Korteweg–de Vries Flows with General Initial Conditions (Mathematical Physics Studies) 9819997372, 9789819997374

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Mathematical Physics Studies

Shinichi Kotani

Korteweg–de Vries Flows with General Initial Conditions

Mathematical Physics Studies Series Editors Giuseppe Dito, Institut de Mathématiques de Bourgogne, Université de Bourgogne, Dijon, France Edward Frenkel, Department of Mathematics, University of California at Berkley, Berkeley, CA, USA Sergei Gukov, California Institute of Technology, Pasadena, CA, USA Yasuyuki Kawahigashi, Department of Mathematical Sciences, The University of Tokyo, Tokyo, Japan Maxim Kontsevich, Institut des Hautes Etudes Scientifiques, Bures-sur-Yvette, France Nicolaas P. Landsman, Chair of Mathematical Physics, Radboud Universiteit Nijmegen, Nijmegen, Gelderland, The Netherlands Bruno Nachtergaele, Department of Mathematics, University of California, Davis, CA, USA Hal Tasaki, Department of Physics, Gakushuin University, Tokyo, Japan

The series publishes original research monographs on contemporary mathematical physics. The focus is on important recent developments at the interface of Mathematics, and Mathematical and Theoretical Physics. These will include, but are not restricted to: application of algebraic geometry, D-modules and symplectic geometry, category theory, number theory, low-dimensional topology, mirror symmetry, string theory, quantum field theory, noncommutative geometry, operator algebras, functional analysis, spectral theory, and probability theory.

Shinichi Kotani

Korteweg–de Vries Flows with General Initial Conditions

Shinichi Kotani Osaka University Toyonaka, Osaka, Japan

ISSN 0921-3767 ISSN 2352-3905 (electronic) Mathematical Physics Studies ISBN 978-981-99-9737-4 ISBN 978-981-99-9738-1 (eBook) https://doi.org/10.1007/978-981-99-9738-1 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 This work is subject to copyright. All rights are solely and exclusively licensed 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, expressed 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 Paper in this product is recyclable.

Preface

Solitary waves which are characteristic in the Korteweg–de Vries (KdV) equation were discovered at a canal in Edinburgh by Scottish engineer J. S. Russell in 1834. The KdV equation was derived by Boussinesq in 1877 and rediscovered by Korteweg–de Vries in 1895 to describe waves of shallow water surfaces; it is a non-linear time evolution equation of the following form: ∂t q = −∂x3 q + 6q∂x q. This equation has a special solution of a form q(x, t) = f (x − ct), which is called a travelling wave solution, if f satisfies an ordinary differential equation −c f  = − f  + 6 f f  ( denotes the derivative in x).

This equation has two kinds of solutions:  f (x) =

√ −2−1 cosh−2 c(x − a)/2  (solitary wave) , −u 0 − 2κ 2 k 2 cn2 κ(x + δ); k 2 (periodic wave)

  where c = 6u 0 + 4 k 2 − 1 κ 2 and cn denotes the Jacobi elliptic function. In 1965 Zabusky–Kruskal [54] showed by computer simulation that if a solution starts from cosx, then it is decomposed into several solitary waves as time evolves and the solitary waves preserve their shapes after collisions, which suggested to them to call the waves solitons. This discovery was a trigger to promote further theoretical investigation of the KdV equation. And in 1967 Gardner–Greene–Kruskal–Miura [13] made a great discovery that the eigenvalues of Schrödinger operators with solutions of the KdV equation as potentials do not depend on time; more precisely, they found a relationship between the KdV equation and scattering theory of Schrödinger

v

vi

Preface

operators. They found also infinitely many invariants for the KdV equation. Their discovery pushed theoretical study and had a crucial influence on the mathematical world, and the mathematical structure of the KdV equation as an infinite-dimensional completely integrable system had been revealed. During the 1970s and 1980s there appeared a large number of papers treating the KdV equation starting from decaying or periodic initial data. In the 1990s and the 2000s almost periodic initial data began to be considered as a natural extension of periodic initial conditions. Egorova [10] treated some limit periodic potentials. Damanik–Goldstein [7], Binder–Damanik–Goldstein–Lukic [1] investigated analytic quasi-periodic potentials with small norm. Eichinger–VandenBoom–Yuditskii [11] solved this problem for almost periodic potentials whose Schrödinger operators have spectra  with a homogeneity. In all these cases, the associated Schrödinger operators have a purely absolutely continuous spectrum. It should be pointed out that there is one paper by Tsugawa [53] treating relatively general quasi-periodic initial data, in which he applied the Fourier restriction method exploited by Bourgain. Unfortunately, his result is on well-posedness local in time. Quite recently, Chapouto–Killip–Visan [5] announced the uniqueness of solutions to the KdV equation if they are bounded. They also provided a counter-example to the Deift conjecture that any solution to the KdV equation with almost periodic initial data is almost periodic in space and time. Rybkin [45, 46] and Grudsky–Rybkin [18] also contributed to the construction of non-decaying nor periodic solutions. They developed the inverse scattering method to step-like potentials which are decaying on the positive axis and applied it to the KdV equation. On the other hand, physicist V. Zakharov [55, 56] considered a solution of the KdV equation starting from a stationary random process, and studied some statistical properties of the solitary waves and his study was succeeded by El–Kamchatnov [12]. If a potential is random enough, then the associated Schrödinger operator is known to have a dense pure point spectrum. Therefore, he guessed that the solution of the KdV equation starting from such random initial data might consist of a soliton cloud, and he called this expected phenomenon soliton turbulence or soliton gas. He insisted that such phenomena often arise in physics and it is important to investigate theoretically the KdV equation starting from random initial data. This was a motivation for the author to study the KdV equation starting from randomly oscillating initial data. It should be mentioned that a solution to the KdV equation with Gaussian white noise initial data has been constructed by Killip–Visan [27], Killip–Murphy–Visan [26]. The derivative of a Brownian motion is called (Gaussian) white noise, which is supposed to be the most random stochastic process, although the individual path is no longer an ordinary function, but a Schwartz distribution. In the process of the construction they employed the invariance in time of the solution with respect to the Gaussian white noise measure whose density is heuristically given by    1 ∞ u(x)2 d x . exp − 2 −∞

Preface

vii

Our aim in this book is to construct solutions starting from wide classes of bounded initial data including almost periodic or more generally ergodic ones. We employ here the method initiated by Sato since it includes decaying or periodic initial data simultaneously, although it is finite dimensional. Later Segal–Wilson [49] (SW for short) gave a firm basis to Sato’s theory so that we can develop it in infinite-dimensional spaces. However, we still have to extend SW’s version, since the resulting solutions obtained by the SW method are meromorphic on the entire plane C and have poles of the form n(x − x0 )−2 with an integer n. Moreover, in their theory the existence or non-existence of the singularities of solutions on the real line is not given attention. This is partially because they treated complex-valued initial data. Therefore our main purpose is to remove these two obstacles. We somewhat believe that any solution to the KdV equation can be obtained by some extension of Sato’s theory. Our results provide a partial affirmative answer to this prediction; however, we had to impose several conditions to initial data q. Among them, one is the lower semi-boundedness of L q = −∂x2 + q, that is, L q ≥ λ0 for some λ0 ∈ R, and another is some smoothness of q. Therefore the Gaussian white noise initial condition is not included in our results. The lower semi-boundedness of L q is used to have a unique self-adjoint extension of L q , which is equivalent to the unique existence of the WTK functions m ± . Smoothness of q is assumed to have smooth solutions. To remove these two conditions might be not so difficult. The author thanks Prof. S. Molchanov for suggesting that he study this subject in 1995. He is grateful also to Profs. Y. Inahama, F. Nakano, T. Kumagai and Y. Wang. They invited the author to give lectures in 2012, 2015 and 2019, which were quite useful in completing this book. Toyonaka, Japan

Shinichi Kotani

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2 Sato’s Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Algebra of Formal Pseudo-differential Operators . . . . . . . . . . . . . . . . 2.2 Flows on Grassmann Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 10 14

3 KdV Flow I: Reflectionless Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Hardy Spaces and Toeplitz Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Characteristic Functions and m-function . . . . . . . . . . . . . . . . . . . . . . . 3.3 Group Action on Ainv and Tau-Function . . . . . . . . . . . . . . . . . . . . . . . 3.4 Non-degeneracy of Tau-Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Identification of m-functions with WTK Functions . . . . . . . . . . . . . . 3.6 KdV Flow on Qr e f l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Boussinesq Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19 19 21 25 32 34 36 39

4 KdV Flow II: Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Hardy Spaces and Toeplitz Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Group Action on Ainv L and Derivation of Equations . . . . . . . . . . . . . . 4.2.1 Estimates of T (ga) and Differentiability . . . . . . . . . . . . . . . . 4.2.2 Derivation of Schrödinger Operator . . . . . . . . . . . . . . . . . . . . . 4.2.3 Derivation of KdV Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Tau-Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Properties of Tau-Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Continuity of Tau-Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Non-negativity Condition of Ainv ............................. L 4.4.1 Non-degeneracy of Tau-Functions for a ∈ Ainv L ,+ . . . . . . . . . 4.4.2 m-function and WTK Function . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 KdV Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Construction of KdV Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Tau-Function Representation of the Flow . . . . . . . . . . . . . . . .

43 43 49 50 54 57 59 62 65 67 69 73 79 79 82

ix

x

Contents

5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Decaying Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 q ∈ L 1 (R) Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Wigner–von Neumann Type Potentials . . . . . . . . . . . . . . . . . . 5.2 Oscillating Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Reflection Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Ergodic Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87 87 88 89 91 91 96

6 Further Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Extension of Remling’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Transfer Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Reflectionless Property on ac Spectrum . . . . . . . . . . . . . . . . . 6.2 Multi-component Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Nonlinear Schrödinger Equation . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Sine–Gordon Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

103 103 103 113 120 120 126

7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Herglotz–Nevanlinna (HN) Functions . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Spectral Theory of 1D Schrödinger Operators . . . . . . . . . . . . . . . . . . 7.2.1 Weyl–Titchmarsh–Kodaira (WTK) Functions . . . . . . . . . . . . 7.2.2 Shift Operation and Its Properties . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Expansion of WTK Functions when z → ∞ . . . . . . . . . . . . . 7.3 Conformal Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Ergodic Schrödinger Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

131 131 136 136 140 141 149 155

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

Chapter 1

Introduction

Our aim is to solve the KdV equation with bounded initial data as generally as possible. We briefly explain several known methods in this chapter. We start with Lax pair formulation, which is a basic tool. [1] Lax Pair Formulation The discovery by Gardner–Greene–Kruskal–Miura [13] (GGKM for short) was beautifully interpreted by Lax [33]. He made an essential contribution to the study of completely integrable systems by giving a unified view. Suppose we have a one-parameter family of operators . L(t) whose eigenvalues are independent of .t. A sufficient condition for it is the existence of unitary operators .U (t) such that .

L(t) = U (t)−1 L(0)U (t)

holds. Here, a sufficient condition for .U (t) to be a unitary operator is that there exist skew-symmetric operators . A(t) .(A(t)∗ = −A(t)) such that ∂ U (t) = U (t)A(t).

. t

Differentiating the above identity yields ∂ L(t) = [L(t), A(t)] .

. t

What he found is that for . L(t), . A(t) defined by .

L(t) = −∂x2 + q(x, t), A(t) = 4∂x3 − (6q(x, t)∂x + 3∂x q(x, t))

it holds that .

( ) [L(t), A(t)] f = −∂x3 q + 6q∂x q f .

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 S. Kotani, Korteweg–de Vries Flows with General Initial Conditions, Mathematical Physics Studies, https://doi.org/10.1007/978-981-99-9738-1_1

1

2

1 Introduction

Hence .q(x, t) is a solution of the KdV equation if and only if .

(∂t L(t)) f (x) = ([L(t), A(t)] f ) (x) ,

which explains the unitary equivalence of . L(t) and . L(0) beautifully. The pair {L(t), A(t)} is called a Lax pair for the KdV equation.

.

[2] Application In the beginning of the research the KdV equation was studied mainly in two categories: decaying case and periodic case. (i) Decaying Solutions. The first stage of theoretical study of the KdV equation began by applying the scattering theory for decaying potentials. Suppose a solution .q(x, t) to the KdV equation satisfies the condition { ∞ . (1 + |x|) |q(x, t)| d x < ∞ for every t. −∞

Then, the scattering data associated with the potential .q(x, t) ⎧ coefficient), ⎨ r{ (k, t) (reflection } 2 −η j (t) 1≤ j≤n (negative eigenvalues), . . } ⎩{ m j (t) 1≤ j≤n (weight) are known to satisfy ⎧ 3 ⎨ r (k, t) = r (k, 0) e8(ik) t , 1≤ j ≤n η j (t) = η j (0), . ⎩ 8η 3j t m j (t) = m j (0)e , 1≤ j ≤n thanks to the Lax pair. Then applying the inverse scattering method one has a solution to the KdV equation ) ( q(x, t) = −2∂x2 log det I + Fx,t ,

.

where the integral operator . Fx,t on . L 2 (R+ ) is defined by ⎧ { ∞ ⎪ ⎪ F f = F(x + r + s, t) f (s) ds with a kernel (r ) ⎪ x,t ⎨ 0 { n . . ∑ 1 ∞ 3 ⎪ 8(ik)3 t+2ikx ⎪ r 0) e dk + 2 m j e8η j t−2η j x F(x, t) = (k, ⎪ ⎩ π −∞ j=1

1 Introduction

3

If the initial potential.q is reflectionless, that is,.r (k, 0) = 0 identically, then.r (k, t) = 0 for any .t and one has the .n-solitons solution { q(x, t) =

.

−2∂x2

log det I +

(√

mi m j 4 e ηi + η j

(

) ) ηi3 +η 3j t−(ηi +η j )x

) .

1≤i, j≤n

(ii) Periodic Solutions and Algebro-geometric Quasi-periodic Solutions. The inverse scattering method works only for decaying potentials; however, the existence of infinitely many invariants and the Lax idea pushed the development of the inverse spectral theory for periodic potentials, and many researchers (e.g. Novikov, Dubrovin, McKean–Trubowitz, Its–Matveev) contributed. In the course of the study solutions represented by .Θ-functions have been found. They are generally quasi-periodic in space and time. The best result at present for periodic solutions can be found in Kappeler–Topalov [25]: The KdV equation is uniquely solvable in s . H (T) (.s ≥ −1).. They understood the KdV equation as an infinite-dimensional integrable Hamiltonian system, and described action-angle variables by the spectral quantities. This result implies that a periodic Gaussian white noise also can be treated as an initial datum, since it belongs to . H s (T) (.s < −1/2). [3] Other Methods On the other hand, there appeared two methods which make it possible to construct solutions starting from non-decaying and non-periodic initial data. One is the method discovered by R. Hirota and developed by M. Sato and his group. The other is the method of V.A. Marchenko. Hirota [22] paid attention to the fact that in several nonlinear equations their solutions take a form of 2 .q = ∂ x log f or = g/ f , and he considered equations satisfied by . f (or . f, g), and introduced a new notion of differentiation of bilinear form (Hirota derivative). His inspiration came from the Riccati equation and the Burgers equation. Sato [48] tried to clarify its mathematical structure, and reached the conclusion that the KdV equation is nothing but a dynamical system with infinitely many time parameters on an infinite-dimensional Grassmann manifold. Sato’s idea made it possible to obtain rational solutions in unified .n-solitons and .Θ-functions; however, his presentation was algebraic. Later, Segal–Wilson [49] modified Sato’s theory analytically. In 1979, Marchenko [37] solved the KdV equation through a Banach space valued linear equation 3 .∂t q = −∂ x q. He found that a solution of the KdV equation can be obtained by a contraction of the above .q. It seems that his approach has not been fully studied yet.

4

1 Introduction

[4] Plan In this book we employ the Segal–Wilson (SW) version of Sato theory to construct the KdV flow. One of the key steps to realize this application is to rewrite the theory in terms of the Weyl–Titchmarsh–Kodaira functions (WTK function for short) 2 .m ± for a 1D Schrödinger operator . L q = −∂ x + q. They are holomorphic functions on the upper half plane .C+ taking the positive imaginary part there, and such a function is called Herglotz–Nevanlinna function (HN function for short). The WTK functions can be defined uniquely for any real-valued potential .q if . L q is uniquely extendable as a self-adjoint operator in . L 2 (R), and describe the spectrum of . L q completely. There is also a one-to-one correspondence between potentials and their WTK functions. Here we mention only the representation of the Green function )−1 ( .gz (x, y) = L q − z (x, y) by .m ± , namely for .x ≥ y (ϕz (x) + m + (z)ψz (x)) (ϕz (x) − m − (z)ψz (x)) g (x, y) = − m + (z) + m − (z) { ∞ ) 1 ( ∑ (dλ) φλ (x) , φλ (y) , = λ0 λ − z

. z

where .ϕz , .ψz are solutions to { .

−ϕ''z + qϕz = zϕz with ϕz (0) = 1, ϕ'z (0) = 0 −ψz'' + qψz = zψz with ψz (0) = 0, ψz' (0) = 1

,

and .φz (x) = .t (ϕz (x) , ψz (x)). The matrix-valued measure .∑ (dλ) is defined as .

with

(∑ ((a, b)) u, u) = lim ∈↓0

⎛

1 π

{

b

Im (M(λ + i∈)u, u) dλ for any u ∈ C2

a

−m + (z) −1 ⎜ m + (z) + m − (z) m + (z) + m − (z) ⎜ . M(z) = ⎜ ⎝ m − (z) 1 m + (z) + m − (z) m + (z)−1 + m − (z)−1

⎞ ⎟ ⎟ ⎟. ⎠

Since .φz (x) is entire with respect to .z, all singularities of .gz (x, y) arise from those of M, thus the spectrum of . L q coincides with the singularity of . M(z). In the Appendix necessary information for the spectral theory of 1D Schrödinger operators is available. Once we can restate Sato’s theory using WTK functions, then the extension of Sato’s theory or its SW version is straightforward. Even the proof of non-existence of singularities can be carried out by using the properties of HN functions. In order to provide to readers an outline of this book we state a conclusion by simplifying the situation. Let ).ω be a smooth positive function .ω on .R satisfying ( −(n−1) .ω (y) = ω (−y) = O y as . y → ∞ for an odd positive integer .n and .

1 Introduction .

5

D+ = {z ∈ C; |Re z| < ω (Im z)} , D− = {z ∈ C; |Re z| > ω (Im z)} .

Set .Cn = ∂ D+ (the boundary), namely Cn = {±ω (y) + i y; y ∈ R} .

.

Then .Cn is a simple closed curve in the Riemann sphere satisfying Cn = −Cn , Cn = C n .

.

Define

{ } ┌(n) = g = eh ; h is a real odd polynomial of degree ≤ n .

.

The .n in .Cn is chosen so that .g ∈ ┌(n) remains bounded on .Cn . For a real potential q with . L q ≥ λ0 (.< 0) let .m ± be the WTK functions and set

.

{ m (z) =

.

( ) −m +( −z 2) if Re z > 0 . m − −z 2 if Re z < 0

Then .m is holomorphic on .C\ (iR ∪ [−μ0 , μ0 ]) with .μ0 = potentials by Q(n)

.

⎧ ⎨ q; m (z) = z +

√

−λ0 . Define a class of

⎫ ( ) m k z −k + O z −L+1 on D− for ⎬ = . 1≤k≤L−2 ( ) ⎩ ⎭ any L ≥ 2 and D− with ω (y) = O y −(n−1) as y → ∞ ∑

The combination of Proposition 4.7 and Theorem 4.1 yields: Theorem For each odd positive integer .n there exists a flow .{K (g)}g∈┌(n) on ( ) n Q (n) such that . K (et z )q (x) satisfies the ).(n + 1) /2-th KdV equation. Especially ( ( ) tz 4t z 3 . K (e )q (x) = q (x + t) and . K (e )q (x) yields a solution to the KdV equation. .

Elements of .Q(n) are known to be infinitely differentiable. Theorem 4.1 treats functions with finite differentiablity. Thus it is crucial to know conditions under which .m has the above asymptotic behavior near the imaginary axis. .m(z) = z holds if .q = 0. Moreover for any .∈ > 0 m (z) = z +

.

∑

) ( π m k z −k + O z −L+1 if |arg (±z)| < − ∈ 2 1≤k≤L−2

is known to be valid if .q ∈ C L−2 (−a, a) for an .a > 0 (see Lemma 7.7), hence the number . L is related with the degree of smoothness of .q. Although several sufficient conditions for .q ∈ Q(n) are known, any necessary and sufficient condition for this is open, hence a direct description of .Q(n) seems hard. If .q ∈ L 1 (R), then .m has a

6

1 Introduction

finite limit .m (i y + 0) on .iR\ {0} and one can show the asymptotic expansion for . L on .iR is valid if .q ( j) ∈ L 1 (R) for . j = 0, .1,.. . ., . L − 2 (see Theorem 5.1), hence the Schwartz class .S is included in .Q(n) for any .n. Also for periodic potentials the WTK functions take finite values on the real line and its behavior is closely related to the smoothness of .q. These two cases can be treated also by inverse spectral problems as mentioned above. The common feature of these two cases is the absolute continuity of the spectra. Generally speaking if the spectral measures contain some singular parts, it becomes harder to analyze the behavior of WTK functions (or equivalently Green functions) near the positive axis. There is another important example of .Q(n) , namely bounded smooth ergodic potentials, which was the main motivation of the author. This class contains smooth almost periodic potentials, and in some cases the associated Schr¯odinger operators . L q have a singular spectrum. In the proof of the theorem the non-existence of singularities of solutions .q(t, x) for real .t and .x is a crucial issue. Since the correspondence between potentials .q and unified WTK functions .m is one-to-one, the action .┌(n) on .Q(n) can be regarded as one on a space of WTK functions, which is denoted by .g · m. Moreover its action can be extended to a certain class of rational functions. With these facts in mind the proof of the absence of singularities is established by noting a formula .

( ) qζ · m (z) =

)−1 ( z2 − ζ 2 . − m (ζ) with qζ (z) = 1 − ζ −1 z m (z) − m (ζ)

The rational function .qζ plays an essential role in the proof since any .g of .┌(n) can be expressed as a limit of products of .qζ s. It should be mentioned that Johnson [24] first introduced a .z 2 -invariant subspace by using WTK functions, which made the author recognize the significance of WTK functions in Sato’s theory. The main results of this book consist of 6 theorems: Theorems 3.1, 4.1, 5.1, 5.2, 5.3, 6.1. To understand these 6 theorems readers can start from Chap. 4. The author prepared Chap. 2 in order to clarify the reason why Sato’s theory or its Segal–Wilson version can work in such a beautiful way. The content of this chapter essentially depends on Segal–Wilson [49]. In Chap. 3 the author tries to reconstruct the paper [49] in terms of WTK functions when .m ± are reflectionless on .[λ1 , ∞) (.λ1 > 0), that is .m + (λ + i0) = −m − (λ + i0) a.e. λ ≥ λ1 , which√is equivalent to the analyticity of .m on .C\ ([−μ0 , μ0 ] ∪ [−iμ1 , iμ1 ]) with μ1 = λ1 . In this case one can replace the curve .Cn by a disc with center at the origin and radius .> max (μ0 , μ1 ), which makes the whole story transparent. This reflectionless property is a generalization of the conventional one in the case .q ∈ L 1 (R). The original part in this chapter is in rewriting [49] using WTK functions .m ± and the proof of the non-existence of singularities of solutions on .R relies on this rewriting. It is not very difficult to remove this reflectionless property and an extension is achieved in Chap. 4. As we mentioned above the main result is stated in terms of .m ± , we have to provide sufficient conditions for it, and this is realized

.

1 Introduction

7

in Chap. 5. Section 6.1 of Chap. 6 is related to global analysis of solutions of the KdV equation when the time .t tends to .∞. The result asserts that the reflectionless property holds in the limit .t → ∞ on the region of the absolute continuous spectrum of the initial . L q under some positivity condition. This is a natural generalization of Remling’s theorem [41]. This reflectionless property not only implies the existence of an absolutely continuous spectrum but also restricts the shapes of potentials very strongly. For instance if the spectrum .∑ consists of finitely many intervals and .m ± are reflectionless on .∑, then the associated potential .q is algebro-geometric, namely .q can be represented by a .Θ-function on a closed Riemannian surface. Moreover in order to exhibit the scope of Sato’s theory the author provides two cases of multicomponent systems in Sect. 6.2, in which the nonlinear Schrödinger equation and Sine–Gordon equation are discussed in the present framework. The completion of this section relies on future work. Chaper 3 is based on [32] and the rest is on [31], but the result on Wigner–von Neumann type initial data in Sect. 5.1.2 and the generalization of Remling’s theorem in Sect. 6.1.2 are new. Throughout the book the following notations will be employed: R = the set of all real numbers C = the set of all complex numbers Z = the set of all integers R− = {x ∈ R, x ≤ 0} . . R+ = {x ∈ R, x ≥ 0} , C+ = {z ∈ C, Im z > 0} , C− = {z ∈ C, Im z < 0} Z+ = {n ∈ Z, n ≥ 0} , Z− = {n ∈ Z, n ≤ 0} z denotes the complex conjugate of z: x + i y = x − i y

Chapter 2

Sato’s Theory

It was a great discovery by GGKM that solutions .q(x, t) to the KdV equation make the spectrum of 1D Schrödinger operators with potential .qt (x) = q(x, t) invariant. This fact was interpreted by Lax as follows. Let .

3 3 L = ∂x2 + q and P = ∂x3 + q∂x + q ' . 2 4

Then .

[P, L] =

1 ''' 3 ' q + qq 4 2

holds. Hence .q is a solution to the KdV equation ∂q=

. t

1 ''' 3 ' q + qq , 4 2

if and only if ∂ L = [P, L]

. t

(2.1)

holds. Since the operator . P is anti-symmetric (. P ∗ = −P), (2.1) yields the unitary equivalence between two operators .∂x2 + qt , .∂x2 + q0 , hence the spectra of the two operators are the same. Let us consider this idea in a more general setting. Let . L be a differential operator .

L = ∂xν + qν−2 ∂xν−2 + · · · + q1 ∂x + q0 ,

(2.2)

and assume the existence of another differential operator . P such that ord.[P, L] ≤ ν − 2. Then, such a . P is essentially unique, and denoting by . f j (.0 ≤ j ≤ ν − 2) the coefficients of .[P, L] one has the equivalence © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 S. Kotani, Korteweg–de Vries Flows with General Initial Conditions, Mathematical Physics Studies, https://doi.org/10.1007/978-981-99-9738-1_2

9

10

2 Sato’s Theory

∂ L = [P, L] ⇐⇒ ∂t q j = f j .

. t

Since . f j are known to be universal differential polynomials of .∂xk q j , this relation { } provides a closed non-linear differential equation of . q j 1≤ j≤ν−2 . Although it is possible to find such a . P by hand calculation, it becomes harder for large .ν. Sato used formal pseudo-differential operators .

Q = ∂x + r1 (x)∂x−1 + r2 (x)∂x−2 + · · ·

with smooth coefficients .r j to reveal the mechanism. With this . Q a Lax pair is obtained for positive integers .n, .ν by setting .

( ) L = (Q ν )+ , P = Q n + ,

where .(·)+ denotes the differential operator part of .·. ord. L for a differential operator L is defined as the order of the highest derivative. Then ord.[P, L] ≤ ν − 2 holds and . L, . P constitute a Lax pair. The detail is explained in the next section.

.

2.1 Algebra of Formal Pseudo-differential Operators Sato’s theory provides a systematic procedure to obtain Lax pairs. In this section we explain his theory based on the paper by Segal–Wilson [49]. We start from . L of (2.2). A very smart way to obtain . P for a given . L is to use fractional powers . L r/n , which leads us to consider the algebra of the formal pseudodifferential operator .

R=

∑

r j (x) ∂xj with N ≥ 1 and smooth functions r j .

j≤N j

j

Here .∂x acts as the usual differential operators for . j ≥ 0. To see the operation .∂x for . j < 0 let ∑ (1) ∑ (2) . R1 = r j ∂xj , R2 = r j ∂xj . j≤N

Since .

R1 R2 =

∑ j≤N

j r (1) j ∂x

∑ j ' ≤N

j≤N

j r (2) j ' ∂x = '

∑

( ) (2) j ' j r (1) j ∂x r j ' ∂x ,

j, j ' ≤N

( ) j j' the operation .∂x r (2) ∂ for . j < 0 should be defined suitably. Noting the identity ' x j ( ) ∑ (k + 1) (k + 2) · · · n ( n−k ) j+k ∂x r ∂x ∂ n r ∂xj = (n − k)! k≤n

. x

(2.3)

2.1 Algebra of Formal Pseudo-differential Operators

11

for .n ≥ 0 we define the left-hand side of (2.3) by the right-hand side also for .n < 0. For instance we define for .n = −1 ( j) ∑ ( ) −1 r ∂x = .∂ x (−1)k ∂xk r ∂x−1−k+ j . k≥0

Then the set .Psd of all pseudo-differential operators forms an associative algebra. Lemma 2.1 Let .

L = ∂xν + qν−2 ∂xν−2 + · · · + q1 ∂x + q0

with smooth functions .q j (x). Then there exists uniquely . Q ∈ .Psd satisfying . L = Q ν such that ∑ . Q = ∂x + r j ∂x− j . j≥1

The coefficients .r j are universal differential polynomials in .q j . Proof We calculate it only for .ν = 2, .3. For .ν = 2 note the identity ⎛ .

⎝∂ x +

∑

⎞2 r j ∂x− j ⎠ = ∂x2 +

j≥1

∑

s j ∂x− j

j≥0

with s = 2r1 , s j =

. 0

r 'j

+ 2r j+1 +

∑

(−1)

1≤l≤k≤ j

k−l

( ) k−1 rlr (k−l) j−k for j ≥ 1, l−1

where .r0 = 0. Then letting . L = ∂x2 + q yields ⎧ 1 ⎪ ⎪ ⎨ r1 = 2 q ) ( . , 1 ' 1 ∑ k−l k − 1 ⎪ rlr (k−l) for j ≥ 1 (−1) ⎪ j−k ⎩ r j+1 = − 2 r j − 2 l − 1 1≤l≤k≤ j hence r =

. 1

1 1 1 1 1 3 q, r2 = − q ' , r3 = q '' − q 2 , r4 = − q ''' + qq ' , . . . . 2 4 8 8 16 8

For .ν = 3 the identity

(2.4)

12

2 Sato’s Theory

⎛ .

⎝∂ x +

∑

⎛

⎞3

r j ∂x− j ⎠ = ⎝∂x +

j≥1

∑

⎞⎛ r j ∂x− j ⎠ ⎝∂x2 +

j≥1

= ∂x3 + (r1 + s0 ) ∂x +

∑

∑

⎞ s j ∂x− j ⎠

j≥0

t j ∂x− j

j≥0

with ⎧ ' ⎪ ⎨ t0 = r 2 + s0 . t = s 'j + s j+1 + r j+2 + ⎪ ⎩ j

∑

) ( k−1 rl s (k−l) j−k if j ≥ 1 l−1

(−1)k−l

1≤l≤k≤ j

implies ⎧ r 1 + s0 = q 1 ⎪ ⎪ ⎪ ⎨ r 2 + s ' + s1 = q 0 0

.

⎪ ⎪ r = −s 'j − s j+1 − ⎪ ⎩ j+2

∑

( (−1)k−l

1≤l≤k≤ j

) k−1 rl s (k−l) j−k for j ≥ 1 l−1

if . L = ∂x3 + q1 ∂x + q0 . Hence r =

. 1

) 1 1( 2 1 1 q1 , r 2 = q0 − q1' , r3 = q1'' − q12 − q0' , . . . . 3 3 9 9 3

(2.5) ∎

For . R =

∑

j

j≤N

r j ∂x ∈ Psd, define R+ =

.

∑

r j ∂xj , R− =

0≤ j≤N

∑

r j ∂xj .

j≤−1

[( ] ) Lemma 2.2 ord. L n/ν + , L ≤ ν − 2 holds for any integer .n ≥ 1 and the coef[( ] ) ficients . f j of . L n/ν + , L are differential polynomials in .qk . The time evolution equation [( ] ) L n/ν + , L .∂t L = is equivalent to ∂ qj = fj.

. t

[

]

Proof Since . L n/ν , L = [Q n , Q ν ] = 0, one has [( .

L n/ν

] [( ] [( ) ] ) n/ν n , L = − L , L = − Q , L , + − −

)

2.1 Algebra of Formal Pseudo-differential Operators

13

which is of the order at most .ν − 2. The rest of the statement is clear. ∎ ( n/ν ) Therefore . P = L satisfies the condition that ord.[P, L] ≤ ν − 2. We cal+ culate the equations .∂t q j = f j for a few cases. If .ν = 2 and . L = ∂x2 + q, then n ∂t L = [P, L] 1 ∂t q = q ' 1 ''' 3 ' . 3 ∂t q = . q + qq 4 2 15 2 ' 5 ' '' 15 ''' 15 ''''' 5 ∂t q = q q + qq − qq − q 8 4 8 16

(2.6)

If .ν = 3 and . L = ∂x3 + q1 ∂x + q0 , then n ∂t L = [P, L] { ∂t q1 = q1' 1 ∂t q0 = q0' . . { ∂t q1 = 2q0' − q1'' 2 2 2 ∂t q0 = q0'' − q1''' − q1 q1' 3 3

(2.7)

In the second case .q1 satisfies 1 4 ( ' )' q1 q1 , ∂ 2 q1 = − q1'''' − 3 3

. t

which is the Boussinesq equation. These countably many Lax pairs include the KdV hierarchy and the Boussinesq hierarchy of nonlinear equations if .ν = 2 and .3 respectively. Each .ν provides a hierarchy of nonlinear equations, which is called the.ν-th KdV hierarchy. The totality of these hierarchies is called the KP hierarchy. When .ν = 2, . L = ∂x2 + q is self-adjoint and the spectrum is lying on .R. In this case the equation .∂t L t = [Pt , L t ] yields the unitary equivalence of . L 0 and . L t , which was a great advantage to have solutions. However, when .ν ≥ 3, . L is not necessarily self-adjoint and the spectrum spreads in .C, which will make generalization difficult. Although the equation .∂t L t = [Pt , L t ] indicates many invariants for the KP hierarchy, they do not provide a direct procedure obtaining solutions. Sato constructed the KdV flow as an action of commutative group on an infinite-dimensional Grassmann manifold through the .τ -function. Segal–Wilson [49] realized Sato’s theory on 2 . L (C), which made the theory accessible at least to the author. In the next section we explain Segal–Wilson’s approach to Sato’s theory.

14

2 Sato’s Theory

2.2 Flows on Grassmann Manifolds For .r > 0 let . H = L 2 (|z| = r ) which is the . L 2 -space{on}the disc { .}{z ∈ C; |z| = r }, and. H+ ,. H− be the closed subspaces of. H generated by. z j j≥0 , . z j j r .

.

(2.9)

The property .gW ∈ Gr (ν) for .W ∈ Gr (ν) , .g ∈ ┌ does not necessarily hold since the condition (G2) is not valid automatically for .gW . Instead .gW is always compatible if so .W is; however, it causes singularities for solutions. In view of this for an integer (ν) .n ≥ 1 assume . W ∈ Gr satisfies .ex−1 W ∈ Gr (ν) for any .x ∈∑R with .ex (z) = e x z ∈ −1 ┌. Then (G2) for .ex W and .1 ∈ H+ implies that there exists . j≥1 κ j (x) z − j ∈ H− such that ⎛ ⎞ ∑ xz ⎝ . f (x, z) ≡ e κ j (x) z − j ⎠ ∈ W (2.10) 1+ j≥1

for any .x ∈ R. This function is called the Baker–Akhiezer function and will play a key role in Sato theory. First we show that . f (x, z) works as a (generalized) eigen function for a certain linear differential operator of order .ν. Define polynomials . pk of .z of deg . pk ≤ .k − 1 by ( ( )) ∂ k f (x, z) = e x z z k + pk (x, z) + O z −1 .

. x

{ ' } The coefficients of these . pk are universal polynomials of . ∂xk κ j , and generally . pk satisfies ) ( k−1 . pk (x, z) = κ1 z + O z k−2 . The first two terms are .

Since

p1 (x, z) = κ1 , p2 (x, z) = κ1 z + κ2 + 2∂x κ1 .

2.2 Flows on Grassmann Manifolds

15

⎛ .

z ν f (x, z) = e x z ⎝z ν + κ1 z ν−1 +

∑

κ j z ν− j

⎞ ) + O z −1 ⎠ , (

2≤ j≤ν

one has

( ( )) ∂ ν f (x, z) − z ν f (x, z) = e x z p (x, z) + O z −1

. x

with a polynomial . p of .z of deg . p ≤ ν − 2. Then one can show that there exist .qk (x) such that ∑ ( ) . p (x, z) = qk (x) z k + pk (x, z) (2.11) 0≤k≤ν−2

holds, and one sees ( −1 .e x

∂xν

ν

f −z f +

)

∑

qk ∂xk

f

( ) = O z −1 ∈ H− .

0≤k≤ν−2

Since .∂xk f , .z ν f ∈ W , the property (G2) for .ex−1 W implies .

with .

L ν f − zν f = 0 ∑

L ν = ∂xν +

(2.12)

qk ∂xk .

0≤k≤ν−2

{

} ' q (x) are universal polynomials of . ∂xk κ j . Generally, one has: { } Lemma 2.3 . κ j j≥1 and .{qk }0≤k≤ν−2 are related as follows: for any . j ≤ ν − 2

. k

0 = qj +

∑

.

0≤k≤ν−2

qk κk− j +

∑

) l ∂ l− j−k κk , l− j −k x

( ql

− j≤k≤l− j−1,l≤ν

(2.13)

where we set .κ j = qk = 0{if . j ≤} 0, .k ≤ −1 and .qν−1 = 0, .qν = 1. Especially .q j is a universal polynomial of . ∂xl κk k+l≤ν− j , and conversely .{κk (x)}k≥1 is determined { } uniquely from . q j (x) ν≤ j≤ν−2 , .{κk (0)}k≥1 . Proof Substituting (2.10) into the identity (2.12) one has (2.13) immediately. For instance the first two terms are 0 = qν−2 + ν∂x κ1 , 0 = qν−3 + qν−2 κ1 +

.

The rest of the proof is clear from this identity.

ν (ν − 1) 2 ∂x κ1 + ν∂x κ2 . 2 ∎

The differential operator . L ν plays a crucial role in the following argument and is called the underlying operator for .Gr (ν) . For .ν = 2 one has a Schrödinger operator

16

2 Sato’s Theory .

L 2 = ∂x2 + 2∂x κ1 . n

Now for an integer .n ≥ 1 let .et,x (z) = e x z+t z with another real variable .t and −1 assume .et,x W ∈ Gr (ν) for any .t, .x ∈ R. Define ⎛ .

n f (t.x.z) = e x z+t z ⎝1 +

∑

⎞ κ j (t, x) z − j ⎠ ∈ W .

(2.14)

j≥1

Taking the derivative with respect to .t, one has ) ( e−1 ∂t f (t, x.z) − z n f (t.x.z) ∈ H− .

. t,x

Since . L n of (2.12) defined by .{qk }0≤k≤n−2 of (2.11) (replace .ν by .n) satisfies ( ) e−1 L n − z n f ∈ H− ,

. t,x

it holds that

e−1 (∂t f − L n f ) ∈ H− .

. t,x

Observing .∂t f − P f ∈ W , one sees ∂ f − L n f = 0.

(2.15)

. t

{ } The Eqs. (2.12), (2.15) yield some identities for . ∂xk κ j , ∂t κ j . For instance if .ν = 2, .n = 3 one has 1 3 3 ∂x κ1 − (∂x κ1 )2 , .∂t κ1 = 4 2 which is the KdV equation of .q = −2∂x κ1 . If .ν, .n are large, the calculation is tedious and does not give any systematic understanding. The mechanism is clarified through the algebra Psd. With the .κ j of (2.14) set .

Kt = I +

∑

κ j (t, ·) ∂x− j ∈ Psd.

j≥1 −j

Since .∂x e x z = z − j e x z , one has ⎛ .

K t e x z = e x z ⎝1 +

∞ ∑ j=1

hence (2.12), (2.15) yield

⎞ κ j z − j ⎠ = e−t z f (t, x, z) , n

(2.16)

2.2 Flows on Grassmann Manifolds

.

17

) xz {( ν xz ( K t ∂x e =n (L ) νx zK t ) e ∂t K t + K t ∂x e = (P K t ) e x z

with . P = L n . It is generally valid that if some . K ∈ Psd satisfies . K e x z = 0, then . K = 0. Therefore { K t ∂xν = L ν K t . (2.17) ∂t K t + K t ∂xn = P K t holds. One can show that there exists .

K −1 = I +

∑

b j ∂x− j ∈ Psd

j≥1

with .b1 = −κ1 for any . K ∈ Psd of (2.16). Set . Q t = K t ∂x K t−1 ∈ Psd. Then . Q t takes a form ∑ ∑ . Q t = ∂ x + κ1 + b1 + r j ∂x− j = ∂x + r j ∂x− j , j≥1

j≥1

and (2.17) implies . Q νt = L ν , hence . Q t = L 1/ν due to Lemma 2.1. Equation (2.17) ν implies also −1 . (∂t K t ) K t + Q nt = P. (2.18) ( ) Observing . (∂t K t ) K t−1 + = 0 one has ( .

Q nt

) +

= P.

(2.19)

Moreover from (2.18) it follows that ( ) ∂ Q t = ∂t K t ∂x K t−1

. t

= (∂t K t ) ∂x K t−1 − K t ∂x K t−1 (∂t K t ) K t−1 ] [ = P − Q nt , Q t = [P, Q t ] . One can show the identity

] [ ∂ Q kt = P, Q kt

. t

for any .k ≥ 1 by induction. Suppose for some .k ≥ 1 one has this identity. Then ( ) ∂ Q k+1 = (∂t Q t ) Q kt + Q t ∂t Q kt t [ ] [ ] = [P, Q t ] Q kt + Q t P, Q kt = P, Q k+1 t

. t

completes the proof. Hence one has ] [ ∂ L ν = ∂t Q νt = P, Q υt = [P, L ν ] ,

. t

18

2 Sato’s Theory

which yields a non-linear equation ∂ q j = f j for 0 ≤ j ≤ ν − 2

. t

(2.20)

[( ] ) with the coefficients . f j of . L n/ν , L . ν ν + Now one can state the core of Sato’s theory. x z+t z (ν) Proposition 2.1 Suppose.W W ∈ Gr (ν) for.t,.x ∈ R. Define { ∈} Gr satisfies.e . f (t, x, z) by (2.14). Then . q j given by (2.13) solves the Eq. (2.20), some 0≤ j≤ν−2 solutions of which are given in (2.6), (2.7). n

This proposition makes it possible to define a flow on .Gr (ν) by an action of .┌, and by choosing one-parameter group .gt on .┌ one can create a nonlinear equation. The only requirement is.e x z gt W ∈ Gr (ν) , and if this property fails at some points, then the resulting solution has a singularity at such points. Moreover Sato described. f (t, x, z) by the tau-function, which is a Fredholm determinant of an operator concerning .p+ of (2.8). The space .Gr (ν) works as a space of initial data. From a point of application it is required to present a way to relate a .W ∈ Gr (ν) with a given initial datum .q and to identify what kind of functions meet the criteria.

Chapter 3

KdV Flow I: Reflectionless Case

In this chapter we rewrite Sato’s theory for .ν = 2 in terms of the spectral quantities of the underlying Schrödinger operator . L(.= L 2 ). One of the features in this case is the self-adjointness of . L for which the spectral property is well investigated. Moreover, Lax observation teaches us the unitary equivalence between two Schrödinger operators with potentials of initial data and the solution. Therefore it is natural to ask if it is possible to state .Gr (2) completely by spectral terminologies of Schrödinger operators. A Schrödinger operator . L q = −∂x2 + q is called reflectionless on a Borel subset . F of .R if its WTK functions .m ± (see Definition 7.1 in Appendix) satisfy m + (λ + i0) = −m − (λ + i0) for a.e. λ ∈ F.

.

In this chapter we treat initial data such that the associated Schrödinger operator is reflectionless on .(λ1 , ∞) for some .λ1 > 0.

3.1 Hardy Spaces and Toeplitz Operators In the previous chapter Sato’s theory is stated in the Hilbert space . H = L 2 (|z| = r ) and .z ν -invariant closed subspaces in . H . For later purposes we replace the circle .{|z| = r } by an arbitrary bounded closed smooth simple curve .C satisfying symmetries .C = −C, C = C. The first property is required due to .ν = 2, namely .(−z)2 = z, and the second one is necessary in order for solutions to be real valued. .C separates the plane .C into two domains . D± where . D+ is the bounded part and . D− is the unbounded part. A © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 S. Kotani, Korteweg–de Vries Flows with General Initial Conditions, Mathematical Physics Studies, https://doi.org/10.1007/978-981-99-9738-1_3

19

20 .

3 KdV Flow I: Reflectionless Case

z 2 -invariant subspace .W is given by a bounded vector-valued function . a = (a1 , a2 ) on .C as { } 2 . W a = a (λ) f (λ) ; f ∈ L (C) ,

where the product . a (λ) f (λ) is defined by { .

a (λ) f (λ) = a1 (λ) f e (λ) + a2 (λ) f o (λ) with . f e (λ) = ( f (λ) + f (−λ)) /2, f o (λ) = ( f (λ) − f (−λ)) /2

(3.1)

Clearly.z 2 Wa ⊂ Wa holds, and the closedness of.Wa is equivalent to a non-degeneracy condition . inf |a1 (λ) a2 (−λ) + a1 (−λ) a2 (λ)| > 0. (3.2) λ∈C

The Hilbert space . H is . H = L 2 (C) and the associated Hardy spaces . H± are ⎧ ⎨ H+ = {the { closure of all rational functions with no poles inD}+ } the closure of all rational functions . ( ) u with no poles ⎩ H− = in D− satisfying u(z) = O z −1 Clearly one has .

H = H+ ⊕ H− (direct sum not necessarily orthogonal),

and projections .p± to . H± are given by { ⎧ f (λ) 1 ⎪ dλ for z ∈ D+ ⎨ (p+ f ) (z) = 2πi λ {C − z . . f (λ) 1 ⎪ ⎩ (p− f ) (z) = dλ for z ∈ D− 2πi C z − λ It should be noted that .

(p± f )e = p± f e , (p± f )o = p± f o

hold. To describe the condition (G2) of (2.8) we introduce a Toeplitz operator. For a bounded function .a on .C the Toeplitz operator .T (a) with symbol .a is defined by .

T (a) u = p+ (au) for u ∈ H+ .

T (a) is a bounded operator on . H+ . For a vector-valued bounded function . a = (a1 , a2 ) define

.

.

T (a) u = p+ (au) = T (a1 ) u e + T (a2 )u o for u ∈ H+ .

3.2 Characteristic Functions and m-function

21

Since the condition (G2) for .Wa is equivalent to the invertibility of .T (a), we define a set .Ainv of symbols by } { Ainv = a = (a1 , a2 ); a j are bounded on C and T (a) is invertible on H+ .

.

Therefore we develop Sato’s theory on .Ainv instead of .Gr (2) . Indeed one can show that any .z 2 -invariant closed subspace .W of . L 2 (|z| = r ) is represented as .W = Wa with bounded . a satisfying (3.2).

3.2 Characteristic Functions and .m-function Since .1, .z ∈ H+ and multiplication of .z 2n to .{1, z} generate . H+ , for . a ∈ Ainv we define ) ( { u (z) = ( T (a)−1 1) (z) ∈ H+ . , (3.3) v(z) = T (a)−1 z (z) ∈ H+ and ⎧ ϕ (z) = a1 (z)u e (z) + a2 (z)u o (z) − 1 = p− (au) (z) ∈ H− ⎪ ⎨ a ψa (z) = a1 (z)ve (z) + a2 (z)vo (z) − z = p− (av) (z) ∈ H− . , ⎪ ⎩ Δa (z) = (1 + ϕa (−z)) (ψa (z) + z) − (1 + ϕa (z)) (ψa (−z) − z) 2z

(3.4)

where the product . au should be understood as (3.1). Lemma 3.1 If . a ∈ Ainv , .{ϕa , ψa } satisfies the following properties: (i) . Δa (b) /= 0 on . D− and .

T (a)−1

1 (ϕa (b) + 1) v − (ψa (b) + b) u ) ( = . z+b Δa (b) z 2 − b2

(3.5)

(ii) . {ϕa , ψa } determines .Wa . ~ be another simple closed smooth curve in . D+ and assumed . a ∈ Ainv to be (iii) Let .C ~+ with the domain. D ~+ associated with.C. ~ If.T (a) restricted on holomorphic on. D+ \ D ~+ (the Hardy space associated with . D ~+ ) is invertible, then .{ϕa , ψa } is analytically .H { } ~a corresponding to the curve .C. ~− and coincides with the . ϕ ~ ~a , ψ continuable in . D Proof Here the suffix . a is omitted. Equation (3.4) implies that for .b ∈ D− we have decompositions

22

3 KdV Flow I: Reflectionless Case

⎧ a1 (z)u e (z) + a2 (z)u o (z) ⎪ ⎪ ⎪ ⎪ ⎪ ( z 2 − b2 ) ( ) ⎪ ⎪ ϕ (b) + 1 ϕ (−b) + 1 1 ϕ (z) − ϕ (b) ϕ (z) − ϕ (−b) 1 ⎪ ⎪ − + − ⎨= 2b z−b z+b 2b z−b z+b . a1 (z)ve (z) + a2 (z)vo (z) ⎪ ⎪ ⎪ ⎪ ⎪ ( z 2 − b2 ) ( ) ⎪ ⎪ 1 ψ (z) − ψ (b) ψ (z) − ψ (−b) 1 ψ (b) + b ψ (−b) − b ⎪ ⎪ ⎩= + − − 2b z−b z+b 2b z−b z+b into elements of . H+ and . H− , hence ⎧ ⎪ ⎪ ⎨ T (a)

( ) u 1 ϕ (b) + 1 ϕ (−b) + 1 − = z 2 − b2 2b ( z − b z+b ) . v 1 ψ (b) + b ψ (−b) − b , ⎪ ⎪ − = ⎩ T (a) 2 z − b2 2b z−b z+b

(3.6)

⎧ Δ (b) (ϕ (b) + 1) v − (ψ (b) + b) u ⎪ ⎨ T (a) = 2 − b2 z z+b . + 1) v − (ψ (−b) − b) u Δ (b) . ⎪ ⎩ T (a) (ϕ (−b) = z 2 − b2 z−b

(3.7)

which yields

If .Δ (b) = 0, then the invertibility of .T (a) implies { .

(ψ (b) + b) u − (ϕ (b) + 1) v = 0 . (ψ (−b) − b) u − (ϕ (−b) + 1) v = 0

Applying .T (a) we have { .

(ψ (b) + b) − (ϕ (b) + 1) z = 0 , (ψ (−b) − b) − (ϕ (−b) + 1) z = 0

which means ψ (b) + b = ϕ (b) + 1 = ψ (−b) − b = ϕ (−b) + 1 = 0,

.

hence (3.6) implies .u = v = 0. This contradicts .T (a) u = 1, .T (a) v = z, and we have .Δ (b) /= 0. Equation (3.5) can be deduced from (3.7). To show (ii) let ϕa (z) =

∑

.

j≥1

Since for any integer .n ≥ 0

κ j z − j , ψa (z) =

∑ j≥1

b j z− j .

3.2 Characteristic Functions and m-function

{ .

one has

23

( ) ( ) z 2n ϕa (z) = a1 (z) (z 2n u)e (z) + a2 (z) ( z 2n u) o (z) − z 2n , z 2n ψa (z) = a1 (z) z 2n v e (z) + a2 (z) z 2n v o (z) − z 2n+1 ⎧ ∑ ( ) ⎪ T (a) z 2n u − z 2n = p+ z 2n ϕa = a j z 2n− j ⎪ ⎨ ∑ ( 2n ) 1≤ j≤2n . . 2n 2n+1 ⎪ z ψa = T z v − z = p b j z 2n− j (a) ⎪ + ⎩ 1≤ j≤2n

Applying .T (a)−1 yields ⎧ ∑ ⎪ T (a)−1 z 2n = z 2n u − κ j T (a)−1 z 2n− j ⎪ ⎨ 1≤ j≤2n ∑ . , −1 2n+1 2n ⎪ T z = z v − b j T (a)−1 z 2n− j (a) ⎪ ⎩

(3.8)

1≤ j≤2n

hence

∑ ⎧ ⎪ aT (a)−1 z 2n = z 2n (ϕ + 1) − κ j aT (a)−1 z 2n− j ⎪ ⎨ 1≤ j≤2n ∑ . , −1 2n+1 2n ⎪ aT z = z b j aT (a)−1 z 2n− j + z) − (a) (ψ ⎪ ⎩ 1≤ j≤2n

{ } which shows . aT (a)−1 z k k≥0 can be obtained inductively from .ϕ, .ψ. Therefore .ϕ, { } −1 k .ψ determine . W a since . aT (a) z k≥0 generates .Wa . (iii) follows easily from an identity .

1 2πi

{

a (λ) u (λ) 1 dλ = λ−z 2πi

C

{ ~ C

a (λ) u (λ) dλ λ−z

~+ . for any .u ∈ H+ and .z ∈ D

∎

The functions .{ϕa , ψa } are called the characteristic functions for .Wa . The .mfunction for .Wa is defined by m a (z) =

.

z + ψa (z) + κ1 (a) 1 + ϕa (z)

(3.9)

with κ (a) = lim zϕa (z) .

. 1

z→∞

κ (a) is added so that we have an asymptotic behavior

. 1

m a (z) = z + o (1) as z → ∞.

.

(3.10)

24

3 KdV Flow I: Reflectionless Case

(i) of (3.1) implies that .1 + ϕa (z) is not identically .0, hence .m a is meromorphic on D− . Later we will see that .m a determines the potential .q and is equal to the WTK function under a certain condition on . a. A concrete subset of.Ainv is given as follows: Let.m = (m 1 , m 2 ) with holomorphic functions .m 1 , .m 2 on .C\ (D+ ∩ (R ∪ iR)) satisfying

.

{ .

(i) m j (z) − 1 ∈ H− . (ii) m 1 (z)m 2 (−z) + m 1 (−z)m 2 (z) /= 0 on C\ (D+ ∩ (R ∪ iR))

(3.11)

For such .m, .n define new elements by ⎧ ) ( (z), m 1 (z)n 2,o (z) + m 2 (z)n 2,e (z) ) ⎪ ⎨ (m · n) (z) ( = m(1 (z)n 1,e (z) + m 2 (z)n ) 1,o ) ( 2 m 2,e (z) − m 1,o (z) 2 m 1,e (z) − m 2,o (z) . . ̂ (z) = , ⎪ ⎩m m 1 (z)m 2 (−z) + m 1 (−z)m 2 (z) m 1 (z)m 2 (−z) + m 1 (−z)m 2 (z) Then we have ̂ also satisfy (i) and (ii), and Lemma 3.2 .m · n, .m ̂=m ̂ · m = (1, 1) m·m

(3.12)

T (m · n) = T (m) T (n) ,

(3.13)

.

holds. Moreover it holds that .

hence .m, n ∈ Ainv is valid. Proof The first statement and (3.12) are clear. To show (3.13) note .m H− ⊂ H− for any bounded holomorphic function .m on . D− . Therefore, if scalers .m 1 , .m 2 satisfy the condition (i) of (3.11), then for .u ∈ H+ p (m 1 m 2 u) = p+ (m 1 p+ m 2 u) + p+ (m 1 p− m 2 u) = p+ (m 1 p+ m 2 u)

. +

holds. With this property in mind we have .

T (m · n) u (( ) ) ) ( = p+ m 1 n 1,e + m 2 n 1,o u e + m 1 n 2,o + m 2 n 2,e u o ( ( )) ( ( )) = p+ m 1 p+ n 1,e u e + n 2,o u o + p+ m 2 p+ n 1,o u e + n 2,e u o = T (m 1 ) (T (n) u)e + T (m 2 ) (T (n) u)o = T (m) T (n) u,

which shows (3.13).

∎

−1 (If−2. a) (z) = m (z) = (m 1 (z) , m 2 (z)), then due to .m (z) = (1, 1) + m1 z + .O z we have

3.3 Group Action on Ainv and Tau-Function .

25

T (m)1 = p+ m 1 (z) = 1, T (m)z = p+ m 2 (z)z = z + m 12

with .m1 = (m 11 , m 12 ), hence .u(z) = 1, .v(z) = z − m 12 . Therefore { .

ϕa (z) = m 1 (z) − 1 ψa (z) = −m 12 m 1 (z) + zm 2 (z) − z

follows, which yields m a (z) =

.

−m 12 m 1 (z) + zm 2 (z) zm 2 (z) + m 11 = + m 11 − m 12 . m 1 (z) m 1 (z)

(3.14)

3.3 Group Action on .Ainv and Tau-Function The KdV flow is described by a group action on .Ainv with { } ┌ = g = eh ; h is holomorphic in a neighborhood of D+ ∪ C .

.

(3.15)

For . a ∈ Ainv , .g ∈ ┌ a natural product .ga is an element of .Ainv if .T (ga) has a bounded inverse. Note that we have slightly changed the type of action since in the previous section this action was .g −1 W . But this does not cause any difficulties. The invertibility of .T (ga) is crucial in our setting and this will be verified by using the tau-function, which is defined by the determinant of the difference between .T (ga) and .T (a), namely .g −1 T (ga)T (a)−1 . The tau-function describes the .┌ action very well. To define the determinant we have to show the relevant operators are of trace class. For .a ∈ Ainv , .g ∈ ┌ define { .

.

Sa u (z) = p− (au) for u ∈ H+ Hg w = p+ (gw)

for w ∈ H−

.

(3.16)

Hg is akin to the Hankel operator if . D+ is the unit disc.

Lemma 3.3 The following are valid: (i) It holds that . T (ga) = gT (a) + Hg Sa .

(ii) . Hg Sa defines a trace class operator from . H+ to . H+ . Proof For .u ∈ H+

(3.17)

26

3 KdV Flow I: Reflectionless Case .

T (ga) u = p+ (gp+ au) + p+ (gp− au) = gT (a) u + p+ (gp− au) = gT (a) u + Hg Sa u

holds, which proves (i). To have (ii) we show . Hg is of trace class. . Hg satisfies .

Hg w (z) =

1 2πi

{ C

1 g (λ) w (λ) dλ = λ−z 2πi

{ C

g (λ) − g (z) w (λ) dλ λ−z

for .w ∈ H− , .z ∈ D+ , which shows . Hg has a smooth kernel. Let . L = ∂λ2 + c, where 2 −1 .c is chosen so that . L has a bounded inverse on . L (C). Then, it is known that . L is of trace class, hence the identity .

( ) Hg = L −1 L Hg

implies that . Hg is of trace class.

∎

Now we define the tau-function by ) ( τ (g) = det g −1 T (ga)T (a)−1 ,

. a

(3.18)

which is possible since g −1 T (ga)T (a)−1 = I + g −1 Hg Sa T (a)−1

.

with a trace class operator .g −1 Hg Sa . Tau-functions satisfy the following properties. For .g ∈ ┌ define | ' | | '' |) ( . ||g|| = sup |g (λ)| , |g (λ)| , |g (λ)| . λ∈C

Lemma 3.4 For . a ∈ Ainv , .g1 , .g2 ∈ ┌ one has: (i) If .τa (g1 ) /= 0, then .T (g1 a) is invertible and .g1 a ∈ Ainv . (ii) If .τa (g1 ) /= 0, then it holds that τ (g1 g2 ) = τa (g1 ) τg1 a (g2 ) (cocycle property).

. a

|| || || || || || (iii) Suppose.||g j ||,.||g −1 j || ≤ c for . j = 1, .2. Then there exists a constant .c1 depending on .c such that .

|| ||) || ( || |τa (g1 ) − τa (g2 )| ≤ c1 ||g1 − g2 || || Sa T (a)−1 || exp c1 || Sa T (a)−1 ||

holds. (iv) Suppose .g1 ∈ ┌ satisfies .g1 (z) = g1 (−z). Then for any .g2 ∈ ┌

3.3 Group Action on Ainv and Tau-Function

27

τ (g1 g2 ) = τa (g1 ) τa (g2 ) .

. a

Proof (i) follows from the property of the determinant. (ii) follows from an identity .

)( ) ( (g1 g2 )−1 T (g1 g2 a)T (a)−1 = g1−1 g2−1 T (g2 g1 a)T (g1 a)−1 g1 g1−1 T (g1 a)T (a)−1 ,

and the property of the determinant, namely .

( ) det (g1 g2 )−1 T (g1 g2 a)T (a)−1 )( )) (( = det g1−1 g2−1 T (g2 g1 a)T (g1 a)−1 g1 g1−1 T (g1 a)T (a)−1 ) ( ) ( = det g1−1 g2−1 T (g2 g1 a)T (g1 a)−1 g1 det g1−1 T (g1 a)T (a)−1 ) ( ) ( = det g2−1 T (g2 g1 a)T (g1 a)−1 det g1−1 T (g1 a)T (a)−1 .

(iii) comes from the inequalities { .

|det(I + A) − det(I + B)| ≤ |A − B|1 exp (|A|1 + |B|1 + 1) |AB|1 ≤ |A|2 |B|2 , |AB|1 ≤ |A|1 |B|

with trace norm .|·|1 and Hilbert–Schmidt norm .|·|2 . Then (3.17) yields .

||( || ) |τa (g1 ) − τa (g2 )| ≤ || g1−1 Hg1 − g2−1 Hg2 Sa T (a)−1 ||1 || || || (|| ) × exp ||g1−1 Hg1 Sa T (a)−1 ||1 + ||g2−1 Hg2 Sa T (a)−1 ||1 + 1 || || || || ≤ ||g1−1 Hg1 − g2−1 Hg2 ||1 || Sa T (a)−1 || || || || ) || || ) ((|| × exp ||g −1 Hg || + ||g −1 Hg || || Sa T (a)−1 || + 1 , 1

1

1

2

2

1

| || || || | | hence we have only to estimate .||g1−1 Hg1 − g2−1 Hg2 ||1 , .||g1−1 Hg1 |1 + |g2−1 Hg2 |1 . This is possible if we note .

|( | | −1 | | | ) |g Hg − g −1 Hg | ≤ | g −1 − g −1 Hg | + |g −1 Hg −g | 1 2 1 1 1 1 2 1 1 2 1 2 2 |( | | | | | | | ) ≤ | g1−1 − g2−1 L −1 |2 | L Hg1 |2 + |g2−1 L −1 |2 | L Hg1 −g2 |2 ,

and we have (iii). To show (iv) note .

T (g1 g2 a)u = p+ (g2 g1 au) = p+ (g2 ag1 u) = T (g2 a) g1 u

for .u ∈ H+ . Therefore an identity .T (g1 g2 a) = T (g2 a) g1 is valid. Then .

( ) (g1 g2 )−1 T (g1 g2 a)T (a)−1 = g1−1 g2−1 T (g2 a)T (a)−1 g1 g1−1 (T (g1 a)) T (a)−1

implies (iv).

∎

28

3 KdV Flow I: Reflectionless Case

For later purposes we compute .τa (r ) for some rational functions { }.r by .ϕa , .m a . Let .r be a rational function with no poles on . D+ and simple poles . ζ j 1≤ j≤ p on . D− . Assume .r (z) = O (1) . Then .r has the expansion r (z) = r0 +

p ∑

.

r j qζ j (z) ,

j=1

where

q (z) = (1 − z/ζ)−1 .

. ζ

Then it holds that .

{

r (λ) v (λ) dλ λ−z C { p ∑ ( ) qζ j (λ) v (λ) 1 dλ = = rj r j v ζ j qζ j (z) , 2πi C λ−z j=1 j=1

Hr v (z) =

1 2πi p ∑

(3.19)

hence .dim Hr (H− ) ≤ p is valid. Since (3.18) implies r −1 T (r a) T (a)−1 − I = r −1 Hr Sa T (a)−1 ,

.

one sees .r −1 T (r a) T (a)−1 − I is of finite rank and .τa (r ) is equal to .

( ) ( )) ( with f i = r −1 qζi . det δi j + r j Sa T (a)−1 f i ζ j

(3.20)

Lemma 3.5 (i) For .ζ, .ζ1 , .ζ2 ∈ D− ⎧ ( ) ⎨ τa qζ = ϕa (ζ) + 1 ( ) m a (ζ1 ) − m a (ζ2 ) . . ⎩ τa qζ1 qζ2 = (ϕa (ζ1 ) + 1) (ϕa (ζ2 ) + 1) ζ1 − ζ2

(3.21)

(ii) For a rational function .r satisfying r (z) = O (1) , r (z)−1 = O (1)

.

{ } { } assume the zeros . η j 1≤ j≤ p ⊂ D− and the poles . ζ j 1≤ j≤ p ⊂ D− are simple. Then

3.3 Group Action on Ainv and Tau-Function

29

⎛

( ) )( ( ) )⎞ p ( ∏ ϕa ζ j + 1 ϕa −η j + 1 ⎠ ( ) ( ) ( ) .τ a (r ) = ⎝ ' η ̂ ' ζ η r r Δ a j j j j=1 ( ( )) ( ) m a (ζi ) − m a −η j 1 × det det ηi − ζ j ζi2 − η 2j

(3.22)

holds with .̂ r (z) = r (z)−1 . Proof Since for .r = qζ .

p = 1, r1 = f 1 = 1,

hence applying (3.19) to .r = qζ one has ( ) ( ) −1 .τ a qζ = 1 + r 1 Sa T (a) f 1 (ζ) = 1 + ϕa (ζ) . For .r = qζ1 qζ2 .

p = 2, r1 = qζ2 (ζ1 ) , r2 = qζ1 (ζ2 ) , f 1 = qζ−1 , f 2 = qζ−1 , 2 1

hence (

) ( ) ( ) 1 + r1 Sa T (a)−1 f 1 (ζ1 ) r2 Sa T (a)−1 f 1 (ζ2 ) . det ( ) ( ) r1 Sa T (a)−1 f 2 (ζ1 ) 1 + r2 Sa T (a)−1 f 2 (ζ2 ) ( ( ) ( ) ) 1 + qζ2 (ζ1 ) ϕa (ζ1 ) − ζ2−1 ψa (ζ1 ) qζ1 (ζ2 ) ϕa (ζ2 ) − ζ2−1 ψa (ζ2 ) = det ( ) ( ) qζ2 (ζ1 ) ϕa (ζ1 ) − ζ1−1 ψa (ζ1 ) 1 + qζ1 (ζ2 ) ϕa (ζ2 ) − ζ1−1 ψa (ζ2 ) = (1 + ϕa (ζ1 )) (1 + ϕa (ζ2 ))

m a (ζ1 ) − m a (ζ2 ) , ζ1 − ζ2

which shows (i). Since . f j (z) is a rational function with poles at .ηi and . f j (∞) = 0, an identity .

f j (z) =

p ∑

ri j qηi

i=1

with r = lim qηi (z)−1 f j (z) = lim qηi (z)−1 r (z)−1 qζ j (z) = −

. ij

z→ηi

z→ηi

qζ j (ηi ) ηi r ' (ηi )

is valid. Then (3.5) yields .

T (a)−1 f j =

∑ i

ri j T (a)−1 qηi =

∑ i

ri j ηi

(ϕa (−ηi ) + 1) v − (ψa (−ηi ) − ηi ) u ) ( Δa (ηi ) ηi2 − z 2

30

3 KdV Flow I: Reflectionless Case

and .

∑

(ϕa (−ηi ) + 1) (ψa + z) − (ψa (−ηi ) − ηi ) (ϕa + 1) ) ( Δa (ηi ) ηi2 − z 2 i ∑ m a − m a (−ηi ) ). ( = (ϕa + 1) ri j ηi (ϕa (−ηi ) + 1) Δa (ηi ) ηi2 − z 2 i

aT (a)−1 f j =

Since

ri j ηi

S T (a)−1 f j (z) = p− aT (a)−1 f j (z) = aT (a)−1 f j (z) − f j (z)

. a

and . f j (ζi ) = ri−1 δi j , we have .τ a

(( ) ( ) ) δi j + ri aT (a)−1 f j (ζi ) − ri f j (ζi ) 1≤i, j≤ p ) ( ( ) −1 = det ri aT (a) f j (ζi ) ⎛ ⎞ p ∑ − m + 1 m ϕ (ζ ) (−η ) (−η ) a a a i k k ⎠ = det ⎝ri (ϕa (ζi ) + 1) rk j ηk Δa (ηk ) ηk2 − ζi2 k=1 ⎛ ( ( ) )( ( ) )⎞ ( )) p ( ∏ ) ( m a (ζi ) − m a −η j ϕa ζ j + 1 ϕa −η j + 1 ⎝ ⎠ ( ) , = det −ri ri j ηi det Δa η j ζi2 − η 2j j=1

(r ) = det

where ⎛ ⎞ ) ) ( ( p ∏ ( ) qζ j (ηi ) 1 1 ⎝ ⎠ ( ) det , = . det −ri ri j ηi = det ri r ' (ηi ) ηi − ζ j r ' (ηi )̂ r' ζj j=1 ∎

which is (3.22).

The transformed .m-function .m qζ a can be computed from .m a , which will be used in the proof of non-degeneracy of tau-functions. ( ) Lemma 3.6 If .τa qζ /= 0, one has ⎧ m a (ζ) − m a (z) ⎪ ⎪ ⎨ ϕqζ a + 1 = (ϕa + 1) ζ−z . . 2 2 − ζ z ⎪ ⎪ m q a (z) = − m (ζ) ⎩ a ζ m a (z) − m a (ζ)

(3.23)

Proof The first identity follows from the cocycle property of Lemma 3.4 and (ii) of Lemma 3.5. Namely ( ) τ a qζ q z m a (z) − m a (ζ) ( ) = (ϕa (z) + 1) .ϕqζ a (z) + 1 = τqζ a (qz ) = . z−ζ τ a qζ

(3.24)

3.3 Group Action on Ainv and Tau-Function

To show the second identity we start from q au = qζ + qζ ϕa = qζ + qζ ϕa (ζ) + qζ (ϕa − ϕa (ζ)) ,

. ζ

which yields p

. +

( ) qζ au = qζ + qζ ϕa (ζ) .

( ) ( ) Multiplying .z 2 − ζ 2 and noting . λ2 − ζ 2 au = a λ2 − ζ 2 u one has ( .

) z 2 − ζ 2 qζ (z) (1 + ϕa (ζ)) (( ) (( ) ) ) = p+ λ2 − ζ 2 qζ au (z) + p+ z 2 − λ2 qζ au (z) { ( ) ) ( 2 1 2 = p+ qζ a λ − ζ u (z) − (z + λ) qζ (λ) a (λ) u (λ) dλ 2πi C { { ) ( )( a (λ) u (λ) ζ z+ζ dλ a (λ) u (λ) dλ − = T qζ a z 2 − ζ 2 u (z) + 2πi C 2πi C ζ − λ ( )( ) = T qζ a z 2 − ζ 2 u (z) + ζκ1 (a) − (z + ζ) ϕa (ζ) ,

( ) which together with . z 2 − ζ 2 qζ (z) = −ζ (z + ζ) yields .

( )( ) T qζ a z 2 − ζ 2 u = −ζ (z + ζ) (1 + ϕa (ζ)) − ζκ1 (a) + (z + ζ) ϕa (ζ) = −ζz − ζ (ζ + κ1 (a)) .

Hence

( .

) ( )−1 ( )−1 z − ζ (ζ + κ1 (a)) T qζ a 1, z 2 − ζ 2 u = −ζT qζ a

and multiplying by .qζ a one has ( .

) ) ) ( ( z 2 − ζ 2 qζ au = −ζ z + ψqζ a − ζ (ζ + κ1 (a)) 1 + ϕqζ a .

On the other hand ( .

z −ζ 2

2

)

) ( ) −ζ z 2 − ζ 2 ( 1 + ϕqζ a qζ au = −ζ (z + ζ) (1 + ϕa ) = m a (z) − m a (ζ)

holds, hence m qζ a (z) =

.

= Equation (3.24) implies

( ) z + ψqζ a (z) + κ1 q ζ a 1 + ϕqζ a (z) ( ) z2 − ζ 2 − (ζ + κ1 (a)) + κ1 qζ a . m a (z) − m a (ζ)

31

32

3 KdV Flow I: Reflectionless Case

) ( ( ) m a (z) − m a (ζ) −1 κ qζ a = lim zϕqζ a (z) = lim z (ϕa (z) + 1) z→∞ z→∞ z−ζ = ζ + κ1 (a) − m a (ζ) ,

. 1

∎

which shows the second identity.

3.4 Non-degeneracy of Tau-Functions In this section we show the invertibility of .T (ga) for a certain class of . a ∈ Ainv and .g ∈ ┌. For a rational function .r = p/q with polynomials . p, .q define .

deg r = deg p − deg q.

Set ⎧ g = g} ⎨ ┌real = {g { ∈ ┌;inv } = a ∈ A ; a = a, τa (r ) ≥ 0 for any rational r} ∈ ┌real , Ainv . + { ⎩ = a ∈ Ainv ; a = a, τa (g) ≥ 0 for any g ∈ ┌real

(3.25)

where . f (z) = f (z). The third identity follows from the continuity of .τa . The property . a = a implies .ϕa = ϕa , .ψ a = ψa , .m a = m a , and .τa takes real values on .┌real . The cocycle property implies .

a ∈ Ainv + , τ a (g) > 0 =⇒ τga (r ) =

τa (r g) ≥ 0 =⇒ ga ∈ Ainv + . τa (g)

(3.26)

Lemma 3.7 Suppose . a ∈ Ainv + . (i) If .g ∈ ┌real has finite numbers of zeros and poles in .C, then .τa (g) > 0, hence .ga ∈ Ainv + . (ii) .m a satisfies .Im m a (z) / Im z > 0 and .1 + ϕa has no zeros in . D− . Proof 1.◦ ) First we show .τa (r ) > 0 for any real rational .r . If .r = qs with .s ∈ D− ∩ R, then (3.21) implies { .

τa (q ( s2)) = ϕa (s) + 1 2 ' . τa qs = (ϕa (s) + 1) m a (s)

Since .qs , .qs2 ∈ ┌real , these two tau-functions should be non-negative. Since .ϕa + 1, .ψ a + z are holomorphic on . D− , the order of each zero on .R of .ϕ a + 1 should be even. Hence the order of the pole .m a is also even (note .ψa (z) + z /= 0 if .ϕa (z) + 1 = 0 due to .Δa (z) /= 0) and .m 'a has an odd order pole at any zero of .ϕa + 1, which contradicts .m 'a (s) ≥ 0. Hence .ϕa + 1 /= 0 on . D− ∩ R, and .τa (qs ) > 0. One has also

3.4 Non-degeneracy of Tau-Functions

τ

. a

33

( −1 ) ( ) qs = τa rs−1 τa (q−s ) = Δa (s)−1 τa (q−s ) > 0 (with rs = qs q−s ).

If .r = qζ qζ with .ζ ∈ D− \R, then (ii) of Lemma 3.5 implies τ

. a

( ) Im m a (ζ) . qζ qζ = |ϕa (ζ) + 1|2 Im ζ

Therefore on the domain.{z ∈ D− \R; ϕa (z) + 1 /= 0}, one has.Im m a (z) / Im z ≥ 0. Since .Im m a is harmonic in this domain, the property .Im m a (z) / Im z ≥ 0 together with the maximum principle( implies that .Im m a has no zeros or is identically ) zero. Since .m a (z) = z + O z −1 , the second possibility does not occur, hence −1 .Im m a (z) / Im z > 0 holds. .m a cannot have poles since .m a (z) satisfies a similar property, which implies that .ϕa + 1 does not vanish on . D− \R. )−1 ( If .r = qζ qζ , then (iv) of Lemma 3.4 implies τ

. a

(( (( ) )−1 ) )−1 ( )−1 ( ) qζ qζ = τa qζ q−ζ qζ q−ζ q−ζ q−ζ = |Δa (ζ)|−2 τa q−ζ q−ζ

(( ( ) )−1 ) > 0. Suppose .τa (r0 ) > 0 for a due to .τa qζ q−ζ = Δa (ζ), one has .τa qζ qζ rational .r0 ∈ ┌real . Then (3.26) implies .r0 a ∈ Ainv + , and the cocycle property together with the above argument shows τ (r0 r1 ) = τa (r0 ) τr0 a (r1 ) > 0

. a

( )−1 for .r1 = qs , .qs−1 , .qζ qζ , . qζ qζ , which completes the proof for general .r by induction. 2.◦ ) .τa (g) > 0 holds for any .g designated in (i). One sees that.g can be approximated by rational functions.rn ∈ ┌real . Since.grn−1 → 1 and( .τa (1) ) = 1, the continuity of .τa implies that there exists .n ≥ 1 such that −1 .τ a gr n > 0. Then applying the cocycle property yields ( ) ( ) τ (g) = τa grn−1rn = τa grn−1 τgrn−1 a (rn ) > 0

. a

due to .grn−1 a ∈ Ainv + . The property (ii) has been proved in the above argument.

∎

A sufficient condition for . a ∈ Ainv + is given by ⎫ ⎧ m; m holomorphic on C\ ([−μ0 , μ0 ] ∪ i [−μ1 , μ1 ]) and fulfills ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ (i) Im m(z)/ Im z > 0, )m = m refl ( −1 .M = ⎪ ⎪ (ii) m(z) = z + O z ⎪ ⎪ ⎭ ⎩ (iii) m (x) > m(−x) for any x > μ0 (3.27) with(.μ0√ , .μ1 )> 0. The superscript .r e f l is added since the associated .m ± (z) = ∓m ± −z satisfy

34

3 KdV Flow I: Reflectionless Case

m + (λ + i0) = −m − (λ + i0) for λ > μ21 (reflectionless on (μ21 , ∞)).

.

μ0 is related to the lower bound of the spectrum of the Schrödinger operator . L q associated with .m ± since in this case .m ± are holomorphic on .C\(−∞, λ0 ] with 2 .λ0 = −μ0 . .

refl Lemma 3.8 It holds that .m = (1, m(z)/z) ∈ Ainv + for .m ∈ M . In this case .ϕm = 0, .ψm = m − z and the .m-function is .m.

Proof One easily sees that.m satisfies (3.11), hence.m ∈ Ainv . Equation (3.14) shows the .m-function is .m. We have to show .τm (r ) ≥ 0 for any rational .r ∈ ┌real . Let ( )−1 −1 .r 0 ∈ ┌real be one of .qs , .qs , .qζ qζ , . qζ qζ . Then .τm (r0 ) > 0 is clearly valid, and refl Lemma 7.2, Lemma 7.3 imply .m r0 m ∈ M . Here note that .m ga = m a if .g ∈ ┌ is even. Suppose .τm (r0 ) > 0 and .m r0 m ∈ Mrefl for an .r0 ∈ ┌real . Then applying this ( )−1 argument to .m r0 m and another one .r1 of .qs , .qs−1 , .qζ qζ , . qζ qζ , one has τ (r0 r1 ) = τm (r0 ) τr0 m (r1 ) > 0,

. m

∎

which completes the proof.

3.5 Identification of .m-functions with WTK Functions Suppose the boundaries .±∞ are of limit point type. Then, it is known that the symmetric operator . L q has a unique self-adjoint extension in . L 2 (R) and the WTK functions .m ± (z) are holomorphic on .C\sp. L q and Herglotz–Nevanlinna, namely .Im m ± (z) / Im z > 0. Necessary information for the spectral theory for . L q can be obtained in the Appendix. Now we consider the relationship between .m a (z) and .m ± (z). For . W = W a the function in (2.10) is described as ) ( f (x, z) = a (z) T (ex a)−1 1 (z) with ex (z) = e x z ,

. a

(3.28)

if .ex a ∈ Ainv . The underlying operator for .Gr (2) was a Schrödinger operator given in (2.12), namely .

L q = −L 2 = −∂x2 + q

and . f a satisfies .

with q (x) = −2∂x κ1 (ex a) ,

L q f a = −z 2 f a .

Taking derivative in (3.28) with respect to .x one has ) ( ∂ f a (x, z) = −a (z) T (ex a)−1 T (zex a) T (ex a)−1 1 (z) ,

. x

(3.29)

3.5 Identification of m-functions with WTK Functions

35

hence

.

( ) ∂x f a (x, z)|x=0 = −a (z) T (a)−1 T (za) T (a)−1 1 (z) ) ( = −a (z) T (a)−1 T (za) u (z) ( ) = −a (z) T (a)−1 (z + κ1 ) (z) = − (z + ψa (z) + κ1 (1 + ϕa (z))) ,

which shows m a (z) = −

.

∂x f a (x, z)|x=0 . f a (0, z)

(3.30)

On the other hand, (3.28) implies f

. ey a

( ( )−1 ) 1 (z) = e y (z) f a (x + y, z) . (x, z) = e y (z) a (z) T ex e y a

Therefore, (3.30) yields | | ∂x f e y a (x, z) | ∂ y f a (y, z) ∂x f a (x + y, z) || | , . − m e y a (z) = = = f e y a (x, z) |x=0 f a (x + y, z) |x=0 f a (y, z)

(3.31)

hence it holds that ( { f (x, z) = f a (0, z) exp −

x

. a

) m e y a (z) dy .

(3.32)

0 inv Since .ex a ∈ Ainv + for . a ∈ A+ is valid from Lemma 3.7, one can define a smooth function .q(x) = −2∂x κ1 (ex a) with no singularity on .R.

Proposition 3.1 Let . a ∈ Ainv + and set .q(x) = −2∂x κ1 (ex a). Then the boundaries .±∞ are of limit point type for the Schrödinger operator . L q . The .m-function .m a and the WTK functions .m ± are connected by { m a (z) =

.

( ) −m +( −z 2) if Re z > 0 . m − −z 2 if Re z < 0

(3.33)

In particular one has .m a ∈ Mrefl for any .μ0 > 0 such that .[−μ0 , μ0 ] ⊂ D+ . Conversely, any .m ∈ Mrefl can be the .m-function of .m = (1, m(z)/z) ∈ Ainv + . Proof The key ingredient for the proof is (3.32), which shows ∂ m ex a (z) = −z 2 − q(x) + m ex a (z)2 ,

. x

since . f a (x, z) satisfies (3.29). Equation (3.34) implies

(3.34)

36

3 KdV Flow I: Reflectionless Case

∂ Im m ex a (z) = − Im z 2 + 2 Re m ex a (z) Im m ex a (z) ,

. x

which together with (3.32) yields ( .

{

| f a (x, z)| = | f a (0, z)| exp −2 2

x

)

Re m e y a (z) dy ( { x ) Im z 2 2 Im m a (z) = | f a (0, z)| exp − dy . Im m ex a (z) 0 Im m e y a (z) 2

0

Then an identity {

b

)) ( ( { b Im m a (z) Im z 2 | f a (x, z)| d x = | f a (0, z)| dy 1 − exp − Im z 2 0 Im m e y a (z) 2

.

0

2

follows. Since .Im z 2 = 2 Re z Im z and .Im m e y a (z) / Im z > 0 hold due to (ii) of Lemma 3.7, if .Re z > 0, we have { ∞ Im m a (z) | f a (x, z)|2 d x ≤ | f a (0, z)|2 . < ∞. (3.35) Im z 2 0 On the other hand, if .z = λ ∈ D− ∩ R, then (3.32) implies that . f a (x, λ) / f a (0, λ) is a positive solution to . L q f = −λ2 f , hence the boundary .+∞ is of limit point type owing to Lemma 7.5. Since . f a (x, z) ∈ L 2 (R+ ) if .Re z > 0 and .Im z /= 0, the 2 uniqueness of such a(solution ) justifies. f a (x, z) = f + (x, −z ), which shows the iden2 tity .m a (z) = −m + −z if (.Re z )> 0. The boundary .−∞ can be treated similarly, and we obtain .m a (z) = m − −z 2 if .Re z < 0, which completes the proof. The last ∎ statement follows from Lemma 3.8. This proposition says that for . a ∈ Ainv + its √.m-function .m a (z) is analytically continuable up to .C\ ([−μ0 , μ0 ] ∪ iR) (.μ0 = −λ0 ) although originally we knew its analyticity only on . D− , which means the non-negativity of the tau-functions is a very strong property.

3.6 KdV Flow on .Qrefl Since the commutative group .┌real acts on .Ainv + by Lemma 3.7, we define a class of potentials by } { Qrefl = q; q (x) = −2∂x κ1 (ex a) with a ∈ Ainv + .

.

Then any one-parameter group .gt ∈ ┌real creates a solution to an equation belonging to the KdV hierarchy. Sato described these solutions by the tau-functions. In this section we define the KdV flow in the context of Sato. The key is (3.21):

3.6 KdV Flow on Qrefl

37

τ

. a

( ) qζ = 1 + ϕa (ζ) .

The coefficient .κ1 is the first one of the Taylor expansion of .ϕa (ζ) at .ζ = ∞, hence ) ( ( ) ( ( ) ) κ = lim ζ τa qζ − 1 = lim ζ τa qζ − τa (1) .

. 1

ζ→∞

ζ→∞

q is close to .e z/ζ for large .ζ in the sense that

. ζ

( ) e z/ζ qζ (z)−1 = e z/ζ (1 − z/ζ) = 1 + δζ (z) with δζ (z) = O ζ −2 .

.

Then (iii) of Lemma 3.4 yields τ

. qζ a

( ) ( ) ( ) 1 + δζ − 1 = τqζ a 1 + δζ − τqζ a (1) = O ζ −2 .

Since the cocycle property implies τ

. a

( ) ( z/ζ ) ( ) e − 1 = τa qζ τqζ a e z/ζ qζ−1 − 1 ( ) ( ( ( ) ) ( ) ) = τa qζ − 1 τqζ a e z/ζ qζ−1 + τqζ a 1 + δζ − 1

uniformly on . D+ , one sees .

( ) ( ( ) ) ) ( ( ) lim ζτa e z/ζ − 1 = lim ζ τa qζ − 1 τqζ a e z/ζ qζ−1 ζ→∞ ζ→∞ ( ) ) ( + lim ζ τqζ a 1 + δζ − 1 ζ→∞

= κ1 . Therefore one has from the cocycle property ( ( ) ) q(x) = −2∂x lim ζτex a e z/ζ − 1 ζ→∞ ) ( ( ) τa ex e z/ζ = −2∂x lim ζ − 1 = −2∂x2 log τa (ex ) . ζ→∞ τa (ex )

.

Define an operation on functions by .

( ) dζ f (z) =

ζ 2 − z2 − f (ζ) . f (ζ) − f (z)

Then the identity .m qζ a = dζ m a holds due to (3.25). We prepare:

(3.36)

38

3 KdV Flow I: Reflectionless Case

Lemma 3.9 Let . a1 , . a2 ∈ Ainv + . Then the followings are valid: (i) The identity .m a1 = m a2 implies .m ga1 = m ga2 for any .g ∈ ┌real . (ii) The identity .∂x2 log τa1 (ex ) = ∂x2 log τa2 (ex ) holds if and only if .m a1 = m a2 . Proof As we have seen in the proof of Lemma 3.8, if we let .r0 be one of .qs , .qs−1 , ( )−1 .qζ qζ , . qζ qζ , then .m r0 a1 = m r0 a2 holds due to m qζ a = dζ m a , m qζ−1 a = m q−ζ a , m qζ qζ a = dζ dζ m a , m (qζ q )−1 a = m q−ζ q−ζ a . ζ

.

Then inductively one can show .m r a1 = m r a2 for any rational .r ∈ ┌real . Equation (3.21) implies ( ) τa qζ2 ' .m a (ζ) = ( )2 , τ a qζ hence the cocycle property yields ( ) ( ) τa gqζ2 τa (g) τga qζ2 ' .m ga (ζ) = ( )2 = ( )2 . τga qζ τa gqζ Then the continuity of .τa (·) yields (i). (ii) follows from the one-to-one correspondence between a potential .q and its WTK functions .m ± , since Proposition 3.1 shows the coincidence of .m a and the ∎ WTK function of . L q with .q = −2∂x κ1 (ex a). For .g ∈ ┌real , .q (x) = −2∂x2 log τa (ex ) ∈ Qrefl define .

( ) (K (g) q) (x) = −2∂x2 log τga (ex ) = −2∂x2 log τa (gex ) .

(3.37)

This is well-defined due to Lemma 3.9. Theorem 3.1 .{K (g)}g∈┌real defines a flow on .Qrefl , that is . K (g1 g2 ) = K (g1 ) K (g2 ) holds for any .g1 , .g2 ∈ ┌real . Especially one has ( ) ) K et z q (x) = q (x + t) ( ( 3) ) 1 3 . q (t, x) = K et z q (x) satisfies ∂t q = q ''' + qq ' 4 2

{( .

The second one is the KdV equation. Proof The flow property is proved by the cocycle property as .

K (g1 g2 ) q (x) = −2∂x2 log τa (g1 g2 ex ) = −2∂x2 log τg1 a (g2 ex ) = K (g2 ) K (g1 ) q. ∎

3.7 Boussinesq Equation

39

One can call this .{K (g)}g∈┌real the KdV flow on .Qrefl . Proposition 3.1 makes it possible to express the space .Qrefl of initial data as {

Q

.

refl

} q ; the WTK (√ functions m ± of q are given by ) ( ) √ = m + (z) = −m −z , m − (z) = m − −z for an m ∈ Mrefl ⎫ ⎧ q ; the WTK functions m ± of q satisfy for λ0 < λ1 ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ (i) m + (λ + i0) = −m − (λ + i0) for a.e. λ > λ1 . (3.38) = (ii) m ± holomorphic on C\[λ0 , ∞) ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ (iii) m + (λ) + m − (λ) < 0 for λ < λ0

It is desirable to have a more direct form of .Qrefl . Necessary conditions for .q ∈ Qrefl are known. (i) . q ∈ Qrefl is meromorphic on the entire plane .C. This is because .τa (ez ) is an entire function. [37] proved .q(z) is holomorphic on a strip (ii){Lundina [35], Marchenko } −1/2 . |Im z| < B (. B = λ1 − λ0 ) and satisfies ( )−2 |q(z) − λ1 | ≤ 2B 1 − B 1/2 |Im z|

.

by assuming the corresponding .m ± (z) are reflectionless on .(λ1 , ∞) and .spL q ⊂ [λ0 , ∞). If .q is in . L 1 ((1 + |x|) d x, R) and reflectionless on .(λ1 , ∞), then .q has to be a multi-soliton potential. If .q is periodic, then the reflectionless property on .(λ1 , ∞) implies that .q is represented by a theta function. In this way the reflectionless condition is very strong and .Qrefl is a very restrictive class of initial data. One can ask naturally if some extension is possible in the framework of Sato’s theory. Our next task is to develop a way to obtain more general class of initial data, which will be carried out later.

3.7 Boussinesq Equation KdV flow has been constructed on .Gr (2) . In this section we try to develop the theory on .Gr (3) and find the difference from the KdV flow. Let .C be a simple smooth closed curve in .C satisfying ϑC = C with ϑ = e2πi/3 (ϑ3 = 1).

.

A closed .z 3 -invariant subspace .W of . L 2 is obtained by a bounded function . a (λ) = t . (a1 (λ) , a2 (λ) , a3 (λ)) on .C. Define

40

3 KdV Flow I: Reflectionless Case

.

a f (λ) =

)) ( 1( a1 (λ) f (λ) + a2 (λ) f (ϑλ) + a3 (λ) f ϑ2 λ . 3

Then λ3 a f =

.

( )3 ( )) 1( 3 a1 λ f + a2 (ϑλ)3 f (ϑλ) + a3 ϑ2 λ f ϑ2 λ = aλ3 f 3

holds, hence .Wa = a H+ satisfies .z 3 Wa ⊂ Wa . Set .

(T (a) u) (z) = (p+ (a f )) (z) .

Assume .T (ex a) (.ex (z) = e x z ) is invertible for any .x ∈ R, and set ⎛ ⎞ ∑ ) −1 . f a (x, z) = a T (e x a) 1 (z) = e−x z ⎝1 + a j z − j ⎠ ∈ Wa (

j≥1

Then .

) ( − ∂x3 f a (x, z) − 3a1' ∂x f a (x, z) − 3 a1'' − a2' + a1' a1 f a (x, z) = z 3 f a (x, z)

and .

) ( 1 − a 'j+2 + a ''j+1 − a '''j + a1' a j+1 − a1' a 'j − a1'' − a2' + a1' a1 a j = 0 3

hold for . j = 1, .2, .· · · . Moreover, assuming .e x z+t z Wa ∈ Gr (3) for any .x, .t ∈ R and setting 2

( ( )−1 ) ( )) 2 2 ( f (t, x, z) = a T e x z+t z a 1 (z) = e−x z−t z 1 + O z −1 ∈ Wa ,

. a

one has

∂ f a (t, x, z) + ∂x2 f a (x, z) + 2a1' f a (x, z) = 0.

. t

Then the coefficients .a j defined by ⎛ f (t, x, z) = e−x z−t z ⎝1 + 2

∑

. a

⎞ a j z− j ⎠

j≥1

satisfies

∂ a j = 2a 'j+1 − a ''j − 2a1' a j

. t

for any . j ≥ 1. Then one sees the functions

3.7 Boussinesq Equation

41

{ .

q1 = 3a(1' ) q0 = 3 a1'' − a2' + a1' a1

fulfills a system of nonlinear equations { .

This implies ∂ 2 q1 = −

. t

{

and .

∂t q1 = −q1'' − 2q0' 2 2 . ∂t q0 = q0'' + q1''' + q1 q1' 3 3

)'' 1 ( '' q + 2q12 (Boussinesq equation), 3 1

L q0 ,q1 = ∂x3 + q1 ∂x + q0 (underlying operator) . L q0 ,q1 f a (x, z) = −z 3 f a (x, z)

The Lax pair induces the invariance of the spectrum of the underlying operator . L q0 ,q1 along solutions to the Boussinesq equation. Therefore, in this case also its spectral structure is supposed to play an important role in the study of the construction of the Boussinesq flow and the properties of the flow. However, on the contrary to the KdV flow the spectrum of the underlying operator . L q0 ,q1 generally spreads on some domain in .C, since the anti-symmetry . L q0 ,q1 = −L q∗0 ,q1 holds if and only if .2q0 − q1' = 0 is valid, which is a quite degenerate case. This makes it difficult to investigate ( )−1 2 the existence of .T e x z+t z a , or equivalently non-existence of singularities of solutions to the Boussinesq equation.

Chapter 4

KdV Flow II: Extension

Since in the last chapter we assumed the reflectionless property, the region of singularities of the .m-function remains bounded and we could choose the bounded curve .C surrounding the region of the singularities. However, when we intend to remove the reflectionless property, naturally we have to treat an unbounded curve and an unbounded domain . D+ . Therefore we have to extend Sato’s theory on the unbounded curve .C. Due to the unboundedness of the curve .C several new problems arise: (i) .z n with .n ≥ 0 are not in . L 2 (C) which complexifies the definition of .ϕa , .ψa , .m a . (ii) Since the element of . H− has no Taylor expansion at .z = ∞, one cannot employ pseudo-differential operators to derive Schrödinger operators, the KdV equation etc. (iii) The( tau-function was essential to deduce the existence of .T (ga)−1 : however, ) −1 −1 cannot be defined without extra conditions on . a. .det g T (ga) T (a) These obstructions will be resolved in this chapter. In the process of the construction of the flow two integer parameters .n, . L appear. h .n is an odd number describing the degree of polynomials .h when .g = e , so it corresponds to the numbering of the equations belonging to the KdV hierarchy. Hence for the KdV equation .n = 3. A positive integer . L is related to the degree of differentiability of initial data and solutions.

4.1 Hardy Spaces and Toeplitz Operators Let .C be a simple smooth closed curve in .C ∪ {∞} defined by C = {±ω (y) + i y; y ∈ R}

.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 S. Kotani, Korteweg–de Vries Flows with General Initial Conditions, Mathematical Physics Studies, https://doi.org/10.1007/978-981-99-9738-1_4

43

44

4 KdV Flow II: Extension

Fig. 4.1 The horizontal axis is imaginary

with a smooth positive function .ω on .R satisfying .ω (y) = ω (−y), hence .C satisfies C = −C, C = C.

.

The curve .C is regarded as a simple closed curve in the Riemann sphere and oriented anti-clockwise. The shape of.C is illustrated below (the. y-axis is horizontal) (Fig. 4.1): . D± are the interior and exterior domains separated by the curve .C defined by .

D+ = {z ∈ C; |Re z| < ω (Im z)} , D− = {z ∈ C; |Re z| > ω (Im z)} .

For a positive odd integer .n let .g = eh with a real odd polynomial .h of degree .≤ n. The curve .C is chosen so that .g remains bounded on . D+ , or more concretely ( −(n−1) ) as |y| → ∞. .ω (y) = O y Only when .n = 1 does the curve .C not have to approach to .iR. The Hardy spaces associated with curve .C are defined by .

H± = the closure in L 2 (C) of rational functions with no poles in D± .

We will use the notation . H (D± ) instead of . H± if it is necessary. It holds that .

L 2 (C) = H+ ⊕ H− .

For . f ∈ L 2 (C) define p f (z) = ±

. ±

and

{

f (λ) dλ for z ∈ D± , λ C −z

1 2πi

1 . (Θf ) (z) = lim ∈↓0 2πi

{

f (λ) dλ for z ∈ C. λ C∩{|λ−z|>∈} − z

It is known that .Θ is a bounded operator on . L 2 (C) (see [4, 6]) and .p± have a finite limit a.e. when .z approaches to an element of .C. They are connected by { .

p+ f (z) = f (z) /2 + (Θf ) (z) for z ∈ C, p− f (z) = f (z) /2 − (Θf ) (z)

4.1 Hardy Spaces and Toeplitz Operators

45

and .p± are projections from . L 2 (C) onto . H (D± ) respectively. It should be noted that .p± are generally not orthogonal projections. As in the previous section we employ the notations f (z) = ( f (z) + f (−z)) /2, f o (z) = ( f (z) − f (−z)) /2

. e

for a function . f on .C. It should be noted also that p : L 2e (C) → H+ ∩ L 2e (C) and p+ : L 2o (C) → H+ ∩ L 2o (C) ,

. +

(4.1)

where . L 2e (C), . L 2o (C) denote the even part and the odd part of . L 2 (C) respectively. To investigate the differentiability of solutions to the KdV equation we have to admit the multiplication operation by rational functions on . H± . To realize such operations some modification of the spaces . H± is necessary. For an . N ∈ Z+ and .b ∈ D− set N . H+,N = (z − b) H+ . (4.2) Clearly, . H+,N does not depend on .b, and .

z k ∈ H+,N for any integer k ≤ N − 1.

Lemma 4.1 . H+,N , . H− are subspaces of the vector space .z N L 2 (C) and satisfy .

H+,N ∩ H− = {0} .

Proof Let .u ∈ H+,N ∩ H− . Since .u ∈ H+,N , for .b ∈ D− one has u (z) 1 . = N 2πi (z − b)

{

u (λ)

C

(λ − z) (λ − b) N

dλ

for .z ∈ D+ . On the other hand, .u ∈ H− implies for .z ∈ D+ .

1 2πi

{ C

u (λ) (λ − z) (λ − b) N

dλ =

u (b) 1 . ∂ N −1 (N − 1)! b z − b

Therefore .u turns out to be a polynomial of degree at most . N − 1, which is possible ∎ only when .u = 0 since .u ∈ H− ⊂ L 2 (C) and .C is unbounded. Define .

( ) HN = H+,N ⊕ H− ⊂ z N L 2 (C) (direct sum),

and extend the definition of the projections on . HN by p f = f + , p− f = f − if f = f + + f − with f + ∈ H+,N , f − ∈ H+ .

. +

We impose a norm on . HN by

46

4 KdV Flow II: Extension

/{ .

|| f || =

−2N

| f + (λ)| |λ| 2

/{ |dλ| +

C

| f − (λ)|2 |dλ|. C

A certain class of bounded functions .a acts on . HN , namely we have: Lemma 4.2 Let .a be a bounded function on .C such that there exists a bounded holomorphic function .a0 on . D+ satisfying .

z N (a(z) − a0 (z)) is bounded on C.

(4.3)

Then, .a f ∈ HN for . f ∈ HN and { .

p+ (a f ) = p+ ((a − a0 ) f + ) + a0 f + + p+ a f − p− (a f ) = p− ((a − a0 ) f + ) + p− (a0 f − )

(4.4)

hold. Proof Let . HN ϶ f = f + + f − with . f + ∈ H+,N , . f − ∈ H− . Since .(a − a0 ) f ± ∈ L 2 (C), one can apply .p± to them and a f = (p+ ((a − a0 ) f + ) + a0 f + + p+ a f − ) + (p− ((a − a0 ) f + ) + p− (a0 f − ))

.

∎

yields the decomposition of .a f into . H+,N and . H− .

Subsequently a space of symbols .A L for . L ∈ Z+ can be introduced as follows: { AL =

.

a = (a1 , a2 ) ; a1 , a2 are bounded functions on C with a j satisfying (4.3) for N = L

} .

(4.5)

Lemma 4.2 enables us to define the Toeplitz operator on . H+,N by T (a)u = p+ (au) ∈ H+,N ,

. N

where . au is as before .

au = a1 u e + a2 u o .

Let . L ≥ N ' ≥ N . Then .{TN (a)} N ≥0 has the property that if . a ∈ A L , then . TN ' (a)| H (D ) = TN (a). Therefore we use the notation N + .

Set

T (a) = TN (a).

} { Ainv L = a ∈ A L ; T (a) is invertible on H+,L .

.

inv ' It should be noted that .Ainv L ⊃ A L ' holds if . L ≥ L.

(4.6)

4.1 Hardy Spaces and Toeplitz Operators

47

Characteristic functions and .m-functions for . a ∈ Ainv L can be defined similarly as in the previous section, but we have to replace . H+ by . H+,N for a suitable . N . .u ∈ H+,1 , .v ∈ H+,2 and .ϕ a , .ψ a can be defined as (3.3) and (3.4) respectively for inv . a ∈ A2 due to .1 ∈ H+,1 , . z ∈ H+,2 , namely { .

ϕa (z) = p− (au) (z) ∈ H− with u = T (a)−1 1 ∈ H+,1 . ψa (z) = p− (av) (z) ∈ H− with v = T (a)−1 z ∈ H+,2

Lemma 3.1 can be stated as follows. Lemma 4.3 If . a ∈ Ainv 2 , .{ϕ a , ψ a } satisfies the following properties: (i) . Δa (b) /= 0 on . D− and .

T (a)−1

1 (ϕa (b) + 1) v − (ψa (b) + b) u ) ( = . z+b Δa (b) z 2 − b2

(4.7)

(ii) . {ϕa , ψa } determines .Wa = a H+,N . (iii) Suppose . a ∈ Ainv L for some . L ≥ 2. Then there exist .κ j (a), .ι j (a) ∈ C and .φ a , .χ a ∈ H− such that ⎧ L−1 ⎪ ∑ ⎪ ⎪ ⎪ ϕ κ j (a) z − j + φa (z) z −L+1 = (z) ⎪ ⎨ a .

j=1

L−2 ∑ ⎪ ⎪ ⎪ ⎪ ψ ι j (a) z − j + χa (z) z −L+2 = (z) ⎪ ⎩ a

.

(4.8)

j=1

Proof The proof for (i), (ii) is exactly the same as that of the previous chapter, so we omit it. (iii) is proved by applying (4.4) of Lemma 4.2 to . a ∈ Ainv L . Namely, we have { 1 (a (λ) − a0 (λ)) u(λ) dλ. .ϕ a (z) = p− (au) = 2πi C z−λ ) ( Since . a (λ) − a0 (λ) = O λ−L and .u ∈ H+,1 , λ L−1 (a (λ) − a0 (λ)) u(λ) ∈ L 2 (C)

.

hold, hence

.

1 2πi

is valid with

{ C

∑ (a (λ) − a0 (λ)) u(λ) dλ = κ j (a) z − j + φa (z) z −L+1 z−λ j=1 L−1

48

4 KdV Flow II: Extension

⎧ { 1 ⎪ ⎪ λ j−1 (a (λ) − a0 (λ)) u(λ)dλ ⎨ κ j (a) = 2πi { C . λ L−1 (a (λ) − a0 (λ)) u(λ) . ⎪ ⎪ φa (z) = 1 dλ ⎩ 2πi C z−λ Similarly, one has

.

1 2πi

{

∑ (a (λ) − a0 (λ)) v(λ) dλ = ι j (a) z − j + χa (z) z −L+2 z−λ j=1 L−2

C

⎧ { 1 ⎪ ⎪ λ j−1 (a (λ) − a0 (λ)) v(λ)dλ ⎨ ι j (a) = 2πi {C . . λ L−2 (a (λ) − a0 (λ)) v(λ) ⎪ ⎪ χa (z) = 1 dλ ⎩ 2πi C z−λ

with

∎ The .m-function for .Wa is defined by m a (z) =

.

z + ψa (z) + κ1 (a) . 1 + ϕa (z)

(4.9)

(i) of (4.3) implies that .1 + ϕa (z) is not identically .0, hence .m a is meromorphic on D− .

.

A subset of .Ainv L is given by .B L : BL

.

=

{

} m (z) = (m 1 (z) , m 2 (z)) ; m is holomorphic on C\ ([−μ0 , μ0 ] ∪ iR) with μ0 > 0 and satisfies (i), (ii) below: (4.10)

(i) .m(z) = 1 +

∑

) ( mk z −k + O z −L on . D− with .1 = (1, 1), .mk ∈ C2 .

1≤k max {n, − deg r }. Proof Let .∈ be .0 < ∈ < 1 and .m = deg r ≤ 0. We first show that there exists a constant .c1 such that .

| | { n−1 | g (z)−1 g (λ) − 1 | | ≤ c1 |z| −1 ( | ) if |z − λ| ≤ ∈ |λ| | | |λ| 1 + |z|−m |λ|m if |z − λ| > ∈ |λ| λ−z

(4.21)

holds for .z, .λ ∈ C. If .|z − λ| ≤ ∈ |λ|, then with .c2 = (1 − ∈)−1 it holds that c−1 |z| ≤ |λ| ≤ c2 |z| .

. 2

(4.22)

Noticing an identity g (z)−1 g (λ) − 1 eh(λ)−h(z) − 1 r (z)−1 r (λ) − 1 = r (z)−1 r (λ) + λ−z λ−z λ−z | | and .|r (z)−1 r (λ)| ≤ c3 on .C 2 ∩ {|z − λ| ≤ ∈ |λ|} due to (4.22), we have only to estimate them separately. .

.

| | | r (z)−1 r (λ) − 1 | | ≤ c4 |z|−1 | | | λ−z

is clear if .|z − λ| ≤ ∈ |λ|. The other estimate can be obtained from .

h (λ) − h (z) eh(λ)−h(z) − 1 = λ−z λ−z

{

1

et(h(λ)−h(z)) dt. 0

4.2 Group Action on Ainv L and Derivation of Equations

53

If .|z − λ| > ∈ |λ|, the second estimate in (4.21) is straightforward, hence we have (4.21). Note .Δ ≤ Δ1 + Δ2 with { ⎧ 2 ⎪ |z|2(n−1) |z|−2N |dz| |dλ| Δ = c ⎪ 1 ⎨ 1 |z−λ|≤∈|λ| { . . ( )2 ⎪ ⎪ ⎩ Δ2 = c12 |λ|−2 1 + |z|−m |λ|m |z|−2N |dz| |dλ| |z−λ|>∈|λ|

Hence .Δ1 < ∞ is valid if . N > n. On the other hand, .Δ2 is finite if . N + m > 1/2. ∎ A similar argument gives the differentiability. Lemma 4.7 Let .gt (z) = eth(z) ∈ ┌n with a polynomial .h of degree .n. For . N ≥ 1 assume .gt a ∈ Ainv N +n for any .t ∈ R. Then, for any .u ∈ H+,N { ∂t T (gt a) : H+,N → H+,N +n and ∂t T (gt a) u = p+ (hgt au) . ∂t T (gt a)−1 u = −T (gt a)−1 ∂t T (gt a) T (gt a)−1 u ∈ H+,N +n holds. A higher derivative .∂tk T (gt a)−1 u exists if .gt a ∈ Ainv N +kn for any .t ∈ R. Proof Let . N1 = N + n. The first identity follows easily from ∂ T (gt a) u = ∂t (gt a0 u + p+ (gτ (a − a0 ) u)) = hgt a0 u + p+ (hgτ (a − a0 ) u) = p+ (hgτ au) .

. t

To show the second identity we first verify the continuity of .T (gt a) in . H+,N1 with −1 −1 .t. Equation (4.19) implies that the HS norm of .gt Hgt − gs Hgs is domirespect to√ nated by . Δ with { Δ=

.

C

| | | gt (λ) gt (z)−1 − gs (λ) gs (z)−1 |2 −2N | |z| 1 |dz| |dλ| . | | | 2 λ−z

Since .

sup

τ ∈I , z, λ∈C

(4.23)

| τ (h(λ)−h(z)) | |e | ∈ |λ|, we have

54

4 KdV Flow II: Extension

{ Δ ≤ c1 (t − s)2 { + c1

.

|λ−z|≤∈|λ|

|λ−z|>∈|λ|

( 2(n−1) ) |z| + |λ|2(n−1) |z|−2N1 |dz| |dλ|

| |2 |λ|−2 |gt (λ) gt (z)−1 − gs (λ) gs (z)−1 | |z|−2N1 |dz| |dλ| .

The first term is dominated by .c2 (t − s)2 if 2 (n − 1) − 2N1 + 1 < −1 =⇒ N1 > n,

.

which is satisfied if . N > 0. The second term tends to .0 as .s → t if .−2N1 < −1, which is always valid since . N1 ≥ 1. Therefore we have the continuity of .gt−1 Hgt −1 in the HS-norm. Since .gt u is continuous ( −1 for )any fixed .u ∈ H+,N1 and .gt Hgt is a compact operator, one sees . Hgt = gt gt Hgt is continuous in the operator norm, and hence the continuity of.T (gt a) in that norm follows, which implies the continuity of .T (gt a)−1 . Consequently, noting the identity (

−1 T (g −1 − T (g a)−1 t+∈ a) t

.∈

)

= T (gt+∈ a)−1 ∈−1 (T (gt a) − T (gt+∈ a)) T (gt a)−1 ,

we have the lemma. The existence of higher derivatives can be shown similarly. ∎

4.2.2 Derivation of Schrödinger Operator First we derive a Schrödinger operator from . a ∈ A L and .g = ex with e (z) = e x z .

. x

The curve .C is chosen so that .ex (z) remains bounded for any fixed .x ∈ R, namely the function .ω (y) defining .C satisfies ω (y) = O (1) as |y| → ∞.

.

For . a ∈ A L with . L ≥ 3 assume .ex a ∈ Ainv L for any . x ∈ R. Define u = T (ex a)−1 1 ∈ H+,1 , wx = p− (ex au x ) ∈ H− .

. x

Then, for a (bounded a (λ) = a (λ) − ) holomorphic vector . a0 (z) on . D+ satisfying .~ a0 (λ) = O λ−L on .C wx (z) =

.

1 2πi

{ C

a (λ) u x (λ) e xλ~ dλ z−λ

4.2 Group Action on Ainv L and Derivation of Equations

55

j

holds. Since Lemma 4.7 implies .∂x u x ∈ H+, j+1 for . j ≤ L − 1, ⎧ { λ~ a (λ) u x (λ) + ~ a (λ) ∂x u x (λ) 1 ⎪ ⎪ e xλ dλ ⎨ ∂x wx (z) = 2πi {C z−λ . . 2 λ~ a (λ) u x (λ) + 2λ~ a (λ) ∂x u x (λ) + ~ a (λ) ∂x2 u x (λ) 1 ⎪ ⎪ dλ e xλ ⎩ ∂x2 wx (z) = 2πi C z−λ (4.24) Since .u x ∈ H+,1 , the expansion .

∑

(z − λ)−1 =

z −k λk−1 + z −M λ M (z − λ)−1

1≤k≤M

shows { 1 a (λ) u x (λ) dλ λk−1 e xλ~ 2πi C 1≤k≤L−1 { L−1 xλ a (λ) λ e ~ −L+1 1 u x (λ) dλ +z 2πi C z−λ ∑ ≡ z −k sk (x) + z −L+1 w ~x (z)

wx (z) =

.

∑

z −k

(4.25)

1≤k≤L−1

with .w ~x ∈ H− . Since .∂x u x ∈ H+,2 , .∂x2 u x ∈ H+,3 , (4.24) shows similarly ⎧ ⎪ ∂ w = ⎪ ⎨ x x (z) .

2 ⎪ ⎪ ⎩ ∂x wx (z) =

∑ 1≤k≤L−2 ∑

z −k sk' (x) + z −L+2 w ~x(1) (z) z −k sk'' (x) + z −L+3 w ~x(2) (z)

∈ H− .

(4.26)

1≤k≤L−3

with .w ~x(1) , .w ~x(2) ∈ H− . Equation (4.8) implies wx (z) = ϕex a (z) , s1 (x) = κ1 (ex a) .

.

Set

( ) f (x, z) = a (z) u x (z) = e−x z 1 + ϕex a (z) .

. a

Proposition 4.1 Let . L ≥ 3 and assume .ex a ∈ Ainv L for any . x ∈ R. Set q(x) = −2s1' (x) = −2∂x κ1 (ex a) .

.

Then, a Schrödinger equation .

− ∂x2 f a (x, z) + q(x) f a (x, z) = −z 2 f a (x, z)

holds, and .{sk (x)}2≤k≤L−2 in (4.25) is determined by a recurrence relation

(4.27)

56

4 KdV Flow II: Extension ' s '' + 2s1' sk − 2sk+1 = 0, (1 ≤ k ≤ L − 3)

. k

(4.28)

for given .s1 (x), .{sk (0)}2≤k≤L−2 . Proof The identity .wx = ex au x − 1 yields { .

which implies

∂x wx = ex (zau x + a∂x u x ) ( ), ∂x2 wx = ex z 2 au x + a∂x2 u x + 2za∂x u x ) ( ∂ 2 wx − 2z∂x wx = ex a ∂x2 u x − z 2 u x .

. x

(4.29)

Here we have used the identity .

z 2 a (z) u (z) = a (z) z 2 u (z) .

Our strategy is to modify (4.29) so that the left-hand side is an element of . H− and the right-hand side is an element of .(ex a) H+,3 . From (4.26) .

z∂x wx = s1' (x) +

∑

z −k+1 sk' (x) + z −L+3 w ~x(1) (z) = s1' (x) + vx

2≤k≤L−2

follows with .vx ∈ H− , which combined with (4.29) yields ∂ 2 wx + 2s1' wx − 2vx ( ) = ex a ∂x2 u x − z 2 u x + 2z∂x wx + 2s1' (ex au x − 1) − 2vx ) ( = ex a ∂x2 u x − z 2 u x + 2s1' u x .

. x

Note

∂ 2 wx + 2s1' wx − 2vx ∈ H− , ∂x2 u x − z 2 u x + 2s1' u x ∈ H+,3 .

. x

Then, applying .p+ to (4.30) we have ) ( 0 = T (ex a) ∂x2 u x − z 2 u x + 2s1' u x ,

.

and the invertibility of .T (ex a) on . H+,3 yields ∂ 2 u x − z 2 u x + 2s1' u x = 0,

. x

which is (4.27) by applying . a (z). Since 0 = ∂x2 wx + 2s1' wx − 2vx ∑ ( ) −k ' sk'' + 2s1' sk − 2sk+1 z + z −L+3~ = vx

.

1≤k≤L−3

(4.30)

4.2 Group Action on Ainv L and Derivation of Equations

57

with .~ vx ∈ H− , we have (4.28).

∎

The .m-function .m a has another representation by . f a (x, z). Corollary 4.1 It holds that .

| ∂x f a (x, z) || = −m a (z) . f a (x, z) |x=0

(4.31)

Proof Set .b(z) = ∂x u x (z)|x=0 , .~ b(z) = ∂x f a (x, z)|x=0 . Then b(z) ∈ H+,2 and a (z) b(z) = ~ b(z)

.

hold. Since .e x z f a (x, z) = 1 + wx (z), (4.26) shows ∑

~ b(z) = −z (1 + w0 (z)) +

.

z −k sk' (0) + z −L+2 w ~0(1) (z)

1≤k≤L−2

= −z − s1 (0) + w (z) with .w ∈ H− . Applying .p+ to the identity .

a (z) b(z) = −z − s1 (0) + w (z)

we have .

hence

T (a) b(z) = −z − s1 (0),

b(z) = −T (a)−1 z − s1 (0)T (a)−1 1,

.

which implies ~ b(z) = a (z) b(z) = − (z + ψa (z)) − s1 (0) (1 + ϕa (z)) .

.

Consequently we have .

| ∂x f a (x, z) || − (z + ψa (z)) − s1 (0) (1 + ϕa (z)) = | f a (x, z) x=0 1 + ϕa (z) = −m a (z) + κ1 (a) − s1 (0) = −m a (z)

due to .s1 (0) = κ1 (a).

4.2.3 Derivation of KdV Equation Our next task is to derive the KdV equation from .g = et,x with

∎

58

4 KdV Flow II: Extension 3

e (z) = e x z+t z .

. t,x

In this case the curve .C is determined by requiring .et,x (z) to be bounded on .C, hence ) ( ω (y) = O y −2 as y → ∞.

.

For . L ≥ 4 let . a ∈ A L and assume .et,x a ∈ Ainv for any .t, .x .∈ R. Let .u t,x = ( ( ) L )−1 T et,x a 1 ∈ H+,1 and .wt,x = p− et,x au t,x ∈ H− . Then, Lemma 4.7 implies ∑

wt,x (z) =

z −k sk (t, x) + z −L+1 w ~t,x (z) with sk ∈ C, w ~t,x ∈ H−

.

1≤k≤L−1

for .t, .x ∈ R. Proposition 4.2 Let . L ≥ 4 and assume .et,x a ∈ Ainv L for any .t, . x ∈ R. Set ( ) q(t, x) = −2∂x s1 (t, x) = −2∂x κ1 et,x a .

.

Then .q satisfies the KdV equation 1 3 3 ∂x q(t, x) − q(t, x)∂x q(t, x). (4.32) 4 2 Proof In this case . N = 1, .n = 3. Our strategy for the proof is similar to that of the last one. Since .wt,x = et,x au t,x − 1 ∈ H− ∂ q(t, x) =

. t

{ .

( ) ∂x wt,x = et,x ( zau t,x + a∂x u t,x ) , ∂t wt,x = et,x z 3 au t,x + a∂t u t,x

which leads us to ) ( ∂ wt,x = z 2 ∂x wt,x + et,x a ∂t u t,x − z 2 ∂x u t,x .

. t

Since .

z 2 ∂x wt,x ≡ zs1' + s2' + s3' z −1 mod z −1 H− ,

substituting .1 = et,x au t,x − wt,x and .

z = zet,x au t,x − zwt,x = ∂x wt,x − et,x a∂x u t,x − zwt,x ≡ s1' z −1 − s1 − s2 z −1 − et,x a∂x u t,x ) ( ) ( ≡ s1' − s2 z −1 − s1 et,x au t,x − wt,x − et,x a∂x u t,x ) ) ( ( ≡ s1' − s2 + s12 z −1 − et,x a s1 u t,x + ∂x u t,x

(.≡ means .mod .z −1 H− ) into the above identity yields

4.3 Tau-Function .

59

z 2 ∂x wt,x (( ) )) ( ≡ s1' s1' − s2 + s12 z −1 − et,x a s1 u t,x + ∂x u t,x + s2' + s3' z −1 ( ) ( ( ) ) ) ( = s2' et,x au t,x − wt,x + s1' s1' − s2 + s12 + s3' z −1 − et,x a s1 s1' u t,x + s1' ∂x u t,x ) ) ) ) ( ( (( = −s2' wt,x + s1' s1' − s2 + s12 + s3' z −1 + et,x a s2' − s1 s1' u t,x − s1' ∂x u t,x .

Therefore ) ) ( ( ∂ wt,x ≡ −s2' wt,x + s1' s1' − s2 + s12 + s3' z −1 ) ) (( + et,x a s2' − s1 s1' u t,x + ∂t u t,x − z 2 ∂x u t,x − s1' ∂x u t,x

. t

holds, and we have ) {( ' ' 2 s2 − s1 s1' u t,x + ∂(t u t,x t,x = 0 ( '− z ∂x u t,x2 )− s1 '∂)x u−1 . . ' ' ∂t wt,x + s2 wt,x − s1 s1 − s2 + s1 + s3 z ≡ 0 ( ) due to the invertibility of.T et,x a on. H+,3 . Since the coefficient of.z −1 for the second identity vanishes, it follows that ) ) ( ( ∂ s + s2' s1 − s1' s1' − s2 + s12 + s3' = 0.

. t 1

3

Here the identities for .k = 1, .2 and .et z a in (4.28) of Proposition 4.1 { .

s1'' + 2s1' s1 − 2s2' = 0 s2'' + 2s1' s2 − 2s3' = 0

allow us to have ( ( ) ) ∂ s = −s2' s1 + s1' s1' − s2 + s12 + s3' ( ) ( )2 = −s1 s1'' /2 + s1' s1 + s1' + s12 s1' − s1' s2 + s2'' /2 + s1' s2 ( ) ( )2 )' ( = −s1 s1'' /2 + s1' s1 + s1' + s12 s1' + s1'' /2 + s1' s1 /2 ( )2 = s1''' /4 + 3 s1' /2,

. t 1

which is (4.32) by substituting .q(t, x) = −2∂x s1 (t, x).

∎

4.3 Tau-Function The tau-function defined in [29] is written in the present context as ) ( τ (g) = det g −1 T (ga) T (a)−1 .

. a

(4.33)

60

4 KdV Flow II: Extension

The operator.g −1 T (ga) T (a)−1 is a map on. H+,N and the determinant is well-defined if the operator .g −1 T (ga) T (a)−1 − I is of trace class on . H+,N . The identity (4.20) implies −1 .g T (ga) T (a)−1 − I = g −1 Hg Sa T (a)−1 , hence it is sufficient for this that . Hg Sa is of trace class. There are two cases where Hg Sa is of trace class:

.

(i) . g −1 Hg is of trace class. (ii) .g −1 Hg and . Sa are of Hilbert–Schmidt class. (i) is the case for .g = r ∈ ┌0 . However, for the other case one has to impose an extra condition on . a. To avoid this inconvenience we use the modified determinant .det 2 , namely −tr A . det 2 (I + A) = det(I + A)e . It is known that this determinant can be extended to any operator . A of Hilbert– Schmidt class. Since . I + A is invertible if and only if .det 2 (I + A) /= 0, this .det2 is sufficient to verify the existence of .T (ga)−1 . Set ( ) h τ (2) (g) = det 2 g −1 T (ga) T (a)−1 for a ∈ Ainv L , g = r e ∈ ┌n .

. a

(4.34)

τ (2) (g) can be defined if .g −1 Hg Sa is of HS class, which is valid if .g −1 Hg is of HSclass as a map from . H− to . H+,N , and Lemma 4.6 implies that .g −1 Hg is of HS class if . N > max {n, −m} , (4.35)

. a

and hence .

L > max {n, −m} .

(4.36)

Conversely, if . L satisfies (4.36), there exists . N satisfying (4.35) and . N ≤ L. In the definitions of .τa (g), .τa(2) (g) the operator .g −1 T (ga) T (a)−1 is a map on (2) . H N (D+ ), hence .τ a (g), .τ a (g) may depend on . N . To show the independence of the tau-functions on . N we need a metric-free nature of determinant. For the necessary facts of determinant refer to [50]. Lemma 4.8 Let . H , . H1 be two Hilbert spaces and . H1 be a subspace of . H as vector spaces. Assume . H1 is dense. Suppose a linear operator . A on . H is of Hilbert–Schmidt class and satisfies .

AH1 ⊂ H1 and A H1 ≡ A| H1 is of Hilbert–Schmidt class in H1 .

Then .

( ) det 2 (I H + A) = det 2 I H1 + A H1

) ( holds. If . A is of trace class, then .det (I H + A) = det I H1 + A H1 holds as well.

4.3 Tau-Function

61

Proof Let .{en }n≥1 be a complete orthonormal basis of . H1 and .{ f n }n≥1 be the orthonormal vectors in . H generated from .{en }n≥1 by the Gram–Schmidt process. Since . H1 is dense in . H , .{ f n }n≥1 turns to be complete in . H . Let .Vn be the .ndimensional subspace generated by .{ek }1≤k≤n and . Pn , . Q n be the orthogonal projections to .Vn from . H1 , . H respectively. Then it is known that { .

( ) det 2 I H1 + Pn A H1 Pn → det 2 (I H1 + A H1 ) as n → ∞. det 2 (I H + Q n AQ n ) → det 2 (I H + A)

(4.37)

Let . B, . E be .n × n matrices whose entries are ∑ (( ) ) ( ) . fi = bi j e j → B = bi j , and E = ei , e j H . 1≤ j≤n

If we write .

AnH1 =

((

Aei , e j

) ) H1 1≤i, j≥n

,

AnH =

((

A fi , f j

) ) H 1≤i, j≤n

,

then the identity . AnH = B AnH1 E B ∗ yields ( ) H . det 2 (I H + Q n AQ n ) = det 2 In + A n ( ) = det 2 In + B AnH1 E B ∗ ( ) = det 2 In + B AnH1 B −1 ( ) ( ) = det 2 In + AnH1 = det2 I H1 + Pn A H1 Pn due to . B E B ∗ = In . This together with (4.37) completes the proof for .det 2 . The proof ∎ for .det is similar. Now we have: Lemma 4.9 For . a ∈ A L , .g ∈ ┌n assume that . N , . N ' ∈ Z+ satisfy (4.35) and the two operators { T (a) : H+,N → H+,N . T (a) : H+,N ' → H+,N ' are bijective. Then the determinants and the modified determinants on . H+,N and H+,N ' in (4.33) and (4.34) are equal, hence .τa (g), .τa(2) (g) do not depend on . N .

.

Proof Note that for . N ' ≥ N an identity .

| g −1 Hg Sa TN ' (a)−1 | H+,N = g −1 Hg Sa TN (a)−1

holds on . H+,N . On the other hand, for .u ∈ H+ the identity .

(b − z) N u (z) = lim (b − z) N (1 − ∈z)−N u (z) ∈→0

(4.38)

62

4 KdV Flow II: Extension

implies . H+ is dense in . H+,N . Then, applying Lemma 4.9 to .

H = H+,N ' , H1 = H+,N , A = g −1 Hg Sa TN ' (a)−1 ∎

yields the lemma.

This lemma admits flexibility in choosing . N , namely . N can be arbitrary if (4.35) is satisfied for given . L, .m, .n. Therefore .τa(2) (g) can be defined for . a ∈ Ainv L , .g ∈ ┌n under the condition (4.36). However, it should be noted that for any rational function .r ∈ ┌0 the operator . Hr : H− → H+ is of finite rank as we have noticed in (3.19). Namely, suppose .r has only simple poles in . D− . Then the expansion r (z) = r0 +

p ∑

.

)−1 ( r j qζ j (z) , (qζ (z) = 1 − ζ −1 z )

j=1

yields .

Hr v (z) =

p ∑

( ) r j v ζ j qζ j (z) ,

(4.39)

j=1

hence .dim Hr (H− ) ≤ p, which shows .r −1 T (r a) T (a)−1 − I = r −1 Hr Sa T (a)−1 is of finite rank on any space . H+,N with . N such that .− deg r ≤ N ≤ L. Therefore, one has: Lemma 4.10 .τa (r ) can be defined for any .r ∈ ┌0 , . a ∈ Ainv L if . L ≥ − deg r , and .τ a (r ) is given by .

( ) ( )) ( with f i = r −1 qζi . det δi j + r j Sa T (a)−1 f i ζ j

4.3.1 Properties of Tau-Function The tau-function is a key tool to study the KdV flow, and in this section we give fundamental properties for the tau-function. Note . a ∈ A L , g ∈ ┌n =⇒ ga ∈ A L , since .g is holomorphic and bounded on . D+ . Assume further .

hj a ∈ Ainv ∈ ┌n with deg r j = m j for j = 1, 2. L , gj = rje

4.3 Tau-Function

63

We consider three tau-functions.τa(2) (g1 g2 ),.τa(2) (g1 ),.τg(2) 1 a (g2 ) simultaneously, which is possible if . L ∈ Z+ satisfies . L > max {n, −m 1 − m 2 } and .g1 a ∈ Ainv L . For simplicity of notation set .

)( )) (( (g1 g2 )−1 T (g2 g1 a) T (g1 a)−1 g1 − I g1−1 T (g1 a)T (a)−1 − I ( ) = tr (g1 g2 )−1 Hg2 Sg1 a T (g1 a)−1 Hg1 Sa T (a)−1 , (4.40)

E a (g1 , g2 ) = tr

which is well-defined under . L > max {n, −m 1 − m 2 }. Lemma 3.4 but the continuity is rewritten as follows. Lemma 4.11 For . a ∈ Ainv L we have the following. (i) For .g = r eh ∈ ┌n let . N be such that .

max {n, − deg r } < N ≤ L.

Then the map .T (ga) is bijective from . H+,N to . H+,N +deg r if and only if .τa(2) (g) /= 0. (ii) Let .g j = r j eh j ∈ ┌n for . j = 1, .2 and assume .

max {n, − deg r1 − deg r2 } < N ≤ L.

If .τa(2) (g1 ) /= 0, then it holds that τ (2) (g1 g2 ) = τa(2) (g1 ) τg(2) (g2 ) exp (−E a (g1 , g2 )) . 1a

. a

(4.41)

Additionally, if .r1 , .r2 ∈ ┌0 (rational functions) and .τa (r1 ) /= 0, then τ (r1r2 ) = τa (r1 ) τr1 a (r2 ) .

. a

(4.42)

(iii) Moreover suppose .g1 satisfies .g1 (z) = g1 (−z). Then, it holds that .T (g2 g1 a) = T (g2 a) g1 and (2) (2) .τg a (g2 ) = τ a (g2 ) . 1 Similarly, for rational functions .r1 ∈ ┌0 , .r2 ∈ ┌0 satisfying .r1 (z) = r1 (−z) we have τ

. r1 a

(r2 ) = τa (r2 ) , τa (r1r2 ) = τa (r1 ) τa (r2 ) .

Proof The identity (4.20) implies g −1 T (ga) T (a)−1 = I + g −1 Hg Sa T (a)−1

.

and .g −1 Hg Sa T (a)−1 is of HS class on . H+,N under the condition .max {n, − deg r } < N ≤ L due to Lemma 4.6. Then general theory of Fredholm determinant shows that the operator .g −1 T (ga) T (a)−1 is bijective on . H+,N if and only if .

( ) det 2 g −1 T (ga) T (a)−1 /= 0,

64

4 KdV Flow II: Extension

which implies the bijectivity of .T (ga) from . H+,N to . H+,N +deg r , which is (i). Recall ( ) −1 (2) .τ a (g1 g2 ) = det 2 (g1 g2 ) T (g1 g2 a) T (a)−1 , and notice (g1 g2 )−1 T (g1 g2 a) T (a)−1 ( ) )( ) ( = g1−1 g2−1 T (g2 g1 a) T (g1 a)−1 g1 g1−1 T (g1 a)T (a)−1 .

.

(4.43)

Generally, identities { .

( ) det 2 G −1 (I + A) G = det 2 (I + A) det 2 ((I + A) (I + B)) = det 2 (I + A) det 2 (I + B) e−tr(AB)

hold for a bounded operator .G having bounded inverse and HS operators . A, . B. Then taking determinant in (4.43) yields ( ) ( ) τ (2) (g1 g2 ) = det 2 g1−1 g2−1 T (g1 g2 a) T (g2 a)−1 g1 det 2 g1−1 T (g1 a)T (a)−1 )) ( ( × exp −tr g1−1 g2−1 Hg2 Sg1 a T (g1 a)−1 g1 g1−1 Hg1 Sa T (a)−1

. a

= τa(2) (g1 ) τg(2) (g2 ) exp (−E a (g1 , g2 )) 1a τ (r ) can be defined for .r ∈ ┌0 , hence taking the determinant in (4.43) we easily have (4.42). Suppose .g1 (z) = g1 (−z). For .u ∈ H+,N , . a = (a1 , a2 ) we have

. a

.

T (g2 g1 a) u = p+ (g1 g2 a1 ) u e + p+ (g1 g2 a2 ) u o = p+ (g2 a1 ) g1 u e + p+ (g2 a2 ) g1 u o = p+ (g2 a1 ) (g1 u)e + p+ (g2 a2 ) (g1 u)o = T (g2 a) (g1 u) ,

which implies .T (g2 g1 a) = T (g2 a) g1 on . H+,N , hence .

T (g2 g1 a) T (g1 a)−1 = T (g2 a) g1 g1−1 T (a)−1 = T (g2 a) T (a)−1

is valid. This shows (2)

.τg a 1

( ) ( ) (2) (g2 ) = det 2 g2−1 T (g2 g1 a) T (g1 a)−1 = det 2 g2−1 T (g2 a) T (a)−1 = τa (g2 ) .

∎ .

Lemmas 3.5 and 3.6 hold in this case as well under an appropriate condition on a. We state (ii) of Lemma 3.5 only for . p = 2.

4.3 Tau-Function

65

Lemma 4.12 Suppose . a ∈ Ainv 2 . Then the following hold: (i) For .ζ, .ζ1 , .ζ2 ∈ D− ⎧ m a (ζ) − m a (z) ⎪ ⎪ ⎨ ϕqζ a + 1 = (ϕa + 1) ζ−z . . ζ 2 − z2 ⎪ ⎪ − m a (ζ) ⎩ m qζ a (z) = m a (ζ) − m a (z) (ii) One has ⎧ ( ) ⎨ τa qζ = ϕa (ζ) + 1 ( ) m a (ζ1 ) − m a (ζ2 ) . . ⎩ τa qζ1 qζ2 = (ϕa (ζ1 ) + 1) (ϕa (ζ2 ) + 1) ζ1 − ζ2

(4.44)

(4.45)

{ } (iii) For a rational function .r ∈ ┌0(0) assume the zeros . η j 1≤ j≤ p ⊂ D− and the poles { } . ζj ⊂ D− are simple. Then the formula (3.22) is valid as it is. 1≤ j≤ p Proof The proof is simply a repetition of that of Lemma 3.5, so we omit it.

∎

4.3.2 Continuity of Tau-Functions Later we approximate .g ∈ ┌n by rational functions and the continuity of the taufunctions is required. Set d (g1 , g2 ) ( )1/2 | { | | g1 (z)−1 g1 (λ) − g2 (z)−1 g2 (λ) |2 −2N | |z| | = |dz| |dλ| | | 2 z−λ

. N

(4.46)

C

for .g1 , g2 ∈ ┌n(0) . Then we have: h1 h2 Lemma 4.13 Let. a ∈ Ainv L and .g1 = r 1 e ( , .g2 = ) r2 e ∈ ┌n . Assume. L satisfies. L ≥ N > max {n, − deg r1 , − deg r2 } and .d N g j , 1 ≤ c1 for . j = 1, .2. Then there exists a constant .ca depending on .c1 , . a, . N such that .

| | |τ (2) (g1 ) − τ (2) (g2 )| ≤ ca d N (g1 , g2 ) . a

(4.47)

a

(

) −1

Proof Recall the definition .τa(2) (g) = det 2 I + g −1 Hg Sa T (a) for .g ∈ ┌n . The −1 −1 HS-norm of .g Hg Sa T (a) is dominated by .d N (g, 1) due to . H1 = 0, hence if (2) .d N (g, 1) < ∞, then .τ a (g) is defined finitely. Generally, if .||A||HS , .||B||HS ≤ c1 there exists a constant .c2 depending only on .c1 such that

66

4 KdV Flow II: Extension .

|det 2 (I + A) − det 2 (I + B)| ≤ c2 ||A − B||HS .

Therefore.τa(2) (g1 ) − τa(2) (g2 ) can be estimated by those of the HS-norms of.g1−1 Hg1 − g2−1 Hg2 on the space . H+,N , which is just denoted by .d N (g1 , g2 ) due to (4.19). ∎ For later purposes we give a sufficient condition for the convergence of .τa(2) (gk ). Lemma 4.14 Assume the following properties for .gk , .g ∈ ┌n(0) : ⎧ ⎨ (i) there exist c1 , c2 > 0 such | that|for z ∈ C . c1 ≤ |gk (z)| ≤ c2 , |gk' (z)| ≤ c2 |z|n−1 . ⎩ (ii) gk (z) → g (z) as k → ∞ for any z ∈ C

(4.48)

Then, for .r ∈ ┌0 and . a ∈ A L with .

it holds that

L > max {n, − deg r } ,

τ (2) (r gk ) → τa(2) (r g) .

. a

Proof Choose an integer . N ≥ 0 such that . L ≥ N > max {n, − deg r }. Set Δk (z, λ) =

.

r (z)−1 g (z)−1 r (λ) g (λ) − r (z)−1 gk (z)−1 r (λ) gk (λ) . z−λ {

Since d (r gk , r g)2 =

. N

C2

|Δk (z, λ)|2 |z|−2N |dz| |dλ| ,

To have.d N (r gk , r g) → 0 as.k → ∞ it is sufficient to show that there exists a function f integrable with respect to .|z|−2N |dz| |dλ| such that

.

.

|Δk (z, λ)|2 ≤ f (z, λ) .

Note r (λ) g (λ) − r (z) g (z) z−λ −1 −1 r (λ) gk (λ) − r (z) gk (z) . − r (z) gk (z) z−λ

Δk (z, λ) = r (z)−1 g (z)−1

.

Then (4.48) implies | ⎧| | r (z) g (z) − r (λ) g (λ) | ⎪ | | ⎪ { m+n−1 ⎨| | |z| ( z − λ | | ) if |z − λ| ≤ ∈ |λ| . | r (z) gk (z) − r (λ) gk (λ) | ≤ c1 |λ|−1 |z|m + |λ|m if |z − λ| > ∈ |λ| . ⎪ | ⎪ ⎩ || | z−λ

4.4 Non-negativity Condition of Ainv L

67

Set ( ( ) ) f (z, λ) = |z|−m |z|m+n−1 I|z−λ|≤∈|λ| + |λ|−1 |z|m + |λ|m I|z−λ|>∈|λ| .

. 1

| | Since .|r (z)−1 gk (z)−1 | ≤ c2 |z|−m , it is sufficient to show the integrability of 2 . f 1 (z, λ) . The proof proceeds just as that of Lemma 4.6. The exponent of the first term is .2 (n − 1) − 2N + 1, which is less than .−1 if . N > n. The integral of the second term is dominated by { .

C2

( ) |λ|−2 |z|−2m |z|2m + |λ|2m |z|−2N |dz| |dλ| ,

which is finite if .

− 2N < −1, − 2m − 2N < −1 =⇒ N > 1/2, 1/2 − m. ∎

4.4 Non-negativity Condition of .Ainv L Generally, potentials arising from . a ∈ Ainv L are complex valued, so to obtain real potentials some sort of realness for . a and .g is required. . a ∈ A L , .g ∈ ┌n are called real if they satisfy .

( ) a (λ) = a λ for λ ∈ C, g(z) = g(z) for z ∈ C.

(4.49)

If . a and .g are real in this sense, then clearly we have { .

ϕa (z) = ϕa (z), ψa (z) = ψa (z), m a (z) = m a (z) , τa (g), τa(2) (g) ∈ R

and the associated potential takes real values. Define a subclass of .Ainv L : { } inv (2) Ainv L ,+ = a ∈ A L ; τ a (r ) ≥ 0 for any real rational r with deg r = 0 { } = a ∈ Ainv L ; τ a (r ) ≥ 0 for any real rational r with deg r = 0 .

.

(4.50)

For a bounded curve .C we have defined .Ainv + for any real rational .r without the restriction.deg r = 0 (see (3.25)). However, if we remove this restriction in the present case, then the condition . L > max {n, − deg r } requires . L to be .∞, which is not realistic. The second identity follows from

68

4 KdV Flow II: Extension

)) ( ( τ (2) (r ) = τa (r ) exp −tr r −1 T (r a) T (a)−1 − I .

. a

τ (r ) is well-defined for any rational .r with .deg r = 0 since the relevant operator is of finite rank. Our strategy to show .τa(2) (g) > 0 for real .g ∈ ┌n(0) is as follows: Note (0) .r ∈ ┌0 is equivalent to the condition that .r is rational with .deg r = 0 (see (4.13)). . a

(i) Show .τa(2) (r ) > 0 for any real .r ∈ ┌0(0) and . a ∈ Ainv L ,+ . (0) (ii) Approximate a general real .g ∈ ┌n by a sequence of real .rk ∈ ┌0(0) . ( ) (iii) Use the continuity of .τa(2) (·) to have .τa(2) grk−1 > 0 for sufficiently large .k and show .grk−1 a ∈ Ainv L ,+ . (iv) Apply the cocycle property of .τa(2) (·) to have .τa(2) (g) > 0, namely ( ) ( ) (2) ( −1 )) ( > 0. τ (2) (g) = τa(2) grk−1rk = τa(2) grk−1 τgr −1 (r k ) exp −E a gr k , r k a

. a

k

This programme will be realized in the next section. In the process of the proofs we use several times the following lemma, which was already applied for a bounded curve in Lemma 3.7. Lemma 4.15 Let . a ∈ Ainv L ,+ . Then one has: (i) For a real .r ∈ ┌0(0) assume .τa (r ) > 0. Then .r a ∈ Ainv L ,+ holds.. (0) (0) (ii) For a real .r ∈ ┌0 assume there exists .rk ∈ ┌0 such that ⎧ ⎨ rk (z) → r (z) for any z ∈ C c ≤ |rk (z)| ≤ c−1 for some c > 0 . . ⎩ τa (rk ) > 0 for any a ∈ Ainv L ,+ Then .τa(2) (r ) > 0 is valid. Proof (i) follows from the cocycle property, namely τ (r1 ) =

. ra

τa (rr1 ) ≥ 0 for any real r1 ∈ ┌0(0) . τa (r )

Under the condition of (ii) { { ( ) 1 rk (λ) r (λ)−1 v (λ) v (λ) 1 dλ → dλ . Hrr −1 v (z) = k 2πi C z−λ 2πi C z − λ for any .z ∈ (D− , .v) ∈ H− holds and the numbers of the poles ( of).rk converge to that of r , hence.τa rrk−1 → τa (1) = 1. Fix.k ≥ 1 such that.τa rrk−1 > 0. Then (i) implies −1 inv .rr k a ∈ A L ,+ , hence .

( ) ( ) τ (r ) = τa rrk−1rk = τa rrk−1 τrrk−1 a (rk ) > 0.

. a

∎

4.4 Non-negativity Condition of Ainv L

69

This lemma will be proved to be valid for general.g ∈ Ainv L ,+ later by approximating (0) .g ∈ ┌n by rational functions.

4.4.1 Non-degeneracy of Tau-Functions for . a ∈ Ainv L,+ To investigate properties of .m a and .τa(2) for . a ∈ Ainv L ,+ we prepare several lemmas. In this section the curve .C is parametrized by C = {±ω (y) + i y; y ∈ R}

.

(

)

with a smooth function .ω (y) > 0 satisfying .ω (y) = ω (−y) and .ω (y) = O y −(n−1) . We would like to show a lemma corresponding to Lemma 3.7. In what follows .τa (r ) for .r ∈ ┌0(0) will be used instead of .τa(2) (r ). We have to assume . a ∈ Ainv L ,+ with . L ≥ 2 since in the proof we use .ϕ a , .m a . Lemma 4.16 Let . a ∈ Ainv L ,+ with . L ≥ 2. (i) .m a satisfies .

Im m a (z) / Im z > 0

and .1 + ϕa has no zeros in . D− . (ii) .τa (r ) > 0 holds for any real .r ∈ ┌0(0) , which in particular means .r a ∈ Ainv L ,+ . Proof 1.◦ ) First we show .1 + ϕa (s) ≥ 0, .(1 + ϕa (s))2 m 'a (s) ≥ 0 for any .s ∈ D− ∩ R. For .s, .t ∈ D− ∩ R (4.45) implies τ

. a

( −1 ) ( ) τa (qs qt ) qs q−t = τa qs qt (qt q−t )−1 = τa (qt q−t ) (1 + ϕa (s)) (1 + ϕa (t)) (m a (s) − m a (t)) = Δa (t) (s − t)

−1 since.qt q−t is even. Noting.Δa (t) > 0 and.qs q−t ∈ ┌0(0) is real, one has.(1 + ϕa (s))2 ' m a (s) ≥ 0 by setting .t = s. Letting .t → ∞ yields

1 + ϕa (s) = lim (1 + ϕa (s)) (1 + ϕa (−t))

.

t→∞

m a (s) − m a (−t) s2 − t 2

= lim τa (qs qt ) ≥ 0, t→∞

due to (4.45). Then the argument of Lemma 3.7 works and we have .1 + ϕa (s) > 0 and .m a (s) ( has )no poles on . D− ∩ R. 2.◦ ) .τa qζ qζ ≥ 0 holds for .ζ ∈ D− . −1 −1 Applying (3.22) to .r = qζ qζ q−η q−η with .ζ, .η ∈ D− one has

70

4 KdV Flow II: Extension

τ

. a

τ

. a

( ) −1 −1 qζ qζ q−η q−η =

| |2 | | |η + ζ|2 |η + ζ | |ϕa (η) + 1|2 |ϕa (ζ) + 1|2

(4 Im η Im ζ) |Δa (η)|2 ⎛| ⎞ | | m (η) − m (ζ) |2 || m (η) − m (ζ) ||2 a a a | a | | ⎠. × ⎝| | − || | 2 2 − ζ2 | | η 2 η −ζ

(4.51)

( ) (0) qζ qζ pη pη ≥ 0 holds if . a ∈ Ainv L ,+ , since .qζ qζ pη pη ∈ ┌0 is real. Since ) ) ( ( ϕa (η) = κ1 (a) η −1 + O η −3/2 , m a (η) = η + O η −1/2

.

due to (4.8) if .η ∈ iR, one has .

|ϕa (ζ) + 1|2

( ) Im m a (ζ) −1 −1 q−η ≥ 0, = lim τa qζ qζ q−η η→∞ Im ζ

which implies .Im m a (ζ) / Im ζ ≥ 0. Then, again the argument of Lemma 3.7 is effective and we have .1 + ϕa (z) /= 0 and .m a (z) has no poles on . D− \R. 1.◦ ) and 2.◦ ) prove the statement (i). Generally, a real .r ∈ ┌0(0) is a product of ⎧ ( )−1 ⎪ with η, ζ ∈ D− \R (1) qζ qζ q−η q−η ⎪ ⎪ ⎨ −1 (2) qs q−t with s, t ∈ D− ∩ R . . (3) qζ qζ (q−s q−t )−1 with ζ ∈ D− \R, s, t ∈ D− ∩ R ⎪ ⎪ ⎪ ( ) −1 ⎩ with η ∈ D− \R, s, t ∈ D− ∩ R (4) qs qt q−η q−η

(4.52)

Therefore, if .τa (r ) > 0 is proved for any .r of these 4 cases and for any . a ∈ Ainv L ,+ , (0) we have .τa (r ) > 0 for any real .r ∈ ┌0 owing to (i) of Lemma 4.15. ( )−1 We begin from the case (1) and let .r = qζ qζ q−η q−η . Then (4.51) implies | |2 | | | m a (η) − m a (ζ) |2 || m a (η) − m a (ζ) || | ≤| .| | | | 2 | | η2 − ζ 2 η2 − ζ

(4.53)

due to (i) of this lemma. Owing to the symmetry of .r with respect to .ζ, .η, one ( ( )−1 ) can assume .Im ζ > 0, .Im η > 0. Assume .τa qζ0 qζ0 q−η0 q−η0 = 0 for some .η0 , .ζ0 ∈ D− and set 2 m a (z) − m a (ζ0 ) z 2 − ζ0 . f (z) = . m a (z) − m a (ζ0 ) z 2 − ζ02 From (4.53) one has .

| f (z)| ≤ 1 for any z ∈ D− ∩ C+ ,

(4.54)

4.4 Non-negativity Condition of Ainv L

71

and the assumption implies the equality at .z = η0 in (4.54), which concludes . f (z) = eiα with .α ∈ R identically on . D− ∩ C+ . Then ( ) 2 m a (ζ0 ) − eiα m a (ζ0 ) z 2 + eiα m a (ζ0 )ζ02 − m a (ζ0 ) ζ0 .m a (z) = ( ) 2 1 − eiα z 2 + eiα ζ02 − ζ0 holds, which contradicts .m a (z) = z + o (1) as .z → ∞. Therefore we have .| f (z)| < 1 always, which is nothing but .τa (r ) > 0. The case (2) is already proved in the argument of (i) if .s /= t due to the analyticity of .m a (z). The case .t = s is proved as a limiting case of .s /= t by applying (ii) of Lemma 4.15. ( ) Similarly, one can show.τa qζ qζ (q−s q−s )−1 > 0 as a limiting case of (1). Setting −1 .r 1 = qζ qζ (q−s q−s ) we have τ

. a

( ) ( ) ( ) −1 −1 qζ qζ (q−s q−t )−1 = τa r1 q−s q−t = τa (r1 ) τr1 a q−s q−t > 0,

which shows the case (3). The case (4) can be shown similarly. The next task is to approximate general .g ∈

∎

┌n(0) by real rational functions of .┌0(0) .

Lemma 4.17 Let .g ∈ ┌n(0) . Then there exists a sequence of rational functions (0) .{r k }k≥1 ⊂ ┌0 such that .r k → g in the sense of (4.48). Proof Let .U be a neighborhood of . D + whose boundary is described by an equation |x| = c |y|−(n−1) for large .|y| with sufficiently large .c > 0. For integer .k ≥ 1 let

.

⎛

z ⎞k ⎜ 2k ⎟ . .φk (z) = ⎝ z ⎠ 1− 2k 1+

Note .limk→∞ φk (z) = e z . For a positive constant .a ≤ k an inequality e−2a ≤ |φk (z)| ≤ e2a

.

if |Re z| ≤ a

holds. If .h(z) = c1 z n + lower degree terms, then c ≡ sup |Re h (z)| < ∞

. 2

z∈U

is valid. Define real rational functions by r (z) = φk (h (z)) .

. k

The zeros and poles of .rk (z) are determined by the equation .h (z) = ±2k. If .a is chosen so that .a > c2 , then clearly there exist a constant .c3 > 1 such that

72

4 KdV Flow II: Extension

c−1 ≤ |rk (z)| ≤ c3 for z ∈ U and k ≥ 1

. 3

holds. Moreover, | | h (z) |k−1 | | | | ' | | ' || ' | | ' | || 1 + 2k || | h (z) ||−2 | | | | | | | | | 1− . r k (z) = h (z) φk (h (z)) = h (z) | | h (z) | | | 2k | | |1 − 2k shows .

| ' | |r (z)| ≤ c4 |z|n−1 for z ∈ U and k ≥ 1. k

Since .limk→∞ rk (z) = eh(z) = g(z), all the conditions of Lemma 4.14 are satisfied. ∎ Now we have Proposition 4.3 Let .g ∈ ┌n(0) be real and . a ∈ Ainv L ,+ with . L ≥ max {n + 1, 2}. Then, (2) inv .τ a (g) > 0 holds, hence .ga ∈ A L ,+ is valid. Proof Let .{rk }k≥1 be the sequence of Lemma 4.17 approximating .g and .r be any real rational function of.┌0(0) . First note if.τa(2) (g) > 0 for a real.g ∈ ┌n(0) , then.ga ∈ Ainv L ,+ . This is because (2) (2) .τ a (gr ) = lim τ a (r k r ) ≥ 0 k→∞

and τ (2) (r ) =

. ga

τa(2) (gr ) τa(2) (g)

exp (E a (g, r )) ≥ 0.

{ } Since . gk = grk−1 k≥1 also satisfies the conditions of (4.48) with .g = 1, Lemma 4.14 shows ( −1 ) (2) grk = τa(2) (1) = 1. . lim τ a k→∞

( ) Fix a sufficiently large .k ≥ 1 such that .τa(2) grk−1 > 0. Then applying the above −1 inv remark to.grk−1 one has.grk−1 a ∈ Ainv L ,+ . Applying (ii) of Lemma 4.16 to.gr k a ∈ A L ,+ (2) and the rational function .rk we have .τgr −1 a (rk ) > 0. The cocycle property of tauk functions implies ( ) ( ) (2) ( −1 )) ( > 0. τ (2) (g) = τa(2) grk−1rk = τa(2) grk−1 τgr −1 (r k ) exp −E a gr k , r k a

. a

If .g = r eh with real .r ∈ ┌0(0) , then

k

4.4 Non-negativity Condition of Ainv L

73

( h) ( ( h )) > 0, τ (2) (g) = τa(2) (r ) τr(2) a e exp −E a r, e

. a

∎

which completes the proof.

4.4.2 .m-function and WTK Function If the curve .C is bounded, Proposition 3.1 states that the .m-function .m a can be identified with the WTK function for the associated Schrödinger operator if. a ∈ Ainv + . For an unbounded curve .C also the exactly same statement is possible if . a ∈ Ainv L ,+ with . L ≥ 3. We consider the relationship between .m a (z) and .m ± (z). Equation (4.8) allows us to define the coefficient .κ1 (a) by ) ( ) ( ϕa (z) = aT (a)−1 1 (z) − 1 = κ1 (a) z −1 + O z −2

.

xz for . a ∈ Ainv ∈ ┌1 . L ,+ with . L ≥ 2. Let .ex (z) = e

Proposition 4.4 Let . a ∈ Ainv L ,+ with . L ≥ 3 and define the potential q (x) = −2∂x κ1 (ex a) .

.

Then the boundaries .±∞ are of limit point type for the Schrödinger operator . L q . The .m-function .m a and the Weyl functions .m ± are connected by { m a (z) =

.

( ) −m +( −z 2) if Re z > 0 . m − −z 2 if Re z < 0

(4.55)

The Baker–Akhiezer function is given by ) ( f (x, z) = aT (ex a)−1 1 (z) .

. a

This proposition says that for . a ∈ Ainv L ,+ its .m-function .m a (z) is analytically continuable up to .C\ ([−μ0 , μ0 ] ∪ iR) (.(−μ0 , μ0 ) = D+ ∩ R) although originally we knew its analyticity only on . D− . The next issue is to show the converse statement of Proposition 4.4. This proposition and Lemma 4.16 implies that .m = m a for . a ∈ Ainv L ,+ satisfies ⎧ Im m (z) ⎪ ⎨ >0 on C\ (R ∪ iR) Im z . . m(x) − m(−x) ⎪ ⎩ > 0 if x ∈ R and |x| > μ0 x

(4.56)

It should be remarked that the analyticity of .m on . D− implies .1 + ϕa (z) /= 0 on D− since .1 + ϕa (z) and .z + ψa (z) do not vanish simultaneously due to .Δa (z) /= 0. Recall the operation .dζ :

.

74

4 KdV Flow II: Extension

.

( ) dζ f (z) =

z2 − ζ 2 − f (ζ) f (z) − f (ζ)

{ } for a function . f on .C as long as they have meaning. Then . dζ ζ∈D− is commutative and .dζ d−ζ = id. Proposition 4.5 Let . L ≥ 2. For . a ∈ Ainv L suppose that .m a is holomorphic on inv .C\ ([−μ0 , μ0 ] ∪ iR) and satisfies (4.56). Then, . a ∈ A L ,+ holds. Proof The proof resembles that of Lemma 4.16. We have to show .τa (r ) ≥ 0 for any real rational function .r ∈ ┌0(0) . Clearly it is sufficient for this to prove .τa (r ) > 0 for any real rational function .r ∈ ┌0(0) with simple zeros and poles in . D− . Since such .r is a product of the 4 types of rational functions of (4.52), first we prove .τa (r ) > 0 for .r of (4.52). If .r is of the type (1) with .η, .ζ ∈ D− ∩ C+ , then .τa (r ) is given by (4.51) and .τa (r ) > 0 is equivalent to .| f (z)| < 1 on . D− ∩ C+ with 2

.

Set .w = ζ 2 . Then

f (z) =

m a (z) − m a (ζ) z 2 − ζ . m a (z) − m a (ζ) z 2 − ζ 2

(√ ) (√ ) z − ma w z − w . f ( z) = (√ ) (√ ) ma z − ma w z − w √

ma

(√ ) (√ ) holds. Since m a ) z > 0, The Schwarz lemma | √.m a| z is analytic on .C+ and .Im(√ z is implies .| f ( z)| < 1 for .z, .ζ ∈ C+ , unless .m a ma

.

(√ ) az + b z = cz + d

with( some constants .a, .b, .c, .d satisfying .ad − bc /= 0, which is impossible since √ ) √ ma z = z + o (1) as .z → ∞. Therefore we have .| f (z)| < 1 if .Re z, .Im z > 0, .Re ζ, .Im ζ > 0. On the other hand, when . z ∈ C− we use the identity .

( √ ) (√ ) ma − z − ma w z − w √ . . f (− z) = ( √ ) (√ ) ma − z − ma w z − w If .Re ζ > 0, .Im ζ > 0, then .Im w > 0 holds and it implies | | |z − w| | | . |z − w| < 1 and .

( √ ) (√ ) Im m a − z > 0, Im m a w > 0

4.4 Non-negativity Condition of Ainv L

due to .Im

√

z < 0, .Im

√

75

w > 0, hence

| | | | | m (−√z ) − m (√w ) z − w | | m (−√z ) − m (√w ) | a a | | a | a | .| | 0, .Re ζ, .Im ζ > 0. The rest of the cases can be proved by the symmetry and we have .τa (r ) > 0. −1 with .s /= t one has For the type (2) .r = qs q−t τ (r ) =

. a

Recall .

(1 + ϕa (s)) (1 + ϕa (−t)) m a (s) − m a (t) . Δa (t) s−t

(1 + ϕa (s)) (1 + ϕa (−t)) > 0 if |s| , |t| > μ0 Δa (t)

is valid due to the analyticity of .m a on . D− . On the other hand, the property Im m a (z) / Im z > 0 implies

.

m a (t + i∈) − m a (t − i∈) Im m a (t + i∈) ≥ 0, = lim ∈→0 ∈→0 Im (t + i∈) 2i∈

m 'a (t) = lim

.

which shows .

m a (s) − m a (t) > 0 if s, t ∈ (−∞, −μ0 ) or (μ0 , ∞) . s−t

If .s ∈ (−∞, −μ0 ) and ., t ∈ (μ0 , ∞), then from .(m (x) − m (−x)) /x > 0 an inequality .m a (s) − m a (t) < 0 follows, hence we have .τa (r ) > 0 for the case (2). If .r = qζ qζ (q−s q−t )−1 , the cocycle property implies ( ) ( ) ( ) −1 −1 τqζ q −1 a q−ζ qζ (q−s q−t )−1 . τ (r ) = τa qζ q−ζ q−ζ qζ (q−s q−t )−1 = τa qζ q−ζ

. a

−ζ

−1 Since the .m-function for .qζ q−ζ a is .dζ dζ m a , which satisfies (4.56) due to Lemma 7.3, we have .τqζ q −1 a ( ps pt ) /= 0 in view of the last argument, and −ζ

τ

. q q −1 a ζ −ζ

( ) ( ) q−ζ qζ ps pt = τqζ q −1 a q−ζ qζ τqζ q −1 a ( ps pt ) −ζ −ζ ( ) = Δqζ q −1 a ζ τqζ q −1 a ( ps pt ) /= 0 −ζ

−ζ

76

4 KdV Flow II: Extension

is valid. Therefore we have .τa (r ) > 0. The case (4) .r = qs qt pη pη can be treated similarly, hence .τa (r ) > 0 for .r of any type of (4.52). Suppose .r1 , .r2 be one of the type of (4.52). We have proved .τa (r1 ) > 0. On the other hand, the .m-function .m r1 a satisfies (4.56), since .m r1 a is obtained by repeating the operation .dζ dζ , .dt to .m a . Then .τr1 a (r2 ) > 0 follows from the above argument. Therefore .τ a (r 1 r 2 ) = τ a (r 1 ) τr1 a (r 2 ) > 0 is valid. Continuing this argument one has .τa (r ) > 0 for any real .r ∈ ┌0(0) .

∎

It is certainly better to prove .τa (r ) > 0 directly by showing ( ( )) m a (ζi ) − m a −η j . det /= 0 (see Lemma 3.5) ζi2 − η 2j for .m a satisfying (4.56), but the author has no such proof. However, it should be 4.17, noted that if a real rational function .r satisfies .r (z) r (−z) = 1 like .rk( in Lemma ) then .η j = −ζ j holds. If .Re ζ j > 0 for any . j, then .m a (ζi ) = −m + −ζi2 , hence the present determinant becomes ( )⎞ ⎛ m + (ξi ) − m + ξ j n ⎠ with ξi = −ζi2 . . (−1) det ⎝ ξi − ξ j Since the representation theorem for HN functions implies ( ) { ∞ m + (ξi ) − m + ξ j σ+ (dλ) ), ( . = ξi − ξ j −∞ (λ − ξi ) λ − ξ j the determinant is a Gram determinant, which is positive. The inverse spectral theory tells us that a potential can be determined uniquely by its WTK function, hence the .m-function .m a of . a ∈ Ainv L ,+ determines the potential .q(x) = −2∂ x κ1 (e x a). Therefore it is significant to obtain a condition for a given function .m to be an .m-function for an . a ∈ Ainv L ,+ . For this purpose we define a class (n) of functions .m. Let .M L be the set of all functions .m satisfying (M.1), (M.2) as follows: (M.1) .m is holomorphic on .C\ ([−μ0 , μ0 ] ∪ iR) and satisfies ⎧ Im m (z) ⎪ ⎨ > 0 and m(z) = m(z) on C\ (R ∪ iR) Im z . . (4.57) m(x) − m(−x) ⎪ ⎩ >0 if x ∈ R and |x| > μ0 x ) ( (M.2) .m has an asymptotic behavior on some .C satisfying .ω (y) = O y −(n−1) in (4.14):

4.4 Non-negativity Condition of Ainv L

m (z) = z +

∑

.

77

) ( m k z −k + O z −L+1 on D− .

(4.58)

1≤k≤L−2

Lemma 4.16 and Proposition 4.4 imply the .m a for . a ∈ Ainv L ,+ should satisfy (4.57). (M.2) comes from Lemma 4.4. Set .m (z) = (1, m (z) /z). One easily see: (n) Proposition 4.6 .m ∈ Ainv L ,+ holds for .m ∈ M L . The arising .m gm for any real .g ∈ ┌n(0) does not depend on .C, namely .m gm remains the same for another curve .C ' as long as .m has the asymptotic behavior on .C ' .

Proof Since .m m = m and .m ∈ B L due to Lemma 4.4, Proposition 4.5 implies .m ∈ Ainv L ,+ . Since Lemma 3.1 remains valid also on an unbounded curve, we have the second statement. ∎ The condition (M.1) is necessary for .m to be the .m-function for an . a ∈ Ainv L ,+ , but (M.2) is not, although it is not far from being necessary. This is because m a = (z + ψa ) / (1 + ϕa ) + κ1 (a)

.

with .ϕa , .ψa ∈ H− satisfying ⎧ L−1 ∑ ⎪ ⎪ ⎪ ⎪ ϕ κ j (a) z − j + φa (z) z −L+1 = (z) ⎪ ⎨ a .

j=1

L−2 ∑ ⎪ ⎪ ⎪ ⎪ ψ ι j (a) z − j + χa (z) z −L+2 = (z) ⎪ ⎩ a

with φa , χa ∈ H−

(4.59)

j=1

due to (4.8). It may be interesting to see to what extent .m a for . a ∈ Ainv L ,+ (C) has inv the property (M.2). Since .A L ,+ depends on the curve .C, we denote it by .Ainv L ,+ (C) temporarily. inv Proposition 4.7 Let .C ' =σC with .σ>1. Then one has .m a ∈ M(n) L ' for . a ∈ A L ,+ (C) ' with . L = L − (n + 1) /2.

Proof .φa ∈ H− satisfies φa (z) =

.

1 2πi

{

φa (λ) dλ C z−λ

for .z ∈ D− . To have a bound for this integral we need to estimate { |dλ| .I = . 2 C |z − λ| Dividing .C = C1 + C2 by { .

C1 = {λ ∈ C; |Im (z − λ)| ≤ δ |Im z|} C2 = {λ ∈ C; |Im (z − λ)| > δ |Im z|}

78

4 KdV Flow II: Extension

one has . I = I1 + I2 , where {

{

|dλ|

I =

. 1

C1

, I2 = |z − λ|2

C2

|dλ| . |z − λ|2

δ ∈ (0, 1) is specified later. Since .C is parametrized as .ω (t) + it, we see

.

( )1/2 1 + ω ' (t)2

{ I =

. 1

C1

where

(t − Im z)2 + (ω(t) − Re z)2

dt ≤ c1 πρ (z)−1 ,

( )1/2 ρ (z) = inf |Re (z − λ)| , c1 = sup 1 + ω ' (t)2 .

.

λ∈C1

t∈R

I is estimated as

. 2

( )1/2 1 + ω ' (t)2 dt

{ I =

. 2

C2

|(ω (t) − Re z) + i (t − Im z)|2 dt ≤ c2 |Im z|−1 . 2 C2 |t − Im z|

{

≤ c1

We have to show .ρ (z) ≥ c3 |Im z|−(n−1) , if .δ is chosen suitably. Assume .Re z,.Im z > 0. Then one has C1 = {λ ∈ C; (1 − δ) Im z ≤ Im λ ≤ (1 + δ) Im z} .

.

Since .ω (t) = t −(n−1) for sufficiently large .t, an inequality .

Re λ = ω (Im λ) ≤ ω ((1 − δ) Im z)

' implies for .z ∈ D− .

|Re (z − λ)| ≥ Re z − Re λ ≥ σω (Im z) − ω ((1 − δ) Im z) ( ) = σ − (1 − δ)−(n−1) |Im z|−(n−1) .

Therefore, choosing .δ so that .σ > (1 − δ)−(n−1) we have .

I ≤ c3 |Im z|n−1 + c2 |Im z|−1

( ) with .c3 = c1 π σ − (1 − δ)−(n−1) . Then Schwarz inequality shows 1 . |φ a (z)| ≤ I 4π 2

{ |φa (λ)|2 |dλ| ≤ c |z|n−1

2

C

(4.60)

4.5 KdV Flow

79

' for .z ∈ D− for a constant .c > 0. Similarly we have .|χa (z)|2 ≤ c |z|n−1 , hence from (4.59) ⎧ ' L −1 ( ') ∑ ⎪ ⎪ ⎪ ⎪ ϕa (z) = κ j (a) z − j + O z −L ⎪ ⎨ j=1 ' . on D− ' L −2 ( ' ) ⎪ ∑ ⎪ ⎪ ⎪ ψa (z) = ι j (a) z − j + O z −L +1 ⎪ ⎩ j=1

follows with . L ' = L − (n + 1) /2, which implies m a (z) = z +

' L −2 ∑

.

( ' ) ' ck z −k + O z −L +1 on D− ,

k=1

and .m a ∈ M(n) L' .

∎

4.5 KdV Flow We are ready to construct the KdV flow in this general setting.

4.5.1 Construction of KdV Flow Since we are going to define the KdV flow by making use of the .m -functions by following the argument of Lemma 3.9, .m 'a should be represented by the tau-function. The identities in (4.44) imply for . a ∈ Ainv L ,+ with . L ≥ 2 ( ) τa qζ2 ' .m a (ζ) = ( )2 τ a qζ ( ) ) (( ) τa(2) qζ2 ( ) ( ) = (2) ( )2 exp tr qζ−2 T qζ2 a − 2qζ−1 T qζ a + T (a) T (a)−1 . τ a qζ The cocycle property shows for . a ∈ Ainv L ,+ with . L ≥ 3 ( ) (2) ) (( ) qζ2 τga ( 2 ) ( ) −2 −1 ' −1 .m ga (ζ) = exp tr q T q ga − 2q T q ga + T T (ga) (ga) ζ ( ) ζ ζ ζ 2 (2) τga qζ ( ) τa(2) gqζ2 = (2) ( )2 τa(2) (g) exp tr B1 τa gqζ

80

4 KdV Flow II: Extension

for .g ∈ ┌n(0) with .

( ) ( ) B1 = qζ−2 T gqζ2 a T (ga)−1 − 2qζ−1 T gqζ a T (ga)−1 + I (( )−1 ( )( ) ) + gqζ2 T gqζ2 a T (ga)−1 g − I g −1 T (ga) T (a)−1 − I (( ) )( ( ) ) −1 − 2 gqζ T gqζ a T (ga)−1 g − I g −1 T (ga) T (a)−1 − I = B2 − g −1 B2 g + g −1 B2 T (ga) T (a)−1 ,

where .

( ) ( ) B2 = qζ−2 T gqζ2 a T (ga)−1 − 2qζ−1 T gqζ a T (ga)−1 + I .

Then we have ( ) ( ) ( ) tr B1 = trg −1 qζ−2 T gqζ2 a − 2qζ−1 T gqζ a + T (ga) T (a)−1 .

.

Since for .v ∈ H+,2 .

( ) ( ) ( ) qζ−2 T gqζ2 a − 2qζ−1 T gqζ a + T (ga) v (z) ( )2 { g (λ) qζ (z)−1 qζ (λ) − 1 ~ a (λ) v (λ) 1 = dλ 2πi C λ−z { 1 a (λ) v (λ) (λ − z) g (λ)~ = dλ, 2πi C (ζ − λ)2

the above operator is of rank .2, and the trace turns to tr B1 =

.

1 2πi

{ C

Θ (g, λ) dλ (ζ − λ)2

with ( ( ) ) ) ( Θ (g, λ) = g (λ) λ~ a (λ) T (a)−1 g −1 (λ) − ~ a (λ) T (a)−1 g −1 z (λ) ,

.

hence

( ) ) ( { τa(2) gqζ2 Θ (g, λ) 1 ' (2) .m ga (ζ) = τ dλ exp (g) ( )2 a 2πi C (ζ − λ)2 τa(2) gqζ

(4.61)

holds. Lemma 4.18 For . L ≥ max {n + 1, 3} let . a1 , . a2 ∈ Ainv L ,+ . (i) Suppose .m a1 = m a2 . Then .m ga1 = m ga2 for any real .g ∈ ┌n(0) . (ii) The identity .∂x κ1 (ex a1 ) = ∂x κ1 (ex a2 ) holds for any .x ∈ R if and only if .m a1 = m a2 .

4.5 KdV Flow

81

Proof Propositions 4.3, 4.4, 4.5 provide the necessary ingredients. Suppose .m a1 = m a2 for . a1 , . a2 ∈ Ainv L ,+ . Then Lemma 4.12 implies ) ( ) ( −1 −1 m qζ q−η a1 (z) = dζ dη m a1 (z) = dζ dη m a2 (z) = m qζ q−η a2 (z) .

.

Repeating this operation finite times one can show .m r a1 = m r a2 for any real rational function .r ∈ ┌0(0) . For a real .g ∈ ┌n(0) we approximate it by real rational functions (0) ' .r k ∈ ┌0 , which is possible by Lemma 4.17. We show the convergence .m r a (ζ) → k ' m ga (ζ) for each fixed .ζ ∈ D− by making use of (4.61). Lemma 4.14 shows ( ) ( ) τa(2) gqζ2 τa(2) rk qζ2 . lim ( ) = (2) ( )2 . k→∞ τ (2) r q 2 τa gqζ a k ζ On the other hand, .Θ (rk , λ) → Θ (g, λ) in . L 2 (C) is valid if . L ≥ max {n + 1, 3}, hence ' ' . lim m r a (ζ) = m ga (ζ) k k→∞

follows. Consequently, one sees m 'ga1 (ζ) = lim m r' k a1 (ζ) = lim m r' k a2 (ζ) = m 'ga2 (ζ)

.

k→∞

k→∞

for .ζ ∈ D− . Since generally .m ga (ζ) = ζ + o (1), we have .m ga1 = m ga2 . (ii) is proved by the uniqueness of the correspondence between the Weyl functions .m ± and the potential .q. That is, .m a 1 = m a 2 follows from .

− 2∂x κ1 (ex a1 ) = −2∂x κ1 (ex a2 ) = q(x).

Then (i) yields .m ga1 = m ga2 , which implies again by the uniqueness .

− 2∂x κ1 (ex ga1 ) = −2∂x κ1 (ex ga2 ) .

We have used the condition . L ≥ max {n + 1, 3} to have the differentiability. Set

} { inv Q(n) L = q; q(x) = −2∂x κ1 (ex a) with real a ∈ A L ,+ ,

.

∎ (4.62)

where .n is the odd integer describing the order the curve .C approaching to .R. Then the above lemma allows us to define .

(K (g) q) (x) = −2∂x κ1 (ex ga) if q (x) = −2∂x κ1 (ex a) ∈ Q(n) L

{ } (0) (0) for any .g ∈ ┌n,real with .┌n,real = g ∈ ┌n(0) ; g = g . The flow property is immediate from the definition, and we have:

82

4 KdV Flow II: Extension

Theorem 4.1 Suppose . L ≥ max {n + 1, 3}. Then .{K (g)}g∈┌(0) defines a flow on n,real

Q(n) (x) is).C 1) in .t L . For a real odd polynomial .h of degree .n the function .(K (eth)(q) ( 3 and.C n in.x and satisfies the.(n + 1) /2th KdV equation. Especially,. K et z q (x) satisfies the KdV equation

.

∂ q(t, x) =

. t

1 3 3 ∂x q(t, x) − q(t, x)∂x q(t, x) 4 2

if .q ∈ Q(3) L for . L ≥ 4. We call the flow .{K (g)}g∈┌(0) KdV flow. Proposition 4.6 shows that .q whose n,real

(n) unified WTK function is in .M(n) L is an element of .Q L . It might be helpful to remark that one can define an equivalent flow on the space of .m-functions. Let

} { ~(n) = m; m(z) = m a (z) for a ∈ Ainv M L ,+ , L

.

(4.63)

where the number .n is the order of the curve .C approaching .R and define ~(n) . g · m a = m ga for real g ∈ ┌n(0) , m a ∈ M L

.

(0) (i) of Lemma 4.18 justifies this definition. Theorem 4.1 implies that .┌n,real acts on (n) (0) ~ if . L ≥ max {n + 1, 3}. For a real rational .r ∈ ┌ this action is given by an .M 0 L iteration of the operation .dζ m. If we define ⊓ (n) ⊓ (n) (n) ~ , ~(n) M .M∞ = ML , M ∞ = L L≥1

then Proposition 4.7 shows

L≥1

(n) ~∞ M = M(n) ∞,

.

(4.64)

) ( as long as .ω describing .C satisfies .ω (y) = O y −(n−1) , and one has .g · m ∈ M(n) ∞ for real .g ∈ ┌n(0) , .m ∈ M(n) . ∞

4.5.2 Tau-Function Representation of the Flow If the curve .C is bounded, we have proved the identity q(x) = −2∂x2 log τa (ex ) .

.

4.5 KdV Flow

83

For an unbounded curve .τa (ex ) has to be replaced by .τa(2) (ex ), however, this procedure does not provide a compact form like the above. In this section imposing an extra condition on . a we define .τa and obtain the same formula. We assume ||2 { || || z~ a (λ) || || |z|−2 |dz| |dλ| < ∞, || a (z) − λ~ . (4.65) || || 2 z − λ C which shows the operator . Sλa is of HS from . H1 (D+ ) to . H (D− ). This condition can be verified in the same manner as Lemma 4.6 for . a(z) = (1, m (z) /z) if .m satisfies (M.1), (M.2) for sufficiently large . L. Let

e (z) = e x z ∈ ┌1(0) , et,x (z) = e x z+t z ∈ ┌3(0) . 3

. x

In Proposition 4.4 for . a satisfying .ex a ∈ Ainv 3 for any . x ∈ R we have introduced the potential .q associated with . a of Schrödinger operator by .q(x) = −2∂x κ1 (ex a). The .κ1 (a) is obtained by the characteristic functions as κ (a) = lim ζϕa (ζ) .

. 1

ζ→∞

Proposition 4.8 Assume . a ∈ Ainv L ,+ and (4.65). Then an identity κ (ex a) = ∂x log τa (ex )

. 1

holds, which yields

q(x) = −2∂x2 log τa (ex ) ,

.

if . L ≥ 2. Generally for .g ∈ ┌n(0) and . L ≥ max {n + 1, 2} .

(K (g) q) (x) = −2∂x2 log τa (gex )

holds. Especially, the solution .q(t, x) to the KdV equation starting from .q(x) is given by ( ) 2 .q(t, x) = −2∂ x log τ a et,x , if . L ≥ 4. The condition . L ≥ 4 is necessary for the differentiability of .q(t, x) in .t (see Proposition 4.2). Proof The definition of .ϕa implies { ϕa (λ) 1 dλ κ (a) = lim ζϕa (ζ) = lim ζ ζ→∞ ζ→∞ 2πi C ζ − λ ) ( { ~ a (λ) T (a)−1 1 (λ) 1 = lim ζ dλ ζ→∞ 2πi C ζ −λ { ) ( 1 ~ = a (λ) T (a)−1 1 (λ) dλ, 2πi C

. 1

(4.66)

84

4 KdV Flow II: Extension

which is finite if .Ainv 2,+ . On the other hand, the formal identity ) ( ) ( τ (e∈ ) = det e∈−1 T (e∈ a) T (a)−1 = det I + e∈−1 He∈ Sa T (a)−1

. a

is justified by reproving below that .e∈−1 He∈ Sa T (a)−1 defines a trace class operator on . H1 (D+ ). For .v ∈ H1 (D+ ) it holds that S v (z) =

. a

with

1 2πi

{ C

( ) ~ a (λ) v (λ) dλ = l1 (a, v) z −1 + z −1 Sa(1) v (z) z−λ

{ ⎧ 1 ⎪ ⎪ ~ l1 (a, v) = a (λ) v (λ) dλ ⎪ ⎪ 2πi C ⎪ ⎪ ⎪ { ⎨ λ~ a (λ) v (λ) 1 Sa(1) v (z) = dλ . , ⎪ 2πi C z−λ ⎪ ⎪ { ⎪ ⎪ ⎪ 1 a (λ) − z~ a (z)) v (λ) (λ~ ⎪ ⎩ = dλ ∈ H (D− ) 2πi C z−λ

) ( a (λ) = O λ−L for . L ≥ 2. Hence for .z ∈ D+ since .~ −1 .e∈ He∈ Sa v (z)

{

e∈(λ−z) dλ + e∈−1 He∈ z −1 Sa(1) v (z) C (λ − z) λ 1 − e−∈z + e∈−1 He∈ z −1 Sa(1) v (z) = l1 (a, v) z

l1 (a, v) = 2πi

(4.67)

holds. Since . Sa(1) defines an HS operator from . H1 (D+ ) to . H (D− ) under (4.65), −1 −1 (1) .e∈ He∈ z Sa becomes a trace class operator on . H1 (D+ ), which makes it possible to define .τa (e∈ ) rigorously. Moreover, in this case for .w ∈ H (D− ) e−1 He∈ z −1 w (z) =

. ∈

1 2πi

{ ( C

) e∈(λ−z) − 1 − ∈ λ−1 w (λ) dλ λ−z

holds due to .

1 2πi

{ C

1 1 λ−1 w (λ) dλ = λ−z 2πi

{

λ−1 w (λ) dλ = 0.

C

Hence the square of the HS-norm of .∈−1 e∈−1 He∈ z −1 is δ ≡ (2π)

. ∈

−2

|2 | ∈(λ−z) | |e −1 −2 −2 | | | ∈ (λ − z) − 1| |λ| |z| |dλ| |dz| . 2 C

{

Since there exists a constant .c such that

4.5 KdV Flow

85

| |{ | | | | e∈(λ−z) − 1 | | | . | ∈ (λ − z) − 1| = |

0

1

| ( t∈(λ−z) ) | e − 1 dt || ≤ c

holds for .∈ ∈ R, .λ, .z ∈ C, the dominated convergence theorem shows .lim∈→0 δ∈ = 0. Consequently, we have .

√ || || ||e−1 He z −1 S (1) T (a)−1 || ≤ c1 ∈ δ∈ = o (∈) . ∈ a ∈ trace

(4.68)

The first term of (4.67) generates a rank .1 operator and 1 1 1 − e−∈z l1 (a,v) = l1 (a,v) = . lim ∈→0 ∈ z 2πi

{ ~ a (λ) v (λ) dλ. C

Noting an identity (see [50]) .

)) ( ( det (I + A) = exp (tr log (I + A)) = exp tr A + O || A||2trace

if .||A||trace < 1, we have ) ) ( ( τ (e∈ ) = exp tr e∈−1 He∈ Sa T (a)−1 + o (∈) ) ( { ( ) 1 ~ a (λ) T (a)−1 1 (λ) dλ + o (∈) = exp ∈ 2πi C { ( ) 1 ~ =1+∈ a (λ) T (a)−1 1 (λ) dλ + o (∈) 2πi C

. a

for sufficiently small .∈ due to (4.68). Then (4.66) implies .

lim

∈→0

τa (e∈ ) − 1 = κ1 (a) , ∈

and the cocycle property shows .

lim

∈→0

τa (ex+∈ ) − τa (ex ) τe a (e∈ ) − 1 = lim x τa (ex ) = κ1 (ex a) τa (ex ) , ∈→0 ∈ ∈

hence .∂x log τa (ex ) = κ1 (ex a) holds. To show the second identity for .g ∈ ┌n(0) the property (4.65) is required, namely ||2 || || zg (z)~ a (λ) || a (z) − λg (λ)~ || |z|−2 |dz| |dλ| < ∞. || || || 2 z − λ C

{ .

This is reduced to show | | | 1 − g (z)−1 g (λ) |2 | ||λ~ | a (λ)||2 |z|−2 |dz| |dλ| < ∞, | | 2 z − λ C

{ .

86

4 KdV Flow II: Extension

which can be verified similarly as Lemma 4.6 under the condition. L ≥ max {n + 1, 2}. Then the cocycle property shows .

(K (g) q) (x) = −2∂x2 log τga (ex ) = −2∂x2 log τa (gex )

due to Proposition 4.3 if . L ≥ max {n + 1, 2}.

∎

Chapter 5

Applications

~(n) ~(n) The definition of .Q(n) L was given in (4.62) in terms of .m-functions in .M L and .M L (n) (n) (n) ~ , hence it is meaningful to find conditions of contains .M L . .M L is close to .M L potentials .q so that their WTK functions .m ± satisfy (M.1), (M.2) (see (4.57), (4.58)). In this section we provide concrete examples of this class including the well-known decaying and periodic cases. Throughout this section we treat .g = eh with real odd polynomial .h of degree .n, hence the curve .C is taken so that .g is bounded on . D+ , namely { C=

.

} ±ω (y) + i y; y ∈ R, ω (y) > 0, (ω (y) = )ω (−y) , . ω is smooth and satisfies ω (y) = O y −(n−1) as y → ∞

5.1 Decaying Potentials For a potential .q satisfying .q(x) → 0 as .x → ±∞ there are a large number of papers on the spectral properties of . L q . If .q satisfies .(1 + |x|) q ∈ L 1 (R), then the { }n−1 scattering theorem works. The spectrum of. L q is equal to. λ j j=0 ∪ R+ and is purely absolutely continuous on .R+ . In this case it is well-known that the KdV equation is solvable by applying the inverse scattering theorem. In this section we investigate the asymptotic behavior of .m ± on .R+ for decaying potentials. The situation changes depending on .q ∈ L 1 (R) or not.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 S. Kotani, Korteweg–de Vries Flows with General Initial Conditions, Mathematical Physics Studies, https://doi.org/10.1007/978-981-99-9738-1_5

87

88

5 Applications

5.1.1 .q ∈ L 1 (R) Case If a potential .q satisfies .q ∈ L 1 (R+ ), it is known that for .0 /= k ∈ C+ ≡ {z ∈ C; . Im z ≥ 0} there exists a unique solution . f + (x, k) of .

− ∂x2 f + (x, k) + q(x) f + (x, k) = k 2 f + (x, k) {

such that .

f + (x, k) = ei xk + o (1) as x → ∞, f +' (x, k) = ikei xk + o (1)

(5.1)

where .' denotes the derivative with respect to .x. . f + (x, k) is called the Jost solution and coincides with the Baker–Akhiezer function. Equation (5.1) shows . f + (·, k) ∈ L 2 (R+ ), hence ( √ ) f +' 0, z ( √ ) .m + (z) = f + 0, z and one can see that .m + (z) is extendable to .C+ \ {0} as a continuous function. f (x, k) is obtained as a unique solution to an integral equation

. +

e−i xk f + (x, k) = 1 +

{

∞

.

x

e2ik(s−x) − 1 q (s) e−iks f + (s, k) ds. 2ik

Rybkin [43] showed e−i xk f + (x, k) = 1 +

N +1 ∑

.

( ) f j (x) (2ik)− j + o k −N −1

(5.2)

j=1

for.q such that.q ( j) ∈ L 1 (R{+ ) for. j} = 0,.1,.. . .,. L. The small.o is uniform with respect to .x ≥ 0. The coefficients . f j (x) are determined inductively by ⎧ { ∞ ⎪ ⎪ q (s) ds ⎨ f 1 (x) = −Q(x) ≡ − {x ∞ . . ⎪ ⎪ ⎩ f j+1 (x) = − f j' (x) − q (s) f j (s) ds, ( j ≥ 1) x

(L− j+1)

Therefore one can show . f j+1 is . L − j times differentiable and . f j+1 ∈ L 1 (R+ ). Since { ∞ ( −i xk )' . e f + (x, k) = e2ik(s−x) q (s) e−iks f + (s, k) ds, x

substituting (5.2) we have asymptotic behavior

5.1 Decaying Potentials

.

89

L+1 ( −i xk ) ( )' ∑ e f + (x, k) = g j (x) (2ik)− j + o k −N −1 , j=1

which leads us to L+1 ∑ √ ) ( .m + (z) = − −z + c j (−z)− j/2 + o z −(L+1)/2 if z → ∞ on C+ .

(5.3)

j=1

An analogous asymptotic behavior for .m − (z) is possible if .q ( j) ∈ L 1 (R− ) for . j = 0, ( j) .1,.. . ., . L. If .q are continuous at .x = 0, then the coefficients of the expansion for j+1 .m − (z) are equal to .(−1) cj. Then we have: Theorem 5.1 If .q ( j) ∈ L 1 (R) for . j = 0, .1,.. . ., . L, then (M.2) is satisfied with . L + 2 for any .n ≥ 1, and we have .q ∈ Q(n) L+2 (C). Therefore one can define the KdV flow (0) . K (g)q for . g ∈ ┌n if . L ≥ max {n − 1, 1}. L q for .q ∈ L 1 (R) has purely absolutely continuous non-degenerate spectrum on .R+ since .m ± are continuous on .R+ . If .(1 + |x|) q ∈ / L 1 (R) but .q ∈ L 1 (R), then the spectrum on .R− may consist of infinitely many eigenvalues accumulating to .0. .

5.1.2 Wigner–von Neumann Type Potentials For potentials .q ∈ / L 1 (R), even if .q (x) → 0 as .x → ±∞, anything can occur on .R+ for the spectrum of . L q from purely absolutely continuous spectrum to dense point spectrum. And depending on the cases we have to take different approaches. There is an interesting case called Wigner–von Neumann type (WvN type). Wigner and von Neumann investigated the potential .

) ( − 8x −1 sin 2x + O x −2 ,

and showed that there exists one eigenvalue on .R+ among the absolutely continuous spectrum on .R+ . In this section we show the .m-function arising from .m ± satisfies the property (M.2) if a potential is of WvN type. There are many papers concerning WvN type potentials and we employ here the work by Gilbert–Harris–Riehl [17]. The following real-valued potential .q is called WvN type: M ∑ .q(x) = h j (x)e2ic j x (5.4) j=−M

with .c j ∈ R\ {0}, .h j ∈ C L+1 (R+ ) satisfying

90

5 Applications .

| | | (k) | |h j (x)| ≤ p(x)k+1 for k = 0, 1, . . . , L + 1

for some . p(x) such that .x p (x) L+2 ∈ L 1 (R+ ). There may be another definition of WvN type potentials. We give the following lemma with sketch of the proof. Lemma 5.1 ([17]) Suppose .q is of WvN type. Then one has ) ( √ ) ( √ m + (z) = i z + r 0, i z + O |z|−(L+2)/2 on Im z ≥ 0,

.

where .r (0, z) is a rational function with poles at .

{ ( )} − c j1 + c j2 + · · · + c jk −M≤ j1 ,..., jk ≤M, 1≤k≤L+1 ⊂ R

and satisfies

( ) r (0, z) = O z −1 .

.

Proof The proof is carried out by combining Theorems 1 and 3 of [17]. Let m + (x, z) = m + (z, θx q) (see (7.14)). Then .m + (x, z) satisfies a Riccati equation

.

∂ m + (x, z) = q(x) − z − m + (x, z)2 .

. x

Set

(5.5)

∞

( √ ) ∑ √ m + (x, z) = i z + r x, i z + m n (x, z) ,

.

n=1

( √ ) where .r x, i z is chosen so that ⎧ √ ' 2 1 ⎨ Q (x, ( z) = q(x) − r −{r t − 2i zr ∈ )L (R+ ) √ √ . r (s, i z)ds ≤ K , (x < t) ⎩ Re 2i z (t − x) + 2

(5.6)

x

for .Re z > 0, .Im z ≥ 0 and .m n (x, z) are defined inductively by ) ⎧ ' ( √ ' ⎨ m 1 + (2i √z + 2r ) m 1 = Q ( denotes ∂x ) ' m + 2i z + 2r ) m 2 = −m 21 , . ∑ ⎩ 2' ( √ m n + 2i z + 2r m n = −m 2n−1 − 2m n−1 n−2 m for n ≥ 3 k k=1 which comes from (5.5), (5.6). Observing {

∞

m 1 (x, z) = −

.

x

( ) { t √ √ exp 2i z (t − x) + 2 r (s, i z)ds Q (t, z) dt x

one has from (5.6) .m 1 (x, z) is well-defined. Successively, one can estimate .m n and obtain

5.2 Oscillating Potentials

91 .

|m n (x, z)| ≤ 2−(n−1) a(x)η (z)

with .a ∈ L 1 (R+ ), .η (z) = o (1). In Theorem 3 of the paper they showed for .q of WvN type that.r( (0, z) can be ) taken as a rational function with the prescribed property ∎ and .η (z) = O |z|−(L+2)/2 holds on .Im z ≥ 0. Theorem 5.2 Suppose.q is of WvN type of (5.4). Then.q ∈ Q(n) L+3 holds for any.n ≥ 1. The above approach can be applied to other slowly decaying potentials like ( √ ) √ q(x) = cx −α (0 < α ≤ 1), q(x) = c sin x / x.

.

Rybkin [47] and Grudsky–Rybkin [19] constructed a solution to the KdV equation with initial data .q(x) = (A/x) sin 2ωx when .γ = |A/ (4ω)| < 1/2 by extending the inverse scattering theorem.

5.2 Oscillating Potentials A typical non-decaying potential is a periodic one and there are many works on the KdV equation with periodic initial data. In this section we show that some class of oscillating potentials can be initial data for the KdV flow. The key tool is the generalized reflection coefficient . R by which one can show in particular that smooth bounded ergodic potentials can be initial data for the KdV. The advantage of the reflection coefficient is in its invariance under the KdV flow.

5.2.1 Reflection Coefficient 1 For a real valued potential .q ∈ L loc (R) let . L q = −∂x2 + q and .m ± be its WTK functions assuming the boundaries .±∞ are of limit point type. For .z ∈ C+ set

⎧ 1 m + (z)m − (z) ⎪ ⎪ m 1 (z) = − , m 2 (z) = ⎪ ⎪ m + (z) + m − (z) m + (z) + m − (z) ⎪ ⎪ ⎨ m + (z) + m − (z) . . R(z) = ⎪ m + (z) + m − (z) ⎪ ( ) ⎪ ⎪ 1 1 ⎪ ⎪ ⎩ ξ j (z) = arg m j (z) = Im log m j (z) , ( j = 1, 2) π π .

(5.7)

R(z), .ξ1 (z) are called the reflection coefficient and xi-function respectively. .m j are HN functions and . R, .ξ j satisfy

92

5 Applications .

|R(z)| ≤ 1,

ξ j (z) ∈ [0, 1] .

Since any HN function has a finite limit on .R a.e., the limits . R(λ + i0), .ξ j (λ + i0) exist for a.e. .λ ∈ R. The reflection coefficient coincides with the conventional one if .q decays sufficiently fast, and describes the absolutely continuous spectrum .∑ac of . L q , namely Remling [42] showed ∑ac = {λ ∈ R; |R (λ + i0)| < 1} modulo measure 0.

.

(5.8)

Actually for periodic potentials or more generally for ergodic potentials including almost periodic potentials it is known that ∑ac = {λ ∈ R; |R (λ + i0)| = 0} ((7.43)).

.

The reason why. R(z) is crucial for the study of ergodic potentials is because . R(z) can be estimated by derivatives of .q, and the asymptotic behavior of the WTK functions .m ± can be calculated well by . R(z). Namely it holds that: { ∞ { } (i) For ergodic potentials bounded q (k) =⇒ λ M |R (λ + i0)| dλ < ∞ 0 { ∞ . . (ii) Generally λ M |R (λ + i0)| dλ < ∞ =⇒ (M.2) ((4.58)) 0

Step (i) will be considered in the next section. Step (ii) is carried out in this section, which is roughly as follows. Lemma 7.8 yields .

hence

| | |ξ j (z) − 1/2| ≤ |R (z)| /2,

) ({ ∞ ⎧ ξ1 (λ + i0) − Iλ>0 /2 1 ⎪ ⎪ dλ exp ⎨ m 1 (z) = √ λ−z 2 √−z ({λ0∞ ) . −z ξ + i0) − Iλ>0 /2 (λ ⎪ 2 ⎪ exp dλ ⎩ m 2 (z) = − 2 λ−z λ0

(5.9)

(5.10)

hold if .|R (λ + i0)| is integrable on .R+ . The estimates for .m j yields those of .m ± , which allow the expansion (M.2). In this argument one can replace the integrability ̂ through a conformal map, which allows a wider on .R+ by that of on the curve .C class of ergodic potentials. We prepare two curves for a positive odd integer .n ⎧ −(n−1) for |y| ≥ 1 ⎨ C = {{±ω (y) + i y; y ∈ R} with } ω (y) = cy 2 ̂ C = −z ; z ∈ C, Re z > 0 = {x ± i ̂ ω (x) ; x ∈ R, x ≥ λ0 } . ⎩ ̂ ω (Re z) , Re z > λ0 } D− = {z; |Im z| > ̂ with . ̂ ω (x) = ̂ cx 1−n/2 for .x ≥ 1.

(5.11)

5.2 Oscillating Potentials

93

1 Proposition 5.1 For an integer . M ≥ n assume .q ∈ L loc (R) satisfies .q ∈ M−1 .C (−a, a) for some .a > 0 and .inf spL q ≥ λ0 . Assume further

{ .

̂ C

| | M |z R(z)| |dz| < ∞.

(5.12)

Then, .q ∈ Q(n) L for any . L ≤ M − n + 2 holds, namely it is valid that ∑

m (z) = z +

) ( m j z − j + O z −L+1 on D− .

.

1≤ j≤L−2

Proof Let .φ be .φ (z) = −φk (−z) in Lemma 7.11 with .k = (n − 1) /2. Then .φ maps ̂− conformaly. Without loss of generality one can assume.−ak2 < λ0 . C\[0, ∞) onto. D (5.12) implies { ∞ | | |φ (λ) M R (φ (λ)))| |dφ (λ)| < ∞, .

.

0

which is equivalent to

{

∞

.

| | M |λ R(φ (λ))| dλ < ∞

(5.13)

0

due to .φ' (λ) = 1 + o (1). Hence applying (7.25) to .m ± (φ (z)) one has ( { ∞ ) ⎧ √ ξ1 (φ (λ)) − Iλ>0 /2 ⎪ −1 ⎪ dλ ⎨ − (2m 1 (φ (z))) = − −z exp − ~ λ−z λ0 ({ ), . ∞ ξ (φ (λ)) + ξ (φ (λ)) − I ⎪ 1 2 λ>0 /2 ⎪ dλ ⎩ 1 + 4m 1 (φ (z)) m 2 (φ (z)) = 1 − exp ~ λ−z λ1

( ) where .φ ~ λ j = λ j for . j = 0, .1. Set { δ (z) =

.

∞

ξ1 (φ (λ)) + ξ2 (φ (λ)) − Iλ>0 /2 dλ. λ−z

~ λ1

Then applying (7.24) to . f (λ) = ξ1 (φ (λ)) + ξ2 (φ (λ)) − Iλ>0 , where .λ M f is integrable due to (5.13), one has δ (z) =

M ∑

.

a j z − j + z −M

{

∞

~ λ1

j=1

∑ ) ( λ M f (λ) dλ = a j z − j + O z −M (Im z)−1 λ−z j=1 M

{

with aj =

∞

.

a

̂− Since Lemma 7.11 implies for .z ∈ D

λ j−1 f (λ) dλ.

94

5 Applications

φ−1 (z) = z − g1 (−z) − (−z)−(n−1)/2+1/2 g2 (−z)

.

with functions .g j holomorphic in a neighborhood of .z = ∞ and taking real values on .R, one has 1 + 4m 1 (z) m 2 (z) ( ) = 1 − exp δ φ−1 (z)

.

=

M ∑

∑

~ a j (−z)− j +

) ( d j (−z)− j−1/2 + O z −M (Im z)−1

(n−1)/2≤ j≤M

j=1

̂− for some .~ on . D a j , .d j ∈ R, which combined with (7.27) implies that the second term vanishes and { a j = b j for j = N , . . . , M if S /= φ ~ a j = 0 for j = 1, . . . , N − 1 and ~ . if S = φ ~ a j = 0 for j = 1, . . . , M ̂− . Therefore due to .{∈ < arg z < π − ∈} ⊂ D √ .

1 + 4m 1 (z) m 2 (z) ( ( )) ( ) {√ b(N (−z)−N /2 1 + )p1 z −1 + O z −M+N /2 (Im z)−1 if S /= φ = O z −M/2 (Im z)−1/2 if S = φ

̂− with holds on . D .

p1 (z) =

∑

(5.14)

( ) b j /b N (−z) j−N .

N +1≤ j≤M

Since (5.13) also implies the integrability of .λ M (ξ1 (φ (λ)) − Iλ>0 /2), one has similarly

.

− (2m 1 (z))−1

⎞ ⎛ M ∑ √ ) ( = − −z ⎝1 + ~ c j z − j + O z −M (Im z)−1 ⎠ j=1

̂− . This combined with (5.14) yields on . D ( ) √ m ± (z) = − (2m 1 (z))−1 1 ± 1 + 4m 1 (z) m 2 (z) ⎞ ⎛ M ∑ √ ) ( ~ c j z − j + O z −M (Im z)−1 ⎠ = − −z ⎝1 +

.

j=1

5.2 Oscillating Potentials

95

⎧√ ( ( −1 )) −N /2 ⎨ b N (−z) ( −M+N /2 1 + p1−1z) × ⎝1 ± +O z if S /= φ (Im ) z) ⎩ ( −M/2 O z if S = φ (Im z)−1 ⎛

⎞ ⎠

̂− , hence on . D {

−m + (−z 2 ) if Re z > 0 m − (−z 2 ) if Re z < 0 ⎧ ( ( ) ) ⎨ z + p2 (z −1 ) + O z −2M+N +1 Im z 2 −1 if S /= φ ( = ( ) ) ⎩ z + p3 (z −1 ) + O z −M+1 Im z 2 −1 if S = φ

m (z) =

.

on . D− with polynomials . p j satisfying . p j (0) = 0. Here we have the oddness of . N and { √ z if Re z > 0 2 . z = . −z if Re z < 0 Since .

| | |Im z 2 | = 2 |Re z| |Im z| ≥ 2 |Im z| ω (|Im z|) > c |Im z|2−n

and .m(z) is holomorphic on . D− , one can apply Phragmén–Lindelöf principle on sectors containing .R± , and we obtain ) ( if S /= φ z + p2 (z −1 ) + O z(−2M+N +1+n−2 ) z + p3 (z −1 ) + O z −M+1+n−2 if S = φ ) ( { if S /= φ z + p2 (z −1 ) + O z(−2M+N +n−1 ) , = z + p3 (z −1 ) + O z −M+n−1 if S = φ {

.

|m (z)| ≤

which is (4.58) for . L ≤ M − n + 2. {

If . R satisfies .

∞

| M | |λ R(λ + i0)| dλ < ∞,

∎

(5.15)

0

Then without using the conformal map .φ one can directly estimate .m ± and one has 1 Corollary 5.1 For an integer . M assume .q ∈ L loc (R) satisfies .q ∈ C M−1 (−a, a) (n) for some .a > 0, and (5.15). Then .q ∈ Q L for any . L ≤ M − n + 2 holds, namely it is valid that ∑ ) ( .m (z) = z + m j z − j + O z −L+1 on D− . 1≤ j≤L−2

| | The modulus .|m j | of the coefficient .m j is bounded from above by the moment (5.15). Proof The proof proceeds just as that of Proposition 5.1.

∎

96

5 Applications

| | We have added the above corollary because .| Rq (λ + i0)| is invariant under the KdV flow, that is .

| | | | | Rq (λ + i0)| = | R K (g)q (λ + i0)| for a.e. λ ∈ R.

(5.16)

This comes from the existence of the transfer matrices for the KdV flow (see Proposition 6.1). This fact was first recognized by Rybkin [44]. (n) Since (7.20) shows .m 1 = q(0)/2, of the shifted .m 1 { and } .m 1 (the .n-th derivative at .x = 0) can be obtained from . m j j≥1 , each derivative .q (n) (0) is expressed by { } a polynomial of . m j j≥1 . Consequently, applying the identity (5.16) to the shift, | | one has a uniform bound for .|q (n) (x)| by the moment (5.15), which means that the condition .q ∈ C M−1 (−a, a) follows from (5.15). The detail can be found in [32]. One can apply this corollary to ergodic potentials. For ergodic potentials .qω it is known that from (7.43) | | } { ∑ac (the absolutely continuous spectrum) = λ ∈ R; | Rqω (λ + i0)| = 0

.

| | and (5.8) implies .| Rqω (λ + i0)| = 1 on .R\∑ac . Therefore (5.15) reads {

{ .

R+ \∑ac

λ M dλ =

∞

| | M |λ R(λ + i0)| dλ < ∞,

0

namely this condition implies the existence of rich ac spectrum, although it allows the existence of singular spectrum. Actually, in a certain case of almost periodic potentials where . L qω has a purely ac spectrum, Damanik–Goldstein [7], Eichinger– VandenBoom–Yuditskii [11] constructed solutions to the KdV equation as a generalization of periodic potentials.

5.2.2 Ergodic Potentials Generally, it is not appropriate to use . R(z) to estimate .m ± , since it is a kind of tautology. However, for ergodic potentials one can estimate . R(z) which leads us to have asymptotics of .m ± near .R. To apply Proposition 5.1 to ergodic potentials we need a lemma. The necessary ̂1 be a simple closed curve containterminologies can be found in the Appendix. Let.C ̂1 = {x + i ̂ as .C ω ω (x) > 0 ing.[λ0 , ∞) parametrized (x)} x∈R with a smooth function. ̂ ) ( satisfying . ̂ ω (x) = O x −(n 1 /2−1) . Lemma 5.2 Suppose the Lyapunov exponent .γ (λ) satisfies {

∞

.

0

λm+1/2 γ (λ) dλ < ∞

(5.17)

5.2 Oscillating Potentials

97

for some integer .m ≥ 4. Then, for a.e. .ω ∈ Ω the condition { .

̂1 C

|z| M |Rω (z)| |dz| < ∞

(5.18)

̂1 (with approaching order .n 1 ) by any integer . M such that is fulfilled on the curve .C {

} n1 m − n1 . M < min − 1, . 2 2

(5.19)

Proof Set ⎧ √ 1√ ⎪ ⎨ ρ (λ) = −λN (λ) I[λ0 ,0] (λ) + λγ (λ) I(0,∞) (λ) π . . (0)/2) c = E (q ω ⎪ ⎩ w (z) = E (m ± (z, ω)) √ Since . −zw(z) is of HN (due to .Re w < 0, .Im w > 0), one has √ c 1 w (z) = − −z − √ +√ −z −z

{

∞

.

λ0

ρ (λ) dλ. λ−z

(5.20)

To make use of the condition (5.17) we decompose .w as follows: w (z) =

m ∑

wk (z) + w ~(z)

.

k=−1

with

⎧ wk (z) = ck (−z)−k−1/2 ⎪ ⎪ { ∞ ⎪ ⎪ ⎨ k−1 c−1 = −1, c0 = −c, ck = (−1) λk−1 ρ (λ) dλ . λ0 { ⎪ ∞ m ⎪ λ ρ (λ) ⎪ ⎪ ~(z) = (−1)m (−z)−m−1/2 dλ ⎩w λ−z λ0

holds due to the assumption (5.17). Set ⎧ − Re w(z) ⎪ ⎪ χ(z) = − Im w ' (z) ⎪ ⎪ ⎪ Im z ⎨ − Re wk (z) . − Im wk' (z) . χk (z) = ⎪ Im z ⎪ ⎪ − Re w ~(z) ⎪ ⎪ ⎩~ χ(z) = − Im w ~' (z) Im z Then χ(z) =

m ∑

.

k=−1

χk (z) + ~ χ (z) ,

(5.21)

98

5 Applications

̂1 is closer and roughly speaking the .χk terms are small according as the curve .C χ term is small for large .Re z if it to .R+ since .Re (−x − i0)1/2 = 0 for .x > 0 and .~ compensates the large .(Im z)−1 . Noting .

(−x − i y)−k−1/2

( )−k−1/2 = i x −k−1/2 (−1)k 1 + i yx −1 ⎛ ⎞ ( −1 )2 1 −1 1 − + 1/2) i yx − + 1/2) + 3/2) yx (k (k (k ⎠ = i x −k−1/2 (−1)k ⎝ 2( ) −1 3 +O yx

for .x ≥ 1, .0 < y < 1, we have c−1 χk (x + i y)

. k

− Re (−x − i y)−k−1/2 − (k + 1/2) Im (−x − i y)−k−3/2 y ( ( ) ) ( ) 1 1 x −k−3/2 + O x −k−1/2 y 2 x −3 − (−1)k+1 k + x −k−3/2 = (−1)k+1 k + 2 2 ( ( )2 ) 1 + (−1)k+1 (k + 1/2) (k + 3/2) (k + 5/2) x −k−3/2 yx −1 + O y 3 x −k−7/2 2( ) = O y 2 x −k−7/2 . =

On the other hand, the estimates |{ ∞ m | { ∞ | λ ρ (λ) || −j | |λ|m ρ (λ) dλ ( j = 1, 2) ≤ y . dλ | | j λ0 (λ − z) λ0 yield a bound for the last term .w ~ of (5.21). Then, we have ) ( ~ χ (x + i y) = O y −2 x −m−1/2 ,

.

hence χ(x + i y) =

m ∑

.

) ( χk (x + i y) + ~ χ (x + i y) = O y 2 x −5/2 + y −2 x −m−1/2 .

k=−1

√ ) ( This together with.Im w (x + i y) = O x 1/2 (due to. N (λ) ∼ λ as.λ → ∞) yields √ .

)1/2 ) ( ( 2χ (z) Im w (z) = O y 2 x −2 + y −2 x −m = O yx −1 + y −1 x −m/2 .

̂1 is parametrized as .x + i x −(n 1 /2−1) near .x = ∞, applying Therefore, if the curve .C (7.42) we have

5.2 Oscillating Potentials

99

) { √ |z| M |R (z, ω)| |dz| ≤ |z| M 2χ (z) Im w (z) |dz| ̂ ̂ C C { ∞ ( ) ≤c x M x −(n 1 /2−1) x −1 + x n 1 /2−1 x −m/2 d x,

({ E

.

1

which is finite for . M such that .

M − (n 1 /2 − 1) − 1 < −1 and M + (n 1 /2 − 1) − m/2 < −1. ∎

Then, Fubini’s theorem implies the condition (5.18). Theorem 5.3 Let .{qω (x)}ω∈Ω be an ergodic potential satisfying sup

.

x∈R, 0≤ j≤m

| ( j) | |q (x)| < ∞ for a.e. ω ∈ Ω. ω

Then, .qω ∈ Q(n) L for a.e. .ω ∈ Ω holds for any integer . L < m/4 − n + 1. Proof Suppose the Lyapunov exponent .γ (λ) satisfies { .

∞

λm+1/2 γ (λ) dλ < ∞.

(5.22)

0

Since Lemma 5.2 and (5.19) require ( .

L + n − 2 ≤ M < min

m − n1 n1 − 1, 2 2

) ,

n should satisfy

. 1

2L + 2n − 2 < n 1 < m − 2L − 2n + 4.

.

(5.23)

If m − 2L − 2n + 4 − (2L + 2n − 2) = m − 4L − 4n + 6 > 2,

.

one can choose an odd integer .n 1 satisfying (5.23). Then applying Lemma 5.2 and Proposition 5.1 we have .qω ∈ Q(n) L for a.e. .ω if . L < m/4 − n + 1. On the other hand, from .qθx ω (y) = qω (x + y) = (K (e x ) qω ) (y) the identity . fg

) ( (θx ω) = K (g)qθx ω (0) = (K (g)K (ex ) qω ) (0) = (K (ex ) K (g)qω ) (0) = K (g)qω (x)

follows. Moreover, Kotani–Krishna [29] showed that .qω ∈ Cbm (R) implies

100

5 Applications

{

∞

.

λm+1/2 γ (λ) dλ < ∞,

(5.24)

0

which is sufficient for (5.22) and completes the proof of Theorem 5.3. (5.24) can be verified by (5.20) and the identity (7.21) m + (z) = −k

L ∑

.

) ( c j k − j + O k −L ,

j=0

{ } where coefficients .c j are described by polynomials of . q (l) (0) 0≤l≤ jˆ2 .

∎

This theorem allows a case that. L qω has purely point spectrum. The above theorem states the possibility of construction of the KdV flow for a.e. .ω ∈ Ω. This is not acceptable especially when .qω is almost periodic. However, Corollary 5.1 removes this obstruction as follows. Set { } q .∑refl = λ ∈ R; Rq (λ + i0) = 0 . θ q

q

x Proposition 6.1 implies .∑refl = ∑refl for any .x ∈ R. For an almost periodic potential .q set .Ωq = the uniform closure of {θ x q} x∈R ,

which is known to be a compact commutative group. (7.43) states ~ q

∑refl = {λ ∈ R; γ (λ) = 0} ≡ Z for a.e. ~ q ∈ Ωq .

.

~ q

Lemma 5.3 For any .~ q ∈ Ωq it holds that .Z ⊂∑refl . θ q

q

x is clear from Proposition 6.1. In [28] (see also [42]) Proof The identity .∑refl = ∑refl

(n) it is proved that if.m (n) ± are reflectionless on a Borel set . F ⊂ R and .m ± (z) → m ± (z) for any .z ∈ C+ , then .m ± is also reflectionless on . F. Since the uniform convergence ∎ of .qn to .q implies .m (n) ± (z) → m ± (z), we have clearly the conclusion. { Therefore, if . R+ \Z λ M dλ < ∞ holds, then

{

{ .

λ M dλ ≤

~ q R+ \∑refl

R+ \Z

λ M dλ < ∞

is valid as well. For an almost periodic potential .q whose vanishing set .Z of the Lyapunov exponent .γ satisfies { .

R+ \Z

for any element .~ q ∈ Ωq

λ M dλ < ∞,

(5.25)

5.2 Oscillating Potentials { .

R+

101

| | | λ M | R~ q (λ + i0) dλ =

{ ~ q

| | | λ M | R~ q (λ + i0) dλ +

~ q

| | | λ M | R~ q (λ + i0) dλ ≤

R+ ∩∑refl

{ =

R+ \∑refl

{ ~ q

{

R+ \∑refl

R+ \Z

| | | λ M | R~ q (λ + i0) dλ

λ M dλ < ∞.

Consequently, if an almost periodic potential .q satisfies (5.25) without ambiguity of q. a.e. statement one can construct the KdV flow starting from .~

Chapter 6

Further Topics

In this chapter two topics are presented. One is related to a global analysis in time .t for solutions to the KdV equation in terms of the absolutely continuous spectrum of the Schrödinger operator . L q . Another is a generalization of the above framework to some multi-component systems, which yields the Nonlinear Schrödinger equation and Sine–Gordon equation. This generalization was already mentioned by Sato [48].

6.1 Extension of Remling’s Theorem Remling [40, 41], based on Breimesser–Pearson [2, 3], showed a remarkable result that a weak limit of shifted potentials with absolutely continuous (ac for short) spectrum has the reflectionless property on the ac spectrum. As we point out in the Appendix the shift .θx of potentials induces transfer matrix. A similar phenomenon is expected for the KdV flow since the shift operation is the solution to the first equation in the KdV hierarchy and there exist transfer matrices also for the KdV flow. In this section we define the transfer matrix and show an analogue of Remling’s theorem for the KdV flow.

6.1.1 Transfer Matrix Set

⎧ ⊓ ⎪ M(n) (see (4.64)) ∞ ⎨ M∞ = n≥1 ⊓ . . ⎪ Q(n) (see (4.62)) ⎩ Q∞ = L

(6.1)

L≥1,n≥1

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 S. Kotani, Korteweg–de Vries Flows with General Initial Conditions, Mathematical Physics Studies, https://doi.org/10.1007/978-981-99-9738-1_6

103

104

6 Further Topics

(0) If .m a ∈ M∞ holds for . a ∈ Ainv L , then .⎡n acts on .M∞ for any odd .n due to Proposition 4.7. In this section we will work on .M∞ for simplicity. For . a ∈ Ainv L let .{ϕ a , ψ a } be its characteristic functions. The odd integer .n is related with the parametrization ) ( of curve .C = {ω (y) + i y} with .ω (y) = O y −(n−1) . Set

( ) ) ( ⎧ ψa,e (z) + κ1 (a) 1 + ϕa,e (z) z + ψa,o (z) + κ1 (a) ϕa,o (z) ⎪ ⎪ ⎨ ⊓a (z) = 1 + ϕa,e (z) ϕa,o (z) ) ( . . ⎪ ge (z) go (z) ⎪ (0) ⎩ G(z) = for g ∈ ⎡n go (z) ge (z) (6.2) Note . det ⊓ a (z) = zΔ a (z) / = 0, det G(z) = g(z)g(−z) / = 0 on D− . Set T (z, g) = ⊓ga

. a

(√ ) √ (√ )−1 −z G( −z)−1 ⊓a −z .

(6.3)

Temporarily we assume .Re z > 0, and later will show .Ta (z, g) can be extended as an entire function. This definition comes from an identity ⎧ ( 2 ) −1 ⎨ Ta −z , g π a (z) = g(z) π ga (z) ) ( . z + ψa (z) + κ1 (a) (1 + ϕa (z)) ⎩ π a (z) = 1 + ϕa (z)

(6.4)

We have the following: Lemma 6.1 Suppose .m a ∈ M∞ . Then .Ta (z, g) satisfies the following properties: (i) .Ta (z, g) has a cocycle property: for .g1 , .g2 ∈ ⎡n(0) T (z, g1 g2 ) = Tg1 a (z, g2 ) Ta (z, g1 ) .

(6.5)

. a

(ii) .Ta (z, g) depends only on .m a , namely if .m a1 = m a2 , then .Ta1 (z, g) = Ta2 (z, g) holds. (iii) .Ta (z, g) is an entire function of .z satisfying .Ta (z, g) = Ta (z, g) if .g = eh with an odd real polynomial .h. ( (√ ) ( √ ))−1 (iv) .Δga (z) = Δa (z), .det Ta (z, g) = g −z g − −z hold for any .g ∈ ⎡n(0) . Especially if .g = eh with an odd polynomial .h, then .det Ta (z, g) = 1. Proof Every property but (i) is shown first for rational functions .r , and for general g ∈ ⎡n(0) by approximating general .g. The cocycle property (6.5) follows from (6.4). −1 . Observe First show (ii) for .r = qζ q−η

.

T (z, r ) = −

. a

η ζ

(

1 + m qζ a (η) f −m a (ζ) − m qζ a (η) (1 + m a (η) f ) −f 1 + m a (η) f

) (6.6)

6.1 Extension of Remling’s Theorem

with .

f (z) =

105

1 ζ 2 − η2 . m a (ζ) − m a (η) −z − η 2

This identity follows from (

)

T z, qζ = ζ

. a

−1

(

m a (ζ) ζ 2 + z − m a (ζ)2 −1 m a (ζ)

) ,

and ( ( )−1 ) ( ) ( ) ( ) = Ta z, qζ qη = Tqζ a z, qη Ta z, qζ . T (z, r ) = Ta z, qζ qη qη q−η

. a

Equation (6.6) implies.Ta (z, r ) depends only on.m a since.m qζ a = dζ m a . For a general rational .r ∈ ⎡0(0) one can show the property inductively by applying the cocycle property. Then approximating .g ∈ ⎡n(0) by a sequence of .rk ∈ ⎡0(0) the continuity of . T (ga) yields (ii). ( ) From (6.6).Ta −z 2 , r is known to be an even function of.z which is meromorphic ( ) on .C whose poles are .±zeros of .r , hence inductively one sees so is .Ta −z 2 , r for any rational .r ∈ ⎡0(0) . If .g = eh ∈ ⎡n(0) , one can approximate it by rational (functions) 2 .r k whose poles are diverging to .∞, hence if we show the convergence of . Ta −z , r k ( 2 ) to .Ta −z , g , we have (iii). From (ii) one can replace . a by .m (z) = (1, m a (z) /z). Then the convergence follows immediately from that of .T (rk m) on any .C. Equation (4.44) shows ( )( ) m qζ a (z) − m qζ a (−z) Δqζ a (z) = 1 + ϕqζ a (z) 1 + ϕqζ a (−z) 2z m a (z) − m a (−z) = Δa (z) . = (1 + ϕa (z)) (1 + ϕa (−z)) 2z

.

Then inductively one can show .Δr a (z) = Δa (z), hence the continuity implies the identity for general .g ∈ ⎡n(0) . The second identity is straightforward from the first one. ∎ We call this matrix .Ta (z, g) the transfer matrix. The reason why we have introduced .Ta (z, g) is in the identity ) ( ( 2 ) z + ψga (z) + κ1 (ga) 1 + ϕga (z) = m ga (z) , . Ta −z , g · m a (z) = 1 + ϕga (z)

(6.7)

which follows from (6.4). Here we have defined the .2 × 2 matrix action on .C by ( .

t11 t12 t21 t22

) ·m =

t11 m + t12 . t21 m + t22

106

6 Further Topics

The existence of a transfer matrix leads us to the following invariance. The WTK q functions .m ± for a potential .q ∈ Q∞ are given by { .

(√ ) q m + (z) = −m a −z for z ∈ C+ ( √ ) q m − (z) = m a − −z for z ∈ C+

(6.8)

with . a ∈ Ainv L associated with the potential .q, namely .q (x) = −2∂x κ1 (ex a). Moreover, { K (g)q (√ ) (√ ) ) ( q m+ (z) = −m ga −z = −Tq (z, g) · m a −z = −Tq (z, g) · −m + (z) . ( √ ) ( √ ) K (g)q q m− (z) = m ga − −z = Tq (z, g) · m a − −z = Tq (z, g) · m − (z) (6.9) holds. We have changed the notation here from .Ta (z, g) to .Tq (z, g). Proposition 6.1 Let. Rq (z) be the reflection coefficient for a potential.q ∈ Q∞ . Then .

| | | | | Rq (λ + i0)| = | R K (g)q (λ + i0)| for a.e. λ ∈ R

for any .g ∈ ⎡n(0) and odd integer .n ≥ 1. Proof For .λ ∈ R set

( T (λ, g) = T =

. q

t11 t12 t21 t22

) .

Then (6.22) implies .ti j ∈ R, hence (6.9) yields | | | K (g)q | K (g)q | | | m+ (λ + i0) + m − (λ + i0) || | | | . R K (g)q (λ + i0) = | K (g)q | K (g)q | m + (λ + i0) + m − (λ + i0) | | | | −T · (−m q (λ + i0)) + T · m q (λ + i0) | | | + − ( ) =| | | −T · −m q+ (λ + i0) + T · m q− (λ + i0) | | || | | t m q (λ + i0) + t | | m q (λ + i0) + m q (λ + i0) | 22 | | + | 21 − | − =| || | | t21 m q (λ + i0) + t22 | | m q+ (λ + i0) + m q− (λ + i0) | − | | = | Rq (λ + i0)| . ∎ The property T (z, g) · m ∈ C+ for m ∈ C+

. q

(6.10)

plays a crucial role to investigate the asymptotic behavior of.Tq (z, gt ) · m a as.t → ∞ for one-parameter group of .gt ∈ ⎡n(0) . For this purpose it is required to define the infinitesimal generator of .Ta (z, gt ).

6.1 Extension of Remling’s Theorem

107

For .gt = eth with an odd real polynomial .h define .

| Aq (z, h) = ∂t Tq (z, gt )|t=0 ,

(6.11)

and call it the generator of .Tq (z, gt ). Then we have ∂ Tq (z, gt ) = A K (gt )q (z, h) Tq (z, gt ) ,

(6.12)

. t

which is derived from the cocycle property T (z, gt+s ) = TK (gt )q (z, gs ) Tq (z, gt ) .

. q

We calculate . Aq (z, h). Since . Aq (z, h) can be described by .m q , we set ) ( m (z) .m = 1, z with .m ∈ M∞ . Then we have: Lemma 6.2 It holds that . A m (z, h) is linear in .h and .tr A m (z, h) = 0. It takes a form .

( ) A m −z 2 , h = p+

(

1 mo

(

−m e m 2e − m 2o −1 me

) ) ( ) 0 c (h) h + 0 0

with a real constant .c(h) given by (

( c (h) = ∂t κ1 (gt m)|t=0 = lim z h + p+

.

z→∞

( ) )) me 1 h − mp+ h . mo mo

( ) ~m −z 2 , g by Proof We use (6.4). Define .~ π m (z), .T ( ~ π m (z) =

.

) ) ( z + ψm (z) ~m −z 2 , g ~ , T π m (z) = g (z)−1 ~ π gm (z) . 1 + ϕm (z)

Then

( T (z, g) =

. m

1 κ1 (gm) 0 1

)

( )−1 1 κ1 (m) ~ Tm (z, g) 0 1

(6.14)

holds. First we compute .∂t ~ π gt m . Since 1 + ϕgt m = gt mT (gt m)−1 1, z + ψgt m = gt mT (gt m)−1 z,

.

taking derivative in .t and setting .t = 0 yields .

| ∂t ~ π gt m |t=0 = h~ πm −

(

mT (m)−1 T (hm) T (m)−1 z mT (m)−1 T (hm) T (m)−1 1

(6.13)

)

108

6 Further Topics

with c = ∂t κ1 (gt m)|t=0 .

.

Observe .

T (m)−1 1 = 1, T (m)−1 z = z

since .

T (m) 1 = p+ (1) = 1, T (m) z = p+ (m) = z.

Note generally .

ha = ~ ah with ~ a = (a2 , a1 )

m) h holds. On the other hand, for any odd function .h. Hence .T (hm) = T (~ .

implies .

~ ∈ B L (see (4.10)) m, m

~) T (m)−1 T (~ m) = T (. m) T (~ m) = T (. m·m

due to Lemma 4.4. Therefore we have ( ) | ~ ) hz m·m mT (. | . ∂t ~ π gt m t=0 − h~ πm = − ~) h m·m mT (. ⎛ ( 2 ( ) )⎞ m e − m 2o zm e m h + p+ h ⎟ ⎜ −p+ z ( m) o ⎟. ( mo ) = −⎜ ⎠ ⎝ −m e z m h + p+ h p+ mo z mo In the above calculation we have used the fact that .p+ preserves the parity. Then (6.13) yields .

| ( )| ~m −z 2 , gt | ~ ∂t T π (z) = −h~ π gt m (z)| y=0 π m (z) + ∂t ~ t=0 m ⎛ ( 2 ( ) )⎞ m e − m 2o zm e m p h + h −p + ⎜ + ⎟ z ( m) o ⎟. ( mo ) = −⎜ ⎝ ⎠ −m e z m h + p+ h p+ mo z mo

Noting an identity .p+ (z f ) = zp+ ( f ) for any even function . f , one has ⎛

( 2 ( ) )⎞ m e − m 2o me h + mp h −p + ⎜ + ⎟ ( )| o ⎟. ~m −z 2 , gt | ~ ( mo ) ( m) . ∂t T π (z) = − ⎜ t=0 m ⎝ ⎠ −m e 1 h + mp+ h p+ mo mo

6.1 Extension of Remling’s Theorem

109

Then, decomposing the both sides to even part and odd part we have ( )| ~m −z 2 , gt | = p+ . ∂t T t=0

(

1 mo

(

−m e m 2e − m 2o −1 me

) ) h . ∎

Then (6.14) yields the conclusion. We compute . A m (z, h) for some .h. For .m ∈ M∞ let m(z) = z +

L−1 ∑

.

) ( m j z − j + O z −L in D−

(6.15)

j=1

for any . L ≥ 1 and . D− . Then .

1 mo

(

−m e m 2e − m 2o −1 me

)

= M−1 z + M1 z −1 + M2 z −3 + · · ·

with ) ) ( ) ( ( ⎧ 2 ⎪ M = 0 −1 , M = 0 −m 1 , M = −m 2 m 1 − m 3 ⎪ 1 2 ⎨ −1 −1 0 m1 m2 0 0 ) ( . . 3 2 ⎪ m 1 m 2 − m 4 −2m 1 + m 2 + 2m 1 m 3 − 2m 5 ⎪ ⎩ M3 = −m 21 + m 3 −m 1 m 2 + m 4 If .m comes from the WTK functions of a potential .q, then the concrete values of .m j are given in (7.20), namely .m j = c j+1 and ⎧ q ' (0)/4 ⎪ ⎨ m 1 = q(0)/2, m 2 =j−1 1 ' 1∑ . . m l−1 m j−l for j ≥ 2 ⎪ ⎩ m j = 2 m j−1 − 2 l=2

(6.16)

For .h n = z n one has .c (h 1 ) = −m 1 , .c(h 3 ) = m 21 , hence ( ⎧ 0 ⎪ ⎪ A h = (z, ) 1 ⎨ m 0 ( . ⎪ 0 ⎪ ⎩ A m (z, h 3 ) = 0

) ( 0 −2m 1 z+ −1 0 ) ( ) . ( ) −m 2 2m 21 − m 3 0 m1 −1 2 z+ z + 1 0 m1 m2 0

1 0

)

(6.17)

Since .Ta (z, gt ) satisfies a linear equation (6.12) and it is related with the .mfunction by (6.7), .m gt a is supposed to fulfill a certain Riccati equation. Lemma 6.3 Let ( 2 ) .m t (z) = m gt m (z) and A gt m −z , h =

(

a b c −a

) .

110

6 Further Topics

Then it satisfies ∂ m t = −cm 2t + 2am t + b.

. t

∎

Proof This is just a simple calculation. Therefore we have { h = z =⇒ ∂t m t = m 2t − z 2 − 2m 1 ) ( . . h = z 3 =⇒ ∂t m t = z 2 − m 1 m 2t − 2m 2 m t − z 4 − m 1 z 2 + 2m 21 − m 3

Here the coefficients.m j are those of (6.15) for.m = m t = m gt m , hence they depend on .t and they form a closed equation. Especially when .m t (z) − z is holomorphic at . z = ∞, which is valid in case that .m ± are reflectionless, the coefficients form an infinite-dimensional Riccati equation. For the KdV equation the Riccati equation was first considered by Rybkin [44]. Now we investigate the property (6.10). Let . A(t) be .2 × 2 matrix with .tr A(t) = 0 and .T (t) be the solution to .

(

Let .

A(t) =

T ' (t) = A(t)T (t), T (0) = I .

(6.18)

) ) ( a(t) b(t) t (t) t12 (t) , , T (t) = 11 t21 (t) t22 (t) c(t) −a(t) (

and ᴧ = −i J ,

.

J=

) 0 −1 . 1 0

ᴧ is a Hermitian matrix with eigenvalues .±1. It is easy to see that a .2 × 2 matrix T maps the upper half plane .C+ into .C+ as a fractional linear transformation if and only if t . (ᴧT z, T z) > 0 for any z = (z, 1) with z ∈ C+ ,

. .

since .

Im (T · z) =

) ( Im (t11 z + t12 ) t21 z + t22 |t21 z + t22 |

2

=

(ᴧT z, T z) . 2 |t21 z + t22 |2

Moreover, a .2 × 2 matrix . A with .tr A = 0 makes . H = J A to be symmetric, namely H = H t . Define a self-adjoint matrix . H (t) by

.

.

H (t) = Im (J A(t)) =

) 1 ( J A(t) − J A(t) . 2i

Lemma 6.4 Assume .

H (t) ≥ 0 for any t ≥ 0.

Then, for .t ≥ 0, .T (t) maps .C+ into .C+ .

(6.19)

6.1 Extension of Remling’s Theorem

111

Proof Set . f (t) = (ᴧT (t) z, T (t) z). Then .

( ) ( ) f ' (t) = ᴧT ' (t) z, T (t) z + ᴧT (t) z, T ' (t) z ) ( ) ( = ᴧJ −1 J A(t)T (t) z, T (t) z + ᴧT (t) z, J −1 J A(t)T (t) z = 2 (H (t)T (t) z, T (t) z) ≥ 0, (6.20)

hold. Since . f (0) = (ᴧz, z) = 2 Im z > 0, we easily see . f (t) > 0 for any .t ≥ 0, which completes the proof. ∎ On .C+ define a pseudo metric .γ by γ (z 1 , z 2 ) = √

.

|z 1 − z 2 | . √ Im z 1 Im z 2

Then it is known that γ (F (z 1 ) , F (z 2 )) ≤ γ (z 1 , z 2 )

(6.21)

.

holds for any HN function . F. Set T = {T ∈ S L (2, C) ; T maps C+ into C+ .}

. +

and for .T ∈ T+ define a norm by ρ (T ) = sup

.

z∈C+

2 Im z , (ᴧT z, T z)

which satisfies .ρ (T ) ≤ 1 due to (6.21). Since / / ⎧ Im z 1 Im z 2 ⎪ γ (T · z 1 , T · z 2 ) ⎪ = ⎨ γ (z 1 , z 2 ) (ᴧT z 1 , T z 1 ) (ᴧT z 2 , T z 2 ) , . ⎪ 2 Im z 2 Im w 2 Im z ⎪ ⎩ = with w = T1 · z (ᴧT2 T1 z, T2 T1 z) (ᴧT2 w, T2 w) (ᴧT1 z, T1 z) we have inequalities for .T , .T j ∈ T+ (. j = 1, .2) { .

γ (T · z 1 , T · z 2 ) ≤ ρ (T ) γ (z 1 , z 2 ) for any z 1 , z 2 ∈ C+ ρ (T2 T1 ) ≤ ρ (T2 ) ρ (T1 )

.

(6.22)

Therefore .ρ (T ) describes the magnitude of the contraction of a matrix .T concerning to the metric .γ. For later purposes we have to estimate .ρ (T (t)) for the solution .T (t) of (6.19). Define a non-decreasing function by φ (t) = inf

.

z∈C+

1 2 Im z

∫

t 0

(H (s)T (s) z, T (s) z) ds.

112

6 Further Topics

For the solution (6.18) we have from (6.20) ρ (T (t)) =

.

1 , 1 + φ (t)

hence .ρ (T (t)) < 1 is equivalent to .φ (t) > 0, and especially if ∫

t

.

T (s)∗ H (s)T (s) ds ≥ cI

0

holds for some .c > 0, then φ (t) ≥ inf

.

z∈C+

|z|2 + 1 c (z, z) = c inf = c. z∈C+ 2 Im z 2 Im z

(6.23)

Lemma 6.5 Assume (6.19) and there exist .t0 , .c > 0 such that ∫

nt0

.

(n−1)t0

T (s)∗ H (s)T (s) ds ≥ cI

(6.24)

holds for any integer .n ≥ 1. Then it holds that for any .t ≥ 0 ) ( ρ T (t) T (t0 )−1 ≤ (1 + c)−n+1 if nt0 ≤ t < (n + 1) t0 .

.

Moreover, . K = T (t0 ) · C+ (.C+ = {z ∈ C; Im z ≥ 0} ∪ {∞}) is a closed disc in .C+ and −n+1 .γ (T (t) z 1 , T (t) z 2 ) ≤ c K (1 + c) γ (T (t0 ) z 1 , T (t0 ) z 2 ) with c = sup γ (w1 , w2 ) < ∞.

. K

w1 ,w2

Proof For .t0 ≥ 0 let .Tt0 (t) be the solution to T ' (t) = A(t)Tt0 (t) with Tt0 (t0 ) = I .

. t 0

Then the identity .

T (t) = Tnt0 (t)T(n−1)t0 (nt0 )T(n−2)t0 ((n − 1) t0 ) · · · Tt0 (2t0 )T (t0 )

for .t such that .nt0 ≤ t < (n + 1) t0 and (6.22) yield ) ( ρ T (t) T (t0 )−1 ( ) ( ) ( ) ( ) ≤ ρ Tnt0 (t) ρ T(n−1)t0 (nt0 ) ρ T(n−2)t0 ((n − 1) t0 ) · · · ρ Tt0 (2t0 ) .

.

6.1 Extension of Remling’s Theorem

113

( ) On the other hand, (6.23), (6.24) show .ρ T(k−1)t0 (kt0 ) ≤ (1 + c)−1 , hence one has the first conclusion. Equation (6.20) implies for .z ∈ C+ .

(ᴧT (t0 ) z, T (t0 ) z) 2 |t21 z + t22 |2 ( ) ∫ t0 1 Im z + = (s)T z, T z) ds (H (s) (s) |t21 z + t22 |2 0 ) ( Im z + c |z|2 + 1 c ) > 0, ( ≥ ≥ 2 |t21 z + t22 | 2 max |t21 |2 , |t22 |2

Im (T (t0 ) · z) =

∎

hence the second conclusion follows.

6.1.2 Reflectionless Property on ac Spectrum In [28] it was shown that for ergodic potentials we have always the reflectionless property on the ac spectrum. This implies a close relation between ac spectrum and the reflectionless property. In this section we generalize Remling’s work [40]. The proof is essentially based on the argument made by Breimesser–Pearson and Remling. Let .H be the set of all HN functions on .C+ , namely H = {m : C+ → C+ ; m is holomorphic} .

.

(

Define ωz (S) = Im

.

1 π

∫ S

1 ds s−z

) =

1 π

∫ S

y dt (t − x)2 + y 2

for .z = x + i y. Pearson [38, 39] introduced the following notion of convergence. Definition 6.1 A sequence of HN functions .{m n } converges to .m ∈ H in value distribution if ∫ ∫ . lim ωm n (x) (S) d x = ωm(x) (S) d x (6.25) n→∞

A

A

for all Borel sets . A, . S of .R such that .|A| < ∞. This convergence is verified by using effectively the metric .γ, since an inequality .

|ωz (S) − ωw (S)| ≤ γ (z, w)

holds for any .z, w ∈ C+ , which is derived from (6.21) and .ωz (S) ≤ 1. The following two lemmas were used by Breimesser–Pearson [2, 3].

114

6 Further Topics

Lemma 6.6 For a sequence .{m n } of .H and .m ∈ H, the following statements are equivalent: .(1) .m n → m compact uniformly on .C+ . .(2) .m n → m in value distribution. .(3) .(6.25) holds for any bounded intervals . A, S. Lemma 6.7 Let . A ∈ B (R) with .|A| < ∞. Then |∫ | ∫ | | | ωm(x+i y) (S) d x − ωm(x) (S) d x | = 0. . lim sup | | y↓0 m∈H,S∈B(R)

A

A

For .q ∈ Q∞ assume ∫

∞

.

| | λ M | Rq (λ + i0)| dλ < ∞ for any M ≥ 1,

(6.26)

0

and .gt = eth with a real odd polynomial .h set { .

Ωq+ (h) = the compact uniform closure of {K (gt ) q ∈ Q∞ ; t ≥ 0} ) ( . Hq (z) = Im J Aq (z, h)

Then Proposition 5.1 shows any .~ q ∈ Ωq+ (h) satisfies (6.26), and ∫

| | λ | R~q (λ + i0)| dλ ≤

∞

∫

M

.

0

| | λ M | Rq (λ + i0)| dλ < ∞ for any M ≥ 1.

∞

0

(6.27)

This is because if .qn → q weakly, namely ∫ .

∫ qn (x) ϕ (x) d x =

lim

n→∞ R

then .

R

q (x) ϕ (x) d x for any ϕ ∈ C0∞ (R) ,

| | | | | Rq (λ + i0)| ≤ lim | Rq (λ + i0)| for a.e. λ ∈ R n n→∞

| (k) | q (x)| has is valid due to Remling [42]. Then from Corollary 5.1 one knows .|~ a uniform upper bound by the moments (6.26), which shows the compactness of + .Ωq (h) endowed with the compact uniform metric. For .α > 0, .a < b let .

Dα = {z ∈ C+ ; 0 < Im z < α, a < Re z < b} .

6.1 Extension of Remling’s Theorem

115

(i) There exists an .α > 0 such that for any .~ q ∈ Ωq+ (h), .z ∈ Dα H~q (z) ≥ 0 (non-negative definite).

.

(6.28)

(ii) There exists .t0 > 0 such that for any .~ q ∈ Ωq+ (h), .z ∈ Dα ∫

t0

.

T~q (z, gs )∗ HK (gs )~q (z) T~q (z, gs ) ds

(6.29)

0

strictly positive definite.. Lemma 6.8 Assume .q ∈ Q∞ satisfies (6.28), (6.29) and ∫

∞

.

| | λ M | Rq (λ + i0)| dλ < ∞ for any M ≥ 1.

0

Let . K be a compact subset of . Dα . Then we have )) ( ( ⎧ δ= sup ρ T~q z, gt0 < 1 ⎪ ⎨ z∈K ,~ q ∈Ωq+ (h) ) ( . . ⎪ lim sup γ Tq (z, gt ) w1 , Tq (z, gt ) w2 = 0 ⎩ t→∞

(6.30)

z∈K ,w1 ,w2 ∈C+

Proof Since .Ωq+ (h) is compact and .T~q (z, gs ), . HK (gs )~q (z) are continuous with q , one sees that there exists a constant .c > 0 such that respect to .z, .~ ∫

t0

.

T~q (z, gs )∗ HK (gs )~q (z) T~q (z, gs ) ds ≥ cI

0

for any .z ∈ K , .~ q ∈ Ωq+ (h). Then (6.23) yields δ ≤ (1 + c)−1 < 1,

.

∎

and Lemma 6.5 completes the proof.

We remark that if a .q satisfies (6.26), then .q ∈ Q∞ [32]. Moreover .q ∈ Qrefl certainly satisfies (6.26). There is one point where we have to be careful to apply Remling’s argument. Remling treated the case .h 1 (z) = z, in which the corresponding . H is always nonnegative definite, hence the transfer matrix .Tq (z, eth 1 ) maps .C+ to .C+ for any .t > 0, th . z ∈ C+ . However, the property . Tq (z, e n ) ∈ T+ holds only for . z of a subdomain . Dα of .C+ if .n ≥ 3. To avoid this obstruction we prepare a conformal map .φ from .C+ to . Dα satisfying .φ (∞) ∈ / (a, b). Let .a1 , .b1 ∈ R such that .φ (a1 ) = a, .φ (b1 ) = b and q .φ ((a1 , b1 )) = (a, b). We define HN functions .m .± by q

q

m .± (z) = m ± (φ (z)) for z ∈ C+ .

.

116

6 Further Topics

.q (z, g) = Tq (φ (z) , g). Then .T .~q (z, gt ) · maps .C+ to .C+ for Similarly, we denote .T q ∈ Ωq+ (h). any .z ∈ C+ , .t ≥ 0, .~ Theorem 6.1 Assume .q ∈ Q∞ satisfies (6.28), (6.29) and ∫

∞

.

| | λ M | Rq (λ + i0)| dλ < ∞ for any M ≥ 1.

0 q Let .∑ac be the ac spectrum for ( . L) q . Let .{tn }n≥1 be a positive sequence diverging to .∞ such that .~ q = limn→∞ K gtn q exists. Then it holds that q m + (λ + i0, ~ ∩ (a, b) , q ) = −m − (λ + i0, ~ q ) for a.e. λ ∈ ∑ac

(6.31) { q .± , Proof We apply the argument of Breimesser–Pearson and Remling to . m } .q (z, g) . The property (6.31) is equivalent to .T .

∫

∫

.

A

ωm.~q+ (λ+i0) (S) dλ =

A

ωm.~q− (λ+i0) (−S) dλ

(6.32)

( q) for any Borel set . A ⊂ φ−1 ∑ac ∩ (a1 , b1 ), . S ∈ B (R). We start from the left-hand ( ) K (gt )q q = limn→∞ K gtn q implies that of .m ± n (z) for side. Since the convergence of .~ K (gt )q ~ q .+ by .m .± n in (6.32) with arbitrary any .z ∈ C+ , Lemma 6.6 allows us to replace .m small difference if .n is large. Observe ∫

∫

.

A

ωm.K (gtn )q (λ+i0) (S) dλ = +

∫

A

= A

q ω−T.q (λ,gtn )·(−. m + (λ+i0)) (S) dλ

ωT.

q

(

(λ,gtn )·

q

)

−. m + (λ+i0)

(−S) dλ.

The first identity follows from (6.9) and the second one is due to .ωz (S) = ω−z (−S). ( ) ( ) .q λ, gtn for .λ ∈ (a1 , b1 ). We would like to treat .q λ, gtn = T We have used also .T q .m .+ (λ + i0) as if it is a constant, so we decompose it as .

A = A0 ∪ A1 ∪ A2 ∪ · · · ∪ A N

disjoint

so that there exists .m j ∈ C+ such that for . j = 1, 2, · · · , N ) ( q γ −. m + (λ + i0), m j ≤ ϵ for any λ ∈ A j , and |A0 | ≤ ϵ,

.

( ) q q .q λ, gtn ∈ which is possible because .m .− (λ + i0) ∈ C+ for a.e. .λ ∈ ∑ac . Since .T S L (2, R) for .λ ∈ R, we have ) ) ( ) ( ( ) ( ( ) q q .q λ, gtn · m j = γ −. .q λ, gtn · −. γ T m + (λ + i0) , T m + (λ + i0), m j ≤ ϵ.

.

6.1 Extension of Remling’s Theorem

117

Applying the inequality .

|ωz (S) − ωw (S)| ≤ γ (z, w)

for any z, w ∈ C+ ,

(6.33)

we see |∫ | ∫ | | | | | | ) (−S) dλ − .| ωT. λ,g ·(−. ω dλ (−S) | ≤ ϵ |A j| . q . λ,g ·m T q( tn ) j | A q ( tn ) m + (λ+i0) | Aj On the other hand, Lemma 6.7 implies that there exists . y > 0 such that for any m∈H | | ∫ |∫ | | | | | .| ωm(λ+i0) (−S) dλ − ωm(λ+i y) (−S) dλ| ≤ ϵ | A j | (6.34) | Aj | Aj

.

holds, which makes it possible to change the estimate ( ) on .R to that on some domain .q z, gtn · m j . Then Lemma 6.8 shows in .C+ if we apply (6.34) to the HN function .T ) ( ) ( ( ) q .q λ + i y, gtn · m j , T .q λ + i y, gtn · m .− (λ + i y) ≤ ϵ, γ T

.

for every sufficiently large .n. Hence, owing to (6.33) |∫ | ∫ | | | || | q .ϵ | A j | ≥ | ωT.q (λ+i y,gtn )·m j (−S) dλ − ωT.q (λ+i y,gtn )·. m − (λ+i y) (−S) dλ| | Aj | Aj |∫ | ∫ | | | | =| ωT.q (λ+i y,gtn )·m j (−S) dλ − ωm.K (gtn )q (λ+i y) (−S) dλ| | Aj | − Aj K (gtn )q

is valid. Again applying (6.34) to .m .−

one has

| | ∫ |∫ | | | | | .| ωm.K (gtn )q (λ+i y) (−S) dλ − ωm.K (gtn )q (λ+i0) (−S) dλ| ≤ ϵ | A j | | Aj − | − Aj Summing up the inequalities, we have .

|∫ | ∫ | | | ω K (gtn )q | ≤ 3ϵ | A| ω dλ − dλ (S) (−S) K (gtn )q | | m .+ m .− (λ+i0) (λ+i0) A

A

for every sufficiently large .n. Since .ϵ > 0 is arbitrary, one has the identity (6.32), which completes the proof. ∎

.

For several odd polynomials .h we verify the conditions (6.28), (6.29). Since Aq (z, h) is linear in .h we have only to compute it for .h n (z) = z n , and (6.17) yields

118

6 Further Topics

( ) ⎧ 00 ⎪ ⎪ Im J A h = y (z, ) q 1 ⎨ 01 ) ( with z = λ + i y. . ⎪ −1 0 ⎪ ⎩ Im J Aq (z, h 3 ) = y 0 −2x + m 1 Hence .Im J Aq (z, h 1 ) ≥ 0 for any .z ∈ C+ .The second property (6.29) is certified without difficulty, and we have Remling’s result. For .h 3 one cannot expect to have (6.28), but .−h 3 has a chance, namely ( Im J Aq (z, −h 3 ) =

.

1 0 0 2λ − m 1

) y>0

if .λ > m 1 /2 = q(0)/4 (see (6.16)). For .h = ch 1 − h 3 ( .

Im J Aq (z, h) =

1 0 0 c + 2λ − m 1

) y>0

( ) if .λ > m 1 /2 − c/2 = q(0)/4 − c/2. If .q (t, x) = K eth 3 q is the solution to the KdV equation 1 3 3 .∂t q = (6.35) ∂ q − q∂x q with q(0, x) = q (x) , 4 x 2 then

( .

( ) ) K eth q (x) = q (−t, x + ct)

holds. Therefore, if 1 1 c c sup q (−t, x + ct) = − + sup q (t, x) ≡ λ (q) , λ>− + 2 4 t≥0,x∈R 2 4 t≤0,x∈R

.

then one can apply the theorem on .(λ (q) , ∞). Since we are assuming | | the integrability of . Rq (λ + i0), .q (t, x) can be estimated by the moments of .| Rq (λ + i0)| as follows. Equation (5.10), (7.21) imply ∫ q (0) = λ0 +

∞

.

∫

hence q (t, x) = λ0 +

λ0 ∞

.

λ0

(1 − 2ξ1 (λ + i0)) dλ,

( ) K (e g )q 1 − 2ξ1 x t (λ + i0) dλ

with the xi-function of the potential . K (ex gt ) q. On the other hand, we have .

| | |ξ1 (λ + i0) − 1/2| ≤ | Rq (λ + i0)| /2,

6.1 Extension of Remling’s Theorem

119

| | and .| Rq (λ + i0)| is invariant under the KdV flow. Consequently, one has ∫ sup q (t, x) ≤ λ0 +

.

t≤0,x∈R

∞

λ0

| | | Rq (λ + i0)| dλ,

which implies ) ( ∫ ∞ | | 1 c | | Rq (λ + i0) dλ . + λ0 + .λ (q) ≤ − 2 4 λ0

(6.36)

Corollary 6.1 Suppose a potential .q satisfies ∫

∞

.

| | λ M | Rq (λ + i0)| dλ < ∞ for any M ≥ 1.

0

Let .q(t, x) be the solution to the KdV equation (6.35) with initial data .q, and let q (x) = lim q (−tn , x + ctn )

.

n→∞

for some positive sequence.tn → ∞. Then the Schrödinger operator. L q with potential q q has the reflectionless property on .∑ac ∩ (λ (q) , ∞) (the ac spectrum of . L q ). .λ (q) has a bound (6.36).

.

The asymptotic behavior of solutions to the KdV equation has been studied intensively. For decaying initial data .q Tanaka [52] showed the following. Let the KdV equation be 3 .∂t q = −∂ x q + 6q∂ x q. Then for any .ϵ > 0 it holds that | | | | n ∑ | | ) ( 2 2 | . sup |q(t, x) − s x − 4η t − δ ; 4η j j j | → 0 as t → ∞, | x>ϵt | | j=1 { } where . −η 2j are the negative eigenvalues for the Schrödinger operator and s (x; c) = −

.

Therefore if .c = 4ηk2 , then

c ) (c > 0). (√ cx/2 2 cosh 2

120

6 Further Topics

q(t, x + ct) ∼

.

n ∑ ) ) ( ) ( ( s x + c − 4η 2j t − δ j ; 4η 2j → s x − δk ; 4ηk2 . j=1

) ( s x − δk ; 4ηk2 is called a single soliton and the associated Schrödinger operator is reflectionless on .(0, ∞) and has single eigenvalue at .−ηk2 . This result is consistent with Corollary 6.1. Generally, the reflectionless property on some set imposes a very strong restriction on potentials. For instance, if . L q is reflectionless on .spL q = [λ0 , ∞), then .q = λ0 identically. If . L q is reflectionless on .spL q = ∪nj=1 I j with intervals . I j , then .q is described by the .Θ-function of the closed Riemann surface for the hyper elliptic function associated with the .spL q .

.

6.2 Multi-component Systems A vector version of the present theory is possible as was pointed out by Sato, and it creates new nonlinear equations. In this section two examples are provided.

6.2.1 Nonlinear Schrödinger Equation The Nonlinear Schrödinger equation (NLS) 1 i∂t q = − ∂x2 q ± 4 |q|2 2

.

(6.37)

is also known to have a Lax pair and be an integrable system. The equation is called a defocusing NLS or focusing NLS depending on the sign .±. Its underlying operator is the Dirac operator ( i

.

( ( ) )( ) ) ( ) f1 0 q f1 1 0 f1 +2 =z . ∂x f2 f2 f2 ±q 0 0 −1

(6.38)

The Dirac operator is self-adjoint for a .+ sign and non-self-adjoint for a .− sign. In this section we provide a framework of constructing “Nonlinear Schrödinger flow” by tracing briefly the procedure for the KdV flow. Let . D+ be a bounded simply connected domain in the complex plane .C satisfying .

} { D+ = D + = λ ∈ C; λ ∈ D+ and 0 ∈ D+ .

Denote by .C the boundary of . D+ and . D− = C\ (D+ ∪ C). Let

6.2 Multi-component Systems .

121

H± = the Hardy space on D± ,

and denote the projections onto . H± by .p± .The vector versions of . H± are denoted by H ± , that is

.

.

( ) H + = H+ × H+ , H − = H− × H− ⊂ L 2 (C) = L 2 (C) × L 2 (C) ,

and the same notations .p± for the { projections } onto . H ± are used. Segal-Wilson’s setting for the KdV flow consists of. Gr (2) , ⎡ where.Gr (2) is a{set of closed subspaces of 2 2 h . L (C) which is. z -invariant and a commutative group.⎡ = e ; h is holomorphic on . D+ }. In this nonlinear Schrödinger case the corresponding objects are { } ⎧ 2 ⎨ Gr = W ; W is a closed subspace of L (C) satisfying zW ⊂ W {( h ) } . . e 0 ⎩⎡ = ; h holomorphic on a neighborhood of D + −h 0 e For a bounded measurable .2 × 2 matrix function . A (λ) on .C define the Toeplitz operator with symbol . A by .

(T (A)u) (z) = p+ (Au) (z) for u ∈ H + .

T (A) defines a bounded operator on . H + . Throughout the section we assume there exists a bounded .T (A)−1 . Everything proceeds as in the case of the KdV flow. For a non-negative integer .n define .2 × 2 matrix-valued function with entries in . H+ by ( ) { U (n) (z) = U A(n) (z) = T (A)−1 z n I . . Φ A = AU (0) − I .

Every entry of .Φ A is an element of . H− . Define 1 . An = 2πi

∫ λn−1 Φ A (λ) dλ. C

Now for .u ∈ H + note .

and

∫

λ (Au) (λ) dλ λ−z C ∫ ∫ 1 1 = zp+ (Au) + (Au) (λ) dλ = zT (A) u + (Au) (λ) dλ 2πi C 2πi C

T (A) zu =

1 2πi

122

6 Further Topics

∫

λ2 (Au) (λ) dλ λ−z C ∫ ∫ 1 1 = z 2 T (A) u + z λ (Au) (λ) dλ, (Au) (λ) dλ + 2πi C 2πi C

1 . T (A) z u = 2πi 2

hence one has .

1 T (A) U (1) = z I = zT (A) U (0) = T (A) zU (0) − 2πi ∫ 1 Φ A (λ) dλ = T (A) zU (0) − 2πi C = T (A) zU (0) − A1 ,

which shows

∫

(

) AU (0) (λ) dλ

C

(6.39)

U (1) = zU (0) − T (A)−1 A1 = zU (0) − U (0) A1 .

.

(6.40)

Here for a .2 × 2 matrix . B and .n ∈ Z≥0 note .

T (A)−1 z n B = U (n) B.

Similarly, one has .

T (A) U (2) = z 2 I = z 2 T (A) U (0) = T (A) z 2 U (0) − z A1 − A2 ,

(6.41)

hence U (2) = z 2 U (0) − U (1) A1 − U (0) A2

.

= z 2 U (0) − zU (0) A1 + U (0) A21 − U (0) A2 .

(6.42)

For .g(z) = eith(z) with a holomorphic function .h on a neighborhood of . D+ set ( .

G(w) =

w 0 0 w −1

) .

(6.43)

Then one has a new symbol .G (g) A, and an action on the set of symbols is obtained. For an one-parameter family .gt = eith one can define .G (gt ) and has ( ∂ G (gt ) = i hσ3 G (gt ) = i hG (gt ) σ3 with σ3 =

. t

) 1 0 . 0 −1

( ) 2 For .gt,x (z) = ei x z+it z set .G t,x = G gt,x . For simplicity we abbreviate .t, .x and write

6.2 Multi-component Systems .

123

G = G t,x , U (n) = UG(n)A , Φ = G AUG(n)A − I .

Then (6.39), (6.40) show ∂ U (0) = −i T (G A)−1 σ3 T (G A)zU (0) ) ( = −i T (G A)−1 σ3 T (G A) U (1) + U (0) A1

. x

= −i T (G A)−1 σ3 (z I + A1 ) = −iU (1) σ3 − iU (0) σ3 A1 = −i zU (0) σ3 − iU (0) [σ3 , A1 ]

(6.44)

and similarly (6.41), (6.42) show ( ) ∂ U (0) = −i T (G A)−1 σ3 T (G A)U (2) + z A1 + A2 ( ) = −i T (G A)−1 σ3 z 2 I + z A1 + A2

. t

= −iU (2) σ3 − iU (1) σ3 A1 − iU (0) σ3 A2 = −i z 2 U (0) σ3 − i zU (0) [σ3 , A1 ] + iU (0) A1 [σ3 , A1 ] − iU (0) [σ3 , A2 ] . (6.45) Now we compute derivatives of .Φ. Equation (6.44) shows ( ) ∂ (I + Φ) = ∂x G AU (0) = i zσ3 G AU (0) + G A∂x U (0) ) ( = i zσ3 (I + Φ) − G A i zU (0) σ3 + iU (0) [σ3 , A1 ] = i zσ3 Φ − i zΦσ3 − i (I + Φ) [σ3 , A1 ] ,

. x

(6.46)

hence comparing the coefficients of .z − j one has ] [ ∂ A j = −i A j+1 , σ3 − i A j [σ3 , A1 ] ,

. x

and (6.45) yields ( ) ∂ (I + Φ) = ∂t G AU (0) = i z 2 σ3 G AU (0) + G A∂t U (0)

. t

= i z 2 σ3 Φ − i z 2 Φσ3 − i z (I + Φ) [σ3 , A1 ] + i (I + Φ) A1 [σ3 , A1 ] − i (I + Φ) [σ3 , A2 ] , hence ] [ ∂ A j = i σ3 , A j+2 − i A j+1 [σ3 , A1 ] + i A j A1 [σ3 , A1 ] − i A j [σ3 , A2 ] .

. t

Especially, for . j = 1

124

6 Further Topics

⎧ ⎪ ⎨ ∂x A1 = −i [A2 , σ3 ] − i A1 [σ3 , A1 ] ∂t A1 = −i [A3 , σ3 ] − i A2 [σ3 , A1 ] + i A21 [σ3 , A1 ] − i A1 [σ3 , A2 ] . ⎪ ⎩ = ∂x A2 + i A21 [σ3 , A1 ] − i A1 [σ3 , A2 ] (

hold. Set .

A1 =

)

a11 a12 a21 a22

( , A2 =

b11 b12 b21 b22

(6.47)

) .

One has from the first identity of (6.47) ⎧ ∂x a12 = 2ib12 − 2ia11 a12 ⎪ ⎪ ⎪ ⎨ ∂x a21 = −2ib21 + 2ia21 a22 . , ⎪ ∂x a11 = 2ia12 a21 ⎪ ⎪ ⎩ ∂x a22 = −2ia12 a21

(6.48)

and the third identity shows ⎧ ∂t a11 = ∂x b11 − 2i (a11 a12 + a12 a22 ) a21 + 2ia12 b21 ⎪ ⎪ ⎪ ⎨ ∂ a = ∂ b + 2i (a 2 + a a ) a − 2ia b t 12 x 12 12 21 12 11 12 ( 11 ) . . 2 ⎪ ∂t a21 = ∂x b21 − 2i a21 a12 + a22 a21 + 2ia22 b21 ⎪ ⎪ ⎩ ∂t a22 = ∂x b22 + 2i (a21 a11 + a22 a21 ) a12 − 2ia21 b12

(6.49)

Consequently, (6.48) and (6.49) imply ⎧ ⎪ i∂t a12 = 1 ∂ 2 a12 − 4a 2 a21 ⎨ 12 2 x . . ⎪ ⎩ i∂t a21 = − 1 ∂ 2 a21 + 4a 2 a12 21 2 x

(6.50)

(0) The underlying operator is obtained from . F (x, z) = AUG(e , namely . F satisfies x )A

i∂x F − F [σ3 , A1 ] = z Fσ3 ,

.

which yields

( iσ3 ∂x f +

.

2a12 0 0 2a21

) f =zf

with . f = .t F e1 . Therefore the underlying operator is ( .

L = iσ3 ∂x +

2a12 0 0 2a21

) .

(6.51)

According to (6.37) the Eq. (6.50) is defocusing if .a21 = a 12 and focusing if .a21 = −a 12 . To have such a property for . A1 we introduce .2 × 2 matrices by

6.2 Multi-component Systems

125

( σ =

. 2

) ( ) 01 0 1 , σ4 = . 10 −1 0

The following are easily shown: { .

A = σ2−1 Aσ2 =⇒ A1 = σ2−1 A1 σ2 =⇒ a21 = a 12 , a22 = a 11 A = σ4−1 Aσ4 =⇒ A1 = σ4−1 A1 σ4 =⇒ a21 = −a 12 , a22 = a 11

.

(6.52)

These properties of symbols are preserved under the group action by .G(g) if .g satisfies ( ) .g (z) g (z) = g (z) g (z) = 1, or equivalently .g = ei h with a real .h. ) )−1 ( ( 2 Proposition 6.2 For a symbol . A on .C assume there exists .T G ei x z+it z A for any .x, .t ∈ R. Let ⎧ ( ) ( ( ) )−1 2 2 ⎪ ⎪ 1 = I + A1 (t, x) z −1 + A2 (t, x) z −2 + · · · ⎨ G ei x z+it z AT G ei x z+it z A ) ( . . a11 a12 ⎪ ⎪ ⎩ A1 (t, x) = a21 a22

Then . A1 (t, x) satisfies ⎧ ⎪ i∂t a12 = 1 ∂ 2 a12 − 4a 2 a21 ⎨ 12 2 x . , ⎪ ⎩ i∂t a21 = − 1 ∂ 2 a21 + 4a 2 a12 21 2 x and one has the NLS equations according as { .

A = σ2−1 Aσ2 =⇒ a21 = a 12 , a22 = a 11 (defocusing NLS) A = σ4−1 Aσ4 =⇒ a21 = −a 12 , a22 = a 11 (focusing NLS)

Since the underlying operator for the defocusing NLS is a self-adjoint Dirac operator, it has the Weyl theory and WTK functions exist [34]. It is supposed that everything proceeds similarly to the case of the KdV flow. The process is ongoing and the results will be published later. However, for the focusing NLS difficult problems related to the spectrum of the underlying Dirac operators arise. This situation is similar to the case of the Boussinesq equation.

126

6 Further Topics

6.2.2 Sine–Gordon Equation The Sine–Gordon equation is a nonlinear equation for .u = u(t, x) ∂ 2 u − ∂x2 u + sin u = 0,

. t

or equivalently θ

. ξη

= 2 sin 2θ,

(θ (ξ, η) = u (2 (ξ − η) , 2 (ξ + η)) (t, x) /2) .

In this case the curve .C consists of two closed simple curves with the origin outside. We assume .C satisfies .C = C, C = −C. D− is outside of the two curves containing the origin .0, and . D+ = C\ (D− ∪ C). Almost everything goes similarly to the NLS case. For .2 × 2 bounded matrix-valued function . A (λ) on .C and .n ∈ Z define a .2 × 2 matrix-valued function with entries in . H + by

.

{ .

U (n) (z) = T (A)−1 z n I . Φ A = AU (0) − I

Note .z n e1 , .z n e2 ∈ H + hold for any .n ∈ Z due to .0 ∈ D− . Every entry of .Φ A is an element of . H− . Define for .n ∈ Z 1 . An = 2πi

∫ λn−1 Φ A (λ) dλ. C

U (1) was already computed in (6.40). .U (−1) is represented by .U (0) as follows. Applying .p+ to the identity −1 (0) . Az U = z −1 I + z −1 Φ A

.

one has .

( ) T (A)z −1 U (0) = T (z −1 A)U (0) = z −1 I + p+ z −1 Φ A = z −1 (I + Φ A (0)) ,

since p

. +

(

) ( ) z −1 Φ A (z) = p+ z −1 (Φ A − Φ A (0)) (z) + z −1 Φ A (0) = z −1 Φ A (0) .

Then .

z −1 U (0) = U (−1) + T (A)−1 z −1 Φ A (0) = U (−1) + U (−1) Φ A (0)

follows, hence

6.2 Multi-component Systems

127

U (−1) = z −1 U (0) (I + Φ A (0))−1 .

(6.53)

.

Set .

) ( ) ( G(z) = G gξ,η (z) (see (6.43) with gξ,η (z) = exp i ξz + ηz −1 . (

Then −1

∂ G = i zσ3 G, ∂η G = i z σ3 G with σ3 =

. ξ

1 0 0 −1

)

( ) holds. Assume .T G ξ,η A is invertible for any .ξ, η ∈ R. Denote .Φ = ΦG A for simplicity. Then .∂ξ (I + Φ) was already computed in (6.46). The derivative with respect to .η is obtained similarly by (6.53), namely ∂ (I + Φ) = i z −1 σ3 (I + Φ) − i z −1 (I + Φ) B

. η

with . B = (I + Φ (0))−1 σ3 (I + Φ (0)). Consequently, we have { .

∂ξ (I + Φ) = −i z [(I + Φ) , σ3 ] − i (I + Φ) [σ3 , A1 ] . ∂η (I + Φ) = i z −1 σ3 (I + Φ) − i z −1 (I + Φ) B

(6.54)

At .z = 0 the first identity of (6.54) yields ∂ (I + Φ (0)) = −i (I + Φ (0)) [σ3 , A1 ] .

(6.55)

. ξ

(

Now assume .

A=

σ4−1 Aσ4

with σ4 =

) 0 1 . −1 0

Then one can show .Φ = σ4−1 Φσ4 , . A1 = σ4−1 A1 σ4 , so set ( .

I + Φ (0) =

Note .

B=

c b −b c

1 |c|2 + |b|2

)

(

( , A1 =

cc − bb 2bc 2cb bb − cc

Since ∂ A1 = iσ3 − i B

. η

from (6.54), one sees

a11 a12 −a 12 a 11 ) .

) .

128

6 Further Topics

( ∂

. η

0 2a12 2a 12 0

) = ∂η [σ3 , A1 ] = −i [σ3 , B] ) ( −2i 0 2bc , = 2 |c| + |b|2 −2cb 0

hence ∂ a

. η 12

=

−2ibc . |c|2 + |b|2

(6.56)

Equation (6.55) shows ( ∂

. ξ

c b −b c

)

( = −i {

hence .

c b −b c

)(

0 2a12 2a 12 0

) ,

∂ξ c = −2ia 12 b . ∂ξ b = −2ia12 c

(6.57)

Further we assume additional symmetry for . A, namely .

( ) A (λ) = A −λ .

Since .G A also satisfies the same property, one has Φ (z) = Φ (−z),

.

hence Φ (0) = Φ (0) =⇒ Im Φ (0) = 0, − A1 = A1 =⇒ Re A1 = 0

.

are valid. Now let b = r cos θ, c = r sin θ, a12 = iu.

.

Then (6.56) and (6.57) imply ⎧ ⎪ ⎨ rξ sin θ + r θξ cos θ = −2ur cos θ rξ cos θ − r θξ sin θ = 2ur sin θ , . ⎪ ⎩ ∂η u = −2 cos θ sin θ hence .rξ = 0, .θξ = −2u hold, and one has the Sine–Gordon equation θ

. ξη

= −2u η = 4 cos θ sin θ = 2 sin 2θ.

The underlying operator is the same as the focused NLS equation, namely a non self-adjoint Dirac operator.

6.2 Multi-component Systems

129

Fig. 6.1 The curve .C consists of 2 closed curves symmetric with respect to the origin

Proposition 6.3 For a symbol . A on .C (see Fig. 6.1) assume .

( ) A = σ4−1 Aσ4 , A (λ) = A −λ ,

) )−1 ( ( −1 A and there exists .T G eiξz+iηz for any .ξ, .η ∈ R. Let ⎧ ( ) ( ( ) )−1 −1 −1 ⎪ ⎪ G eiξz+iηz AT G eiξz+iηz A 1 = I + Φ (ξ, η, z) ⎪ ⎪ ⎨ ) ( c b . . with b, c ∈ R I + Φ (ξ, η, 0) = ⎪ ⎪ c −b ⎪ ⎪ ⎩ b = r cos θ, c = r sin θ Then .θ satisfies Sine–Gordon equation: θ

. ξη

= 2 sin 2θ.

The underlying operator is a non-self-adjoint Dirac operator.

Chapter 7

Appendix

7.1 Herglotz–Nevanlinna (HN) Functions A holomorphic function .m mapping .C+ into .C+ is called a Herglotz–Nevanlinna function (HN function for short). A Möbius transformation yields a holomorphic function ( ) 1+w .m ~ (w) = m i 1−w on .Δ = {|w| < 1}. Since .Im m ~ (w) is a positive harmonic function on .Δ, there exists a non-negative measure .μ on the unit circle such that ) ( ∫ 2π ∫ 2π 1 1 1 − |w|2 eiθ + w μ (dθ) = . Im m ~ (w) = Re iθ | | μ (dθ) . 2π 0 2π 0 |eiθ − w |2 e −w

Then m ~ (w) = α +

.

−1 2πi

∫

2π 0

eiθ + w μ (dθ) eiθ − w

holds with some .α ∈ R. Setting .

σ (dλ) 1 1 λ−i = μ (dθ) with = eiθ and β = μ ({0}) . λ2 + 1 2π λ+i 2π

one has z−i λ−i ∫ + −1 −1 1 + w λ + i z + i σ (dλ) μ ({0}) + .m (z) = α + z − i λ2 + 1 2πi 1 − w i R λ−i − λ+i z+i ) ∫ ( λ 1 − 2 σ (dλ) . = α + βz + λ +1 R λ−z © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 S. Kotani, Korteweg–de Vries Flows with General Initial Conditions, Mathematical Physics Studies, https://doi.org/10.1007/978-981-99-9738-1_7

131

132

7 Appendix

Lemma 7.1 Let .m be an HN function. Then the following are valid. (i) .m has a representation ∫ ( m (z) = α + βz +

.

R

) λ 1 − 2 σ (dλ) λ−z λ +1

with .α ∈ R, .β ≥ 0 and a non-negative measure .σ on .R satisfying ∫ .

σ (dλ) < ∞. 2 Rλ + 1

(ii) .m has a non-tangential finite limit .

lim m (λ + iϵ) = m (λ + i0) ϵ↓0

for a.e. .λ ∈ R. In particular .Im m (λ + i0) = πσ ' (λ) (the absolutely continuous part of .σ). For the proof of (ii) refer [9]. A holomorphic function .m on .C\iR satisfying m (z) = m (z), Im m(z)/ Im z > 0 for z ∈ C\i R

.

(7.1)

yields two HN functions .m ± by m + (z) = −m

.

(√

) ( √ ) −z ,m − (z) = m − −z for z ∈ C+ .

m ± are extended to .C− as .m ± (z) = m ± (z). Due to Lemma 7.1 .m ± are represented as ) ∫ ∞( λ 1 .m ± (z) = α± + β± z + − 2 σ± (dλ) . λ−z λ +1 λ0

.

The original .m is recovered by .m ± by m (z) =

⎧ ( ) ⎨ −m + −z 2 if Re z > 0 ⎩

. ( ) m − −z 2 if Re z < 0

d m (z) =

z2 − ζ 2 − m (ζ) . m (z) − m (ζ)

.

Define an operation . ζ

This transformation is introduced here since we have m qζ a = dζ m a

.

(7.2)

7.1 Herglotz–Nevanlinna (HN) Functions

133

for . a ∈ Ainv due to (3.25). This operation is commutative, namely .dζ1 dζ2 = dζ2 dζ1 . Lemma 7.2 Suppose .m satisfies the property (7.1). Then. dζ dζ m also satisfies (7.1) for .ζ ∈ C\ (R ∪ iR). Proof Since .dζ dζ m = dζ dζ m = dζ dζ m holds, one can assume .Im z > 0, .Im ζ > 0. From the definition .dζ dζ m can be computed as d dζ m (z) =

. ζ

z2 − ζ

(

2 2

z2 − ζ 2 ζ2 − ζ − m (z) − m (ζ) m (ζ) − m (ζ)

Set

−

ζ2 − ζ

)

2

m (ζ) − m (ζ)

− m (ζ) .

2

w=

.

ζ2 − ζ z2 − ζ 2 ,a = (∈ R) . m (z) − m (ζ) m (ζ) − m (ζ)

Then d dζ m (z) =

. ζ

(7.3)

m (z) w − m (ζ)a − a, w−a

hence

.

Im dζ dζ m (z) = Im

) ( m (z) w − m (ζ)a (w − a)

|w − a|2 |w| Im m(z) − a Im w (m(z) − m (ζ)) − a 2 Im m (ζ) = |w − a|2 ( ) |w|2 Im m(z) − a Im z 2 − ζ 2 − a 2 Im m (ζ) = . |w − a|2 2

Suppose .Re z, .Re ζ > 0. Then .

) ( |w|2 Im m(z) − a Im z 2 − ζ 2 − a 2 Im m (ζ) ( | |2 ) | | Im v Im u u−v | | Im m + (u) = −| Im m + (v) Im m + (u) m + (u) − m + (v) |

with .u = −z 2 , .v = −ζ 2 ∈ C− . The Herglotz–Nevanlinna representation for .m + shows ⎧ ∫ ∞ m + (u) − m + (v) 1 ⎪ ⎪ + = β σ+ (dλ) ⎨ + u−v − u) (λ (λ − v) λ0 ∫ ∞ . , 1 Im m + (u) ⎪ ⎪ + σ = β (dλ) ⎩ + 2 + Im u λ0 |λ − u| which implies

134

7 Appendix

| |2 | | Im v Im u u−v | | >0 . −| Im m + (v) Im m + (u) m + (u) − m + (v) | if .u, .v ∈ C− . The case .Re z > 0, .Re ζ < 0 is computed similarly, that is .

) ( |w|2 Im m(z) − a Im z 2 − ζ 2 − a 2 Im m (ζ) | |2 | | u−v | | Im m + (u) − Im v Im u > 0, = −| m + (u) + m − (v) | Im m − (v)

since .Im m + (u) < 0, .Im m − (v) > 0 if .Im u < 0, .Im v > 0, which completes the proof. ∎ If .m is holomorphic on some part of .R, one sees that .dζ dζ m may have a pole on that region. To avoid this situation we need an additional condition. Note .m is holomorphic on .C\ [−μ0 , μ0 ] for .μ0 > 0 if and only if the associated .m ± are holomorphic on .C\(−∞, λ0 ] with .λ0 = −μ20 < 0. In addition to this property we impose on .m a condition m (x) > m(−x) for any x ∈ (μ0 , ∞) .

.

(7.4)

Lemma 7.3 Assume .m is holomorphic on .C\ [−μ0 , μ0 ] and satisfies (7.4). Then, d m,.dζ dζ m are holomorphic on.C\ [−μ0 , μ0 ] if.s ∈ R\ [−μ0 , μ0 ],.ζ ∈ C\ [−μ0 , μ0 ], and they satisfy (7.1), (7.4).

. s

Proof One can assume .Im z > 0. The proof is similar to the previous one. Assume s > μ0 , .Re z > 0. Then, setting .w ≡ −z 2 ∈ C− , .u = −s 2 < λ0 we have

.

d m (z) =

. s

−m +

( ) w−u z2 − s2 ) ( ) + m + −s 2 = + m + (u) . m + (w) − m + (u) −z 2 + m + −s 2

(

Since .m + is Herglotz–Nevanlinna, m + (w) − m + (u) . = β+ + w−u

∫

∞

λ0

1 σ+ (dλ) (λ − w) (λ − u)

(7.5)

and we see .Im (m + (w) − m + (u)) / (w − u) > 0 due to .Im w < 0, .u < λ0 , which implies .Im ds m (z) > 0. If .s < −μ0 , .Re z > 0, then d m a (z) =

. s

−m +

(

( ) w−u z2 − s2 ) ( ) − m − −s 2 = − m − (u) . m + (w) + m − (u) −z 2 − m − −s 2

Since .

m + (w) + m − (u) m + (w) − m + (u) m + (u) + m − (u) = + ∈ C− w−u w−u w−u

7.1 Herglotz–Nevanlinna (HN) Functions

135

due to .m + (u) + m − (u) = −m (−s) + m (s) < 0, we have .ds m a (z) ∈ C+ . The cases (.s < −μ0 , .Re z < 0), (.s > μ0 , .Re z < 0) can be treated similarly. Under the condition, .m is holomorphic on .C\ [−μ0 , μ0 ] and takes real values on .R\ [−μ0 , μ0 ]. The proof of analyticity of .ds m on .R\ [−μ0 , μ0 ] for .s ∈ R\ [−μ0 , μ0 ] proceeds in the same way to the above, so we omit the proof. The property (7.4) for .ds m is proved as follows: If . x, .s > μ0 , then x 2 − s2 x 2 − s2 − m(x) − m(s) m(−x) − m(s) ( 2 ) x − s 2 (m(−x) − m(x)) = . (m(x) − m(s)) (m(−x) − m(s))

d m (x) − ds m (−x) =

. s

Observing .m ± are strictly increasing on .(−∞, λ0 ) (see (7.5)) one sees .m is also strictly increasing on .R\ [−μ0 , μ0 ], hence .

x 2 − s2 < 0, (m(x) − m(s)) (m(−x) − m(s))

which shows .ds m (x) − ds m (−x) > 0. For the proof of the analyticity of .dζ dζ m it is sufficient to have 2

z2 − ζ 2 ζ2 − ζ /= 0 for z ∈ R\ [−μ0 , μ0 ] . . − m (z) − m (ζ) m (ζ) − m (ζ)

(7.6)

For .ζ ∈ C\R we know that for .z ∈ R\ [−μ0 , μ0 ] ( .

Im

2

z2 − ζ 2 ζ2 − ζ − m (z) − m (ζ) m (ζ) − m (ζ)

) = Im (dz m) (ζ) /= 0

holds from Lemma 7.2, hence (7.6) is valid. For the proof of (7.4) for .dζ dζ m note ) ( 2 ( ) ( ) x − ζ 2 (w (−x) − w (x)) . dζ dζ m (x) − dζ dζ m (−x) = (w (x) − a) (w (−x) − a) with .w(x) =

x2 − ζ2 and .a as in (7.3). Since m (x) − m (ζ) ( w (−x) − w (x) =

.

x2 − ζ

2

)

(m (x) − m (−x))

(m (−x) − m (ζ)) (m (x) − m (ζ))

,

136

7 Appendix

one has .

( ) ) ( dζ dζ m (x) − dζ dζ m (−x) | | 2 |x − ζ 2 |2 (m (x) − m (−x)) = (w (x) − a) (w (−x) − a) (m (−x) − m (ζ)) (m (x) − m (ζ))

and we know .Im w (±x) /= 0, .Im m (ζ) /= 0, so one has ( ) dζ dζ m (−x) > 0.

( ) dζ dζ m (x) − ∎

.

7.2 Spectral Theory of 1D Schrödinger Operators As we have seen in the introduction the KdV equation is closely related to 1D Schrödinger operators. This section provides necessary knowledge on it, which will be required in the subsequent sections. The role of Weyl–Titchmarsh–Kodaira (WTK) functions is stressed in this book. The WTK function was introduced by H. Weyl in 1910 to classify the boundary types for Sturm-Liouville operators and later it was applied independently by Titchmarsh and Kodaira to obtain spectral representation of the operators. There appeared two movements concerning WTK functions. In 1982 JohnsonMoser used them to develop the spectral theory of 1D Schödinger operators with almost periodic potentials, which was a generalization of Floquet theory for periodic differential operators. This triggered a vast number of studies of 1D Schödinger operators with ergodic potentials. On the other hand, two notions related to WTK functions were introduced. In 1996 Gesztesy–Simon [15] defined the xi-function through WTK functions and applied it to the inverse spectral theory of 1D Schödinger operators. In 1997 Gesztesy–Nowell–Pötz [14] considered a generalization of reflection coefficients for potentials having non-trivial ac spectra and it was followed by Rybkin [44] and Remling [42]. These quantities are defined for arbitrary potentials and are very helpful to know asymptotic behaviors of WTK functions near .R. In this section we focus on WTK functions, xi-functions and reflection coefficients. As for general spectral theory of 1D Schrödinger operators Marchenko [36] is recommended.

7.2.1 Weyl–Titchmarsh–Kodaira (WTK) Functions For a real-valued .q ∈ L 1 (R+ ) and .z ∈ C let .ϕz , .ψz be solutions to { .

−ϕ''z + qϕz = zϕz with ϕz (0) = 1, ϕ'z (0) = 0 . −ψz'' + qψz = zψz with ψz (0) = 0, ψz' (0) = 1

7.2 Spectral Theory of 1D Schrödinger Operators

137

For .z ∈ C+ define { .

∫

a

Dz (a) = m ∈ C+ ;

} Im m |ϕz (x) + mψz (x)| d x ≤ . Im z 2

0

Then .m ∈ Dz (a) if and only if .

A + mC + mC + Bmm ≤ 0,

which is equivalent to

| | | | | A C ||2 || C ||2 | . m + ≤| | − . | B| B B

The identity .ϕz (a) ψz' (a) − ϕ'z (a) ψz (a) = 1 shows ⎧ ⎪ ϕz (a) ψz' (a) − ϕ'z (a) ψz (a) C ⎪ ⎪ = ⎪ ⎪ ⎨B ψz (a) ψz' (a) − ψz' (a) ψz (a) .

⎪ | |2 ⎪ |−2 ⎪ |C | A || ⎪ | ⎪ ⎩ || || − = |ψz (a) ψz' (a) − ψz' (a) ψz (a)| B B

,

which implies that . Dz (a) forms a non-degenerate disc in .C+ with center and radius as follows: ⎧ ⎪ ϕ (a) ψz' (a) − ϕ'z (a) ψz (a) ⎪ ⎪ the center = − z ⎪ ⎨ ψz (a) ψz' (a) − ψz' (a) ψz (a) . . (7.7) ⎪ ⎪ | |−1 ⎪ ⎪ ⎩ the radius = ||ψz (a) ψ ' (a) − ψ ' (a) ψz (a)|| z z We call . Dz (a) a Weyl disc. From the definition . Da (z) ⊂ Db (z) holds if .a < b, hence ⊓ . Dz (∞) = Dz (a) a>0

turns to be a closed disc or one point, which is equivalently stated as follows: Let .Wz be a subspace of . L 2 (R+ ) defined by .

{ } Wzq = u ∈ L 2 (R+ ) ; − u '' + qu − zu = 0 . q

It is clear that .dim Wz ≤ 2. q

(LC) . Dz (∞) is a disc if and only if .dim Wz = 2. q (LP) . Dz (∞) degenerates to one point if and only if .dim Wz = 1. .

To know that the properties (LC) and (LP) occur simultaneously with respect to z ∈ C+ we define a Volterra integral operator .Vz by

138

7 Appendix

∫ .

x

Vz f (x) =

(ϕz (x) ψz (y) − ϕz (y) ψz (x)) f (y) dy.

0

Then, we immediately obtain an equivalence: u = Vz f ⇐⇒ −u '' + qu − zu = f , u(0) = u ' (0) = 0,

.

{

and .

)−1 ( ϕ ϕz2 = I − (z 2 − z 1 ) Vz1 )−1 z1 ( ψz2 = I − (z 2 − z 1 ) Vz1 ψz 1

for any .z 1 , .z 2 ∈ C. Therefore, if .ϕz1 , .ψz1 ∈ L 2 (R+ ) are valid, then ( .

I − (z 2 − z 1 ) Vz1

)−1

turns out to be a bounded operator in . L 2 (R+ ), hence .ϕz2 , .ψz2 ∈ L 2 (R+ ) hold, which implies: Lemma 7.4 (LC) and (LP) are equivalent to the following statements respectively: q

.

(LC) dim Wz = 2 holds for any z ∈ C. q (LP) dim Wz = 1 holds for any z ∈ C\R.

(7.8)

The boundary .+∞ is said to be of limit circle type if (LC) holds and of limit point type if (LP) occurs. The boundary .−∞ has also a similar classification. A simple sufficient condition for .q under which . L q satisfies (LP) is known from [21]. Lemma 7.5 Suppose . L q f = λ f has a positive solution . f on .(0, ∞) for some .λ ∈ R. Then the boundary .+∞ is of limit point type. Proof Define

∫

x

u(x) = f (x)

.

f (y)−2 dy.

1

/ L 2 ((1, ∞)), which implies that .+∞ is Then .u satisfies . L q u = λ1 u. We show .u ∈ of limit point type due to Lemma 7.4. For this purpose set ∫

x

s(x) =

.

)' ( f (y)−2 dy and t (x) = −s(x)−1 .

1

Then .t (x) > 0 and .

− s(x)−1 = c +

∫

x

t (y) dy 1

with some constant .c. Since .s(x)−1 > 0, we have

7.2 Spectral Theory of 1D Schrödinger Operators

∫

x

.

139

t (y) dy < −c,

1

which implies

∫

∞

.

t (y) dy ≤ −c < ∞.

(7.9)

1

On the other hand, ∫ ∞ ∫ 2 . u(x) d x = 1

∞

'

∫

−1

∞

s (x) s(x) d x = 2

1

1

∫

dx

∞

( )' = −s(x)−1

1

∫

∞

dx t (x)

holds. Since for any .t (x) > 0 ∫ .

∞

∫

∞

t (y) dy +

1

t (y)−1 dy =

1

∫

∞

(

) t (y) + t (y)−1 dy ≥

1

(7.9) shows

∫

∞

.

1

2dy = ∞,

1

∫

∞

u(x) d x = 2

t (x)−1 d x = ∞,

1

∎

which completes the proof.

The existence of a positive solution follows if the operator . L q is lower semibounded on .(0, ∞), namely if there exists a constant .λ0 ∈ R such that ∫

∞

.

0

(

) L q f (x) f (x)d x ≥ λ0

∫

∞

f (x) f (x)d x

(7.10)

0

holds for any . f ∈ C0∞ ((0, ∞)) (.=the set of all infinitely differentiable functions . f with compact support in .(0, ∞)). Especially if .q(x) ≥ λ0 for any .x > 0, then . L q is of limit point type at .+∞. A more general condition for (LP) is “There exists c > 0 such that q(x) ≥ −cx 2 for every sufficiently large x.”

.

It is known that if .q(x) = −cx α with some .c > 0, .α > 2, then the boundary .∞ is of limit circle type. Suppose the boundaries .±∞ are of limit point type for . L q , then it is known that 2 . L q can be defined as a self-adjoint operator on . L (R) uniquely without imposing any boundary condition at .±∞. We consider only the case when the boundaries .±∞ are of limit point type in what follows. In this case, for any .z ∈ C+ there exists a unique .m + (z) ∈ Dz (∞) satisfying ∫ .

0

∞

|ϕz (x) + m + (z) ψz (x)|2 d x ≤

Im m + (z) . Im z

(7.11)

140

7 Appendix

Due to .−ϕz (a) /ψz (a) ∈ Dz (a) (observe .ψz (a) /= 0), we see .

− lim

a→∞

ϕz (a) = m + (z) ψz (a)

for any .z ∈ C+ , which shows analyticity of .m + (z) on .C+ and .m + is of HN. Set f (x, z) = ϕz (x) + m + (z) ψz (x) .

. +

Then, . f + (x, z) satisfies { .

− f +'' (x, z) + q(x) f + (x, z) = z f + (x, z) . f + (0, z) = 1, f + (x, z) ∈ L 2 (R+ )

(7.12)

The above argument shows that there exists a unique solution to the Eq. (7.12) and f ' (0, z) = m + (z) holds. At the boundary.−∞ one can define the HN function.m − (z) similarly.

. +

Definition 7.1 .m ± (z) are called Weyl–Titchmarsh–Kodaira (WTK for short) functions at .±∞ respectively. These .m ± (z) play a crucial role in this book. The WTK functions .m ± are HN functions, hence 1 m + (z) m − (z) .− + m + (z) + m − (z) m + (z) + m − (z) is Herglotz–Nevanlinna as well. The spectral measure .σ of this HN function is called the spectral measure of . L q . Especially, supp.σ coincides with the spectrum of . L q . The Green function .gz (x, y) for the operator . L q , which is the integral kernel of ( )−1 . Lq − z , is given by g (x, y) = gz (y, x) =

. z

f + (x, z) f − (y, z) f + (x, z) f − (y, z) =− Wrons ( f + , f − ) m + (z) + m − (z)

(7.13)

for .x ≥ y.

7.2.2 Shift Operation and Its Properties Since we treat ergodic potentials later, we define the shift operation .θx for .q by .

(θx q) (·) = q (· + x) .

We designate .m ± (z, q), . f ± (x, z, q) for .m ± (z), . f ± (x, z) respectively if it is necessary to indicate the dependence of the quantities on .q explicitly.

7.2 Spectral Theory of 1D Schrödinger Operators

141

Lemma 7.6 .m ± (z, q) satisfy { .

( ) ) ( ∫x f ± (x, z, q) = exp ± 0 m ± z, θ y q dy . ±∂x m ± (z, θx q) = q(x) − z − m ± (z, θx q)2

(7.14)

Proof We prove the identities only for . f + (x, z, q), .m + (z, θx q). First note f (· + y, z, q) ∈ L 2 (R+ ) and

. +

.

− f +'' (x + y, z, q) + q (x + y) f + (x + y, z, q) = z f + (x + y, z, q) , θ q

θ q

q

hence . f + (· + y, z, q) ∈ Wz y . The identity .dim Wz y = dim Wz = 1 implies ( ) f (x + y, z, q) = f + (y, z, q) f + x, z, θ y q ,

. +

which in particular shows . f + (y, z, q) /= 0 and ( ( ) ) f ' (y, z, q) m + z, θ y q = f +' 0, z, θ y q = + . f + (y, z, q)

.

(7.15)

Therefore, we have the first identity of (7.14). The Riccati equation in (7.14) can be obtained from (7.15) also, since . f + satisfy the second-order linear differential equation. ∎ This argument using the shift operation was employed by [23] very effectively in the study of Schrödinger operators with almost periodic potentials.

7.2.3 Expansion of WTK Functions when . z → ∞ In the construction of the KdV flow we use an asymptotic expansion of .m ± (z) at z = ∞. We begin with well-known results. Atkinson √ observed the following fact in 1981. For the simplicity of notation we use .k = −z. If .z ∈ C+ , then .Re k > 0 and .Im k < 0. Since the radius of . Dz (a) has been computed in (7.7), .−ϕz (a) /ψz (a), .m + (z) ∈ Dz (a) imply

.

| | | ϕz (a) || 2 | |. . m + (z) + ≤| | | | | ψz (a) |ψz (a) ψz' (a) − ψz' (a) ψz (a)|

(7.16)

Therefore, to obtain an asymptotic behavior of .m + (z) as .z → ∞, we have only to know the asymptotic behavior of .ϕz (a), .ψz (a) for a fixed .a > 0. To this end we consider the following integral equation: ∫ U (x, k) = ekx +

.

0

x

sinh k (x − y) q (y) U (y, k)dy, k

(7.17)

142

7 Appendix

which determines the .ϕz , .ψz : ϕz (x) =

.

U (x, k) − U (x, −k) U (x, k) + U (x, −k) , ψz (x) = . 2 2k

U (x, k) can be solved by an iteration:

.

⎧ ∫ x sinh k (x − y) ⎪ ⎪ q (y) Un−1 (y) dy for n ≥ 1, U0 (x) = ekx ⎪ ⎨ Un (x) = k 0 ∞ . . ∑ ⎪ ⎪ U (x, k) = U (x) n ⎪ ⎩ n=0

| | Since .|1 − e−2k(x−y) | ≤ 2 if .x − y > 0 and .Re k > 0, we have .

∫ x | | | | | | |e−kx Un (x)| ≤ |2k|−1 |1 − e−2k(x−y) | |q (y)| |e−ky Un−1 (y)| dy ∫ x0 | | −1 |q (y)| |e−ky Un−1 (y)| dy, ≤ |k| 0

⎧| | ) ( ⎪ |e−kx Un (x)| ≤ 1 |k|−1 Q(x) n ⎪ ⎪ ⎪ ⎪ | ∫n! x | ⎨| | | 1 + e−2k(x−y) | | | |e−kx U ' (x)| ≤ | |q (y)| |e−ky Un−1 (y)| dy | . n | | ⎪ 2 0 ⎪ ⎪ ⎪ )n ( −1 |k| ⎪ ⎩ ≤ |k| Q(x) n! ∫x with . Q(x) = 0 |q (y)| dy, hence

and

⎧ ∞ | ∑ | |( | ( ) ) ⎪ ⎪ |U (x, k) − ekx | ≤ ⎪ |Un (x)| ≤ |ekx | exp |k|−1 Q(x) − 1 ⎪ ⎨ .

n=1

∞ | ∑ | ' | | kx | ( | ' ( ) ) ⎪ ⎪ kx | |U (x)| ≤ |e | |k| exp |k|−1 Q(x) − 1 | ⎪ ≤ (x, k) − ke U ⎪ n ⎩

.

n=1

Therefore, we have ⎧ ( −1 kx ) |ϕ ⎪ z (x) − cosh kx| |= O k( e) | ⎪ ⎨| ' sinh kx | =| O ekx |ϕz (x) − k −1 ) ( . |ψz (x) − k sinh kx | = O k −2 ekx , ⎪ ⎪ | | ) ( ⎩| ' ψz (x) − cosh kx | = O k −1 ekx which implies

(7.18)

7.2 Spectral Theory of 1D Schrödinger Operators

143

) ) ( ( ψz (a) ψz' (a) − ψz' (a) ψz (a) = 2i Im k −1 sinh ka cosh ka + O k −2 e2a Re k ) ( = 2−1 e2a Re k i Im k −1 + O k −2 e2a Re k .

.

Therefore, if .z → ∞ in a sector .{ϵ < arg z < π − ϵ}, which is equivalent to .k → ∞ in the sector .{−π/2 + ϵ/2 < arg k < −ϵ/2} for some .ϵ ∈ (0, π), then 2 4e−2a Re k | |, |∼ | | |Im k −1 | | |ψz (a) ψz' (a) − ψz' (a) ψz (a)|

.

which means the exponentially fast decay of the right-hand side of (7.16). This observation makes it possible to reduce the asymptotic problem of .m + (z) to that of .−ϕz (a) /ψz (a), and this is possible essentially by getting more precise estimate of (7.18) if we assume the smoothness of .q (see [8, 20]). Their result is m + (z) = −k

L ∑

.

( ) c j k − j + O k −L ,

(7.19)

j=0 L−1 if [0, a] for { .q }∈ C { some}.a > 0. To obtain a concrete expression for the coefficients . cj we define . c j (x) by j≥0 j≥0

m + (z, θx q) = −k

L ∑

.

) ( c j (x)k − j + O k −L .

j=0

Substituting this expansion in the Riccati equation (7.14) yields

.

−k

L ∑

⎛ c'j (x)k − j + ⎝k

j=0

L ∑

⎞2

) ( c j (x)k − j ⎠ = q(x) + k 2 + O k −L ,

j=0

and we have ⎧ 2 ' ' 2 ⎪ c0 = 1, − c0 + 2c0 c1 = 0, − c1 + c1 + 2c0 c2 = q ⎨ j−1 ∑ . , ' + 2c + cl c j−l = 0 for j ≥ 3 −c ⎪ j ⎩ j−1 l=1

which is equivalent to ⎧ = q/2 ⎪ ⎨ c0 = 1, c1 = 0, c2j−1 ∑ 1 ' 1 . . cl c j−l for j ≥ 3 ⎪ ⎩ c j = 2 c j−1 − 2 l=1

(7.20)

144

7 Appendix

{ } Then,. c j j≥0 in (7.20) is obtained by setting.c j = c j (0). This formula was shown by [16, 51]. The other WTK function .m − (z) has also similar asymptotics, since .m − (z) q (x) = q(−x). Setting .. c j (x) = (−1) j c j (−x) is nothing but .m + (z) with potential .. c0 (x) = 1, .. c1 (x) = 0, .. c2 (x) = . q (x)/2 and for . j ≥ 3 one sees .. 1∑ 1 cl (−x) c j−l (−x) (−1) j c'j−1 (−x) − (−1) j 2 2 l=1 j−1

. c (x) = (−1) j c j (−x) =

. j

1∑ 1 ' . cl. c j−l (x) , = . c j−1 (x) − 2 2 l=1 j−1

{ { } } c j (x) j≥0 becomes the . c j (x) j≥0 for the potential .. hence . . q . Therefore, we have m − (z) = −k

L ∑

.

) ( (−1) j c j k − j + O k −L .

j=0

Lemma 7.7 Suppose .q ∈ C asymptotic expansions

L−1

(−a, a) for some .a > 0. Then, for any .ϵ ∈ (0, π)

⎧ L ⎪ ∑ ) ( ⎪ ⎪ ⎪ c j k − j + O k −L m (z) = −k ⎪ ⎨ + .

j=0

L ∑ ⎪ ) ( ⎪ ⎪ ⎪ m − (z) = −k (−1) j c j k − j + O k −L ⎪ ⎩

(7.21)

j=0

as .z = −k 2{ → ∞} in a sector .{ϵ < arg z < π − ϵ} hold, where .c j is a real number .c j (0) with . c j (x) determined by (7.20). Each .c j (0) − q ( j−2) (0)/2 j−1 is a unij≥0{ } versal polynomial of . q(0), q ' (0), · · · , q ( j−4) (0) , which implies .c1 (0) = c3 (0) = c5 (0) = · · · = c j (0) = 0 for an odd . j is equivalent to .q (1) (0) = q (3) (0) = · · · = q ( j−2) (0) = 0. Later,.m ± are required to have this expansion in a domain. D of.C+ whose boundary ∂ D approaches to .R+ at .+∞. The above procedure ceases to work when the distance between .z and .R converges to .0 due to the singularity of .m ± (z) on .R. To avoid this obstacle we consider two quantities: xi-function and reflection coefficient. Let .m ± be HN functions and define new HN functions by

.

m 1 (z) = −

.

1 m + (z)m − (z) , m 2 (z) = . m + (z) + m − (z) m + (z) + m − (z)

(7.22)

Note that .m 1 coincides with .gz (0, 0) (see (7.13)) when .m ± are the WTK functions of . L q . Generally .log m is of HN since .0 < Im log m = arg m < π if .m is Herglotz– Nevanlinna. Therefore

7.2 Spectral Theory of 1D Schrödinger Operators

.

log m j (z) = α j +

1 π

∫

∞ −∞

(

) λ 1 − 2 ξ j (λ) dλ λ−z λ +1

145

(7.23)

holds with .α j ∈ R, .ξ j (λ) ∈ [0, 1]. In the expression of HN functions the coefficient β vanishes and the measure .σ has a density due to .0 < Im log m < π. If .m ± are the WTK functions of . L q . It is known by [15] that

.

∫ q(x) = λ0 +

∞

.

λ0

(1 − 2ξ1 (λ, x)) dλ (trace formula),

where .ξ1 (λ, x) is the .ξ1 -function for the shifted potential .q (· + x), if .1 − 2ξ1 (λ, x) is integrable and . L q is lower semi-bounded on .R with bound .λ0 . Another quantity is defined by .

R(z) =

m + (z) + m − (z) . m + (z) + m − (z)

The properties of HN functions imply that . R(z) can be extended for a.e. point in .R and .|R(z)| ≤ 1 holds for .z ∈ C+ . The modulus .|R(λ)| for .λ ∈ R is equal to that of the conventional reflection coefficient if .(1 + |x|) q ∈ L 1 (R). Moreover it will be shown that .|R(λ)| for .λ ∈ R is invariant under the KdV flow. Definition 7.2 (xi-function, reflection coefficient) 1. .ξ1 (λ) ∈ [0, 1] is called the xi-function of . L q . 2. . R(z) is called the reflection coefficient for .q or . L q . In the sequel we call .ξ2 (λ) xi-function as well. . R(λ) can control .ξ j (λ) by the lemma below. Lemma 7.8 For two numbers .m ± ∈ C+ let m1 = −

.

) ( m+m− 1 , m2 = ∈ C+ , ξ j = arg m j /π ∈ (0, 1) m+ + m− m+ + m−

and . R = (m + + m − ) / (m + + m − ) ∈ {|z| < 1}. Then it holds that for . j = 1, .2 .

Proof Set .

Note

| | | | |ξ j − 1 | ≤ 1 |R| . | 2| 2

−1 = r eiπξ1 = ir eiπ(ξ1 −1/2) . m+ + m−

−1 4r cos π (ξ1 − 1/2) m+ + m− 2 . |R| = 1 − =1− . 1 1 1 1 + + Im m + Im m − Im m + Im m − 4 Im

146

7 Appendix

Observing .|ξ1 − 1/2| ≤ 1/2 and .

we have

4r ≤ 1, 1 1 + Im m + Im m −

| | . |R| ≥ 1 − cos π (ξ1 − 1/2) ≥ 4 |ξ1 − | 2

| 1 ||2 . 2|

To show the second inequality for .ξ2 , note .

m+m− 1 =− m+ + m− m .+ + m .−

with .m .± = (−m ± )−1 ∈ C+ and .

.− m− .+ + m (−m + )−1 + (−m − )−1 .= m R. = = R −1 −1 m .+ + m .− m− (−m + ) + (−m − )

Then, one can apply the previous argument to .m .± .

∎

Our strategy to obtain (7.21) when .lim z→∞ dist (z, R) = .0 is as follows. Suppose ∫

∞

.

λ M |R (λ)| dλ < ∞

0

| | for some . M ∈ Z+ . Then the functions .|ξ j (λ) − 1/2| also satisfy the same condition by Lemma 7.8, and an identity ∫ .

a

∞

∑ f (λ) z− j dλ = − λ−z j=1 M

∫ a

∞

λ

j−1

f (λ) dλ + z

−M

∫ a

∞

λ M f (λ) dλ λ−z

(7.24)

yields asymptotics for .m 1 , .m 2 , from which the expected estimates for .m ± follow. First we obtain a more explicit expression of .m j from the xi-functions under additional conditions on .m ± . Lemma 7.9 Let .m ± be HN functions satisfying the following conditions for .λ0 ≤ 0: (i) . m ± are holomorphic on .C\[λ0 , ∞) and take real values on .(−∞, λ0 ). (ii) . m ± (λ) → −∞ as .λ → −∞. ∫∞ (iii) . 0 |R (λ)| dλ < ∞. Then there exist .c1 , .c2 > 0 and .λ1 ≤ λ0 such that

7.2 Spectral Theory of 1D Schrödinger Operators

147

(∫ ∞ ) ⎧ ξ1 (λ) − Iλ>0 /2 c1 ⎪ ⎪ exp dλ = m √ (z) ⎨ 1 λ−z −z λ(∫ 0 ). . ∞ √ ξ (λ) − 1 + Iλ>0 /2 2 ⎪ ⎪ −z exp dλ m = −c (z) ⎩ 2 2 λ−z λ1

(7.25)

Proof Under the condition (iii) Lemma 7.8 implies ∫

∞

.

| | |ξ j (λ) − 1/2| dλ < ∞ for j = 1, 2.

0

Since .m 1 (λ) → 0 as .λ → −∞ and .∂λ m 1 (λ) > 0 (due to the Herglotz–Nevanlinna representation), we have.m 1 (λ) > 0 on.(−∞, λ0 ) and.ξ1 (λ) = 0 on.(−∞, λ0 ). Then ) λ 1 − 2 ξ1 (λ) dλ λ +1 −∞ λ − z ∫ ∞ ξ1 (λ) − Iλ>0 /2 dλ + = a1 + λ−z λ0 ∫ ∞ ξ1 (λ) − Iλ>0 /2 = a2 + dλ − λ−z λ0

∫ .

∞

(

) ∫ ( 1 ∞ λ 1 − 2 Iλ>0 dλ 2 λ0 λ−z λ +1 1 log (−z) , 2

with real constants .a1 , .a2 , which yields c1 exp .m 1 (z) = √ −z

(∫

∞

λ0

ξ1 (λ) − Iλ>0 /2 dλ λ−z

)

with a positive constant.c1 , hence the first equality of (7.25) holds. The formula.m 2 (z) can be obtained similarly if we notice .m 2 (λ) → −∞ as .λ → −∞. Namely, letting .λ1 be the unique zero of .m 2 (λ) on .(−∞, λ0 ) and .λ0 if .m 2 (λ) has no zero there, one has .m 2 (λ) < 0 on .(−∞, λ1 ) and .m 2 (λ) > 0 on .(λ1 , λ0 ), hence the second equality ∎ of (7.25) follows. From the estimates for .m 1 , .m 2 one can obtain asymptotics of .m ± . We have to be careful that .m ± can be recovered from .m 1 , .m 2 by m± = −

.

) √ 1 ( 1 ± 1 + 4m 1 m 2 , 2m 1

(7.26)

√ but there is a freedom of choice of . ·. Keeping this in mind we know that given holomorphic functions .m ± (z) coincide with one of .

( ) √ − 1 ± 1 + 4m 1 (z) m 2 (z) / (2m 1 (z))

√ on a domain . D if . 1 + 4m 1 (z) m 2 (z) is defined as a holomorphic function on . D.

148

7 Appendix

1 Lemma 7.10 For .q ∈ L loc (R) assume .q ∈ C L−1 (−a, a) for some .a > 0 and .inf spL q ≥ λ0 . Set

{ .

inf S if S /= φ with L if S = φ

N=

} { S = 1 ≤ j ≤ L − 1, odd j; q ( j) (0) /= 0 . Let .m ± be its WTK functions. Then ⎧ ∑ ) ( ⎨ b j (−z)− j + O z −L−1 if S /= φ .1 + 4m 1 (z)m 2 (z) = N ≤ j≤L ⎩ ( −L−1 ) O z if S = φ

(7.27)

on any sector .{ϵ < arg z < π − ϵ} (.ϵ ∈ (0, π)) with .b N = c N /= 0. Proof From (7.21), ⎛ ⎞ ⎧ ⎪ ∑ ( −L/2 ) √ ⎪ −j⎠ ⎪ ⎝1 + ⎪ (c0 = 1) −z c (z) + m (z) = −2 m + O z (−z) + − 2 j ⎪ ⎨ .

1≤ j≤L/2

⎪ ⎪ ⎪ ⎪ ⎪ ⎩ m + (z) − m − (z) = −2

L ∑

) ( c2 j+1 (−z)− j + O z −L/2 (c1 = 0)

(N −1)/2≤ j≤(L−1)/2

is valid as .z → ∞ keeping .ϵ < arg z < π − ϵ, since Lemma 7.7 implies c = 0 for any odd j ≤ N − 2 and c N /= 0.

. j

Hence, if . S /= φ, (m + (z) − m − (z))2 (m + (z) + m − (z))2 (∑ ) )2 ( −j + O z −L/2 1 (N −1)/2≤ j≤(L−1)/2 c2 j+1 (−z) ) ( = ∑ −z 1 + 1≤ j≤L/2 c2 j (−z)− j + O z −L/2−1/2 ∑ ) ( = b j (−z)− j + O z −L−1

1 + 4m 1 (z)m 2 (z) =

.

N ≤ j≤L

holds with .b N = c N /= 0. If . S = φ, then ) ( m + (z) − m − (z) = O z −L/2 ,

.

) ( hence .1 + 4m 1 (z)m 2 (z) = O z −L−1 .

∎

7.3 Conformal Maps

149

7.3 Conformal Maps Although the Riemann mapping theorem says that every simply connected domain on .C can be an image of a conformal map on .C+ , in our situation a quantitative estimate of it is necessary. In this section we provide a model of conformal map from .− of (5.11). .C\(−∞, 0] to . D A conformal map .ψ on .C+ is easily obtained if .Im ψ ' (z) has a definite sign on √ z, and a more general conformal map in this .C+ . A simple example is .ψ∞ (z) = framework can be constructed for an integer .k ≥ 1 by an integral ψk (z) =

.

k − 1/2 √ z+ k + 1/2

∫

∞

√

z + t (1 + t)−k−3/2 dt.

0

This .ψk satisfies .

Re ψk (z) > 0, Im ψk (z) > 0, Re ψk' (z) > 0, Im ψk' (z) < 0

for .z ∈ C+ , hence .ψk maps .C+ to .ψk (C+ ) (⊂ C+ ), and .φk (z) = ψk (z)2 maps .C+ to .φk (C+ ) (⊂ C+ ) conformaly. Since .ψk (z) takes real values on .[0, ∞), .ψk (z) and .φk (z) can be extended as conformal maps from.C\(−∞, 0] to a domain in.{Re z > 0} and a domain in .C respectively. Set

.

⎧ ∫ ⎪ ⎪ ⎨ ak = 2

1

⎪ ⎪ ⎩

(

0

( )k−1 s2 1 − s2 ds =

√

π⎡ (k) 2⎡ (k + 3/2)

)−1/2 bk = 2ak 2ak2 k (k + 1/2)2 / (k − 1/2)2 + 1

.

Lemma 7.11 The image .φk (C\(−∞, 0]) is described as follows: } { φ (C\(−∞, 0]) = C\ z ∈ C; |Im z| ≤ ω (Re z) , Re z ≤ ak2

. k

with positive smooth function .ω (x) on .(−∞, ak2 ) such that { ω (x) =

.

( ( )) 2ak (−x)−k+1/2 1 + O x −1 as x → −∞ ( 2 ( 2 )1/2 ( )) . 1 + O ak − x as x → ak2 − 0 bk ak − x

(7.28)

Moreover, .φk takes a form of φ (z) = z + f 1 (z) + z −k+1/2 f 2 (z)

. k

(7.29)

with some real rational functions . f 1 , . f 2 (that is, . f j (z) = f j (z) for . j = 1, .2) satisfying { ( )−1 f 1 (∞) = k 2 − 1/4 . . f 2 (∞) = 2 (−1)k ak

150

7 Appendix

Conversely, .φ−1 k (w) has an expression φ−1 (w) = w + g1 (w) + w −k+1/2 g2 (w)

. k

(7.30)

with real .g1 , .g2 analytic in a neighborhood of .∞. Moreover, it holds that )−1 ( g (∞) = − k 2 − 1/4 , g2 (∞) = −2 (−1)k ak . √ Proof Setting .s = (z + t) / (1 + t), we have . 1

k − 1/2 √ z + 2 (z − 1)−k .ψk (z) = k + 1/2

∫

√

z

( )k−1 s2 s2 − 1 ds.

1

∫z ( )k−1 Since the integral . 0 s 2 s 2 − 1 ds is an odd polynomial of degree .2k + 1, the integral ∫ √z ( )k−1 √ −1 k − 1/2 k . p (z) = s2 s2 − 1 ds (z − 1) + 2 z k + 1/2 0 defines a polynomial of degree .k, and ψk (z) = (z − 1)−k

.

(√

z p (z) − p (1)

)

(7.31)

√ holds. It should be noted that . z p (z) − p (1) has a zero of degree .k at .z = 1, so .ψk (z) has no singularity at . z = 1. Set s(x) = Re ψk (x + i0) , t (x) = Im ψk (x + i0)

.

for .x ∈ R. Then, (7.31) implies s(x) =

.

⎧ ⎨ − p(1) (x − 1)−k

for x < 0

, (√ ) ⎩ x p (x) − p (1) for x ≥ 0 (x − 1)−k { √ (x − 1)−k −x p (x) for x < 0 , t (x) = 0 for x ≥ 0

and their asymptotics are s(x) =

⎧ ( ( )) ⎨ ak 1 + kx + O x 2 as x → −0

( )) ( ⎩ ak (−x)−k 1 + kx −1 + O x −2 as x → −∞ ⎧ √ ⎪ ⎪ k − 1/2 −x (1 + O (x)) as x → −0 ⎨ k + 1/2 t (x) = , ⎪ ( ( −1 )) ⎪√ ⎩ −x 1 + O x as x → −∞

.

(7.32)

7.3 Conformal Maps

151

where we have used ⎧ k − 1/2 ⎪ ⎪ (−1)k ⎨ p (1) = (−1)k−1 ak , p(0) = k + 1/2 ( ) . . 2 ⎪ ⎪ z k−1 + · · · ⎩ p(z) = z k − k − 2 4k − 1 From (7.4)

(7.33)

φ (z) = ψk (z)2 = z + f 1 (z) + z −k+1/2 f 2 (z)

. k

follows, which yields (7.29) with ⎧ ( ) ⎨ f 1 (z) = (z − 1)−2k p (1)2 + zp(z)2 − z .

⎩

f 2 (z) = −2 (z − 1)−2k p (1) p (z) z k

,

⎧ { −k+1/2 x if x > 0 ⎪ ⎪ ⎨ Re φk (x + i0) = x + f 1 (x) + f 2 (x) × 0 if x 0 ⎪ ⎪ ⎩ Im φk (x + i0) = √ −x x −k f 2 (x) if x < 0

and

is valid, hence (7.33) shows ⎧ )−1 ( ) ( ⎨ Re φk (x + i0) = x + k 2 − 1/4 + O (−x)−1 .

⎩

Im φk (x + i0) = 2ak (−x)

−k+1/2

(

+ O (−x)

−k−1/2

)

(7.34)

as .x → −∞, and ( ⎧ ( ) ) ( ) k − 1/2 2 ⎪ 2 2 ⎪ ⎪ x + O x2 Re φk (x + i0) = ak + 2kak + ⎪ ⎨ k + 1/2 .

⎪ ⎪ ⎪ ) ( √ ⎪ ⎩ Im φk (x + i0) = 2 k − 1/2 ak −x + O (−x)3/2 k + 1/2

(7.35)

as .x → −0. Since .Re φk (x + i0) = s(x)2 − t (x)2 , .Im φk (x + i0) = 2s (x) t (x), (7.32) implies ⎧ ⎨ Re φk (x + i0) is increasing and moving from − ∞ to ∞ .

⎩

. Im φk (x + i0) > 0 on (−∞, 0) and 0 on [0, ∞)

(7.36)

152

7 Appendix

Therefore, .ω can be defined by an equation ω (Re φk (x + i0)) = Im φk (x + i0) .

.

due to (7.36), and (7.34), (7.35), (7.36) show .ω (x) satisfies (7.28). We show (7.30). Set .ϑ (z) = z 2 . .ϑ is a conformal map from .{Re z > 0} to ~k (s) = ψk (ϑ (s)). Then the function .C\(−∞, 0] and define .ψ .

~k (s) − s F(s) = ψ ) (( ( )−k )−k ( 2 ) = − p (1) s 2 − 1 + s s2 − 1 p s −1 .

is a rational function whose poles only at .s = ±1 and has expansion .

F(s) = c1 s −1 + c2 s −3 + · · · + ck s −2k+1 + ck+1 s −2k + · · ·

( )−1 at .s = ∞ with .c1 = 2k 2 − 1/2 and .ck+1 = − p(1), namely the first coefficient of even order starts from .2k. We consider an equation for a given .t: s + F(s) = t

.

and find a solution of a form s = t + G(t).

.

Since the even coefficients of the power series of . F vanish up to .2 (k − 1), Lemma 7.12 shows that there exists uniquely such a .G that is real and analytic near .t = ∞ ~k (z) is one-to-one on.{|z| > r1 } and the odd coefficients of.G vanish up to.2 (k − 1)..ψ and its inverse is given by .w + G(w) on .{|w| > r2 }. Since .φk (z) = .ϑ (ψk (z)) is a conformal map from ) .C\(−∞, 0] to .φk (C\(−∞, 0]), its inverse is given by ( −1 −1 ~ ϑ−1 (w) for .w ∈ φm (C\(−∞, 0]). Let .φk (w) = ϑψ ⎧ (√ )) ( √ )) ( 1 ( (√ ) ⎪ ⎨ G 1 (t) = G t +G − t = Ge t 2 ( ) . (√ ) . ( √ )) 1 1 ( (√ ) ⎪ ⎩ G 2 (t) = √ G t − G − t = √ Go t 2 t t ( ) Then .G (t) = G 1 t 2 + t G 2 (t 2 ), and we have .

( −1 −1 ) ~ ϑ ϑψ (w) (√ )2 √ = w + G 1 (w) + wG 2 (w) ( ) √ = w + G 1 (w)2 + w (G 2 (w) + 1)2 − 1 + 2 wG 1 (w) (G 2 (w) + 1) .

7.3 Conformal Maps

153

Since .G has the even coefficients vanishing up to .2 (k − 1), .w k G 1 (w) is analytic near .w = ∞. Therefore, setting ⎧ ⎨ g1 (w) = G 1 (w)2 + wG 2 (w) (G 2 (w) + 2) .

we have

⎩

, g2 (w) = 2wk G 1 (w) (G 2 (w) + 1) φ−1 (w) = w + g1 (w) + w−k+1/2 g2 (w)

. k

with some .g1 , .g2 analytic in a neighborhood of .∞ satisfying ( )−1 g (∞) = − k 2 − 1/4 , g2 (∞) = −2 (−1)k ak ,

. 1

∎

which completes the proof.

∑ −j Lemma 7.12 Let . F be a power series of .s −1 given by . F(s) = ∞ and j=1 a j s assume it has the positive radius of convergence and consider an equation: t = s + F(s).

(7.37)

.

| | (i) This equation is uniquely solvable if .|t −1 | is sufficiently small and it has a form: s = t + G(t)

.

with a convergent power series of .t −1 given by

.

G(t) =

∞ ∑

x j t−j .

(7.38)

j=1

{ }n (ii) .xn is determined from . a j j=1 for each .n ≥ 1. The first three coefficients are x = −a1 , x2 = −a2 , x3 = −a12 − a3 .

. 1

(iii) Suppose . F(s) has a form

.

F(s) =

k ∑ j=1

a2 j−1 s −2 j+1 +

∞ ∑

a j s− j

j=2k

for an .k ≥ 1. Then, the coefficients .x j of .G(t) vanish for even . j up to .2 (k − 1). Moreover, if .a2 j /= 0, then .x2 j = −a2 j .

154

7 Appendix

Proof Replacing .s by .s −1 and .t by .t −1 we see the Eq. (7.37) is equivalent to t=

.

s . 1 + s F(s −1 )

(7.39)

The condition on . F implies t (0) = 0,

.

dt d 2t (0) = 1, 2 (0) = 0, ds ds

hence the complex function theory shows the existence of the solution .s(t) of (7.39) in a neighborhood of .0 satisfying s(0) = 0,

.

ds d 2s (0) = 1, 2 (0) = 0, dt dt

which implies the existence of .G of the form { of }n (7.38). One can show inductively that the coefficient .xn is determined from . a j j=1 . To show (iii) one can assume .a j = 0 for every . j ≥ 2k owing to (ii). The relation between . F, . G is rewritten as .

F(s) + G (F(s) + s) = 0.

If we define . . f (s) = − f (−s), then the above equation becomes .

( ) . (s) + G . F(s) . + s = 0. F

. (s) = F(s), the uniqueness implies .G . (s) = G(s), which shows the Therefore, if . F first part of (iii). To show the second part we note that if ⎧ k−1 ⎪ ∑ ⎪ ⎪ ⎪ a j s − j + ak s −k ≡ F1 (s) + ak s −k F(s) = ⎪ ⎪ ⎪ ⎨ j=1 k−1 ∞ . , ∑ ∑ −j −k ⎪ ⎪ G(s) = x s + x s + x j s− j j k ⎪ ⎪ ⎪ ⎪ j=1 j=k+1 ⎪ ⎩ ≡ G (1) (s) + xk s −k + G (2) (s) and with some .bm .

) ( F1 (s) + G (1) (s + F1 (s)) = bk s −k + O s −k−1

holds, which is verified by induction, then the identity 0 = F1 (s) + ak s −k + G (1) (s + F1 (s) + ak s −k ) ( )−k + xk s + F1 (s) + ak s −k + G (2) (s + F1 (s) + ak s −k )

.

7.4 Ergodic Schrödinger Operators

155

together with .

) ( G (1) (s + F1 (s) + ak s −k ) = G (1) (s + F1 (s)) + O s −k−2

implies .xk = −ak − bk . Since, if .k is even and .a2 j = 0 for any . j ≤ k/2, then {(ii) }implies .xk = 0, and hence .bk = 0. However, clearly .bk is determined from . aj , hence .bk = 0 is valid if .a2 j = 0 for any . j ≤ k/2 − 1 regardless of 1≤ j≤k−1 the value .ak . Consequently, we have .xk = −ak if .k is even and .a2 j = 0 for any . j ≤ k/2 − 1 holds. ∎

7.4 Ergodic Schrödinger Operators This section provides several basic facts on 1D Schrödinger operators with ergodic potentials, which are necessary in this book. Let .(Ω, F, P) be a probability space and .{θx }x∈R be a one-parameter group of .F-measurable transformations on .Ω which satisfies .

( ) P θx−1 A = P (A) for any x ∈ R and A ∈ F (stationarity).

( .

) Ω, F, P, {θx }x∈R is called ergodic if it satisfies .

( ) P θx−1 A Θ A = 0 for any x ∈ R =⇒ P (A) = 0 or 1.

For an .F-measurable real valued function . Q on .Ω set q (x) = Q (θx ω) , ω ∈ Ω.

. ω

Then we obtain an ergodic potential .{qω }ω∈Ω . Example 7.1 A simple but important example is quasi-periodic potentials. Set Ω = Rn /Zn and for .α ∈ Rn

.

θ ω = xα + ω, P = the Lebesgue measure on Rn /Zn .

. x

( ) This . Ω, F, P, {θx }x∈R is ergodic if .α is rationally independent and the resulting .qω (x) is a quasi-periodic function. If .n = 1, we have a periodic function and for .n = ∞ in a certain sense we have an almost periodic function. One has more random ergodic potentials. For a technical reason we assume ∫ E (|Q|) =

.

Ω

|Q (ω)| P (dω) < ∞ and Q (ω) ≥ λ0 for any ω ∈ Ω.

(7.40)

156

7 Appendix

E denotes the expectation by . P. Then one can consider the associated Schrödinger operator 2 . L ω = −∂ x + qω .

.

Under the condition (7.40) it is known that .inf sp . L ω ≥ λ0 and the boundaries .±∞ are of limit point type for . L ω for a.e. .ω ∈ Ω. One can apply the Weyl spectral theory to each . L ω . The Floquet exponent is defined by w(z) = E (m ± (z, ω)) (the two expectations coincide),

.

(7.41)

by which the Lyapunov exponent and integrated density of states are defined by γ (z) = − Re w(z) (≥ 0) , N (λ) =

.

.

1 Im w(λ) (λ ∈ R) . π

N (λ) is non-negative, continuous and non-decreasing on .R. Kotani [28] found an identity .

γ (z) 1 − Im w ' (z) = E Im z 4

((

) ) 1 1 |R (z, ω)|2 + Im m + (z, ω) Im m − (z, ω)

for .z ∈ C+ . This identity as well as (7.41) are deduced from the Riccati equations .

± ∂x m ± (z, θx ω) = qω (x) − z − m ± (z, θx ω)2 (see (7.14)),

and the trivial identity E (∂x f (θx ω)) = ∂x E ( f (θx ω)) = 0.

.

Set χ (z) =

.

γ (z) − Im w ' (z) ≥ 0. Im z

Then applying the Schwarz inequality we have / ( √ .E (|R (z, ω)|) ≤ 4χ (z) E

1

+

1

)−1

Im m + (z, ω) Im m − (z, ω) / ( ) √ Im m + (z, ω) + Im m − (z, ω) ≤ 4χ (z) E 4 √ = 2χ (z) Im w (z) (due to (7.41)).

(7.42)

It is also known [28] that ω ∑ac = {λ ∈ R; γ (λ) = 0} = {λ ∈ R; R (λ, ω) = 0} for a.e. ω ∈ Ω.

.

(7.43)

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Index

A A-space inv .A , 21 inv .A+ , 32 .A L , 46 inv .A L , 46 inv .A L ,+ , 67 B Baker–Akhiezer function . f (x, z), 14 . f a (x.z), 34 .B L , 48 Boussinesq equation, 13

.⎡,

25

.⎡real ,

32 49 (0) .⎡n , 49 (0) .⎡n,real , 81 .⎡n ,

H Hardy space . H± , 20, 44 . H+,N , 45 . H N , 45 HN function, 131 K .κ1

C characteristic functions .ϕa , ψa , 21, 47 D . D± ,

44 ) . dζ f (z), 73 .Δ a (z), 21 .det 2 , 60 (

F . f e (z), . f o (z),

45 Floquet exponent .w(z), 156

G (z, h), 106 14 group

. Aq

.Gr

(ν) ,

(a), 23, 83 97 KdV equation, v KdV hierarchy, 13

.χ(z),

L Lax pair, 2 Lyapunov exponent .γ (z), 156 M m-function .m a (z), 23 M-space refl , 33 .M (n) .M L , 76 ~(n) , 82 .M L .M∞ , 103 N NLS, 120

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 S. Kotani, Korteweg–de Vries Flows with General Initial Conditions, Mathematical Physics Studies, https://doi.org/10.1007/978-981-99-9738-1

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162

Index .∑ac ,

P .p± ,

20

Q Q-space refl , 36, 39 .Q (n) .Q L , 81 .Q∞ , 103 .qζ (z), 28 R reflection coefficient . R(z), 91 Reflectionless, 19

q

92 100

.∑refl ,

T Tau function .τ a (g), 26 (2) .τ a (g), 60 Toeplitz operator . T (a), 20 . TN (a), 46 transfer matrix .Ta (z, g), 104 W .Wa ,

20 WTK function .m ± (z), 140

S . Sa , . Hg ,

25 Sine–Gordon equation, 128 Spectrum

X xi-function . ξ j (z), 91