL0 (γs

) = 1, j = 0, 1, · · · , k − 1, and νL0 (γsk ) = 0, s = 0

where j +1 γs (t)



Cj 0 := Rhj (stθ0 ) 0 Cj

"

 j γs (t)

Cj−1 0 0 Cj−1

# , γs0 = γ .

(5.20)

We denote by γs (t) = γsk (t), k = νL0 (γ ). By Theorem 1.9, we have the following iL0 (γs ) = constant for s ∈ (0, 1] or s ∈ [−1, 0).

(5.21)

104

5 The L-Index Theory

Theorem 1.11 ([189]) Suppose γ ∈ Pτ (2n)0L0 and γs ∈ Pτ (2n)∗L0 defined in Lemma 1.10. We have iL0 (γs ) − iL0 (γ−s ) = νL0 (γ ), ∀ s ∈ (0, 1].

(5.22)

Proof We follow the ideas of the proof of Theorem 5.4.1 of [223]. For simplicity, we omit the constant matrix from (5.20), so we write γs in following form γs (t) = Rhk (stθ0 )Rhk−1 (stθ0 ) · · · Rh1 (stθ0 )γ C (t). Fix 0 < s ≤ 1. We define symplectic paths by ξj (t) = Rhk (stθ0 ) · · · Rhj +1 (stθ0 )Rhj (−stθ0 ) · · · Rh1 (−stθ0 )γ C (t), j = 0, · · · , k. (5.23) We note that ξj ∈ Pτ (2n)∗L0 and ξ0 = γs , ξk = γ−s . We now only need to prove iL0 (ξj ) − iL0 (ξj +1 ) = 1.

(5.24)

We take a path ηj (t) = Rhk (sθ0 ) · · · Rhj +1 (sθ0 )Rhj +1 (−2stθ0 )Rhj (−sθ0 ) · · · Rh1 (−sθ0 )γ C (1). We have ηj (0) = ξj (1), ηj (1) = ξj +1 (1), ηj (1/2) ∈ Sp(2n)0L0 and furthermore dim kerL0 ηj (1/2) = 1, ηj (t) ∈ Sp(2n)∗L0 for all t ∈ [0, 1] \ {1/2}, where the dimension of the L0 kernel is defined in (5.2). So by definition, we have ξj +1 ∼L0 ηj ∗ ξj , the joint path α ∗ β is defined in (5.5). So by Theorem 1.9, we need to prove iL0 (ξj ) − iL0 (ηj ∗ ξj ) = 1.

(5.25)

We define a path ξ¯ ∈ Pτ (2n)0L0 by ξ¯ (t) =



ξj (2t), t ∈ [0, 1/2], ηj ( 2t−1 2 ), t ∈ [1/2, 1].

1 The path ξj can be taken as γs1 in Lemma 1.10 for ξ¯ , and ηj ∗ ξj can be taken as γ−s ¯ for ξ . So the two paths ξj and ηj ∗ ξj is located in different connected component. Note that s, θ0 > 0, so we have

iL0 (ξj ) − iL0 (ηj ∗ ξj ) > 0.

(5.26)

But by definition and θ0 > 0 is small, it is clear that iL0 (ξj ) − iL0 (ηj ∗ ξj ) ≤ 1.

(5.27)

5.1 Definition of L-Index

105

 

Equation (5.25) follows from (5.26) and (5.27). Remark In fact there holds iL0 (γ−s ) = inf{iL0 (γ˜ )| γ˜ ∈ Pτ (2n)∗L0 , and it is sufficiently close to γ },

(5.28)

iL0 (γs ) = sup{iL0 (γ˜ )| γ˜ ∈ Pτ (2n)∗L0 , and it is sufficiently close to γ }.

(5.29)

We will prove these statements in (5.143). Theorem 1.12 ([189]) For any symplectic path γ ∈ P1 (2n), by using the notations in Proposition 1.3, there holds iL0 (γ ) =

n 

 E

j =1

 θj (1) − θj (0) , π

(5.30)

where E(a) = max{k ∈ Z| k < a}. Proof From the arguments of the proof of Proposition 1.3 and the result of Theorem 1.11, we see that there exist exact k = νL0 (γ ) eigenvalues θj (t) with θj (1) = 2pπ for p ∈ Z, and iL0 (γ−s ) =

  n  n    θj (1) − θj (0) θj (1) − θj (0) −k, iL0 (γs ) = , 0 < s ≤ 1. π π j =1

j =1

(5.31) It is easy to see  n   θj (1) − θj (0) j =1

π

−k =

n  j =1

 E

 θj (1) − θj (0) . π

Equation (5.30) follows from (5.10), (5.28), (5.31) and (5.32).

(5.32)  

Theorem 1.13 (Symplectic addivity, [189]) For two path γj ∈ P(nj ), j = 1, 2, there holds iL0 (γ1  γ2 ) = iL0 (γ1 ) + iL0 (γ2 ). Proof (5.33) follows from definition and (5.28), (5.29). We omit the details.

(5.33)  

Theorem 1.14 ([189]) For two path γj ∈ Pτ (2n), j = 0, 1 with γ0 (1) = γ1 (1), it holds that iL0 (γ0 ) = iL0 (γ1 ) if and only if γ0 ∼L0 γ1 with fixed endpoints. Proof The proof follows from (5.28) and (5.29) and the definition of the index theory. We note that γ0 ∼L0 γ1 with fixed endpoints if and only if γs,0 ∼L0 γs,1 with fixed endpoints.  

106

5 The L-Index Theory

In the Sect. 5.5 below (see the proof of (5.145)), we shall prove the following result. Theorem 1.15 (Homotopy invariant, [189]) For two path γj ∈ Pτ (2n), j = 0, 1, if γ0 ∼L γ1 , there hold iL (γ0 ) = iL (γ1 ), νL (γ0 ) = νL (γ1 ).

(5.34)



 C(t) E(t) Suppose γ (t) = ∈ P1 (2n) is a symmetry positive definite E T (t) D(t) symplectic path. From C(t) > 0 and D(t) > 0 (positive definite matrices), we have √ √ det(D(t) − −1E(t))(D(t) + −1E(t))−1 √ √ (5.35) = det(I − −1E(t)D −1 (t))(I + −1E(t)D −1 (t))−1 . Since D T (t)E(t) = E T (t)D(t), E(t)D −1 (t) is a symmetric n × n matrix path. Suppose λ1 (t) ≤√ · · · ≤ λn (t) are the eigenvalues of E(t)D −1 (t). We set 1 + √ λj −1 = rj (t)e −1θj (t)/2 for j = 1, · · · , n. So we have n √  √ √ det(I − −1E(t)D −1 (t))(I + −1E(t)D −1 (t))−1 = e2 −1θ(t) , θ (t) = θj (t). j =1

It is easy to see that −

π π < θj (t) − θj (0) < , t ∈ [0, 1]. 2 2

(5.36)

So we have  nπ nπ < . (θj (t) − θj (0)) < 2 2 n

−

j =1

By an easy computation(see (5.30)), we have − n ≤ iL0 (γ ) ≤ 0.

(5.37)

Thus we have the following result. Proposition 1.16 For a symmetrical positive definite path γ ∈ Pτ (2n), we have iL (γ ) ∈ [−n, 0], ∀ L ∈ (n).

(5.38)

5.1 Definition of L-Index

107

Proof γ is a symmetrical positive definite path implies that γc defined in Definition 1.7 is also a symmetrical positive definite path. So (5.38) follows from (5.37) and Definition 1.7.  

5.1.2

The Relations of iL (γ ) and i1 (γ )

For a symplectic path γ ∈ Pτ (2n), there exist an orthogonal symplectic path O ∈ Pτ (2n) and a symmetric positive definite path P ∈ Pτ (2n) such that γ = OP . It is called the polar decomposition of γ . Suppose  γ =

S V T U



 , O=

A B −B A



 , P =

C E ET D

 .

Then we have V = AE + BD, U = −BE + AD. So by direct computations we have √ √ det(U − −1V )(U + −1V )−1 √ √ √ √ = det(A − −1B)(A + −1B)−1 det(D − −1E)(D + −1E)−1 . (5.39) Theorem 1.17 ([189]) For γ ∈ Pτ (2n) ∩ O(2n), there holds 0 ≤ i1 (γ ) − iL0 (γ ) ≤ n.

(5.40)

For any γ ∈ Pτ (2n) there holds − n ≤ i1 (γ ) − iL0 (γ ) ≤ n.

(5.41)

Proof When γ ∈ Pτ (2n) ∩ O(2n), the functions θj in (5.30) and (2.5) are the same. By the inequality 2E(x) ≤ E(2x) ≤ 2E(x) + 1 we have that i1 (γ ) − n ≤ iL0 (γ ) ≤ i1 (γ ). This is exactly (5.40) and (5.41) follows from (5.39) and the estimate (5.36).

 

As in the Morse theory of closed geodesics on Riemannian or Finsler Manifolds (cf. [161]), we have

108

5 The L-Index Theory

Definition 1.18 ([189]) For any γ ∈ Pτ (2n) and L ∈ (n), we call i1 (γ ) − iL (γ ) the concavity of γ w.r.t. the Lagrangian subspace L and denoted it by concavL (γ ) = i1 (γ ) − iL (γ ).

(5.42)

If c is a closed geodesic on a Riemannian manifold M, we know from [190] that ind(c) = i1 (c). In this case, the energy functional is defined in the loop space  = {x ∈ W 1,2 ([0, 1], M)| x(0) = p1 , x(1) = p2 } for p1 = p2 , by linearized the geodesic equation, and translate it to a Hamiltonian system, then the boundary condition is that z(0), z(1) ∈ L0 . So iL0 (c) = ind (c). In this case concavL0 (c) is just the concavity defined by Morse as in [242] (see also [161]). Theorem 1.19 ([189]) The concavity of γ ∈ Pτ (2n) ∩ O(2n) w.r.t. L satisfies 0 ≤ concavL (γ ) ≤ n.

(5.43)

The concavity of γ ∈ Pτ (2n) w.r.t. L satisfies − n ≤ concavL (γ ) ≤ n.

(5.44)

Proof From Definition 1.7, we have iL (γ ) = iL0 (γc ). By the homotopic invariant of Maslov-type index theory, there holds i1 (γ ) = i1 (γc ).   We know that any Lagrangian path β(t) starting from L0 can be written as β(t) = γ (t)L0 for γ ∈ Pτ (2n) ∩ O(2n). iL0 (γ ) in some sense is the algebraic intersection number (with signs determined by orientation) of the Lagrangian path β with the constant path L0 (cf. [77]). So the special case γ ∈ Pτ (2n) ∩ O(2n) is important in our index theory. For a solution z = z(t) of a nonlinear Hamiltonian system with periodic boundary condition: 

z˙ (t) = J H  (t, z(t)), z(t) ∈ R2n z(1) = z(0),

(5.45)

by choosing a Lagrangian subspace L such that z(0) = z(1) ∈ L, we can understand it as a special solution of the following problem with Lagrangian boundary condition

5.1 Definition of L-Index

109



z˙ (t) = J H  (t, z(t)), z(t) ∈ R2n z(1), z(0) ∈ L.

(5.46)

Linearizing (5.45) at z, we get a linear Hamiltonian system with periodic boundary condition  y˙ = J B(t)y (5.47) y(0) = y(1), where B(t) = H  (t, z(t)). Suppose the fundamental solution of the linear system y = J B(t)y

(5.48)

is γ (t) ∈ Pτ (2n). The Maslov-type index i1 (z) = i1 (γ ) of z as a periodic solution is just an index theory of the problem (5.47). The L-index iL (z) = iL (γ ) of z as a special solution of (5.46) with Lagrangian boundary condition is just an index theory of the following problem 

y˙ = J B(t)y y(0), y(1) ∈ L,

(5.49)

which is the linearized form of the problem (5.46). For the periodic solution z of the Hamiltonian system (5.45), γ ∈ Pτ (2n) is the fundamental solution of (5.48). The concavity of z w.r.t. the Lagrangian subspace L with z(0) ∈ L is defined by concavL (z) = concavL (γ ).

(5.50)

Corollary 1.20 ([189]) (1) For any γ ∈ Pτ (2n) ∩ O(2n) and L ∈ (n), there holds |iL0 (γ ) − iL (γ )| ≤ n.

(5.51)

(2) For any γ ∈ Pτ (2n) and L ∈ (n), there holds |iL0 (γ ) − iL (γ )| ≤ 2n.

(5.52)

(3) For any γ ∈ Pτ (2n) and L ∈ (n), |iL0 (γ ) − iL (γ )| depends only on the end matrix γ (τ ) and L. Proof From (5.40), (5.41) and (5.43), (5.44) one gets (5.51), (5.52) directly. The third part is a direct consequence of the following result.  

110

5 The L-Index Theory

Theorem 1.21 ([189]) For any L ∈ (n), and γ ∈ Pτ (2n), concavL (γ ) depends only on the end matrix γ (τ ) and the Lagrangian subspace L. Thus for any symplectic matrix M ∈ Sp(2n), we can define the L-concavity of M by concavL (M) := concavL (γ ), ∀ γ ∈ Pτ (2n), γ (τ ) = M. Proof We fix τ = 1. Firstly, assume L = L0 and γ ∈ Pτ (2n)∗ ∩ Pτ (2n)∗L0 . By definition there exist β : [0, 1] → Sp∗ (2n) and βL0 : [0, 1] → Sp∗ (2n)L0 such that i1 (γ ) =

θγ (1) − θγ (0) θβ (1) − θβ (0) + , π π

(5.53)

where θγ and θβ is defined as θ˜ just before Definition II.1.1 with respect to γ and β, respectively. iL0 (γ ) =

γ (1) − γ (0) βL0 (1) − βL0 (0) n + − , π π 2

(5.54)

¯ in (5.7) with respect to γ and βL0 , respectively. where γ and βL0 is defined as  We note that (by (5.39)) θγ (1) − θγ (0) γP (1) − γP (0) γ (1) − γ (0) = + , π π π where γP is the positive definite actor of γ in its polar decomposition. By (5.36), we γ (1)−γ (0) know that P π P depends only on its end γP (1). Thus we have the concavity βL (1)−βL (0) θβ (1)−θβ (0) 0 0 − π π γP (1)−γP (0) n − + 2. π

i1 (γ ) − iL0 (γ ) =

(5.55)

Since the symplectic path β and βL0 is determined only by γ (1) and L0 , we see that the result of Theorem 1.21 is true in this case. For the degenerate cases, by the homotopic invariant, we can choose a very short rotation path η and ηL0 depending only on γ (1) such that η ∗ γ ∈ Pτ (2n)∗ , ηL0 ∗ γ ∈ Pτ (2n)∗L0 and i1 (γ ) = i1 (η ∗ γ ), iL0 (γ ) = iL0 (ηL0 ∗ γ ). Then by the same computations as in (5.53) and (5.54), we still get i1 (γ ) − iL0 (γ ) = something determined only by γ (1) and L0 . For the general case, for any P ∈ Sp(2n) ∩ O(2n) such that L = P L0 , there holds i1 (γ ) = i1 (P −1 γ P ), iL (γ ) = iL0 (P −1 γ P ).  

5.1 Definition of L-Index

111

In the Chap. 7 below, we will revisit the topic of concavity by using Hörmander index theory.

5.1.3 L-Index for General Symplectic Paths For a general continuous symplectic path ρ : [a, b] → Sp(2n) and the Lagrangian subspace L0 , we define its Maslov type L0 -index as follow. Definition 1.22 We define iˆL0 (ρ) = iL0 (γb ) − iL0 (γa ),

(5.56)

where γa ∈ P(2n) is a symplectic path ended at ρ(a) and γb ∈ P(2n) is the composite of symplectic path γa and ρ, i.e., γb = ρ ∗ γa . In general, for any Lagrangian subspace L, we define iˆL (ρ) = iL (γb ) − iL (γa ).

(5.57)

We remind that for the constant path γ = I there holds iL0 (I ) = −n and iL1 (I ) = −n, so iˆL0 (γ ) = iL0 (γ ) + n and iˆL1 (γ ) = iL1 (γ ) + n for γ ∈ P(2n). Lemma 1.23 The index iˆL (ρ) is well defined, i.e., it is independent of the choice of γa . Proof We only prove the case L = L0 . For two symplectic paths γa , γa ∈ P(2n), we denote γb , γb ∈ P(2n) correspondingly as in Definition 1.22. We should prove iL0 (γb ) − iL0 (γb ) = iL0 (γa ) − iL0 (γa ). We suppose    Sb Vb Sa Va , γa = , γb = T b Ub T a Ua 

and γb =

    Sb Vb Sa Va  , γ . = a Tb Ub Ta Ua



For these four matrices, as in Definition 1.2, we suppose det(U¯ a − and

√ √ √ √ ¯ ¯ −1V¯a ) = ρa e −1a , det(U¯ b − −1V¯b ) = ρb e −1b

(5.58)

112

5 The L-Index Theory

det(U¯ a −

√

−1V¯a ) = ρa e

√

¯ a −1

, det(U¯ b −

√

−1V¯b ) = ρb e

√

¯ −1 b

.

¯ have the path additivity, we prove the equality (5.58) By definition, the functions ’s in the non-degenerate cases. For the degenerate cases, since the two paths have the same end points, we can prove the equality (5.58) by taking non-degenerate paths close to the degenerate ones and taking the infimum in every terms.   Theorem 1.24 The index iˆL0 has the following properties (1) (Affine Scale Invariance). For k > 0, l ≥ 0, we have the affine map ϕ : [a, b] → [ka + l, kb + l] defined by ϕ(t) = kt + l. For a given continuous path ρ : [ka + l, kb + l] → Sp(2n), there holds iˆL0 (ρ) = iˆL0 (ρ ◦ ϕ).

(5.59)

(2) (Homotopy Invariance rel. End Points). If δ : [0, 1] × [a, b] → Sp(2n) is a continuous map with δ(0, t) = ρ1 (t), δ(1, t) = ρ2 (t), δ(s, a) = ρ1 (a) = ρ2 (a) and δ(s, b) = ρ1 (b) = ρ2 (b) for s ∈ [0, 1], then iˆL0 (ρ1 ) = iˆL0 (ρ2 ).

(5.60)

(3) (Path Additivity). If a < b < c, and ρ[a,c] : [a, c] → Sp(2n) is concatenate path of ρ[a,b] and ρ[b,c] , then there holds iˆL0 (ρ[a,c] ) = iˆL0 (ρ[a,b] ) + iˆL0 (ρ[b,c] ).

(5.61)

(4) (Symplectic Additivity). Let Lk0 , ρk : [a, b] → Sp(2nk ), k = 1, 2, L0 = L10 L20 , ρ = ρ1  ρ2 . Then we have iˆL0 (ρ) = iˆL1 (ρ1 ) + iˆL2 (ρ2 ). 0

0

(5.62)

Here the symplectic direct sum of two Lagrangian subspaces L and L is defined by L  L = {(x  , x  , y  , y  )T | (x  , y  )T ∈ L , (x  , y  )T ∈ L }. (5) (Symplectic Invarince) Let matrix P ∈ Sp(2n) be a symplectic matrix. We have iˆP L0 (PρP −1 ) = iˆL0 (ρ).

(5.63)

(6) (Normalization). For L0 = R and ρ : [−ε, ε] → Sp(2) with ε > 0 small and ρ(t) = eJ t , we have (i) iˆL0 (ρ) = 1; (ii) iˆL0 (ρ[−ε,0] ) = 0; (iii) iˆL0 (ρ[0,ε] ) = 1.

(5.64)

5.1 Definition of L-Index

113

Proof We prove the statements (5) and (6) only. The remainders are direct consequence of the definition. The proof of the statement (6) is very easy. By definition, it is easy to see iL0 (γ−ε ) = 0 and iL0 (γε ) = 1. This proves statement (i). (ii) and (iii) are similar. For the proof of the statement (5), we first prove the following result. Lemma 1.25 If P ∈ Sp(2n) is a symplectic matrix satisfying P L0 = L0 , then we have iL0 (γ ) = iL0 (P γ P −1 )

(5.65)

for any γ ∈ P1 (2n). It is easy to see that we can write the matrix P in the form 

A 0 C (AT )−1

P =



with AT C = C T A and so P

−1



A−1 0 = −C T AT

 .

So the set of all P ∈ Sp(2n) satisfying P L0 = L0 is a subgroup of Sp(2n), we denote it by F(L0 ). Suppose the end point γ (1) of γ : [0, 1] → Sp(2n) is 

 S V . T U

Thus we have P γ (1)P

−1



ASA−1 AV AT = a21 a22



where a21 = CSA−1 +(A−1 )T T A−1 −CV C T −(A−1 )T U C T and a22 = CV AT + T (A−1 )T U AT . It is  clear thatdim ker(V  ) = dimker(AV A ). We can connect the C± 0 ±1 0 , C± = and P in F(L0 ) by a continuous two matrices I± = 0 C± 0 In−1 path P (s) with P (0) = I± and P (1) = P (in fact, we can get this path in two steps. We first connect C in the presentation of matrix P to 0 by sC, correspondingly we get  a path A(s)  with A(s) ≡ A. Since A is nondegenerate, then we can connect A to ±1 0 ). So δ(s, t) = P (s)γ (t)P (s)−1 is a homotopy between the two paths γ 0 In−1 and P γ P −1 . The statement (5.65) follows from Theorem 1.15.  

114

5 The L-Index Theory

Proof of the statement (5) Suppose L = P L0 . For L, there is M ∈ Osp(2n) such that ML0 = L. By definition, there holds iL (γ ) = iL0 (M −1 γ M) for γ ∈ P(2n). Since P −1 ML0 = L0 , by Lemma 1.25 there holds iL (γ ) = iL0 (M −1 γ M) = iL0 (P −1 MM −1 γ MM −1 P ) = iL0 (P −1 γ P ). Now (5.63) follows from the definition.

(5.66)  

For the index iˆL , we have a similar result as Theorem 1.24. The only difference is that the statement (6) in Theorem 1.24 should replaced by (6) For dimension 1 subspace L of R2 , and ρ : [−ε, ε] → Sp(2) with ε > 0 small and ρ(t) = P eJ t P −1 with P ∈ Osp(2n) satisfying P L0 = L, we have (i) iˆL (ρ) = 1; (ii) iˆL (ρ[−ε,0] ) = 0; (iii) iˆL (ρ[0,ε] ) = 1.

(5.67)

We will see that the six axioms in Theorem 1.24 character the index iˆL0 uniquely. In the next chapter, we will define an index μL0 (γ ) as the Maslov type index of a pair of Lagrangian paths and prove that this index also satisfies the six axioms of Theorem 1.24. So we have iˆL0 (γ ) = iL0 (γ ) + n = μL0 (γ ) for every γ ∈ P(2n). Remark 1.26 A path ρ : [0, 1] → Sp(2n) is called L-nondegenerate if νL (ρ(0)) = νL (ρ(1)) = 0. For any path ρ, there exists ε > 0 such that e−θJ ρ with 0 < |θ | < ε is L-nondegenerate and there holds iˆL (ρ) = iˆL (e−θJ ρ), 0 < θ < ε. For any L0 -nondegenerate path ρ, we can connect by a path α from ρ(0) (and β from ρ(1) respectively) to M+ or M− in the connected component of Sp(2n)∗L0 . Then we get a composite path ρ˜ = β ∗ ρ ∗ α −1 . We have ˜ ˜ (1) − (0) , ˜ = iˆL0 (ρ) = iˆL0 (ρ) π √

˜ ˜ ˜ ˜ is defined by det Q(t) where  = e2 −1(t) with Q(t) defined from ρ˜ in (5.6) and (5.7). We omit the proofs of these statements and leave them to the readers.

5.1 Definition of L-Index

115

We are ready to define an index for a pair of a continuous Lagrangian path f : [0, 1] → (n) × (n) with f (t) = (L1 (t), L2 (t)), 0 ≤ t ≤ 1. According to (1.20), we know that there are U1 (t), U2 (t) ∈ Osp(2n) such that Lj (t) = Uj (t)L0 , then we have the following definition. Definition 1.27 iˆ0 (f ) = iˆL0 (γ12 ),

(5.68)

where γ12 (t) = U1 (t)−1 U2 (t), 0 ≤ t ≤ 1. For a general symplectic space V = (R2n , ω) and a path of Lagrangian pair f : [a, b] → Lag(V ) × Lag(V ), by choosing a linear symplectic map T : (R2n , ω) → (R2n , ω0 ), we can define ˆ ) = iˆ0 (Tf T −1 ). i(f

(5.69)

ˆ ) has the Theorem 1.28 The definition (5.68) is well defined. Furthermore i(f following properties (1) For k > 0, l ≥ 0, we have the affine map ϕ : [a, b] → [ka + l, kb + l] defined by ϕ(t) = kt + l. For a given continuous path f : [ka + l, kb + l] → Lag(V ) × Lag(V ), there holds ˆ ) = i(f ˆ ◦ ϕ). i(f

(5.70)

(2) If δ : [0, 1] × [a, b] → Lag(V ) × Lag(V ) is a continuous map with δ(0, t) = f1 (t), δ(1, t) = f2 (t), δ(s, a) = f1 (a) = f2 (a) and δ(s, b) = f1 (b) = f2 (b) for s ∈ [0, 1], then ˆ 2 ). ˆ 1 ) = i(f i(f

(5.71)

(3) If a < b < c, and f[a,c] : [a, c] → Lag(V ) × Lag(V ) is concatenate path of f[a,b] and f[b,c] , then there holds ˆ [a,b] ) + i(f ˆ [b,c] ). ˆ [a,c] ) = i(f i(f

(5.72)

(4) Let fk : [a, b] → Lag(Vk ) × Lag(Vk ), k = 1, 2, V = V1 ⊕ V2 f = f1 ⊕ f2 . Then we have ˆ 2 ). ˆ ) = i(f ˆ 1 ) + i(f i(f

(5.73)

(5) Let P (t) ∈ Sp(2n) be a symplectic path and f (t) = (L1 (t), L2 (t)) ∈ (n) × (n). We define (P∗ f )(t) = (P (t)L1 (t), P (t)L2 (t)). Then we have iˆ0 (P∗ f ) = iˆ0 (f ).

(5.74)

116

5 The L-Index Theory

(6) Let V0 = (R2 , ω0 ). Define f : [−ε, ε] → (1) × (1) with ε > 0 small as a pair Lagrangian path: f (t) = (R, R(cos t, sint)), t ∈ [−ε, ε]. Then (i) iˆ0 (f ) = 1, (ii) iˆ0 (f[−ε,0] ) = 0, (iii) iˆ0 (f[0,ε] ) = 1.

(5.75)

Proof We only need to prove the case of V = (R2n , ω0 ). For that the definition (5.68) is well defined, if L1 (t) = U1 (t)L0 = U1 (t)L0 and L2 (t) = U2 (t)L0 = U2 (t)L0 , there exist two orthogonal matrices Cj (t) ∈ O(n), j = 1, 2   Cj (t) 0  such that Uj (t) = Uj (t)Oj (t) with Oj (t) = . Then we have 0 Cj (t)   S(t) V (t) U1 (t)−1 U2 (t) = O1 (t)−1 U1 (t)−1 U2 (t)O2 (t). Set U1 (t)−1 U2 (t) = . T (t) U (t) By direct computation, we arrive at U1 (t)−1 U2 (t) = O1 (t)−1 U1 (t)−1 U2 (t)O2 (t)   C1 (t)−1 S(t)C2 (t) C1 (t)−1 V (t)C2 (t) . = C1 (t)−1 T (t)C2 (t) C1 (t)−1 U (t)C2 (t) dim ker C1 (t)−1 V (t)C2 (t) = dim ker V (t). We define the homotopy δ(s, t) = O1 (st)−1 U1 (t)−1 U2 (t)O2 (st), s ∈ [0, 1] and note that iˆL0 (U1−1 U2 ) = iˆL0 (O1 (0)−1 U1−1 U2 O2 (0)) = iˆL0 (U1−1 U2 O2 (0)O1 (0)−1 ).   D± 0 in the subgroup We can connect the matrix P = O2 (0)O1 (0)−1 to I± = 0 C±     C 0 ±1 0 O(n) := {O; O = . , C is orthogonal matrix} with C± = 0 C 0 In−1 So we have iˆL0 (U1−1 U2 ) = iˆL0 (U1−1 U2 C± ) = iˆL0 (C± U1−1 U2 ).

(5.76)

5.2 The (L, L )-Index Theory

117

We of C− . We recall that M+ = −J =  only   need to consider  the situation 0 I 0 Dn , Dn = C− . It is clear that M− = C− M+ and M− = −Dn 0 −I 0 and M+ = C− M− . In Remark 1.26, we take ρ(t) = eθJ U1 (t)−1 U2 (t) and C− ρ(t) = C− eθJ U1 (t)−1 U2 (t), 0 < θ < ε. Then we get the corresponding path ρ˜ of ρ and ρˆ = C− ρ˜ of C− ρ. So we have ˜ ˜ (1) − (0) iˆL0 (ρ) = π and ˜ ˜ ˆ ˆ (1) − (0) (1) − (0) = = iˆL0 (ρ). iˆL0 (C− ρ) = π π Thus we have iˆL0 (U1−1 U2 ) = iL0 (U1−1 U2 C− ) = iL0 (C− U1−1 U2 ), Therefore there holds iˆL0 ((U1 )−1 U2 ) = iˆL0 (U1−1 U2 ). The Properties (1)–(6) can be proved as in that of Theorem 1.24. We omit the details.  

5.2 The (L, L )-Index Theory 

 S(t) V (t) For any γ ∈ P1 (2n), we write it in the form γ (t) = , where T (t) U (t) S(t), T (t), V (t), U (t) are n × n continuous matrix functions. We note that for the symplectic matrix P linear Lagrangian subspaces L0 , L there exists  an orthogonal  √ A −B such that P L0 = L and P has the form P = with A ± −1B ∈ U (n). B A   T T B A . Here P is uniquely determined by L up to an orthogonal So P −1 = −B T AT matrix C ∈ O(n). Also we know the fact that the symplectic group is a Lie group and its Lie algebra is sp(2n) = {M ∈ L(R2n , R2n )|J M + M T J = 0}. It is easy T to see that for M ∈ sp(2n) there holds J etM = e−tM J . It is well known that for a connected compact Lie group G with its Lie algebra G, there holds exp G = G.

118

5 The L-Index Theory

For any M ∈ sp(2n), it is well known that the one parameter curve exp(tM) = etM in Sp(2n) is a Lie subgroup of Sp(2n), and SO(2n) ∩ Sp(2n) ⊆ exp(sp(2n)). So for the above P ∈ Sp(2n), there is a matrix M ∈ sp(2n) such that M + M T = 0, P = eM and etM ∈ Sp(2n). ∈ P1 (2n) with γ (t) =

Definition 2.1   ([207]) For any symplectic path γ S(t) V (t) , we define the (L0 , L )-nullity to be T (t) U (t)

   νLL0 (γ ) ≡ νL0 (γ M ) = dim ker(AT V (1) + B T U (1)) = dim γ (1)L0 ∩ L , (5.77) where γ M (t) = e−tM γ (t) and νL0 (γ ) was defined in Definition 1.1 for any γ ∈ Pτ (2n). Remark 2.2 The (L0 , L )-nullity in (5.77) is well defined. Since for another choice P  , we have 

P =P



C 0 0 C



 =

A −B B A



C 0 0 C



 =

AC −BC BC AC

 ,

(5.78)

where C ∈ O(n), then dim ker((AC)T V (1) + (BC)T U (1)) = dim ker(AT V (1) + B T U (1)).

(5.79)

In general, iL0 (γ M ) depends on the choice of P . It is well known that O(n) has components, one contains C1 = In and another contains C2 =  two connected  −1 0 . 0 In−1   A −B Let P = , we can choose C = C1 (or C2 ) such that det(AC + B A √ √ −1BC) = e −1ϑ with ϑ ∈ (− π2 , π2 ]. In the following we always choose P √ √ satisfying the condition det(A + −1B) = e −1ϑ with ϑ ∈ (− π2 , π2 ]. √ For P and P√ satisfy the relation (5.78) and det(A + −1B) = det(AC + √ −1BC) = e −1ϑ , with ϑ ∈ (− π2 , π2 ]. Then there holds det(C) = 1, so we have C ∼ C1 , i.e., there exists η ∈ C([0, 1], O(n)) such that η(0) = C1 , η(1) = C. Then there exists M ∈ C([0, 1], Sp(2n)), satisfies M(s) + M T (s) = 0 and P (s) = P η(s) = eM(s) . So from Theorem 1.15, we have iL0 (γ M(0) ) = iL0 (γ M(1) ). We now choose M satisfying P = eM . We write the orthogonal symplectic path in the following form  γ0 (t) = etM =

A(t) B(t) −B(t) A(t)



5.2 The (L, L )-Index Theory

119

√ with w0 (t) := A(t) − −1B(t) ∈ U (n). If replacing M by M + 2kπ J, k ∈ Z, we √ t (M+2kπ J ) 2 −1kπ t w (t). We with wk (t) = e get another symplectic path γk (t) = e 0 choose continuous function k : [0, 1] → R, k (0) = 0 such that det wk (t) = e

√

−1π k (t)

.

So by choosing a suitable k ∈ Z, we can assume that k (1) ∈ (−n, n]. In the following we will always choose the matrix M such that 0 (1) ∈ (−n, n]. For example if L = L0 , we have M = 0, P = I2n and 0 (1) = 0. In the following definition, the matrices P and M are chosen in this way. Definition 2.3 ([207]) For the symplectic path γ ∈ Pτ (2n), we use the L0 -index to define the (L0 , L )-index of γ by 

iLL0 (γ ) ≡ iL0 (γ M ), γ M (t) = e−tM γ (t).

(5.80)



From the discussion above, we see that the index iLL0 (γ ) is well defined and if L = 

L0 , we have iLL0 (γ ) = iL0 (γ ). We now give another description of the index via the following definition.

Definition 2.4 For γ ∈ Pτ (2n), L ∈ (2n), the (L0 , L )-index of γ is defined by 

−1 μL γ ∗ ξ ) − iL0 (ξ ), L0 (γ ) = iL0 (P

(5.81)

where ξ ∈ Pτ (2n) satisfies ξ(τ ) = P −1 . 

We note that μL L0 (I ) = 0 for the constant path γ (t) = I . 



Theorem 2.5 The indices iLL0 (γ ) and μL L0 (γ ) have the following relation 





L L μL L0 (γ ) = iL0 (γ ) − iL0 (I ).

(5.82)

We will restate the result in (5.82) as a special case of Theorem VII.1.3 and prove it later. For any Lagrangian pair (L, L ) ∈ (n) × (n), we now define the index pair   L (iL (γ ), νLL (γ )) for any γ ∈ Pτ (2n). We can choose an orthogonal symplectic matrix O satisfying OL0 = L, then we define a new symplectic path γO (t) = O −1 γ (t)O. Denoting by L˜  = O −1 L

120

5 The L-Index Theory 



Definition 2.6 ([207]) The (L, L )-index pair (iLL (γ ), νLL (γ )) for any γ ∈ Pτ (2n) is defined by    ˜ νLL (γ ) = νLL0 (γO ) = dim γ (1)L ∩ L , ˜



iLL (γ ) = iLL0 (γO ), O −1 L = L˜  .

(5.83) (5.84)

Remark 2.7 It is easy to see that the index pair is well defined by the homotopic  invariance of the L0 -index theory, and we note that iLL (γ ) can be defined directly   as that of iLL0 (γ ) in Definition 2.3 with the L-index, i.e., iLL (γ ) = iL (γ M ). We also note that the (L, L )-index theory reserves some properties as the L-index theory in [189], we derive some of them such as the Galerkin approximation and the saddle point reduction in the next two sections. The homotopic invariance of this index theory is still true, but we don’t discuss it here. If L = L , we have 



(iLL (γ ), νLL (γ )) = (iL (γ ), νL (γ )). We denote by 







iLL (B) = iLL (γB ), νLL (B) = νLL (γB ),

(5.85)

where B ∈ L2 ([0, 1], Ls (R2n )), and γB is the fundamental solution of the linear system z˙ (t) = J B(t)z(t). If B1 , B2 ∈ L2 ([0, 1], Ls (R2n )), with B1 (t) < B2 (t), a.e.t ∈ [0, 1], we can show as in [189] that 





iLL (B2 ) − iLL (B1 ) = ILL (B1 , B2 ), 

where ILL (B1 , B2 ) = 



(5.86)



νLL ((1 − s)B1 + sB2 ) is well defined, that is to say

s∈[0,1)

νLL ((1 − s)B1 + sB2 ) > 0 only happens at finite points s ∈ [0, 1). As in Sect. 1.2, the general symplectic space (R2n , ω) is accompanied by the symplectic transform group which is the J -symplectic matrix group SpJ (2n). We denote by Pτ,J (2n) = {γ ∈ C 0 ([0, τ ], SpJ (2n))|γ (0) = I } the set of J symplectic paths. From Proposition I.2.2 of Chap. 1, we have T Pτ,J (2n)T −1 = Pτ (2n), where the symplectic transform T : (R2n , ω) → (R2n , ω0 ) defined in Sect. 1.1 corresponds to a matrix T satisfying T T J T = J −1 . The set of Lagrangian subspaces of the symplectic space (R2n , ω) is denoted by (2n, ω). Then we have T : (2n, ω)  (2n, ω0 ) = (2n).



5.3 Understanding the Index i P (γ ) in View of the Lagrangian Index iLL (γ )

121

Definition 2.8 For L, L ∈ (2n, ω), γ ∈ Pτ,J (2n), the (L, L )-index of γ is defined as 







iLL (γ ) = iTT LL (γ0 ), νLL (γ ) = νTT LL (γ0 ),

(5.87)

where γ0 (t) = T γ (t)T −1 .

5.3 Understanding the Index i P (γ ) in View of the  Lagrangian Index iLL (γ ) From the symplectic space (R2n , ω0 ), we can construct a dimensional 4n symplectic ˜ 0 ) = (R2n ⊕ R2n , ω0 ⊕ (−ω0 )) in the sense that space (R4n ,  ω0 ⊕ (−ω0 )(x1 ⊕ x2 , y1 ⊕ y2 ) = ω0 (x1 , y1 ) − ω0 (x2 , y2 ).

(5.88)

For a symplectic matrix P ∈ Sp(2n), the generated dimensional 2n subspace ˜ 0 ). Particularly, LP = {(x ⊕ P x)|x ∈ R2n } is a Lagrangian subspace of (R4n ,  for γ ∈ Pτ (2n), the graph of γ in R4n , L(t) = {(x ⊕ γ (t)x)|x ∈ R2n }, t ∈ [0, τ ] is ˜ 0 ). If we write x ⊕ y ∈ R2n ⊕ R2n in the form (x, y)T , a Lagrangian path of (R4n ,  then any two symplectic matrices M1 , M2 ∈ Sp(2n) determine a symplectic ˜ 0 ) as transformation of (R4n ,  

M1 0 0 M2

 .

(5.89)

˜ 0 ) as Particularly, from γ ∈ Pτ (2n), we get a symplectic path of (R4n ,   (t) =

 I2n 0 . 0 γ (t)

˜ 0 ), we Taking LI = {(x, x)T | x ∈ R2n }, which is a Lagrangian subspace of (R4n ,  4n ˜ see that (t)LI is a Lagrangian path of (R , 0 ). Definition 3.1 For P ∈ Sp(2n) and γ ∈ Pτ (2n), we define μP (γ ) = iLLIP (), nP (γ ) = νLLIP ().

(5.90)

Theorem 3.2 For any P ∈ Sp(2n), γ ∈ Pτ (2n), there hold μP (γ ) = i P (γ ), nP (γ ) = ν P (γ ).

(5.91)

We will restate this theorem as a special case of Theorem VII.1.3 and prove it later.

122

5 The L-Index Theory

5.4 The Relation with the Morse Index in Calculus Variations We consider the following problem 

[P (t)x  (t) − Q(t)x(t)] + QT (t)x  (t) + R(t)x = 0, x(0) = x(1) = 0,

(5.92)

where P and R are symmetrial n × n matrix function, we suppose −P > 0 (positive definite). For simplicity, we assume P , Q are smooth and R is continuous. The equations in (5.92) was studied by M.Morse in chapter IV of [242]. We turn it into a first order system with Lagrangian boundary condition by setting z(t) = (x(t), y(t))T ∈ R2n with y = P (t)x  (t) − Q(t)x(t): 

z˙ = J B(t)z z(0), z(1) ∈ L0 ,

(5.93)

where B = B(t) is defined by  B(t) =

−R(t) − QT (t)P −1 (t)Q(t) −P −1 (t)Q(t)

 −QT (t)P −1 (t) . −P −1 (t)

(5.94)

We take the space W = W01,2 ([0, 1], Rn ), the subspace of W 1,2 ([0, 1], Rn ) with the elements x satisfying x(0) = x(1) = 0. Define the following functional on W '1 ϕ(x) = − 12 0 P −1 (t)(P (t)x  (t) − Q(t)x(t)), P (t)x  (t) − Q(t)x(t)

 − (R(t) + QT (t)P −1 (t)Q(t))x(t), x(t) dt. (5.95) The critical point of ϕ is a solution of the problem (5.92), and so we get a solution of the problem (5.93). Denote the Morse index of the functional ϕ at x = 0 by mL0 (B), which is the total multiplicity of the negative eigenvalues of the Hessian of ϕ at x = 0, and the nullity of ϕ at x = 0 by nL0 (B). Assume that the fundamental solution of the linear system in (5.93) is γB . We denote by iL0 (B) = iL0 (γB ), νL0 (B) = νL0 (γB ). Theorem 4.1 ([189]) There holds iL0 (B) = mL0 (B), νL0 (B) = nL0 (B).

(5.96)

Proof It is easy to see for any x, ξ ∈ W there holds ϕ  (x)ξ =



1 0

(P (t)x  (t)−Q(t)x(t)) +QT (t)x  (t)+R(t)x(t), ξ(t) dt.

(5.97)

5.4 The Relation with the Morse Index in Calculus Variations

123

and for ξ, ζ ∈ W , (ϕ  (0)ξ, ζ ) =



1

(P (t)ξ  (t) − Q(t)ξ(t)) + QT (t)ξ  (t) + R(t)ξ(t), ζ (t) dt,

0

(5.98) where ·, · is the Euclidean inner product in R2n . From (5.98), we see that the nullity space of this quadratic form is just the space of the solution of problem (5.92), which is one to one corresponding to the space of solutions of problem (5.93). A solution z of (5.93) should satisfy z(t) = γ (t)z(0), z(0) ∈ L0 , z(1) ∈ L0 .

(5.99)

From (5.99) we get that nL0 (B) = dim ker V (1) = νL0 (B),

(5.100)

V (t) is defined in (5.1). So we get the second result of (5.96). In order to prove the first result of (5.96), following the ideas of [125], we set L(x) = (P (t)x  (t) − Q(t)x(t)) + QT (t)x  (t) + R(t)x(t), x ∈ W.

(5.101)

Then L : W ⊂ L2 → L2 := L2 ([0, 1], R2n ) is self-adjoint in L2 . The negative definite subspace W − is a finite dimensional space. By definition, there holds mL0 (B) = The number of negative eigenvalues of L counting by multiplicity. Let ej = (a1 , · · · , an )T ∈ Rn with aj = 1 and ak = 0 for k = j . Let xj be a solution of the first equation of (5.92) satisfying the initial condition xj (0) = 0 ∈ Rn and x˙j (0) = P −1 ej . Then by setting y(t) = P (t)x  (t) − Q(t)x(t) and z = (xj , yj )T ∈ R2n we get z a solution of the linear Hamiltonian system in (5.93) with the initial condition   0 (z1 (0), · · · , zn (0)) = . (5.102) I 

 S(t) V (t) ∈ Pτ (2n) be the fundamental solution of the linear T (t) U (t) Hamiltonian system in (5.93). There holds

Let γ (t) =

 (z1 (t), · · · , zn (t)) =

 V (t) . U (t)

(5.103)

The solution space of the differential equation in (5.92) with the initial data x(0) = 0   is that spaned by the special solutions x1 , · · · , xn . We first prove the following result.

124

5 The L-Index Theory

Lemma 4.2 For V (t) defined in (5.103), let ν(s) = dim ker V (s) = n − rankV (s). There are only finitely many points s ∈ (0, 1) such that ν(s) > 0 and the following result is true  ν(s). (5.104) mL0 (B) = 0 0, and ν(s) equals the number of negative eigenvalues of (5.105) with s = 1.

0 0. So by continuity, we have θj (s1 ) < θj (s0 ) < θj (s2 ), 0 ≤ s1 < s0 < s2 ≤ 1. From the proof of Lemma 4.3, ν(s) > 0 only when detV (s) = 0, and this happens when some phase angle θj (s) is a multiple of π . The multiplicity ν(s) equals the number of phase angles θj (s) that are multiples of π . So the number

126

5 The L-Index Theory

 E

θj (1) π



 =E

θj (1) − θj (0) π



equals the number of times that is a multiple of π , for 0 < s < 1. This implies that 

ν(s) =

n  j =1

0 0. There holds iL (γ ) ≥ 0.

(5.111)

5.5 Saddle Point Reduction Formulas

127

Proof There exists an orthogonal symplectic matrix M such that iL (γ ) = iL0 (Mγ M T ). ˙ = Equation (5.111) follows from that Mγ M T is the fundamental solution of w(t) J MB(t)M T w(t), and MB(t)M T > 0.  

5.5 Saddle Point Reduction Formulas We consider the Hamiltonian system with lagrangian boundary conditions 

z˙ = J H  (t, z), z ∈ R2n , z(0) ∈ L, z(1) ∈ L,

(5.112)

where L is a Lagrangian subspace of R2n , H ∈ C 2 ([0, 1] × R2n , R) satisfies the following condition |H  (t, z)| ≤ C(H ), ∀ (t, z) ∈ [0, 1] × R2n , C(H ) is a constant .

(5.113)

The linear Hamiltonian system 

z˙ = J B(t)z, x ∈ R2n , z(0) ∈ L, z(1) ∈ L

(5.114)

is a special case of (5.112) with H (t, z) = 12 (B(t)z, z), for any symmetrical matrix function B ∈ C([0, 1], Ls (R2n )). Here we use Ls (R2n ) to denote the set of symmetrical matrices. We note that there is an orthogonal symplectic matrix P ∈ Sp(2n) ∩ O(2n) such that P L0 = L. By changing the variables in (5.112) by z = P w, we get the following problem 

z˙ = J HL (t, z), z ∈ R2n , z(0) ∈ L0 , z(1) ∈ L0 ,

(5.115)

where HL (t, z) = H (t, P z). HL still satisfies the condition (5.113). In the special case (5.114), we get 

z˙ = J BL (t)z, z ∈ R2n , z(0) ∈ L0 , z(1) ∈ L0

with BL (t) = P T B(t)P . We consider the following functional

(5.116)

128

5 The L-Index Theory

f (z) =

1 1

2

0

 (−J z˙ , z) − H (t, z) dt, z ∈ WL ,

(5.117)

where WL = {z = (x, y)T ∈ W 1,2 ([0, 1], R2n )| z(0), z(1) ∈ L} ⊂ L2 . It is a classical result that any critical point of f is a solution of the problem (5.112). We denote the norm and inner product in L2 by # · #2 and ·, · 2 , respectively. In L2 we define a self-adjoint operator A by

1

Az, z 2 =

(−J z˙ , z) dt, ∀ z ∈ domA = WL ,

(5.118)

0

and define a functional g by

1

g(z) =

H (t, z) dt.

(5.119)

0

Thus there holds f (z) =

1 Az, z 2 − g(z), ∀ z ∈ domA = WL . 2

(5.120)

It is clear that g  (z) = H  (t, z(t)), dg  (z)ξ = H  (t, z(t))ξ, and there exists a constant c(H ) > 0 such that #dg  (z)#2 := #dg  (x)#L2 ≤ c(H ).

(5.121)

The kernel of A is the space E0 = L of the constant curves. The range of A is closed and its resolution is compact. We defined an invertible operator A0 in L2 by A0 z = Az + P0 z, z ∈ WL ,

(5.122)

where P0 : L2 → E0 = L is the projection map. The spectrum σ (A0 ) of A0 is a point spectrum. σ (A) = {kπ | k ∈ Z} = π Z. For every eigenvalue kπ , the eigensubspace Ek are a n-dimensional subspace. If L = L0 , the eigensubspace Ek = span{α1 , · · · , αn }, αj = (a1 , · · · , an , b1 , · · · , bn )T , ai = bi = 0, i = j, aj = sin kπ t, bj = − cos kπ t.

(5.123)

σ (A0 ) = 1 ∪ π Z \ {0}. The eigensubspace of 1 ∈ σ (A0 ) is dimensional n space E0 , and any other eigenvalue kπ possesses an n-dimensional eigensubspace Ek . For the general L, the numbers of dimensions are the same, but the eigensubspces should be transformed by a suitable orthogonal symplectic matrix.

5.5 Saddle Point Reduction Formulas

129

We choose c(H ) > π satisfies c(H ) ∈ / σ (A0 ). Denote by Eλ the spectral resolution of the self-adjoint operator A0 , we define the projections on L2 by P=

c(H )

−c(H )

dEλ , P

+

=

+∞

dEλ , P

c(H )

−

=

−c(H ) −∞

dEλ

The Hilbert space L2 possesses an orthogonal decomposition L2 = L+ ⊕ L− ⊕ X, where L± = P ± L2 , and X = PL2 is a finite dimensional space with dim X = 2d + n = (2m + 1)n for some m ∈ N ∪ {0}. Theorem 5.1 There exist a function a ∈ C 2 (X, R) and an injection map u ∈ C 1 (X, L2 ) such that u : X → WL satisfies the following conditions: 1o The map u has the form u(x) = w(x) + x with Pw(x) = 0. 2o The function a satisfies a(x) = f (u(x)) = 12 Au(x), u(x) 2 − g(u(x)), a  (x) = Ax − Pg  (u(x)) = Au(x) − g  (u(x)), a  (x) = AP − Pdg  (u(x))u (x) = [A − dg  (u(x))]u (x). And a  is globally Lipschitz continuous. x ∈ X is a critical point of a, if and only if z = u(x) is a critical point of f , i.e., z = u(x) is a solution of the problem (5.112). 4o If g  (z) = Bz for all z ∈ L2 , where B is the induced linear operator on L2 from a constant symmetrical matrix B(t) ≡ B defined on R2n , then a(x) = 1 2 (A − B)x, x 2 . 5o If  is a topological space, for any σ ∈  the functional g :  × L2 → R satisfies g(σ, ·) ∈ C 1 (L2 , R), g  ∈ C( × L2 , R), and the inequality (5.121) with the constant c(H ) being independent from σ ∈ . Then the corresponding map u = u(σ, x) and its derivative ux (σ, x) with respect to x are all continuous. 6o There holds 3o

dim ker a  (x) = the nullity of f  (u(x)) = νL (γ ), where γ is the fundamental solution of the linearized system (5.114) with B(t) = H  (t, z(t)) with z(t) = u(x)(t). Proof Up to the author’s knowledge, the saddle point reduction method was introduced by H.Amann in [4]. For the periodic boundary condition case, a theorem like Theorem 5.1 was proved by H. Amann and E. Zehnder in their celebrated paper [5] by using the monotone operator theory. Then K.C. Chang gave a simple and more direct proof in [33]. A proof of combination of their ideas was given in [223]

130

5 The L-Index Theory

and [227]. The key ingredient of the proof is to use the contraction mapping theorem and the implicit function theorem. Define operators by +

S =



+∞

λ

−1/2

−



dEλ , S =

−c(H )

−∞

c(H )

(−λ)

−1/2

dEλ , R =

+c(H )

−c(H )

|λ|−1/2 dEλ .

By noting that S ± g0 (v) are contraction mappings, where g0 (z) = g(z)+ 12 P0 z, z 2 , one can solve the equations z± = ±S ± g0 (S + z+ + S − z− + Rx), ∀ x ∈ X and get the mappings z± = z± (x). Define u(x) = w(x) + x, w(x) = S + z+ (R −1 x) + S − z− (R −1 x). The mapping u : X → WL satisfies all conditions 1o − 6o in Theorem 5.1. We omit the details here. One can find the details in the references mentioned above (for example, see [223]).   For the special case (5.114), it induces a symmetric operator on L2 by

1

Bz, w 2 =

(B(t)z, w) dt, ∀ z, w ∈ L2 .

0

Then the functional f (z) in (5.117) is the following one: f (z) =

1 (A − B)z, z 2 , ∀ z ∈ WL . 2

(5.124)

By Theorem 5.1, we obtain an injection map u : X → WL and a smooth functional a ∈ C ∞ (X, R) defined by a(x) = f (u(x)), x ∈ X.

(5.125)

Let dim X = 2d + n. Note that the origin of X as a critical point of a corresponds to the origin of WL as a critical point of f . Denote by m∗ (B) for ∗ = +, 0, − the positive, null, and negative Morse indices of the functional a at the origin respectively, i.e., the total multiplicities of positive, zero, and negative eigenvalues of the matrix a  (0) respectively. The following is the main result of this section. Theorem 5.2 ([189]) For any L ∈ (n), ⎧ − ⎨ m (B) = d + iL (B) + n, m0 (B) = νL (B), ⎩ + m (B) = d − iL (B) − νL (B).

(5.126)

5.5 Saddle Point Reduction Formulas

131

Proof We only prove the first result in (5.126) in the special case L = L0 . For any C 1 curve γ ∈ Pτ (2n), the matrix J γ˙ γ −1 is symmetric. We denote by Ls (R2n ) the set of symmetric matrices. We let B(t) = −J γ˙ (t)γ (t)−1 ∈ Ls (R2n ). Then γ is a fundamental solution of the following linear Hamiltonian system z˙ = J B(t)z.   By the same arguments as in the proof of Lemma 5.2.2 of [223], we have the following result. Lemma 5.3 Suppose γ0 , γ1 ∈ P1 (2n) possess common end point γ0 (1) = γ1 (1). Suppose γ0 ∼L0 γ1 in the sense of Definition 1.8 via a L0 -homotopy δ : [0, 1] × [0, 1] → Sp(2n) with δ(s, t) = δ(s)(t) as defined in Definition 1.8 such that δ(·, 1) is contractible in Sp(2n). Then the homotopy can be modified to fix the end points all the time, i.e., δ(s, 1) = γ0 (1) for all 0 ≤ s ≤ 1. Lemma 5.4 For the matrix function Q(t) defined in (5.3) with γ ∈ Pτ (2n), let  : [0, 1] → R be a continuous function satisfying √ detQ(t) = exp(2 −1(t)), ∀ t ∈ [0, 1].

(5.127)

We define the rotation number of γ by r(γ ) = (1) − (0).

(5.128)

Then r(γ ) depends only on γ but not on the choice of the function . If γ0 , γ1 ∈ Pτ (2n) possess common end point γ0 (1) = γ1 (1), then r(γ0 ) = r(γ1 ) if and only if γ0 ∼L0 γ1 with fixed end points. Proof We only prove the second part of Lemma 5.4. We note that γ0 ∼L0 γ1 with fixed end points if and only if γ = γ1−1 ∗ γ0 is contractible in Sp(2n). The latter holds if and only if (γ ) = 0. Here we define the path γ1−1 by γ1−1 (t) = γ1 (1 − t).   The following result was proved in [223]. Lemma 5.5 Let B ∈ C([0, 1] × [0, 1], Ls (R2n )) and denote by Bs (·) = B(s, ·). Denote the Morse indices of the functional as on X corresponding to the system (5.114) with B(t) replaced by Bs (t) at x = 0 by m− (Bs ), m0 (Bs ) and m+ (Bs ), where as is defined in Theorem 5.1 with the same X. Suppose m0 (Bs ) = m0 (B0 ), ∀ s ∈ [0, 1]. Then

(5.129)

132

5 The L-Index Theory

m− (Bs ) = m− (B0 ), m+ (Bs ) = m+ (B0 ), ∀ s ∈ [0, 1].

(5.130)

Remark By Theorem 5.1(5o ) and the compactness, we can choose the same X for all as with 0 ≤ s ≤ 1 in Lemma 5.5. Continue the proof of Theorem 5.2 Firstly, we assume νL0 (γ ) = 0 and iL0 (γ ) = k. By choosing a path β in Sp(2n)∗L0 connects γ (1) with M + or M − , we get a path γ0 = β ∗ γ , and there holds γ ∼L0 γ0 . By definition, we have iL0 (γ ) = iL0 (γ0 ) =

r(γ0 ) n − . π 2

(5.131)

Assume δs (·) ∈ Pτ (2n) is the homotopy map between γ0 and γ . We can perturb it slightly so that δs (t) is C 1 . Then by setting Bs (t) = −J δ˙s (t)δs (t)−1 as in Lemma 5.5, we have m0 (B) = m0 (B0 ) = 0 and m− (B) = m− (B0 ), m+ (B) = m+ (B0 ),

(5.132)

where B and B0 corresponding to γ and γ0 respectively. We construct a path γk ∈ Pτ (2n)∗ with iL0 (γk ) = k by  cos(kπ + π2 )t − sin(kπ + π2 )t  = sin(kπ + π2 )t cos(kπ + π2 )t     cos π2t − sin π2t cos π2t − sin π2t   · · ·  . sin π2t cos π2t sin π2t cos π2t 

γ k (t)

Corresponding to this path, the symmetric coefficient matrix of the linear Hamiltonian system is 1 π π B k (t) = (k + )π I2  I2  · · ·  I2 . 2 2 2 By Lemma 5.4, we can construct a C 1 homotopy map δsk (t) with fixed end points between γ0 and γ k . Then by setting Bsk (t) = −J δ˙sk (t)δsk (t)−1 as in Lemma 5.5, we get m0 (B k ) = m0 (B0 ) = 0 and m− (B k ) = m− (B0 ), m+ (B k ) = m+ (B0 ).

(5.133)

We now only need to consider the case when n = 1 and B(t) = μI2 with constant μ = (k + 1/2)π for k ∈ Z. By Theorem 5.1(4o ), we have a μ (x) = We can choose

1 (A − μI2 )x, x 2 , x ∈ X. 2

(5.134)

5.5 Saddle Point Reduction Formulas

133

X=

d1 

Ej

j =−d1

with Ej = (sin j π t, cos j π t)T R. By direct computations, we have m− (B μ ) = d1 + k + 1.

(5.135)

Now by Theorem 1.13 and (5.135), we have m− (B k ) = d + iL0 (γ ) + n.

(5.136)

Equation (5.126) follows from (5.132), (5.133) and (5.136). For the general case, we assume νL0 (γ ) > 0. Then by Lemma 1.10 and Theorem 1.11, we have the perturbed paths γs ∈ Pτ (2n) with γs ∈ Pτ (2n)∗L0 for s = 0. Setting Bs (t) = −J γ˙s (t)γs (t)−1 . Note that B0 (t) = B(t). By Theorem 5.1, we get a functional as on a finite dimensional space X. It is easy to see as → a under the C 2 topology. By Lemma 5.5 and the above case, we have m+ (Bs ) = d − iL0 (γs ), m0 (Bs ) = 0, m− (Bs ) = d + iL0 (γs ) + n, s = 0. By the perturbed theory and Theorem 1.11, if 0 < s ≤ 1, m+ (B) ≥ m+ (B−s ) − m0 (B) = d − iL0 (γ−s ) − νL0 (γ ),

(5.137)

m+ (B) ≤ m+ (Bs ) = d − iL0 (γs ) = d − iL0 (γ−s ) − νL0 (γ ),

(5.138)

m− (B) ≤ m− (B−s ) = d + iL0 (γ−s ) + n,

(5.139)

m− (B) ≥ m− (Bs ) − m0 (B) = d + iL0 (γs ) + n − νL0 (γ ) = d + iL0 (γ−s ) + n. (5.140) From (5.137), (5.138), (5.139) and (5.140), we have m− (B) = d + iL0 (γ−s ) + n, m+ (B) = d − iL0 (γ−s ) − νL0 (γ ).

(5.141)

If we replace γ−s by any path γ˜ ∈ Pτ (2n)∗L0 sufficiently closed to γ and γs by any path γ¯ ∈ Pτ (2n)∗L0 , by the estimates (5.139) and (5.140), we get |iL0 (γ¯ ) − iL0 (γ˜ )| ≤ νL0 (γ ). But by Theorem 1.11, we have iL0 (γs ) − iL0 (γ−s ) = νL0 (γ ).

(5.142)

134

5 The L-Index Theory

Thus there holds iL0 (γ−s ) = min{iL0 (γ˜ ); γ˜ ∈ Pτ (2n)∗L0 sufficiently closed to γ }, iL0 (γs ) = max{iL0 (γ¯ ); γ¯ ∈ Pτ (2n)∗L0 sufficiently closed to γ }.

(5.143)

So we have proved (5.28), (5.29) and iL0 (γ ) = iL0 (γ−s ).

(5.144)

Equation (5.126) follows from (5.141) and (5.144).

 

Proof of Theorem 1.15 We only need to prove the first result of (5.34). We recall that the condition is that γ0 ∼L γ1 . We shall to prove iL (γ0 ) = iL (γ1 ).

(5.145)

By perturbed slightly with fixed end points, we assume γj ∈ C 1 ([0, 1], Sp(2n)) and the homotopy δs (·) ∈ C 1 ([0, 1], Sp(2n)) for all 0 ≤ s ≤ 1. As in the proof of Theorem 5.2, we take Bs (t) = −J δ˙s (t)δs (t)−1 . Then by the condition and Lemma 5.5, we have m0 (Bs ) = m0 (B0 ) for all s ∈ [0, 1]. Thus we have m− (Bs ) = m− (B0 ), ∀ s ∈ [0, 1]. So by the result of Theorem 5.2, we have d + n + iL (Bs ) = d + n + iL (B0 ). Take s = 1, we get (5.145). The proof is completed.

 

5.6 Galerkin Approximation Formulas for L-Index d The eigenspace Ek of the operator A = −J dt in the domain WL1,2 ([0, 1], R2n ) := 0 1,2 2n {z ∈ W ([0, 1], R ) : z(0) ∈ L0 , z(1) ∈ L0 } can be written as

Ek = −J exp(kπ tJ )ak = −J (cos(kπ t)I2n + J sin(kπ t))ak , ak = (ak1 , · · · , akn , 0, · · · , 0) ∈ R2n . 1/2,2

We define a Hilbert space WL0 = WL0

([0, 1], R2n ) ⊂



Ek with L0 boundary

k∈Z

conditions by WL0 = {z ∈ L2 | z(t) =

 k∈Z

−J exp(kπ tJ )ak , #z#2 :=

 k∈Z

(1 + |k|)|ak |2 < ∞}.

5.6 Galerkin Approximation Formulas for L-Index

135

We denote its inner product by ·, · . By the well-known Sobolev embedding theorem, for any s ∈ [1, +∞), there is a constant Cs > 0 such that #z#Ls ≤ Cs #z#, ∀ z ∈ WL0 . For any Lagrangian subspace L ∈ (n), suppose P ∈ Sp(2n) ∩ O(2n) such that L = P L0 . Then we define WL = P WL0 . ( m m   m Ek = z|z(t) = −J exp(kπ tJ )ak the finite We denote by WL0 = k=−m

k=−m

dimensional truncation of WL0 , and WLm = P WLm0 . Let P m = PLm : WL → WLm the orthogonal projection for m ∈ N. Then  = {P m ; m ∈ N} is a Galerkin approximation scheme with respect to A defined in (5.148) below, i.e., there hold P m → I strongly as m → ∞ and P m A = AP m . In this section we still consider the following problem 

z˙ = J H  (t, z), z ∈ R2n , z(0) ∈ L, z(1) ∈ L,

(5.146)

with H satisfying |H  (t, z)| ≤ a(1 + |z|p ), ∀ (t, z) ∈ R × R2n , for some a > 0, p > 1. (5.147) We consider the functional on WL f (z) = 0

1 1

 1 (−J z˙ , z) − H (t, z) dt = Az, z − g(z), z ∈ WL , 2 2

(5.148)

A critical point of f on WL is a solution of (5.146). For a critical point z = z(t), we denote B(t) = H  (t, z(t)) and define an operator B on WL by Bz, w =

1

(B(t)z, w)dt. 0

Using the Floquet theory we have νL (B) = dim ker(A − B).

(5.149)

136

5 The L-Index Theory

For δ > 0, we denote by m∗δ (·), ∗ = +, 0, − the dimension of the total eigenspace corresponding to the eigenvalue λ belonging to [δ, +∞), (−δ, δ) and (−∞, −δ] resp, and denote by m∗ (·), ∗ = +, 0, − the dimension of the total eigenspace corresponding to the eigenvalue λ belonging to (0, +∞), {0} and (−∞, 0) resp. For any adjoint operator Q, we denote Q = (Q|I mQ )−1 , and we also denote P m QP m = (P m QP m )|WLm . The following result is adapted from [95] where the periodic boundary condition was considered (see also [227]). Theorem 6.1 ([192]) For any B(t) ∈ C([0, 1], Ls (R2n )) with the L-index pair (iL (B), νL (B)) and any constant 0 < δ ≤ 14 #(A − B) #, there exists m0 > 0 such that for m ≥ m0 , we have m m m+ δ (P (A − B)P ) = mn − iL (B) − νL (B) − m md (P (A − B)P m ) = mn + iL (B) + n m0δ (P m (A − B)P m ) = νL (B).

(5.150)

Proof We follow the ideas of [95]. Step 1.

There is m1 > 0 such that for m ≥ m1 dim ker(P m (A − B)P m ) ≤ dim ker(A − B).

(5.151)

In fact, by contradiction it is easy to show that there is a constant m2 > 0 such that for m ≥ m2 dim P m ker(A − B) = dim ker(A − B).

(5.152)

Since B is compact, there is m1 ≥ m2 such that for m ≥ m1 #(I − P m )B# ≤ 2δ. Take m ≥ m1 , let WLm = P m ker(A − B) ⊕ Y m , then Y m ⊂ I m(A − B). For y ∈ Y m we have y = (A − B) (A − B)y = (A − B) (P m (A − B)P m y + (P m − I )By). This implies #y# ≤

1 #P m (A − B)P m y#, ∀ y ∈ Y m . 2δ

By (5.152) and (5.153) we have (5.151). Step 2. We distinguish two cases. Case 1. νL (B) = 0. By (5.149) and step 1, for m ≥ m1 we obtain that m0 (P m (A − B)P m ) = dim ker(A − B) = 0.

(5.153)

5.6 Galerkin Approximation Formulas for L-Index

137

Since B is compact, there exists m3 ≥ m1 such that for m ≥ m3 #(I − P m )B# ≤

1 #(A − B) #−1 . 2

Then P m (A − B)P m = (A − B)P m + (I − P m )BP m implies that #P m (A − B)P m z# ≥

1 #(A − B) #−1 #z#, ∀ z ∈ WLm . 2

Thus the eigen-subspace Mδ∗ (P m (A − B)P m ) corresponding to the eigenvalue λ belonging to the intervals defined as the symbol m∗δ (P m (A − B)P m ) mentioned above, and the eigen-subspace M ∗ (P m (A − B)P m ) satisfy Mδ∗ (P m (A − B)P m ) = M ∗ (P m (A − B)P m ), for ∗ = +, 0, −. By Theorem 5.2, there is m0 ≥ m3 such that for m ≥ m0 the relation (5.150) holds. Case 2. νL (B) > 0. By step 1, it is easy to show that there exists m4 > 0 such that for m ≥ m4 m0δ (P m (A − B)P m ) ≤ νL (B).

(5.154)

Let γ ∈ Pτ (2n) be the fundamental solution of the linear Hamiltonian system z˙ = J B(t)z. Let γs , 0 ≤ s ≤ 1 be the perturbed path defined by Lemma 1.10. Define Bs (t) = −J γ˙s (t)γs (t)−1 , t ∈ [0, 1]. Let Bs be the compact operator defined as B corresponding to Bs (t). For s = 0, there holds m0 (A − Bs ) = 0 and #Bs − B# → 0 as s → 0. If s ∈ (0, 1], we have iL (γs ) − iL (γ−s ) = νL (γ ) = νL (B), iL (γ−s ) = iL (B) = iL (γ ).

(5.155)

Choose 0 < s < 1 such that #B − B±s # ≤ 2δ . By case 1, (5.154) and (5.155) and the fact that P m (A − B± )P m = P m (A − B)P m + P m (B − B± )P m there exists m0 ≥ m4 such that for m ≥ m0 m m + m m m+ δ (P (A − B)P ) ≤ m (P (A − Bs )P ) = mn − iL (B) − νL (B)

138

5 The L-Index Theory 0 m m + m m m m m+ δ (P (A − B)P ) ≥ m (P (A − B−s )P ) − mδ (P (A − B)P ) ≥ mn − iL (B) − νL (B).

Hence, m0δ (P m (A − B)P m ) = νL (B) and m m m+ δ (P (A − B)P ) = mn − iL (B) − νL (B).

Note that dim WLm = (2m + 1)n, so m m m− δ (P (A − B)P ) = mn + n + iL (B).

 

Remark From Theorem III.4.3, Theorem III.4.5, and Theorem 5.2, we can get a new proof of Theorem 6.1 easily. Corollary 6.2 ([192]) Suppose Bj (t) ∈ C([0, 1], Ls (R2n )), j = 1, 2 satisfying B1 (t) < B2 (t), i.e., B2 (t) − B1 (t) is positive definite for all t ∈ [0, 1]. Then there holds iL (B1 ) + νL (B1 ) ≤ iL (B2 ).

(5.156)

Proof Just as done in Theorem 6.1, corresponding to Bj (t), we have the operator Bj . Let  = {P m } be the approximation scheme with respect to the operator A. Then by (5.150), there exists m0 > 0 such that if m ≥ m0 there holds m m m− δ (P (A − B1 )P ) = mn + n + iL (B1 ), m m m− δ (P (A − B2 )P ) = mn + n + iL (B2 ),

where we choose 0 < δ < 12 #B2 − B1 #. Since A − B2 = (A − B1 ) − (B2 − B1 ) and B2 − B1 is positive definite in WLm = P m WL and (B2 − B1 )x, x ≥ 2δ#x#. Thus (P m (A − B2 )P m )x, x ≤ −δ#x# with x ∈ Mδ− (P m (A − B1 )P m ) ⊕ Mδ0 (P m (A − B1 )P m ). It implies that mn + n + iL (B1 ) + νL (B1 ) ≤ mn + n + iL (B2 ).   Remark From the proof of Corollary 6.2, it is easy to show that if B1 (t) ≤ B2 (t) for all 0 ≤ t ≤ 1, there hold iL (B1 ) ≤ iL (B2 ), iL (B1 ) + νL (B1 ) ≤ iL (B2 ) + νL (B2 ).

(5.157)

5.6 Galerkin Approximation Formulas for L-Index

139

Definition 6.3 ([192]) For any two matrix functions Bj ∈ C([0, 1], Ls (R2n )), j = 0, 1 with B0 (t) < B1 (t) for all t ∈ R, we define IL (B0 , B1 ) =



νL ((1 − s)B0 + sB1 ).

(5.158)

s∈[0,1)

Theorem 6.4 ([192]) For any two matrix functions Bj ∈ C([0, 1], Ls (R2n )) with B0 (t) < B1 (t) for all t ∈ R, we have IL (B0 , B1 ) = iL (B1 ) − iL (B0 ).

(5.159)

So we call IL (B0 , B1 ) the relative L-index of the pair (B0 , B1 ). Proof Step 1. By Corollary 6.2, if we denote iL (λ) = iL ((1−λ)B0 +λB1 ), νL (λ) = νL ((1 − λ)B0 + λB1 ), there holds iL (λ2 ) ≥ iL (λ1 ) + νL (λ1 ), for λ2 > λ1 .

(5.160)

So the function iL (λ) is a monotonic increasing function in [0, 1]. Step 2. We prove that for any λ ∈ [0, 1) there holds iL (λ + 0) = iL (λ) + νL (λ),

(5.161)

where iL (λ + 0) is the right-hand limit of iL (s) at λ. In fact, by (5.160), we have iL (λ) + νL (λ) ≤ iL (λ + 0). We now use the saddle point reduction methods to prove the opposite inequality iL (λ) + νL (λ) ≥ iL (λ + 0). Denote by Bλ (t) = (1 − λ)B0 (t) + λB1 (t). We define in L2 ([0, 1], R2n )

1

fλ (x) =

[(−J x(t), ˙ x(t)) − (Bλ (t)x(t), x(t))] dt, ∀x ∈ dom(A) = WL .

0

Then by the saddle point reduction methods, we can reduce the functional fλ in L2 ([0, 1], R2n ) to a finite dimensional subspace X of L2 ([0, 1], R2n ) by aλ (x) = fλ (uλ (x)), uλ : X → L2 ([0, 1], R2n ) is injection. aλ is continuously depending on λ. Denote the Morse indices of aλ on X at x = 0 − 0 by m− λ , mλ and mλ . If dim X = 2d +n large enough, there holds (see Theorem 5.2) + 0 m− λ = d + n + iL (λ), mλ = νL (λ), mλ = d − iL (λ) − νL (λ).

(5.162)

For any fixed λ ∈ [0, 1), choosing μ ∈ (λ, 1) ∪ [0, λ) sufficiently close to λ, we obtain ± ± m± λ ≤ mμ ≤ mλ + νL (λ).

(5.163)

140

5 The L-Index Theory

Then by (5.162), we have iL (λ) ≤ iL (μ) and iL (λ) + νL (λ) ≥ iL (μ). It implies iL (λ) + νL (λ) ≥ iL (λ + 0) and iL (λ) ≤ iL (λ − 0). But by (5.160), we have iL (λ) ≥ iL (λ − 0), so iL (λ) = iL (λ − 0). That is to say the function iL (λ) is left continuous at (0, 1]. Moreover if m0λ = m0 is constant in some interval [λ1 , λ2 ], then m− λ = + are constant in this interval. Thus the function i (λ) is locally m− and m+ = m L λ constant at its continuous points, whose all discontinuous points are the points with νL (λ) > 0, and there holds iL (1) = iL (0) +



νL (λ),

0≤λ 0, there holds iL (γ ) =



dim(γ (t)L ∩ L).

(5.164)

0 1}, m0 (Qm ) = {μj | 1 ≤ j ≤ l, μj = 1}. By Theorem 6.1, we have for m > 0 large enough, m− (Qm ) = mn + n + iL (B), m0 (Qm ) = νL (B).

(5.169)

We denote by Q∗k,m the restriction of the quadratic Q∗k to the subspace W m , and ∗ (B) = m− (Q∗ ), ν ∗ (B) = m0 (Q∗ ). By the same argument in [GM], ik,m k,m k,m k,m ∗ (B) → i ∗ (B), ν ∗ (B) → ν ∗ (B) as m → ∞. Let v  = A v for we have ik,m m j k k,m k j j = 1, 2, · · · , l. It is a basis of W m and Q∗k,m (vi , vj ) = 0 for i = j, Q∗k,m (vj ) = μj (μj − 1). Q∗k,m (vj ) is negative if and only if 0 < μj < 1. We now deduce the total multiplicity of the negative eigenvalues μj < 0. If one replaces the inner product ·, · m by the usual one, i.e., replaces the matrix Bk by the identity I , the eigenvalues μj should be replaced by the eigenvalues ηj of Am with respect to the standard inner product. It is easy to check that μj and ηj possesses the same signs. So the total multiplicity of negative μj ’s equals the total multiplicity of negative ηh ’s. But we have ηh = hπ + k, −m ≤ h ≤ m,

(5.170)

each of them has multiplicity n. Therefore, the total multiplicity of the negative ηh is n(m−[k/π ]). So the total multiplicity of μj ∈ (0, 1) is m− (Qm )−n(m−[k/π ]). By definition we have ∗ (B) = m− (Qm ) − n(m − [k/π ]). ik,m

So for m > 0 large enough, from (5.167) we get (5.168).

 

144

5 The L-Index Theory

Corollary 7.2 ([192]) Under the condition of Theorem 6.4, there holds IL (B0 , B1 ) = ik∗ (B1 ) − ik∗ (B0 ).

(5.171)

5.8 The (L, ω)-Index Theory √

For ω = e −1θ with θ ∈ R, we define a Hilbert space E ω = ELω0 consisting of those x(t) in L2 ([0, 1], C2n ) such that e−θtJ x(t) has Fourier expending 

e−θtJ x(t) =

 ej π tJ

j ∈Z

with #x#2 :=



0 aj

 , aj ∈ Cn

(1 + |j |)|aj |2 < ∞.

j ∈Z

For x ∈ E ω , we can write x(t) = e



θtJ

 e

j π tJ

j ∈Z

=



e

√ (θ+j π )t −1

0 aj



√

j ∈Z

=



 e

(θ+j π )tJ

j ∈Z

0 aj



 √   √ −1aj /2 −(θ+j π )t −1 − −1aj /2 +e . aj /2 aj /2 (5.172)

So we can write x(t) = ξ(t) + Nξ(−t), ξ(t) =



e

√ (θ+j π )t −1

√

j ∈Z

For ω = e by

√

−1θ ,

 −1aj /2 . aj /2

(5.173)

θ ∈ [0, π ), we define two self-adjoint operators Aω , B ω ∈ L(E ω )

(Aω x, y) =

1

−J x(t), ˙ y(t) dt, (B ω x, y) =

0

1

B(t)x(t), y(t) dt

0

on E ω . Then B ω is also compact. Definition 8.1 ([209]) We define the index function iωL0 (B) = I (Aω , Aω −B ω ), νωL0 (B) = m0 (Aω −B ω ), ∀ ω = e

√

−1θ

, θ ∈ (0, π ). (5.174)

5.8 The (L, ω)-Index Theory

145

We remind that the relative index I (Aω , III.1.5. From Lemma III.1.7, we see that

Aω − B ω ) is defined in Definition

iωL0 (B) = −sf{Aω − sB ω , 0 ≤ s ≤ 1}.

(5.175)

By the Floquet theory, we have M 0 (Aω , B ω ) is isomorphic to the solution space of the following linear Hamiltonian system x(t) ˙ = J B(t)x(t) satisfying the following boundary condition x(0) ∈ L0 , x(1) ∈ eθJ L0 . If m0 (Aω , B ω ) > 0, there holds γ (1)L0 ∩ eθJ L0 = {0} which is equivalent to ω2 = e2θ

√

−1

% $ √ √ ∈ σ [U (1) − −1V (1)][U (1) + −1V (1)]−1 .

This claim follows from the fact that if γ (1)L0 ∩ eθJ L0 = {0}, there exist a, b ∈ Cn \ {0} such that [U (1) +

√ √ −1V (1)]a = ω−1 b, [U (1) − −1V (1)]a = ωb.

So we have νωL0 (B) = dim(γ (1)L0 ∩ eθJ L0 ), ∀ ω = e

√

−1θ

, θ ∈ (0, π ).

(5.176)

Lemma 8.2 ([209]) The index function iωL0 (B) is locally constant. For ω0 = √ −1θ 0 , θ ∈ (0, π ) is a point of discontinuity of i L0 (B), then ν L0 (B) > 0 and e 0 ω ω0 so dim(γ (1)L0 ∩ eθ0 J L0 ) > 0. Moreover there hold |iωL00+ (B) − iωL00− (B)| ≤ νωL00 (B), |iωL00+ (B) − iωL00 (B)| ≤ νωL00 (B), L0 (B)| ≤ νL0 (B), (5.177) |iωL00− (B) − iωL00 (B)| ≤ νωL00 (B), |iL0 (B) + n − i1+

where iωL00+ (B), iωL00− (B) are the limits on the right and left respectively of the index

function iωL0 (B) at ω0 = e

√

−1θ0

as a function of θ .

146

5 The L-Index Theory

Proof For x(t) =

eθtJ u(t), u(t)

=



 e

j π tJ

j ∈Z



1

((A − B )x, x) = ω

ω



1

−J u(t), ˙ u(t) dt +

0

 0 , we have aj (θ − e−θtJ B(t)eθtJ )u(t), u(t) dt.

0

So we have ((Aω − B ω )x, x) = (qω u, u) with (qω u, u) =

1



1

−J u(t), ˙ u(t) dt +

0

(θ − e−θtJ B(t)eθtJ )u(t), u(t) dt.

0

Since dim(γ (1)L0 ∩ eθJ L0 ) > 0 at only finite (up to n) points θ ∈ (0, π√), for the point θ0 ∈ (0, π ) such that νωL00 (B) = 0, then νωL0 (B) = 0 for ω = e −1θ , θ ∈ (θ0 − δ, θ0 + δ), δ > 0 small enough. By using the notation as in Lemma III.1.4, we have (Pmω (Aω − B ω )Pmω x, x) = (Pm qω Pm u, u). By Lemma III.1.2, we have m0d (Pmω (Aω − B ω )Pmω ) = m0 (Aω − B ω ) = νωL0 (B) = 0. So by the continuity of the eigenvalue of a continuous family of operators we have that ω ω ω ω m− d (Pm (A − B )Pm ) √

ω ω ω must be constant for ω = e −1θ , θ ∈ (θ0 − δ, θ0 + δ). Since m− d (Pm A Pm ) is √ L −1θ , θ ∈ (θ − δ, θ + δ), we have i 0 (B) is constant for constant 0 0 ω √ for ω = e −1θ ω=e , θ ∈ (θ0 − δ, θ0 + δ). The results in (5.177) now follow from some standard arguments.  

It is easy to see that i1L0 (B) = I (A1 , A1 − B 1 ) = iL0 (B) + n. By Definition 8.1 and Lemma 8.2, we see that for any ω0 = e there holds

(5.178) √

−1θ0 ,

θ0 ∈ (0, π ),

5.9 The Bott Formulas of L-Index

147

iωL00 (B) ≥ iL0 (B) + n −

 √ ω=e −1θ ,

νωL0 (B).

(5.179)

0≤θ≤θ0

We note that  √ ω=e −1θ ,

νωL0 (B) ≤ n.

(5.180)

0≤θ≤θ0

So we have iL0 (B) ≤ iωL00 (B) ≤ iL0 (B) + 2n.

(5.181)

5.9 The Bott Formulas of L-Index In this section, we establish the Bott-type iteration formula for the Lj -index theory with j = 0, 1. Without loss of generality, we assume τ = 1. Suppose the continuous symplectic path γ : [0, 1] → Sp(2n) is the fundamental solution of the following linear Hamiltonian system z˙ (t) = J B(t)z(t),

t ∈R

(5.182)

with B(t) satisfying B(t + 2) = B(t) and B(1 + t)N = N B(1 − t) for t ∈ R. This implies B(t)N = NB(−t) for t ∈ R. By the unique existence theorem of the linear differential equations, we get γ (1 + t) = Nγ (1 − t)γ (1)−1 Nγ (1), γ (2 + t) = γ (t)γ (2).

(5.183)

For j ∈ N, we define the j -times iteration path γ j : [0, j ] → Sp(2n) of γ by γ 1 (t) = γ (t), t ∈ [0, 1],  γ (t) = 2

γ (t), t ∈ [0, 1], Nγ (2 − t)γ (1)−1 Nγ (1), t ∈ [1, 2],

(5.184)

and in general, for k ∈ N, we define γ (2) = Nγ (1)−1 N γ (1) and ⎧ ⎪ γ (t), t ∈ [0, 1], ⎪ ⎪ ⎪ ⎪ ⎨ Nγ (2 − t)γ (1)−1 Nγ (1), t ∈ [1, 2], γ 2k−1 (t) = · · · · · · ⎪ ⎪ ⎪ Nγ (2k − 2 − t)Nγ (2)k−1 , t ∈ [2k − 3, 2k − 2], ⎪ ⎪ ⎩ γ (t − 2k + 2)γ (2)k−1 , t ∈ [2k − 2, 2k − 1],

(5.185)

148

5 The L-Index Theory

⎧ ⎪ γ (t), t ∈ [0, 1], ⎪ ⎪ ⎪ ⎪ ⎨ Nγ (2 − t)γ (1)−1 Nγ (1), t ∈ [1, 2], γ 2k (t) = · · · · · · ⎪ ⎪ ⎪ γ (t − 2k + 2)γ (2)k−1 , t ∈ [2k − 2, 2k − 1], ⎪ ⎪ ⎩ Nγ (2k − t)Nγ (2)k , t ∈ [2k − 1, 2k].

(5.186)

For γ ∈ Pτ (2n), we define γ k (τ t) = γ˜ k (t) with γ˜ (t) = γ (τ t).

(5.187)

For the L0 -index of the iteration path γ k , we have the following Bott-type formulas. Theorem 9.1 ([209]) Suppose ωk = eπ

√

−1/k .

iL0 (γ k ) = iL0 (γ 1 ) +

For odd k we have

(k−1)/2 

iω2i (γ 2 ), k

i=1

νL0 (γ k ) = νL0 (γ 1 ) +

(k−1)/2 

νω2i (γ 2 ), k

i=1

and for even k, we have L0 (γ 1 ) + iL0 (γ k ) = iL0 (γ 1 ) + i√ −1

k/2−1 

iω2i (γ 2 ), k

i=1

L0 (γ 1 ) + νL0 (γ k ) = νL0 (γ 1 ) + ν√ −1

k/2−1 

νω2i (γ 2 ). k

i=1

√ L0 L0 We remind that (i√ (γ ), ν√ (γ )) is the (L0 , ω) index of γ with ω = −1. −1 −1 Before proving Theorem 9.1, we give some notations and definitions. We define the Hilbert space ELk 0 =

⎧ ⎨ ⎩

x ∈ L2 ([0, k], C2n ) | x(t) =

 j ∈Z

 ej tπ/kJ

0 aj



⎫ ⎬ , aj ∈ Cn , #x#2 < ∞ , ⎭

where we still denote L0 = {0} × Cn ⊂ C2n which is the Lagrangian subspace of

the linear complex symplectic space (C2n , ω˜ 0 ) and |x#2 := j ∈Z (1 + |j |)|aj |2 . For x ∈ ELk 0 , we can write

5.9 The Bott Formulas of L-Index

x(t) =



e

j tπ/kJ

j ∈Z

=

 j ∈Z

149

    0 − sin(j tπ/k)aj = cos(j tπ/k)aj aj j ∈Z

√  √   √ √ −1aj /2 − −1aj /2 ej π t −1/k + e−j π t −1/k (5.188) . aj /2 aj /2

On ELk 0 we define two self-adjoint operators and a quadratical form by

k

(Ak x, y) =

−J x(t), ˙ y(t) dt,

k

(Bk x, y) =

0

B(t)x(t), y(t) dt,

0

QkL0 (x, y) = ((Ak − Bk )x, y),

(5.189) (5.190)

where in this section ·, · is the standard Hermitian inner product in C2n . Lemma 9.2 ([209]) ELk 0 has the following natural decomposition ELk 0 =

k−1 

ωl

EL0k ,

(5.191)

l=0 ωl

here we have extended the domain of functions in EL0k from [0, 1] to [0, k] in the obvious way, i.e., ωl EL0k

⎧ ⎫  ⎬ ⎨  0 = x ∈ ELk 0 | x(t) = elπ tJ /k ej π tJ . ⎩ aj ⎭ j ∈Z

Proof Any element x ∈ ELk 0 can be written as x(t) =

√  √   √ √  −1aj /2 − −1aj /2 ej π t −1/k + e−j π t −1/k aj /2 aj /2 j ∈Z

=

k−1 



l=0 j ≡l (modk)

 √  √   √ √ −1aj /2 j π t −1/k −j π t −1/k − −1aj /2 e +e aj /2 aj /2

√  k−1   √ √  −1bj /2 lπ t −1/k j π t −1 e e = bj /2 l=0 j ∈Z

+e−lπ t

√

√ −1/k −j π t −1

e

:= ξx (t) + ηx (t),

 √  − −1bj /2 bj /2 (5.192)

150

5 The L-Index Theory

where ξx (t) =

k−1

l=0

j ∈Z

√ √ elπ t −1/k ej π t −1

and bj = aj k+l . By setting ωk = eπ obtain (5.191).

√

−1/k ,

√  −1bj /2 , ηx (t) = N ξx (−t) bj /2

and comparing (5.172) and (5.192), we  

Note that the natural decomposition (5.191) is not orthogonal under the quadratical form QkL0 defined in (5.190). So the type of the iteration formulas in Theorem 9.1 is somewhat different from the original Bott formulas in [28] of the Morse index theory for closed geodesics and of Maslov-type index theory for periodic solutions of Hamiltonian systems and the Bott-type formulas in [78]. This is also our main difficulty in the proof of Theorem 9.1. However, after recombining the terms in the decomposition of Lemma 9.2, we can obtain an orthogonal decomposition under the quadratical form QkL0 . For 1 ≤ l < k2 and l ∈ N, we set ωk−l

ωl

ELω0k ,l = EL0k + EL0k . Then we have the following Lemma 9.3 ELk 0 has the following QkL0 orthogonal decompositions: ELk 0 = EL1 0 ⊕

(k−1)/2 

ELω0k ,l

∀k ∈ 2N + 1,

(Codd )

∀k ∈ 2N.

(Ceven )

l=1

ELk 0

=

EL1 0

√

k

−1

⊕ EL0

⊕

2 −1 

ELω0k ,l

l=1

Proof For any x ∈ ELk 0 , in the following argument we always extend it naturally to  

0 , t ∈ [0, 2k]. By an element in L2 ([0, 2k], C2n ) by x(t) = j ∈Z ej tπ/kJ aj direct computation, for any x, y ∈ ELk 0 we have 1 ((Ak − Bk )x, y)= 2

0

2k

1 −J x(t), ˙ y(t) dt− 2



2k

B(t)x(t), y(t) dt.

(5.193)

0

We denote simply by 12 Qk (x, y) = ((Ak − Bk )x, y). For any 0 ≤ l ≤ k, define the Hilbert space ⎧ ⎫ ⎨ ⎬  Ekl = x ∈ L2 ([0, 2k], C2n ) | x(t) = eltπ/kJ ej tπ J cj , cj ∈ C2n , #x#2 < ∞ . ⎩ ⎭ j ∈Z

5.9 The Bott Formulas of L-Index

151

Thus for any x ∈ Ekl we have x(t + 2) = wk2l x(t). Then by the argument of j Lemma I.5.2 of [78], we have Eki and Ek are Qk orthogonal for any i = j . ωl

For any x ∈ EL0k with 1 ≤ l ≤ k − 1, by the definition of ξx and ηx in the proof of Lemma 9.2 we have ξx ∈ Ekl and ηx ∈ Ekk−l . Hence x ∈ Ekl + Ekk−l . So for any x ∈ ELω0k ,i with 1 ≤ i < k2 we have x ∈ Eki + Ekk−i . Note that for any x ∈ ELω0k ,0 ω ,k/2

k/2

we have x ∈ Ek0 , if k is even, then for any x ∈ EL0k we have x ∈ Ek . Hence the QkL0 orthogonal decompositions (Codd ) and (Ceven ) hold by (5.193) and the Qk j

orthogonality of Eki and Ek with i = j . The proof of Lemma 9.3 is complete. ωl

  ωk−l

Let 1 ≤ l < k2 , l ∈ N. For any z = x + y ∈ ELω0k ,l with x ∈ EL0k and y ∈ EL0k , by the arguments in the proofs of Lemmas 9.2 and 9.3 we have z = ξx + ηx + ξ y + ηy with ξx + ηy := z1 ∈ Ekl and ηx + ξy := z2 ∈ Ekk−l . Then by B(−t) = N B(t)N and B being 2-periodic we have

2k

2k

=

2k

2k

N B(t)N (y(−t) + N x(t)), (y(−t) + N x(t)) dt

0 2k

=

B(2k − t)(y(2k − t) + Nx(t − 2k)), (y(2k − t) + N x(t − 2k)) dt

0

=

B(t)(y(t) + Nx(−t)), (y(t) + Nx(−t)) dt

0

=

B(t)z2 (t), z2 (t)

0

B(t)(Ny(−t) + x(t)), N(y(−t) + x(t)) dt

0 2k

=

B(t)z1 (t), z1 (t)

(5.194)

0 j

So by the Qk orthogonality of Eki and Ek with i = j , (5.193), (5.194), and formula (34) on page 37 of [78] we have 1 (Bk z, z) = 2 1 = 2



2k

B(t)(z1 (t) + z2 (t)), (z1 (t) + z2 (t))

0 2k 0

1 B(t)z1 (t), z1 (t) + 2

0

2k

B(t)z2 (t), z2 (t)

152

5 The L-Index Theory



2k

=

B(t)z1 (t), z1 (t)

0



2

=k

B(t)z1 (t), z1 (t) dt.

(5.195)

0

Similarly we have

2

(Ak z, z) = k

−J z˙ 1 (t), z1 (t) dt.

(5.196)

0 √

Note that for z ∈ EL1 0 and z ∈ EL0−1 we have k (Bk z, z) = 2 (Ak z, z) =

0

k 2

2



1

B(t)z(t), z(t) dt = k

B(t)z(t), z(t) dt,

(5.197)

0



2



1

−J z˙ (t), z(t) dt = k

0

−J z˙ (t), z(t) dt.

(5.198)

0

We also note that u(t) = ξx (t) + ηy (t) = =



e



e

√ √ lπ −1t/k j π −1t

j ∈Z √ √ lπ −1t/k j π −1t

e

√

e

−1(αj − βj ) (αj + βj )

uj , uj ∈ C2n .



(5.199)

j ∈Z

We set Eω2l = k

⎧ ⎨ ⎩

u ∈ L2 ([0, 2], C2n ) | u(t)

= elπ

√

−1t/k



ej π

√

−1t

uj , #u#2 :=

j ∈Z

 j ∈Z

⎫ ⎬ (1 + |j |)|uj |2 < +∞ . ⎭

Proof We define self-adjoint operators on Eω2l by k

(Aω2l u, v) = k

2 0

−J u(t), ˙ v(t) dt, (Bω2l u, v) = k

2

B(t)u(t), v(t) dt

0

and a quadratic form Qω2l (u) = ((Aω2l − Bω2l )u, u), u ∈ Eω2l . k

k

k

k

5.9 The Bott Formulas of L-Index

153

Here Qω is just the quadratic form fω defined on p.133 of [223]. In order to complete the proof of Theorem 9.1, we need the following result.   Lemma 9.4 For a symmetric 2-periodic matrix function B and ω ∈ U \ {1}, there hold I (Aω , Aω − Bω ) = iω (γ 2 ),

(5.200)

m0 (Aω − Bω ) = νω (γ 2 ).

(5.201)

Proof In fact, (5.200) follows directly from Definition 2.3 and Corollary 2.1 of [234] and Lemma III.1.7, (5.201) follows from the Floquet theory. We note also that (5.200) is the eventual form of the Galerkin approximation formula. We can also prove it step by step as the proof of Theorem 3.1 of [192] by using the saddle point reduction formula in Theorem 6.1.1 of [223].   Continue the proof of Theorem 9.1 By Lemma 9.4, for 1 ≤ l < k2 , l ∈ N we have I (Aω2l , Aω2l − Bω2l ) = iω2l (γ 2 ), m0 (Aω2l − Bω2l ) = νω2l (γ 2 ). k

k

k

k

k

k

k

(5.202)

By Definition 8.1, we have √

I (A

−1

√

,A

−1

−B

√

−1

L0 ) = i√ (γ ), −1

√

m0 (A

−1

−B

√

−1

L0 ) = ν√ (γ ). −1

(5.203) We also have I (A1 , A1 − B 1 ) = iL0 (γ ) + n,

m0 (A1 − B 1 ) = νL0 (γ ),

(5.204)

I (Ak , Ak − Bk ) = iL0 (γ k ) + n,

m0 (Ak − Bk ) = νL0 (γ k ).

(5.205)

and

By (5.195)–(5.199), Lemma III.1.4, Definition III.1.5, Remark III.1.6 and Lemma 9.3, for odd k, sum the first equality in (5.202) for l = 1, 2, · · · , k−1 2 and the first equality of (5.204) correspondingly. By comparing with the first equality of (5.205) we have k−1

iL0 (γ k ) = iL0 (γ ) +

2 

iω2l (γ 2 ), k

(5.206)

l=1

and for even k, sum the first equality in (5.202) for l = 1, 2, · · · , k2 − 1 and the first equalities of (5.203) and (5.204) correspondingly. By comparing with the first equality of (5.205) we have

154

5 The L-Index Theory k

iL0 (γ k ) =

L0 iL0 (γ ) + i√ (γ ) + −1

2 −1 

iω2l (γ 2 ).

(5.207)

if k is odd,

(5.208)

k

l=1

Similarly we have k−1

νL0 (γ ) = νL0 (γ ) + k

2 

νω2l (γ 2 ), k

l=1

k

νL0 (γ ) = k

L0 νL0 (γ ) + ν√ (γ ) + −1

2 −1 

l=1

νω2l (γ 2 ), k

if k is even. (5.209)  

Then Theorem 9.1 holds from (5.206)–(5.209). From the formulas in Theorem 9.1, we note that L0 L0 iL0 (γ 2 ) = iL0 (γ 1 ) + i√ (γ 1 ), νL0 (γ 2 ) = νL0 (γ 1 ) + ν√ (γ 1 ). −1 −1

Definition 9.5 The mean L0 -index of γ is defined by iL0 (γ k ) . i¯L0 (γ ) = lim k→+∞ k ¯ 2 )(cf. [223] for example), the following result is By definitions of i¯L0 (γ ) and i(γ obvious. Proposition 9.6 The mean L0 -index of γ is well defined, and 1 i¯L0 (γ ) = 2π



√

π

iB (e

−1θ

0

)dθ =

¯ 2) i(γ , 2

(5.210)

here we have written iB (ω) = iω (B) = iω (γB ). For L1 = Rn × {0}, we have√the L1 -index theory established in [189]. Similarly as in Definition 8.1, for ω = eθ −1 , θ ∈ (0, π ), we define ⎧ ⎫   ⎨ ⎬  aj , aj ∈ Cn , #x# < + ∞ . ELω1 = x ∈ L2 ([0, 1], C2n ) | x(t) = eθ tJ ej π tJ ⎩ ⎭ 0 j ∈Z

In ELω1 we define two operators AωL1 and BLω1 by the same way as the definitions of operators Aω and B ω in Sect. 5.8, but the domain is ELω1 . We define iωL1 (B) = I (AωL1 , AωL1 − BLω1 ), νωL1 (B) = m0 (AωL1 − BLω1 ). Theorem 9.7 ([209]) Suppose ωk = eπ

√

−1/k .

For odd k we have

5.9 The Bott Formulas of L-Index

155 k−1

iL1 (γ k ) = iL1 (γ 1 ) +

2 

iω2i (γ 2 ), k

i=1 k−1

νL1 (γ ) = νL1 (γ ) + 1

k

2 

νω2i (γ 2 ).

(5.211)

k

i=1

For even k, we have L1 (γ 1 ) + iL1 (γ k ) = iL1 (γ 1 ) + i√ −1

k/2−1 

L1 (γ 1 ) + νL1 (γ k ) = νL1 (γ 1 ) + ν√ −1

iω2i (γ 2 ), k

i=1 k/2−1 

νω2i (γ 2 ). k

i=1

Proof The proof is almost the same as that of Theorem 9.1. The only thing different   from that is that the matrix N should be replaced by N1 = −N . Proposition 9.8 ([209]) There hold i(γ 2 ) = iL0 (γ 1 ) + iL1 (γ 1 ) + n,

(5.212)

ν1 (γ 2 ) = νL0 (γ 1 ) + νL1 (γ 1 ),

(5.213)

i−1 (γ 2 ) =

L1 L1 i√ (γ 1 ) + i√ (γ 1 ), −1 −1

L1 L1 ν−1 (γ 2 ) = ν√ (γ 1 ) + ν√ (γ 1 ). −1 −1

(5.214) (5.215)

Proof As in the proof of Theorem 9.1, we set E1 = {z ∈ L2 ([0, 2], C2n )| z(t) =



ej tπ J αj , αj ∈ C2n , #z# < ∞}.

j ∈Z

Then we have E1 = WL0 ⊕ WL1 . So from Theorem 6.1 we have I (A1 , A1 − B1 ) = i(γ 2 ) + n = iL0 (γ 1 ) + n + iL1 (γ 1 ) + n, which implies (5.212) and (5.213) is also true. By the Bott-type index theory of the ω-index theory for symplectic paths we have i(γ 4 ) = i(γ 2 ) + i−1 (γ 2 ), ν(γ 4 ) = ν(γ 2 ) + ν−1 (γ 2 ).

156

5 The L-Index Theory

Hence (5.214) and (5.215) hold from Theorems 9.1–9.7.  

5.10 Iteration Inequalities of L-Index 5.10.1 Precise Iteration Index Formula From the Bott-type formulas in Theorem 9.1, we prove the abstract precise iteration index formula of iL0 . Theorem 10.1 Let γ ∈ Pτ (2n), γ k be defined by (5.185)–(5.187), and M = γ 2 (2τ ). Then for every k ∈ 2N − 1, there holds k−1 + (1) − C(M)) iL0 (γ k ) = iL0 (γ 1 ) + (i(γ 2 ) + SM 2   √  kθ − SM E (e −1θ ) − C(M), + 2π

(5.216)

θ∈(0,2π )

where C(M) is defined by 

C(M) =

− SM (e

√

−1θ

)

θ∈(0,2π )

and ± SM (ω) = lim iωexp(±√−1ε) (γ 2 ) − iω (γ 2 ) ε→0+

is the splitting number of the symplectic matrix M at ω for ω ∈ U. For every k ∈ 2N, there holds $ % k + − 1 i(γ 2 ) + SM (1) − C(M) − C(M) 2   √ √  kθ − − −1θ SM SM (e )+ E (e −1θ ). (5.217) 2π 

iL0 (γ k ) = iL0 (γ 2 ) + −

 θ∈(π,2π )

θ∈(0,2π )

Proof By the definition of the splitting number, we have iω0 (γ 2 ) = i(γ 2 ) +

 0≤θ 0, ∀ v ∈ L(t0 ) \ {0}.

(6.5)

  γ11 (t) γ12 (t) . By Lemma 1.3(2), we only need to prove Proof We set γB (t) = γ21 (t) γ22 (t) n the result  for L= L0 = {0} ×  R and t0 = 0. In this case, we have the frame of γ12 (t) 0 L(t) = and L(0) = . So γ22 (t) I   0 QL(0),L(0) . ˙ (v) = − u, γ˙12 (0)u = u, B22 (0)u , v = u   Remark 1.6 Together with Remark 1.4 and Lemma 1.5, we know that if we orient the manifold (n) \ 0 (R2n , L) = {L ∈ (n)| dim L ∩ L > 0} in eθJ L , θ > 0, L ∈ (n) \ 0 (R2n , L), then the path L(t) = γB (t)L crosses the manifold (n) \ 0 (R2n , L) transversally and in positive direction provided B(t) > 0 for t ∈ [0, 1].

6.2 Maslov Type Index for a Pair of Lagrangian Paths Let V = (R2n , ω) ˜ be a symplectic vector space. The set of its Lagrangian space is Lag(V ). A path of Lagrangian pair f is a continuous map: f : [a, b] → Lag(V ) × Lag(V )

6.2 Maslov Type Index for a Pair of Lagrangian Paths

165

for some interval [a, b], a < b. So f (t) = (L1 (t), L2 (t)) ∈ Lag(V ) × Lag(V ). The set of all paths of Lagrangian pair is denote by P (V ). Theorem 2.1 ([32]) There is a unique function μV : P (V ) → Z satisfying the following axioms. (1) (Affine Scale Invariance). For k > 0, l ≥ 0, we have the affine map ϕ : [a, b] → [ka + l, kb + l] defined by ϕ(t) = kt + l. For a given continuous path f : [ka + l, kb + l] → Lag(V ) × Lag(V ), there holds μV (f ) = μV (f ◦ ϕ).

(6.6)

(2) (Homotopy Invariance rel. End Points). If δ : [0, 1] × [a, b] → Lag(V ) × Lag(V ) is a continuous map with δ(0, t) = f1 (t), δ(1, t) = f2 (t), δ(s, a) = f1 (a) = f2 (a) and δ(s, b) = f1 (b) = f2 (b) for s ∈ [0, 1], then μV (f1 ) = μV (f2 ).

(6.7)

(3) (Path Additivity). If a < b < c, and f[a,c] : [a, c] → Lag(V ) × Lag(V ) is concatenate path of f[a,b] and f[b,c] , then there holds μV (f[a,c] ) = μV (f[a,b] ) + μV (f[b,c] ).

(6.8)

(4) (Symplectic Additivity). Let fk : [a, b] → Lag(Vk ) × Lag(Vk ), k = 1, 2, V = V1 ⊕ V2 f = f1 ⊕ f2 . Then we have μV (f ) = μV1 (f1 ) + μV2 (f2 ).

(6.9)

(5) (Symplectic Invarince) Let matrix P (t) ∈ Sp(2n) be a symplectic path. We define (P∗ f )(t) = (P (t)L1 (t), P (t)L2 (t)). Then we have μV (P∗ f ) = μV (f ).

(6.10)

(6) (Normalization). Let V = (R2 , ω˜ 0 ). Define f : [−ε, ε] → (1) × (1) with ε > 0 small by the formula f (t) = (R, R(cos t, sint)), t ∈ [−ε, ε]. Then (i) μV (f ) = 1; (ii) μV (f[−ε,0] ) = 0; (iii) μV (f[0,ε] ) = 1.

(6.11)

166

6 Maslov Type Index for Lagrangian Paths

Proof The existence is established in Definition V.1.27 and Theorem V.1.28. Now we follow the ideas of [32] to prove the uniqueness. We say that a path of Lagrangian pair f (t) = (L1 (t), L2 (t)), t ∈ [a, b] is proper if L1 (a) ∩ L2 (a) = {0} and L1 (b) ∩ L2 (b) = {0}. Suppose μV (f ) is the index for f (t) = (L1 (t), L2 (t)), a ≤ t ≤ b satisfying the ˆ ). By choosing a symplectic six axioms in Theorem 2.1. We prove that μV (f ) = i(f 2n basis of V , we can present the data as V = (R , ω˜ 0 ) with the standard symplectic matrix J as the complex structure. Taking L = Rn × {0}, we suppose L1 (t) = φ1 (t)L and L2 (t) = φ2 (t)L with φ1 (t), φ2 (t) ∈ Osp(2n) by (1.20) and the bijection φ(L) defined after that. Then by the axiom (5) μV (f ) = μV (f  ), f  (t) = (L, φ1−1 (t)φ2 (t)L). So we can in further  suppose  f (t) = (L, L2 (t)), a ≤ t ≤ b. Up to a symplectic AB matrix of the form , we can assume L ∩ L2 (a) = Rα and L2 (a) = Rα ⊕ 0 D J Rn−α . We introduce the path γ1 (t) = (Rn , eJ t Rα ⊕ J Rn−α ), t ∈ [0,

π ]. 4

Then we define the “tail” consists of first traveling along γ1 (t −(a − π2 )) for a − π2 ≤ t ≤ a − π4 and then along the reverse path for a − π4 ≤ t ≤ a. Denote by  the ˆ composite of the “tail” with the original path f (t) = (L, L2 (t)), a ≤ t ≤ b, and  the part of  from a − π4 to b. Then we have ˆ μV (f ) = μV () = μV (γ1 ) + μV (). ˆ is proper. Note that at the beginning point t = a − π4 the path  In a similar manner, at t = b, we may arrange L2 (b) = Rβ ⊕ J Rn−β , and consider the path γ2 (t) = (Rn , eJ t Rβ ⊕ J Rn−β ), 0 ≤ t ≤

π 4

˜ of  ˆ and γ2 ( π − t + b), b ≤ t ≤ b + π . Then we have and the composite path  4 4 ˜ μV (f ) = μV (γ1 ) + μV (γ2 ) + μV (), ˜ is proper. where  ˜ Next we modify (t) = (L, L˜ 2 (t)), a − π4 ≤ t ≤ b + π4 by continuously deforming L˜ 2 (t) to a smooth path which intersects L transversally. At these finite ˜ is locally isomorphic to one of the intersection points ti , we may assume that  following two types:

6.2 Maslov Type Index for a Pair of Lagrangian Paths

167

(1) γ + (t) = {Rn , eJ (t−ti ) R1 ⊕ Rn−1 , |t − ti | < δ}, (2) γ − (t) = {Rn , e−J (t−ti ) R1 ⊕ Rn−1 , |t − ti | < δ}. Outside these intervals |t − ti | < δ, the two Lagrangian paths have trivial intersection. We assume μV (γ + |[ti −δ,ti ] ) = y and μV (γ + |[ti ,ti +δ] ) = x. So μV (γ + ) = x + y, μV (γ − ) = −(x + y) and furthermore μV (γ1 ) = αx, ˜ transverse to L there are p intersection points μV (γ2 ) = βy. So if after making  of type (1) and q intersection points of type (2), then it yields the computation: μV (f ) = (p − q)(x + y) + αx + βy.

(6.12)

By using conditions in (6.11), we see that x = 1 and y = 0. So μV (f ) = p + q − α. ˆ we also have i(f ˆ ) = p +q −α. From the definition and the properties of the index i, ˆ Thus μV (f ) = i(f ) for any path f .   In [32] the authors defined four indices for paths of Lagrangian pair satisfying the six axioms in Theorem 2.1, two of them are geometrical ones μgeo,1 and μgeo,2 , the others are analytical ones μanal,1 and μanal,2 . In the following we introduce the first geometrical Maslov index μgeo,1 and simply denote it by μCLM (f ). We first define the Maslov index for a proper path of Lagrangian pair (cf. [127]). For a proper path, say h(t) = (Lˆ 1 (t), Lˆ 2 (t)), we will define the Maslov index μproper (h) ∈ Z. This index is to count with signs and multiplicities the number of times that Lˆ 1 (t) ∩ Lˆ 2 (t) = {0} at t ranges from t = a to t = b. More precisely, we first assume (Lˆ 1 (t), Lˆ 2 (t)) is a smooth path. Let Z be the subspace in [a, b]×Lag(V ) consisting of all pairs (t, L) which has the property Lˆ 1 (t) ∩ L = 0.

(6.13)

It is well known that Z ∩ ({t} × Lag(V )) is a codimension one subvariety of {t} × Lag(V ) and has singularities of codimension 3 in {t} × Lag(V ) (see Theorem I.3.3). The top stratum of Z ∩ ({t} × Lag(V )) has a canonical transverse orientation (see [10], and Lemma I.3.1). Indeed, if {t} × L is a point on this top stratum, then the path of Lagrangian {t} × eJ θ L crosses the stratum transversally as θ increases. It defines the desired transverse orientation. Hence for the path h(t) = (Lˆ 1 (t), Lˆ 2 (t)), a ≤ t ≤ b, we may, by a slight perturbation keeping the endpoints fixed, modify the oriented path γ = (t, Lˆ 2 (t)), a ≤ t ≤ b to a new path γ  intersecting Z only at points of top smooth stratum and crossing them transversely.

168

6 Maslov Type Index for Lagrangian Paths

Definition I Define μproper (h) as μproper (h) = (Z ∩ γ  ) in [a, b] × Lag(V ).

(6.14)

This intersection number is counted with signs. Since h(t) and so L1 (t) is assumed to be smooth, the union of strata . {top strata ofZ ∩ ({t} × Lag(V ))} a 0. Note that sgnD(t) remains constant.

6.3 Hörmander Index Theory

175

Finally we remark that γ (t) = {x + A(t)x|x ∈ L1 }, A(t) : L1 → L2 ˜ ˜ : γ (t0 ) → L2 . = {y + A(t)y| y ∈ γ (t0 )}, A(t) ˜ So A(t) = A(t0 ) + A(t)(I + A(t0 )) on L1 , hence d d ˜ ω(A(t)x, ˜ y)t=t0 = ω( ˜ A(t)x, y)t=t0 dt dt

(6.37)

if x, y ∈ γ (t0 ) ∩ L1 . So B  (t0 ) > 0 if and only if γ (t) crosses 1 (V , L1 ) in the positive direction.   Lemma 3.6 ([143]) sgn(L1 , L2 ; M) = −sgn(L1 , M; L2 ) if L1 , L2 , M are mutually transversal. Proof We can write M = {x + Ax| x ∈ L1 }, A : L1 → L2 , L2 = {y + By| y ∈ L1 }, B : L1 → M. If x ∈ L1 , then (x + Ax) + Bx = (x + Bx) + Ax belongs to both M and L2 , hence x + Ax + Bx = 0. So ω(Ax, ˜ y) = −ω(Bx, ˜ y), ∀ x, y ∈ L1 .   Lemma 3.7 ([143]) If L1 , L2 , M1 , M2 ∈ Lag(V ), Lj transverses to Mk for j, k = 1, 2, then s(L1 , L2 ; M1 , M2 ) =

1 [sgn(L1 , M2 ; L2 ) − sgn(L1 , M1 ; L2 )]. 2

(6.38)

Proof Note that the right hand side is well defined and (6.38) holds if L1 , L2 are transversal in view of Lemmas 3.5 and 3.6. Now assume dim L1 ∩ L2 = k. Let γ (t) be a curve in Lag(V ) such that γ (0) = L2 and γ  (0), regarded as a quadratic form on L2 , is positive definite on L1 ∩ L2 . Then, using formulas analogous to (6.36) and (6.37) we obtain that γ (t) is transversal to L1 , sgn(L1 , M2 ; γ (t)) = sgn(L1 , M2 ; L2 ) + k, sgn(L1 , M1 ; γ (t)) = sgn(L1 , M2 ; L2 ) + k

176

6 Maslov Type Index for Lagrangian Paths

if t > 0 is small. This proves (6.38) because it holds when L2 is replaced by γ (t), t > 0, small, and the left-hand side is locally constant in L2 .   For any four Lagrangian subspaces L1 , L2 , M1 , M2 , there is an ε > 0 such that Lj is transversal to eJ θ Mk , 0 < |θ | < ε, j, k = 1, 2. Definition 3.8 For L1 , L2 , M1 , M2 ∈ Lag(V ), we define the Hörmander index as s(L1 , L2 ; M1 , M2 ) = s(L1 , L2 ; e−J θ M1 , e−J θ M2 ), 0 < θ < ε,

(6.39)

the right hand side of (6.39) is defined in Definition 3.2. It is easy to see that the Hörmander index s(L1 , L2 ; M1 , M2 ) is well-defined, i.e., it is independent of θ ∈ (0, ε) and the Eqs. (6.31), (6.32), (6.33) and (6.34) hold in this case.

Chapter 7

Revisit of Maslov Type Index for Symplectic Paths

7.1 Maslov Type Index for Symplectic Paths We recall that (R2n , ω) ˜ is a symplectic space, and Sp(2n, ω) ˜ is the symplectic group ˜ That is of (R2n , ω). Sp(2n, ω) ˜ = {M ∈ L(R2n )| M ∗ ω˜ = ω}. ˜ We denote by P(2n, ω) ˜ = {γ ∈ C([0, 1], Sp(2n, ω))| ˜ γ (0) = I } the set of continuous and piecewise smooth symplectic paths starting from I and (n, ω) ˜ the ˜ We recall that P(2n) = P(2n, ω˜ 0 ). We set of Lagrangian subspaces of (R2n , ω). ˜ also denote by P(2n, ω) ˜ = {γ | γ ∈ C([a, b], Sp(2n, ω))} ˜ the set of continuous and piecewise smooth symplectic paths. ˜ Suppose L ∈ (n, ω). ˜ For γ ∈ P(2n, ω), ˜ we now define the L-index μL (γ ) of γ . Definition 1.1 We define μL (γ ) = μCLM (f ), f (t) = (L, γ (t)L), 0 ≤ t ≤ 1.

(7.1)

From Lemma VI.1.5 and Remark VI.1.6, we see that (compare with Corollary V.6.5) μL (γB ) =



dim(γB (t)L ∩ L)

(7.2)

t∈[0,1)

for B(t) > 0, t ∈ [0, 1], and specially, we have μL (γB ) ≥ n. We note that the symplectic structure ω˜ may not be the standard one. We will unify various indices as spacial cases of μˆ Lˆ (γ ) as defined in (7.3). For this purpose, we consider the linear symplectic spaces (R2n ⊕ R2n , −ω˜ 0 ⊕ ω˜ 0 ) with −ω˜ 0 ⊕ © Springer Nature Singapore Pte Ltd. 2019 C. Liu, Index theory in nonlinear analysis, https://doi.org/10.1007/978-981-13-7287-2_7

177

178

7 Revisit of Maslov Type Index for Symplectic Paths

ω˜ 0 ((x1 , y1 ), (x2 , y2 )) = −ω˜ 0 (x1 , x2 ) + ω˜ 0 (y1 , y2 ) for all (xj , yj ) ∈ R2n ⊕ R2n , j = 1, 2. For a symplectic matrix M ∈ Sp(2n), the graph Gr(M) of M : R2n → R2n is defined by Gr(M) = {(x, Mx)| x ∈ R2n }. It is obvious that Gr(M) is a Lagrangian subspace of V = (R2n ⊕ R2n , −ω˜ 0 ⊕ ω˜ 0 ). Let Lˆ ∈ Lag(V ), and γ : [a, b] → Sp(2n) be a continuous symplectic paths. Definition 1.2 We define ˆ Gr(γ (t)), a ≤ t ≤ b. μˆ Lˆ (γ ) = μCLM (f ), f (t) = (L,

(7.3)

We have the following four spacial cases: (1) (2) (3) (4)

Lˆ = L ⊕ L for L ∈ (n), we have the L-index: μˆ L (γ ).  Lˆ = L ⊕ L for L, L ∈ (n), we have the (L, L )-index: μˆ L L (γ ). Lˆ = Gr(P ) for P ∈ Sp(2n), we have the P -index: μˆ P (γ ). Lˆ = Gr(I ), I is the 2n × 2n identity matrix, we have the I -index: μˆ I (γ ).

˜ We recall that the L-index iˆL (γ ), γ ∈ P(2n), is defined in (5.57), the index i P (γ ) is defined in Definition IV.1.7, μP (γ ) is defined in Definition V.3.1, and iLL0 (γ ) defined in Definition V.2.6, μL L0 (γ ) defined in Definition V.2.4. The following result shows that indices defined in different manner in fact are the same. Theorem 1.3 We have the following equalities (1) (2) (3) (4)

˜ iˆL (γ ) = μˆ L (γ ) for γ ∈ P(2n); iP (γ ) = μP (γ ) = μˆ P (γ ), γ ∈ P(2n), P ∈ Sp(2n); L iLL0 (γ ) − iLL0 (I ) = μL ˆL L0 (γ ) − μL0 (I ) = μ L0 (γ ), γ ∈ P(2n); i1 (γ ) + n = iI (γ ) = μˆ I (γ ), γ ∈ P(2n).

Proof By Theorem VI.2.1, we only need to prove the various indices satisfy the axioms (1)–(6). The equality in (1) above is a consequence of Theorem V.1.24. The equality (2) is a consequence of Theorem IV.1.21. And equality (3) can be proved in the same manner. Equality (4) is a special case of equality (2).   Let B ∈ C([0, 1], Ls (R2n )), and γB : [0, 1] → Sp(2n) be the fundamental solution of the following linear Hamiltonian system y(t) ˙ = J B(t)y(t). We take a Lagrangian subspace Vω = L0 ⊕ (eθJ L0 ) for ω = e

√

−1θ ,

θ ∈ [0, π ),.

0 Definition 1.4 We define the ω-index μL ω (B) by

CLM 0 (Vω , Gr(γB )). μL ω (B) = μ

(7.4)

Lemma 1.5 There holds 0 iωL0 (B) = μL ω (B), ω = e

√

−1θ

, θ ∈ [0, π ).

Proof It is a direct consequence of Theorem 1.3(3).

(7.5)  

7.2 The ω-Index Function for P -Index

179

7.2 The ω-Index Function for P -Index Definition 2.1 A complex symplectic space V = C2n is a complex vector space, together with a prescribed symplectic form , namely a sesqui-linear (or conjugate bilinear) complex-valued function  : V × V → C u, v → (u, v)

(7.6)

satisfying the following axioms, for all u, v, w ∈ V and all complex numbers c1 , c2 ∈ C, (1) (c1 u + c2 v, w) = c1 (u, w) + c2 (v, w) (linearity property in first argument), (2) (u, v) = −(v, u) (skew-Hermitian property), (3) (u, V ) implies u = 0 (non-degeneracy property). Properties (1) and (2) of Definition 2.1 imply (u, c1 v + c2 w) = c¯1 (u, v) + c¯2 (u, w). The subspace L with maximum dimension is called Lagrangian subspace if |L = 0. As an example, we complexification the symplectic space (R2n , ω˜ 0 , J ) as 2n (C , ω˜ 0 , J ). In (R2n , ω˜ 0 , J ), we have ω˜ 0 (u, v) = J u, v = v T J u where ·, ·

is the standard inner product. In (C2n , ω˜ 0 , J ), we have ω˜ 0 (u, v) = (J u, v) = v¯ T J u where (·, ·) is the standard Hermitian inner product. We know that (C2n , ω˜ 0 , J ) is a complex symplectic vector space. The linear bijection M : C2n → C2n satisfying ω˜ 0 (Mu, Mv) = ω˜ 0 (u, v) for all u, v ∈ C2n is called complex symplectic map, the corresponding matrix M satisfies M ∗ J M = J , where M ∗ is the conjugate transpose of M, which is called complex symplectic matrix. We denote by Sp(2n, C) the set of all complex symplectic matrices. It is a Lie group and Sp(2n) = Sp(2n, R) is √ its a Lie subgroup. For any M ∈ Sp(2n, C), we have e −1θ M ∈ Sp(2n, C). The dimension n subspace L of C2n is called Lagrangian subspace if ω˜ 0 (x, y) = 0 for all x, y ∈ L. We note that the complexification of any Lagrangian subspace L of (R2n , ω˜ 0 , J ) is a Lagrangian subspace.   −J 0 We now define (C2n × C2n , , J ) with  = −ω˜ 0 ⊕ ω˜ 0 and J = . For 0 J any M ∈ Sp(2n, C), the graph Gr(M) = {(x, Mx)| x ∈ C2n } of M : C2n → C2n is a Lagrangian subspace of (C2n × C2n , , J ). For P ∈ Sp(2n, R), ω =eθJ and γ ∈ P(2n, R), we  have two Lagrangian x x subspaces Gr(ωP ) = { : x ∈ C2n } and Gr(γ ) = { : x ∈ C2n , t ∈ ωP x γ (t)x [0, 1]}. Definition 2.2 For P ∈ Sp(2n, R), ω = eθJ and γ ∈ P(2n, R), we define (Gr(ωP ), Gr(γ )). μPω (γ ) = μCLM C

(7.7)

180

7 Revisit of Maslov Type Index for Symplectic Paths

Theorem 2.3 For P ∈ Sp(2n, R), ω = eθJ and γ ∈ P(2n, R), we have iωP (γ ) = μPω (γ ).

(7.8)

Proof The proof is the same as uniqueness arguments as before, we omit it here.

 

7.3 The Concavity of Symplectic Paths and (ε, L0 , L1 )-Signature From Theorem V.1.21, we know that the concavity defined by concavL (M) := concavL (γ ) = i1 (γ ) − iL (γ ) depends only on the end matrix M = γ (1) and the Lagrangian subspace L. Definition 3.1 ([189, 209]) For two Lagrangian subspaces L, L ∈ (n), we define the (L, L )-concavity of γ by concav(L,L ) (γ ) = iL (γ ) − iL (γ ), γ ∈ P(2n),

(7.9)

and the ∗-(L, L )-concavity of γ by concav∗(L,L ) (γ ) = iL (γ ) + νL (γ ) − iL (γ ) − νL (γ ), γ ∈ P(2n).

(7.10)

Theorem 3.2 The concavity concav(L,L ) (γ ) depends only on the end matrix M = γ (1). So we denote it by concav(L,L ) (M) and ˜ concav0(L,L ) (γ ) = iˆL (γ ) − iˆL (γ ), γ ∈ P(2n)

(7.11)

depends only on the end matrices M1 = γ (0) and M2 = γ (1). So we denote it by concav0(L,L ) (M1 , M2 ). Proof From Theorem V.1.21, we see that iL (γ ) − iL (γ ) = concavL (γ ) − concavL (γ ) depends only on the end matrix M = γ (1). And iˆL (γ ) − iˆL (γ ) = iL (γ ) − iL (γ ) − iL (I ) + iL (I ) depends only on the end matrix M = γ (1) for γ ∈ P(2n). Now by definition, if ˜ γ ∈ P(2n), we have γ0 and γ1 ∈ P(2n) such that γ1 = γ ∗ γ0 . Thus

7.3 The Concavity of Symplectic Paths and (ε, L0 , L1 )-Signature

181

iˆL (γ ) − iˆL (γ ) = iL (γ1 ) − iL (γ1 ) − iL (γ0 ) + iL (γ0 ) depends only on M1 = γ0 (1) = γ (0) and M2 = γ1 (1) = γ (1).

 

From Remark VI.3.3(3), Definition VI.3.8, Theorem 1.3 and Theorem 3.2, we see that concav0(L,L ) (M1 , M2 ) = s(L ⊕ L, L ⊕ L ; Gr(M1 ), Gr(M2 )).

(7.12)

As a special case, for γ ∈ P(2n), there holds iL (γ ) − iL (γ ) = s(L ⊕ L, L ⊕ L ; Gr(I ), Gr(M)) + iL (I ) − iL (I ).

(7.13)

If L0 = {0} × Rn and L1 = Rn × {0}, then we have iL1 (γ ) − iL0 (γ ) = s(L0 ⊕ L0 , L1 ⊕ L1 ; Gr(I ), Gr(M))

(7.14)

since there hold iL0 (I ) = iL1 (I ) = n in this case. We further have the following results. Theorem 3.3 For γ ∈ P(2n) with M = γ (1) ∈ Osp(2n), there holds concav(L0 ,L1 ) (M) = s(L0 ⊕ L0 , L1 ⊕ L1 ; Gr(I ), Gr(M)) = 0.

(7.15)

Proof Since M = γ (1) ∈ Osp(2n) can be written as M = eP with J P = P J . So J M = MJ and by choosing γ (t) = eP t , we obtain the result (7.15) from Definition III.1.7 of iL1 (γ ) with L1 = J L0 .     −I 0 For any P ∈ Sp(2n) and ε ∈ R, we set N = and 0 I    0 I εJ 0 I −εJ P e − P T eεJ e I 0 I 0 = P T J e2εJ NP − J e−2εJ N     sin 2ε I − cos 2ε I sin 2ε I cos 2ε I = PT P+ . − cos 2ε I − sin 2ε I cos 2ε I − sin 2ε I

Mε (P ) = e−εJ



(7.16)

Theorem 3.4 ([307]) For γ ∈ P(2n), there holds concav(L0 ,L1 ) (γ (1)) =

1 sgnMε (γ (1)) 2

(7.17)

for ε > 0 being sufficiently small. Moreover, there holds concav∗(L0 ,L1 ) (γ (1)) =

1 sgnMε (γ (1)) 2

(7.18)

182

7 Revisit of Maslov Type Index for Symplectic Paths

for ε < 0 and |ε| being sufficiently small. We call sgnMε (A) the of the symplectic matrix A. Proof We set V0 = L0 ⊕ L0 and V1 = L1 ⊕ L1 . By definition and above arguments, we have ˜

˜

iL1 (γ ) − iL0 (γ ) = s(V0 , V1 ; Gr(I ), Gr(M)) = s(V0 , V1 ; e−J ε Gr(I ), e−J ε Gr(M)),   −J 0 ˜ here J = −J ⊕ J = and M = γ (1). By (6.38), we have 0 J ˜

˜

s(V0 , V1 ; e−J ε Gr(I ), e−J ε Gr(M)) =

1 ˜ ˜ [sgn(V0 , e−J ε Gr(M); V1 ) − sgn(V0 , e−J ε Gr(I ); V1 )]. 2

(7.19)

Claim 1. For ε > 0 small enough, ˜

sgn(V0 , e−J ε Gr(I ); V1 ) = 0.

(7.20)

In fact, ˜

e−J ε Gr(I ) =

 eεJ x 2n . ; x ∈ R e−εJ x



˜

−J ε Gr(I ). Suppose V1 = (Rn ×{0})×(Rn ×{0}) = {y+Ay|  εJ y∈ V0 }, A: V0 → e e x p ,x = If y = (0, a, 0, b)T ∈ V0 satisfies Ay = , then we have e−εJ x q a+b p = 2b−a sin ε and q = − 2 cos ε . So there holds

β β α α Ay = ( a + b, −a, − a − b, −b), 2 2 2 2 where α =

sin ε cos ε

−

cos ε sin ε ,

β=

sin ε cos ε

+

cos ε sin ε .

Setting  = −ω˜ 0 ⊕ ω˜ 0 , then we have

0 / α α β β α (Ay, y) = − a + b, a − − a − b, b = − (xj2 + yj2 ) − βxj yj , 2 2 2 2 2 n

j =1

where a = (x1 , · · · , xn ), b = (y1 , · · · , yn ). Now it is easy to see that α < 0, β > 0 and the signature of the above quadratic form is zero. Thus we complete the proof of Claim 1. Claim 2. For ε > 0 small enough, ˜

sgn(V0 , e−J ε Gr(γ (1)); V1 ) = sgn(Mε (γ (1))).

(7.21)

7.3 The Concavity of Symplectic Paths and (ε, L0 , L1 )-Signature

183

   x p ˜ 2n and e−J ε Gr(γ (1)) = ;x = ∈R γ (1)x q

 Proof of Claim 2. Gr(γ (1)) =

  eεJ x 2n ;x ∈ R . e−εJ γ (1)x

˜

−J ε Gr(γ (1)). If y = (0, a, 0, b)T ∈ Suppose V1 = {y+Ay|  εJ y∈ V0 }, A : V0 → e   e x −P2 eεJ x 2n V0 satisfies Ay = , x ∈ R , then y = , where P2 is the e−εJ x −P2 e−εJ x   00 projection P2 : R2n → L0 , i.e., P2 = . So A is a bijection and the quadratic 0I form (Ay, y) is non-degenerate.

   −P2 eεJ x eεJ x , (Ay, y) =  e−εJ x −P2 e−εJ x εJ εJ = ω˜ 0 (e x, P2 e x) − ω˜ 0 (e−εJ γ (1)x, P2 e−εJ γ (1)x). 

Since ω˜ 0 (u, v) = v T J u, we have (Ay, y) = x T e−εJ P2T J eεJ x − x T γ (1)T eεJ P2T J e−εJ γ (1)x = x T M˜ ε (γ (1))x = 12 x T [M˜ ε (γ (1)) + M˜ ε (γ (1))T ]x,   sin ε cos εI − sin2 εI ˜ where Mε (γ (1)) = cos2 εI − sin ε cos εI   sin ε cos εI sin2 εI γ (1). We see that M˜ ε (γ (1)) + M˜ ε (γ (1))T = + γ (1)T − cos2 εI − sin ε cos εI Mε (γ (1)). Equation (7.17) follows from (7.19), (7.20) and (7.21). For ε < 0, |ε| small enough, it is clear that ˜

iL0 (γ ) = μCLM (V0 , e−εJ Gr(γ )) − νL0 (γ ), ˜

iL1 (γ ) = μCLM (V1 , e−εJ Gr(γ )) − νL1 (γ ). By the same arguments as above, we see that (7.18) holds.

 

If P ∈ Osp(2n), then J P = P J , P T J = J P T , and so we have J Mε (P )J T = −Mε (P ). Thus we have sgn(Mε (P )) = sgn(J Mε (P )J T ) = −sgn(Mε (P )), it implies sgn(Mε (P )) = 0 and give a new proof of Theorem 3.3.

184

7 Revisit of Maslov Type Index for Symplectic Paths

Remark 3.5 We list some basic facts about the signature of Mε (P ) and leave the proofs to readers as exercises. (1) sgn(Mε (P (s))) is constant for continuous path P : [0, 1] → Sp(2n) with νLj (P (s)) = constant for j = 0, 1 (cf. Lemma 2.2 of [307] for a proof); (2) We have Mε (P1  P2 ) = Mε (P1 )  Mε (P2 ), sgnMε (P1  P2 ) = sgnMε (P1 ) + sgnMε (P2 ) :         1b cos θ − sin θ 1 0 a 0 , ± or ± , , 01 sin θ cos θ −b 1 0 a −1 θ ∈ [0, 2π ], a = 0, b > 0;     1 −b 2 −1 (4) sgn(Mε (P ) = 2 for P = ± or ± , b > 0; 0 1 −1 1   10 , b > 0. (5) sgn(Mε (P ) = −2 for P = ± b1

(3) sgn(Mε (P ) = 0 for P =

Definition 3.6 ([308]) We call two symplectic matrices M1 and M2 in Sp(2n) are s special homotopic(or (L0 , L1 )-homotopic) and denote by M1 ∼ M2 , if there are Pj ∈ Sp(2n) with Pj = diag(Qj , (QTj )−1 ), where Qj is a n × n invertible real matrix, and det(Qj ) > 0 for j = 1, 2, such that M1 = P1 M2 P2 . s

It is clear that ∼ is an equivalent relation.   s Ai Bi ∈ Sp(2ki ), i = 0, 1, 2, k1 = k2 and M1 ∼ Remark 3.7 Let Mi = Ci D i M2 (in this time), then AT1 C1 , B1T D1 are congruent to AT2 C2 , B2T D2 respectively. So m∗ (AT1 C1 ) = m∗ (AT2 C2 ) and m∗ (B1T D1 ) = m∗ (B2T D2 ) for ∗ = ±, 0. Furthermore, if M0 = M1  M2 (here k1 = k2 is not necessary), then m∗ (AT0 C0 ) = m∗ (AT1 C1 ) + m∗ (AT2 C2 ), m∗ (B0T D0 ) = m∗ (B1T D1 ) + m∗ (B2T D2 ).

(7.22)

So m∗ (AT C) and m∗ (B T D) are (L0 , L1 )-homotopic invariant. The following formula will be used frequently Nk M1−1 Nk M1 = I2k + 2



B1T C1 B1T D1 AT1 C1 C1T B1

 .

(7.23)

We recall that 0 (M) is the path connected component of (M) containing M (cf. Definition I.5.2).

7.3 The Concavity of Symplectic Paths and (ε, L0 , L1 )-Signature

185

Definition 3.8 For any M1 ,M2 ∈ Sp(2n), we call M1 ≈ M2 if M1 ∈ 0 (M2 ). Remark 3.9 It is easy to check that ≈ is an equivalent relation. If M1 ≈ M2 , we have M1k ≈ M2k for any k ∈ N and M1  M3 ≈ M2  M4 for M3 ≈ M4 . Also we have P MP −1 ≈ M for any P , M ∈ Sp(2n). Lemma 3.10 Assume M1 ∈ Sp(2(k1 + k2 )) , M2 ∈ Sp(2k3 ) and M3 ∈ Sp(2(k1 + k2 + k3 )) have the following block forms ⎛

⎛

A1 A2 ⎜ A3 A4 M1 = ⎜ ⎝ C1 C2 C3 C4

0 A5 0 0 C5 0

A1 ⎞ ⎜ 0 B1 B2 ⎜   ⎜ B3 B4 ⎟ B A ⎜ A3 5 5 ⎟ , M2 = , M = ⎜ 3 ⎜ C1 D1 D2 ⎠ C5 D 5 ⎜ ⎝ 0 D3 D4 C3

A2 0 A4 C2 0 C4

B1 0 B3 D1 0 D3

0 B5 0 0 D5 0

⎞ B2 0 ⎟ ⎟ ⎟ B4 ⎟ ⎟ D2 ⎟ ⎟ 0 ⎠ D4

with A1 , B1 , C1 , D1 ∈ L(Rk1 ), A4 , B4 , C4 , D4 ∈ L(Rk2 ), A5 , D5 ∈ L(Rk3 ). Then M3 ≈ M1  M 2 .

(7.24)

⎛⎛

⎞ ⎛ ⎞⎞ Ik1 0 0 Ik1 0 0 Proof Let P = diag ⎝⎝ 0 0 Ik2 ⎠ , ⎝ 0 0 Ik2 ⎠⎠. It is east to verify that 0 Ik3 0 0 Ik3 0 P ∈ Sp(2(k1 + k2 + k3 )) and M3 = P (M1  M2 )P −1 . Then (7.24) holds from Remark 3.9 and the proof of Lemma 3.10 is completed.     Ik 0 . Then Lemma 3.11 ([308]) Let k ∈ N and any symplectic matrix P = C Ik p P ≈ I2  N1 (1, 1)q  N1 (1, −1)r with p = m0 (C), q = m− (C), r = m+ (C). Proof It is clear that  P ≈

Ik 0 B Ik

 ,

where B = diag(0, −Im− (C) , Im+ (C) ). Since J1 N1 (1, ±1)(J1   1 0 Remark 3.9 we have N1 (1, ±1) ≈ . Then ∓1 1 P ≈ I2m

0 (C)

− (C)

 N1 (1, 1)m

By Lemma II.3.14 we have

+ (C)

 N1 (1, −1)m



)−1

=

.

 1 0 , by ∓1 1

186

7 Revisit of Maslov Type Index for Symplectic Paths

SP+ (1) = m0 (C) + m− (C) = p + q.

(7.25)

By the definition of the relation ≈, we have 2p + q + r = ν1 (P ) = 2m0 (C) + m+ (C) + m− (C).

(7.26)

Also we have p + q + r = m0 (C) + m+ (C) + m− (C) = k.

(7.27)

By (7.25)–(7.27) we have m0 (C) = p,

m− (C) = q,

m+ (C) = r.  

The proof of Lemma 3.11 is complete. s Lemma 3.12 ([308]) For M˜ 1 , M˜ 2 ∈ Sp(2n), if M˜ 1 ∼ M˜ 2 , then

sgnMε (M˜ 1 ) = sgnMε (M˜ 2 ),

0 ≤ |ε| * 1,

(7.28)

N M˜ 1−1 N M˜ 1 ≈ N M˜ 2−1 N M˜ 2 .

(7.29)

Proof By Definition 3.6, there are Pj ∈ Sp(2n) with Pj = diag(Qj , (QTj )−1 ), Qj being n × n invertible real matrix, and det(Qj ) > 0 such that M˜ 1 = P1 M˜ 2 P2 . Since det(Qj ) > 0 for j = 1, 2, we can joint Qj to In by invertible matrix path. Hence we can joint P1 M˜ 2 P2 to M˜ 2 by symplectic path preserving the nullity νL0 and νL1 . By the basic fact (1) in Remark 3.5, (7.28) holds. Since Pj N = N Pj for j = 1, 2. Direct computation shows that N M˜ 1−1 N M˜ 1 = N(P1 M˜ 2 P2 )−1 N(P1 M˜ 2 P2 ) = P2−1 N M˜ 2−1 N M˜ 2 P2 .

(7.30)

Thus (7.29) holds by Remark 3.9. The proof of Lemma 3.12 is complete.     AB Lemma 3.13 ([308]) Let P = ∈ Sp(2n), where A, B, C, D are all n × n CD matrices. Then ≤ n−νL0 (P ), for 0 < ε * 1. If B = 0, we have 12 sgnMε (P ) ≤ 0 for 0 < ε * 1. (ii) Let m+ (AT C) = q, we have (i)

1 2 sgnMε (P )

1 sgnMε (P ) ≤ n − q, 2

0 ≤ |ε| * 1.

(7.31)

7.3 The Concavity of Symplectic Paths and (ε, L0 , L1 )-Signature

187

≥ dim ker C − n for 0 < ε * 1, If C = 0, then 12 sgnMε (P ) ≥ 0 for 0 < ε * 1 (iv) If both B and C are invertible, we have

(iii)

1 2 sgnMε (P )

sgnMε (P ) = sgnM0 (P ),

0 ≤ |ε| * 1.

Proof Since P is symplectic, so is for P T . From P T J P = J and P J P T = J we get AT C, B T D, AB T , CD T are all symmetric matrices and AD T − BC T = In ,

AT D − C T B = In .

(7.32)

We denote by s = sin 2ε and c = cos 2ε. By definition of Mε (P ), we have  Mε (P ) =  =

AT C T B T DT



sIk −cIk −cIk −sIk



AB CD



 +

sIk cIk cIk −sIk



sAT B − 2cC T B − sC T D sAT A − 2cAT C − sC T C + sIk T T T T sB B − 2cB T D − sD T D − sIk sB A − 2cB C − sD C

 .

So 

AT C C T B M0 (P ) = −2 BT C BT D





CT 0 = −2 0 BT



AB CD

 ,

where we have used AT C is symmetric. So if both B and C are invertible, M0 (P ) is invertible and symmetric, its signature is invariant under small perturbation, so (iv) holds. If νL0 (P ) = dim ker B > 0, since B T D = D T B, for any x ∈ ker B ⊆ Rn , x = 0, and 0 < ε * 1, we have     0 0 Mε (P ) · = (sB T B − 2cD T B − sD T D − sIn )x · x x x = −s(D T D + In )x · x < 0.

(7.33)

So Mε (P ) is negative definite on (0⊕ker B) ⊆ R2n . Hence m− (Mε (p) ≥ dim ker B which yields that 12 sgnMε (P ) ≤ n − dim ker B = n − νL0 (P ), for 0 < ε * 1. Thus (i) holds. Similarly we can prove (iii). If m+ (AT C) = q > 0, let AT C is positive definite on E ⊆ Rn , then for 0 ≤ |s| * 1, similar to (7.33) we have Mε (P ) is negative on E ⊕ 0 ⊆ R2n . Hence m− (Mε (P ) ≥ q, which yields (7.31).   Lemma 3.14  ([309]) Let 2k ×2k symmetric real matrix E have the following block 0 E1 form E = . Then E1T E2

188

7 Revisit of Maslov Type Index for Symplectic Paths

m± (E) ≥ rankE1 .

(7.34)

Proof We only need to prove Lemma 3.14 in the case rankE1 > 0. Suppose rankE1 > 0, there is a linear subspace F of Rk with dim F = rankE1 and a number δ > 0 such that for any y ∈ F with |y| = 1 there holds |E1 y| ≥ δ.

(7.35)

So for 0 < ε * 1 and any y ∈ F with |y| = 1 we have  E

E1 y εy

   E1 y · = 2εE1 y · E1 y + ε2 E2 y · y = ε(2|E1 y|2 + εE2 y · y) εy ≥ ε(2δ 2 − ε||E2 ||) > 0.

(7.36)

E2 on Rk . So E is positive definite on the where ||E2 || is the operator norm of  E1 y |y ∈ F } of R2k . Hence we have rankE1 -dimensional linear subspace εy m+ (E) ≥ rankE1 . By the same argument we have m− (E) = m+ (−E) ≥ rank(−E1 ) = rankE1 . The proof of Lemma 3.14 is complete.   Lemma 3.15 ([210]) Let A1 and A3 be k × k real matrices. Assume both A1 and A1 A3 are symmetric and σ (A3 ) ⊂ (−∞, 0). Then sgnA1 + sgn(A1 A3 ) = 0.

(7.37)

Proof It is clear that A3 is invertible. We prove Lemma 3.15 in the following two steps. Step 1. We assume that A1 is invertible and proceed by induction on k ∈ N. If k = 1, then A1 , A3 ∈ R and (7.37) holds obviously. Now assume (7.37) holds for 1 ≤ k ≤ l. If we can prove (7.37) for k = l + 1, then by the mathematical induction (7.37) holds for any k ∈ N and Lemma 3.15 is proved in the case A1 is invertible. In view of the real Jordan canonical form decomposition of A3 , we only need to prove (7.37) for k = l + 1 in the following Case 1 and Case 2. Case 1. There is an invertible (l + 1) × (l + 1) real matrix such that Q−1 A3 Q is the ⎞ ⎛ λ 1 0 ··· 0 0 ⎜0 λ 1 ··· 0 0⎟ ⎟ ⎜ ⎟ ⎜ (l + 1)-order Jordan form ⎜ ... ... ... · · · ... 0 ⎟ := A˜ 3 with λ < 0. ⎟ ⎜ ⎝0 0 0 ··· λ 1⎠ 0 0 0 ··· 0 λ Denoting by A˜ 1 = QT A1 Q. We have A˜ 1 A˜ 3 = QT A1 Q Q−1 A3 Q = QT A1 A3 Q.

7.3 The Concavity of Symplectic Paths and (ε, L0 , L1 )-Signature

189

Hence both matrices A˜ 1 and A˜ 1 A˜ 3 are symmetric and sgnA1 + sgn(A1 A3 ) = sgnA˜ 1 + sgn(A˜ 1 A˜ 3 ).

(7.38)

Since A˜ 1 = (ai,j )1≤i,j ≤l+1 and A˜ 1 A˜ 3 = (ci,j )1≤i,j ≤l+1 are symmetric, ai,j = aj.i and ci,j = cj,i for 1 ≤ i, j ≤ l + 1. Claim C.1. ai,j = 0 for i + j ≤ l + 1 and ai,j = al+1,1 for i + j = l + 2 with 1 ≤ i, j ≤ l + 1. For 2 ≤ j ≤ l + 1, since c1,j = cj,1 , λa1,j + a1,j −1 = λaj,1 = λa1,j . Thus a1,j −1 = 0,

2 ≤ j ≤ l + 1.

(7.39)

For 2 ≤ i, j ≤ l + 1, since ci,j = cj,i we have λai,j + ai,j −1 = λaj,i + aj,i−1 = λai,j + ai−1,j . So ai,j −1 = ai−1,j ,

2 ≤ i, j ≤ l + 1.

(7.40)

By (7.39) and (7.40) we have ai,j = ai−1,j +1 = · · · = a2,i+j −2 = a1,i+j −1 = 0, 1 ≤ i, j and i + j ≤ l + 1,

(7.41)

al+1,1 = al,2 = al−1,3 = · · · = a2,l = a1,l+1 .

(7.42)

Hence, by (7.41) and (7.42), Claim C.1 is proved. By Claim C.1, let a = a1,l+1 , then ⎛

0 ⎜0 ⎜ ⎜0 ⎜ ⎜ A˜ 1 = ⎜ 0 ⎜ ⎜0 ⎜ ⎝0 a

0 0 0 0 0 a ∗

0 0 0 0 · ∗ ∗

0 0 0 · ∗ ∗ ∗

0 0 · ∗ ∗ ∗ ∗

0 a ∗ ∗ ∗ ∗ ∗

⎞ a ∗⎟ ⎟ ∗⎟ ⎟ ⎟ ∗⎟, ⎟ ∗⎟ ⎟ ∗⎠ ∗

⎛

0 ⎜ 0 ⎜ ⎜ 0 ⎜ ⎜ A˜ 1 A˜ 3 = ⎜ 0 ⎜ ⎜ 0 ⎜ ⎝ 0 λa

0 0 0 0 0 λa ∗

000 0 0 0 0 λa 00 · ∗ 0 · ∗ ∗ · ∗∗ ∗ ∗∗∗ ∗ ∗∗∗ ∗

It is easy to see that A˜ 1 A˜ 3 is congruent to λA˜ 1 . Since λ < 0,

⎞ λa ∗ ⎟ ⎟ ∗ ⎟ ⎟ ⎟ ∗ ⎟. ⎟ ∗ ⎟ ⎟ ∗ ⎠ ∗

(7.43)

190

7 Revisit of Maslov Type Index for Symplectic Paths

sgn(A˜ 1 A˜ 3 ) = sgn(λA˜ 1 ) = −sgn(A˜ 1 ), sgn(A˜ 1 A˜ 3 ) + sgnA˜ 1 = 0.

(7.44)

Equations (7.38) and (7.44) imply (7.37). Hence Step 1 is proved in Case 1. Case 2. There exists an invertible (l+1)×(l+1) real matrix Q such that Q−1 A3 Q = diag(A4 , A5 ), where A4 is a k1 × k1 real matrix with σ (A4 ) ⊂ (−∞, 0) and A5 is a k2 -order Jordan form ⎛

⎞ 00 0 0⎟ ⎟ .. ⎟ . 0⎟ ⎟ 0 ··· λ 1⎠ 0 0 0 ··· 0 λ

λ1 ⎜0 λ ⎜ ⎜ A5 = ⎜ ... ... ⎜ ⎝0 0

0 ··· 1 ··· .. . ···

with λ < 0, 1 ≤ k1 , k2 ≤ l and k1 + k2 = l + 1. We still denote by A˜ 1 = QT A1 Q, then A˜ 1 A˜ 3 = QT A1 Q Q−1 A3 Q = QT A1 A3 Q. So both A˜ 1 and A˜ 1 A˜ 3 are symmetric and sgnA1 + sgn(A1 A3 ) = sgnA˜ 1 + sgn(A˜ 1 A˜ 3 ).

(7.45)

Correspondingly we can write A˜ 1 in the block form decomposition A˜ 1 =   E1 E2 , where E1 is a k1 × k1 real symmetric matric and E4 is a k2 × k2 E2T E4 real symmetric matrix. Then A˜ 1 A˜ 3 =



E1 A4 E2 A5 E2T A4 E4 A5



is symmetric. Subcase 1. E4 is invertible. In this case we have     0 Ik1 E1 E2 Ik1 −E2 E4−1 0 Ik2 E2T E4 −E4−1 E2T Ik2   E1 − E2 E4−1 E2T 0 = 0 E4

(7.46)

7.3 The Concavity of Symplectic Paths and (ε, L0 , L1 )-Signature

191

and   0 E1 A4 E2 A5 Ik1 E2T A4 E4 A5 −E4−1 E2T Ik2   E1 A4 − E2 E4−1 E2T A4 0 = 0 E4 A5   (E1 − E2 E4−1 E2T )A4 0 . = 0 E4 A5 

Ik1 −E2 E4−1 0 Ik2



(7.47)

Since the matrices A˜ 1 and A˜ 1 A˜ 3 are symmetric and invertible, by (7.46) and (7.47), both E1 − E2 E4−1 E2T and (E1 − E2 E4−1 E2T )A4 are symmetric and invertible. Hence from 1 ≤ k1 ≤ l, σ (A4 ) ⊂ (−∞, 0) and our induction hypothesis we obtain sgn((E1 − E2 E4−1 E2T )A4 ) + sgn(E1 − E2 E4−1 E2T ) = 0.

(7.48)

By (7.47), E4 A5 is symmetric. Since E4 is symmetric and invertible, σ (A5 ) ⊂ (−∞, 0) and 1 ≤ k2 ≤ l, by our induction hypothesis we have sgn(E4 A5 ) + sgnE4 = 0.

(7.49)

sgnA˜ 1 = sgn(E1 − E2 E4−1 E2T ) + sgnE4 .

(7.50)

From (7.46) we obtain

By (7.47) there holds sgn(A˜ 1 A˜ 3 ) = sgn((E1 − E2 E4−1 E2T )A4 ) + sgn(E4 A5 ).

(7.51)

Then by (7.48)–(7.51) we have sgn(A˜ 1 A˜ 3 ) + sgnA˜ 1 = 0. Therefore, (7.45) and (7.52) imply (7.37). Subcase 2. E4 is not invertible. In this case we define k2 -order real invertible matrix ⎛

0 ⎜0 ⎜ ⎜0 ⎜ ⎜ E0 = ⎜ 0 ⎜ ⎜0 ⎜ ⎝0 1

⎞ 000001 0 0 0 0 1 0⎟ ⎟ 0 0 0 · 0 0⎟ ⎟ ⎟ 0 0 · 0 0 0⎟. ⎟ 0 · 0 0 0 0⎟ ⎟ 1 0 0 0 0 0⎠ 000000

(7.52)

192

7 Revisit of Maslov Type Index for Symplectic Paths

Then it is easy to verify  that E0 A5 is symmetric and E4 + εE0 is invertible for  E2 E1 . Since A˜ 1 and A˜ 1 A˜ 3 are invertible, we 0 < ε * 1. Define Aε = E2T E4 + εE0 have both Aε and Aε A˜ 3 are symmetric and invertible. Thus sgnA˜ 1 = sgnAε ,

sgn(A˜ 1 A˜ 3 ) = sgn(Aε A˜ 3 ),

for 0 < ε * 1.

(7.53)

By the proof of Subcase 1, we have sgn(Aε A˜ 3 ) + sgnAε = 0.

(7.54)

sgn(A˜ 1 A˜ 3 ) + sgnA˜ 1 = 0.

(7.55)

So from (7.53) we obtain

Then (7.37) holds from (7.55). So in Case 2 (7.37) holds for k = l + 1. Hence in the case A1 is invertible Lemma 3.15 holds and Step 1 is finished. Step 2. We assume that A1 is not invertible. If A1 = 0, (7.37) holds obviously. If 1 ≤ rankA1 = m ≤ k − 1, there is a real orthogonal matrix G such that  GT A1 G =

0 0 0 Aˆ 1

 (7.56)

,

where Aˆ 1 is an m-order invertible real symmetric matrix. Correspondingly we write G−1 A3 G =



F1 F2 F3 F4

 ,

where F1 is a (k − m) × (k − m) real matrix and F4 is an m × m real matrix. Since A1 A3 is symmetric, from GT A1 A3 G = GT A1 GG−1 A3 G =



0 0 Aˆ 1 F3 Aˆ 1 F4



we get Aˆ 1 F3 = 0. Hence F3 = 0 by the invertibility of Aˆ 1 . Therefore we can write G−1 A3 G =



F1 F2 0 F4

 .

(7.57)

7.3 The Concavity of Symplectic Paths and (ε, L0 , L1 )-Signature

193

Hence  G A1 A3 G = T

T

0 0 0 Aˆ 1 F4

 ,

(7.58)

where Aˆ 1 F4 is symmetric. Also by (7.57) the matrix F4 is invertible and σ (F4 ) ⊂ (−∞, 0). Thus by the proof of Step 1, there holds sgn(Aˆ 1 F4 ) + sgnAˆ 1 = 0.

(7.59)

Identities (7.56) and (7.58) give sgn(A1 A3 ) + sgnA1 = sgn(Aˆ 1 F4 ) + sgnAˆ 1 .

(7.60)

Then (7.59) and (7.60) give (7.37). Hence the Step 2 is proved. By Step 1 and Step 2 Lemma 3.15 holds.

 

We recall that the elliptic hight e(P ) of P is the total algebraic multiplicity of all eigenvalues of P on U for any P ∈ Sp(2n).   A1 Ik Lemma 3.16 ([210]) Let R = ∈ Sp(2k) with A3 being invertible. If A3 A2 e(Nk R −1 Nk R) = 2m, where 0 ≤ m ≤ k, then m−k ≤

1 sgnMε (R) ≤ k − m, 2

0 ≤ |ε| * 1.

(7.61)

Proof Since e(Nk R −1 Nk R) = 2m, there exists a symplectic matrix P ∈ Sp(2k) such that P −1 (Nk R −1 Nk R)P = Q1  Q2

(7.62)

with σ (Q1 ) ∈ U, σ (Q2 ) ∩ U = ∅, Q1 ∈ Sp(2m), and Q2 ∈ Sp(2k − 2m). By (ii) of Lemma 3.13, since A3 is invertible we only need to prove (7.61) for ε = 0. Step 1. Assume A1 is invertible. Since R is symplectic, we conclude from R T Jk R = Jk that AT1 A3 and A2 are symmetric and AT1 A2 − AT3 = Ik . Because R T is also symplectic, A1 is symmetric. Hence A1 A3 is symmetric and A1 A2 − AT3 = Ik . By definition we have

(7.63)

194

7 Revisit of Maslov Type Index for Symplectic Paths

 M0 (R) = R

T

 = −2

0 −Ik −Ik 0



 R+

A1 A3 AT3 A3 A2



0 Ik Ik 0



(7.64)

.

Since A1 is invertible, there holds   A1 A3 AT3 Ik −A−1 1 A3 A2 0 Ik   0 A1 A3 = T +A A 0 −A−1 2 1 3   A1 A3 0 , = 0 A−1 1 

Ik 0 −A−1 1 Ik



(7.65)

where in the last equality we have used the equality (7.63). From (7.65) we obtain   1 1 A1 A3 0 . sgnM0 (R) = − sgn 0 A−1 2 2 1

(7.66)

By the Jordan canonical form decomposition of complex matrix, there exists a complex invertible k-order matrix G1 such that ⎛

u1 ⎜0 ⎜ ⎜ G−1 1 A3 G1 = ⎜ ⎜0 ⎝0 0

∗ ∗ u2 ∗ . 0 ..

∗ ∗ ∗

0 0 uk−1 0 0 0

⎞ ∗ ∗⎟ ⎟ ⎟ ∗⎟ ⎟ ∗⎠ uk

with u1 , u2 , . . . , uk ∈ C. Equation (7.23) gives Nk R −1 Nk R = I2k + 2



A3 A2 A1 A3 AT3

 (7.67)

.

Since 

Ik 0 −A1 Ik

by (7.67) we have



A3 A2 A1 A3 AT3



Ik 0 A1 Ik



 =

Ik + 2A3 A2 −A1 −Ik

 ,

7.3 The Concavity of Symplectic Paths and (ε, L0 , L1 )-Signature



Ik 0 A1 Ik

−1

(Nk R −1 Nk R)



Ik 0 A1 Ik



 =

195

3Ik + 4A3 2A2 −2A1 −Ik

 := R1 . (7.68)

By (7.68), for any λ ∈ C we get  λI2k − R1 =

(λ − 3)Ik − 4A3 −2A2 2A1 (λ + 1)Ik

 (7.69)

.

Since A1 is invertible, by (7.63) there holds 

Ik − 12 ((λ − 3)Ik − 4A3 )A−1 1 0 Ik



(λ − 3)Ik − 4A3 −2A2 2A1 (λ + 1)Ik   0 − 12 ((λ2 − 2λ + 1)Ik − 4λA3 )A−1 1 = . 2A1 (λ + 1)Ik



(7.70)

Then by (7.69)–(7.70) we have det(λI2k − R1 ) = det((λ2 − 2λ + 1)Ik − 4λA3 ).

(7.71)

Denote by u1 , u2 , . . . , uk the k complex eigenvalues of A3 , (7.71) gives det(λI2k − R1 ) = "ki=1 (λ2 − 2λ + 1 − 4λui ) = "ki=1 (λ2 − (2 + 4ui )λ + 1).

(7.72)

Thus from (7.68) and (7.72) we get det(λI2k − Nk R −1 Nk R) = "ki=1 (λ2 − 2λ + 1 − 4λui ) = "ki=1 (λ2 − (2 + 4ui )λ + 1).

(7.73)

It is easy to check that the equation λ2 − (2 + ui )λ + 1 = 0 has two solutions on U if and only if −4 ≤ ui ≤ 0 for i = 1, 2, 3 . . . , k. So by (7.62) without loss of generality we assume uj ∈ [−4, 0) for 1 ≤ j ≤ m and uj ∈ / [−4, 0) for m + 1 ≤ j ≤ k. Then there exists a real invertible matrix k-order Q such that −1

Q

 A3 Q =

A4 0 0 A5



:= A˜ 3

and σ (A4 ) ⊂ [−4, 0), σ (A5 ) ∩ [−4, 0) = ∅, where A4 is an m-order real invertible matrix and A5 is a (k − m)-order real matrix. Denote by A˜ 1 = QT A1 Q. We have A˜ 1 A˜ 3 = QT A1 Q Q−1 A3 Q = QT A1 A3 Q.

196

7 Revisit of Maslov Type Index for Symplectic Paths

Hence both A˜ 1 and A˜ 1 A˜ 3 are symmetric and we conclude that sgnA1 + sgn(A1 A3 ) = sgnA˜ 1 + sgn(A˜ 1 A˜ 3 ).

(7.74)

Correspondingly we can write A˜ 1 in the block form decomposition A˜ 1 =   E1 E2 , where E1 is an m-order real symmetric matric and E4 is a (k −m)-order E2T E4 real symmetric matrix. Then A˜ 1 A˜ 3 =



E1 A4 E2 A5 E2T A4 E4 A5



is symmetric. By the same argument as the proof of Subcase 2 of Lemma 3.15 without loss of generality we can assume E1 is invertible (otherwise we can perturb it slightly such that it is invertible). So as in Subcase 1 of the proof of Lemma 3.15 we obtain   =

Im 0 −E2T E1−1 Ik−m



0 E1 0 E4 − E2T E1−1 E2

E1 E2 E2T E4 



Im −E1−1 E2 0 Ik−m



(7.75)

and 

Im 0 −E2T E1−1 Ik−m



E1 A4 E2 A5 E2T A4 E4 A5   0 E1 A4 . = 0 (E4 − E2T E1−1 E2 )A5



Im −E1−1 E2 0 Ik−m



(7.76)

By (7.76) we also have that E1 A4 is symmetric. Since E1 is symmetric and invertible, σ (A4 ) ⊂ [−4, 0), by Lemma 3.15 we have sgn(E1 A4 ) + sgnE1 = 0.

(7.77)

By (7.75) and (7.75), there hold sgnA˜ 1 = sgn(E4 − E2T E1−1 E2 ) + sgnE1 ,

(7.78)

sgn(A˜ 1 A˜ 3 ) = sgn((E4 − E2T E1−1 E2 )A5 ) + sgn(E1 A4 ).

(7.79)

Equations (7.77)–(7.79) give

7.3 The Concavity of Symplectic Paths and (ε, L0 , L1 )-Signature

sgn(A˜ 1 A˜ 3 ) + sgnA˜ 1 = sgn((E4 − E2T E1−1 E2 )A5 ) + sgn(E4 − E2T E1−1 E2 ) ∈ [−2(k − m), 2(k − m)].

197

(7.80)

Then (7.61) holds from (7.66), (7.74) and (7.80). Step 2. Assume A1 is not invertible.  = k. It is easy to check that M0 (R) = If A1 =0, then A3 = −Ik and m 0 Ik 0 Ik is congruent to 2 , so sgnM0 (R) = 0 and (7.61) holds. 2 Ik −A2 Ik 0 If 1 ≤ rankA1 = r ≤ k − 1, there is a k × k invertible matrix G with det G > 0 such that (G−1 )T A1 G−1 = diag(0, ),

(7.81)

where  is an r × r real invertible matrix. Hence  Ik (GT )−1 A1 G−1 GA3 G−1 GA2 GT ⎞ ⎛ 0 0 Ik−r 0 ⎜ 0  0 Ir ⎟ ⎟ := R2 = ⎜ ⎝ B1 B2 D1 D2 ⎠ , (7.82)

diag((GT )−1 , G) · R · diag(G−1 , GT ) =



B3 B4 D 3 D 4 where B1 and D1 are (k − r) × (k − r) matrices, B4 and D4 are r × r matrices. Since R2 is symplectic and  is invertible, there holds R2T Jk R2 = Jk . It implies that B3 = 0, D3 = D2T , B1 = −Ik−r , and D1 , D4 are symmetric. Thus ⎛

0 ⎜ 0 R2 = ⎜ ⎝ B1 0

0  B2 B4

Ik−r 0 D1 D2T

⎞ 0 Ir ⎟ ⎟. D2 ⎠ D4

For t ∈ [0, 1], we define ⎛

0 ⎜ 0 β(t) = ⎜ ⎝ B1 0

⎞ 0 Ik−r 0  0 Ir ⎟ ⎟. tB2 tD1 tD2 ⎠ B4 tD2T D4

It is easy to check that β is a symplectic path and νLj (β(t) = 0 for all t ∈ [0, 1] and j = 0, 1. We also have β(1) = R2 and

198

7 Revisit of Maslov Type Index for Symplectic Paths

⎛

0 ⎜ 0 β(0) = ⎜ ⎝ B1 0

0  0 B4

Ik−r 0 0 0

⎞ 0   Ir ⎟ ⎟ = −Jk−r   Ir := R3 . 0 ⎠ B4 D 4 D4

Then by (1) of Remark 3.5 we have 1 1 1 sgnM0 (R2 ) = sgnM0 (−Jk−r ) + sgnM0 2 2 2   1  Ir . = sgnM0 B4 D 4 2



 Ir B4 D 4



(7.83)

s

Since R2 ∼ R, by (7.83) we have 1 1 sgnM0 (R) = sgnM0 2 2



 Ir B4 D 4

 .

(7.84)

By (7.23), there holds ⎛

Nk R2−1 Nk R2 = I2k

B1 B2 ⎜ 0 B4 + 2⎜ ⎝ 0 0 0 B4

D1 D2T B1T B2T

⎞ D2 D4 ⎟ ⎟. 0 ⎠

(7.85)

B4T

By (7.85) for any λ ∈ C, we obtain det(λI2k − Nk R2−1 Nk R2 ) = det((λ − 1)Ik−r − 2B1 ) det((λ − 1)Ik−r − 2B1T ) ·   −2D4 (λ − 1)Ir − 2B4 · det −2B4 (λ − 1)Ir − 2B4T = det(λI2k − Nk R3−1 Nk R3 ),

(7.86)

where ⎛

Nk R3−1 Nk R3 = I2k

So (7.86) gives

B1 0 ⎜ 0 B4 + 2⎜ ⎝ 0 0 0 B4

0 0 B1T 0

⎞ 0 D4 ⎟ ⎟. 0 ⎠ B4T

7.3 The Concavity of Symplectic Paths and (ε, L0 , L1 )-Signature

σ (Nk R −1 Nk R) = σ (Nk R2−1 Nk R2 ) = σ (Nk R3−1 Nk R3 ).  Since B1 = −Ik−r and R3 = (−Jk−r )  " e Nr



 Ir B4 D 4

−1

 Nr

199

(7.87)

  Ir , (7.87) gives B4 D 4

 Ir B4 D 4

# = 2(m − (k − r)).

(7.88)

Step 1 implies that     1   Ir  ≤ r − (m − (k − r)) = k − m. sgnM 0   B4 D 4 2

(7.89)

Then (7.61) follows from (7.84) and (7.89). This finishes the proof of Step 2. With Step 1 and Step 2, the proof of Lemma 3.16 is completed.     AB Lemma 3.17 ([309]) Let R ∈ Sp(2k) has the block form R = with 1 ≤ CD rankB = r < k. We have ⎞ ⎛ A1 B1 Ir 0 s ⎜ 0 D1 0 0 ⎟ ⎟ (i) R ∼ ⎜ ⎝ A3 B3 A2 0 ⎠, where A1 , A2 , A3 are r × r matrices, D1 , D2 , D3 are C3 D 3 C2 D 2 (k − r) × (k − r) matrices, B1 , B3 are r × (k − r) matrices, and C2 , C3 are (k − r) × r matrices. (ii) If A3 is invertible, we have s

R∼



A1 Ir A3 A2

   D1 0  ˜ , D3 D2

(7.90)

where D˜ 3 is a (k − r) × (k − r) matrix. (iii) If 1 ≤ rankA3 = λ ≤ r − 1, then

s

R∼



U Iλ V



⎛ ˜ A1 ⎜ 0 ⎜ ⎝ 0 C˜ 3

B˜ 1 D1 B˜ 3 D˜ 3

Ir−λ 0 ˜ A2 C˜ 2

⎞ 0 0 ⎟ ⎟, 0 ⎠ D˜ 2

(7.91)

where A˜ 1 , A˜ 2 are (r − λ) × (r − λ) matrices, B˜ 1 , B˜ 3 are (r − λ) × (k − r) matrices, C˜ 2 , C˜ 3 are (k −r)×(r −λ) matrices, D1 , D˜ 2 , D˜ 3 are (k −r)×(k −r) matrices, U, V ,  are λ × λ matrices, and  is invertible. (iv) If A3 = 0, then rankB3 = rankC3 and σ (Nk R −1 Nk R) = {1}. Moreover, if B3 = 0 and C3 = 0 we have

200

7 Revisit of Maslov Type Index for Symplectic Paths

s



R∼

A1 Ir 0 A2



 

D1 0 D3 D2

 ;

(7.92)

if k ≥ 2r and rankB3 = rankC3 = r, then ⎛

Ir ⎜ 0 −1 Nk R Nk R ≈ ⎜ ⎝ 0 2A1  0 m∗ (AT C) = m∗ A1

⎞ 2Ir 2A2 0   Ir 0 0 ⎟ ⎟  Ik−2r 0 , 2A1 Ir 0 ⎠ U4 Ik−2r 0 2Ir Ir  A1 + m∗ (U4 ), ∗ = +, −, 0, 0

(7.93)

(7.94)

where U4 is a (k − 2r) × (k − 2r) symmetric matrix, when k = 2r, the term  Ik−2r 0 will not appear in the right hand side of (7.93) and m∗ (U4 ) = U4 Ik−2r 0 for ∗ = +, −, 0 in (7.94). Remark 3.18 Note that R2 in (iii) satisfies the condition of (iv) above, so there hold s

R2 ∼



A˜ 1 Ir 0 A˜ 2



 

D1 0 D¯ 3 D˜ 2



and rankB˜ 3 = rankC˜ 3 . Proof of Lemma 3.17 Since rankB = r, there are two invertible k × k matrices U and V with det U > 0 and det V > 0 such that   I 0 U BV = r . 0 0 So there holds ⎛

A1 ⎜ C1 s T −1 T −1 R ∼ diag(U, (U ) ) R diag((V ) , V ) = ⎜ ⎝A3 C3

B1 D1 B3 D3

Ir 0 A2 C2

⎞ 0 0⎟ ⎟ := R˜ 1 , B2 ⎠ D2

(7.95)

where for j = 1, 2, 3, Aj is an r × r matrix, Dj is a (k − r) × (k − r) matrix, Bj is an r × (k − r) matrix, and Cj is (k − r) × r matrix. Since R˜ 1 is still a symplectic matrix, we have C1 = 0, B2 = 0. So

7.3 The Concavity of Symplectic Paths and (ε, L0 , L1 )-Signature

⎛ A1 ⎜0 R˜ 1 = ⎜ ⎝A3 C3

B1 D1 B3 D3

Ir 0 A2 C2

201

⎞ 0 0⎟ ⎟. 0⎠ D2

This proves the statement (i). Suppose A3 is invertible. By R˜ 1 is symplectic, we have 

AT1 0 B1T D1T

   T T   A2 0 A3 C3 Ir 0 − = Ik . B3T D3T C2 D 2 0 0

(7.96)

D1T D2 = Ik−r .

(7.97)

Hence

By direct computation we have ⎛ A1 ⎜0 ⎜ ⎝A3 C3

B1 D1 B3 D3

Ir 0 A2 C2

⎞⎛ 0 0 Ir −A−1 3 B3 ⎟ ⎜ 0 ⎟ ⎜ 0 Ik−r 0 0 ⎠⎝0 0 Ir D2 0 0 B3T (AT3 )−1

⎞ ⎛ 0 A1 ⎟ ⎜ 0 ⎟ ⎜0 = 0 ⎠ ⎝A3 Ik−r C3

B˜ 1 D1 0 D˜ 3

Ir 0 A2 C˜ 2

⎞ 0 0⎟ ⎟. 0⎠ D2 (7.98)

So by (7.97) we have ⎛ ⎞⎛ Ir −B˜ 1 D2T 0 0 A1 ⎜ 0 Ik−r ⎟⎜ 0 0 0 ⎜ ⎟⎜ ⎝0 0 ⎠ ⎝A3 0 Ir 0 0 D2 B˜ 1T Ik−r C3

B˜ 1 D1 0 ˜ D3

Ir 0 A2 C˜ 2

⎞ ⎛ 0 A1 ⎜0 0⎟ ⎟=⎜ 0 ⎠ ⎝A3 D2 C˜ 3

0 D1 0 ˜ D3

Ir 0 A2 C˜ 2

⎞ 0 0⎟ ⎟ := R˜ 2 . 0⎠ D2 (7.99)

Then we have s s R˜ 2 ∼ R˜ 1 ∼ R.

(7.100)

Since R˜ 2 is a symplectic matrix, we have R˜ 2T Jk R˜ 2 = Jk , then it is easy to check that C˜ 3 = 0, C˜ 2 = 0. Hence we have     D 0 A1 Ir  ˜1 . R˜ 2 = A3 A2 D3 D2 This proves the statement (ii).

(7.101)

202

7 Revisit of Maslov Type Index for Symplectic Paths

Suppose A3 = 0 and A3 is not invertible. In this case, suppose rankA3 = λ, then 0 < λ < r. There is an invertible r × r matrix G with det G > 0 such that GA3 G−1 =



 0 0 0

(7.102)

where  is a λ × λ invertible matrix. Then we have ⎞⎛ ⎞ ⎛ T −1 0 0 0 A1 B1 Ir 0 (G ) ⎟ ⎜ ⎜ 0 Ik−r 0 0 ⎟ ⎟ ⎜ 0 D1 0 0 ⎟ ⎜ ⎠ ⎝ ⎝ 0 0 G 0 A3 B3 A2 0 ⎠ C3 D 3 C2 D 2 0 0 0 Ik−r ⎛ ⎞ B˜ 1 Ir 0 A˜ 1 ⎜ 0 D1 0 0 ⎟ ⎟ ˜ =⎜ ⎝GA3 G−1 B˜ 3 A˜ 2 0 ⎠ := R3 . D3 C˜ 2 D2 C˜ 3

⎛

G−1 ⎜ 0 ⎜ ⎝ 0 0

⎞ 0 0 ⎟ 0 Ik−r 0 ⎟ T ⎠ 0 0 G 0 0 I −k−r 0

(7.103)

By (7.102) we can write R˜ 3 as the following block form ⎛

U1 ⎜U ⎜ 3 ⎜ ⎜0 R˜ 3 = ⎜ ⎜ ⎜ ⎝0 G1 ⎛

U2 U4 0 0 0 G2

F1 F2 D1 E1 E2 D3

Iλ 0 0 W1 W3 K1

⎞ 0 Ir−λ 0 ⎟ ⎟ ⎟ 0 0⎟ ⎟. W2 0 ⎟ ⎟ W4 0 ⎠ K2 D 2 0

(7.104)

⎛ ⎞ ⎞ Iλ Iλ 0 −−1 E1 0 0 ⎠. By (7.104) Let Q1 = ⎝ 0 0 Ir−λ 0 ⎠ and Q2 = ⎝ 0 Ir−λ −1 0 Ik−r −G1  0 0 Ik−r we have ⎞ ⎛ U1 U2 F˜1 Iλ 0 0 ⎟ ⎜U U F˜ 0 I r−λ 0 ⎟ ⎜ 3 4 2 ⎟ ⎜ ⎜ 0 0 D1 0 0 0 ⎟ diag((QT1 )−1 , Q1 ) R˜ 3 diag(Q2 , (QT2 )−1 ) = ⎜ ⎟ := R˜ 4 . ⎜  0 0 W1 W2 0 ⎟ ⎟ ⎜ ⎝ 0 0 E2 W3 W4 0 ⎠ 0 G2 D˜ 3 K˜ 1 K˜ 2 D2 (7.105) Since R˜ 4 is a symplectic matrix we have R˜ 4T J R˜ 4 = J . Then by direct computation we have U2 = 0, U3 = 0, W2 = 0, W3 = 0, F˜1 = 0, K˜ 1 = 0, and U1 , U4 , W1 , W4 are all symmetric matrices, moreover there hold

7.3 The Concavity of Symplectic Paths and (ε, L0 , L1 )-Signature

203

U4 W4 = Ir−λ ,

(7.106)

D1 D2T = Ik−r ,

(7.107)

U4 E˜ 2 = GT2 D1 .

(7.108)

So ⎛

U1 ⎜0 ⎜ ⎜ ⎜0 R˜ 4 = ⎜ ⎜ ⎜ ⎝0 0

0 U4 0 0 0 G2

0 F˜2 D1 0 ˜ E2 D˜ 3

⎞ Iλ 0 0 ⎛ ˜ 0 Ir−λ 0 ⎟ A1 ⎟   ⎟ ⎜ 0 U Iλ 0 0 0⎟ ⎜ ⎟= ⎝ 0 V W1 0 0 ⎟ ⎟ ⎠ 0 W4 0 C˜ 3 0 K2 D 2

B˜ 1 D1 B˜ 3 D˜ 3

Ir−λ 0 A˜ 2 C˜ 2

⎞ 0 0 ⎟ ⎟, 0 ⎠ D˜ 2

(7.109)

where U = U1 , V = W1 , A˜ 1 = U4 , B˜ 1 = F˜2 , B˜ 3 = E˜ 2 , C˜ 3 = G2 , A˜ 2 = W4 , C˜ 2 = K2 , D˜ 2 = D2 . This proves statement (iii). ⎞ ⎛ A1 B1 Ir 0 s ⎜ 0 D1 0 0 ⎟ ⎟ Lastly we prove (iv). If A3 = 0, then R ∼ ⎜ ⎝ 0 B3 A2 0 ⎠ := R3 . Since C3 D 3 C2 D 2 T R3 Jk R3 = Jk , it is easy to check that A1 , A2 and B3T B1 + D3T D1 are symmetric, A1 A2 = Ir , D1 D2T = Ik−r , AT1 B3 = C3T D1 and B1T A2 + D1T C2 = B3T . So rankB3 = rankC3 . By direct computation we have ⎛

Nk R3−1 Nk R3 − I2k

0 B3 ⎜ 0 0 = 2⎜ ⎝ 0 AT1 B3 B3T A1 B1T B3 + D1T D3

A2 0 0 B3T

⎞ 0 0⎟ ⎟. 0⎠

(7.110)

0

Then it is easy to check that (Nk R3−1 Nk R3 − I2k )4 = 0.

(7.111)

Hence by Lemma 3.12 σ (Nk R −1 Nk R) = σ (Nk R3−1 Nk R3 ) = {1}. Furthermore if B3 = 0 and C3 = 0, we have  diag

⎛ A1    ⎜ 0 0 Ir −B1 D1−1 Ir ⎜ , ⎝ 0 0 Ik−r (D1T )−1 B1T Ik−r 0

B1 D1 0 D3

Ir 0 A2 C2

⎞ 0 0 ⎟ ⎟ 0 ⎠ D2

204

7 Revisit of Maslov Type Index for Symplectic Paths

⎛

A1 ⎜ 0 =⎜ ⎝ 0 0

0 D1 0 D3

Ir 0 A2 C˜ 2

⎞ 0 0 ⎟ ⎟ := R. ˜ 0 ⎠ D2

(7.112)

Since R˜ is symplectic, it is easy to check that C˜ 2 = 0, hence by the conclusion of (i) and (7.112), (7.92) holds. If k > 2r and rankB3 = rankC3 = r, then there are (k − r) × (k − r) matrices G1 and G2 with det(Gi ) > 0 for i = 1, 2 such that G1 C3 = (Ir , 0)T ,

B3 G2 = (Ir , 0).

(7.113)

Then we have         0 0 Ir 0 Ir 0 Ir Ir , R3 , diag 0 (GT1 )−1 0 G1 0 G2 0 (GT2 )−1 ⎞ ⎛ A1 Bˆ 1 Ir 0 ⎜ 0 Dˆ 1 0 0 ⎟ ⎟ (7.114) =⎜ ⎝ 0 B3 G2 A2 0 ⎠ := R4 . G1 C3 Dˆ 3 Cˆ 2 Dˆ 2 By (7.113) there exist (k − r) × r matrix G3 and an r × (k − r) matrix G4 such that 

Ir 0 G3 Ik−r



0 B3 G2 G1 C3 Dˆ 3



Ir G4 0 Ik−r



⎛

⎞ 0 Ir 0 = ⎝ Ir 0 0 ⎠ , 0 0 D˜ 3

(7.115)

where D˜ 3 is a (k − 2r) × (k − 2r) matrix. Then we have  diag ⎛ ⎜ ⎜ ⎜ ⎜ =⎜ ⎜ ⎜ ⎝

A1 0 0 0 Ir 0

Ir −GT3 0 Ik−r

U1 E1 E3 Ir 0 0

U2 E2 E4 0 0 D˜ 3

Ir 0 0 A2 V1 V2

       0 Ir 0 Ir Ir G4 , R4 diag , 0 Ik−r G3 Ik−r −GT4 Ik−r ⎞ 0 0 0 0 ⎟ ⎟ ⎟ 0 0 ⎟ (7.116) ⎟ := R5 , 0 0 ⎟ ⎟ F1 F2 ⎠ F3 F4

7.3 The Concavity of Symplectic Paths and (ε, L0 , L1 )-Signature

205

where E1 and F1 are r × r matrices, E4 , F4 are (k − 2r) × (k − 2r) matrices, others are corresponding matrix blocks. Since R5 is symplectic, R5T Jk R5 = Jk . Then we can check that E1 = A1 , E2 = 0, F3 = 0, F1 = A2 , U2 = E3T D˜ 3 , U1 , V1 and E4T D˜ 3 are symmetric. So we have ⎛

A1 ⎜ 0 ⎜ ⎜ ⎜ 20 R5 = ⎜ ⎜ 0 ⎜ ⎝ Ir 0

U1 A1 E3 Ir 0 0

U2 0 E4 0 0 D˜ 3

Ir 0 0 A2 V1 V2

0 0 0 0 A2 0

⎞ 0 0 ⎟ ⎟ ⎟ 0 ⎟ ⎟. 0 ⎟ ⎟ F2 ⎠ F4

(7.117)

Note that by the above construction ⎛

⎞T ⎛ ⎞ ⎛ ⎞ 0 Ir 0 0 A1 A1 U1 U2 0 ⎝ Ir 0 0 ⎠ ⎝ 0 A1 0 ⎠ = ⎝ A1 U1 U2 ⎠ . 0 U2T D˜ 3T E4 0 0 D˜ 3 0 E3 E4 Hence by the Remark 3.7 we have ⎛

⎞ 0 A1 0 m∗ (AT C) = m∗ ⎝ A1 U1 U2 ⎠ , 0 U2T D˜ 3T E4

∗ = +, −, 0.

(7.118)

Since ⎞⎛ ⎞⎛ ⎞ −1 T 0 0 Ir 0 A1 Ir − 12 A−1 0 1 U1 −A1 U2 ⎠ ⎝ − 1 U1 A−1 Ir 0 ⎠ ⎝ A1 U1 U2 ⎠ ⎝ 0 Ir 0 1 2 −1 T T T ˜ 0 U2 D3 E4 0 0 Ik−2r −U2 A1 0 Ik−2r ⎛ ⎞ 0 A1 0 = ⎝ A1 0 0 ⎠, T 0 0 D˜ 3 E4 ⎛

by (7.118) we have ∗

m (A C) = m T

∗



0 A1 A1 0

By direct computation we have



+ m∗ (D˜ 3T E4 ),

∗ = +, −, 0.

(7.119)

206

7 Revisit of Maslov Type Index for Symplectic Paths

⎛

Nk R5−1 Nk R5 = I2k

0 ⎜ 0 ⎜ ⎜ ⎜ 0 + 2⎜ ⎜ 0 ⎜ ⎝ A1 0

Ir 0 0 0 0 0 0 A1 U1 U2 U2T D˜ 3T E4

A2 0 0 0 Ir 0

⎞ 00 0 0⎟ ⎟ ⎟ 0 0⎟ ⎟. 0 0⎟ ⎟ 0 0⎠ 00

(7.120)

Since A1 and A2 is invertible, it is easy to see that ν1 (Nk R5−1 Nk R5 ) = k − r + dim ker(D˜ 3T E4 ). For any t ∈ R, define ⎛

Ir ⎜ 0 ⎜ ⎜ ⎜ 0 β(t) = ⎜ ⎜ 0 ⎜ ⎝ 2A1 0

2Ir 0 0 Ir 0 Ik−2r 0 2A1 2tU1 2tU2 2tU2T (1 + t)D˜ 3T E4

2A2 0 0 Ir 2Ir 0

0 0 0 0 Ir 0

⎞ 0 0⎟ ⎟ ⎟ 0⎟ ⎟. 0⎟ ⎟ 0⎠ Ir

Since Nk R5−1 Nk R5 is a symplectic matrix, we can check that β(t) is a symplectic matrix for each t ∈ R. Since A1 is invertible and rank(β(t) − I2k ) is independent of t ∈ [0, 1]. So we have σ (β(t)) = {1} and ν1 (β(t)) = k − r + dim ker(D˜ 3T E4 ) = constant.

(7.121)

So by the definition of R3 , R4 , R5 , Lemma 3.12, and (7.121) we have ⎛

Ir ⎜ 0 Nk R −1 Nk R ≈ Nk R5−1 Nk R5 = β(1) ≈ β(0) = ⎜ ⎝ 0 2A1   0 I .  ˜k−2r D3T E4 Ik−2r

2Ir Ir 2A1 0

2A2 0 Ir 2Ir

⎞ 0 0⎟ ⎟ 0⎠ Ir (7.122)

Denote by U4 = D˜ 3T E4 . Then (7.93) and (7.94) hold from (7.119) and (7.122). So (iv) is proved. The proof of Lemma 3.17 is complete.   The following result is about the (L0 , L1 )-normal forms of L0 -degenerate symplectic matrices. Lemma 3.19 ([210]) Using the same notations in Lemma 3.17. If A3 = 0, then A1 , A2 are symmetric and A1 A2 = Ir . Suppose m+ (A1 ) = p, m− (A1 ) = r − p and 0 ≤ rankB3 = λ ≤ min{r, k − r}, then

7.3 The Concavity of Symplectic Paths and (ε, L0 , L1 )-Signature

Nk R

−1

 Nk R ≈

11 01



p+q − 

1 −1 0 1

(r−p+q + )

207

q 0

 I2

 D(2)λ , (7.123)

m+ (AT C) = λ + q + ,

(7.124)

m0 (AT C) = r − λ + q 0 ,

(7.125)

−

−

m (A C) = λ + q , T

(7.126)

where q ∗ ≥ 0 for ∗ = ±, 0, q + + q 0 + q − = k − r − λ, M 0 means correspondent component does not appears at all for M being one of the four matrices on the right hand side of (7.123). Proof By (i) of Lemma 3.17 and A3 = 0 we have ⎛

A1 s ⎜ 0 R∼⎜ ⎝ 0 C3

B1 D1 B3 D3

Ir 0 A2 C2

⎞ 0 0 ⎟ ⎟ := R1 . 0 ⎠ D2

(7.127)

Since R1 is symplectic we have R1T Jk R1 = Jk . Then we have A1 , A2 are symmetric and A1 A2 = Ir . D1 D2T = Ik−r and AT1 B3 = C3T D1 . Equation (7.23) yields ⎞ 2B3 2A2 0 Ir ⎜ 0 0 0 ⎟ Ik−r ⎟. Nk R1−1 Nk R1 = ⎜ T ⎝ 0 Ir 0 ⎠ 2A1 B3 2B3T A1 2B1T B3 + 2D1T D3 2B3T Ik−r ⎛

(7.128)

By Remark 3.7 we obtain ∗

m (A C) = m T

∗



0 AT1 B3 T T B3 A1 B1 B3 + D1T D3

 ,

∗ = +, −, 0.

(7.129)

Since 0 ≤ rankB3 = λ ≤ min{r, k − r}, there exist r × r and (k − r) × (k − r) real invertible matrices G1 and G2 such that  G1 B3 G2 =

Iλ 0 0 0

 := F.

(7.130)

Note that ifλ  = 0 then B3 = 0, if λ = min{r, k − r} then G1 B3 G2 =   Iλ , if λ = r = k − r then G1 B3 G2 = Iλ . The proof below can Iλ 0 or 0 still go through by corresponding adjustment.

208

7 Revisit of Maslov Type Index for Symplectic Paths

By (7.130) we have 

  −1 T  0 AT1 B3 A1 G1 0 B3T A1 B1T B3 + D1T D3 0 G2 ⎞ ⎛ 0 0 Iλ 0   ⎜0 0 0 0 ⎟ 0 G1 B3 G2 ⎟ = =⎜ ⎝ Iλ 0 U1 U2 ⎠ . GT2 B3T GT1 U 0 0 U2T U4 G1 A−1 0 1 0 GT2



(7.131)

Then ⎛

⎞⎛ Iλ 0 0 0 0 ⎜ 0 Ir−λ 0 ⎟ ⎜ 0 ⎟⎜ 0 ⎜ ⎝ − 1 U1 0 Iλ 0 ⎠ ⎝ Iλ 2 T 0 −U2 0 0 Ik−r−λ ⎛ ⎞ 0 0 Iλ 0 ⎜0 0 0 0 ⎟ ⎟ =⎜ ⎝ Iλ 0 0 0 ⎠ . 0 0 0 U4

0 0 0 0

Iλ 0 U1 U2T

⎞⎛ ⎞ 0 Iλ 0 − 12 U1 −U2 ⎜ 0 ⎟ 0 ⎟ ⎟ ⎜ 0 Ir−λ 0 ⎟ ⎠ ⎝ U2 0 ⎠ 0 0 Iλ U4 0 0 0 Ik−r−λ

(7.132)

Set q ∗ = m∗ (U4 ),

∗ = ±, 0.

(7.133)

Then q + + q 0 + q − = k − r − λ and (7.124)–(7.126) hold from (7.129), (7.131) and (7.132). Also by (7.132) and Lemma 3.11 we have 

0 Ik−r−λ 2U4 Ik−r−λ



 ≈

11 01

q −

q 0  I2

 

1 −1 0 1

q + .

(7.134)

By (7.131), there holds −1 −1 T diag((GT1 )−1 A1 , G−1 2 , G1 A1 , G2 )(Nk R1 Nk R1 ) −1 T T −1 diag(A−1 1 G1 , G2 , A1 G1 , (G2 ) ) ⎛ ⎞ Ir 2E 2A˜ 1 0 ⎜ 0 Ik−r 0 0 ⎟ ⎟ := M, =⎜ ⎝ 0 2F Ir 0 ⎠ 2F T 2U 2E T Ik−r T −1 ˜ where A˜ 1 = (GT1 )−1 A1 G−1 1 , E = (G1 ) A1 B3 G2 = A1 F .

(7.135)

7.3 The Concavity of Symplectic Paths and (ε, L0 , L1 )-Signature

209

Since M is symplectic, we have M T Jk M = Jk . Then we have E = A˜ 1 F . Since A˜ 1 = (GT1 )−1 A1 G−1 1 , it is congruent to diag(a1 , a2 , . . . , ar ) with ai = 1, 1 ≤ i ≤ p

and

aj = −1, p + 1 ≤ j ≤ r for some 0 ≤ p ≤ r.

(7.136)

Then there is an invertible r × r real matrix Q such that det Q > 0 and QA˜ 1 QT = diag(a1 , a2 , . . . , ar ) = diag(diag(a1 , a2 , . . . , aλ ), diag(aλ+1 , . . . , ar )) := diag(1 , 2 ). (7.137) Since det Q > 0 we can joint it to Ir by invertible continuous matrix path. So there is a continuous invertible symmetric matrix path α1 such that α1 (1) = A˜ 1 and α1 (0) = diag(a1 , a2 , . . . , ar ) with m∗ (α1 (t)) = m∗ (A˜ 1 ) = m∗ (A1 ),

t ∈ [0, 1], ∗ = +, −.

Define symmetric matrix path  α2 (t) =

2tU1 2tU2 2tU2T 2U4

 ,

t ∈ [0, 1].

For t ∈ [0, 1], define ⎛

⎞ Ir 2α1 (t)F 2α1 (t) 0 ⎜ 0 0 0 ⎟ Ik−r ⎟. β(t) = ⎜ ⎝ 0 0 ⎠ 2F Ir 2F T α2 (t) 2F T α1 (t)T Ik−r Then since M is symplectic, it is easy  to check that β is a continuous path Iλ 0 of symplectic matrices. Since F = , and α1 (t) is invertible, by direct 0 0 computation, we have rank(β(t) − I2k ) = 2λ + rank(α1 (t)) + rank(U4 ) = 2λ + r + m+ (U4 ) + m− (U4 ). Hence ν1 (β(t)) = ν1 (β(1)) = ν1 (M),

t ∈ [0, 1].

Because σ (β(t)) = {1}, by Definition 3.8 and Lemma 3.10

210

7 Revisit of Maslov Type Index for Symplectic Paths

M = β(1) ≈ β(0) ⎞ ⎛ Iλ 0 21 21 0 0 ⎜ 0 I 0 22 0 ⎟ r−λ 0 ⎟  ⎜  ⎟ ⎜ Ik−r−λ 0 0⎟ 0 ⎜ 0 0 Iλ 0  =⎜ ⎟ ⎜ 0 0 2Iλ Iλ 0 0 ⎟ 2U4 Ik−r−λ ⎟ ⎜ ⎝ 0 0 0 0 Ir−λ 0 ⎠ 0 21 0 Iλ 2Iλ 0 ⎛ ⎞ Iλ 21 21 0     ⎜ 0 Iλ 0 0 ⎟ 0 ⎟  Ir−λ 22  Ik−r−λ ≈⎜ ⎝ 0 2Iλ Iλ 0 ⎠ 0 Ir−λ 2U4 Ik−r−λ 2Iλ 0 21 Iλ ⎞ ⎛ Iλ 21 21 0     ⎜ 0 Iλ 0 0 ⎟ 0 1 2aj Ik−r−λ ⎟  ♦r .  =⎜ j =λ+1 ⎝ 0 2Iλ Iλ 0 ⎠ 0 1 2U4 Ik−r−λ 2Iλ 0 21 Iλ

(7.138)

We define continuous symplectic matrix path ⎛

2(1 − t 2 )1 21 Iλ ⎜ 0 0 (1 + t)I λ ψ(t) = ⎜ 2 ⎝ Iλ 0 2(1 − t )Iλ 0 2(1 − t)1 2(1 − t)Iλ

0 0 0

⎞ ⎟ ⎟, ⎠

t ∈ [0, 1].

1 1+t Iλ

Since 1 is invertible, ν(ψ(t)) ≡ λ for t ∈ [0, 1]. So by σ (ψ(t)) ∩ U = {1} for t ∈ [0, t] and Definition 3.8 we obtain ⎛

Iλ ⎜ 0 ⎜ ⎝ 0 2Iλ

1 Iλ 2Iλ 0

21 0 Iλ 21

⎞ 0     0⎟ ⎟ = ψ(0) ≈ ψ(1) = Iλ 21  2Iλ 0 0⎠ 0 Iλ 0 12 Iλ Iλ   1 2aj = ♦λj=1  D(2)λ . (7.139) 0 1

Thus by (7.138), (7.139) and Remark 3.9 we get      0 1 aj I .  D(2)λ  k−r−λ M ≈ ♦rj =1 0 1 U4 Ik−r−λ So by (7.134), (7.136) and Remark 3.9, there holds

(7.140)

7.4 The Mixed (L0 , L1 )-Concavity

 M≈

11 01

211



(p+q − ) 

1 −1 0 1

(r−p+q + )

q 0

 I2

 D(2)λ .

(7.141)

By Lemma 3.12, (7.127) and (7.135), we have Nk R −1 Nk R ≈ M.

(7.142)

Then (7.123) holds from (7.141) and (7.142). The proof of Lemma 3.19 is completed.  

7.4 The Mixed (L0 , L1 )-Concavity Definition 4.1 ([210]) The mixed (L0 , L1 )-concavity and mixed (L1 , L0 )concavity of a symplectic path γ ∈ Pτ (2n) are defined respectively by μ(L0 ,L1 ) (γ ) = iL0 (γ ) − νL1 (γ ), μ(L1 ,L0 ) (γ ) = iL1 (γ ) − νL0 (γ ). Proposition V.9.8 and Theorem 3.4, imply the following result. Proposition 4.2 There hold μ(L0 ,L1 ) (γ ) + μ(L1 ,L0 ) (γ ) = i(γ 2 ) − ν(γ 2 ) − n, μ(L0 ,L1 ) (γ ) − μ(L1 ,L0 ) (γ ) =

(7.143)

∗ concav(L (γ ) 0 ,L1 )

=

1 sgnMε (γ (τ )), 0 < −ε * 1. 2

(7.144)

Theorem 4.3 ([210]) For γ ∈ Pτ (2n), let P = γ (τ ). If iL0 (γ ) ≥ 0, iL1 (γ ) ≥ 0, i(γ ) ≥ n, γ 2 (t) = γ (t − τ )γ (τ ) for all t ∈ [τ, 2τ ], then μ(L0 ,L1 ) (γ ) + SP+2 (1) ≥ 0,

(7.145)

μ(L1 ,L0 ) (γ ) + SP+2 (1) ≥ 0.

(7.146)

We recall that γ 2 is the 2-times iteration path γ 2 : [0, 2τ ] → Sp(2n) of γ defined in (5.184). Proof The proofs of (7.145) and (7.146) are almost the same. We only prove (7.146). Claim 1. Under the conditions of Theorem 4.3, if  P ≈ 2

11 01

p1

 D(2)p2  P˜ ,

(7.147)

212

7 Revisit of Maslov Type Index for Symplectic Paths

then i(γ 2 ) + 2SP+2 (1) − ν(γ 2 ) ≥ n + p1 + p2 .

(7.148)

Proof of Claim 1. By Theorem 7.8 of [226] we have P ≈

q I2 1

 

11 01



q2 

1 −1 0 1

q3  (−I2 )

q4

 

−1 1 0 −1



q5 

−1 −1 0 −1

q6

R(θ1 )  · · ·  R(θq7 )  · · ·  R(θq7 +q8 )  N2 (ω1 , b1 )  · · ·  N2 (ωq9 , bq9 ) D(2)q10  D(−2)q11 ,

(7.149)

where qi ≥ 0 for 1 ≤ i ≤ 11 with q1 + q2 + · · · + q8 + 2q9 + q10 + q11 = n, θj ∈ (0, π ) for 1 ≤ j ≤ q7 , θj ∈ (π, 2π ) for q7 + 1 ≤ j ≤ q7 + q8 , ωj ∈ (U \ R)   bj 1 bj 2 satisfying bj 2 = bj 3 for 1 ≤ j ≤ q9 . for 1 ≤ j ≤ q9 and bj = bj 3 bj 4 By (7.149) and Remark 3.9 we obtain P ≈ 2

(q +q ) I2 1 4

 

11 01



(q2 +q6 ) 

1 −1 0 1

(q3 +q5 )

R(2θ1 )  · · ·  R(2θq7 )  · · ·  R(2θq7 +q8 ) N2 (ω1 , b1 )2  · · ·  N2 (ωq9 , bq9 )2  D(2)(q10 +q11 ) .

(7.150)

By Theorem 7.8 of [226], (7.147) and (7.150), there hold q2 + q6 ≥ p1 ,

q10 + q11 ≥ p2 .

(7.151)

Since γ 2 (t) = γ (t − τ )γ (τ ) for all t ∈ [τ, 2τ ] we have γ 2 is also the twice iteration of γ in the periodic boundary value case, so by the Bott-type formula (2.13), the proof of Lemma 4.1 of [232], and Lemma II.3.14 we have i(γ 2 ) + 2SP+2 (1) − ν(γ 2 ) √  = 2i(γ ) + 2SP+ (1) + (SP+ (e −1θ ) −(



θ∈(0,π )

(SP− (e

√

−1θ

) + (ν(P ) − SP+ (1)) + (ν−1 (P ) − SP− (−1)))

θ∈(0,π )

= 2i(γ ) + 2(q1 + q2 ) + (q8 − q7 ) − (q1 + q3 + q4 + q5 ) ≥ 2n + q1 + 2q2 + (q8 − q7 ) − (q3 + q4 + q5 ) = n + (2q1 + 3q2 + q6 + 2q8 + 2q9 + q10 + q11 )

7.4 The Mixed (L0 , L1 )-Concavity

213

≥ n + 2q2 + q6 + q10 + q11 ≥ n + p1 + p 2 ,

(7.152)

where in the first equality we have used SP+2 (1) = SP+ (1) + SP+ (−1) and ν(γ 2 ) = ν(γ ) + ν−1 (γ ), in the first inequality we have used the condition i(γ ) ≥ n, in the third equality we have used that q1 + q2 + · · · + q8 + 2q9 + q10 + q11 = n, in the last inequality we have used (7.151). By (7.152) Claim 1 holds. We continue with the proof of Theorem 4.3. We set A = μ(L1 ,L0 ) (γ ) + SP+2 (1) and B = μ(L0 ,L1 ) (γ ) + SP+2 (1). By (5.212) and (5.213) we have iL0 (γ ) + iL1 (γ ) = i(γ 2 ) − n,

νL0 (γ ) + νL1 (γ ) = ν(γ 2 ).

(7.153)

From (7.153) or (7.143) we obtain A + B = i(γ 2 ) + 2SP+2 (1) − ν(γ 2 ) − n.

(7.154)

Case 1. νL0 (γ ) = 0. In this case iL1 (γ ) + SP+2 (1) − νL0 (γ ) ≥ 0 + 0 − 0 = 0 and (7.146) holds. Case 2. νL0 (γ ) = n.   A 0 In this case P = , so A is invertible and CD m0 (AT C) = νL1 (P ) = νL1 (γ ).

(7.155)

By Lemma 3.11 we have N P −1 NP = ≈ I2m

0 (AT C)



In 0 2AT C In



− (AT C)

 N1 (1, 1)m

+ (AT C)

 N1 (1, −1)m

.

(7.156)

By Claim 1, (7.156) and (7.154), there holds A + B ≥ m− (AT C). By Theorem 3.4, Lemma 3.13 and (7.155) we obtain

(7.157)

214

7 Revisit of Maslov Type Index for Symplectic Paths

A − B ≥ m+ (AT C) + m0 (AT C) − n.

(7.158)

Then (7.157) and (7.158) give 2A ≥ m− (AT C) + (m+ (AT C) + m0 (AT C)) − n = 0 which yields A ≥ 0 and (7.146) holds. Case 3. 1 ≤ νL0 (γ ) = νL0 (P ) ≤ n − 1. In this case by (i) of Lemma 3.17 we have  P :=

AB CD



⎛

A1 ⎜ s 0 ∼⎜ ⎝ A3 C3

B1 D1 B3 D3

Ir 0 A2 C2

⎞ 0 0 ⎟ ⎟, 0 ⎠ D2

where A1 , A2 , A3 are r × r matrices, D1 , D2 , D3 are (n − r) × (n − r) matrices, B1 , B3 are r × (n − r) matrices, and C2 , C3 are (n − r) × r matrices. We divide Case 3 into the following 3 subcases. Subcase 1. A3 = 0. In this subcase let λ = rankB3 . Then 0 ≤ λ ≤ min{r, n − r}, A1 is invertible, A1 A2 = Ir and D1 D2T = Ik−r , so we have that A is invertible, furthermore there holds m0 (AT C) = dim ker C = νL1 (P ). Suppose m+ (A1 ) = p, m− (A1 ) = r − p, then by Lemma 3.19 we have Nk R

−1

 Nk R ≈

11 01



p+q − 

1 −1 0 1

(r−p+q + )

q 0

 I2

 D(2)λ , (7.159)

m+ (AT C) = λ + q + ,

(7.160)

m0 (AT C) = r − λ + q 0 ,

(7.161)

−

−

m (A C) = λ + q , T

(7.162)

where q ∗ ≥ 0 for ∗ = +, −, 0 and q + + q 0 + q − = n − r − λ. By (7.159) and Claim 1, there holds i(γ 2 ) + 2SP+2 (1) − ν(γ 2 ) ≥ n + p + q − + λ ≥ n + q − + λ.

(7.163)

Equation (7.163) and (7.154) give A + B ≥ q − + λ. By Theorem 3.4, Lemma 3.13, and (7.160)–(7.162), we have

(7.164)

7.4 The Mixed (L0 , L1 )-Concavity

215

A−B ≥ m+ (AT C) + m0 (AT C) − n = q+ + λ + r − λ + q0 − n = r + q + + q 0 − n.

(7.165)

Since q + + q 0 + q − = n − r − λ, (7.164) and (7.165) imply 2A ≥ q − + λ + (r + q + + q 0 ) − n = (q − + q + + q 0 ) − (n − r − λ) =0 which yields (7.146). Subcase 2. A3 is invertible. In this case by (ii) of Lemma 3.17 there holds 

s

P ∼

A1 Ir A3 A2

   D1 0  ˜ := P1  P2 , D3 D2

(7.166)

where D˜ 3 is a (k − r) × (k − r) matrix. Then by (7.166) and Lemma 3.12 we obtain P 2 ≈ (Nr P1−1 Nr P1 )  (Nn−r P2−1 Nn−r P2 ).

(7.167)

Let e(Nr P1−1 Nr P1 ) = 2m, by Lemma 3.16 we have 0 ≤ m ≤ r and 1 sgnMε (P1 ) ≤ r − m, 2

0 < −ε * 1.

(7.168)

Also by (7.167) and (7.150), there exists P˜1 ∈ Sp(2m) such that Nr P1−1 Nr P1 ≈ D(2)(r−m)  P˜1 .

(7.169)

By Lemma 3.11, there holds Nn−r P2−1

 Nn−r P2 = 

≈

11 01

0 In−r T ˜ 2D1 D3 In−r

m− (D1T D˜ 3 )



m0 (D1T D˜ 3 )  I2

 

1 −1 0 1

m+ (D1T D˜ 3 )

So by Claim 1 and (7.169), (7.170), (7.167) and (7.154) we have

. (7.170)

216

7 Revisit of Maslov Type Index for Symplectic Paths

A + B ≥ m− (D1T D˜ 3 ) + r − m.

(7.171)

By Theorem 3.4 and Lemma 3.13 together with Lemma 3.16, for 0 < −ε * 1 we get 1 1 A − B = − sgnMε (P1 ) − sgnMε (P2 ) 2 2 ≥ −r + m − (n − r) + m+ (D1T D˜ 3 ) + m0 (D1T D˜ 3 ) = m + m+ (D1T D˜ 3 ) + m0 (D1T D˜ 3 ) − n,

(7.172)

where we have used the fact m0 (D1T D˜ 3 ) = ker(D˜ 3 ) = νL1 (P2 ). Note that m+ (D1T D˜ 3 ) + m0 (D1T D˜ 3 ) + m− (D1T D˜ 3 ) = n − r.

(7.173)

Then by (7.171), (7.172) and (7.173) we have 2A ≥ m− (D1T D˜ 3 ) + r − m + (m + m+ (D1T D˜ 3 ) + m0 (D1T D˜ 3 )) − n = m+ (D1T D˜ 3 ) + m0 (D1T D˜ 3 ) + m− (D1T D˜ 3 ) − (n − r) =0 which yields (7.146). Subcase 3. 1 ≤ rankA3 = l ≤ r − 1. In this case by (iii) of Lemma 3.17 there holds

s

P ∼



U Il V



⎛ ˜ ˜ A1 B1 ⎜ 0 D1 ⎜ ⎝ 0 B˜ 3 C˜ 3 D˜ 3

Ir−l 0 A˜ 2 C˜ 2

⎞ 0 0 ⎟ ⎟ := P3  P4 , 0 ⎠ D˜ 2

(7.174)

where A˜ 1 , A˜ 2 are (r − l) × (r − l) matrices, B˜ 1 , B˜ 3 are (r − l) × (n − r) matrices, C˜ 2 , C˜ 3 are (n − r) × (r − l) matrices, D1 , D˜ 2 , D˜ 3 are (n − r) × (n − r) matrices, U, V ,  are l × l matrices, and  is invertible.   A˜ B˜ ˜ ˜ B, ˜ C, ˜ D˜ are (n − l)-order Let λ = rankB3 and denote P4 = ˜ ˜ , where A, CD real matrices. Assume m+ (A˜ 1 ) = p, m− (A˜ 1 ) = r − l − p, then by Lemma 3.19 we have

7.4 The Mixed (L0 , L1 )-Concavity

Nk P4−1 Nk P4

 ≈

11 01

217



(p+q − ) 

1 −1 0 1

(r−l−p+q + )

q 0

 I2

 D(2)λ , (7.175)

˜ = λ + q +, m+ (A˜ T C) ˜T

(7.176)

˜ =r −l−λ+q , m (A C)

(7.177)

˜ = λ + q −, m− (A˜ T C)

(7.178)

0

0

where q ∗ ≥ 0 for ∗ = +, −, 0 and q + + q 0 + q − = n − r − λ. Let e(Nl P3−1 Nl P3 ) = 2m, by Lemma 3.16 we obtain 0 ≤ m ≤ l and 1 sgnMε (P3 ) ≤ l − m, 2

0 < −ε * 1.

(7.179)

By similar argumet as in the proof of Subcase 2, there exists P˜3 ∈ Sp(2m) such that Nr P3−1 Nr P3 ≈ D(2)(l−m)  P˜3 .

(7.180)

So by Claim 1, (7.174), (7.175), (7.180), and (7.154) we have A + B ≥ q − + l − m + λ.

(7.181)

By Theorem 3.4, Lemma 3.13, (7.176), (7.177) and (7.179), for 0 ≤ −ε * 1 we obtain 1 1 A − B = − sgnMε (P3 ) − sgnMε (P4 ) 2 2 1 ˜ + m0 (A˜ T C) ˜ ≥ − sgnMε (P3 ) − (n − l) − m+ (A˜ T C) 2 ≥ −l + m − (n − l) + (λ + q + ) + (r − l − λ + q 0 ) = (q + + q 0 + r) − n − (l − m).

(7.182)

Since q + + q 0 + q − = n − r − λ, by (7.181) and (7.182) we have 2A ≥ q − + l − m + λ + (q + + q 0 + r) − n − (l − m) = (q + + q 0 + q − ) − (n − r − λ) =0 which yields (7.146). Hence (7.146) holds in Cases 1–3 and the proof of Theorem 4.3 is completed.  

218

7 Revisit of Maslov Type Index for Symplectic Paths

Remark 4.4 Both the estimates (7.145) and (7.146) in Theorem 5.1 are optimal. In fact, we can construct a symplectic path satisfying the conditions of Theorem 5.1 such that the equalities in (7.145) and (7.146) hold. Let τ = π and γ (t) = R(t)n , t ∈ [0, π ]. It is easy to see that iL0 (γ ) = νL0 (γ (t)) = 0 and also iL1 (γ ) =



0 0 such that |V  (t, x)| ≤ C, (t, x) ∈ [0, 1] × Rn .

(V∞ )

There exists a continuous symmetric matrix function C∞ (t) such that V  (t, x) = C∞ (t)x + o(|x|)

232

8 Applications of P -Index

uniformly in t ∈ R as |x| → ∞. Let C0 (t) = V  (t, 0). For m ∈ 2N ⎛ C0 (t) 0 · · · ⎜ 0 C0 (t) · · · ⎜ B0 (t) = ⎜ . .. ⎝ .. . ··· 0

0

0 0 .. .

⎞ ⎟ ⎟ ⎟, ⎠

(8.35)

· · · C0 (t)

and ⎛ ⎞ C∞ (t) 0 · · · 0 ⎜ 0 C∞ (t) · · · 0 ⎟ ⎜ ⎟ B∞ (t) = ⎜ . .. .. ⎟ . . ⎝ . . ··· . ⎠ 0 0 · · · C∞ (t)

(8.36)

J J J J Set (i0 , ν0 ) = (iM (B0 ), νM (B0 )), (i∞ , ν∞ ) = (iM (B∞ ), νM (B∞ )) with J = Am , M = Tm−1 defined before. For m ∈ 2N + 1,

⎞ 2C0 (t) −C0 (t) C0 (t) · · · C0 (t) −C0 (t) ⎜−C0 (t) 2C0 (t) −C0 (t) · · · −C0 (t) C0 (t) ⎟ ⎟ ⎜ B0 (t) = ⎜ . ⎟ .. .. .. .. . ⎠ ⎝ . . . . ··· . −C0 (t) C0 (t) −C0 (t) · · · −C0 (t) 2C0 (t) ⎛

(8.37)

and ⎛

⎞ 2C∞ (t) −C∞ (t) C∞ (t) · · · C∞ (t) −C∞ (t) ⎜−C∞ (t) 2C∞ (t) −C∞ (t) · · · −C∞ (t) C∞ (t) ⎟ ⎜ ⎟ B∞ (t) = ⎜ ⎟. .. .. .. .. .. ⎝ ⎠ . . . ··· . . −C∞ (t) C∞ (t) −C∞ (t) · · · −C∞ (t) 2C∞ (t)

(8.38)

J J J J Set (i0 , ν0 ) = (iM (B0 ), νM (B0 )), (i∞ , ν∞ ) = (iM (B∞ ), νM (B∞ )) with J = −1 Am−1 , M = B˜ m−1 defined before. If

/ [i0 , i0 + ν0 ], i∞ ∈

(8.39)

the delay differential system (8.24) possesses at least one nontrivial 2mτ -periodic solution x with x(t − mτ ) = −x(t). Furthermore, if the trivial solution z0 = 0 of problem (8.16), with H (t, z) = V (t, x1 ) + · · · + V (t, xm ) for even m and H (t, z) =

8.2 The Existence of Periodic Solutions for Delay Differential Equations

233

V (t, x1 ) + · · · + V (t, xm−1 ) + V (t, −x1 + x2 − · · · + xm−1 ) for odd m, is not pseudo-degenerated, and / [i0 − 2N, i0 + ν0 + 2N ], i∞ ∈

(8.40)

the delay differential system (8.24) possesses at least two nontrivial 2mτ -periodic solutions as above. Here and in the sequel, the number N in (8.40) is the dimension of the space where the corresponding first order Hamiltonian systems are considered. Proof Since V (t, −x) = V (t, x), z0 = 0 is the trivial solution of problem (8.16). Here H (t, z) = V (t, x1 ) + · · · + V (t, xm ), z = (x1 , · · · , xm ) for even m and H (t, z) = V (t, x1 ) + · · · + V (t, xm−1 ) + V (t, −x1 + x2 − · · · + xm−1 ), z = (x1 , · · · , xm−1 ) for odd m. It is clear that the condition (V1) implies (H2) in Theorem 1.1. For m ∈ 2N, |H  (t, z) − B∞ (t)z|  |V  (t, xj ) − C∞ (t)xj | |xj | = . |z| |xj | |z| m

j =1

We only need to deal with all xj = 0. Since V ∈ C 2 , (V1) implies that |V  (t,xj )−C∞ (t)xj | |xj |

|z| → ∞. Since

is bounded. So when |xj | |z|

|V  (t,xj )−C∞ (t)xj | |xj |

 0, then

|xj | |z|

→ 0 as

≤ 1, there holds |H  (t, z) − B∞ (t)z| → 0, as |z| → 0. |z|

The case m ∈ 2N + 1 is similar, we note that |z| → ∞ if and only if (x1 , · · · , xm−1 , −x1 + x2 − · · · + xm−1 ) → ∞. So the condition (V∞ ) implies   (H∞ ) in Theorem 1.1. Now the result follows from Theorem 1.1. Remark 2.6 (i) Since V (t, −x) = V (t, x), so the solutions of equation (8.24) appear in pairs. That is to say if x is a solution of (8.24), so is −x. (ii) When m = n = 1, the problem (8.24) was considered by Kaplan and Yorke in [15] with V  (t, x) = −f (x). Hence M = J = −J . In this case, for a constant J ¯ ¯ matrix function B(t) = aI d2 , the index iM (B) = 2j if (4j + 3)π (4j − 1)π 0 for x = 0 with f  (x) is bounded and with all other conditions in Theorem 1.1 of [15], the result is still true, i.e., there exists a pair nontrivial periodic solution x, −x of equation )π )π (8.19) with x(t + 2τ ) = −x(t). If α < (1−4j and β > (5−4j , then it is easy 2 2 to see (8.40) holds, so there exist at least two pair nontrivial periodic solutions. We note that the function H (t, z) is even in z since V (t, x) is even in x, so the solutions appear in pairs. Moreover, if ν0 = 0, Eq. (8.24) may possesses at least |i∞ − i0 | − 1 pairs nontrivial solutions under some conditions on the Hessian H  (t, z). Similarly, as consequences of Theorem 1.3, we have the following two results. Theorem 2.7 ([188]) Suppose V satisfies the condition (V∞ ) and the following conditions (V2)

There exist constants a > 0 and p > 1 such that |V  (t, x)| ≤ a(1 + |x|p ), ∀ (t, x) ∈ [0, 1] × Rn .

(V3)

The matrix function C0 (t) := V  (t, 0) satisfies V  (t, x) = C0 (t)x + o(|x|) as |x| → 0 uniformly in t ∈ [0, 1].

(V4)

For v(t, x) = V (t, x) − 12 (C∞ (t)x, x) with (t, x) ∈ R × Rn , either v(t, x) → 0, |v  (t, x)| → 0 as |x| → +∞ uniformly in t,

(8.41)

v(t, x) → ±∞, |v  (t, x)| = 0 as |x| → ∞ uniformly in t.

(8.42)

or

Then the delay differential system (8.24) possesses a nontrivial 2mτ -periodic solution x with x(t − mτ ) = −x(t), provided that J J J J J J (B0 ), iM (B0 )+νM (B0 )]∩[iM (B∞ ), iM (B∞ )+νM (B∞ )] = ∅, [iM

(8.43)

where in (8.43) when m ∈ 2N, J = Am , M = Tm−1 , B0 (t) and B∞ (t) are −1 defined as in (8.35) and (8.36), when m ∈ 2N+1, J = Am−1 , M = B˜ m−1 , B0 (t) and B∞ (t) are defined as in (8.37) and (8.38) with C0 (t) and C∞ (t) defined in (V3) and (V∞ ) respectively. Proof Take H (t, z) = V (t, x1 ) + · · · + V (t, xm ), z = (x1 , · · · , xm ) for even m and H (t, z) = V (t, x1 ) + · · · + V (t, xm−1 ) + V (t, −x1 + x2 − · · · + xm−1 ), z = (x1 , · · · , xm−1 ) for odd m in Theorem 1.3. It is easy to see that V satisfying the condition (V∞ ) implies H satisfying the condition (H∞ ). Also (V2) implies (H1), (V3) implies (H3) and (H4) follows from (V4).  

8.2 The Existence of Periodic Solutions for Delay Differential Equations

8.2.7.2

235

First Order Delay Hamiltonian Systems

For a function G ∈ C 2 (R × R2n , R) with G(t + τ, x) = G(t, x), we consider the periodic solutions of following first order delay Hamiltonian system x(t) ˙ = Jn (∇G(t, x(t − τ )) + ∇G(t, x(t − 2τ )) + · · · + ∇G(t, x(t − (m − 1)τ ))). (8.44) Let z = (x1 , · · · , xm ) and H (t, z) = G(t, x1 ) + · · · + G(t, xm ). The following results are also consequences of Theorems 1.1 and 1.3. Theorem 2.8 Suppose G satisfies the conditions (V1) and (V∞ ) in Theorem 2.5. Then the system (8.44) possesses an mτ -periodic solution x0 . Suppose z0 (t) is the solution of (8.32) corresponding to x0 . Let B0 (t) = H  (t, z0 (t)) and ⎛ ⎞ C∞ (t) 0 · · · 0 ⎜ 0 C∞ (t) · · · 0 ⎟ ⎜ ⎟ B∞ (t) = ⎜ . .. .. ⎟ . . ⎝ . . ··· . ⎠ 0 0 · · · C∞ (t) J J J J Set (i0 , ν0 ) = (iM (B0 ), νM (B0 )), (i∞ , ν∞ ) = (iM (B∞ ), νM (B∞ )) with J = −1 defined before. If (8.7) holds, the system (8.44) possesses at least Jn,m , M = Pn,m two mτ -periodic solutions. Furthermore, if z0 is not pseudo-degenerated, and (8.40) holds, then the system (8.44) possesses at least three an mτ -periodic solutions.

Theorem 2.9 ([188]) Suppose G satisfies the conditions (V2),(V3), (V4) and (V∞ ) in Theorem 2.7. Then the system (8.44) possesses a nontrivial solution, provided the −1 and the matrix functions twisted condition (8.43) holds with J = Jn,m , M = Pn,m B0 (t), B∞ (t) are defined as in (8.35), (8.36) with C0 (t) and C∞ (t) defined in (V3) and (V∞ ) respectively. The proofs of above two results are the same as in that of Theorems 2.5 and 2.7, respectively. We omit the details.

8.2.7.3

Second Order Delay Hamiltonian Systems

For a function V ∈ C 2 (R × Rn , R) with V (t + τ, x) = V (t, x), we consider the mτ -periodic solutions of following second order delay Hamiltonian system x(t) ¨ = −[∇V (t, x(t − τ )) + ∇V (t, x(t − 2τ )) + · · · + ∇V (t, x(t − (m − 1)τ ))]. (8.45) Let z = (x, y) = (x1 , · · · , xm , y1 , · · · , ym ), and H(t, z) = − 12 (Am y, y) − V (t, x1 ) − · · · − V (t, xm ). From Theorems 1.1 and 1.3, we obtain the following results.

236

8 Applications of P -Index

Theorem 2.10 ([188]) Suppose V satisfies the conditions (V1) and (V∞ ) in Theorem 2.5. Then the system (8.45) possesses an mτ -periodic solution x0 . Suppose w0 (t) is the solution of (8.34) corresponding to x0 . Let B0 (t) = H (t, w0 (t)), ⎛ ⎞ C∞ (t) 0 · · · 0   ⎜ 0 C∞ (t) · · · 0 ⎟ −R∞ (t) 0 ⎜ ⎟ , R∞ (t) = ⎜ . B∞ (t) = . .. ⎟ .. 0 −Am ⎝ .. . ··· . ⎠ 0 0 · · · C∞ (t) mn×mn J J J J Set (i0 , ν0 ) = (iM (B0 ), νM (B0 )), (i∞ , ν∞ ) = (iM (B∞ ), νM (B∞ )) with J = −1 Jmn , M = Pn,m defined in Sect. 8.2 after (8.34). If (8.7) holds, the system (8.45) possesses at least two solutions. Furthermore, if w0 is not pseudo-degenerated, and (8.40) holds, then the system (8.45) possesses at least three solutions.

Theorem 2.11 ([188]) Suppose V satisfies the conditions (V2),(V3), (V4) and (V∞ ) in Theorem 2.7. Then the system (8.45) possesses a nontrivial solution, provided the −1 , twisted condition (8.43) holds with J = Jmn , M = Pn,m ⎛ C0 (t) 0 · · ·   ⎜ 0 C0 (t) · · · −R0 (t) 0 ⎜ , R0 (t) = ⎜ . B0 (t) = .. 0 −Am ⎝ .. . ··· 0

0

0 0 .. .

· · · C0 (t)

⎞ ⎟ ⎟ ⎟ ⎠ mn×mn

and B∞ (t) defined as in Theorem 2.10.

8.3 The Minimal Period Problem for P -Symmetric Solutions In this section, we apply the P -index theory and its iteration theory to the P boundary problem of the following autonomous Hamiltonian system 

x˙ = J H  (x), x ∈ R2n , x(τ ) = P x(0),

(8.46)

where P ∈ Sp(2n) satisfying P k = I , here k is assumed to be the smallest positive integer such that P k = I (this condition for P is called (P )k condition in the sequel). And H (x) ∈ C 2 (R2n , R) satisfying H (P x) = H (x), H  denote the gradient of H . We note that the matrix P ∈ Sp(2n) satisfying P k = I is not necessary orthogonal " # a b symplectic, P = is an example with k = 3 and n = 1. A 2 − a +a+1 −a −1 b solution (τ, x) of the problem (8.46) is called P -solution of the Hamiltonian system

8.3 The Minimal Period Problem for P -Symmetric Solutions

237

(in [293] it was called a P -periodic orbit). Since P k = I , so the P -solution (τ, x) can be extended as a kτ -periodic solution (kτ, x k ). We say that a T -periodic solution (T , x) of a Hamiltonian system in (8.46) is P -symmetric if x( Tk ) = P x(0). T is the P -symmetric period of x. We say T to be the minimal P -symmetric period of x if T = min{τ > 0 | x(t + τk ) = P x(t), ∀t ∈ R}. For the minimal P -symmetric periodic problem, we have the following result. Theorem 3.1 ([194]) Suppose P ∈ Sp(2n) satisfying the (P )k condition, and the Hamiltonian function H satisfies the following conditions: (H1) (H2)

H ∈ C 2 (R2n , R) satisfying H (P x) = H (x), ∀x ∈ R2n . There exist constants μ > 2 and R0 > 0 such that 0 < μH (x) ≤ H  (x) · x, ∀ |x| ≥ R0 .

(H3) (H4)

H (x) = o(|x|2 ) at x = 0. H (x) ≥ 0 for all x ∈ R2n .

Then for every τ > 0, the system (8.46) possesses a non-constant P -solution (τ, x) satisfying dim kerR (P − I ) + 2 − ν P (x) ≤ i P (x) ≤ dim kerR (P − I ) + 1,

(8.47)

Moreover, if this solution x further satisfies the following conditions: (HC) H  (x(t)) > 0 for every t ∈ R. kτ Then the minimal P -symmetric period of x is kτ or k+1 . Suppose γ¯ (t) ∈ Pτ (2n) is the fundamental solution of the Hamiltonian system z˙ (t) = J B(t)z(t) with B(t) = H  (x(t)). If γ¯ ∈ / P Pτe (2n) = {γ¯ ∈ −1 e Pτ (2n)|P γ¯ (τ ) ∈ Sp (2n)}, then the minimal P -symmetric period of x is kτ , i.e., the P -symmetric periodic solution (kτ, x k ) generated from x possesses minimal P -symmetric period.

In order to get the information about the Maslov-type indices of a critical point, we need the following result which was proved in [110, 163, 274]. Theorem 3.2 Let E be a real Hilbert space with orthogonal decomposition E = X ⊕ Y , where dim X < +∞. Suppose f ∈ C 2 (E, R) satisfies the (PS) condition, and the following conditions: (F1)

There exist ρ and α > 0 such that f (w) ≥ α, ∀w ∈ ∂Bρ (0) ∩ Y.

(F2)

There exist e ∈ ∂B1 (0) ∩ Y and R > ρ such that f (w) < α, ∀w ∈ ∂Q.

238

8 Applications of P -Index

where Q = (BR (0) ∩ X) ⊕ {re | 0 ≤ r ≤ R}. Then (1) f possesses a critical value c ≥ α, which is given by c = inf max f (h(w)), h∈ w∈Q

where  = {h ∈ C(Q, E) | h = id on ∂Q}. (2) If f  (w) is Fredholm for w ∈ Kc (f ) ≡ {w ∈ E : f  (w) = 0, f (w) = c}, then there exists an element w0 ∈ Kc (f ) such that the negative Morse index m− (w0 ) and nullity m0 (w0 ) of f at w0 satisfy m− (w0 ) ≤ dim X + 1 ≤ m− (w0 ) + m0 (w0 ).

(8.48)

(3) Suppose that there is an S 1 action on E, f is S 1 -invariant, and for w0 defined in (2) the set S 1 ∗ w0 is not a single point. Then (8.48) can be further improved to m− (w0 ) ≤ dim X + 1 ≤ m− (w0 ) + m0 (w0 ) − 1.

(8.49)

Let WP = γP W 1/2,2 (Sτ , R2n ), it is the space of all W 1/2,2 functions z defined in R satisfying z(t + τ ) = P z(t). Its is an inner product space with inner product ·, ·

and norm # · #. We will denote the Ls norm by # · #s for s ≥ 1. The space WP can be continuously embedded into Ls ([0, τ ], R2n ), i.e. there is αs > 0 such taht #z#s ≤ αs #z#, ∀z ∈ WP .

(8.50)

Let A and B be the self-adjoint operators defined on WP by the following bilinear forms: τ τ Ax, y = (−J x(t), ˙ y(t))dt, Bx, y = (B(t)x(t), y(t))dt. (8.51) 0

0

Suppose that · · · ≤ λ−k ≤ · · · ≤ λ−1 < 0 < λ1 ≤ · · · ≤ λk ≤ · · · are all nonzero eigenvalues of the operator A(count with multiplicity), and correspondingly, ej is the eigenvector of λj satisfying ej , el = δj l . We denote the kernel of the operator A by WP0 which is exactly the space kerR (P − I ). We define the subspaces of WP by WPm = Wm− ⊕ WP0 ⊕ Wm+

m + with Wm− = {z ∈ WP | z(t) = j =1 a−j e−j (t), a−j ∈ R} and Wm = {z ∈

m WP | z(t) = j =1 aj ej (t), aj ∈ R}. For z ∈ WP , we define

8.3 The Minimal Period Problem for P -Symmetric Solutions

1 f (z) = 2



kτ



kτ

(−J z˙ (t), z(t))dt −

0

239

 H (z)dt = k

0

1 Az, z − 2



τ

 H (z)dt .

0

(8.52)

It is well known that f ∈ C 2 (WP , R) whenever H ∈ C 2 (R2n , R) and |H  (z)| ≤ a1 |z|s + a2

(8.53)

for some s ∈ (1, +∞) and all z ∈ R2n . Looking for solutions of (8.46) is equivalent to looking for critical points of f on WP . Proof of Theorem 3.1. We carry out the proof in several steps. Step 1. Since the growth condition (8.53) has not been assumed for H , we need to truncate the function H suitably to get a function HK satisfying the condition (8.53). We follow the method in Rabinowitz’s pioneering work [254] (cf. also [96] and [255]). Let K > R0 and select χ ∈ C ∞ (R, R) such that χ (y) ≡ 1 if y ≤ K, χ (y) ≡ 0 if y ≥ K + 1, and χ  (y) < 0 if y ∈ (K, K + 1), where K is free for now. Set HK (z) = χ (|z|)H (z) + (1 − χ (|z|))RK |z|4 ,

(8.54)

where the constant RK satisfies RK ≥

H (z) . K≤|z|≤K+1 |z|4 max

Then HK ∈ C 2 (R2n , R), satisfies (H3), (H4) and (8.53) with s = 2. Moreover a straightforward computation shows (H2) hold with μ replaced by ν = min{μ, 4}, i.e., there exist R0 > 0 such that 0 < νHK (z) ≤ HK (z) · z, ∀ |z| ≥ R0 .

(8.55)

Since HK ∈ C 2 (R2n , R), then HK (z) is bounded for |z| ≤ R0 . Thus for K > R0 there exist positive constant K1 , K2 independent of K such that RK ν|z|4 − K1 ≤ νHK (z) ≤ HK (z) · z + K2 , ∀z ∈ R2n

(8.56)

via (8.54) the form of HK and (8.55). Integrating (8.55) then yields HK (z) ≥ a3 |z|ν − a4

(8.57)

for all z ∈ R2n , where a3 , a4 > 0 are independent of K. Define a functional fK on WP by ' kτ ' kτ fK (z) = 12 0 (−J z˙ (t), z(t))dt − 0 H (z)dt ' kτ = k2 Az, z − 0 HK (z)dt, ∀z ∈ WP ,

(8.58)

240

8 Applications of P -Index

then fK ∈ C 2 (WP , R). Step 2. For m > 0, let fK,m = fK |WPm . We will show that fK,m satisfies the hypotheses of Theorem 3.2. In fact, by (H3), for any  > 0, there is a δ > 0 such that HK (z) ≤ |z|2 for |z| ≤ δ. Since HK (z)|z|−4 is uniformly bounded as |z| → +∞, there is an M1 = M1 (, K) such that HK (z) ≤ M1 |z|4 for |z| ≥ δ. Hence HK (z) ≤ |z|2 + M1 |z|4 , ∀z ∈ R2n .

(8.59)

Therefore by (8.59) and the Sobolev embedding theorem,

kτ 0

$ % HK (z)dt ≤ CK #z#22 + M1 #z#44 ≤ CK (α2 +M1 α4 #z#2 )#z#2 ,

(8.60)

where CK is a constant depending on K. Let Xm = Wm− ⊕ WP0 , Ym = Wm+ .

(8.61)

Consequently for z ∈ Ym , we have k fK,m (z) = Az, z − 2



kτ

HK (z)dt 0

(8.62)

kλ1 ≥ #z#2 − CK (α2 + M1 α4 #z#2 )#z#2 . 2 So there are constants ρ = ρ(K) > 0 and α = α(K) > 0, which are sufficiently small and independent of m, such that fK,m (z) ≥ α, ∀z ∈ ∂Bρ (0) ∩ Ym .

(8.63)

Let e = e1 ∈ ∂B1 (0) ∩ Ym and set Qm = {re | 0 ≤ r ≤ r1 } ⊕ (Br1 ∩ Xm ), where r1 is free for the moment. Let z = z− + z0 ∈ Wm− ⊕ WP0 , then fK,m (z + re)

kτ k k HK (z + re)dt = Az− , z− + r 2 Ae, e − 2 2 0 kτ kλ−1 − 2 kλ1 2 #z # + r − HK (z + re)dt. ≤ 2 2 0 If r = 0, from condition (H4), there holds

(8.64)

8.3 The Minimal Period Problem for P -Symmetric Solutions

fK,m (z + re) ≤

241

kλ−1 − 2 #z # ≤ 0. 2

(8.65)

If r = r1 or #z# = r1 , by (8.57), there holds

kτ



τ

HK (z + re)dt ≥

0

0



τ

HK (z + re)dt ≥ a3



≥ a5 = a5 (

0 τ

τ

ν/2 |z + re|2 dt

|z + re|ν dt − τ a4

0

− a6

(8.66)

(|z0 |2 + |z− |2 + r 2 |e|2 )dt)ν/2 − a6

0

≥ a7 (|z0 |ν + r ν ) − a6 . Combining (8.66) with (8.64) yields fK,m (z + re) ≤

kλ−1 − 2 kλ1 r 2 + #z # − a7 (#z0 #ν + r ν ) + a6 . 2 2

So we can choose r1 large enough which is independent of K and m such that fK,m (z + re) ≤ 0, ∀z ∈ ∂Qm .

(8.67)

Next we will show that fK,m satisfies (P.S) condition on WPm for m > 0, i.e., any sequence {zj } ⊂ WPm possesses a convergent subsequence in WPm , provided  (zj ) → 0 as j → ∞. We suppose #fK,m (zj )# ≤ C, fK,m (zj ) is bounded and fK,m then for large j : 1  C + #zj # ≥ fK,m (zj ) − fK,m (zj )zj 2 kτ 1 = [ HK (zj ) · zj − HK (zj )]dt 2 0 kτ −1 −1 ≥ ν(2 − ν ) HK (zj )dt − C1

(8.68)

0

≥ ν(2−1 − ν −1 )



τ

HK (zj )dt − C1

0

≥ C2 #zj #44 − C3 via (8.56). In (8.68), C1 is independent of K, but both C2 and C3 depend on K. So {zj } is bounded in WPm . Since WPm is finite dimensional, the sequence {zj } has a convergent subsequence.

242

8 Applications of P -Index

We have already verified all the conditions of Theorem 3.2, then fK,m has a critical value cK,m ≥ α, which is given by cK,m = inf max fK,m (g(w)), g∈m w∈Qm

(8.69)

where m = {g ∈ C(Qm , WPm ) | g = id on ∂Qm }. Note that there is a natural S 1 -invariant on WP and WPm defined by θ ∗ x(t) = x(t + θ ), ∀x ∈ WP , θ ∈ [0, kτ ]/{0, kτ } = S 1 .

(8.70)

 (x) is Fredholm for any critical Now since WPm is finite dimensional and then fK,m 1 point x, and fK,m is S -invariant under the above S 1 -action (8.70) on WPm . So there is a critical point xK,m of fK,m which satisfies

m− (xK,m ) ≤ dim Xm +1 = m+dim kerR (P −I )+1 ≤ m− (xK,m )+m0 (xK,m )−1. (8.71) Step 3. We prove that there exists a nonconstant P -solution (τ, xK ) of the following problem 

x˙ = J HK (x), x(τ ) = P x(0).

(8.72)

On the one hand, since id ∈ m , by (8.64) and (H4) we have cK,m ≤ sup fK,m (w) ≤ w∈Qm

kλ1 2 r . 2 1

(8.73)

kλ1 2 r . 2 1

(8.74)

Then in the sense of subsequence we have cK,m → cK , α ≤ cK ≤

On the other hand, we need to prove that fK satisfies the (P.S)∗ condition on WP , i.e., any sequence {zm } ⊂ WP satisfying zm ∈ WPm , fK,m (zm ) is bounded and  fK,m (zm ) → 0 possesses a convergent subsequence in WP . It is a well-known result in the case of general periodic solution. For the reader’s convenience, we give the proof following the idea in the appendix of [14].  fK,m (zm ) → 0 as m → +∞ implies − J z˙ m − Pm HK (zm ) = m , with

(8.75)

8.3 The Minimal Period Problem for P -Symmetric Solutions

243

#m #(WPm ) → 0, as m → +∞. 0 + z+ + z− . We remind that the dual space of W is denoted by W  . Writing zm = zm m m Using the same arguments as (8.68) and by some direct estimates, we see that {zm } is bounded in WP . Thus by passing to a subsequence, we may assume that

zm → z in WP weakly, zm → z in Lp strongly for 1 ≤ p < +∞, 0 zm → z0 in R2n .

By (8.54) the form of HK , there exists constant M2 such that |HK (z)| ≤ M2 |z|3 + M2 , ∀z ∈ R2n . This implies HK (zm ) → HK (z) strongly in L2 . Thus Pm HK (zm ) → HK (z) strongly in L2 , and thus in WP . Therefore (8.75) implies that z˙ m = ςm + m

(8.76)

holds in WP , where ςm → ς = J HK (z) in L2 . This implies z˙ = ς

(8.77)

in WP . Since ς ∈ L2 , then z ∈ W 1,2 and thus z ∈ C 2 , i.e., (8.77) holds in the classical sense. Because WPm is a subspace of WP , Pm : WP → WPm is projection. 0 #zm − Pm z#2W m = #˙zm − Pm z˙ #2(W m ) + |zm − z 0 |2 . P

P

Then 0 − z 0 |2 . #zm − Pm z#2W m ≤ (#ςm − Pm ς #(WPm ) + #m #(WPm ) )2 + |zm P

From #ςm − Pm ς #(WPm ) ≤ M3 #ςm − Pm ς #L2 → 0 for some M3 > 0 independent of m, we then obtain #zm − Pm z#2 = #zm − Pm z#2W m → 0. P

This proves zm → z in W strongly. We have thus proved that fK satisfies (P.S)∗ condition. Hence in the sense of the subsequence we have

244

8 Applications of P -Index

xK,m → xK , fK (xK ) = cK , fK (xK ) = 0.

(8.78)

From above we conclude that fK possesses a critical value cK ≥ α = α(K) > 0 with a corresponding critical point xK . By the standard arguments as (6.35)–(6.37) in [255], xK is a classical nonconstant P -solution of (8.72). Indeed, if xK (t) is a constant solution of (8.72), then it should belong to kerR (P − I ) and k fK (xK ) = AxK , xK − 2



kτ

HK (xK )dt ≤ 0.

0

This contradicts to fK (xK ) = cK ≥ α > 0. Step 4. We show that there is a K0 > 0 such that for all K ≥ K0 , #xK #L∞ < K. Then HK (xK ) = H  (xK ) and x = xK is a nonconstant P -solution of (8.46). By (8.74), cK ≤ kλ2 1 r12 independently of K. By (8.56), we obtain λ1 2 1 r1 ≥ fK (xK ) − fK (xK )xK 2 2 τ −1 −1 ≥ (2 − ν ) HK (xK ) · xK dt − C.

(8.79)

0

with C = ν −1 K2'τ is independent of K. Therefore (8.79) provides a K independent τ upper bound for 0 HK (xK ) · xK dt. By (8.56), HK (ζ ) ≤ ν −1 HK (ζ ) · ζ + C/τ, ∀ζ ∈ R2n .

(8.80)

Recalling that HK (xK ) ≡ constant since xK satisfies an autonomous Hamiltonian system, so replacing ζ by xK , integrating (8.80) over [0, τ ], (8.80) yields kτ HK (xK ) ≤ ν

−1

0

kτ

HK (xK ) · xK + C.

(8.81)

The right hand side of (8.81) is bounded from above independently of K. Then (8.57) and (8.81) yield a K independent L∞ bound for xK . So choose K large such that #xK #L∞ < K thus xK is a P -solution of the problem (8.46). We denote it simply by x. Step 5. We prove that dim kerR (P − I ) + 2 − ν P (x) ≤ i P (x) ≤ dim kerR (P − I ) + 1. Let B = H  (x(t)) and B be the operator defined by (8.51) corresponding to B(t). By direct computation, we get

8.3 The Minimal Period Problem for P -Symmetric Solutions

245

fK (z)w, w − k (A − B)w, w

kτ [(HK (xK (t))w, w) − (HK (z(t))w, w)]dt, ∀w ∈ W. = 0

Then by the continuity of HK , #fK (z) − k(A − B)# → 0 as # z − xK #→ 0.

(8.82)

Let d = 14 #(A − B) #−1 . By (8.82), there exists r0 > 0 such that #fK (z) − k(A − B)#
min{λ > 0 | x(t + λ) = P x(t), ∀t ∈ R}, then there exists some l such that T ≡

τ = min{λ > 0 | x(t + λ) = P x(t), ∀t ∈ R}. l

Thus x(τ − T ) = x(0), both (l − 1)T and kT are the period of x. Since kT is the minimal P -symmetric period, we obtain kT ≤ (l − 1)T and then k ≤ l − 1. Note that x|[0,kT ] is the k-th iteration of x|[0,T ] . Suppose γ ∈ PT (2n) is the fundament solution of the following linear Hamiltonian system z˙ (t) = J B(t)z(t) with B(t) = H  (x|[0,T ] (t)). Suppose ξ be any symplectic path in PT (2n) such that ξ(T ) = P −1 , since P k = I , then ν(ξ, 1) = ν(ξ, k + 1) = ν(ξ, l).

(8.86)

All eigenvalues of P and P −1 are all on the unit circle, then the elliptic height e(P −1 ) = e(P ) = 2n.

(8.87)

Since the system (8.46) is autonomous, we have ν1 (x|[0,kT ] ) ≥ 1 and ν P

l−1

(γ , l − 1) = ν1 (x|[0,(l−1)T ] ) ≥ 1.

By Corollary IV.6.15, P l−1 = I and (8.86)–(8.87), we have

(8.88)

8.3 The Minimal Period Problem for P -Symmetric Solutions

i I (γ , l − 1) = i P

l−1

(γ , l − 1)

l

≤ i P (γ , l) − i P (γ , 1) + ν(ξ, 1) − ν(ξ, l) + +

247

e(P −1 γ (T )) 2

e(P −1 ) l−1 − ν P (γ , l − 1) 2 l

≤ i P (γ , l) − i P (γ , 1) +

(8.89)

e(P −1 γ (T )) +n−1 2

l

≤ i P (γ , l) − i P (γ , 1) + 2n − 1. l

P (x ) ≤ dim ker (P − I ) + 1, here we write i P (x ) for Note that i P (γ , l) = i[0,τ R ] K [0,τ ] K i P (xK ) to remind the solution xK is defined in the interval [0, τ ]. By the definition of Maslov P -index,

i I (γ , l − 1) = i1 (γ , l − 1) + n. So we get i1 (γ , l − 1) ≤ dim kerR (P − I ) − i P (γ , 1) + n.

(8.90)

By the condition (HC) and (4.39) in Lemma IV.2.4, we have i P (γ , 1) = i P (B) =

 s∈[0,1)

ν P (sB) =



dim kerR (γB (sT ) − P ).

(8.91)

s∈[0,1)

Here we remind that B(t) = H  (x|[0,T ] (t)) and γB is the fundamental solution of the linear Hamiltonian system z˙ (t) = J B(t)z(t). Since γB (0) = I , so dim kerR (γB (sτ ) − P ) = dim kerR (P − I ) when s = 0. Thus we have i P (γ , 1) ≥ dim kerR (P − I ).

(8.92)

i1 (γ , l − 1) ≤ n.

(8.93)

From (8.90), it implies

By the convex condition (HC) We also have i1 (x|[0,kT ] ) ≥ n and i1 (x|[0,(l−1)T ] ) ≥ n.

(8.94)

248

8 Applications of P -Index

We set m = l−1 k . Note that x|[0,(l−1)T ] is the m-th iteration of x|[0,kT ] . By (8.88), (8.94), (8.93) and Lemma 4.1 in [197], we obtain m = 1 and then k = l − 1. From the process of the proof, we see that only if e(P −1 γ (T )) = 2n, we kτ can obtain k = l − 1. In this case, the minimal P -symmetric period of x is k+1 . Note that γ¯ (τ ) = P l−1 γ (T )(P −1 γ (T ))l−1 = P (P −1 γ (T ))l . So we have e(P −1 γ (T )) = e((P −1 γ (T ))l ) = e(P −1 γ¯ (τ )). If γ¯ ∈ / P Pτe (2n), then −1 e(P γ (T )) ≤ 2n−2. We get i1 (γ , l−1) < n by taking the same process as (8.89)– (8.90). It contradicts to the second inequality of (8.94). At the moment, the minimal P -symmetric period of x is kτ . We set Sp(2n)k ≡ {P ∈ Sp(2n) | P satisf ies (P )k condition}, In the follows we consider for what P ∈ Sp(2n)k the minimal P -symmetric period of x is kτ .   j1  Lemma 3.3 ([216]) If P ∈ Sp(2n)k , then there exists a matrix I2p  R( 2π k )

r 2rπ jr −1 · · ·  R( k ) ∈ 0 (P ), with p + m=1 jm = n.

Proof For P ∈ Sp(2n)k , we have σ (P −1 ) = σ (P ) ⊆ {1, e

√ 2π −1 k

,e

√ 4π −1 k

,··· ,e

√ 2(k−1)π −1 k

} ⊆ U.

By the Theorem 1.8.10 of [223], there exists M1 (ω1 )  M2 (ω2 )  · · ·  Ms (ωs ) ∈ 0 (P −1 ) where Mi (ωi ) is a basic normal form of a eigenvalue ωi of P −1 , 1 ≤ i ≤ s. And the following are the basic normal forms for eigenvalues in U.   λb Case 1. N1 (λ, b) = , λ = ±1, b = ±1, 0. 0λ = 0 and λ ∈ {−1, 1} ∩ σ (P −1 ). Since P ∈ Sp(2n)  k , we have b  cos θ − sin θ Case 2. R(θ ) = , θ ∈ (0, π ) ∪ (π, 2π ). sin θ cos θ , 4π , · · · 2(k−1)π }. Since P ∈ Sp(2n)k , we have θ ∈ { 2π k   k k  R(θ ) b b1 b2 , bi ∈ Case 3. N2 (ω, b) = , θ ∈ (0, π ) ∪ (π, 2π ), b = b3 b4 0 R(θ ) R, b2 = b3 . ⎛ ⎞ √ √1

2 From direct computation, it is easy to check that the matrix S = ⎝ √−1

" satisfies that SR(θ )S −1 =

√

e

−1θ

0

# 0

√

e−

−1θ

. Then there holds

√

2

√−1 2 √1 2

⎠

8.3 The Minimal Period Problem for P -Symmetric Solutions



S 0 0S





S −1 0 N2 (ω, b) 0 S −1



 =

249

SR(θ )S −1 SbS −1 0 SR(θ )S −1

 ,

where " SbS

−1

=

1 2 (b1 1 2 (b2

+ b4 ) − + b3 ) +

√

−1 (b2 √2 −1 2 (b1

− b3 ) 12 (b2 + b3 ) − − b4 ) 12 (b1 + b4 ) +

√ −1 (b1 √2 −1 2 (b2

− b4 ) − b3 )

# . (8.95)

Denoted by 

X(i) SR(iθ )S −1 0 SR(kθ )S −1 

x1 (i) x2 (i) x3 (i) x4 (i) computation, we have

where X(i) =





 =

SR(θ )S −1 SbS −1 0 SR(θ )S −1

and X(1) = SbS −1 =

x1 (k) = ke x4 (k) = ke

√

−1(k−1)θ



i , i ∈ N,

 x1 (1) x2 (1) . By direct x3 (1) x4 (1)

x1 (1),

√ − −1(k−1)θ

(8.96)

(8.97)

x4 (1).

Thus from P k = I we have X(k) = 0, so x1 (1) = x4 (1) = 0, i.e. √

−1 (b2 − b3 ) = 0, 2 √ 1 −1 (b1 + b4 ) + (b2 − b3 ) = 0. 2 2

1 (b1 + b4 ) − 2

Therefore we have b2 = b3 , which is contradict to the definition of the basic normal form N2 (ω, b). 2(k−1)π 4π From Case1 - Case3, we get Mi (ωi ) = R(θi ) where θi ∈ {0, 2π }, k , k ,··· k 1 ≤ i ≤ s. And the lemma is proved.   For the notations in Lemma 3.3, we define the set of k-admissible symplectic matrices by Sp(2n)ad k

≡ P ∈ Sp(2n)k | k − 2

r  m=1

( k m · jm > 1, r < . 2

Theorem 3.4 ([216]) Suppose P ∈ Sp(2n)ad k ∪ {I }, and the Hamiltonian function H satisfies (H1)–(H4) and (HC), then for every τ > 0, the system (8.72) possesses a non-constant P -solution (τ, x) such that the minimal P -symmetric period of the extended kτ -periodic solution (kτ, x k ) is kτ .

250

8 Applications of P -Index

We remind that when P = I , it implies k = 1. This is the case of periodic solution which was proved in [66] by D. Dong and Y. Long, and in [197] by C. Liu and Y. Long. So we only need to prove the case when P ∈ Sp(2n)ad k . −1 , there Lemma 3.5 ([216]) For P ∈ Sp(2n)ad k and ξ ∈ Pτ (2n) with ξ(τ ) = P holds

(k + 1)i(ξ ) − i(ξ, k + 1) =

r 

(k − 2m)jm − kp.

(8.98)

m=1

Proof By Theorem 9.3.1 of [223], we have i(ξ, k + 1) = (k + 1)(i(ξ ) + SP+−1 (1) − C(P −1 )) 

+2

E(

θ∈(0,2π )

√ (k + 1)θ − )SP −1 (e −1θ ) − (SP+−1 (1) + C(P −1 )) 2π

= (k + 1)i(ξ ) + kSP+−1 (1) − (k + 2)C(P −1 ) 

+2

θ∈(0,2π )

E(

√ (k + 1)θ − )SP −1 (e −1θ ), 2π

(8.99) ± (ω) denote the splitting number of M ∈ Sp(2n) at ω ∈ U, C(M) = where SM √

− −1θ ) and E(a) = min{m ∈ Z | m ≥ a}. S (e θ∈(0,2π ) M 2π j1 jr ∈ 0 (P −1 ) with 2r < k.  · · ·  R( 2rπ For P ∈ Sp(2n)ad k , I2p  R( k ) k ) By direct computation, for θ ∈ (0, π ) ∪ (π, 2π ), we have SP+−1 (1) = p ; C(P −1 ) =

r 

(8.100)

jm ;

m=1

√ SP−−1 (e −1θ )

=

jm , if θ = 0,

2mπ k ,1

≤ m ≤ r, e

√ 2mπ −1 k

∈ σ (P −1 ) ;

otherwise. (8.101)

So there holds 

E(

θ∈(0,2π )

= E(

√ (k + 1)θ − )SP −1 (e −1θ ) 2π

2(k + 1) r(k + 1) k+1 )j1 + E( )j2 + · · · + E( )jr k k k

= 2j1 + 3j2 + · · · + (r + 1)jr .

(8.102)

8.3 The Minimal Period Problem for P -Symmetric Solutions

251

Then we have i(ξ, k + 1) = (k + 1)i(ξ ) + kp − (k + 2)(j1 + j2 + · · · + jr ) + 2(2j1 + 3j2 + · · · + (r + 1)jr ) = (k + 1)i(ξ ) + kp − (k − 2)j1 − (k − 4)j2 − · · · − (k − 2r)jr , (8.103) r  thus (k + 1)i(ξ ) − i(ξ, k + 1) = (k − 2m)jm − kp.   m=1

Proof of Theorem 3.4 Following the proof of Theorem 3.1. If the minimal P kτ symmetric period of x is k+1 , then there hold e(P −1 γ (T )) = 2n, and i P (γ , 1) = dim kerR (P − I ); i P (γ , l) = i P (γ , k + 1) = dim kerR (P − I ) + 1,

(8.104)

k

ν P (γ , k) = ν P (γ , 1) = 1. Here we remind that the left inequality in (4.62) of Proposition IV.6.13 holds independent of the choice of ξ ∈ Pτ (2n), then for any ξ ∈ Pτ (2n) we have i P (γ , k + 1) ≥ (k + 1)(i P (γ , 1) + ν P (γ , 1) − n) + n − 1 + (k + 1)i1 (ξ ) − i(ξ, k + 1). (8.105) By the condition P ∈ Sp(2n)ad , we get k dim kerR (P − I ) = 2p, k−2

r 

m · jm > 1.

(8.106) (8.107)

m=1

Applying (8.104), (8.106) and Lemma 3.5 to (8.105), we get k−2

r 

m · jm ≤ 1.

(8.108)

m=1

It is contradict to the inequality (8.107). So the minimal P -symmetric period of   (kτ, x k ) is kτ . " # a b We note that the matrix P = ∈ Sp(2)ad 3 for b > 0 with a 2 +a+1 − b −a − 1

0 −1 ). Thus for the solution x found by (r, p; j1 ) = (1, 0; 1), i.e., R( 2π 3 ) ∈  (P variational method with the saddle point theorem as in Theorem 3.1, its three times iteration (3τ, x 3 ) is the minimal P -symmetric periodic solution.

Chapter 9

Applications of L-Index

In this chapter, we apply the L-index theory developed in Chap. 5 to study the existence and multiplicity of L-solutions of nonlinear Hamiltonian systems. In Sect. 9.1, we consider the existence of brake solution of asymptotically linear Hamiltonian system via the variational method and the L-index theory. In Sect. 9.2, by using the iteration theory of the L-index, we consider the minimal periodic problem for brake solutions of super-quadratic autonomous Hamiltonian systems. In Sect. 9.3, we obtain an infinitely many brake solutions of super-quadratic nonautonomous Hamiltonian systems.

9.1 The Existence of L-Solutions of Nonlinear Hamiltonian Systems In this section, we consider the solutions of the following nonlinear Hamiltonian systems with Lagrangian boundary condition 

x(t) ˙ = J H  (t, x(t)), x(t) ∈ R2n , x(0) ∈ L, x(1) ∈ L,

(9.1)

where L ∈ (n), here (n) is the set of all Lagrangian subspaces of (R2n , ω0 ). The Hamiltonian function H ∈ C 2 ([0, 1] × R2n , R) satisfying condition (H0 ) H  (t, 0) ≡ 0, t ∈ [0, 1]. (H∞ ) There exist continuous symmetric matrix functions B1 (t), B2 (t) with iL (B1 ) = iL (B2 ), νL (B2 ) = 0 such that B1 (t) ≤ H  (t, x) ≤ B2 (t), ∀(t, x) with |x| ≥ r for some large r > 0, ∀ t ∈ [0, 1].

© Springer Nature Singapore Pte Ltd. 2019 C. Liu, Index theory in nonlinear analysis, https://doi.org/10.1007/978-981-13-7287-2_9

253

254

9 Applications of L-Index

For two symmetric matrices A and B, A ≤ B means that A − B is a semi-positive definite matrix, and A < B means that A − B is a positive definite matrix similarly. We have the following result. Theorem 1.1 Let H satisfy conditions (H0 ) and (H∞ ). Suppose J B1 (t) = B1 (t)J and B0 (t) = H  (t, 0) satisfying one of the following twisted conditions B1 (t) + kI ≤ B0 (t),

(9.2)

B0 (t) + kI ≤ B1 (t),

(9.3)

or

with the constant k ≥ π . Then (9.1) possesses at least one nontrivial solution. Further more if νL (B0 ) = 0, the system (9.1) possesses at least two nontrivial solutions. We use the following result to prove Theorem 1.1. Lemma 1.2 (Theorem 5.1 and Corollary II.5.2 of [33]) Suppose f ∈ C 2 (L, R) satisfies the (PS) condition, f  (0) = 0 and there is r ∈ / [m− (f  (0)), m− (f  (0)) + 0  m (f (0))] with Hq (L, fa ; R) ∼ = δq,r R, then f has at leat one nontrivial critical point u1 = 0. Moreover, if m0 (f  (0)) = 0 and m0 (f  (u1 )) ≤ |r − m− (f  (0))|, then f has one more nontrivial critical point u2 = u1 . Without loss any generality we can suppose H (t, 0) = 0 and L = L0 . By the condition (H∞ ) and the remark after Corollary V.6.2, we get that iL (B1 )+νL (B1 ) ≤ iL (B2 ) + νL (B2 ), so we have νL (B1 ) = 0. We shall first prove that under the above conditions (9.2) or (9.3), there holds / [iL (B0 ), iL (B0 ) + νL (B0 )]. iL (B1 ) ∈

(9.4)

More clearly, under the condition (9.2), it is claimed iL (B1 ) = iL (B1 ) + νL (B1 ) < iL (B0 ),

(9.5)

and under the condition (9.3), it is claimed iL (B0 ) + νL (B0 ) < iL (B1 ).

(9.6)

We first prove (9.5). By Corollary 2.2 and condition (9.2), we have iL (B1 ) ≤ iL (B1 + kI ) ≤ iL (B0 ). We shall prove iL (B1 ) < iL (B1 + kI ).

(9.7)

9.1 The Existence of L-Solutions of Nonlinear Hamiltonian Systems

255

 S1 (t) V1 (t) ∈ Pτ (2n) is a symplectic path which is In fact, suppose γ1 (t) = T1 (t) U1 (t) the fundamental solution of the linear Hamiltonian system associate with the matrix function B1 (t). Since J B1 (t) = B1 (t)J , one can show that exp(J kt)γ1 (t) is the fundamental solution of the linear Hamiltonian system 

z˙ = J (B1 (t) + kI )z.

(9.8)

One has  S (t) cos kt − T1 (t) sin kt exp(J kt)γ1 (t) = 1 S1 (t) sin kt + T1 (t) cos kt

 V1 (t) cos kt − U1 (t) sin kt . V1 (t) sin kt + U1 (t) cos kt

The associate unitary n × n matrix Q(t) defined by (5.6) with respect to the above matrix is Q(t) = [U1 (t) −

√

−1V1 (t)][U1 (t) + √ = Q1 (t) exp(2k −1t).

√

√ −1V1 (t)]−1 exp(2k −1t)

In (5.7) and Definition V.1.2, j = θj (1) − θj (0) and 1j = θj1 (1) − θj1 (0) associate to Q(t) and Q1 (t) respectively satisfy j = θj (1) − θj (0) = 1j + 2k = θj1 (1) − θj1 (0) + 2k. Since k ≥ π , there holds iL (B1 ) + n ≤ iL (B1 + kI ).

(9.9)

Thus we have proved (9.5). Equation (9.6) can be proved similarly. By the condition (H∞ ), H  (t, x) is bounded and there exist μ1 , μ > 0 such that I ≤ H  (t, x) + μI ≤ μ1 I, ∀(t, x).

(9.10)

We define a convex function N(t, x) = H (t, x) + 12 μ|x|2 . Its Fenchel dual defined by N ∗ (t, x) = sup {(x, y) − N(t, y)} satisfies N ∗ ∈ C 2 ([0, 1] × R2n , R) and (cf. y∈R2n

[78]) N ∗  (t, y) = N  (t, x)−1 , for y = N  (t, x). From (9.10) we have

(9.11)

256

9 Applications of L-Index ∗  μ−1 1 I ≤ N (t, y) ≤ I, ∀(t, y).

(9.12)

So we have |x| → ∞ if and only if |y| → ∞ with y = N  (t, x). Thus there exists r1 > 0 such that (B2 (t) + μI )−1 ≤ N ∗  (t, y) ≤ (B1 (t) + μI )−1 , ∀(t, y) with |y| ≥ r1 .

(9.13)

We choose μ > 0 satisfying (9.10) and μ ∈ / σ (A), and recall that (μ x)(t) = −J x(t) ˙ + μx(t). We define the following functional

1

f (u) = − 0

1 [ (−1 u(t), u(t)) − N ∗ (t, u(t))] dt, ∀u ∈ L. 2 μ

(9.14)

It is easy to see that f ∈ C 2 and satisfies (PS) condition (cf. [78]). There is a one to one corresponding from the critical points of f to the solutions of Hamiltonian systems (9.1). We note that 0 is a trivial critical point of f and N ∗  (t, 0) = 0. At every critical point u0 , the second variation of f defines a quadratical form on L by (f  (u0 )u, u) = −



1 0

∗  [(−1 μ u(t), u(t)) − (N (t, u0 (t))u(t), u(t))] dt, ∀u ∈ L.

Its Morse index and nullity are both finite. We denote the index pair by (iμ∗ (u0 ), νμ∗ (u0 )). The critical point u0 corresponding to a solution x0 = −1 μ u0 of (9.1), and N ∗  (t, u0 (t)) = N  (t, x0 (t))−1 . So by Theorem V.7.1, we have iμ∗ (u0 ) = iL (x0 ) + n + n

/μ0 π

, νμ∗ (u0 ) = νL (x0 ).

The index pair (iL (x0 ), νL (x0 )) is the L-index of the following linear Hamiltonian system y(t) ˙ = J H  (t, x0 (t))y(t). By condition (9.2) and the result (9.9), we have iL (B1 ) + νL (B1 ) + n ≤ iL (B0 ).

(9.15)

By condition (9.3), similarly we have iL (B0 ) + νL (B0 ) + n ≤ iL (B1 ). From (9.15) and the above inequality, we have that |iL (B0 ) − iL (B1 )| ≥ n, and |iμ∗ (B0 ) − iμ∗ (B1 )| ≥ n.

(9.16)

9.1 The Existence of L-Solutions of Nonlinear Hamiltonian Systems

257

In the following, we need to prove that the Homological groups satisfy Hq (L, fa ; R) ∼ = δqr R, q = 0, 1, · · · ,

(9.17)

for some a ∈ R and r = iμ∗ (B1 ). fa = {x ∈ L|f (x) ≤ a} is the level set below a. In the following we follow the ideas of the proof of Lemma II.5.1 in [33] to prove (9.17). See [67] and [187] for some similar arguments. Step 1. Under the condition (H∞ ), there holds + L = L− μ (B1 ) ⊕ Lμ (B2 ),

(9.18)

where L∗μ (B) for ∗ = ±, 0 is defined in Sect. 5.6. In fact, it is clear that L− μ (B1 ) ∩ ∗ (B ) = ν (B ) = 0, we have L = L− (B ) ⊕ L+ (B ). By L+ (B ) = {0}. By ν 2 L 2 2 2 μ μ 2 μ μ condition (H∞ ), we have iμ∗ (B1 ) = iμ∗ (B2 ) = r. Suppose ξ1 , ξ2 , · · · , ξr is a base − + ± of L− μ (B1 ). Decompose ξj by ξj = ξj + ξj with ξj ∈ Lμ (B2 ). It is clear that − − − ξ1 , · · · , ξr is linear independent, so it is a base of Lμ (B2 ). For any ξ ∈ L, there − − − holds ξ = ξ − + ξ + with ξ ± ∈ L± μ (B2 ). Suppose ξ = a1 ξ1 + · · · + ar ξr . Then r r   + ξ= aj ξj + (ξ + − aj ξj+ ) = ξ1 + ξ2 with ξ1 ∈ L− μ (B1 ) and ξ2 ∈ Lμ (B2 ). j =1

j =1

Step 2. For sufficiently small s > 0, from the structure of the symplectic group and the definition of the Maslov-type index, we know that νL (B1 − sI ) = νL (B1 ) = 0, νL (B2 + sI ) = νL (B2 ) = 0, and so iL (B1 − sI ) = iL (B1 ) = iL (B2 ) = iL (B2 + + sI ). Denote the so called deformation space by DR = L− μ (B1 −sI )⊕{u ∈ Lμ (B2 + sI )| #u# ≤ R}. For R > 0 and −a > 0 large, we have the following deformation result Hq (L, fa ; R) = Hq (DR , DR ∩ fa ; R).

(9.19)

The proof of (9.19) is standard in the Morse theory [29]. We only need to use the negative gradient flow of f to deform (L, fa ) to (DR , DR ∩ fa ). For any u = + u1 + u2 ∈ L with u1 ∈ L− μ (B1 − sI ) and u2 ∈ Lμ (B2 + sI ), by the self-adjointness, we have '1

[(−1 u, u2 − u1 ) − (N ∗  (t, u), u2 − u1 )] dt '1 '1 = 0 (−1 u1 , u1 )dt − 0 (−1 u2 , u2 )dt + '1 '1 + 0 ( 0 N ∗  (t, τ u)dτ (u1 + u2 ), u2 − u1 )dt '1 '1 '1 = 0 (−1 u1 , u1 )dt − 0 ( 0 N ∗  (t, τ u)dτ u1 , u1 )dt '1 '1 '1 − 0 (−1 u2 , u2 )dt + 0 ( 0 N ∗  (t, τ u)dτ u2 , u2 )dt.(9.20)

(f  (u), u2 − u1 ) = −

0

By (9.12) and (9.13), we have

258



9 Applications of L-Index



1

( 0

1

N ∗  (t, τ u)dτ u1 , u1 )dt

0

=

1 0

h(t,u)

0

(N ∗  (t, τ u)dτ u1 , u1 )dt+

1 0



1

≤ c0 #u# +

1

(N ∗  (t, τ u)dτ u1 , u1 )dt

h(t,u)

((B1 (t) + μI − sI )u1 , u1 )dt,

(9.21)

0

where h(t, u) =

r1 |u(t)| .

Similarly, We have

' 1 ' 1 ∗  '1'1 ∗  0 ( 0 N (t, τ u)dτ u2 , u2 )dt ≥ 0 h(t,u) (N (t, τ u)dτ u2 , u2 )dt '1 ≥ 0 ((B2 (t) + μI + sI )u2 , u2 )dt − c#u#, for some c > 0 (9.22) So by (9.20), (9.21), and (9.22), we have (f  (u), u2 − u1 ) ≥ c1 #u1 #2 + c2 #u2 #2 − c3 (#u1 # + #u2 #).

(9.23)

Thus for large R with #u1 # ≥ R or #u2 # ≥ R, we have (−f  (u), u2 − u1 ) < −1.

(9.24)

We know from (9.24) that f has no critical point outside DR , and that −f  (u) points inward to DR on ∂DR . So we can define the deformation by the negative gradient flow of f . In fact, for any u = u1 + u2 ∈ / DR , let σ (θ, u) = eθ u1 + e−θ u2 , and du = log #u2 # − log R. We define the deformation map η : [0, 1] × L → L by  η(θ, u1 + u2 ) =

u1 + u2 , #u2 # ≤ R, σ (du θ, u), #u2 # > R.

η satisfies the following properties η(0, ·) = id, η(1, L) ⊂ DR , η(1, fa ) ⊂ DR ∩ fa η(θ, fa ) ⊂ fa , η(θ, ·)|DR = id|DR . Thus the pair (DR , DR ∩ fa ) is a deformation retract of the pair (L, fa ). Step 3. For large R, −a > 0, there holds Hq (DR , DR ∩ fa ) ∼ = δq,r R. In fact, similarly to the above computation, for large number m > 0, we have

(9.25)

9.1 The Existence of L-Solutions of Nonlinear Hamiltonian Systems

'1 0

N ∗ (t, u(t))dt =

≤ '

' |u(t)|≥mr1

dt ''

'1 0

+ ''

dt '1 0

''

[0,1]×[0,1] τ (N

259

∗  (t, τ su(t))u(t), u(t))dτ ds

N ∗ (t, 0)dt

[0,1]×[0,1] τ (N

∗  (t, τ su(t))u(t), u(t))dτ ds

+ cm

≤ |u(t)|≥mr1 dt |sτ u(t)|≥r1 ,τ,s∈[0,1] τ (N ∗  (t, τ su(t))u(t), u(t))dτ ds + ' '' + |u(t)|≥mr1 dt |sτ u(t)|≤r1 ,τ,s∈[0,1] τ (N ∗  (t, τ su(t))u(t), u(t))dτ ds + cm '1 ≤ 12 0 ((B1 (t) + μI )−1 u(t), u(t))dt + km #u# + cm , where cm and km are constants depending only on m, and km → 0 as m → +∞. So for the small s in the step 2 above, we can choose a large number m such that 0

1

1 N (t, u(t))dt ≤ 2 ∗



1

((B1 (t) + μI − sI )−1 u(t), u(t))dt + C, ∀u ∈ L

0

(9.26) (B − sI ) and for some constant C > 0. Thus for any u = u1 + u2 with u1 ∈ L− 1 μ u2 ∈ L+ (B + sI ) with #u # ≤ R, there holds 2 2 μ f (u) ≤ −C1 #u1 #2 + C2 #u1 # + C3 ,

(9.27)

where Cj , j = 1, 2, 3 are constants and C1 > 0. It implies that f (u) → −∞ if and only if #u1 # → ∞ uniformly for u2 ∈ L+ μ (B2 +sI ) with #u2 # ≤ R. In the following we denote Br = {x ∈ L| #x# ≤ r} the ball with radius r in L. Therefore for −a1 > −a2 sufficiently large, there exist three numbers with R < R1 < R2 < R3 satisfying − (L+ μ (B2 + sI ) ∩ BR3 ) ⊕ (Lμ )(B1 − sI ) \ BR2 ) ⊂ fa1 ∩ DR3 − ⊂ (L+ μ (B2 + sI ) ∩ BR3 ) ⊕ (Lμ )(B1 − sI ) \ BR1 ) ⊂ fa2 ∩ DR3 .

Recall that σ (θ, u) = eθ u1 + e−θ u2 . By definition, we have f (σ (0, u)) = f (u) > a1 and f (σ (θ, u)) → −∞ as θ → ∞ if u = u1 + u2 ∈ DR3 ∩ (fa2 \ fa1 ). It implies that there exists θ0 = θ0 (u) > 0 such that f (σ (θ0 , u)) = a1 . But by (9.24), there holds d f (σ (θ, u)) ≤ −1, at any point θ > 0. dθ By the implicit function theorem, θ0 (u) is continuous with respect to the variable u. We define another deformation map η0 : [0, 1] × fa2 ∩ DR3 → fa2 ∩ DR3 by  u, u ∈ fa1 ∩ DR3 , η0 (θ, u) = σ (θ0 (u)θ, u), u ∈ DR3 ∩ (fa2 \ fa1 ). It is clearly that η0 is a deformation from fa2 ∩ DR3 to fa1 ∩ DR3 . We now define

260

9 Applications of L-Index

η(u) ˜ = d(η0 (1, u)) with d(u) =

u, #u1 # ≥ R1 , u2 + #uu11 # R1 , 0 < #u1 # < R1 .

This map defines a strong deformation retract: − η˜ : DR3 ∩ da2 → (L+ μ (B2 + sI ) ∩ BR3 ) ⊕ (Lμ (B1 − sI ) ∩ {u ∈ L| #u# ≥ R1 }).

Proof Now we can compute the Homological groups Hq (DR3 , DR3 ∩ fa2 ; R) − ∼ = Hq (DR3 , (L+ μ (B2 + sI ) ∩ BR3 ) ⊕ (Lμ (B1 − sI ) ∩ {u ∈ L| #u# ≥ R1 }); R) − ∼ ∼ = Hq (L− μ (B1 − sI ) ∩ BR3 , ∂(Lμ (B1 − sI ) ∩ BR3 ); R) = δqr R.

From (9.16), (9.17), and by using Lemma 1.2, we complete the proof.

 

Corollary 1.3 Let H satisfy the conditions (H0 ) and (H∞ ), and suppose B0 (t) = H  (t, 0) satisfying one of the following twisted conditions B1 (t) < B0 (t), there exists λ ∈ (0, 1) such that νL ((1 − λ)B1 + λB0 ) = 0 (9.28) and B0 (t) < B1 (t), there exists λ ∈ (0, 1) such that νL ((1 − λ)B0 + λB1 ) = 0. (9.29) Then (9.1) possesses at least one nontrivial solution. Furthermore, if ν (B ) L 0 = 0  ν((1 − λ)B1 + λB0 ) ≥ n, and in (9.28), we replace the second condition by or in (9.29), we replace the second condition by

λ∈(0,1)  ν((1 − λ)B0 + λB1 ) ≥ n, the λ∈(0,1)

Hamiltonian systems (9.1) possesses at least two non-trivial solutions. Proof It follows from Theorem V.6.4, the above proof of Theorem 1.1 and Lemma 1.2. In the first case, we have r = iL (B1 ) ∈ / [iL (B0 ), iL (B0 ) + νL (B0 )].   In the second case we have |iL (B0 ) − iL (B1 )| ≥ n. The proof of the Theorem 1.1 in fact proves the following result. Theorem 1.4 Let H satisfy conditions (H0 ) and (H∞ ). Suppose B0 (t) = H  (t, 0) satisfying the following twisted conditions / [iL (B0 ), iL (B0 ) + νL (B0 )]. iL (B1 ) ∈

(9.30)

Then the problem (9.1) possesses at least one nontrivial solution. Moreover, if νL (B0 ) = 0 and |iL (B1 ) − iL (B0 )| ≥ n, the problem (9.1) possesses at least two nontrivial solutions.

9.2 The Minimal Period Problem for Brake Solutions

261

Remark 1.5 The condition B1 (t) < B2 (t) in Theorem V.6.4 can be replaced by B1 (t) ≤ B2 (t) for all t and B2 − B1 ≥ δ > 0 for some constant δ as an operator in L. So the condition in (9.28) and (9.29) can be replaced by this kind of conditions. The condition J B1 (t) = B1 (t)J in (H∞ ) can be replaced by that J B0 (t) = B0 (t)J .

9.2 The Minimal Period Problem for Brake Solutions We now apply Theorem V.10.3 to the brake orbit problem of autonomous Hamiltonian system ⎧ x ∈ R2n , ⎨ −J x˙ = Bx + H  (x), x(τ/2 + t) = Nx(τ/2 − t), ⎩ x(τ + t) = x(t), t ∈ N,

(9.31)

  B1 0 is a 2n × 2n symmetric semi-positive 0 B2 definite matrix whose operator norm is denoted by #B#, B1 and B2 are n × n symmetric matrices. A solution (τ, x) of the problem (9.31) is a brake orbit of the Hamiltonian system, and τ is the brake period of x. To find a brake orbit of the Hamiltonian system in (9.31), it is sufficient to solve the following problem where H (N x) = H (x) and B =



−J x(t) ˙ = Bx + H  (x(t)), x ∈ R2n , t ∈ [0, τ/2], x(0) ∈ L0 , x(τ/2) ∈ L0 .

(9.32)

Any solution x of problem (9.32) can be extended to a brake orbit (τ, x) via the mirror symmetry about L0 by x(τ/2 + t) = Nx(τ/2 − t), t ∈ [0, τ/2] and x(τ + t) = x(t), t ∈ R. Theorem 2.1 Suppose the Hamiltonian function H satisfies the conditions: (H1) (H2)

H ∈ C 2 (R2n , R) satisfying H (Nx) = H (x), ∀x ∈ R2n . there are constants μ > 2 and r0 > 0 such that 0 < μH (x) ≤ H  (x) · x,

(H3) (H4)

∀|x| ≥ r0 .

H (x) = o(|x|2 ) at x = 0. H (x) ≥ 0 ∀x ∈ R2n .

Then for every 0 < τ < (τ, x) satisfying

2π #B# , the system (9.31) possesses a non-constant brake orbit

iL0 (x, τ/2) ≤ 1. Moreover, if x further satisfies the following condition:

(9.33)

262

(HX)

9 Applications of L-Index

H  (x(t)) ≥ 0 ∀t ∈ R and

' τ/2 0

H  (x(t)) dt > 0.

Then the minimal brake period of x is τ or τ/2. We remind that if B = 0, then

2π #B#

= +∞.

Proof We divide the proof into two steps. Step 1. Show that there exists a brake orbit (τ, x) satisfying (9.33) for 0 < τ
0 and χ ∈ C ∞ (R, R) such that χ (t) = 1 if t ≤ K, χ (t) = 0 if t ≥ K + 1, and χ  (t) < 0 if y ∈ (K, K + 1). The number K will be determined later. Set

1 Hˆ K (z) = (Bz, z) + HK (z), 2 with HK (z) = χ (|z|)H (z) + (1 − χ (|z|))RK |z|4 , where the constant RK satisfies RK ≥

H (z) . K≤|z|≤K+1 |z|4 max

We set L2 = L2 ([0, 1], R2n ) and define a Hilbert space E := WL0 = with L0 boundary conditions by

1/2,2 WL0 ([0, 1], R2n )

WL0 = {z ∈ L2 | z(t)   exp(kπ tJ )ak , ak ∈ L0 , #z#2 := (1 + |k|)|ak |2 < ∞}. = k∈Z

k∈Z

We denote its inner product by ·, · . By the well-known Sobolev embedding theorem, for any s ∈ [1, +∞), there is a constant Cs > 0 such that #z#Ls ≤ Cs #z#, ∀ z ∈ WL0 . Define a functional fK on E by

9.2 The Minimal Period Problem for Brake Solutions

fK (z) =

1

0

263

1 ( z˙ · J z − Hˆ K (z)) dt, ∀z ∈ E. 2

(9.34)

For m ∈ N, define E 0 = L0 , Em = { z ∈ E | z(t) =

m 

exp(kπ tJ )ak , ak ∈ L0 },

k=−m

E ± = { z ∈ E | z(t) =



exp(kπ tJ )ak , ak ∈ L0 },

±k>0 + = E ∩ E + , E − = E ∩ E − . We have E = E − ⊕ E 0 ⊕ E + . Let P be and Em m m m m m m m the projection Pm : E → Em . Then {Em , Pm }m∈N form a Galerkin approximation scheme of the operator −J d/dt on E. Denote by fK,m = fK |Em . Set Qm = {re : − ⊕ E 0 )} with some e ∈ ∂B (0) ∩ E + . Then for 0 ≤ r ≤ r1 } ⊕ {Br1 (0) ∩ (Em 1 m + form a topological (in fact large r1 > 0 and small ρ > 0, ∂Qm and Bρ (0) ∩ Em homologically) link (cf. P84 of [34]). By the condition #B# < π , we obtain a constant β = β(K) > 0 such that

(I) (II)

+, fK,m (z) ≥ β > 0, ∀z ∈ ∂Bρ (0) ∩ Em fK,m (z) ≤ 0, ∀z ∈ ∂Qm .

In fact, by (H3), for any ε > 0, there is a δ > 0 such that HK (z) ≤ ε|z|2 if |z| ≤ δ. Since Hˆ K (z)|z|−4 is uniformly bounded as |z| → +∞, there is an M1 = M1 (K) such that Hˆ K (z) ≤ M1 |z|4 for |z| ≥ δ. Hence Hˆ K (z) ≤ ε|z|2 + M1 |z|4 ,

∀z ∈ R2n .

+ , we have For z ∈ ∂Bρ (0) ∩ Em



1 0

HK (t, z)dt ≤ ε#z#2L2 + M1 #z#4L4 ≤ (εC22 + M1 C44 #z#2 )#z#2 .

So we have 1 1 fK,m (z) = Az, z − Bz, z − 2 2



1

HK (z(t))dt 0

π #B# #z#2 − #z#2 − (εC22 + M1 C44 #z#2 )#z#2 2 2 π #B# 2 ρ − (εC22 + M1 C44 ρ 2 )ρ 2 . = ρ2 − 2 2 ≥

Since #B# < π , we can choose constants ρ = ρ(K) > 0 and β = β(K) > 0, +, which are sufficiently small and independent of m, such that for z ∈ ∂Bρ (0) ∩ Em

264

9 Applications of L-Index

fK,m (z) ≥ β > 0. Hence (I) holds. + ∩ ∂B and z = z− + z0 ∈ E − ⊕ E 0 . We have Let e ∈ Em 1 m fK,m (z + re) =

1 1 Az− , z− + r 2 Ae, e

2 2

1 1 Hˆ K (z + re)dt − B(z + re), z + re − 2 0 1 π − 2 π 2 ≤ − #z # + r − Hˆ K (z + re)dt, 2 2 0 If r = 0, from condition (H4), there holds π fK,m (z + re) ≤ − #z− #2 ≤ 0. 2 If r = r1 or #z# = r1 , then from (H2), We have HK (z) ≥ b1 |z|μ − b2 , where b1 > 0, b2 are two constants independent of K and m. Then there holds

1

Hˆ K (z + re)dt ≥ b1

0



1

|z + re|μ dt − b2

0

 ≥ b3

1

|z + re| dt 2

 μ2

− b4

0

$ % ≥ b5 #z0 #μ + r μ − b4 , where b3 , b4 are constants and b5 > 0 independent of K and m. Thus there holds $ % π π fK,m (z + re) ≤ − #z− #2 + r 2 − b5 #z0 #μ + r μ + b4 , 2 2 So we can choose large enough r1 independent of K and m such that ϕm (z + re) ≤ 0,

on ∂Qm .

Then (II) holds. Now define  = { ∈ C(Qm , Em ) | (x) = x for x ∈ ∂Qm }, and set cK,m = inf

sup

∈ x∈ (Qm )

fK,m (x).

9.2 The Minimal Period Problem for Brake Solutions

265

It is well known that fK satisfies the usual (P.S)∗ condition on E, i.e. a sequence {xm } with xm ∈ Em possesses a convergent subsequence in E, provided  fK,m (xm ) → 0 as m → ∞ and |fK,m (xm )| ≤ b for some b > 0 and all m ∈ N (see [170] for a proof). Thus by the saddle point theorem (cf. [255], or Theorem VIII.3.2), we see that cK,m ≥ β > 0 is a critical value of fK,m , we denote the corresponding critical point by xK,m . The Morse index of xK,m satisfies m− (xK,m ) ≤ dim Qm = mn + n + 1. By taking m → +∞, we obtain a critical point xK such that xK,m → xK , m → +∞ and m− d (xK ) ≤ dim Qm = 1 + n + mn, 0 < cK ≡ fK (xK ) ≤ M1 , where  the d-Morse index m− d (xK ) is defined to the total number of the eigenvalues of fK belonging to (−∞, −d] for d > 0 small enough, and M1 is a constant independent of K. Moreover, by the Galerkin approximation method, Theorem V.6.1, we have the d-Morse index satisfying m− d (xK ) = mn + n + iL0 (xK , 1) ≤ 1 + n + mn. Thus we have iL0 (xK , 1) ≤ 1. Now the similar arguments as in the section 6 of [255] yields a constant M2 independent of K such that #xK #∞ ≤ M2 . Choose K > M2 . Then x ≡ xK is a non-constant solution of the problem (9.32) satisfying (9.33). By extending the domain with mirror symmetry of L0 , we obtain a 2-periodic brake orbit (2, x) of problem (9.31). Step 2. Estimate the brake period of (2, x). Denote the minimal period of the brake orbit x by 2/k for some k ∈ N, i.e., (x, 1/k) is a solution of the problem (9.32). By the condition (HX) and B being semi-positive definite, using (9.17) of [66], we have that i1 (x, 2/k) ≥ n for every 2/k-periodic solution (x, 2/k), and by (5.110) we see that iL0 (x, 1/k) ≥ 0 for the L0 -solution (x, 1/k). Together with (5.181), we obtain L0 i√ (x, 1/k) ≥ iL0 (x, 1/k) ≥ 0, i1 (x, 2/k) ≥ n. −1

(9.35)

Since the system (9.31) is autonomous, we have ν1 (x, 2/k) ≥ 1.

(9.36)

Therefore, by Theorem V.10.3, (9.33) and (9.35)–(9.36), we obtain k = 1, 2, 3, 4.

266

9 Applications of L-Index

If k = 3, by (9.33) and (9.35)–(9.36), and by using Theorem V.10.3 again we find the left equality of (5.218) holds for k = 3 and iL0 (x, 1/3) = 0, i1 (x, 2/3) = n, and ν1 (x, 2/3) = 1. The left side hand equality in the inequality (5.218) holds if and only if I2p  N1 (1, −1)q  K ∈ 0 (γ (2/3)) for some non-negative integers p and q satisfying p + q ≤ n and some K ∈ Sp(2(n − p − q)) satisfying σ (K) ⊂ U \ R. If r = n−p −q > 0, then by List in Lemma II. 3.14, we have R(θ1 )· · ·R(θr ) ∈ 0 (K) for some θj ∈ (0, π ). In this case,√all eigenvalues of K on√U+ (on U− ) are located on the arc between 1 and exp(2π −1/k) (and exp(−2π −1/k)) on U+ (in U− ) and are all Krein negative (positive) definite. We remind that γ (t) is the fundamental solution of the linearized system at (2/3, x). By the condition ν1 (x, 2/3) = 1, we have p = 0, q = 1. By Lemma II.2.7, there are paths α ∈ P2/3 (2), β ∈ P2/3 (2n−2) such that γ ∼ α  β, α(2/3) = N1 (1, −1), β(τ ) = K. By the locations of the end point matrix α(2/3) and β(2/3), there are two integers k1 , k2 such that (see the proof of Theorem 4.3 in [197], specially (4.18) and (4.19) there). i1 (α, 2/3) = 2k1 , i1 (β, 2/3) = 2k2 + n − 1. From this result, we see that if n = 1, then N1 (1, −1) ∈ 0 (γ (2/3)), and i1 (x, 2/3) must be even, so i1 (x, 2/3) = n = 1 is impossible. If n > 1, we have n − 1 > 0 and i1 (x, 2/3) = 2(k1 + k2 ) + n − 1. But i1 (x, 2/3) = n, so k1 + k2 = 12 . It is also impossible. If k = 4, the solution (1/2, x) itself is a brake orbit. Thus i1 (x, 1/2) and i1 (x, 1) are well defined and by Theorem V.10.3, we have that the left hand side equality in (5.219) holds for k = 4 and L0 i1 (x, 1/2) = n, ν1 (x, 1/2) = 1, iL0 (x, 1/4) = i√ (x, 1/4) = 0. −1

By the same arguments as above, we still get i1 (x, 1/2) = 2(k1 + k2 ) + n − 1. This is also impossible.   Remark 2.2 If B = 0, the results of Theorem 2.1 hold for every τ > 0. The following condition is more accessible than (HX) but it implies the condition (HX). (H6) H  (x) ≥ 0 for all x ∈ R2n , the set D = {x ∈ R2n |H  (x) = 0, 0 ∈ σ (H  (x))} is hereditarily disconnected, i.e. every connected component of D contains only one point.   U11 (t) U12 (t) , it was For the brake orbit (τ, x) in (9.33), writing H  (x(t)) = U21 (t) U22 (t) proved in [307] that the minimal brake period of x is τ or τ2 provided U22 (t) > 0, t ∈ [0, τ ]. Similarly, we consider the brake orbit minimal periodic problem for the following autonomous second order Hamiltonian system

9.2 The Minimal Period Problem for Brake Solutions

267

⎧ x ∈ Nn , ⎨ x¨ + V  (x) = 0, x(0) = x(τ/2) = 0 ⎩ x(τ/2 + t) = −x(τ/2 − t), x(τ + t) = x(t).

(9.37)

A solution (τ, x) of (9.37) is a kind of brake orbit for the second order Hamiltonian system. In this paper, we consider the following conditions on V : (V1) (V2)

V ∈ C 2 (Rn , R). There exist constants μ > 2 and r0 > 0 such that 0 < μV (x) ≤ V  (x) · x,

(V3) (V4) (V5) (V6)

∀|x| ≥ r0 .

V (x) ≥ V (0) = 0 ∀x ∈ Rn . V (x) = o(|x|2 ), at x = 0. V (−x) = V (x), ∀x ∈ Rn . V  (x) > 0, ∀x ∈ R.

Theorem 2.3 Suppose V satisfies the conditions (V1)–(V6). Then for every τ > 0, the problem (9.37) possesses a non-constant solution (τ, x) such that the minimal period of x is τ or τ/2. Proof Without loss generality, we suppose τ = 2. We define a Hilbert space W which is a subspace of W 1,2 ([0, 1], Rn ) by W = {x ∈ W 1,2 ([0, 1], Rn )| x(t) =

∞ 

sin kπ t · ak , ak ∈ Rn }.

k=1

The inner product of W is still the W 1,2 inner product. We consider the following functional

1

ψ(x) = 0

1 2 ˙ − V (x)) dt, ∀x ∈ W. ( |x| 2

(9.38)

A critical point x of ψ is a solution of the problem (9.37) by extending the domain to R via x(1+t) = −x(1−t) and x(2+t) = x(t). The condition (V3) implies ψ(0) = 0. The condition (V4) implies ψ(∂Bρ (0)) ≥ α0 with ∂Bρ (0) = {x ∈ W | #x# = ρ} for some small ρ > 0 and α0 > 0. In fact, there exists a constant c1 > 0 such that 0

1

|x| ˙ 2 dt ≥ c1 #x#2W .

If #x#W → 0, then #x#∞ → 0. So by condition (V4), for any 0 < ε < exists small ρ > 0 such that

(9.39) c1 2,

there

268

9 Applications of L-Index



1

0

V (x(t))dt ≤ ε#x#22 ≤ ε#x#2W , #x#W = ρ.

Thus we have ψ(x) = 0

1

1 2 c1 ˙ − V (x)) dt ≥ ( − ε)ρ 2 := α0 > 0. ( |x| 2 2

The condition (V2) implies that there exists an element x0 ∈ W with #x0 # > ρ, such that ψ(x0 ) < 0. In fact, we take an element e ∈ W with #e# = 1 and by (V3) '1 we assume 0 V (e(t))dt > 0. Consider x = λe for λ > 0. Condition (V2) implies that there is a constant c2 > 0 such that V (λe) ≥ λμ V (e) − c2 for λ large enough, and there holds

1

ψ(λe) ≤ λ2 0

1 2 |e| ˙ dt − λμ 2



1

V (e(t))dt + c2 < 0.

0

Then we take x0 = λe for large λ such that the above inequalities holds. We define  = {h ∈ C([0, 1], W ) | h(0) = 0, h(1) = x0 } and c = inf sup ψ(h(s)). h∈ s∈[0,1]

By using the Mountain pass theorem (cf. Theorem 2.2 of [255]), from the conditions (V2)–(V4) it is well known that there exists a critical point x ∈ W of ψ with critical value c > 0 which is a Mountain pass point such that its Morse index satisfying m− (x, 1) ≤ 1. If we set y = x˙ and z = (x, y) ∈ R2n , the problem (9.37) can be transformed into the following problem 

z˙ = −J H  (z), z(0) ∈ L0 , z(1) ∈ L0

with H (z) = H (x, y) = 12 |y|2 + V (x). We note that (V5) implies H (Nz) = H (z), so (2, z) is a brake orbit with brake period 2. We remind that in this case the complex structure is −J , but it does not cause any difficult to apply the index theory. By Theorem V.4.1, the Morse index m− (x, 1) of x is just the L0 -index iL0 (z, 1) of (1, z). I.e., there holds m− (x, 1) = iL0 (z, 1), m0 (x, 1) = νL0 (z, 1).

9.2 The Minimal Period Problem for Brake Solutions

269

We can suppose the minimal period of x is 2/k for k ∈ N. But iL0 (z, 1/k) = m− (x, 1/k) ≥ 0, and from the convexity condition (V6), we have i1 (z, 2/k) ≥ n. With the same arguments as in the proof of Theorem 2.1, we get k ∈ {1, 2}.   We note that the functional ψ is even, there may be infinite many solutions (τ, x) satisfying Theorem 2.3. We also note that Theorem 2.3 is not a special case of Theorem 2.1, since the Hamiltonian function H (x, y) = 12 |y|2 + V (x) is quadratic   0 0 in the variables y, in this case B = with #B# = 1. Thus when applying 0 In Theorem 2.1 to this case, we can only get the result of Theorem 2.3 for 0 < τ < 2π . We now consider the following problem ⎧ x ∈ Nn , ⎨ x¨ + V  (x) = 0, x(0) ˙ = x(τ/2) ˙ = 0, ⎩ x(τ/2 + t) = x(τ/2 − t), x(τ + t) = x(t).

(9.40)

A solution of (9.40) is also a kind of brake orbit for the second order Hamiltonian system. By set y = x, ˙ z = (y, x) and H (z) = H (y, x) = 12 |y|2 + V (x), the problem (9.40) can be transformed into the following L0 -boundary value problem 

z˙ = J H  (z) z(0) ∈ L0 , z(τ/2) ∈ L0 .

In this case the condition H (Nz) = H (z) is satisfied automatically. Set B =  In 0 , then #B# = 1. The following result is a direct consequence of Theorem 2.1. 0 0 Corollary 2.4 Suppose V satisfies the conditions (V1)–(V4) and (V6). Then for every 0 < τ < 2π , the problem (9.40) possesses a non-constant solution (τ, x) such that x has minimal period τ or τ/2. We note that if we directly solve the problem (9.40) by the same way as in the proof of Theorem 2.3, the formation of the functional is still ψ as defined in (9.38), but the domain should be W1 = {x ∈ W 1,2 ([0, 1], Rn )| x(t) =

∞ 

cos kπ t · ak , ak ∈ Rn }.

k=0

In this time, it is not able to apply the Mountain pass theorem to get a critical point directly due to the fact Rn ⊂ W1 , so the inequality (9.39) is not true.

270

9 Applications of L-Index

9.3 Brake Subharmonic Solutions of First Order Hamiltonian Systems We consider the first order non-autonomous Hamiltonian systems z˙ (t) = J ∇H (t, z(t)), ∀z ∈ R2n , ∀t ∈ R,

(9.41)

where H ∈ C 2 (R × R2n , R). ˆ Suppose that H (t, z) = 12 (B(t)z, z) + Hˆ (t, z) and H ∈ C 2 (R × R2n , R) satisfies the following conditions: (H1) (H2) (H3) (H4) (H5) (H6)

Hˆ (T + t, z) = Hˆ (t, z), for all z ∈ R2n , t ∈ R,

  −In 0 2n ˆ ˆ H (t, z) = H (−t, Nz), for all z ∈ R , t ∈ R, N = , 0 In Hˆ  (t, z) > 0, for all z ∈ R2n \{0}, t ∈ R, Hˆ (t, z) ≥ 0, for all z ∈ R2n , t ∈ R, Hˆ (t, z) = o(|z|2 ) at z = 0, There is a θ ∈ (0, 1/2) and r¯ > 0 such that 0
0, and (H7) B(t) is a symmetrical continuous matrix, |B| ˆ B(t) is a semi-positively definite for all t ∈ R, ˆ + t) = B(t) ˆ ˆ ˆ ˆ (H8) B(T = B(−t), B(t)N = N B(t), for all t ∈ R. Recall that a T -periodic solution (z, T ) of (9.41) is called brake solution if z(t + T ) = z(t) and z(t) = Nz(−t), the later is equivalent to z(T /2 + t) = N z(T /2 − t), in this time T is called the brake period of z. Theorem 3.1 ([172]) Suppose that H ∈ C 2 (R × R2n , R) satisfies (H1)–(H8), then for each integer 1 ≤ j < 2π/β0 T , there is a j T -periodic nonconstant brake solution zj of (9.41) such that zj and zkj are distinct for k ≥ 5 and kj < 2π/β0 T . Furthermore, {zk p |p ∈ N} is a pairwise distinct brake solution sequence of (9.41) for k ≥ 5 and 1 ≤ k p < 2π/β0 T . ˆ Especially, if B(t) ≡ 0, then 2π/β0 T = +∞. Therefore, one can state the following theorem. ˆ Theorem 3.2 ([172]) Suppose that H ∈ C 2 (R × R2n , R) with B(t) ≡ 0 satisfies (H1)–(H6), then for each integer j ≥ 1, there is a j T -periodic nonconstant brake solution zj of (9.41). Furthermore, given any integers j ≥ 1 and k ≥ 5, zj and zkj are distinct brake solutions of (9.41), in particularly, {zk p |p ∈ N} is a pairwise distinct brake solution sequence of (9.41).

9.3 Brake Subharmonic Solutions of First Order Hamiltonian Systems

271

The first result on subharmonic periodic solutions for the Hamiltonian systems z˙ (t) = J ∇H (t, z(t)), where z ∈ R2n and H (t, z) is T -periodic in t, was obtained by P. Rabinowitz in his pioneer work [258]. Since then, many mathematician made their contributions. See for example [78, 81, 184, 223, 269] and the references therein. Especially, in [81], I. Ekeland and H. Hofer proved that under a strict convex condition and a superquadratic condition, the Hamiltonian system z˙ (t) = J ∇H (t, z(t)) possesses subharmonic solution zk for each integer k ≥ 1 and all of these solutions are pairwise geometrically distinct. In [184], C. Liu obtained a result of subharmonic solutions for the non-convex case by using the Maslov-type index iteration theory. We first consider the following Hamiltonian systems 

z˙ (t) = J ∇H (t, z(t)), ∀z ∈ R2n , ∀t ∈ [0, j T /2], z(0) ∈ L0 , z(j T /2) ∈ L0 ,

(9.42)

where j ∈ N. In order to prove Theorem 3.1, as in Theorem 2.1 and Theorem VIII.3.1, by using saddle point theorem (Theorem VIII.3.2), we have the following result. The proof is omitted. Lemma 3.3 Suppose H (t, z) ∈ C 2 (R × R2n , R) satisfies (H4)–(H7), then for 1 ≤ j < 2π/β0 T , (9.42) possesses at least one nontrivial solution zj whose L0 -index pair (iL0 (zj ), νL0 (zj )) satisfies iL0 (zj ) ≤ 1 ≤ iL0 (zj ) + νL0 (zj ). For a solution (zk , k) of the problem (9.42), we define (˜zk , 2k) as  z˜ (t) =

z(t), t ∈ [0, kT 2 ], , Nz(T − t), t ∈ ( kT 2 kT ].

It is clear that (˜zk , 2k) is a brake solution of (9.41) with period kT . We are ready to give a proof of Theorem 3.1. Proof of Theorem 3.1. For 1 ≤ k < π/β0 , by Lemma 3.3, we obtain that there is a nontrivial solution (zk , k) of the problem (9.42) and its L0 -index pair satisfies iL0 (zk , k) ≤ 1 ≤ iL0 (zk , k) + νL0 (zk , k).

(9.43)

Then (˜zk , 2k) is a nonconstant brake solution of (9.41). For k ∈ 2N − 1, we suppose that (˜z1 , 2) and (˜zk , 2k) are not distinct. By (9.43), Proposition V.9.8 and Theorem V.10.3, we have

272

9 Applications of L-Index

1 ≥ iL0 (zk , k) ≥ iL0 (z1 , 1) +

k−1 (i1 (˜z1 , 2) + ν1 (˜z1 , 2) − n) 2

 k−1 iL0 (z1 , 1) + iL1 (z1 , 1) + n + νL0 (z1 , 1) + νL1 (z1 , 1) − n 2  k−1 iL0 (z1 , 1) + iL1 (z1 , 1) + νL0 (z1 , 1) + νL1 (z1 , 1) , (9.44) = iL0 (z1 , 1) + 2 ≥ iL0 (z1 , 1) +

where L1 = Rn ⊕ {0} ∈ (n). By (H3), (H7) and Corollary V.4.5, we have iL1 (z1 , 1) ≥ 0. We also know that νL1 (z1 , 1) ≥ 0 and iL0 (z1 , 1) + νL0 (z1 , 1) ≥ 1. Then from (9.44) we deduce that 1 ≥ iL0 (z1 , 1) +

k−1 . 2

(9.45)

By 0 ≤ iL0 (z1 , 1) ≤ 1, from (9.45) we have k−1 2 ≤ 1, i.e., k ≤ 3. Since we have the condition k ≥ 5, (˜z1 , 2) and (˜zk , 2k) must be distinct. Similarly, we have that for each k ∈ 2N−1, k ≥ 5 and kj < βπ0 , 1 ≤ j < βπ0 , (˜zj , 2j ) and (˜zkj , 2kj ) are distinct brake solutions of (9.41). Furthermore, (˜z1 , 2), (˜zk , 2k), (˜zk 2 , 2k 2 ), (˜zk 3 , 2k 3 ), · · · , (˜zk p , 2k p ) are pairwise distinct brake solutions of (9.41), where k ∈ 2N − 1, k ≥ 5 and 1 ≤ k p < βπ0 with p ∈ N. For k ∈ 2N, as above, we suppose that (˜z1 , 2) and (˜zk , 2k) are not distinct. By (9.43), Proposition V.9.8 and Theorem V.10.3, we have  1 ≥ iL0 (zk , k) ≥

L0 iL0 (z1 , 1) + i√ (z , 1) + −1 1

 k − 1 (i1 (˜z1 , 2) + ν1 (˜z1 , 2) − n) 2

L0 ≥ iL0 (z1 , 1) + i√ (z , 1) −1 1     k − 1 iL0 (z1 , 1) + iL1 (z1 , 1) + n + νL0 (z1 , 1) + νL1 (z1 , 1) − n + 2 L0 = iL0 (z1 , 1) + i√ (z , 1) −1 1     k − 1 iL0 (z1 , 1) + iL1 (z1 , 1) + νL0 (z1 , 1) + νL1 (z1 , 1) . + 2

(9.46)

Similarly, we also know that iL1 (z1 , 1) ≥ 0, νL1 (z1 , 1) ≥ 0, iL0 (z1 , 1) + L0 νL0 (z1 , 1) ≥ 1. By Lemma V.8.2 and (5.181), we have i√ (z , 1) ≥ iL0 (z1 , 1) ≥ −1 1 0. Then from (9.46) we have  1 ≥ iL0 (z1 , 1) +

 k −1 . 2

(9.47)

9.3 Brake Subharmonic Solutions of First Order Hamiltonian Systems

273

By 0 ≤ iL0 (z1 , 1) ≤ 1, from (9.47) we have k2 − 1 ≤ 1, i.e., k ≤ 4. It contradicts to the condition k ≥ 5. Similarly we have that for each k ∈ 2N, k ≥ 6 and kj < βπ0 , 1 ≤ j < βπ0 , (˜zj , 2j ) and (˜zkj , 2kj ) are distinct brake solutions of (9.41). Furthermore, (˜z1 , 2), (˜zk , 2k), (˜zk 2 , 2k 2 ), (˜zk 3 , 2k 3 ), · · · , (˜zk p , 2k p ) are pairwise distinct brake solutions of (9.41), where k ∈ 2N, k ≥ 6 and 1 ≤ k p < βπ0 with p ∈ N. In all, for any integer 1 ≤ j < βπ0 , z˜ j and z˜ kj are distinct brake solutions of (9.41) for k ≥ 5 and kj < βπ0 . Furthermore, {˜zk p |p ∈ N} is a pairwise distinct brake solution sequence of (9.41) for k ≥ 5 and 1 ≤ k p < βπ0 . The proof of Theorem 3.1 is complete.  

Chapter 10

Multiplicity of Brake Orbits on a Fixed Energy Surface

10.1 Brake Orbits of Nonlinear Hamiltonian Systems For standard symplectic space (R2n , ω0 ) with ω0 (x, y) = J x, y , where J =  the  0 −I is the standard symplectic matrix and I is the n × n identity matrix, an I 0   −I 0 involution matrix defined by N = is clearly anti-symplectic, i.e., N J = 0 I −J N . The fixed point set of N and −N are the Lagrangian subspaces L0 = {0}×Rn and L1 = Rn × {0} of (R2n , ω0 ) respectively. Suppose H ∈ C 2 (R2n \ {0}, R) ∩ C 1 (R2n , R) satisfies the following reversible condition H (Nx) = H (x),

∀ x ∈ R2n .

(10.1)

We consider the following fixed energy problem of nonlinear Hamiltonian system with Lagrangian boundary conditions x(t) ˙ = J H  (x(t)), H (x(t)) = h, x(0) ∈ L0 , x(τ/2) ∈ L0 .

(10.2) (10.3) (10.4)

Here we require that h ∈ R is a regular value of H . It is clear that a solution (τ, x) of (10.2), (10.3), and (10.4) is a characteristic chord on the contact submanifold  := H −1 (h) = {y ∈ R2n | H (y) = h} of (R2n , ω0 ) and it can be extended to the whole real number set satisfying

© Springer Nature Singapore Pte Ltd. 2019 C. Liu, Index theory in nonlinear analysis, https://doi.org/10.1007/978-981-13-7287-2_10

275

276

10 Multiplicity of Brake Orbits on a Fixed Energy Surface

x(−t) = N x(t),

(10.5)

x(τ + t) = x(t).

(10.6)

In this paper this kind of τ -periodic characteristic (τ, x) is called a brake orbit on the hypersurface . We denote by Jb (, H ) the set of all brake orbits on . Two brake orbits (τi , xi ) ∈ Jb (, H ), i = 1, 2 are equivalent if the two brake orbits are geometrically the same, i.e., x1 (R) = x2 (R). We denote by [(τ, x)] the equivalence class of (τ, x) ∈ Jb (, H ) in this equivalence relation and by J˜b (, H ) the set of [(τ, x)] for all (τ, x) ∈ Jb (, H ). In fact J˜b (, H ) is the set of geometrically distinct brake orbits on , which is independent on the choice of H . So from now on we simply denote it by J˜b () and in the notation [(τ, x)] we always assume x has minimal period τ . We also denote by J˜ () the set of all geometrically distinct closed characteristics on . The number of elements in a set S is denoted by # S. It is well known that # J˜b () (and also # J˜ ()) is only depending on , that is to say, for simplicity we take h = 1, if H and G are two C 2 functions satisfying (10.1) and H := H −1 (1) = G := G−1 (1), then # Jb (H ) =# Jb (G ). So we can consider the brake orbit problem in a more general setting. Let  be a C 2 compact hypersurface in R2n bounding a compact set C with nonempty interior. Suppose  has non-vanishing Gaussian curvature and satisfies the reversible condition N ( − x0 ) =  − x0 := {x − x0 |x ∈ } for some x0 ∈ C. Without loss of generality, we may assume x0 = 0 the origin. We denote the set of all such hypersurfaces in R2n by Hb (2n). For x ∈ , let n (x) be the unit outward normal vector at x ∈ . Note that here by the reversible condition there holds n (N x) = N n (x). We consider the dynamics problem of finding τ > 0 and a C 1 smooth curve x : [0, τ ] → R2n such that x(t) ˙ = J n (x(t)), x(−t) = N x(t),

x(t) ∈ , x(τ + t) = x(t),

(10.7) for all t ∈ R.

(10.8)

A solution (τ, x) of the problem (10.7) and (10.8) determines a brake orbit on . Definition 1.1 We denote by Hbc (2n) = { ∈ Hb (2n)|  is strictly convex }, Hbs,c (2n) = { ∈ Hbc (2n)| −  = }. We will prove the following multiplicity result for brake orbits on a hypersurface  ∈ Hbs,c (2n). Theorem 1.2 ([210]) For any  ∈ Hbs,c (2n), there holds #

J˜b () ≥ n.

10.1 Brake Orbits of Nonlinear Hamiltonian Systems

277

Remark 1.3 Theorem 1.2 is a kind of multiplicity result related to the Arnold chord conjecture. The Arnold chord conjecture is an existence result which was prove by K. Mohnke in [241]. Another kind of multiplicity result related to the Arnold chord conjecture was proved in [128].

10.1.1 Seifert Conjecture Let us introduce the famous conjecture proposed by H. Seifert in his pioneer work [268] concerning the multiplicity of brake orbits of certain Hamiltonian systems in R2n . As a special case of (10.1), we assume that H ∈ C 2 (R2n , R) possesses the following form H (p, q) =

1 A(q)p · p + V (q), 2

(10.9)

where p, q ∈ Rn , A(q) is a positive definite n × n matrix for any q ∈ Rn and A is C 2 on q, V ∈ C 2 (Rn , R) is the potential energy. It is clear that a solution of the following Hamiltonian system x˙ = J H  (x), x = (p, q), τ p(0) = p( ) = 0. 2

(10.10) (10.11)

is a brake orbit. Moreover, if h is the total energy of a brake orbit (q, p), i.e., ¯ ≡ {q ∈ H (p(t), q(t)) = h, then V (q(0)) = V (q(τ )) = h and q(t) ∈  n R |V (q) ≤ h} for all t ∈ R. In 1948, H. Seifert in [268] studied the existence of brake orbit for system (10.10) and (10.11) with the Hamiltonian function H in the form of (10.9) and proved ¯ is bounded and that Jb () = ∅ provided V  = 0 on ∂, V is analytic and  n n homeomorphic to the unit ball B1 (0) in R . Then in the same paper he proposed the following conjecture which is still open for n > 2 now (that it is true for the case n = 2 was proved recently in [114] by R. Giambò, F. Giannoni and P. Piccione): #

J˜b () ≥ n under the same conditions.

We note that for the Hamiltonian function 1 2  2 2 |p| + aj qj , 2 n

H (p, q) =

q, p ∈ Rn ,

j =1

/ Q for all i = j and q = (q1 , q2 , . . . , qn ). There are exactly n where ai /aj ∈ geometrically distinct brake orbits on the energy hypersurface  = H −1 (h).

278

10 Multiplicity of Brake Orbits on a Fixed Energy Surface

10.1.2 Some Related Results Since 1948 As a special case, letting A(q) = I in (10.9), the problem corresponds to the following classical fixed energy problem of the second order autonomous Hamiltonian system q(t) ¨ + V  (q(t)) = 0,

for q(t) ∈ ,

1 |q(t)| ˙ 2 + V (q(t)) = h, 2 τ q(0) ˙ = q( ˙ ) = 0, 2

∀t ∈ R,

(10.12) (10.13) (10.14)

where V ∈ C 2 (Rn , R) and h is constant such that  ≡ {q ∈ Rn |V (q) < h} is nonempty, bounded and connected. ¯ A solution (τ, q) of (10.12), (10.13), and (10.14) is still called a brake orbit in . Two brake orbits q1 and q2 : R → Rn are geometrically distinct if q1 (R) = q2 (R). ˜ We denote by O(, V ) and O() the sets of all brake orbits and geometrically ¯ respectively. distinct brake orbits in  Remark 1.4 It is well known that via H (p, q) =

1 2 |p| + V (q), 2

x = (p, q) and p = q, ˙ the elements in O(, V ) and the solutions of (10.2), (10.3), and (10.4) are one to one correspondent. Definition 1.5 For  ∈ Hbs,c (2n), a brake orbit (τ, x) on  is called symmetric if x(R) = −x(R). Similarly, for a C 2 convex symmetric bounded domain  ⊂ Rn , a brake orbit (τ, q) ∈ O(, V ) is called symmetric if q(R) = −q(R). Note that a brake orbit (τ, x) ∈ Jb (, H ) with minimal period τ is symmetric if x(t + τ/2) = −x(t) for t ∈ R, a brake orbit (τ, q) ∈ O(, V ) with minimal period τ is symmetric if q(t + τ/2) = −q(t) for t ∈ R. After 1948, many studies have been carried out for the brake orbit problem. In 1978, S. Bolotin proved in [23] the existence of brake orbits in general setting. K. Hayashi in [136], H. Gluck and W. Ziller in [123], and V. Benci in [19] proved # O() ˜ ¯ = {V ≤ h} is compact, and V  (q) = 0 for all q ∈ ≥ 1 if V is C 1 ,  ∂. P. Rabinowitz in [253] proved that if H satisfies (10.1),  ≡ H −1 (h) is starshaped, and x · H  (x) = 0 for all x ∈ , then # J˜b () ≥ 1. V. Benci and F. Giannoni gave a different proof of the existence of one brake orbit in [21]. It has been pointed out in [112] that the problem of finding brake orbits is equivalent to find orthogonal geodesic chords on manifold with concave boundary. R. Giambó, F. Giannoni and P. Piccione in [113] proved the existence of an orthogonal geodesic chord on a Riemannian manifold homeomorphic to a closed disk and with concave boundary. For multiplicity of the brake problems, A. Weinstein in [293] proved a

10.1 Brake Orbits of Nonlinear Hamiltonian Systems

279

localized result: Assume H satisfies (10.1). For any h sufficiently close to H (z0 ) with z0 is a nondegenerate local minimum of H , there are n geometrically distinct brake orbits on the energy surface H −1 (h). In [24] and in [123], under assumptions of Seifert in [268], it was proved the existence of at least n brake orbits while a very strong assumption on the energy integral was used to ensure that different minimax critical levels correspond to geometrically distinct brake orbits. A. Szulkin in [278] proved that # J˜b (H −1 (h)) ≥ n, if H satisfies conditions in [253] of Rabinowitz √ and the energy hypersurface H −1 (h) is 2-pinched. E. van Groesen in [126] and ˜ ≥ n under different A. Ambrosetti, V. Benci, Y. Long in [6] also proved # O() pinching conditions. Without pinching condition, in [232] Y. Long, C. Zhu and D. Zhang proved that: For any  ∈ Hbs,c (2n) with n ≥ 2, # J˜b () ≥ 2. C. Liu and D. , Zhang in [209] proved that # J˜b () ≥ n2 + 1 for  ∈ Hbs,c (2n). Moreover it was proved that if all brake orbits on  are nondegenerate, then # J˜b () ≥ n + A(), where 2A() is the number of geometrically distinct asymmetric brake orbits on . in [308] the authors improved the results of [209] to that # J˜b () ≥ 0 / Recently, n+1 + 1 for  ∈ Hbs,c (2n), n ≥ 3. In [309] the authors proved that # J˜b () ≥ / 2 0 n+1 + 2 for  ∈ Hbs,c (2n), n ≥ 4. 2

10.1.3 Some Consequences of Theorem 1.2 and Further Arguments As direct consequences of Theorem 1.2 we have the following two important Corollaries. Corollary 1.6 ([210]) If H (p, q) defined by (10.9) is even and convex, then Seifert conjecture holds. Remark 1.7 If the function H in Remark 1.1 is convex and even, then V is convex and even, and  is convex and central symmetric. Hence  is homeomorphic to the unit open ball in Rn . Corollary 1.8 ([210]) Suppose V (0) = 0, V (q) ≥ 0, V (−q) = V (q) and V  (q) is positive definite for all q ∈ Rn \ {0}. Then for any given h > 0 and  ≡ {q ∈ Rn |V (q) < h}, there holds #

˜ O() ≥ n.

It is interesting to ask the following question: whether all closed characteristics on any hypersurfaces  ∈ Hbs,c (2n) are symmetric brake orbits after suitable time translation provided that # J˜ () < +∞? In this direction, we have the following result.

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10 Multiplicity of Brake Orbits on a Fixed Energy Surface

Theorem 1.9 ([210]) For any  ∈ Hbs,c (2n), suppose #

J˜ () = n.

Then all of the n closed characteristics on  are symmetric brake orbits after suitable time translation. For n = 2, it was proved in [138] that # J˜ () is either 2 or +∞ for any C 2 compact convex hypersurface  in R4 . Hence Theorem 1.9 gives a positive answer to the above question in the case n = 2. We also note that for the hypersurface x 2 +y 2  = {(x1 , x2 , y1 , y2 ) ∈ R4 | x12 + y12 + 2 4 2 = 1} we have # J˜b () = +∞ and # J˜ s () = 2, where we have denoted by J˜ s () the set of all symmetric brake orbits b

b

on . We also note that on the hypersurface  = {x ∈ R2n | |x| = 1} there are some non-brake closed characteristics.

10.2 Proofs of Theorems 1.2 and 1.9 In this section we prove Theorems 1.2 and 1.9. For  ∈ Hbs,c (2n), let j :  → [0, +∞) be the gauge function of  defined by j (0) = 0,

and

j (x) = inf{λ > 0 |

x ∈ C}, λ

∀x ∈ R2n \ {0},

where C is the domain enclosed by . Define Hα (x) = (j (x))α , α > 1,

H (x) = H2 (x), ∀x ∈ R2n .

(10.15)

Then H ∈ C 2 (R2n \{0}, R) ∩ C 1,1 (R2n , R). We consider the following fixed energy problem x(t) ˙ = J H (x(t)), H (x(t)) = 1,

(10.17)

x(−t) = N x(t), x(τ + t) = x(t),

(10.16)

(10.18) ∀ t ∈ R.

(10.19)

Denote by Jb (, 2) (Jb (, α) for α = 2 in (10.15)) the set of all solutions (τ, x) of problem (10.16), (10.17), (10.18), and (10.19) and by J˜b (, 2) the set of all geometrically distinct solutions of (10.16), (10.17), (10.18), and (10.19). By

10.2 Proofs of Theorems 1.2 and 1.9

281

Remark 1.2 of [209] or discussion in [232], elements in Jb () and Jb (, 2) are in one to one correspondence. So we have # J˜b ()=# J˜b (, 2). For readers’ convenience in the following we introduce some results which will be used in the proof of Theorem 1.2. In the following we write (iL0 (γ , k), νL0 (γ , k)) = (iL0 (γ k ), νL0 (γ k )) for any symplectic path γ ∈ Pτ (2n) and k ∈ N, where γ k is defined by (5.185) and (5.186). We have the following jumping formulas for the L0 -index. Lemma 2.1 ([209]) Let γj ∈ Pτj (2n) for j = 1, · · · , q. Let Mj = γj2 (2τj ) = Nγj (τj )−1 N γj (τj ), for j = 1, · · · , q. Suppose i¯L0 (γj ) > 0,

j = 1, · · · , q.

Then there exist infinitely many (R, m1 , m2 , · · · , mq ) ∈ Nq+1 such that (i) (ii) (iii) (iv) (v)

νL0 (γj , 2mj ± 1) = νL0 (γj ), + (1) − νL0 (γj )), iL0 (γj , 2mj − 1) + νL0 (γj , 2mj − 1) = R − (iL1 (γj ) + n + SM j iL0 (γj , 2mj + 1) = R + iL0 (γj ), ν(γj2 , 2mj ± 1) = ν(γj2 ), + (1) − ν(γj2 )), i(γj2 , 2mj − 1) + ν(γj2 , 2mj − 1) = 2R − (i(γj2 ) + 2SM j

(vi) i(γj2 , 2mj + 1) = 2R + i(γj2 ), 2nj

where we have set i(γj2 , nj ) = i(γj

2nj

), ν(γj2 , nj ) = ν(γj

) for nj ∈ N.

In order to prove Lemma 2.1, we need the following jumping formulas for the Maslov-type index. Proposition 2.2 (Theorem 4.3 in [233]) Let γj ∈ Pτj (2n) for j = 1, · · · , q be a finite collection of symplectic paths. Extend γj to [0, +∞) by γj (t + τj ) = γj (t)γj (τj ) and let Mj = γ (τj ), for j = 1, · · · , q and t > 0. Suppose ¯ j ) > 0, i(γ

j = 1, · · · , q.

Then there exist infinitely many (R, m1 , m2 , · · · , mq ) ∈ Nq+1 such that (i) ν(γj , 2mj ± 1) = ν(γj ), + (ii) i(γj , 2mj − 1) + ν(γj , 2mj − 1) = 2R − (i(γj ) + 2SM (1) − ν(γj )), j (iii) i(γj , 2mj + 1) = 2R + i(γj ), where we have set i(γj , nj ) = i(γj , [0, nj τj ]), ν(γj , nj ) = ν(γj , [0, nj τj ]) for nj ∈ N. We divide our proof in three steps. Step 1. Application of Proposition 2.2. By Corollary V.10.2, we have ¯ j2 ) = 2i¯L0 (γj ) > 0. i(γ

(10.20)

282

10 Multiplicity of Brake Orbits on a Fixed Energy Surface

So we have ¯ j2 ) > 0, i(γ

j = 1, · · · , q,

(10.21)

where γj2 is the 2-times iteration of γj defined by (5.186). Hence the symplectic paths γj2 , j = 1, 2, · · · , q satisfy the condition in Proposition 2.2, so there exist infinitely (R, m1 , m2 , · · · , mq ) ∈ Nq+1 such that ν(γj2 , 2mj ± 1) = ν(γj2 ),

(10.22)

+ (1) − ν(γj2 )), i(γj2 , 2mj − 1) + ν(γj2 , 2mj − 1) = 2R − (i(γj2 ) + 2SM j

(10.23) i(γj2 , 2mj

+ 1) = 2R

+ i(γj2 ).

(10.24)

Step 2. Verification of (i). By Theorems V.9.1 and V.9.7, we have νL0 (γj , 2mj ± 1) = νL0 (γj ) +

ν(γj2 , 2mj ± 1) − ν(γj2 )

νL1 (γj , 2mj ± 1) = νL1 (γj ) +

2

,

ν(γj2 , 2mj ± 1) − ν(γj2 ) 2

.

(10.25) (10.26)

Hence (i) follows from (10.22) and (10.25). Step 3. Verifications of (ii) and (iii). Proof By Theorems V.9.1 and V.9.7, we have iL0 (γ m ) − iL1 (γ m ) = iL0 (γ ) − iL1 (γ ),

∀m ∈ 2N − 1,

(10.27)

∀m ∈ 2N.

(10.28)

2iL0 (γj , 2mj ± 1) = i(γj2 , 2mj ± 1) − n + iL0 (γj ) − iL1 (γj ).

(10.29)

iL0 (γ m ) − iL1 (γ m ) = iL0 (γ 2 ) − iL1 (γ 2 ), By Proposition V.9.8 and (10.27) we have

By (10.22), (10.23), and (10.29) we have + (1) + n − iL0 (γj ) + iL1 (γj )). 2iL0 (γj , 2mj − 1) = 2R − (i(γj2 ) − 2SM j

(10.30)

So by Proposition V.9.8 we have + iL0 (γj , 2mj − 1) = R − (iL1 (γj ) + n + SM (1)). j

(10.31)

10.2 Proofs of Theorems 1.2 and 1.9

283

Together with (i), this yields (ii). By (10.24) and (10.29) we have 2iL0 (γj , 2mj + 1) = 2R + i(γj2 ) − n + iL0 (γj ) − iL1 (γj ).

(10.32)

By Proposition V.9.8 and (10.32) we have iL0 (γj , 2mj + 1) = R + iL0 (γj ).

(10.33)

Hence (iii) holds and the proof of Lemma 2.1 is complete.

 

For any (τ, x) ∈ Jb (, 2), there is a corresponding path γx ∈ Pτ (2n). For m ∈ N, we denote by iLj (x, m) = iLj (γxm ) and νLj (x, m) = νLj (γxm ) for j = 0, 1. Also we denote by i(x, m) = i(γx2m ) and ν(x, m) = ν(γx2m ). We remind that the 2m symplectic path γxm is defined in the interval [0, mτ 2 ] and the symplectic path γx is defined in the interval [0, mτ ]. If m = 1, we denote by i(x) = i(x, 1) and ν(x) = ν(x, 1). For S 1 = R/Z, as in [232] we define the Hilbert space E by    E = x ∈ W 1,2 (S 1 , R2n ) x(−t) = N x(t),



1

for all t ∈ R and

x(t)dt = 0 .

0

The inner product on E is given by (x, y) =

1

x(t), ˙ y(t) dt. ˙

(10.34)

0

The C 1,1 Hilbert manifold M ⊂ E associated to  is defined by    M = x ∈ E 

0

1

H∗ (−J x(t))dt ˙

= 1 and

1

J x(t), ˙ x(t) dt < 0 ,

0

where H∗ is the Fenchel conjugate function of the function H defined by H∗ (y) = max{(x · y − H (x))| x ∈ R2n }.

(10.35)

(10.36)

Let Z2 = {−id, id} be the usual Z2 group. We define the Z2 -action on E by −id(x) = −x,

id(x) = x,

∀x ∈ E.

Since H∗ is even, M is symmetric to 0, i.e., Z2 invariant. M is a paracompact Z2 -space. We define

284

10 Multiplicity of Brake Orbits on a Fixed Energy Surface

(x) =

1 2



1

J x(t), ˙ x(t) dt,

(10.37)

0

then is a Z2 invariant function and ∈ C ∞ (E, R). We denote by  the restriction of to M , we remind that and  here are the functionals A and A in [232] respectively. Lemma 2.3 ([209, 232]) If # J˜b () < +∞, there is a sequence {ck }k∈N , such that − ∞ < c1 < c2 < · · · < ck < ck+1 < · · · < 0, ck → 0

as k → +∞.

(10.38) (10.39)

For any k ∈ N, there exists a brake orbit (τ, x) ∈ Jb (, 2) with τ being the minimal period of x and m ∈ N satisfying mτ = (−ck )−1 such that for −1

z(x)(t) = (mτ )

1 x(mτ t) − (mτ )2



mτ

x(s)ds,

t ∈ S1,

(10.40)

0

z(x) ∈ M is a critical point of  with  (z(x)) = ck and iL0 (x, m) ≤ k − 1 ≤ iL0 (x, m) + νL0 (x, m) − 1,

(10.41)

where we denote by (iL0 (x, m), νL0 (x, m)) = (iL0 (γx , m), νL0 (γx , m)) and γx the associated symplectic path of (τ, x). We refer [232] for a complete proof of Lemma 2.3. Definition 2.4 (τ, x) ∈ Jb (, 2) with minimal period τ is called (m, k)variational visible, if there is some m and k ∈ N such that (10.40) and (10.41) hold. We call that (τ, x) ∈ Jb (, 2) with minimal period τ infinite variationally visible, if there exist infinitely many (m, k) such that (τ, x) is (m, k)-variationally visible. We denote by V∞,b (, 2) the subset of J˜b () in which a representative (τ, x) ∈ Jb (, 2) of each [(τ, x)] is infinite variationally visible. It is clear that if # J˜b () < +∞, then V∞,b (, 2) = ∅. Lemma 2.5 ([209]) Suppose # J˜b () < +∞. Then there exist an integer K ≥ 0 and an injection map φ : N + K → V∞,b (, 2) × N such that (i) For any k ∈ N + K, [(τ, x)] ∈ V∞,b (, 2) and m ∈ N satisfying φ(k) = ([(τ , x)], m), there holds iL0 (x, m) ≤ k − 1 ≤ iL0 (x, m) + νL0 (x, m) − 1, where x has minimal period τ .

10.2 Proofs of Theorems 1.2 and 1.9

285

(ii) For any kj ∈ N + K, k1 < k2 , (τj , xj ) ∈ Jb (, 2) satisfying φ(kj ) = ([(τj , xj )], mj ) with j = 1, 2 and [(τ1 , x1 )] = [(τ2 , x2 )], there holds m1 < m2 . Proof Since # J˜b () < +∞, there is an integer K ≥ 0 such that all critical values ck+K with k ∈ N come from iterations of elements in V∞,b (, 2). Together with Lemma 2.3, for each k ∈ N, there is a (τ, x) ∈ Jb (, 2) with minimal period τ and m ∈ N such that (10.40) and (10.41) hold for k + K instead of k. So we define a map φ : N + K → V∞,b (, 2) × N by φ(k + K) = ([(τ, x)], m). For any k1 < k2 ∈ N, if φ(kj ) = ([(τj , xj )], mj ) for j = 1, 2. Write [(τ1 , x1 )] = [(τ2 , x2 )] = [(τ, x)] with τ being the minimal period of x, then by Lemma 2.3 we have mj τ = (−ckj +K )−1 ,

j = 1, 2.

(10.42)

Since k1 < k2 and ck increases strictly to 0 as k → +∞, we have m 1 < m2 .

(10.43)

So the map φ is injective, also (ii) is proved. The proof of this Lemma 2.5 is complete.   Lemma 2.6 ([209]) Let γ ∈ Pτ (2n) be extended to [0, +∞) by γ (τ + t) = ˜ γ (t)γ (τ ) for all t > 0. Suppose γ (τ ) = M = P −1 (I2  M)P with M˜ ∈ Sp(2n − 2) and i(γ ) ≥ n. Then we have + i(γ , 2) + 2SM 2 (1) − ν(γ , 2) ≥ n + 2.

Proof By the Bott formula (2.13), (2.21), and (2.23), we have + i(γ , 2) + 2SM 2 (1) − ν(γ , 2) √  + + = 2i(γ ) + 2SM (1) + SM (e −1θ )

−(



θ∈(0,π ) − SM (e

√

−1θ

− − ) + (ν(M) − SM (1)) + (ν−1 (M) − SM (−1)))

θ∈(0,π ) + (1) − n ≥ 2n + 2SM + = n + 2SM (1)

≥ n + 2,

(10.44)

˜ where in the last inequality we have used γ (τ ) = M = P −1 (I2  M)P and the fact + SI2 (1) = 1.  

286

10 Multiplicity of Brake Orbits on a Fixed Energy Surface

Lemma 2.7 ([209]) For any (τ, x) ∈ Jb (, 2) and m ∈ N, there hold iL0 (x, m + 1) − iL0 (x, m) ≥ 1,

(10.45)

iL0 (x, m + 1) + νL0 (x, m + 1) − 1 ≥ iL0 (x, m + 1) > iL0 (x, m) + νL0 (x, m) − 1. (10.46) Proof Let γ be the associated symplectic path of (τ, x) and we extend γ to [0, +∞) by γ |[0, kτ ] = γ k with γ k defined in (5.187) for any k ∈ N. Due to the autonomous 2 Hamiltonian systems, for any m ∈ N, we have νL0 (x, m) ≥ 1,

∀m ∈ N.

(10.47)

Since H is strictly convex, H (x(t)) is positive for all t ∈ R. So by Theorem V.4.1 and Lemma V.4.2, we have iL0 (x, m + 1) =



νL0 (γ (t))

0 K + n and

iL0 (xk , 2mk + 1) = R + iL0 (xk ),

(10.53)

iL0 (xk , 2mk − 1) + νL0 (xk , 2mk − 1) + = R − (iL1 (xk ) + n + SM (1) − νL0 (xk )), k

(10.54)

for k = 1, · · · , p + q, Mk = γk2 (τk ), and iL0 (xk , 4mk + 2) = R + iL0 (xk , 2),

(10.55)

iL0 (xk , 4mk − 2) + νL0 (xk , 4mk − 2) + = R − (iL1 (xk , 2) + n + SM (1) − νL0 (xk , 2)), k

for k = p + q + 1, · · · , p + 2q and Mk = γk4 (2τk ) = γk2 (τk )2 . By Lemma 2.1, we also have

(10.56)

288

10 Multiplicity of Brake Orbits on a Fixed Energy Surface

i(xk , 2mk + 1) = 2R + i(xk ),

(10.57)

i(xk , 2mk − 1) + ν(xk , 2mk − 1) + = 2R − (i(xk ) + 2SM (1) − ν(xk )), k

(10.58)

for k = 1, · · · , p + q, Mk = γk2 (τk ), and i(xk , 4mk + 2) = 2R + i(xk , 2),

(10.59)

i(xk , 4mk − 2) + ν(xk , 4mk − 2) + = 2R − (i(xk , 2) + 2SM (1) − ν(xk , 2)), k

(10.60)

for k = p + q + 1, · · · , p + 2q and Mk = γk4 (2τk ) = γk2 (τk )2 . From (10.52), we can set φ(R − (s − 1)) = ([(τk(s) , xk(s) )], m(s)),

∀s ∈ S := {1, 2, · · · , n},

where k(s) ∈ {1, 2, · · · , p + q} and m(s) ∈ N. We continue our proof to study the symmetric and asymmetric orbits separately. Let S1 = {s ∈ S|k(s) ≤ p},

S2 = S \ S1 .

We shall prove that # S1 ≤ p and # S2 ≤ 2q. These estimates together with the definitions of S1 and S2 yield Theorem 1.2.   Claim 3.1

#S 1

≤ p.

Proof of Claim 3.1. By the definition of S1 , ([(τk(s) , xk(s) )], m(s)) is symmetric when k(s) ≤ p. We further prove that m(s) = 2mk(s) for s ∈ S1 . In fact, by the definition of φ and Lemma 2.5, for all s = 1, 2, · · · , n we have iL0 (xk(s) , m(s)) ≤ (R − (s − 1)) − 1 = R − s ≤ iL0 (xk(s) , m(s)) + νL0 (xk(s) , m(s)) − 1.

(10.61)

By the strict convexity of H and (5.110) or (5.111), we have iL0 (xk(s) ) ≥ 0, so that iL0 (xk(s) , m(s)) ≤ R − s < R ≤ R + iL0 (xk(s) ) = iL0 (xk(s) , 2mk(s) + 1),

(10.62)

for every s = 1, 2, · · · , n, where we have used (10.53) in the last equality. Note that the proofs of (10.61) and (10.62) do not depend on the condition s ∈ S1 .

10.2 Proofs of Theorems 1.2 and 1.9

289

It is easy to see that γxk satisfies conditions of Theorem VII.4.3 with τ = τ2k . Note that by definition iL1 (xk ) = iL1 (γxk ) and νL0 (xk ) = νL0 (γxk ). So by Theorem VII.4.3 we have + iL1 (xk ) + SM (1) − νL0 (xk ) ≥ 0, k

∀k = 1, · · · , p.

(10.63)

Hence by (10.61) and (10.63), if k(s) ≤ p, it follows that iL0 (xk(s) , 2mk(s) − 1) + νL0 (xk(s) , 2mk(s) − 1) − 1 + = R − (iL1 (xk(s) ) + n + SM (1) − νL0 (xk(s) )) − 1 k(s)

1−n −1−n 2 < R−s ≤R−

≤ iL0 (xk(s) , m(s)) + νL0 (xk(s) , m(s)) − 1.

(10.64)

Thus by (10.62) and (10.64) and Lemma 2.7 we obtain 2mk(s) − 1 < m(s) < 2mk(s) + 1. Hence m(s) = 2mk(s) and φ(R − s + 1) = ([(τk(s) , xk(s) )], 2mk(s) ),

∀s ∈ S1 .

Then the injectivity of the map φ induces an injective map φ1 : S1 → {1, · · · , p}, s → k(s). Therefore, # S1 ≤ p and Claim 3.1 is proved. Claim 3.2

#S 2

 

≤ 2q.

Proof of Claim 3.2. By the formulas (10.57), (10.58), (10.59), and (10.60), and (59) of [198] (also Claim 4 on p. 352 of [223]), we have mk = 2mk+q

for k = p + 1, p + 2, · · · , p + q.

(10.65)

By Theorem VII.4.3 there holds + (1) − νL0 (xk , 2) ≥ 0, iL1 (xk , 2) + SM k

p + 1 ≤ k ≤ p + q.

(10.66)

290

10 Multiplicity of Brake Orbits on a Fixed Energy Surface

By (10.56), (10.61), (10.65), and (10.66), for p + 1 ≤ k(s) ≤ p + q we have iL0 (xk(s) , 2mk(s) − 2) + νL0 (xk(s) , 2mk(s) − 2) − 1 = iL0 (xk(s) , 4mk(s)+q − 2) + νL0 (xk(s) , 4mk(s)+q − 2) − 1 + = R − (iL1 (xk(s) , 2) + n + SM (1) − νL0 (xk(s) , 2)) − 1 k(s) + (1) − νL0 (xk , 2)) − 1 − n = R − (iL1 (xk , 2) + SM k

≤ R−1−n < R−s ≤ iL0 (xk(s) , m(s)) + νL0 (xk(s) , m(s)) − 1.

(10.67)

Thus (10.62), (10.67) and Lemma 2.7 imply 2mk(s) − 2 < m(s) < 2mk(s) + 1,

p < k(s) ≤ p + q.

So m(s) ∈ {2mk(s) − 1, 2mk(s) },

for p < k(s) ≤ p + q.

Especially this yields that for any s0 and s ∈ S2 , if k(s) = k(s0 ), then m(s) ∈ {2mk(s) − 1, 2mk(s) } = {2mk(s0 ) − 1, 2mk(s0 ) }. Then, in view of the injectivity of the map φ from Lemma 2.5, we have #

{s ∈ S2 |k(s) = k(s0 )} ≤ 2.

This proves Claim 3.2.

 

By Claims 3.1 and 3.2, we obtain #

J˜b () =# J˜b (, 2) = p + 2q ≥# S1 +# S2 = n.

The proof of Theorem 1.2 is completed. Definition 2.8 We call a closed characteristic x on  a dual brake orbit on  if x(−t) = −N x(t). Lemma 2.9 For  ∈ Hb (2n), and (τ, x) ∈ J (, 2), we have the following statements: (i) if x(R) = N x(R), then x is a brake orbit after suitable time translation; (ii) if x(R) = −N x(R), then x is a dual brake orbit after suitable time translation.

10.2 Proofs of Theorems 1.2 and 1.9

291

Proof Since x ∈ J (, 2), we have x(t) ˙ = J H2 (x(t)). So let y(t) = N x(−t) for t ∈ R, we have y(t) ˙ = −N J H2 (x(−t)) = J NH2 (x(−t)) = J H2 (N x(−t)) = J H2 (y(t)). Hence (τ, y) ∈ J (, 2). If x(R) = Nx(R), (τ, x) and (τ, y) are geometrically the same closed characteristics on . By the existence and uniqueness of first order ordinary differential equations, there exists a unique t0 ∈ [0, τ ) such that x(t +t0 ) = y(t) for all t ∈ R. i.e. x(t + t0 ) = Nx(−t). Let z(t) = x(t +

t0 2)

for all t ∈ R, then we have

z(−t) = x(−t −

t0 t0 t0 + t0 ) = y(−t − ) = N x(t + ) = N z(t). 2 2 2

Hence (τ, z) is a brake orbit. This complete the proof of statement (i). The proof of statement (ii) is similar, so we omit it.   Proof of Theorem 1.9 By Lemma 2.9, if a closed characteristic x on  can both become brake orbits and dual brake orbits after suitable translation, then x(R) = Nx(R) = −N x(R), Thus x(R) = −x(R). Since we also have −N = , (−N)2 = I2n and (−N )J = −J (−N ), dually by the same proof of Theorem 1.2 (with the estimate (7.145) in Theorem VII.4.3), there are at least n geometrically distinct dual brake orbits on . If there are exactly n closed characteristics on , then Theorem 1.2 implies that all of them are brake orbits on  after suitable time translation. By the same argument all the n closed characteristics must be dual brake orbits on . Then by the argument in the first paragraph of the proof of this theorem, all these n closed characteristics on  must be symmetric. Hence all of them are symmetric brake orbits after suitable time translation. The proof of Theorem 1.9 is completed.  

Chapter 11

The Existence and Multiplicity of Solutions of Wave Equations

In this chapter, we apply the index theories defined in Chap. 3 to study the existence and multiplicity of solutions of wave equations. We will use the same concepts and notations as in Sect. 3.3.

11.1 Variational Setting and Critical Point Theories 11.1.1 Critical Point Theorems in Case 1 and Case 2 In this subsection, we will consider the operator equation Au = F  (u), u ∈ D(A) ⊂ H,

(O.E.)

where H is an infinite-dimensional separable Hilbert space, A is a self-adjoint operator on H with its domain D(A), F is a nonlinear functional on H with the self-adjoint operator A ∈ Oe∓ (λ∓ ) which are defined in Sect. 3.3. Firstly, assume F ∈ C 2 (H, R). In order to use the Morse theory to find the solutions of (O.E.), in addition to the index theory developed in Sect. 3.3, we need some further preparations. Assume F satisfying the following conditions according to A ∈ Oe∓ (λ∓ ). (F0∓ )

∓ F  (0) = 0, and there exists B0 ∈ L∓ s (H, λ ) satisfying

F  (z) = B0 z + o(#z#H ), #z#H → 0. (F1∓ )

There exist c ∈ R and δ > 0 satisfying ±c · I < ±F  (z) < ±(λ∓ ∓ δ) · I, ∀z ∈ H.

© Springer Nature Singapore Pte Ltd. 2019 C. Liu, Index theory in nonlinear analysis, https://doi.org/10.1007/978-981-13-7287-2_11

293

294 ∓ (F∞,1 )

11 The Existence and Multiplicity of Solutions of Wave Equations ∓ There exists B∞ ∈ L∓ s (H, λ ) with νA (B∞ ) = 0, such that

F  (z) = B∞ z + o(#z#H ), #z#H → ∞. ∓ ∓ ) There exist B∞,α , B∞,β ∈ L∓ (F∞,2 s (H, λ ) with ±B∞,α ≤ ±B∞,β , ∓ ∓ iA (B∞,α ) = iA (B∞,β ) and νA (B∞,β ) = 0, such that

±B∞,α ≤ ±F  (z) ≤ ±B∞,β , #z#H > R for a constant R > 0. In the following, we assume A ∈ Oe∓ (λ∓ ), then (O.E.) is equivalent to Ak u = Fk (u), u ∈ D(A) ⊂ H,

(11.1)

where Ak = ±(A − k · I ) and k ∈ / σ (A) as defined in Sect. 3.3 with ±k < ±c, k Fk (u) := ±(F (u) − (u, u)H ). 2 Since F ∈ C 2 (H, R) satisfying condition (F1∓ ), Fk ∈ C 2 (H, R) and its Hessian is positive definite everywhere, that is (Fk (u)v, v)H > 0, ∀u, v ∈ H and v = 0. Let Fk∗ be the Fenchel conjugate or Legendre transform of Fk , defined by Fk∗ (z) := sup {(z, u)H − Fk (u)}, ∀z ∈ H. u∈H

The readers can refer [78] for more information about the Fenchel conjugate of a convex functional on Banach space. From the property of Fk∗ , we have that u ∈ D(A) is a solution of (11.1) if and only if z = Ak u

(11.2)

is a solution of the following equation 

∗ A−1 k z = Fk (z), z ∈ H.

(11.3)

So there exists a one-to-one corresponding between the solutions of (11.1) and solutions of (11.3). Define the functional $ on H by 1 $(z) = Fk∗ (z) − (A−1 z, z)H , ∀z ∈ H. 2 k

(11.4)

11.1 Variational Setting and Critical Point Theories

295

The critical points of $ are the solutions of (11.3). In order to use the Morse theory to find the critical points of $ we need the following lemmas. Lemma 1.1 ([284]) Assume A ∈ Oe∓ (λ∓ ) and F ∈ C 2 (H, R), satisfying F1∓ and ∓ F∞,1 , then the functional $ satisfies the (PS) condition. That is to say for any sequence {zn } ⊂ H satisfying that {$(zn )} is bounded and $  (zn ) → 0 in H , there is a convergent subsequence. Proof We only prove the case A ∈ Oe− (λ− ). Assume {zn } ⊂ H is a (PS) sequence of $, i.e., {$(zn )} is bounded and $  (zn ) → 0 in H . From the definition of $,   $  (zn ) = Fk∗ (zn ) − A−1 k zn , by setting yn = −$ (zn ), we have 

∗ A−1 k zn − Fk (zn ) = yn → 0. 

∗ Let A−1 k zn − yn = un , that is Fk (zn ) = un and

Ak (un + yn ) = Fk (un ).

(11.5)

Since νA (B∞ ) = 0, Ak − B∞,k is invertible on H , and we have the following decomposition H = H + ⊕ H −, where Ak − B∞,k is positive definite on H + and negative define on H − , for any u ∈ H , u = u+ + u− with u+ ∈ H + and u− ∈ H − . Then we have ((Ak − B∞,k )(un + yn ), (un + yn )+ − (un + yn )− )H = (Fk (un ) − B∞,k un , (un + yn )+ − (un + yn )− )H − (B∞,k yn , (un + yn )+ −(un + yn )− )H . It is easy to see ((Ak − B∞,k )(un + yn ), (un + yn )+ − (un + yn )− )H ≥ c1 #un + yn #2H , − ) and the fact y → 0, we for some c1 > 0. On the other hand, from condition (F∞ n have

|(Fk (un ) − B∞,k un , (un + yn )+ − (un + yn )− )H − (B∞,k yn , (un + yn )+ −(un + yn )− )H | < (o(#un #H ) + c2 )#un + yn #H , for c2 > 0. That is to say {un } are bounded in H . Further more, since F satisfies conditions (F0− ) and (F1− ), consider the Eq. (11.5), with the similar idea used in Lemma 3.1, concretely speaking, by the implicit function theorem, there exists u ∈ C 1 (H0 ⊕ H1 , H1 ), such that

296

11 The Existence and Multiplicity of Solutions of Wave Equations

un,1 = u(un,0 , yn,1 ), where un = un,0 + un,1 , yn = yn,0 + yn,1 with un,∗ , yn,∗ ∈ H∗ (∗ = 0, 1) and H∗ is defined in Sect. 3.3. Since {un } are bounded in H , {un,0 } are bounded in H0 . Recall that H0 is a finite dimensional space, so {un,0 } has a convergent subsequence, and from the fact that u is C 1 continuous and yn → 0, we have {un,1 } has a convergent subsequence, so {un } has a convergent subsequence. From equation zn = Fk (un ) and the C 1 smooth of Fk , we have {zn } has a convergent subsequence and the proof is complete.   Similarly, we have the following lemma. Lemma 1.2 ([284]) Assume A ∈ Oe∓ (λ∓ ) and F ∈ C 2 (H, R), satisfying (F1∓ ) and ∓ (F∞,2 ), then the functional $ satisfies the (PS) condition. Proof We only prove the case A ∈ Oe− (λ− ). The proof is similar to the proof of Lemma 1.1. Assume {zn } ⊂ H is a (PS) sequence of $, that is {$(zn )} are bounded  and $  (zn ) → 0 in H . Recall that yn = −$  (zn ) → 0, Fk∗ (zn ) = un and Ak (un + yn ) = Fk (un ).

(11.6)

As in the proof of Lemma 1.1, what we need to do is to prove the boundedness of {un }. Arguing indirectly, assume #un #H → ∞, as n → ∞, thus for any ε > 0, there exists N (ε) > 0 such that #un #H >

R , ∀n > N (ε), ε

(11.7)

− with R defined in condition (F∞,2 ). Let



1

Cn = 0



ε

Consider Cn (1) = (F1− ) we have

0

Fk (sun )ds

Fk (sun )ds.

and Cn (2) = ε

1

Fk (sun )ds respectively. From

εc · I ≤ Cn (1) ≤ ελ− · I, − ) and (11.7), we have where the constant c is defined in condition (F1− ). From (F∞,2

(1 − ε)B∞,α ≤ Cn (2) ≤ (1 − ε)B∞,β , thus from Cn = Cn (1) + Cn (2), we have B∞,α − ε(B∞,α − c · I ) ≤ Cn ≤ B∞,β − ε(B∞,β − λ− · I ), ∀n > N (ε).

11.1 Variational Setting and Critical Point Theories

297

Since B∞,α < B∞,β and iA− (B∞,α ) = iA− (B∞,β ), by the property of our index stated in Lemma 3.3, we have νA (B∞,α ) = 0, and since νA (B∞,β ) = 0, we can choose ε small enough such that iA− (B∞,α − ε(B∞,α − c · I )) = iA− (B∞,β − ε(B∞,β − λ− · I )) = iA− (B∞,β ), and νA (B∞,α − ε(B∞,α − c · I )) = νA (B∞,β − ε(B∞,β − λ− · I )) = 0. So from the properties of our index pair, we have iA− (Cn ) = iA− (B∞,β ), and νA (Cn ) = 0. On the other hand, from condition (F0− ) and the definition of Cn , we have Fk (un ) = Cn un , ∀ n = 1, 2, · · · . Now, as done in Lemma 1.1, if we replace B∞ by Cn , the rest part of the proof will be the same. We omit the details here. For the case A ∈ Oe+ (λ+ ), the proof is similar.   Now we are ready to state the following abstract critical point theorem. One of the key points in this theorem is the index twist conditions about origin and infinity, i.e. (11.8) and (11.9) below. It is well known that these twist conditions are related to the famous Poincaré-Birkhoff Theorem on the existence of fixed points of area preserving homeomorphisms on an annulus under twist conditions on the opposite sides of the boundary. The readers can refer [222] for more information about this topic. Theorem 1.3 ([284]) Assume A ∈ Oe∓ (λ∓ ) and F ∈ C 2 (H, R), satisfying (F0∓ ), ∓ ∓ ) (or (F∞,2 )). We divided the result into the following two parts. (F1∓ ) and (F∞,1 Part 1.

If the indices satisfying

iA∓ (B∞ ) ∈ [iA∓ (B0 ), iA∓ (B0 )+νA (B0 )](or iA∓ (B∞,a ) ∈ [iA∓ (B0 ), iA∓ (B0 )+νA (B0 )]), (11.8) then the problem (O.E.) possesses at least one nontrivial solution. Moreover, denote this nontrivial solution by u1 , and denote νA (u1 ) := νA (B1 ), B1 = F  (u1 ). If νA (B0 ) = 0, and

νA (u1 ) < |iA∓ (B1 ) − iA∓ (B0 )|, then the problem (O.E.) possesses one more nontrivial solution.

298

Part 2.

11 The Existence and Multiplicity of Solutions of Wave Equations

If F is even in u and iA∓ (B0 ) = iA∓ (B∞ ),

(11.9)

then the problem (O.E.) possesses at least |iA∓ (B0 )−iA∓ (B∞ )| pairs of nontrivial solutions. We only consider the case A ∈ Oe− (λ− ), and from the proof of Lemma 1.2, − − condition (F∞,2 ) will be similar to (F∞,1 ). So we assume F satisfies conditions − − − (F0 ), (F1 ) and (F∞,1 ). Without loss of generality, we can assume F (0) = 0. / σ (A). Since F satisfies condition (F1− ), choose k ∈ R satisfying k < c and k ∈ k Then Fk (u) := F (u) − 2 (u, u)H is a convex functional on H satisfying Fk (0) = 0 and Fk (0) = 0, so it’s dual Fk∗ is well defined and also a convex functional on H satisfying 

Fk∗ (0) = 0, Fk∗ (0) = 0,

(11.10)

and 

−1 . Fk∗ (0) = [Fk (0)]−1 = B0,k

(11.11)

Proof of Part 1. Recall that the solutions of (O.E.) correspond to the critical points of the functional $ defined in (11.4). From Lemma 3.2 and condition (F1− ), for any critical point z of $, its Morse index m− $ (z) and nullity ν$ (z) are finite. From (3.32), (11.10) and (11.11), 0 is a critical point of $ with − m− $ (0) = iA,k (B0 ), and ν$ (0) = νA (B0 ). 



(11.12) 

For any z ∈ H , let u = Fk∗ (z), that is Fk (u) = z, and Fk∗ (z) = [Fk (u)]−1 . From condition (F1− ) the boundedness of F  , we have #u#H → ∞ ⇔ #z#H → ∞. And − from (F∞,1 ), we have 

−1 z + o(#z#H ). Fk∗ (z) = B∞,k

(11.13)

As in [34], for any pair of topological spaces (X, Y ) with Y ⊂ X, let Hq (X, Y ; R) denote the singular q-relative homology group. Since $ satisfies (P S) condition, from (3.32) and (11.13), with similar arguments as in Lemma II 5.1 of [34], we have Hq (H, $c ; R) ∼ = δqγ R

(11.14)

− (B∞ ). Now, by Definition 3.6, (11.12) for −c large enough and γ = iA,k and (11.14), with Morse inequality, as done in Theorem II 5.1 and Corollary II 5.2 in [34], we will get a proof of the part 1. We omit the details here.

11.1 Variational Setting and Critical Point Theories

299

In order to prove part 2 of Theorem 1.3, instead of Morse theory we make use of the minimax arguments for multiplicity of critical points for even functional, actually the following lemmas. Assume φ ∈ C 2 (H, R) is an even functional, satisfying the (PS) condition and φ(0) = 0. Denote Sa = {u ∈ H | #u# = a}. Lemma 1.4 (See [111, Corollary 10.19] and [222, Lemma 3.4]) Assume Y and Z are subspaces of H satisfying dim Y = j > k = codimZ. If there exist R > r > 0 and α > 0 such that inf φ(Sr ∩ Z) ≥ α, sup φ(SR ∩ Y ) ≤ 0, then φ has j − k pairs of nontrivial critical points. Lemma 1.5 (See [34, Theorem II 4.1] and [222, Lemma 3.5]) Assume Y and Z are subspaces of H satisfying dim Y = j > k = codimZ. If there exist r > 0 and α > 0 such that inf φ(Z) > −∞, sup φ(Sr ∩ Y ) ≤ −α, then φ has j − k pairs of nontrivial critical points. Proof of Part 2 of Theorem 1.3. Let φ = $, φ is an even functional satisfying φ(0) = 0. From Lemma 1.1, φ satisfies the (PS) condition. Since iA− (B0 ) = iA− (B∞ ), firstly, if iA− (B0 ) > iA− (B∞ ), define Y = HT−−

B0 ,k

and Z := HT+−

,

B∞ ,k

− is defined in (3.31). Since our dual functional $ has the where the operator TB,k following two forms

$(z) =

1 −1 1 1 −1 ∗ (B0,k z, z)H − (A−1 k z, z)H + (Fk (z) − (B0,k z, z)H ), ∀z ∈ H, 2 2 2

and $(z) =

1 −1 1 1 −1 (B z, z)H − (A−1 z, z)H + (Fk∗ (z) − (B∞,k z, z)H ), ∀z ∈ H, 2 ∞,k 2 k 2

−1 where B∗,k = (B∗ − k)−1 with ∗ = 0, ∞. That is to say φ has the following two forms

φ(z) = and

1 − 1 −1 (T z, z)H + (Fk∗ (z) − (B0,k z, z)H ), ∀z ∈ H, 2 B0 ,k 2

300

11 The Existence and Multiplicity of Solutions of Wave Equations

φ(z) =

1 − 1 −1 (T z, z)H + (Fk∗ (z) − (B∞,k z, z)H ), ∀z ∈ H. 2 B∞ ,k 2

From Lemma 3.2, we have 1 − (T z, z)H ≤ −c#z#2H , ∀z ∈ Y, 2 B0 ,k and 1 − z, z)H ≥ c#z#2H , ∀z ∈ Z, (T 2 B∞ ,k for some constant c > 0. So we only need to prove the following two equalities 1 −1 Fk∗ (z) − (B0,k z, z)H = o(#z#2H ), z ∈ Y and #z#H → 0, 2

(11.15)

1 −1 Fk∗ (z) − (B∞,k z, z)H = o(#z#2H ), z ∈ Z and #z#H → ∞. 2

(11.16)

and

From (11.10) and (11.11), we have (11.15). From (11.13), we have (11.16). Thus by Lemma 1.4, (O.E.) has iA− (B0 ) − iA− (B∞ ) pairs of nontrivial solutions. If iA− (B0 ) < iA− (B∞ ), define Y = HT−−

B∞ ,k

, and Z = HT+− . B0 ,k

By the same discussion and Lemma 1.5, (O.E.) has iA− (B∞ ) − iA− (B0 ) pairs of nontrivial solutions. Thus we have proved Theorem 1.3.   If F ∈ C 1 (H, R), then the above theorem will not work. To deal with this situation, we first make the following assumptions. ∓ ∓ ∓ (F3∓ ) There exist B1 , B2 ∈ L∓ s (H, λ ) with ±B1 < ±B2 , iA (B1 ) = iA (B2 ), 1 νA (B2 ) = 0, such that ±(F (z) − 2 (B1 z, z)H ) is convex and

1 ±(F (z) − (B2 z, z)H ) ≤ c, ∀ z ∈ H, 2 where c ∈ R is a constant. ∓ (F4∓ ) F (0) = 0, F  (0) = 0 and there exists B3 ∈ L∓ s (H, λ ) with ±B3 > ±B1 , such that 1 ±(F (z) − (B3 z, z)H ) ≥ 0, for #z#H < r, 2

11.1 Variational Setting and Critical Point Theories

301

where r > 0 is a small number. We now consider the case A ∈ Oe− (λ− ), the case A ∈ Oe+ (λ+ ) is similar. As claimed in Remark 3.7, we can replace the number k by some suitable operator B − with B ∈ L− s (H, λ ), denote Bε := B1 − ε · I for some ε > 0, from condition − B1 < B2 , iA (B1 ) = iA− (B2 ) and νA (B2 ) = 0, we have νA (B1 ) = 0, so we can choose ε small enough, such that νA (Bε ) = 0 and (B1 − Bε )−1 − (A − Bε )−1 > 0.

(11.17)

Now, define Fε (z) := F (z) − 12 (Bε z, z)H ∈ C 1 (H, R) and Aε := A − Bε . It is easy to see Fε is convex and Aε is invariable on H , define the functional $ by 1 $(z) = Fε∗ (z) − (A−1 z, z), ∀z ∈ H. 2 ε Then the critical points of $ correspond to the solutions of (O.E.). With the idea of Theorem 3.1.7 in [69], we have the following result. Theorem 1.6 ([284]) Suppose A ∈ Oe∓ (λ∓ ), and F ∈ C 1 (H, R) satisfying (F3∓ ). If $ satisfies (PS) condition, then the problem (O.E.) has a solution. Further more, if F satisfying (F4∓ ) and iA∓ (B3 ) > iA∓ (B2 ),

(11.18)

then the problem (O.E.) has a nontrivial solution. Proof Only consider the case A ∈ Oe− (λ− ). Firstly, since F (z) ≤ 12 (B2 z, z)H + c, we have $(z) ≥

1 1 ((B2 − Bε )−1 z, z) − (A−1 z, z) + c. 2 2 ε

From the definition of the index, condition iA− (B1 ) = iA− (B2 ) and (11.17), we have (B2 − Bε )−1 − A−1 ε > 0, so $(z) → +∞, as #z#H → ∞ and $ is bounded from below, then by Ekeland’s variational principle and the (PS) condition, we have $ gets its minimal value at some point z0 . Of cause, z0 is a critical point of $. Thus we have proved the first part of the theorem. Secondly, if F  (0) = 0, then 0 is a trivial solution of (O.E.). We will prove z0 = 0. In fact, from (11.18), we have 1 1 ((B3 − Bε )−1 z, z) − (A−1 z, z) 2 2 ε

302

11 The Existence and Multiplicity of Solutions of Wave Equations

has an iA− (B3 ) − iA− (B2 )- dimensional negative define space, denote it by Z. From condition (F4∓ ), we have $(z) ≤

1 1 ((B3 − Bε )−1 z, z) − (A−1 z, z), z ∈ Z and #z#H small enough. 2 2 ε

Thus 0 is not a minimal value point of $, and z0 = 0. The proof is complete.

 

11.1.2 Critical Point Theorems in Case 3 In this subsection, we will consider the operator equation (O.E.) on H with the self-adjoint operator A ∈ Oe0 (λa , λb ). Assume F ∈ C 2 (H, R) satisfying the following conditions (F0 ) F  (0) = 0, and there exists B0 ∈ L0s (H, λa , λb ) satisfying F  (z) = B0 z + o(#z#H ), #z#H → 0. (F1 ) There exists δ > 0 satisfying (λa + δ) · I < F  (z) < (λb − δ) · I, ∀z ∈ H. (F∞,1 ) There exists B∞ ∈ L0s (H, λa , λb ) with νA (B∞ ) = 0, such that F  (z) = B∞ z + o(#z#H ), z → ∞. (F∞,2 ) There exist B∞,α , B∞,β ∈ L0s (H, λa , λb ) with B∞,α ≤ B∞,β , iA0 (B∞,α ) = iA0 (B∞,β ) and νA (B∞,β ) = 0, such that B∞,α ≤ F  (z) ≤ B∞,β , #z#H > R, where R > 0 is a constant. b in this case and Pk the projection map on the eigenspace of k if Let k = λa +λ 2 k ∈ σ (A). Recall that (O.E.) is equivalent to (11.1) the following equation Ak u = Fk (u), u ∈ D(A) ⊂ H, with Ak , Fk defined by  Ak := and

A − k · I, k∈ / σ (A), A − k · I + Pk , k ∈ σ (A),

11.1 Variational Setting and Critical Point Theories

 Fk (u) :=

303

k∈ / σ (A), F (u) − k2 (u, u)H , k F (u) − 2 (u, u)H + (Pk u, u)H , k ∈ σ (A).

Instead of using dual variational method to find the critical points, we consider the saddle point reduction here. As done in Sect. 3.3, let P0 =

(λb −λa )/2 −(λb −λa )/2

1dE(z),

with E(z) the spectrum measure of Ak and P1 = I − P0 . Let H∗ = P∗ H, ∗ = 0, 1. Decompose the space E = D(|A|1/2 ) as follows E = E0 ⊕ E1 ,

(11.19)

where E∗ = E ∩ H∗ , ∗ = 0, 1. Then for any z ∈ E, we have the decomposition z = x + y with x ∈ E0 and y ∈ E1 . Define a functional on E as follows: (z) =

1 (Ak z, z)H − Fk (z), ∀z ∈ E. 2

The critical points of are the solutions of (O.E.), since F is C 2 continuous and satisfies condition (F1 ), by standard argument of saddle point reduction, there exists ξ ∈ C 1 (E0 , E1 ), denote z(x) = x + ξ(x), then we have the following functional a on finite dimensional space E0 a(x) = (z(x)), ∀x ∈ E0 , and the following theorem duo to Amann and Zehnder [4], Chang [34] and Long [223]. Theorem 1.7 ([284]) Suppose A ∈ Oe0 (λa , λb ). If F is C 2 continuous and satisfies condition (F1 ), then there is a one-to-one correspondence between the critical points of the C 2 -function a ∈ C 2 (E0 , R) and the solutions of the operator equation (O.E.). Moreover, the functional a satisfies a  (x) = Az(x) − F  (z(x)) = Ax − P0 F  (z(x)), a  (x) = [A − F  (z(x))]z (x) = AP0 − P0 F  (z(x))z (x).

304

11 The Existence and Multiplicity of Solutions of Wave Equations

Since E0 is a finite dimensional space, for every critical point x of a in E0 , the Morse index m− a (x) and nullity νa (x) are finite, and by the definition of the index, 0   it is easy to see m− a (x) = iA (F (z(x))) and νa (x) = νA (F (z(x))). Similar to Lemmas 1.1 and 1.2, we have the following lemma. Lemma 1.8 Assume F ∈ C 2 (H, R), satisfying (F1 ) and (F∞,1 ) (or F∞,2 ), then the functional a satisfies the (PS) condition. The proof is similar to Lemmas 1.1 and 1.2, so we omit it here. Similar to Theorem 1.3, we have the following theorem. Theorem 1.9 ([284]) Assume (F0 ), (F1 ) and (F∞,1 ) (or (F∞,2 )) hold. We state the result as two parts. Part 1.

If

iA0 (B∞ ) ∈ [iA0 (B0 ), iA0 (B0 )+νA (B0 )](or iA0 (B∞,α ) ∈ [iA0 (B0 ), iA0 (B0 )+νA (B0 )]), (11.20) then the problem (O.E.) has at least one nontrivial solution. Moreover, denote this nontrivial solution by u1 , and denote νA (u1 ) := νA (F  (u1 )). If νA (B0 ) = 0, and νA (u1 ) < |iA0 (B1 ) − iA0 (B0 )|, then the problem (O.E.) has one more nontrivial solution. Part 2. If F is even in u and iA0 (B0 ) = iA0 (B∞ ),

(11.21)

then the problem (O.E.) has at least |iA0 (B0 ) − iA0 (B∞ )| pairs of nontrivial solutions.

11.2 Applications: The Existence and Multiplicity of Solutions for Wave Equations 11.2.1 One Dimensional Wave Equations In this subsection, we will consider the following one dimensional wave equation ⎧ ⎨ u ≡ utt − uxx = f (x, t, u), ∀(x, t) ∈ [0, π ] × R, u(0, t) = u(π, t) = 0, ⎩ u(x, t + T ) = u(x, t),

(W.E.)

11.2 Applications: The Existence and Multiplicity of Solutions for Wave. . .

305

where T > 0, f ∈ C 1 ([0, π ] × R2 , R) and T -periodic in variable t. In what follows we assume systematically that T is a rational multiple of π . So, there exist coprime integers a, b, such that T = 2πa b . Let L2 := where i =

√

⎧ ⎨ ⎩

u, u =

 j >0,k∈Z

⎫ a ⎬ uj,k sin j x exp ik t , b ⎭

−1 and uj,k ∈ C with uj,k = u¯ j,−k , its inner product is 

(u, v)2 =

(uj,k , v¯j,k ), u, v ∈ L2 ,

j >0,k∈Z

the corresponding norm is #u#22 =



|uj,k |2 u, v ∈ L2 .

j >0,k∈Z

Consider  as an unbounded self-adjoint operator on L2 . It’s spectrum set is σ () = {(a 2 k 2 − b2 j 2 )/b2 |j > 0, k ∈ Z}. It is easy to see  has only one essential spectrum λ0 = 0. Take the working space H = L2 and the operator A = . As defined in Sect. 3.3, we can define the index 2 ∞ = L∞ ([0, π ] × S 1 , R) pair (iA∓ (B), νA (B)) for any B ∈ L∓ s (L , λ0 ). Denote L 1 the set of all essentially bounded functions, where S = R/T . For any g ∈ L∞ , it is easy to see g determines a bounded self-adjoint operator on L2 , by u(x, t) → g(x, t)u(x, t), ∀u ∈ L2 .

(11.22)

Without confusion, we still denote this operator by g, that is to say we have the 2 continuous embedding L∞ %→ Ls (L2 ). Thus for any g ∈ L∞ ∩ L∓ s (L , λ0 ) we ∓ ∞ have the index pair (iA (g), νA (g)). Besides, for any g1 , g2 ∈ L , g1 ≤ g2 means that g2 (x, t) − g1 (x, t) ≥ 0, a.e.(x, t) ∈ [0, π ] × S 1 . Denote  = [0, π ] × S 1 for simplicity, and assume f ∈ C 1 ( × R, R) satisfying the following conditions. (f0∓ ) (f1∓ )

2 f (x, t, 0) ≡ 0, and denote g0 :=fu (x, t, 0) ∈ L∞ ∩ L∓ s (L , λ0 ). There exists a constant δ > 0 such that

±fu (x, t, u) < −δ, ∀(x, t, u) ∈  × R.

306

11 The Existence and Multiplicity of Solutions of Wave Equations

∓ 2 (f2∓ ) There exist g1 , g2 ∈ L∞ ∩L∓ s (L , λ0 ) with ±g1 < ±g2 , iA (g1 )+νA (g1 ) = ∓ iA (g2 ) and νA (g2 ) = 0, such that

±fu (x, t, u) ≥ ±g1 (x, t), ∀(x, t, u) ∈  × R and ±fu (x, t, u) ≤ ±g2 (x, t), ∀(x, t) ∈  and |u| > R for some constant R > 0. Define F(x, t, u) :=

u

f (x, t, s)ds, ∀u ∈ R,

0

and F(x, t, u)dxdt, ∀u ∈ H.

F (u) := 

If f ∈ C 1 ( × R, R) satisfies (f1∓ ), we have F ∈ C 1 (H, R), the solutions of the operator equation Au = F  (u), u ∈ D(A) are the solutions of (W.E.). And we have the following result. Theorem 2.1 For A =  ∈ Oe∓ (λ∓ ), λ∓ = 0, assume f ∈ C 1 ( × R, R) satisfies conditions (f1∓ ) and (f2∓ ), then (W.E.) has a weak solution. Further more, if f satisfies condition (f0∓ ) and iA∓ (g0 ) > iA∓ (g2 ), then (W.E.) has a nontrivial weak solution. Proof Only consider A =  ∈ Oe− (λ− ), by virtue of the similar reason as stated before Theorem 1.6, we can choose ε > 0 small enough, such that (g1 − gε )−1 − A−1 ε > 0, where g1,ε := g1 − ε · I and Aε := A − g1,ε . Denote Fε (z) := F (z) − 12 (g1,ε z, z)H . In order to prove this theorem, we only need to check the conditions in Theorem 1.6 step by step. Firstly, by condition (f2− ), we have F (z)− 12 (g1 z, z)H is convex. Since νA (g2 ) = 0, we can choose η > 0 small enough, such that iA− (g2 + η · I ) = iA− (g2 ) and νA (g2 + η · I ) = 0, denote g2,η := g2 + η · I , then we have F (z) ≤ 12 (g2,η z, z)H + c, that is to say F satisfies (F3− ).

11.2 Applications: The Existence and Multiplicity of Solutions for Wave. . .

307

Secondly, from (f0− ) and condition iA− (g0 ) > iA− (g2 ), we can choose ζ > 0 small enough, such that F(x, t, u) ≥

1 g0,ζ u2 , for |u| small enough, 2

(11.23)

and iA− (g0,ζ ) = iA− (g0 ) > iA− (g2 ), where g0,ζ := g0 − ζ . Consider the proof of Theorem 1.6, in stead of verifying (F4− ), we only need to prove the following inequality Fε∗ (z) ≤

1 ((g0,ζ − g1,ε )−1 z, z)H , ∀z ∈ Z, with #z# small enough, 2

(11.24)

where Z is an arbitrary finite dimensional subspace of H . By the definition of Fε∗ , it is easy to verify that Fε∗ (z)

= 

Fε∗ (x, t, z)dxdt,

where Fε (x, t, u) := F(x, t, u) − 12 g1,ε u2 and Fε∗ is its Fenchel conjugate corresponding to u. From (11.23), we have Fε∗ (x, t, z) ≤ 12 (g0,ζ − g1,ε )−1 z2 for |z| small enough, and since Z is a finite dimensional space, we have proved (11.24). Now, what we need to prove is the (PS) condition. Recall the functional $ on H defined by 1 $(z) = Fε∗ (z) − (A−1 z, z)H , ∀z ∈ H. 2 ε Recall the constant R in condition (f2− ), define η ∈ C([0, ∞), [0, 1]) by  η(r) :=

1, r > R + 1, 0, 0 ≤ r ≤ R.

Define b ∈ C( × R, R) by b(x, t, u) := η(|u|)fu (x, t, u) + (1 − η(|u|))g2 (x, t), ∀(x, t, u) ∈  × R. From conditon (f2− ), we have g1 ≤ b ≤ g2 and b(x, t, u) ≡ fu (x, t, u), ∀|u| > R + 1. Define B ∈ C( × R, R) by

(11.25)

308

11 The Existence and Multiplicity of Solutions of Wave Equations



1

B(x, t, u) :=

b(x, t, su)ds. 0

Of course, we have g 1 ≤ B ≤ g2 .

(11.26)

For any  > 0, if |u| > (R + 1)/, then   1     [fu (x, t, su) − b(x, t, su)]dsu |f (x, t, u) − B(x, t, u)u| = f (x, t, 0) + 0

       ≤|f (x, t, 0)| +  [fu (x, t, su) − b(x, t, su)]ds  |u| 0  1      +  [fu (x, t, su) − b(x, t, su)]dsu . 

Since  is compact set, from the continuity of f , the first term |f (x, t, 0)| of the above inequality is bounded by some constant c1 . From boundedness condition in (f1∓ ) and the boundedness of function b, the second term satisfies       [f  (x, t, su) − b(x, t, su)]ds  |u| ≤ c2 |u| u   0

for some constant c2 . From (11.25), the third term satisfies 

1

[fu (x, t, su) − b(x, t, su)]dsu ≡ 0

for |u| > (R + 1)/. So we have the following estimate |f (x, t, u) − B(x, t, u)u| ≤ c1 + c2 |u|, ∀ > 0, |u| > (R + 1)/. That is to say f (x, t, u) − B(x, t, u)u = o(|u|), as |u| → ∞.

(11.27)

Therefore we have #F  (u) − B(u)u#2H =

|f (x, t, u) − B(x, t, u)u|2 dxdt 

= o(#u#2H ), #u#H → ∞.

(11.28)

11.2 Applications: The Existence and Multiplicity of Solutions for Wave. . .

309



Let {zn } be a (PS) sequence of $, and denote un = Fε∗ (zn ) (or equivalent Fε (un ) = zn ), we have yn := −$  (zn ) → 0 in H and Aε (un + yn ) = Fε (un ). or equivalent A(un + yn ) = F  (un ) + g1,ε yn .

(11.29)

With the similar discussion as in Lemmas 1.1 and 1.2, we can prove that {un + yn } is bounded in E = D(|A1/2 |). For details, firstly, from (f2− ) and (11.26), for any u ∈ H , A − B(u) is invertible and there exists a constant c > 0, such that c#un + yn #2E ≤ ((A − B(un ))(un + yn ), (un + yn )+ − (un + yn )− )H , where the decomposition u = u+ + u− corresponds to the positive and negative space of A − B(u). Secondly, from (11.28) and yn → 0, we have ((A − B(un ))(un + yn ), (un + yn )+ − (un + yn )− )H =((F  (un ) − B(un ))un , (un + yn )+ − (un + yn )− )H + ((g1,ε + B(un ))yn , (un + yn )+ − (un + yn )− )H ≤(o(#un #H ) + o(1))#un + yn #H . thus {un + yn } is bounded in E. Now, we can not use saddle point reduction method to receive the convergent subsequence of {un }, because F is not C 2 continuous on H . But we can use the property of the spectrum of A in this specific situation. Since 0 is the unique isolate essential spectrum of A, we have the following orthogonal decomposition E = ker A ⊕ E1 with the compact embedding E1 %→ H . Assume un +yn = vn +wn with vn ∈ ker A and wn ∈ E1 , from the boundedness of {un + yn }, without loss of generality, we have vn & v ∈ ker A and wn & w ∈ E1 , thus wn → w in H . Let δ > 0 be the constant in (f1− ), then we have lim

n→∞

δ δ #vn − v#2H = lim (vn − v, vn − v)H n→∞ 2 2 δ δ = lim (vn , vn − v)H − lim (v, vn − v)H . n→∞ 2 n→∞ 2 δ (v, vn − v)H = 0. And since ker(A) ⊥ E1 , we have n→∞ 2

Since vn & v in H , lim

310

11 The Existence and Multiplicity of Solutions of Wave Equations

lim

n→∞

δ δ #vn − v#2H = lim (vn , vn − v)H n→∞ 2 2 δ = lim ((A + I )(vn + wn ), vn − v)H . n→∞ 2

Next, by (11.29), the property yn → 0, wn → w, vn & v in H and boundedness of fu , we have the following equalities lim

n→∞

δ δ #vn − v#2H = lim (F  (vn + wn − yn ) + g1,ε yn + (vn + wn ), vn − v)H n→∞ 2 2 = lim (F  (vn + wn − yn ) n→∞

δ + (vn + wn ), vn − v)H 2 = lim (f (x, t, vn + wn − yn )) n→∞ 

δ + (vn + wn − yn ), vn − v)dxdt 2 = lim (fδ/2 (x, t, vn + wn − yn ) n→∞ 

− fδ/2 (x, t, v + w − y), vn − v)dxdt  = lim (fδ/2 (x, t, ξn ) n→∞ 

(vn − v + wn − w − yn + y), vn − v)dxdt  (fδ/2 (x, t, ξn )(vn − v), vn − v)dxdt, = lim n→∞ 

 := f  + where fδ/2 (x, t, u) := f (x, t, u) + 2δ u, fδ/2 u  < − δ , so we have and yn . From (f1∓ ), we have fδ/2 2

lim

n→∞

δ 2

and ξn depends on vn , wn

δ #vn − v#2H ≤ 0. 2

That is to say vn → v in H , un → v + w in H and zn → Fε (v + w). So, we have proved that $ satisfied (PS) condition, and the proof of this theorem is complete.   Further more, in order to use Theorem 1.3 to receive the solutions of (W.E.), we can use the idea of [58], corresponding to the parity of b, we restrict the wave operator  on an invariant subspace L2b of L2 , such that  is reversible on L2b , thus  can be regard as an unbounded self-adjoint operator on L2b with compact resolvent. For details, recall the space

11.2 Applications: The Existence and Multiplicity of Solutions for Wave. . .

L2 :=

⎧ ⎨ ⎩

u, u =

 j >0,k∈Z

311

⎫ a ⎬ uj,k sin j x exp ik t , b ⎭

defined above, we consider the following two situations. Situation 1. b is odd. In this case, we assume the nonlinear term f satisfying the following condition (fodd ) R2 .

f is

T 2 -periodic in t

and f (x, t, u) = f (π −x, t, u), ∀(x, t, u) ∈ [0, π ]×

Let L21 be the closed subspace of L2 defined by L21 = {u ∈ L2 , u(π −x, t) = u(x, t), u(x, t +

T ) = u(x, t), a.e.x ∈ [0, π ], t ∈ R} 2

Then L21 is invariant by f , that is to say that u ∈ L21 ⇒ f (x, t, u) ∈ L21 and we have u ∈ L21 ⇔ uj,k = 0 if j is even or k is odd.

(11.30)

Let 1 ≡ |L2 , similarly we have 1 is self-adjoint in L21 and L21 ∩ N () = {0}. 1

In this case, we can consider the problem (W.E.) for u ∈ L21 , that is ⎧ 2 u ≡ utt − uxx = f (x, t, u), ⎪ ⎪ ⎨ u(0, t) = u(π, t) = 0, ⎪ u(x, t + T2 ) = u(x, t), ⎪ ⎩ u(π − x, t) = u(x, t)

(W.E.1)

Situation 2. b is even. In this case, we need the nonlinear term f satisfying the following condition (feven )

f is

T 2 -periodic

in t and odd in u.

Let L22 be the closed subspace of L2 defined by L22

 T 2 = u ∈ L , u(x, t) = −u(x, t + ), a.e.x ∈ [0, π ], t ∈ R . 2

Then L22 is invariant by f , and we have u ∈ L22 ⇔ uj,k = 0 for any even k.

(11.31)

Let 2 ≡ |L2 , then we have 2 is self-adjoint in L22 and L22 ∩ N () = {0}, where 2 N() is the kernel of .

312

11 The Existence and Multiplicity of Solutions of Wave Equations

In this case, we can consider the problem (W.E.) for u ∈ L22 , that is ⎧ ⎨ 1 u ≡ utt − uxx = f (x, t, u), u(0, t) = u(π, t) = 0, ⎩ u(x, t + T2 ) = −u(x, t),

(W.E.2)

In these two cases, the key point is that i has compact resolvent on L2i for i = 1, 2. For the sake of simplicity we shall write L instead of L2i (i = 1, 2), correspondingly we write  instead of i (i = 1, 2), and we shall renumber that  has compact resolvent on L. In order to use Theorem 1.3, we can not regard L as the working space, since the functional F defined above will not be C 2 continuous on L. So, we need to find a suitable working space to protect the smoothness of F . Assume σ () = {λn }, satisfying −∞ ← λ−l ≤ · · · ≤ λl → +∞, l → +∞, and for any n ∈ Z, λn is a point spectrum of  with its eigenvector en . That is to say en = λn en . Thus for any u ∈ L, we can rewrite it as +∞ 

u=

un en , un ∈ R,

n=−∞

and it’s norm #u#2L2 =

+∞ 

|un |2 . Since λn = 0, we define a new Hilbert space H

n=−∞

by    H = u ∈ L 

(

+∞ 

|λn |1/2 |un |2 < ∞ ,

(11.32)

n=−∞

with its inner product defined by (u, v)H :=

+∞ 

|λn |1/2 un vn , ∀u, v ∈ H,

n=−∞

where u =

+∞  n=−∞

un en and v =

+∞ 

vn en . It is easy to see the embedding τ :

n=−∞

H %→ L2 is compact. Now, define a self-adjoint operator A on H by (Au, v)H := (u, v)L2 ,

(11.33)

it is easy to see A is unbounded, its spectrum σ (A) = {λn /|λn |1/2 } satisfying

11.2 Applications: The Existence and Multiplicity of Solutions for Wave. . .

Aen =

313

λn en . |λn |1/2

− Thus A has compact resolvent. By Remark 3.7, L− s (H, λ ) = Ls (H ), for any g ∈ ∞ L , by (11.22), g will define a bounded self-adjoint operator τ ∗ g on H by

(τ ∗ gu, v)H := (gu, v)L2 , ∀u, v ∈ H, where τ ∗ is the dual operator of τ . Thus we have the index pair (iA− (τ ∗ g), νA (τ ∗ g)) for any g ∈ L∞ . For simplicity, we denote it by (iA (g), νA (g)). In addition to Theorem 2.1, assume f ∈ C 1 ( × R, R) satisfying the following conditions. (f0 ) (f1 )

f (x, t, 0) ≡ 0, and denote g0 := fu (x, t, 0) ∈ L∞ . There exists cf > 0 such that |fu (x, t, u)| ≤ cf , ∀(x, t, u) ∈  × R.

(f∞,1 )

There exists g∞ ∈ L∞ with νA (g∞ ) = 0 such that f (x, t, u) = g∞ (x, t)u + o(|u|), as |u| → ∞, ∀ (x, t) ∈ .

(f∞,2 ) There exists a number R > 0, and g∞,a , g∞,b ∈ L∞ with iA (g∞,a ) = iA (g∞,b ) and νA (g∞,b ) = 0 satisfying g∞,a (x, t) ≤ fu (x, t, u) ≤ g∞,b (x, t), ∀(x, t) ∈ , |u| > R, The condition (f0 ) means that u ≡ 0 is a trivial solution of (W.E.). Corresponding to Theorem 1.3, we have the following result. Theorem 2.2 ([284]) Assume f ∈ C 1 ( × R, R) satisfying (f0 ), (f1 ) and (f∞,1 ) (or (f∞,2 ) ). Further more, corresponding to the parity of b, assume f satisfying (fodd ) or (feven ). For the results, we have the following two parts. Part 1.

If

iA (g∞ ) ∈ [iA (g0 ), iA (g0 ) + νA (g0 )](or iA (g∞,a ) ∈ [iA (g0 ), iA (g0 ) + νA (g0 )]), (11.34) then the problem (W.E.) has at least one nontrivial weak solution. More over, denote this nontrivial solution by u1 , and denote νA (u1 ) := νA (fu (x, t, u1 )). If νA (g0 ) = 0, and νA (u1 ) < |iA (g1 ) − iA (g0 )|, then the problem (W.E.) has one more nontrivial weak solution.

314

Part 2.

11 The Existence and Multiplicity of Solutions of Wave Equations

If f is odd in u and iA (g0 ) > iA (g∞ ),

(11.35)

then the problem (W.E.) has at least iA (g0 ) − iA (g∞ ) pairs of nontrivial weak solutions. Proof Since the proof of this theorem is similar Theorem 2.1, we only give a brief proof here. Corresponding to the parity of b, let the working space H defined in (11.32) and operator A defined in (11.33). Recall the function F and functional F defined as above with f satisfying (fodd ) or (feven ), then the solutions of (W.E.1) and (W.E.2) are the solutions of (O.E.). Since the embedding τ : H %→ L2 is compact, f ∈ C 1 ([0, π ] × R2 , R) and fu is bounded, we have F ∈ C 2 (H, R) satisfying (F  (u), v)H = (f (x, t, u), v)L2 , ∀u, v ∈ H, and (F  (u)v, w)H = (fu (x, t, u)v, w)L2 , ∀u, v, w ∈ H. That is to say F  (u) = τ ∗ f (x, t, u) and F  (u) = τ ∗ fu (x, t, u). Now we only need to check the conditions of Theorem 1.3 step by step. Firstly, from condition (f0 ), we have F  (0) = 0 and F  (0) = τ ∗ g0 . Secondly, from condition (f1 ), we have F  is bounded on H . Thirdly, from condition (f∞,1 ), we have F  (u) = τ ∗ g∞ u+o(#u#H ) ∓ ), with the same method as #u#H → ∞. Lastly, instead of verifying condition (F∞,2 in Theorem 2.1, from condition (f∞,2 ), we can get a similar estimate as (11.28), so Lemma 1.2 and then Theorem 1.3 will keep valid.  

11.2.2 n-Dimensional Wave Equations In this subsection, we consider the existence and multiplicity of radially symmetric solutions for the n-dimensional wave equation: ⎧ ⎨ u ≡ utt − x u = h(x, t, u), t ∈ R, x ∈ BR , u(x, t) = 0, t ∈ R, t ∈ R, x ∈ ∂BR , ⎩ u(x, t + T ) = u(x, t), t ∈ R, x ∈ BR ,

(n–W.E.)

where BR = {x ∈ Rn , |x| < R}, ∂BR = {x ∈ Rn , |x| = R}, n > 1 and the nonlinear term h is T -periodic in variable t. Restriction of the radially symmetry allows us to know the nature of spectrum of the wave operator. Let r = |x| and S 1 := R/T , if h(x, t, u) = h(r, t, u) then the n-dimensional wave equation (n–W.E.) can be transformed into:

11.2 Applications: The Existence and Multiplicity of Solutions for Wave. . .

⎧ n−1 ⎨ A0 u := utt − urr − r ur = h(r, t, u), u(R, t) = 0, ⎩ u(r, 0) = u(r, T ), ut (r, 0) = ut (r, T ),

315

(r, t) ∈  := [0, R] × S 1 . (RS–W.E.)

A0 is symmetric on L2 (, ρ), where ρ = r n−1 and 

L2 (, ρ) := u|#u#2L2 (,ρ) := |u(t, r)|2 r n−1 dtdr < ∞ . 

By the method of separation of variables, we obtain the eigenvalues of the operator A0 are λj k =

$ γ %2 j

R

 −

2kπ T

2 , j ∈ Z+ , k ∈ Z,

where γj is the j -th positive zero of Jν (x), ν = (n − 2)/2, and Jν is the Bessel function of the first kind of order ν. Denote the corresponding eigenfunctions by {ψj k (t, r)}. We can check that {ψj k } form a complete orthonormal sequence in L2 (, ρ). By the asymptotic properties of the Bessel functions (see [290]), the spectrum of the wave operator can be characterized as follows. Lemma 2.3 (See [266, Theorem 2.1]) Assume that 8R/T = a/b, (a, b) = 1, then A0 has a self-adjoint extension A having no essential spectrum other than the point λ0 = −(n − 3)(n − 1)/4R 2 . If n = 3(mod(4, a)), then A has no essential spectrum. If n = 3(mod(4, a)), the essential spectrum of A is precisely the point λ0 = −(n − 3)(n − 1)/4R 2 . Thus for n = 3(mod(4, a)), the operator A has only one essential spectrum λ0 = −(n − 3)(n − 1)/4R 2 , that is to say λ∓ = λ0 , we can define the index 2 2 pair (iA∓ (B), νA (B)) for any B ∈ L∓ s (L (, ρ), λ0 ) ⊂ Ls (L (, ρ)) and with the same discussion in above subsection, we have the index pair (iA∓ (g), νA (g)) for any 2 1 g ∈ L∞ (, R) ∩ L∓ s (L (, ρ), λ0 ). Assume h ∈ C ( × R, R) satisfying the following conditions. (h∓ 0) (h∓ 1)

2 h(r, t, 0) ≡ 0, and denote g0 :=hu (r, t, 0) ∈ L∞ (, R)∩L∓ s (L (, ρ), λ0 ). There exists a constant δ > 0 such that

±hu (r, t, u) < ±(λ0 ∓ δ), ∀(r, t, u) ∈  × R. 2 There exist g1 , g2 ∈ L∞ (, R) ∩ L∓ (h∓ s (L (, ρ), λ0 ) with ±g1 < ±g2 , 2) ∓ ∓ iA (g1 ) + νA (g1 ) = iA (g2 ) and νA (g2 ) = 0, such that

±hu (r, t, u) ≥ ±g1 (r, t), ∀(r, t, u) ∈  × R and

316

11 The Existence and Multiplicity of Solutions of Wave Equations

±hu (r, t, u) ≤ ±g2 (r, t), ∀(r, t) ∈  and |u| > K for some constant K > 0. We have the similar result as Theorem 2.1 for (n–W.E.). Theorem 2.4 ([284]) For n = 3(mod(4, a)) and the operator A defined above, ∓ assume h ∈ C 1 ( × R, R) satisfying conditions (h∓ 1 ) and (h2 ), then (n–W.E.) has a radially symmetrical weak solution. Further more, if h satisfies condition (h∓ 0) and iA∓ (g0 ) > iA∓ (g2 ), then (n–W.E.) has a nontrivial radially symmetrical weak solution. For n = 3(mod(4, a)), since A has no essential spectrum, we have λ− = +∞ 2 − 2 and L− s (L (, ρ), λ ) = Ls (L (, ρ)) as claimed in Remark 3.7. In addition to Theorem 2.4, we have the similar result as Theorem 2.2. Assume h ∈ C 1 ( × R, R) satisfying the following conditions (h0 ) (h1 )

h(r, t, 0) ≡ 0, and denote g0 := hu (r, t, 0) ∈ L∞ (, R). There exists ch > 0 such that |hu (r, t, u)| ≤ ch , ∀(r, t, u) ∈  × R.

(h∞,1 )

There exists g∞ ∈ L∞ with νA (g∞ ) = 0 such that h(r, t, u) = g∞ (r, t)u + o(|u|), as |u| → ∞, ∀ (r, t) ∈ .

(h∞,2 ) There exists a number K > 0, and g∞,a , g∞,b ∈ L∞ (, R) with iA (g∞,a ) = iA (g∞,b ) and νA (g∞,b ) = 0 satisfying g∞,a (r, t) ≤ hu (r, t, u) ≤ g∞,b (r, t), ∀(r, t) ∈ , |u| > K. Theorem 2.5 ([284]) For n = 3(mod(4, a)) and the operator A defined above, assume h ∈ C 1 ( × R, R) satisfying (h0 ), (h1 ) and (h∞,1 ) (or (h∞,2 )). We have the following two parts of results. Part 1.

If

iA (g∞ ) ∈ [iA (g0 ), iA (g0 ) + νA (g0 )](or iA (g∞,a ) ∈ [iA (g0 ), iA (g0 ) + νA (g0 )]), (11.36) then the problem (n–W.E.) has at least one nontrivial radially symmetrical weak solution. More over, denote this nontrivial solution by u1 , and denote νA (u1 ) := νA (hu (r, t, u1 )). If νA (g0 ) = 0, and νA (u1 ) < |iA (g1 ) − iA (g0 )|,

11.2 Applications: The Existence and Multiplicity of Solutions for Wave. . .

317

then the problem (n–W.E.) has one more nontrivial radially symmetrical weak solution. Part 2. If h is odd in u and iA (g0 ) > iA (g∞ ),

(11.37)

then the problem (n–W.E.) has at least iA (g0 ) − iA (g∞ ) pairs of nontrivial radially symmetrical weak solutions. Remark 2.6 A. Generally, it is hard to compute the index of a bounded self-adjoint operator though we have given its definition, but in some special conditions we can do it easily. For example, from Lemma 2.3, let σ (A) = {λ∓ n } ∪ {λ0 } satisfying − −∞ ← λ− −l < · · · < λl → λ0 , l → +∞,

and + λ0 ← λ + −l < · · · < λl → +∞, l → +∞,

where λ0 is the unique essential spectrum of A (if A has essential spectrum). If ∓ the functions g0 , g∞ in conditions (f0∓ ) and (f∞,1 ), for example, satisfying the following inequality − − λ− k−1 < g∞ (x, t) < λk · · · ≤ λl < g0 (x, t) < λ0 , ∀(x, t) ∈ ,

or + + λ0 < g0 (x, t) < λ+ k · · · ≤ λl < g∞ (x, t) < λl+1 , ∀(x, t) ∈ ,

for some k, l ∈ Z, it is easy to see νA (g∞ ) = 0 and iA− (g0 ) − iA− (g∞ ) > l − k + 1 (or iA+ (g0 ) − iA+ (g∞ ) > l − k + 1) satisfying the twisted conditions. B. The discussion in above two subsections can also be used in the existence and multiplicity of nontrivial solutions of the nonlinear beam equation ⎧ utt + uxxxx = f (x, t, u), ⎪ ⎪ ⎨ u(0, t) = u(π, t) = 0, ∀(x, t) ∈ [0, π ] × R, ⎪ u (0, t) = uxx (π, t) = 0, ⎪ ⎩ xx u(x, t + T ) = u(x, t),

(B.E.)

where T > 0, f ∈ C 1 ([0, π ] × R2 , R) and T -periodic in variable t. If we also assume systematically that T is a rational multiple of π , then the spectrum of 4 operator ∂tt2 + ∂xxxx will be similar to the spectrum of one dimensional wave operator, so all of the results will keep valid in this problem.

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Index

Symbols (L0 , L1 )-homotopic, 184 (ε, L0 , L1 )-signature, 182 (m, k)-variationally visible, 284 D(a), 57 Dω,P (M), 56 E ⊥ω˜ , 2 I (A, A − B), 38 M k , 12 Mn± , 57 M1  M2 , 11 P -periodic orbit, 237 P -solution, 236 R(θ), 57 SpJ (2n), 4 i¯L0 (γ ), 154 i¯P (γ ), 89 i¯1 (γ ), 29 exp(Q), 6 ˆ ), 115 i(f iˆL (ρ), 111 μˆ Lˆ (γ ), 178 L(K n ), 1 Ls (K n ), 1 Oe+ (μ), 41 Oe− (μ), 41 Oe0 (a, b), 41 Pτ (2n), 24 + L+ s (H, λ ), 42 − ), 42 L− (H, λ s 0 Ls (H, λa , λb ), 42 V∞,b (, 2), 284 Osp(2n), 7 Sp(2n)∗ω,P , 56 Sp(2n)0ω,P , 56 © Springer Nature Singapore Pte Ltd. 2019 C. Liu, Index theory in nonlinear analysis, https://doi.org/10.1007/978-981-13-7287-2

Sp(2n)± ω,P , 56 concav∗(L,L ) (γ ), 180 concav0(L,L ) (γ ), 180 concav(L,L ) (γ ), 180 concavL (γ ), 108 μcA (B), 52 μCLM (f ), 168 μA (B), 53 μL (γ ), 177 μV (f ), 165 μF (T ), 40 μ(L0 ,L1 ) (γ ), 211 μproper (h), 168 νA0 (B), 49 νA (B), 52 νA− (B), 46 νA∓ (B), 47 νL0 (γ ), 96 νL (γ ), 100 νω (γ ), 24, 28 νωL0 (B), 144 νωP (γ ), 55 υF (T ), 40 (n), 12 (M), 20 0 (M), 20 e(M), 20 − iA,k (B), 46 iA0 (B), 49  iLL (γ ), 121 i1 (γ ), 25 iA∓ (B), 47 iL0 (γ ), 97 iL (γ ), 99 331

332 iω (γ ), 24, 29 iωP (γ ), 56 iωL0 (B), 144 s(L1 , L2 ; M1 , M2 ), 172 L0 -nullity, 96 Lag(V ), 12 ∗P P τ,ω (2n), 56 ∗ P ∗ (2n), 56 P τ,ω ∗ P 0 (2n), 56 P τ,ω 0 P ∗ (2n), 56 P τ,ω 0 P 0 (2n), 56 P τ,ω ± P SM (ω), 88 ∗ P Pτ,ω (2n), 56 0 P Pτ,ω (2n), 56 (L0 , L )-index, 119 (L0 , L )-nullity, 118

B Bott-type formula, 23 for index pair (i1 , ν1 ), 29 for L-index, 148 for the (P , ω)-index, 83

Index L Linear Hamiltonian system, 17 elliptic, 20 fundamental solution, 17 hyperbolic, 20 Linear symplectic map, 3 Linear symplectic structure, 1

M Maslov type index, 23 (P , ω)-index, 56, 58 L-index, 24, 97, 99, 101 for general continuous symplectic path, 111 P -index, 24, 59 for general continuous symplectic path, 64 ω-index, 23, 29 for a pair of Lagrangian paths, 164 for periodic boundary condition, 23 Mean index for L-boundary, 154 for periodic boundary, 29 for P -index, 89 Mixed (L0 , L1 )-concavity, 211

C Concavity, 108, 180

E Eigenvalue algebraic multiplicity, 19 geometric multiplicity, 19

N Nullity (P , ω)-nullity, 55 ω-nullity, 24

P Positive definite path connectivity, 19 H Homotopy component, 20 Hörmander index, 172

I Infinite variationally visible, 284 Iteration path for P -boundary, 79 for periodic boundary, 29

K Krein type number, 33 Krein negative, 33 Krein positive, 33

R Relative index via dual method, 47, 49 via saddle point reduction, 53 Relative Morse index, 38 via orthogonal projection, 40

S Special homotopic, 184 Spectral flow, 39 Splitting number for (P , ω)-index, 88 for ω-index, 33

Index Symplectic direct sum, 11 Symplectic matrix, 3 autonomous hyperbolic, 20 elliptic, 20 elliptic hight, 20 hyperbolic, 20 polar decomposition, 5 symplectic group, 4

333 Symplectic path, 18 Symplectic vector space, 1 coisotropic subspace, 2 isotropic subspace, 2 Lagrangian subspace, 2 symplectic basis, 2 symplectic isomorphic, 3 symplectic subspace, 2