Operational Methods

Citation preview

operational methods V. P. MASLOV

=

MIR PUBLISHERS MOSCOW

B A

1

ee = 2 n

(A+B) n!

operational

methods

V. P. MASLOV _ _ __

Translated from the Russian by V. GOLO, N. KULMAN, G. VOROPAEVA

Mir Publishers

Mosco111

First published 1976 Revised from the 1973 Russian edition

This translation has been read and approved by the author Professor V.P. Maslov, D.Sc. (Math.)

H a ane.n.uuc"oM Habme

© English translation, Mir Publishers, 1976

CONTENTS

Sec.

1.

Sec. Sec.

2. 3.

Sec.

4.

Sec. 5. Sec. 6. Sec. 7. Sec. 8. Sec.

Sec. Sec. Sec. Sec. Sec.

9.

1. 2. 3. 4. 5.

Sec. 6. Sec. 7. Sec. 8. Sec. 9.

Sec. Sec. Sec. Sec. Sec. Sec. Sec.

1. 2. 3. 4. 5. 6. 7.

Sec. 8. Sec. 9. Sec. 10.

Preface . . . . . . . . . . . . . . . . . . . . Introduction to Operational Calculus . . . . . . . Solution of Ordinary Differential Equations by the Heaviside Operational Method . . . . . . . . . . . . . . . . . . . Difference Equations . . . . . . . . . . . . . . . . . . • Solution of Systems of Differential Equations by the Heaviside Operational Method . . . . . . . . . . . . . . . . . . . Algebra of Convergent Power Series of Noncommutative Ope.............. . rators Spectrum of a Pair of Ordered Operators . . . . . ·. Algebras with ~-t-Structures . . . . . . . . . . . . An Example of a Solution of a Differential Equation Passage of the Equation of Oscillations of a Crystal Lattice into a Wave Equation . . . . . • . . . . . . . . . . . . The Concept of a Quasi-Inverse Operator and Formulation of the Main Theorem . . . . . . . . . . . . . . . . . . . . Chapter I Functions of a Regular Operator Certain Spaces of Continuous Functions and Related Spaces . . . . . . . . . . . . . Embedding Theorems The Algebra of Functions of a Generator . . . . . . . . . The Extension of the Class of Possible Symbols . . . . . Homomorphism of Asymptotic Formulas. The Method of Stationary Phase . . . . . . . The Spectrum of a Generator . . . . . . . . . . . . . . . Regular Operators . . . . . . . . . . . . . . . . . . . . The Generalized Eigenfunctions and Associated Functions Self-Adjoint Operators as Transformers in the Schmidt Space Chapter II Calculus of Noncommutative Operators Preliminary Definitions . . . . . . . . . . • . . . . . . The Functions of Two Noncommutative Self-Adjoint Operators The Functions of Noncommutative Operators The Spectrum of a Vector-Operator Theorem on Homomorphism . . Problems . . . . . . . . . . . Differentiation of the Functions of an Operator Depending on a Parameter . . . . . Formulas of Commutation . . . . . . . . . Growing Symbols The Factor-Spectrum . . • . • . .

7

13 13 20 22 24

35 40 56 58

100

147 149 154

158 173 181 188

194 198 205 210 210 224 228 231 239 242

251 256 261 265

6

CONTEN'IB

Sec. 11. The Functions of Components of a Lie Nilpotent Algebra and Their Representations . . . . . . . . . . . . . . .

Sec. Sec. Sec.

1. 2. 3.

Sec. 4. Sec. 5. Sec. 6.

Chapter III Asymptotic Methods Canonical Transformations of Pseudodifferential Operators The Homomorphism of Asymptotic Formulas The Geometrical Interpretation of the Method of Stationary Phase The Canonical Operator on an Unclosed Curve . . The Method of Stationary Phase • The Canonical Operator on the Unclosed Curve Depending on Parameters Defined Correct to 0 (

!)

Sec. 7. Sec. 8. Sec. 9. Sec. 10.

V-Objects on the Curve The Canonical Operator on the Family of Unclosed Curves The Canonical Operator on the Family of Closed Curves An Example of Commutation of a Canonical Operator with a Hamiltonian Sec. 11. Commutation of a Hamiltonian with a Canonical Operator Sec. 12. The General Canonical Transformation of the Pseudodifferential Operator

Sec. 1. Sec. 2. Sec. 3. Sec. 4. Sec. 5. Sec.

Sec. Sec.

6.

1. 2.

Sec. 3. Sec. 4. Sec. 5. Sec. Sec.

6. 7.

Chapter IV Generalized Hamilton-Jacobi Equations Hamilton-Jacobi Equations with Dissipation The Lagrangean Manifold with a Complex Germ y-Atlases and the Dissipativity Inequality Solution of the Hamilton-Jacobi Equation with Dissipation Preservation of the Dissipativity Inequality. Bypassing Focuses Operation Solution of Transfer Equation with Dissipation Chapter V Canonical Operator on a Lagrangean Manifold with a Complex . . . . . . . . . . Germ and Proof of the Main Theorem Quantum Bypassing Focuses Operation . . . . . . . . . Commutation Formulas for a Complex Exponential and a Hamiltonian . . . . . . . . . . . . . . . . • • . C-Lagrangean Manifolds and the Index of a Complex Germ Canonical Operator Proof of the Main Theorem . . . . Appendix to Sec. 5 . . . . . . . Cauchy Problem for Systems with Complex Characteristics Quasi-Inverse of Operators with Matrix Symbols Appendix. Spectral Expansion of T-products Index . . . . . . . • • . • . . . . . . . . . .

266 273 273 294 301 303 312 315 321 327 333 339 346 348 355 356 360 372 378 386 401

419 419 440 452 469 482 493 503 519 545 557

PREFACE

Operational methods are such methods, which make it possible to reduce differential problems to algebraic problems. This is why these methods are of particular use to specialists dealing with applied mathematics. This book is devoted to one, but sufficiently general operational method, which absorbs many operational methods known to date and allows for the uniform solution of both classical problems, involving differential equations with partial derivatives, and the absolutely new problems of mathematical physics, including those connected with non-linear equations in partial derivatives. We shall proceed to describe this general method after studying the methods well-known in mathematics for calculating operators (mainly self-adjoint operators), for which Heaviside's method served as a source. The main theorem stated in this book belongs to the theory of operators and is proved in the last chapter, but its formulation is given in Introduction as well. This theorem may lead, in particular, to the existence and uniqueness theorems for hyperbolic, elliptic and parabolic equations with variable coefficients and allows to reduce them to integral equations of the second kind with smooth kernels, i.e., to provide an effective solution for these equations. We have in mind that by detaching from the solution the non-smooth (or rapidly oscillating) components, we can thus reduce the problem to the one that is easily solved on an electronic computer. When the rapidly oscillating part detached from the solutions was compared with the exact solution by mearis of a numerical experiment in the one-dimensional example of Sec. 8 of Introduction, their near coincidence was revealed. It turns out, however, that the computer is unable to perform such an experiment even in a two-dimensional case because of the enormous number of operations. Under these circumstances the only thing to do is using suitable asymptotics and reducing the initial problem to an integral equation. Thus, the approximate solution constructed in the main theorem,

8

OPERATI'ONAL METHODS

detaching the non-smooth or rapidly oscillating part, serves as a kind of natural supplement to the electronic computer: together they provide a numerical answer. This book on operational methods should be accessible to seniorcourse students of mathematics and physics faculties at universities and departments of applied mathematics. This means that only a knowledge of classical analysis is required of the reader. The book provides explanations in sufficient volume of such concepts as the theory of Banach algebras of distributions (Chapter I), the theory of linear differential and difference equations (Sees. 1, 2, and 3 of Introduction), the theory of non-linear equations of the first order with partial derivatives (Chapter IV). This material may be also of use to the reader who is already familiar with these questions, because rather often it is not presented in traditional style, and adapted for further reference. The reader who studies the book thoroughly will be equipped to carry on independent research in the modern theory of linear, non-linear differential and differential-difference equations with partial derivatives. Besides, a study of the concrete problems presented here may serve as an excellent springboard for further investigation, although from a certain viewpoint, of the theory of representations, topology and the theory of sheafs. An example of the latter is the theory of V-objects (Chapter III). As to applying the described methods in physics, quite obviously, it is not limited to the examples given in the book. Additional mention should be made of the asymptotics of solutions in the band solid-state theory, in problems of molecular collisions, in the theory of laser resonators, in chainreaction equations in chemistry, in problems of refraction and diffraction, in derivation of integral equations of the LippmannSchwinger and Faddeev type, in calculating quasi-classical amendments to the Thomas-Fermi equation, amendments to electronic plasma equations, asymptotic solutions of Hartree's equations, in electronic optics, in problems of above-barrier reflection and in many other problems of modern mathematical physics. This book has been written in such a way as to serve the widest possible circle of readers. It is suitable for two methods of study. The reader, who seeks to avoid fine assessments and passing to the limit and only wishes to master the practical techniques for obtaining asymptotic solutions, may omit that part of the book which is devoted to functional analysis. The "Introduction to Operational Calculus" has been written with this purpose in view; having studied it, the reader can tackle operator techniques (omitting Chapters I-II) learning how to reduce concrete problems to an integral equation, detaching the non-smooth part of the solution. In following the second method of learning, which consists in the gradual and deeper study of operational methods, it is best

PHEFACE

9

to begin with the first chapter and to read Introduction only after Chapter II. The most effective way of mastering the subject, however, consists rather in first reading Introduction and then reading all the book in succession. The reader should nevertheless be warned that all these methods are not at all easy, because the book provides a new operational calculus-the calculus of ordered operators. This book is actually a synopsis of the course of lectures delivered by the author in the duration of three years at the Department of Applied Mathematics at the Moscow Institute of Electronic MachineBuilding, consecutively to third-, fourth- and fifth-year students. The material of the last chapters was also given in lectures (in the duration of three years) to fifth-year (graduating) students specializing in mathematics at the Physics Department of the Moscow State University. Besides, this course of lectures (an abridged version) was also delivered by the author at several mathematical schools (at the International Mathematics School at Sopot in 1971, in Pushchino in 1971, in Voronezh in 1972, and other places). The experience gained in delivering this course of lectures showed that, despite expectations, operational calculus, with the parallel examination of comparatively few examples at seminar sessions, is grasped more easily by students than questions connected with traditional functional analysis, fine assessments, functional spaces and passing to the limit. The students of senior courses and post-graduates can quickly learn to solve complex problems connected with operational methods. In this book, for instance, the reader will find important formulas obtained by M. V. Karasev (Theorems 4.4 and 6.6 of Introduction) and V. G. Danilov (Theorem 1.1 in Chapter III). If we should draw an analogy between the exposition of operational calculus given in the "Introduction to Operational Calculus" and the hypothetical exposition of differential calculus (see the left column) the approximate result would consist in the following: Sees. 1-3 deal with the ring of polynomials. Sees. 4-5. The rules of the formal differentiation of polynomials are given; various formulas of differentiation for polynomials are derived: formulas for differentiating the product; the composite function, expansion in a Taylor series.

Sees. 1-3 deal with the Heaviside method and operator calculus with constant coefficients. Sees. 4-5. The rules of the new calculus are derived on the example of formal power series of ordered operators. Formulas are derived for commutators of series, composite function and expansion in a Newton series.

10

In Sec. 6 the system of axioms is introduced and the same formulas are derived for arbitrary functions in the form of theorems, thus determining all formulas of the techniques of differentiation. Sec. 7 gives a simple example. Sec. 8 shows how a new solution for a classical physics problem is obtained with the use of introduced techniques.

Sec. 9 sets a problem on the simplest differential equation. The fundamental concept of a characteristic polynomial is introduced and the physical meaning of initial conditions is discussed. The main theorem of existence is formulated and the formula for the solution of an equation in a particular case is given.

OPERATIONAL METHODS

In Sec. 6 the system of axioms is introduced and the formulas for arbitrary functions in the form of theorems are the same, thus determining all formulas of the techniques of the calculation of ordered operators. Sec. 7 gives a simple example. Sec. 8 shows how new physical effects are obtained with the use of introduced techniques in studying the classical problem of deriving the wave equation from equations of the oscillations of a crystal lattice. The main problem is formulated in Sec. 9, the fundamental concept of characteristics for functions of an ordered set of operators is introduced, and the physical meaning of absorption conditions is discussed. The main theorem is formulated and the explicit formula is given for the solution of the main problem in a particular case (the general formula is given in the last chapter).

Further (if we are to continue the analogy) the theory of the derivative as a limit (Chapter I) and partial derivatives as multiple limits (Chapter II) is conducted in succession. The theory of functions in Chapter I is built by systematic use of the conception of "completing with limits", just as real numbers in Cantor's theory are elements of the completion of a set of sequences of rational numbers. This point of view is in accord with the initial conception of physicists with regard to Dirac's delta function as the limit of bell-shaped functions. Sees. 3-9 of Chapter I are devoted to calculating the functions of one operator. These sections introduce the concepts of generating and regular operators generalizing the concepts of self-adjoint and normal operators, respectively. The theorem is proved for regular operators which shows their generality in the case of a discrete spectrum: the regularity of an operator is a necessary and sufficient

PREFAC.E

11

condition for the completeness of eigen and associated elements. The known calculation of self-adjoint operators is derived as a consequence. At the beginning of Chapter II the calculation of functions of two noncommutative ordered operators, their joint spectrum and spectral expansion are studied in detail. Then the functions of several regular operators are examined and formulas for them are derived, which had been obtained from axioms in Introduction. All the techniques for calculating noncommutative operators are thus built, but on a functional basis now. Operational calculus as such ends with this. The rest of the book is devoted to a special transformation making it possible to prove the main theorem. This transformation is called the canonical operator. A great deal of preparatory work is conducted in Chapter III. A canonical operator is introduced in the simplest real one-dimensional case (depending on ordered operators). Chapter IV calls for an examination in greater detail. This part of the book is required for the ultimate construction of a complex canonical operator, but, generally speaking, the chapter is quite detached and in no way connected with the techniques of ordered operators. It may even be read all at once. The chapter is devoted to constructing the solution, in the large, of equations generalizing Hamilton-Jacobi equations. We introduce the notion of Lagrangean manifold with a complex germ which results in a geometric interpretation for solutions of equations of the Hamilton-Jacobi type. The following physical analogy may be cited to illustrate the point. When a stone is dropped into water it causes the waves to spread over the surface in even circles at first; when the wave is reflected the picture that is produced can hardly be given a geometric interpretation. The same may be said of the HamiltonJacobi equation (a particular solution of a certain Hamilton-Jacobi equation will be precisely the one that causes these circles) which on intervals of time not exceeding a certain t1 has a simple and smooth solution. Further, when t > t 1 there occurs a similarity of imposition of waves reflected many times. In order to gain a clear understanding of multi-valued functions (branching of the solution), it is necessary to make them uniform in much the same way as this is done by means of Riemann's sheets for analytical functions with branching. It is precisely the construction of the object in the phase space of the Lagrangean manifold with a complex germ that makes it possible to "unravel" multi-valued solutions of Hamilton-Jacobi type equations (with absorption). The concept "index"-as a whole number-introduced here makes it possible to designate these solutions (sheets) on the Lagrangean manifold with a complex germ.

12

OPERATJ'ONAL METHODS

Besides, the indices of closed paths on the Lagrangean manifold with a complex germ constitute an important characteristic of this object (the characteristic class of an object). After this the complex canonical operator is built in Chapter V and the main theorem is proved. The results of Chapters 1-V and Sees. 4-9 of Introduction mainly belong to the author (with the exception of the theorems in Chapter I concerning self-adjoint operators, and the theorems of Karasev and Danilov mentioned above). The starting point of this research was Feinman's remark* to the effect that if the order of action of the operators is determined by indices, then the operators become as if commuting. The theory of the Lagrangean manifold with a complex germ originated as a result of the study of Leray's works on the Cauchy problem. A. A. Kirillov, who edited the book, made a number of valuable suggestions in principle concerning the structure of the book as a whole. I benefited largely from my consultations with A. A. Samarsky, discussions with D. V. Anosov and V. V. Kucherenko, for which I am deeply grateful to them. I am also very grateful to P. P. Mosolov, G. A. Voropaeva, V. L. Dubnov who made several valuable remarks pertaining to the author's manuscript. This book, however, would never have been written as a school aid if not for my pupils, who made notes and worked at this course of lectures. Moreover, they did not even have detailed synopses of Chapters III and V at their disposal. The first two chapters were jotted down by V. L. Dubnov, the third by M. V. Karasev and V. L. Dubnov, the fourth by V. G. Danilov, the fifth by V. G. Danilov and M. V. Karasev. The chapters IV and V were edited by G. A. Voropayeva; A. G. Prudkovsky made all the calculations on a computer. Besides this, our numerous discussions with V. L. Dubnov, M. V. Karasev, V. G. Danilov, S. Yu. Dobrokhotov, A. G. Prudkovsky and G. A. Voropayeva rendered me invaluable assistance. In preparing the manuscript for the publisher I was also helped by A. G. Davtyan and G. Yu. Malysheva. My gratitude to them all is boundless. V. P. Maslov

* In the article "Operational Calculus Relating to Quantum Electrodynamics" (Phys. Rev., 84, 1951) in the section "Description of Method of Designation". By the recommendation of the editor, Feinman's calculus has not been included in the book, just as the applications to physics enumerated on page 8.

INTRODUC TION TO OPERATIO NAL CALCULUS

Sec. 1. Solution of Ordinary Differential Equations hy the Heaviside Operational Method Let Coo (R) (or C00 ) denote the set of infinitely differentiable oo. The operation (operator), d/dx, functions

belonging to K, i.e., (zH z2 E (J (P (zH z2) = 0) for all P (z1 , z2) E K. Note. Theorem 4.2 states that if Q (z1, z2) E K then Q (z11 z2) X X P (z11 z2) E K, where P (zH z2) is a polynomial.

Let A, B, C be elements of the algebra of operators. Consider a set ,JJ' (zH z2) of polynomials P (z1, z2) with such coefficients in C that 2

( 1

3 )

CP A, B =0.

Definition. The spectrum ac 2

(.A, B) c

C2 of a pair

A, B rela-

tive to C is an intersection of sets of zeroes of polynomials belonging to r!f (z1 , z2); i.e.,

(Z

1,

z2 EO'c

'1 B3)) (P (z 1, (A,

z2) = 0)

for any P (z1, z2) E r!f (z1, Z2). Note, that if Q (z10 z2) E r!f (z11 z2), then Q (z1, z2) P (zH z2) E E r!f (z1, z2), where P (z11 z2) is a polynomial.

Consider an important example of the spectral expansion of the 2

( 1

2 )

function Cf A; B . We shall use formula (4.3) of the commutation of Hamiltonian and exponential. Take in this formula X=x, P= - i

:x,

S(X, P)=S(x),

g(X, P)=g(x)

(cf. (4.31)). Thus, in the right-hand side we obtain

e;s(~) f (~, [P+ ~~ ]) g U) = eiB(x>[f ( ~.

[P+ ~: ]) ] g (x).

38

OPERATIONAL METHIODS

By virtue of Theorem 4.5 we get 2

as )

1

f ( x,[P+ax-] g(x)=

{4 a~) -f (4x 2!\x=f (x, ~!) g(x)+P , ax 1 3, ax g(x)+ as as ax-ax

a1 )

a~

2 4 fJ2f ( 6

a}

as )

5

ax' [P+ax

+PP 6P2 x;Tx,

]g(x).

Calculate the middle term of the equation. Consider to the effect i

2

( 1

3)2

~ x-x = i {

~

.

cp(x)= x 2 cp(x)- 2x

:x xcp (x) + x :x cp (x)} = 0. 2

(1 s)

2

a Hence the spectrum x, x with respect to ih ax is located in R2 by the formula x 1 = x 2 • Hence i

2 (1 3)n =0

~ x-x

forn::;>2

by Theorem 4.2. We have the following "spectral expansion" as a corollary:

2 (13)

(33) 2

2

3) i~P x,x "'iP x,x :x+i:xp~ (33)(1 x,x X - X " ' "'iP(x, x) :x-i[P~(x, x)x :x-P~(x,x) :x x]"' "'iP (x, x) :x

+ iP~ (x, x).

Hence .

P2 f

(~, a~;ax) -! (;, 1 3

a1;ax)

( )

Cj)X"'

as;ax-aSjiix

~- i [

:; lp=astax

:x + ; :~ IP=apas ~:~ Jcp (x).

An operator P such that p ~ +_!_ a2S a2j cp Ql- iip as ax 2 ax2 ap2 as P=ax P-ax

_!}_I

I

INTROIDUCTllON TO OPERATIONAL CAIJCULUS

39

is called a transfer operator (for the Hamiltonian f (x, p) and the given S (x)). Note that we do not use only the concept of spectrum in spectral expansions, the number of terms in the Taylor expansion being important as well. This number is connected with a multiplicity of spectrum in the case of matrices. In the example considered above it is natural to define the multiplicity of spectrum located on the bisector of the first coordinate angle as equal to two. The problem of the multiplicity of spectrum (i.e., the problem of the number of terms in the Taylor expansion) is very hard in general. This problem is reduced to the study of a "subsidiary" Banach space Bmld in Chapters I and II. We shall point out some analogies between the given example and the spectral expansion of matrices to clarify the direction taken in investigating the properties of operators in Chapters I and II; these are the following: 1

1. The operator x acts in the space of differentiable functions Since the following inequality because the second operator is i is valid

:x.

I

:x (eixtg (x))

I+ Ieixtg (x) 1~(1 +It I) max (I g' I+ Igj)

the operator of multiplication by eixt increases as the first power of t in the space of differentiable functions. 2. A matrix A, with a maximal length of adjoint elements equal to 1, satisfies the condition

I ew g I ~ (1

+ It

I)

I g 1.

where g is a vector, I g I is its modulus. 3. The Taylor formula in both cases is reduced to two terms of the spectral expansion. In Chapter II we shall see that the number of terms in spectral expansion is closely connected with the estimates of growth of the operator eiAt. Note. In the sequel it is important to bear in mind the following properties of matrices: (1) any matrix A may be put in the form

(5.2) where A 1 , A 2 commute, A 2 and A 1 have a real spectrum. (2) We can define the matrix spectrum in the following way; let K be a set of polynomials P (x, y), such that P (A 1 , A 2 ) = 0, the spectrum of the matrix P (x, y) = 0 for any P (x, y) E K, the eigenvalues of A in this case are equal to x iy, where x, y E a (A). Thus we may consider only real roots of polynomials of two variables. Therefore, though formula (5.2) is itself a kind of spectral expansion, it is worthwhile remembering when studying a number of non-commuting operators, that each of the operators may be put in the form of a function of two operators with a real spectrum. We shall return to the problem in Chapter I.

+

Sec. 6. Algebras with 1-t -Structures The introduced calculus is still insufficient to solve differential equations with variable coefficients. We shall introduce the necessary algebraic constructions to the effect. We shall first consider the most simple case and then turn to more complicated cases (cf. axioms (~-t 4), (~-t 6 ) below). We shall prove a number of theorems and, on their basis, demonstrate in Sec. 7 how they can be used. Next in Sec. 8 we shall demonstrate the power of the operational calculus by the classical example of the deduction of the wave equation. We shall see that the ·operational calculus provides an adequate means for studying mathematical and physical effects proper to the transition from the system of equations of oscillations of a lattice to the wave equation. The operational calculus of convergent series of operators is introduced in the axiomatic way, i.e., the main points are formulated as axioms, then formulas are derived and the axioms are verified to be valid for the basic operators necessary for the solution of differential equations. The method is convenient as well from the didactic point of view, i.e., it makes easy to become familiar with the ordered operators calculus techniques. We shall work within the framework of the basic algebraic structures. Let .}{; be an algebra with a unit over R, .}{; is generally noncommutative. The elements of .}{; will be called operators. Let cffoo be a set of infinite differentiable functions I (x), x E Rk (k is not fixed) growing togethel' with all their derivatives as I x 11 or less at infinity (l is defined for every function I separately). Functions belonging to cffoo are called symbols. A symbol is called of rank k if the corresponding function depends on k variables. An algebra .Jl; is provided with a I-t-structure if for any finite set A 1 , A 2 , • • • , A k of operators belonging to a set M c: .Jt and any set of numbers n 1 , n 2 , • • • , nk (so that n 1 =!= ni if A 1 , Ai do not commute), the following operation is defined 1

11: (x1 -+AI> ... , xk-+ Ak),

INTROcr:JUGT]ON TO OPERATIONAL GAIJGULUS

41

which substitutes an operator A E A written in the form nk ) ( ni A=[! Ail ..• , Ak ]

for the symbol f (x1 , • • • , xk)· We shall cancel out the brackets [ ], where it will not lead to ambiguity. The operation 11 satisfies the following axioms: (~.t 1 ) The homogeneity axiom: if a E R, then nk ) (ni nk) ( ni [af Ail ... , Ak ] =a[/ A1, ... , Ak, ];

in particular, if f (x1 ,

xk)

••• ,

= 0, then

nk ) [/ Ail ... , Ak ]=0. ( ni

(~.t 2 ) The shifting of indices axiom. Let n 1 , n 2 , • • • , nk and m1 , m 2, ..• , mk be such two sets of indices that, if i =t= j and (ni ... , Ak

... , Bz ],

(6.2)

where ni, mj are such indices that mi+t

> ni+t > mi > n;.

Besides, there is an equality

nt nk \ (nJ. nit ) f ( At, ... , Ak} =! At, ... , Ak ,

(mi g B1o

mz)

... , B 1

(m1 = g Bt.

(6.3)

mz )

... , Bz

for any term of the sum as well. Then the sum (6.2) is equal to

n1

mz )

(mi

[/ ( A1o ... , Ak ]+[g Bh ... , B 1 nit )

by the sum axiom since from (6.2) and (6.3).

n; =I= mj for

(6.4)

]

any i, j. The theorem follows

Theorem 6.2. (The second sum theorem.)

ni

def n;

where A;+B;= [A; +B;].

Proof. We may assume I n 1 - n; I> 2 for j =I= i without loss of generality by virtue of the axiom {f1 2). Hence ni

)

(ni

[ ( A; +B; f

ni-i

ni+i

nk )

At. ... , A;-t. Ai+t. ... , Ak ] -

n.

( ni

ni-1

ni+i

n.

(ni

ni-1

ni+t

nk )

-[Ad Ab ... , A;-t. A;+t. ... , Ak ] nk )

-[Ed Ab ... , A;-t. Ai+b Ak ]= ni

n;+i

=[A;+B;- A; -

n;+2)

B;

(n 1

n;_ 1

n;+ 1

nk )

f At. ... , A;-t. A;+t. ... , Ak ].

44

OPERATIONAL METHODS

We obtain the following formula with the help of the axiom of sum, the axiom of shifting of indices and the axiom of correspondence: n;

(

)

n;+1

ni

n;+2

n;+2

n;+1

B; ]=[A;+B;]-[ A;]-[ B; ]=

[ A;+B; -A; 1

1

1

=[A; +B;]-[A;]-[B;]= A; +B;-A;-B; =0. Then we apply the zero axiom and the theorem is proved. Theorem 6.3. (The theorem of product.) Let n1 , n 2 , • • • , nk, p 1 , • • • , p 1, r 1 , • • . , rm be integers such that p; ni for n1

nk )

all i, j and [f ( Au ... , Ak ] =F. Then for any symbol g (x1 , • • • , Xz, Yr. ... , Ym) there is a relation (

nk )

n1

( P1

Pl

r1

[f A11 ... , Ak g B11 •• , , Bz, C1o n1

(p

1

Pl

r1

rm )

••• ,

Cm ] =

rm )

=[Fg Bb .. . , Bz, Cb ... , Cm ]. Proof. We may assume without loss of generality by axiom (~-t 2 }, that there exists such a number n that n =F n 1 , • • • , nk and Pt < roof. Let k the equality [t

=

2, A 1 =A, A 2 =B. It is necessary to prove

(A, .B h- [t Ct .B h= =[(A B) f U. ~) -! U. M+! (1 ~) -! u. ~) 3

U-~) (1-1)

'

].

We shall put the right-hand side into this form with the help of axioms (~-t 1 ), (~-t 2 ), (~-t 4) and Theorem 6.3

[(AB-sA) 1 U. 1) -t U. 1) +t U. 1) -t U. 1) ]= U-~) (~-1)

3

=[A

I

U. 1) -/ U. 1) +I U. 1) -/ (11) ]= 1

5

A-A

3 (12) - / (52) A,B

=[A f A,B

1

5

3

]+[A I

U. ~)-/ U. 1) ]= 1

A-A

5 (12) - / (52) A,B

=[At A,B

1

5

A-A

5

1 (52) - / (12) A,B ]=

]+[At A,B

A-A

1

5

A-A

=d~-;t)' U. 1~ -~ UJ) ]=[-t (A, h) +t (~. hh= A-A

=[t (l sh-[t L~. hh.

Q.E.D.

The proof in the general case is the same. We shall indicate a formula for a composite function. Theorem 6.5. (K-formula.) Let f, g be symbols of rank 1 and 2, respectively, A, B operators belonging to M such that [A, B] EM,

[g

(A,

shE M.

1 ([g (A,

Then

.Bh) =[/ (g (A . .B) h+[[A~ B] 6~ (A. •.4. s) x

ilg ( 3 4 6 ) 62f X-,- A, B, B 72 uxz ux

x (6.6)

OPERATr0NAL METHODS

46

Indeed~

The note made to Theorem 6.4 is also true of this theorem. the following equality is true (.1 2 [/ Cb C 2 ,

••• ,

1

=[/ ( Ct.

s+1\ k+3 k+2 k+1 (1 2) k Ck, [ g A, B ], Ck+t• Ck+z, ... , Cs I]=

2 C2 ,

k+5

k

••• ,

J 6g

A, A, B

xt

k+B

( 1

2

k

,Ct. Cz, ... , Ck, g (

1 2)

[ g A, B ],

s+2) (k+i k+2) k+3 A, B , Ck+~> ... , Cs ]+

(k+3 k+7 k+9) bg (k+3 k+t. k+6)

+[ [ A, B -6 X

Ck, g

-6 x2

A, B, B

-

62f 2 -

X

6xk+i

(k+1 k+9) k+2 ( 1 2) A, B , [ g A, B ],

s+9) Ck+t. ... , Cs ].

k+10

(6.7)

We shall call formula (6.7), as well as a more particular formula (6.6), K-formula. Proof of Theorem 6.5. There exists an equality 6! . f(z)=f(g(x 1 , x2 ))+(z-g(xt. x 2)) ilx (g(xt. x2 ); z).

(

2

1

3)

Applying the operation f.t: z-+ g, x 1 -+ A, x 2 -+ B of the equality, where g = [g of sum we get

to both sides

Lt Bh, and using the first theorem

[t(g)]=[t(g()t,.B)h+[(~-g(.4,.8)) ~~ (g()t,.B), ~h. We further obtain [t (g)] = [t ( g ( .4,

[t

8) h+ [ (~- g (A, 1) ) g~

(g (A, 8) , ; h=

(g (~. 1) h+ [; ~~ (g(~ •.B), ~) 3 5 ) 6! ( ( 1 5 ) 2) -:-g ( A, B & g A, B , g ]+

"

+[ ( A, g

)

6g (

6x 1

6) 2 ( 1 7 ) 7 ) 62/ ( 3 5 A, A, B 6x2 g A, B , g, g,]

2

3

with the help of axiom (~-t 2 ) and changing the indices of A and g by Theorem 6.4. According to axiom (f.t 2) and Theorem 6.3 3 4 ) 6! ( ( 1 5 ) " 6! ( ( 1 5 ) ,g2) ]=[g ( A,B Tx g A,B ,g2) ]. g A,B [gTx

47

INTRODUCT]ON TO OPERATIONAL CAIJCULUS

By the latter formula and Theorems 6.3 and 6.4 we obtain

The proof is completed by axiom (f1 1) since g (xu x 2) = 0.

-

g (xu x 2 )

=

Theorem 6.6. (The Newton e'Xpansion.) Let A, (A +B), B EM, f be a symbol of rank 1. Then ·

m-1

2k+ 1 ) 2k fN ( 1 2 [/(A+B)]=[f(A)]+ ~[B ... B - A, ... , A ]+ .

6xk

k=1

2m+1)

2m-1

3

2m fjmf (1

2

A +B ].

A, A, ... , A,

+[B ... B oxm

(6.8)

The note cited in the cases of two preceding theorems is valid here as well; its statement is left to the reader. Proof. First obtain the following formula: 2r

2

[T 1

•••

2r

2

=[T 1 2

2r-12r+1)

(1

T,cp A 1 , •••

••• ,

A,, A+B ]= 2r-1 2r+ 1 )

( 1

T,cp A1 , 2r 2r+2

••• ,

A,,

( 1

6cp

A

]+

2r-1 2r+1

2r+3 )

+[T ... T, B -.,- At. ... , A,, A, A+B ], . (6.9) uXr+f

where B, A 1 , . • . , An A, A + B, Tu ... , Tr EM, cp is a symbol of rank r + 1. We cancel out T11 • • • , T r; A 11 • • • , A r for the sake of simplicitry, i.e., we shall prove [f(A+B)]-[f(A)]=[B

g~ (A_, A.f-sh.

(6.10)

The proof of general formula (6.9) is different from the proof of formula (6.10) only by more cumbersome calculations. There exists an equality

48

OPERATIONAL METHODS

Apply the operation f.t: ( x 2 -+ A

-f- B, x -+A); by axiom (~-t 1 ) we get 1

[/ (A-.f-B) -I(.~)- (A+B-~U ~: (A-.f-B, 1h=o. Hence, by axioms (~-t 1 ), (~-t 2 ) and Theorem 6.2, we get [/(A +B)]-[/ (A)]=

=dA.f-B) ~~ (A.f-B, 1)-A ~~ (A.f-B, 1h= {jf (

=[ ( A+B Tx 2

)

3 2 {jf ( 3 A+B, A1 ) ]-[ATx A+B, A1 ) ].

Applying Theorem 6.2 once more, we get the equation [f(A+B)]-[f(A)]

=

[B ~: (A-t-B, Ah.

Formula (6.9) is proved. Now applying successively formula (6.9) to the last term of the right-hand side of (6.8), we obtain the proof by induction. The theorem is proved. We shall define the spectrum of pairs of operators of an algebra with the ~-t-structure along the same lines as we defined it in the case of convergent power series of ordered operators. Definition. Let A, BE M. Let K be such a set of symbols f (x11 x 2 ) that (f (x 1 , x 2)

EK) ~([/(.A,

B}] =0),

1

i.e., the symbol transformed into zero by the operation 2

1

~-t:

(x1

2

-+ A,

x 2 -+ B). The spectrum a of a pair A, B is an intersection of sets of zeroes of functions belonging to K: ((xu· x 2 ) E a) ~ (f (x11 x 2 ) = 0 for any f E K).

Theorem 6.7. The set K is an ideal of an algebra of symbols of rank 2. Proof. Let f (xu x 2 ) E K. Prove that g (xu· x 2 ) f (xlt X 2 ) E K for any g E r{foo (Rn), i.e., it must be proved that if [/

(J., B) f (.A, .Bh = 0. There exists [g (1, s) t (A, sh =[g (1, B) t (~ . .Bh

then [g

.Bh

=

0,

an equality

by the axiom of shifting of indices. We see that

[/ (~ . .B h=[/ (1, sh =0

(1,

(6.11)

49

INTRODUCTION TO OPERATIONAL CADCULUS

by the same axiom. Thus the right-hand side of (6.11) is zero by the zero axiom, Q.E.D. We derive a number of corollaries from the last theorem. We shall consider the two most important corollaries. First reconsider the following version of axiom (f.1 4). (f.L~). Let {fi (x 1, ... , xk)} be such a sequence of symbols that 00

there is only a finite number of non-zero terms of the series LJ fi (xu ... ... , xk) at any fixed point

of operators [/i ~ [ ( LJ

li

('A

1,

.•. , nk )

) ( nl

(x~,

;r~h

Aj, .. . , Ak ]=

j=l

~

LJ

... ,

x~)

i=1

and only a finite number

is not equal to zero; then ( nl

nk ')

[/j AI, ... , Ak ].

j=1

Theorem 6.8. If axiom (f.L~) is true, then the spectrum of any pair of operators belonging to M is not vacant. Proof. Assume the contrary: let a = 0. Then for any fixed point E R 2 there exists such a function f (x1 , x 2) E K that f (xu x 2 ) > 0 in neighborhood U of the point (x~, x~). Any compact set in R 2 can be covered by a finite number of such neighborhoods, hence there exists a function belonging to K strictly positive at points of the set. Cover the plane R 2 by squares qii qii = {(xu X 2) E R 2 ; -e i ~ x1 ~ i 1 e, j - e ~ ~ x2 ~ j 1 e}, (x~, x~)

+ +

+

+ +

> 0 is a fixed number. For every square qii consider a function Iii (x11 x 2 ) E K, strictly positive in qii. Let qJ (x), x E R be an infinitely differentiable nonnegative function such that qJ (x) = 1 when 0 ~ x ~ 1 and qJ (x) = == 0 when x < - ~ and when x > 1 + ~ . Then a function

i, j = 0, ±1, +2, ... , e

00

LJ

f(x 1 , x 2)= i,

i=-00

fii(xp x 2)qJ(x 1-i)qJ(x2-j)

is everywhere positive and fu (x1 ~ x 2 )/f (x11 x 2) E K by virtue of Theorem 6.7. Since LJ Iii (x1 , x 2 )/f (xu x 2 ) = 1, we have by axioms (t-ta), (f.L~) 1

0=[~

(1 j

2

2) (1A,BJ]=1, 2) )/u (1A,B]=[1

A,B

where 1 (x11 x 2 ) is a function equal to 1, and 1 in the right-hand side is a unit operator. This is a contradiction but we thus obtain the statement of the theorem. 4-01225

50

OPERATIONAL METH·ODS

Definition. A complement of the spectrum a is a resolvent set of 1

2

a pair A, B; it is denoted by p

( 1

2)

A, B . We shall now indicate a criterium which will enable us to determine the spectrum using only the functions of operators of one argument.

(.A,

Theorem 6.9. A point (A., !l) belongs to p s} if and only if there exist positive numbers e and 6 and finite junctions p~ (x1 ), p~ (x 2) positive when xi = A., x 2 = 11 and equal to zero when I x 1 -A. I> e, I x 2 - 11 I> 6 (respectively) such that p~ (B) p~ (A) = 0. Proof. The test of sufficiency. We have

p~(B)p~(A)=[p~(B)p~ (A_h hy axiom (!1 5). Hence p~ (x 2 ) p~ (x1 ) E K and the function does not vanish at the point (A., !l)· Therefore, (A., !l) ~a by the definition of the spectrum.

(.A, B);

The test of necessity. Let (A., !l) E p then there exists a.function qJ E K positive at the point (A., !l) and, therefore, positive in its. (2e, 26)-neighborhood. Let P~O,

u(x-t-2n, t)=u(x, t).

(8.2)

We assume that, generally, the sound velocity c depends on x and c (x) E C"" is a 2n-periodic function, c (x) =/= 0. It is natural to consider the smoothest class of functions generated by the values ui at the knots, since only the values of u (x, t) at the knots are important. Let M N be a vector space generated by the vectors eikX, k = 0, +1, ... , +(N - 1), +N. It is easy to see that for any set {uh j = 0, ... , +N, uN = u_N} there exists a function u EM N taking on the values ui at the knots of the lattice. Therefore the initial values for problem (8.2) may be put in the form fJu

u (x, 0) = v1 (x) EM N• Tt (x, 0) = v2 (x) EM N·

(8.3)

Further, applying the equation

e;( -ih

:x )u (x,

t) = u (x+h, t),

we rewrite problems (8.2), (8.3) in the form of pseudodifferential equation 1

def h2 ---at2+ fJ2u 4c2( x2) sm2( - ih Tx fJ ) -0 Lhuu- , 2

t> 0 ,

0

(8 .4)

fJu 0)=v2 (x), u(x+2n, t)=u(x, t). u(x, O)=vt(x), Tt(x,

Assume n

II t II= ( ~ It (x) 12 ax -n

f

12

and take cp E CJ;,> t if the function cp = cp (x, t, h) has s derivatives with respect to x, t, 2n-periodic in x, and there is sup Ot

U. o) +

± k=O

acp£= ( x,2 +---ar

)

0,

J(-zA.

1-1) k

(8.22)

.

The first terms in (8.22) are cancelled out by (8.20) and the conditions a!: (x, 0) = -H ± (x, 0) = +c (x) and we have s

aG (

at

1

acpt (

2) _

k 2

0, A, x - ~ ~ ---;it x, 0

) (

. 1 _ 1)

-zA

.

± k=O

We find the derivatives satisfy the condition

:il

a

+

by (8.11). It is easy to see that they

iicpk acpk at (x, 0) = - a t (x, 0) at the initial moment. Therefore, the operator G ( t, ~) satisfies the second initial condition of (8.12) as well. Thus the reg1:1larizer for (8.6), (8.6') is constructed, Q.E.D.

A,

(3) Asymptotic solutions of the system of lattice oscillations. We shall distinguish between the two problems: (1) finding an asymptotics of an exact solution; (2) finding an asymptotic solution, i.e., the construction of a function which, substituted into the right-hand side, is equal asymptotically to the given one. We shall study problem (1), after we have constructed an asymptotic solution (i.e., after problem (2)). There is the following almost obvious lemma.

DITRQ.DUCTJ:ON TO OPERATIONAL CAlJCULUS

69

Lemma 8.2. Let m > 1, k ~ 2 be integer numbers and j E Ch~ l be a function, where p = p (m, k) is sufficiently great. Then, there exist such functions '¢ i E C~~> t, j = 0, . . . , m - 2 that a linear combination m-2

1

j

U m = h2 R 0 (f)+ ~ h 'l'i i=O

is an asymptotic solution of problem (8.5), i.e.,

[h2

::z

+4c2

(~) sin

2 (-

i~

fx)] Um=f+hmFmf,

um (x+ 2:rt, t) = um (x, t), Um(x, 0)= iJ~m (x, 0)=0, and F mf satisfies an estimate {jj f p JJ

FmfiJ~~

i=1

Cj

I TxT II·

Proof. Substitute the function Um into the equation and apply Lemma 8.1. On equating to zero the coefficients of the powers of h we obtain equations for the functions 'i'h j = 0, ... , m - 2. For example, assuming that the coefficient of h2 equals zero, we obtain an equation for '¢ 0 :

0 c'i'o

+ a~,.c2 (x) ( - i :x) 4 R

0

(f)= 0.

Hence we find by Theorem 8.1 \jl 0 =

- a~,.R 0 [ c2

(-

i

a~

r

R 0 (f)] ,

and the function '¢ 0 satisfies the initial conditions with the righthand sides equal to zero and is 2:rt-periodic in x. In the same way we get all other functions '¢i· The estimate of F mf is obtained from the corresponding estimates of Lemma 8.1 and Theorem 8.1. The lemma is proved. Thus we see that the solution of the wave equation is an asymptotic solution of the system of equations of lattice oscillations. But, firstly, it is an asymptotic solution for a sufficiently smooth righthand side, and, secondly, it is necessary to prove that it is an asymptotics of the sought-for solution. We shall apply the method of the regularizer, studied before in connection with the wave equation, to find a general asymptotic solution of the equation of lattice oscillations and to prove that it is an asymptotics of the sought solution.

70

OPERATIONAL METHODS

A, B)

In this case the regularizer is a usual operator G ( t, depending also on h, the definition is the same as before but in addition to (8.7) we demand that (8.23) I /1 (x11 a, t, s, h) I~ c 8 h 8 +2 (x~ + 1)- 812 • We put the operator

2)

i)2 2 ( x sin 2 h2 -+[4c ~

(

ih

1)

---

2 h

]

into the right-hand side of (8.7'). If the regularizer is constructed, we may obtain an integral equation of the Volterra type similarly to item 2, and prove that its solution exists and obtain as well an estimate for the solution uniform in h as h -+ 0. Note that it is sufficient to demand only condition (8.7) for / 1 • Indeed, if such a weakened regularizer is constructed, we can apply the results of this item, i.e. Lemma 8.2, which enables us to obtain a necessary regularizer correct to kNF, where FE Ch~'tt i.e. to satisfy (8.23). Thus we have reduced the problem to the construction of such a weakened regularizer, that the corresponding right-hand side / 1 satisfies condition (8.7). We shall fulfil this task in the present item with the help of ordered operators. Construct the regularizer for the system of equations of lattice oscillations. Consider problem (8.4) for any arbitrary initial conditions v11 v2 E M N· It is obviously sufficient to consider the case v2 (x) = 0. Next, put the function v1 in the form Vt (x)

=

ao

+

N

2}

aneinx .

n=-N+1

The constant a 0 obviously satisfies problem (8.2). Therefore it is sufficient to consider the initial conditions j u (x, 0)

=

v 0 (i)

=

~

aneinx;

au at (x,

0) =0.

n=±1, ... , ±(N-1), +N

(8.24)

:x.

Let A = -i Define the operator A - 1 (as it was done before) for the function v 0 by the formula A-lvo

=

A-1 (

~ aneinx) ni=O

dcf

~

a:

einx EM N·

njoO

The action of A - 1 transforms functions of this type into functions of the same type, therefore all the powers A-~~TRODUCTI'ON

71

T·O OPERATIONAL CAIJCULUS

:x (A - v

is obviously true. Then

1 0)

=

iv 0 (x, h). Therefore

II

iJ·' I ox·'

(A -k v0 ) N are of order 0 (h 00 ) by the stationary phase method, therefore the formulas obtained in the previous items may be applied in the case of initial conditions (8.62) as well. Now we want to compare the results obtained by the formulas of item 3 and the corresponding computer calculations. In the computer case we took: the length of a circle 3l, l = 10-5 em, the moments of time ti = 2i -10- 10 sec (i = 0, 1, ... , 5), in the units l S (x) =x2 4x when x E [0; 0.25] 1 when x E [0.25; 2) 4(2-x)+1 whenxE[2; 2.25] 0 when x E [2.25; 3]. A solution of the crystal equation for c = const may be calculated with the help of formulas (8.37) and (8.38) along the same lines as it was done in the two previous items. The term corresponding to the sign "+" has no stationary point and therefore may be cancelled out. The stationary phase method is obviously true for ct < L The condition 8 2(J)/8y 2 =1= 0 of Lemma 8.3 is violated at the point ct = 1, x = ~ . At this point, called focal, the crystal vibrations have an amplitude bigger than in the neighboring points. Note that the focal point can be defined directly by the equal ions of bicharacteristics (similar to (8.36)) for c = consL: (8.62') The initial conditions for the system of equations (8.62') are x (0) = x 0 , p (0) = Po = 2x 0 corresponding to initial conditions (8.62) with S = x 2 • Consider a phase plane p, x and there the points x = 2:rtn. If we identify these points we shall obtain a cylinder. The line p = 2x, corres-

89

INTRODUCTION TO OPERATIONAL CALCULUS

ponding to the initial condition p 0 = 2x0 , moves along trajectories of the system of bicharacteristics: indeed, we obtain a curve

p

z

Fig. 5

{x (x 0 , t), p (x 0 , t)} at the fixed moment t. This change of the curves at the moments of time t = tH . . . , t 5 is shown in Fig. 5.

3

(J·to-5cm) .:::::

3

Za~

(H0-5cm}

t=O.'N0-'0s

0

2 (2·10-5 cm}

-Za~

2ae

= J

(8-!0-5t:m)

0 J

-2ae

(3-!o-5cm)

t=o. B·to-70s

z

(Z·to-5cm)

J {J·to-scm)

Fig. 6

The curve is projected on the axis p at the moment t = t5 and when = n the tangent of the curve is parallel to the axis p. The

p

HO

OPERATIONAL METHODS

point at which the tangent is parallel to the axis p- is called a focal point. The graphs max U and min U (with the domain between them blackened) for the exact solution of equation (8.4) with. initial data

Fig. 7 (8.62) and t = t 0 , • • • , t4 are shown in Fig. 6 and the corresponding graphs for the solution obtained with the asymptotic formulas (a = 1) are shown in Fig. 7.

Zae

o~=====• 1

-2ae

(J0-5 cm)

Fig 8

The graphs corresponding to t = t5 are shown in Figs. 8 and 9 (Fig. 8 corresponds to the exact solution). We see that the amplitude of the solution grows rapidly at the focal point. We shall now consider a nonlinear equation of the lattice vibrations.

INTRODUCTION TO OPERATIONAL CAIJCULUS

91

(8) Nonlinear equation of a crystal. We have confined ourselves to the harmonic approximation of the system of Newlon's equations for the atoms of a lattice. In fact, we assumed the amplitude of the initial displacement to be small and ignored the non-linear

Fig. 9

terms. Now we shall take into consideration the amendment introduced into the equations of lattice oscillations by the quadratic terms. It means that a term

is added to equation (8.1), where a is a parameter. On substituting Un Wn into this new equation, where Un is a solution of the linear equation, we obtain the following nonhomogeneous equation

+

i'V n -c2

Wn+t-2Wn+Wn-1 =c2a[(U n+i - Un )2-(U n - U n-1 )2] h2

for I Wn - Wn-l I ~ I Un - Un-I I· For the sake of simplicity we consider the case, where c and a are constants. We shall consider the general case in Appendix. We assume that the dependence of the initial data Un (0) and Un (0) on the parameter h satisfies a substantial condition. We shall introduce the following definition to the effect. A system of functions Vn (h), -N + 1 ~ n ~ N corresponds to a limit distribution g (x, p), -n ~ p ~ n, where g is an even distribution in p, if for any function a E C"" ( -n, n) the following

92

OPERATIONAL METHODS

limit condition is true n n

N

~

limh h-.O

a(nh)vn(vn+k!vn-11.)=~ ~ a(x)g(x,p)coskpd:rdp,

n=-N+1

-n -n

(8.63)

where The physical sense of this definition is especially clear when k = 0, since in this case it is necessary that the mean value a (nh) should converge to a mean value a (x) of distribution. We shall consider, as we did before, a space of functions Jl!JN

instead of the functions on a lattice; moreover, we shall formulate the final theorem in terms of the space functions M N· Note, that the arguments of this item can be extended to a wide class of nonlinear pseudodifferential equations with non-linear terms considered small. The first approximation of the theory of perturbations in this case is a solution of equation (8.4) and the second one satisfies the equation Lhu= -8ic2 r:xh2 sin

~ ( (sin~

u

r),

~

(!)

= (!) = -

a

ih -ax ' (8.fi4)

where u satisfies the linear equation Lhu = 0. For the sake of simplicity we take one term of the sum ~ of formula (8.37) corres±

ponding to the minus sign as a solution u of the linear equation. We assume that the initial value u 0 E M n corresponds to a condition

l~~

n

J

n

u 0a(x)(coskw)u0 dx=

-n

~

n

) a(x)g(x,ro)coskrodxdro.

-n -n

(8.65)

First note that a more natural for the continuous situation and a more general condition follows from condition (8.65). Denote

r

-

n dcf

(f, rp) =

J f (x) cp (x) dx. -n

The following lemma is true. If

Lemma 8.4. Let f (x, ro) E Coo be an even periodic function ro. 'Ph (x) corresponds to the limit distribution g (x, ro), then 2

lim h-+0

1

n

n

( 1\Jh, f ( x, ro) 1\Jh) = ~ ~ f (x, ro) g (x, ro) dx dro. -n -n

INTRODUCTION TO OPERATIONAL CAIJCULUS

Proof. Expand the function We have

93

I (x, ro) in a Fourier series in ro.

00

~ an (x) e-inw,

f (x, (I))=

(8.66)

n=-oo

where an (x) =

n

J f (x, (I)) einw d(l).

2~

(8.67)

-n

By the conditions of the ]emma it follows that for any n there is an inequality max., an ()1_,-C~; x """= - , x

(8.68)

nit

> 0 is any integer, C k = const. The estimate can be easily proved integrating equation (8.67) by parts. We prove that

k

E~ I/(fh (X, (I)) - f (X, (I))) lJln (x) I

2

1

2

1

where we have denoted by

II= 0,

lk (x, ro) a partial sum of series

(8.66)

fk (x, (I))= ~ an (x) einw. In i:S:k

Note that the function lk (x, (I)) uniformly converges to as k-+ oo. We have for any smooth function u (x) 2

1

2

[! (x, (I))- In ( x, =

;rr

J

1

-

(I)) u (x) =

n=oo

~

n

[.\ u(y)e-inyayJeinxJf(x,nh)-f~t(x,nh)]=

n=-oo

-:rt

n

oo

=

I (x, (I))

2~ ~ n=-oo

[

J u (y) e-iny dy]

~ am (x) ei(mnh+nx). !ml>h

-n

Finally, we obtain -21

21

2

1

[t(x,(l))-!n (x,I(I))Ju(x)=( ~an (x)einw)u(x).

(8.69)

JnJ>h

Estimate the nth term in sum (8.69). The inequality is obviously true

II an ( ~) eino;u (x) I ~ max I an (x) I II u 11, X

94

OPERATJONAL METHODS

or by (8.68)

I

an { ~)

I

ein~u (x) ~ ~Z

\1

(8. 70)

u \\,

where k>O is an integer. We have by (8.69), (8.70)

II[!(~. ~)-ht(~.~)Ju(x)ll~ ~ +zcl\\u\\,

Vl,

JnJ>k

!~~~~[! (~, ~)-ht (~, ~)Ju(x)II=O,

as k-+oo.

Hence for any 8 > 0 there exists such a k0 , that

I{

¢h,

(! {:, ~)- fR { :, ~)) 'i'h ) I< 8 when

k';;;-k0,

IJ I [f(x, w)-fk(x, w)]g(x, w)dxdwl ... , An, B

n

( 1

n+ 1 )

]=('A(T)cp) A1, ... , An, B

.

(9.4)

Indeed, take for example T = AkAi, then by (9.3) we get n

1

A~tAi[cp (A 11

••• ,

n+1

1

n

n+i

A,11 B )]=Ak[(Licp) (At, ... , An, B)]= n

1

=(L"Licp) (A 1 ,

•• • ,

1

n+i)

A 11 , B n

=

n+i

=('A (A"Ai) cp) ( A1, ... , An, B ) .

Equality (9.4) for all elements T of the algebra J/1' is verified in the similar way. Thus the operators Li are themselves the generators of the Lie nilpotent algebra, which we shall denote by 11. The constructed mapping 'A: ufl' -+ IT is called an ordered representation of the nilpotent algebra J/('.

107

INTRODUCTION TO OPERATIONAL CALCULUS

It follows from (9.4) that the operators Lk, k = 1, 2, ... , n; a satisfy the same commutating conditions as the operators Ak, B. Particularly the operators L 11 L 2 , • • • , Ln, a are generators of the nilpotent algebra D. Hence if P (x, a) is a polynomial in x with coefficients depending on a, then n

1

n+ 1 )

[P ( A1, .. . , An, B 1

=¢(A 1 ,

,An, B

]= (9.5)

, n+1)

n

1

n+ 1 )

A1, .. . , An, B

n+1)

n

•••

n

( 1

]·[G

where ¢ (x, a) = P ( L 11 • • • , Ln, a G (x, a) for any G (x, a) E E c:;;. We can ask, whether this equality holds for P (x, a) E r!/ and G(x, a) E Cz. In all probability the answer is negative if we limit ourselves to axioms ftdte (without topological axioms). To give a positive answer to the question we shall formulate two additional algebraic axioms, which are useful and, in addition, permit to obtain an explicit formula for any function of non-commuting operators via a composition of functions of commuting operators. Consider symbols f (x, a, t) depending on the parameters t E E Rk (k is not fixed); all the derivatives with respect to the parameters belong to r!/ Axiom (ft 7) (the parameter axiom). Let f (x, a, t) E Cz, t E Rk verify the equation 00

00

n+1

n

1

•

[/(At, ... , An, B, t)]=O for all t E Rk. Then for any function

n+1 n+2)

An, B

where the junction R 11 (x0 , x 1 , faster than R11

[i~l xJ J-h/

= 0z

( c~1

2

as I x

••• ,

1-+ oo,

,

x;, x;+t, ... ,

Xn,

a)

decreases

i.e.,

12

xJ) _, ).

The left quasi-inverse sequence for the operator f

n+1) B

(Au ... , An,

is defined along the same lines (cf. 9.1). We shall investigate the method of reduction .of the main probl~m to a solution of the Cauchy problem for a function of differential operators. Let Gh (

~, ~'

-i

~,

-i

~)

be a differential operator with

Cz

partial derivatives defined in the space (Rn X Mm). Denote this operator by Gh. We define G, by the following equation:

(1

n

n+

1)

(1

n

n+

(Ghcp) A11 ... , An, B =[cp A11 ... , A,, .8 Further, we define Gn+t as follows: n

1) ]A~t,

n+

. . . , An, B

1) JB .

r16

OPERATIONAL METHODS

The operators Gk are generators of the Lie nilpotent algebra which will be denoted by II'. The proof is the same as in Sec. 9. The mapping 'A' : J1f'-+- II' is called the (right) ordered representation of the algebra J1f'. An analogue of (9.5) is true [ 0 and a (y) is subordinate to fo (y). Under these conditions F (y) is called asymptotically p-quasi-homogeneous (p = p1 , • • . , p.), N

fo (y) is called the leading term of the function F (y) and

2J /; (y) ()

is called an essential part of the function F (y). For the time being tak~ X; = Y;, i ~ n, i uah .}!-- = Yn+h• -i uX -%- = 'll· Let the obtained function be asymptotically p-quasi-homogeneous in y and Pn+i = 1, 0 < i ~ m (the variables Y), a are taken as parameters; Y) belongs to a small neighborhood of zero in Rn, a E Mm). The leading term of the function will be called the Hamiltonian function corresponding to the given Hamiltonian, and will be denoted by :rt (y, Y), a). Thus the construction of the Hamiltonian function of an operator A E .Jl; consists of 4 steps: (1) (f.L- 1) the choice of A 1 , •• • ,An, BE X, 2

r

n+1 n+2 1

= r (x, a), f (x, x 0 , a) E r!foo such that f (At. ... , An, B, r) =A;

(2) (A -+ L) the construction of the Hamiltonian by the ordered representation; (3) (p) the pointing out of the leading term :rt (y, YJ) of the Hamiltonian; (4) ~ (p, q) the determining of the Hamiltonian arguments in the function :rt (y, YJ). Thus 2

1-t-1

(

-+ :rt

(y,

A--7 j

n+1 n+2 1)

A1o .. . , An, B, r

-+

Y),

A-+L

---7

p, q

a) ---7 H (p, q, co).

The operations (f.L - 1 ) and (p) are defi.ned not uniquely generally and do not exist occasionally for a given A E .fl. Now we pass over to the last step of the indicated construction. For the sake of definiteness let p; be equal to a unity for the fi.rst s arguments, s ~ n, Pn+i = 1 for 0 < i ~ k.

129

INTRODUCTION TO OPERATIONAL CAI.!CULUS

Take new variables Yt=COt-Pm+b ···•Ys=COs-Pm+B;

YsH=(OsH• ···• Yn=COn;

Yn+k+i = · · · = Yn+m = 0; a1 = qj, ... 'am= qm.

1/n+i = Pt• · · · • Yn+k = Pk• 111 = qm+i• ... '11n = qm+n;

Denote by n(y,

'Y],

a)=Q/t(p, q, co),

H=ReQ/8,

H=lmQJt.

Besides this, denote by JJ8 0 (p, q, co) the essential part of the symbol f (LH ... , Ln, a). Assume that Im JJ8 0 ~ 0 in this item. Let Q e be the manifold in the space of arguments p, q, co defined by the conditions n

2] (qmi-i?
- i = = A 2 , t = B 10 y = B 2 , B = (B 10 B 2), then we obtain the equation

:c

(AI -

A 2) u

=

1.

We shall construct a quasi-inverse sequence for the operator AI- A 2 within the framework of the introduced definitions. The representations of the operators AI, A 2 are . {) L ! =x~-~-­

aat '

Hence the Hamiltonian of the operator AI- A 2 has the form

f (L 1, Lz) = ( X t - i

a!

1 ) -

(

Xz- i

a!

2 ) •

Thus

f (LI (y), L2 (y)) = (YI + Ya) - (y2 Let PI = P2 = p 3 = p4 = 1, then n (y, t], a) = n (y) = f (L (y)).

+ Y4).

The absorption condition is not satisfied in this case, since Im n (y) = 0. It looks like the operator AI - A 2 has no quasiinverse. Indeed, it has not. The operator AI- A 2 , however, can be modified in a way that the obtained operator will have a quasiinverse and its symbol will coincide with the symbol A 1 - A 2 in a closed domain d c R1 X Rt. Consider the operator

1 1 2)

(2)1

f (At, Az, B =A 1 -A 2 -i

0, and e > 0. Let M be the lower boundary of such constants c2 for which the dissipation inequality is true. Then

···•Yn, CG)=

gi(Yh

T

- i -

f

p (qO)

~ Vll(qO, t)l

ei{(qO, t)+tMD3, 2-e(qo, t)) lqO=qO(a, yt)

dt, (9 •75)

where q0 (a, y, t) is a solution of the equation qi (q 0 , t) = ait m > n; qi (q 0 , t) = Yi> i ~ n.

+ n ;;;;=:

;;;::: i

Example. Consider an operator

1

( 2) 1

A 1 -icp B A 2

(9.76)

with the symbol Y1 -

icp (a) Y2•

where cp (a) ;;;;:: 0 equal szero for I a I < ~ and is strictly positive for a> T/2. If the operators A 1 , A 2 , B commute then the quasi-inverse element obviously does not exist (cf. p. 102). Take

140

OPERATIONAL METffiOD {) > 0. In this case it is sufficient that the symbol should belong to C';; only for y 2 > 6.

:x,

For example, let A 1 = - i

B = x, A 2 = k, where k is a

= def

•

•

parameter{) < k < oo and let F (x, k) F (x) be a functiOn w1th a support in Q, such that F (x, k) P (x) = F (x, k). The function P (a) was defined earlier in the remark to the main theorem. Consider the equation for u (x, k):

(A1

c

(

1 -

2) A1 ) u (x, k)

icp B

2

=

P (B) F (x, k),

I F (x, k) I < const.

(9.77)

We can tell by formula (9.77), that (see Sec. 6 of Ch. V) an asymptotic solution of the problem is obtained by a "quasi-inverse" sequence applied to F (x), which satisfies the equation within the accuracy to any function, tending to zero ask~ oo faster than k-N for any N. We shall verify this by the given example, using formula (9.75). The Hamiltonian of the operator (9.76) is of the form y1 - i

8~

-icp(ct)IYzl·

(9.78)

The Hamiltonian function, corresponding to the Hamiltonian (9.78), is of the form cf!t = ffi 1 - P1 + P 3 - icp (ql) I ffi2 - Pz + P4 I, ffi 2 > 0. The systems of Hamilton and of the germ are the following in this case

.

.

.

qz =--= q3 = q4 = 0; p=O; Z 1 =Z 3 Wt

q (0) = qo,

p(O)=O. =0,

(9. 79)

Zz= Z4

= icp' (q 1) ffi 2 ,

.

= - i(p (q 1);

w2 =

.

W3

.

Z

= w4 = 0;

(0) = 0 w (0) = 0.

Calculate the values entering formula (9.75). We have t

0 when t = = T + 1. Then the absorption condition of each trajectory is fulfilled, since p 2 =1= 0 is on a trajectory as it follows from equation (9.85). Therefore, the operator if + iii is quasi-inverse. It is not difficult to see that the solution of the problem

[if

+ ill u =

F (x, t),

where F (x, t) is finite, does not depend on the form of cp (t2), when t 2 < T 2 and if, ii are defined above. Thus the solution of the problem for sufficiently small t (t < T) does not change, when we introduce the absorption term. Note. An addition of the summand -icp (x) p 2 cp (p), where cp (x) = = 0 when I x I < M and cp (x) = a > 0 when I x I = M + 1 provides the absorption condition for every trajectory as well, since for sufficiently large 1: I x (x 0 , 1:) I > M + 1 by equations (9.84). But in this case only the asymptotic of the non-smooth part of the solution does not depend on the form of the absorption term. Nonetheless the addition of the term makes sense (from a physical viewpoint) for some problems. (4) The main theorem. Consider a one-parameter family of symbols • • . , Xn, a,~) and symbols of order n, r (xlt ... , Xn, a, ~), where· is a parameter, 0 < ~ ~ oo. Let limf(xh ... , Xn, a, ~)=f0 (x 1 , •• • , Xn, a,)EJ"'"",

f (x1 , ~

~; .... oo

limr(xit ... ,

s-oo

Xn,

a,

~)=r 0 (x!t

... ,

Xn,

a)EJ"'"",

fo =I= 0, r 0 =I= 0.

Let P 0 (x) be a function of the class CO' (Rn) equal to unity in the domain I x I ~ d, where d is a constant.

OPERATrGNAL METHOD 0, a number 1:' = 1:' (q 0 , w, p 0 , v), 0 ••. , xn, a, x 0 , ~) = 0 z (I x 1-N) for any N uniformly in ~ and a.* n+1 n+2 ) 2 The construction of gN ( A 1 , . . . , An, B, ~ is contained in

Chapter V. In the case ~ = oo we obtain the simplest version of the main theorem formulated earlier. Though in the right-hand side of (9 .85) there is the function P 0 instead of 1, which was in the main problem, this theorem gives a direct answer to the question stated by the main problem. Indeed, it is possible to take a unity instead of the function P 0 in the right-hand side of (9.86), if the common 1 n n+1 spectrum of the sequence A 1 , • • • , An, B belongs to the domain Q; X Q. This condition is not fulfilled as we saw in the problem on the crystal (there is the spectrum of ih ;: on M N in the segment

*

Under

the same conditions there is a left quasi-inverse sequence

2 1 n+2 ) ( n+1 g}v=g]v A1, ... , An, B, r, ~ :

g]vf-Po=Oz

(J x 1-N).

See the Main Theorem in the general form in Ch. V. 10-01225

146

OPERATIONAL l\IETHODS

[-n, n] and

~ =

! ).But it is true for general difference schemes

if a convenient space of smooth functions is taken instead of 111 N· There is a similar situation in the case, when B is the Hamiltonian of the oscillator considered in Sec. 8 and therefore has a discrete spectrum located on a lattice of a step h, where his Planck's constant. 1

n+1n+2)

The construction of gN ( A 1 , . . . , An, B in these problems permits to obtain the quasi-classical asymptotics of the secondary quantized equations. Indeed, all the results obtained for the equations of vibrations of a crystal lattice can be extended to this general case, moreover, if the absorption term in the given equation is missing, it must be introduced in the same way as it was done in the example with the wave operator. The expansion in a power series in ~ can be obtained in just the same way in the general case. Here the parameter v has the function of w in the crystal equation. Problem. Find the effect of Cherenkov's type for a difference scheme approximating, as h ~ 0, the wave equation:

h

"T = d = const, c (x) E Coo, F (n, m) = 0, when n = -1, -2, Find an asymptotic of the solution correct to kN. Hint. The cited Hamiltonian function has the form sin 2 vp 1 = c2 (x 2 ) d 2 ·sin2 vp 2 • Use a space of integer functions of the first order of the type ~: as an analogue of M N (Kotelnikov's theorem). Note. There is no operation "prime" in the statement of the main theorem. Indeed, we have used the operation proving the theorem x} ; the rule of the reduction of the main proonly for M = { i blem remains true without the operation "prime". Therefore, the main theorem is true, if the structures satisfy axioms (f-! 1 )-(f-! 8). If the axioms (f-! 1)-(f-! 6) are true, then the main theorem can be proved only in the case, when the symbol f (x, ~. a) is a polynomial in x. In conclusion, the following statement can be taken as an axiom instead of f.! (8): (9.5) holds for any P E c!/ (so to say, the f.!structure agrees with the ordered representation of the nilpotent Lie algebra).

:x,

00

I. FUNCTIONS OF A REGULAR OPERATOR

In this chapter we shall develop an operational calculus corresponding to a 11-structure (see Introduction) based on the commutative unbounded operator algebra in a Banach space B. It will be our purpose to determine how wide an operator class constitutes a set M. In addition we shall clarify which symbols correspond to bounded operators on B. This will facilitate the forming of a closed extension of the C"" -symbol class. For simplicity our presentation will not depend on the It-structure notion. The symbol concept introduced in this chapter, however, proves to be coincident with a similar notion within the 11-structure framework. To illustrate the function of an operator concept we shall consider a function of multiplication operator A by an independent variable x defined on a Sobolev space W~ (Rn). It is easy to see that if cp E E Ck (Rn) then one can readily define the function of multiplication operator A by x, this operator being in turn a multiplication operator by cp (x). It will be seen that cp (A) is a bounded operator on W~. If the kth derivative of cp (x) is unbounded then the multiplication operator by (jl (x) is also unbounded in w~. On the other hand a multiplication operator by x defined on W~ has the following properties: (1) it generates a one-parameter group of multiplication operators by the exp {ixt} family; (2) the multiplication operator satisfies the condition: for any g E W~ (Rn), where s ;;;;:: k, the following inequality is valid: c (1 + /t lk) II g (x) llws' II eixtg (x) llwk:::;;; 2 2

that is,

II eixt 11:~ :::;;;c (1 +It lk), 2

c = const. 10*

148

OPERATIONAL METHJODS

In terms of the corresponding evolution equation we can formulate the conditions (1), (2) as follows: there exists a unique solution u of the evolution equation i

d~t(t) =Au (t),

u(O)=g,

gEB 2 ,

(*) B 2 =W~,

u being a continuous function of t with values in B 1 satisfying the condition

II u (t)

I!BJ~c (1

+It J•) ·II u (0) lln

=

W~,

k

~

l

2

for all u (0) E B 2 = W~. As we shall show, a wide operator class is covered by the relation between the number s, featured in the estimate on the solution of evolution equation (*) (where, as a given operator, we have a multiplication operator by x), and the boundedness of a multiplication operator by cp (x). The condition (**)for the growth rate of the solution of the evolution equation generated by unbounded operator A becomes necessary, if it is required that the finite k times differentiable function of A should be a bounded operator. If we skip this requirement and restrict ourselves only to the infinitely differentiable funcLions of an operator, then the growth rate condition (**) may be essentially relaxed. Consider a differentiation operator in a space C (R) of continuous functions bounded in infinity (or in a space C N of continuous functions increasing less than I x IN). As it is easily seen, the totality of functions of such an operator form an algebra .9fl 0 (or an algebra$!/ N• respectively). This example shows the necessity of using the algebra $!/ 0 (the algebra .'lfJ N is the case of ·unbounded continuous functions) if we wish to consider an algebra of functions of operators drawn from a class which includes a differentiation operator in a continuous function space. With these two examples we wish to emphasize "the necessity" of using a growth rate condition of type (* *) as well as the algebra $!/ N· Finally, let us consider a multiplication operator A by x iy defined on W~(R 2 ). It is natural to defme a k times differentiable function of x iy as a function cp (x, y) of two arguments x and y. It is evident that each operator, x = ReA and y = Im A, will satisfy the condition (**), ReA being commutative with Im A. We shall introduce operators satisfying a condition of type (**) and call them generators of degree s. Then we shall consider operators iA 2 , where A 1 and A 2 are generators of degrees, A 1 of type A 1 being commutative with A 2 • Such operators will be called regular operators and functions of them will be considered as functions of A 1 and A 2 •

+

+

+

CH. I. FUNCTJ10NS OF A REGULAR OPERATOR

149

In Sec. 7 we shall prove the theorem which will show the sufficiency of a condition of type (**). Specifically, we shall prove that in the case of a purely discrete operator spectrum the existence of a complete system of eigenelements and associated elements (the number of the latter not exceeding N for a given eigenvalue) is equivalent to the regularity of the operator (of degree N). Thus, for a discrete spectrum at least, we shall show how wide the regular operator class is. When the spectrum is not discrete the concept of a system of eigenvectors and associated vectors is introduced in the case of the regular operator in Sec. 8. The theorem of completeness of such a system will be proved there. A similar concept in the case of an arbitrary operator, however, has not yet been involved. Therefore, there is no inverse theorem as yet. Sec. 1. Certain Spaces of Continuous Functions and Related Spaces We shall now consider certain Banach algebras of continuous functions. First of all, consider the set of continuous bounded complex-valued functions with the field of definition Q c: R". As usual, we introduce the structure of a vector space on this set. Define the norm of function f by the formula

II t II= sup It (x) I·

(1.1)

xEQ

Denote such normed space of functions by C (Q). The space C (Q) is a Banach space. In analogy with C (Q) we define the Banach space C (Q), k > 0 being an integer. The norm in C (Q) is defined by the formula

II f llck(Q) = max sup I Dif (x) I, O~c II

Cf! llw~(Rn)'

c = const.

It remains to be shown tha l if the sequence {Cf!i} of functions 0 be a fixed number. Choose a number j 0 so that for j > j 0 the inequality II Cpj0 llc'JV

0 satisfies the

estimate

II

[A

-

(

cl li

Ck + a + z"b)]-k IIB2< B! ~ Tbj"k I b lk+N '

where ck and ck are constants.

Proof. Obviously, for Im z =1= 0 and 0::;;; j < k the function cpi (x) = x 3 (x - z)-k belongs to ~ N (R). Since 1 E !iJ N (R), it follows that the function

=

cpk(x)=xk(x-ztk=1- (x-z)k-xk (x-z)k k

= 1- ~

Ck (-1)i zixn-i (x-ztk

i=1

belongs to .9!! N (R). Therefore, using the corollary of Lemma 3.4 we have cp 0 * (A) (A - z)k g = g for any g E E. Consequently, (A - z)-k

II (A- zrk II~~:::; II cpk* (A) II~~:::;

=

cpo* (A) IR(A-z>''' whence

v12n II cro llssN(R):::;

1 :::; V12JT II -cro llssN max { lb"lh,

1

1b IN+k

}

'

where cp 0 (x) = (x- i)-k. Thus the lemma is proved. Note that the statement of this lemma is not directly derived from Lemma 3.5.

165

CH. I. FUNGTmN'S OF A REGULAR OPERATOR

The operator h: .98 N(R)-+ B 2 corresponds to each vector h E E by means of the formula hqJ

= '11* (A) h.

The operator h is determined in $

N

(R) and is bounded:

sup

II ~ll h llmtd·

Consider next for any IE C N (Rn, B) the operator I 0 [I]: F-1 JO>~ (Rn)-+-+- B defined by the formulas 10 [fJ 6~ =I(£),

Io [/] cp = ) cp (x) I (x) dx, cp EC0 (Rn). Rn

Problem. Prove that

II lo [/]

II~ II

I llcN(Rn, B)'

Note. For any cp EC0 (Rn) and any non-equal vectors from Rn we have k

00

cr+ ~ a;{)~i i=l

= C"JV(Rn)

s~o

... , Sk

k

~ lcr(x)j;(1+1xltdx+ ~latl(1+1sd)N. -oo

i=1

Extend the operator I 0 [I] to the homomorphism I [j] : Cty. (Rn) -+-+- B. We shall use the notation

I

[fl cp = J"

cp (x) f (x) dx,

Rn

which provides the necessary interpretation of equality (3.18). Obviously, both definitions of the operator Cf!mtd (A) are in fact identical.

172

OPERATIONAL l\IETHJODS

Let f E C N (Rn, B) and let the function !F defined by the formula !F (x) = Cf (x), where C is a closed operator on B, belong to C N (Rn, B). Then for any cp E F-1§!'Jv (Rn)

i cp

(x) Cf (x) dx =

C ) cp (x) f

(x) dx.

(3.19}

If cp is an arbitrary element of C"Jv (Rn) then, by choosing the sequence {cpk} c: F-1§!'Jv (Rn), which converges to cp and going to the limit k ~ oo in the equality

JIJlk (x) Cf (x) dx = C ) cpk (x) f (x) dx, Rn

Rn

we achieve that formula (3.19) is also valid in this case. Lemma 3.16. Let Im z =F 0. Then

(A-zt 1 =

[(A -zt 1lmid·

Proof. We have to prove the following two formulas:

[(A - z)- 1 lm 1d (A - z) h = h, (A - z) [(A - z)-1 1m!d = 1.

hE DA.,

(3.20) (3.21)

Formula (3.20) is valid for h E E according to Lemma 3.5; if hn ~ h, hn E E, h E DA, then in (3.20) the limit can be reached because A - z is a closed operator and [(A - z)- 1 1m!d is a homomorphism. This completes the proof of formula (3.20). Now let h E E again. Denote by z (t) the Fourier transform of the function rz (x) = (x- z)- 1 • We have

r

h= [(A-zt 1 ]mid (A-z) h= 00

=

V 2n JI 1

~

r2 (t) U (t) (A-z) hdt=

-00 00

1

r

=,,- \ v

-

2n -oo "'

=(A-z)

~

rz(t)(A-z)U(t)hdt= 1

-"'

I

V 2n J

~

r2 (t)U(t)hdt=

-oo

=

(A-z) [(A -zt 1lmid h.

Here we have made use of formula (3.19). Thus, for h E E (A- z) [(A - z)- 1 lm 1d h = h.

173

CH. I. FUNGTmNlS OF A REGU:LAR OPERAT"OR

From the fact that A is closed and E is dense in Bmict• it follows that the equality is valid for any hE Bmid• i.e., formula (3.21) is valid. This completes the proof of the lemma. We shall summarize the main results obtained in this section in the form of the following theorem. Theorem 3.1. (1) The generator A as an operator on Bmid has a closure A. (2) There exists a homomorphism M of the Banach algebra !}g N (R) into the algebra Op (Bm 1ct) of homomorphisms Bmld-+ Bmld• besides the operator (A - z)- 1 corresponding to the function 1

x-+-x-z'

Imz:foO.

Notation. Let the homomorphism e

N+t

IV 2

(R)

=/(d).

The theorem is proved. Theorem 6.2. Let /.., E R. If (A 'A E p (A). 1 Proof. Let 0 < 6
2t) ,

0 0 e-i7-.2t so that A is a generator of first degree but not a generator of zero degree. For any function cp E !B 1 (R) we have rp (A. 1) 0 0 ) 0 cp (A.z) cp' (A.z) • 0 0 (jl (A. 2) The spectrum a = a (A) of the operator A consists of two points: "- 1 and A2 • Let cp E COO (R), A2 ~ supp cp, cp (A.l) = 0, cp' (A- 1 ) =I= 0. Then cp (A) = 0, but II cp llfE 1 > 0 (here the norm of the class of .equivalency, which the function cp belongs to, is denoted by II cp 11$ 1 as usual). Despite the fact that the homomorphism Ma: !B N (a (A))-+ -+ ~N is not a monomorphism, we obtain a sufficiently good estimate for II cp (A) II from above. cp (A)= (

Definition. An eigenelement of the operator A at the point A. is called such a vector g ED-;.. that (A - A.) g = 0. The associated element of degree k of the operator

A at

the point is called such a vector g

E D-h A

that (A- A.)h g = 0 (but (A - A.)h-t g =1= 0). The eigen and associated elements of the operator A will be called the e.a. elements of the operator A. If the operator A has .only isolated points of the spectrum and

the number of linearly independent e.a. elements corresponding to each point of the spectrum is finite, then the operator A is said to possess a discrete spectrum. Problem. Let h be an eigenvector of the operator A at the point A.. Prove that e-iAt h = e-i'Mh, where h E D:4. Lemma 6.1. Let A be an isolated point of the spectrum of the generator A of degree N. If g is an e.a. element of the operator A at point A., then (A- A.)N+i g = 0. Proof. Let g be an e. a. element of the operator A at the point A.. Then for some k (A -

A.)h g = 0.

Show that if k

>

(6.2)

+

1, then for (6.2) it follows that (A - 'V)h-t g = k > N 1. Let { Xn} be such ·a sequence of infinitely differentiable functions that Xn (A.) = 1, and that supp Xn

= 0. Suppose that

N

+

192

OPE.RATIONAL METHODS

(A- - A+ -

1- , 1-) . is contained in the interval Therefore, the n n support of a function Xn contains only one point of the spectrum of the operator A for sufficiently large n. Besides, let

lim II

l¥n llssN = 0,

ll-->00

where '¢n (x) = (x - A) 11 - 1 Xn (x). It is easily seen that such a sE~­ quence exists. We have

II Xn (A) (A- A) 11 - 1g II-+ o for

n-+ oo.

On the other hand, the vector h

=

(A -

A)k-1 g

is an eigenelement of the operator (A-A)h=(A-A) 11 g=

A

since

o.

This means that 00

Xn (A) h = V12rt

)

Xn (t) e-iAth dt =

-CO> 00

-1- V2rt

) ~ (t)e-,f..thdt-x . (A)h-h

xn

-

n

-

'

-00

i.e., Xn (A) (A- A)k-1 g =(A- A)k-1 g.

From (6.3) it follows that (A- lv) 11 - 1 g that (A - lv)N+ 1 g = 0.

=

0. By induction we obtain

Theorem 6.4. Let the spectrum of the generator A consist of isolated points Az, i = 1, 2, .... Then the set of all e.a. elements of the operator A is complete, i.e., the linear span of the set is dense in Bmid• Proof. Let xi be an arbitrary function of C'i) (R) with a support containing only one point A; of _the spectrum of the operator A and let x' be a function such that x' (x) = 1 in some neighborhood of a point A;. Then (A - A;)N+ 1 Xi (A) = 0. Indeed, let {X~} at a fixed i be the sequence considered in Lemma 6.1 (fork= N 2, }, •= A1), where x~ (x) = 1 in some neighborhood (dependent on n) of t lw point A;. With sufficiently large n we have

+

x~ (A)= xi

(A),

193

CH. I. FUNCTl'ONIS OF A REGULAR OP.ERA'l10R

and \\(A-Ai)N+tx~(.A)II--+0

for

n--+oo,

Q.E.D. In Sec. 3 it was proved that the linear span of the set of all vectors of the form q> (A) h, q> E Co (R), hE Bmid is dense in Bmid· This means that for the proof of the theorem it suffices to verify that for any h E Bmid, q> E Co (R) the vector q> (.A) h can be represented in the form of a finite sum ~ gh where gi satisfies the equality i

Ai)N+t gi = 0. Let {Qi} be the open covering of the sul!port of the function cp, where Ai ~ Qi for i =I= j, Ai E Qi. And let {x'} be a C"' -partition of unity which is subordinate to the covering* {Q;}. Then cp (A) h = 21 cp (A) x; (A) h. Set gi = cp (A) xi (A) h. We have (A- Aj)N+t gi =

= .\ \jJ (A,) 6 (x- /.,) dA,

(8.2)

or, in a more general form, the following expansion is written: \jJ (x)

= ) c (A) l!J>- (x) dt.,,

(8.3)

where {'\JA (x)} is the family of generalized eigenfunctions of the . operator of some physical value. Since \jJA (x) is feaLured in (8.3) as a function of the parameter A, it is quite natural to assume that the operator (A - /.,) acts on the space of functions of the argument A, i.e., to consider A as a multiplication operator by an independent variable. in the present section we shall obtain the generalization of (8.2) for the case of the expansion of an arbitrary vector of Bmid in generalized associated eigenfunctions of a regular operator. Let us proceed to the definitions and notation. Consider Hom (.93' N (Rn), C). By definition, Hom (.93' N (Rn), C) is the space ofboundedfunctionalsin !B N (Rn), i.e., the space S6''N (Rn) for whose elements the following notation was agreed upon: f[g]=) f(A,)g(A,)dA,

fEHom[!BN(Rn), C],

gES6'N·

(8.4)

200

OPERATIONAL METH10•DS

As it was agreed above, this integral which has meaning only for continuous f (A.) is formally written down for any functional f, f (A.) being called a generalized function. Let G (A.) be a continuous function of A. with values in B. Then the integral ) G (A.) f (A.) dA.,

f (A.) E ~N (Rn)

Rn

determines the element of B so that the operator G Gfdet IG(A.)f(A.)dA. Rn

is an element of the space Hom

(~ N

(Rn), B). For any element

G E Hom (~ N (Rn), B) we agree to write formally

Gf = ) G (A.) f (A.) dA..

By analogy with (8.4) ~e shall call G (A.) a generalized function of A. with values in B. In accordance with this definition, ordinary generalized functions are generalized functions with values in the complex plane. Let T be a regular operator of degree N and Bmid be the corresponding intermediate Banach space, and let A. = At iA- 2 E C. Consider the operator T = (T- A.)k,

+

which acts from Bmid to Bmid· We shall consider it as an operator acting from the space of generalized functions of A, with values in Bmid to the space of continuous functions of 'A with values in Bmid so that the values of the operator (T - 'A)k should belong to Bmid for any fixed 'A. Thus, we shall considet (T - 'A)k cter T as an operator acting from Hom (~ N (R 2), Bmtd) to C (R 2, Bmtct), where 'A = At + i'A 2 = = ('AI> 'A 2). We shall make the meaning of this statement more precise. Denote by means of T~k> Hom(~ N (R2), Bm!d)-+ C (R2 , Bm!d) the operator defined by such continuous functions with values in Bmid that the mapping A.-+ (T- A.)k f ('A) is a continuous function in R 2 with values in Bmid· The operator is acting according to the formula T~f (f..) = (T- A.)k f (A.).

CH. I. FUNCTOONIS OF A REGULAR OP.ERA'IiOR

201

We shall extend the operator T~k> up to the operator T as follows. Let {/n} be a sequence of functions belonging to D T(k) such 0 that there exists a distrib11tion f with values in Bmld possessing the property to make for any function '\j) E C';' (R 2) lim ) fn (A-) \jJ (A-)"dA,

=) f (A-) \jJ (A-) dA,.

n-+oo Rn

R2

Let lim T'ok>t n =g. Then set rt =g. Verify the correctness of the given definition. Let for any '\j) E C';' (R 2)

In E DT(k) 0

and

lim \ fn (A-) ljJ (A-) dA, =0. n-+OQ

R2

Then allow for the existence of lim

T~k>fn

n-.oo

= g.

We must show that g = 0. For any '\j) E C';' (R 2):

J

lim ) (T- A_)k fn (A-) ljJ (A-) dA, = (A-) '\j) (A.) dA-. -00~ ~ Let P be a smooth sufficiently rapidly decreasing function, which. is strictly positive, and let T = A 1 iA 2 , where (A 11 A 2 ) = A is the generating set, which determines the regular operator T. Set

+

k

q (x) =P (x) (x 1 + ix 2 - A-)k = ~ qi (x) J..k. i=O

Then the operator [P (A)l - 1 q (A) is a closed extension of the operator (T - A-) 11 , the operator [P (A)] - 1 being closed as an operator inverseto the homomorphism. For this reason

lim n-+oo

J(T- A)k fn (A-) \jJ (A-) dA, = R2 k

=[P(A)r lim 1

.2}

Jqi(A)A_kfn(A.)lp(A.)dA-=0

n-+oo j=O R2

for any '¢ E C';' (R 2 ). Hence, it follows that g

=

0.

Definition. The kernel of the operator T will be called the system of generalized e.a. functions of the order ~ k of the operator T. ~ of generalized e.a. functions of the order of the operator T will be called complete if for any vector h E Bmid there exists a generalized function G E ~ and a function f E $ N (R2}

Definition. The system

~k

OPERATIONAL METHIO!DS

202

.such that h= ~

f ('A) G ('A) d'A.

R2

Theorem 8.1. The system of the generalized e.a. functions of the .order ::::;;;; N + 2 of a regular operator T of degree N is complete. Lemma. Let 0 such that n supp (1 - '¢n) j = 0 whenever n >A. Hence

supp T mo

n

IT ((1-'IJn) /) l~e II f llcN(Rn) for any n

>

A. Lemma 1.1 is proved.

Corollary. For any function {cpn}n:>o c C'((' (Rn) such that lim L (cp 11 ) = L (f)

f E C N (Rn) there exists a sequence (1.4)

n-+oo

for any L E CF. (Rn). Iff =I= 0 then there exists T 1 E CF. (Rn) such that T f (f) =I= 0. Proof. Let L E c-;. (Rn). For any f E c N (Rn) we have L ((1 - '¢n) /). L (f) = L ('¢n/)

+

(1.5)

The function '¢nf is finite and continuous for any n > 0. Consequently, for any n there exists cpn E C~(Rn) such that ll'¢n/- (jln llcN(Rn)~ ~1hz.

From Lemma 1.1 it follows that for any e > 0 there exists M > 0 such that I L ((1 - '¢n) /) I < e II f lieN for n > M. By virtue of (1.5) we obtain

IL

(f)- L ( (jln) I~ II L llcJV(Rn) ·1/n + e II

f llcN(Rn)"

Relation (1.4) follows from this inequality. If 7=I= 0 then there exists x 0 E Rn such that functional llxo defined by the formula . llxo (/) = f (xo) for any f E C N (Rn)

(1.6)

7 (x

0)

=I= 0. The

belongs to X N and llxo (f) = f (x 0 ) =I= 0. Hence the proof is obtained. We shall introduce the operation of convolution of functionals from C"jy (Rn) with functions from C N (Rn). But first we define some continuous linear mappings 1:h and "~'' of the space CN (Rn) into Cx (Rn) (which will be convenient for future usage) in the following way: for any f E C N (Rn) let def

Thf (x) = f (x-h)

OPERATIONAL METH10·DS

214

and def

~

f (x) = f (- x). Definition. The function F defined by the formula F (h) = T

((•hi)~),

where T E C"Fr (Rn) and f E C N (Rn), will be called the convolution of the functional T E Cf,; (Rn) with the function f E C N (Rn) and denoted by ,

1

dcf

*f(h)=T((•hf)~).

(1.7)

The function (• hf) v E C N ( Rn) for any h E Rn. Hence the convolution is defined on Rn everywhere. From (1. 7) we hrJVe

IT* f (h) I::::;;; II T llcj\(11 f lieN (1 +I h I)N.

(1.8)

Moreover, the convolution belongs to C N (Rn). In fact let E Cf,; (Rn), I E C N (Rn) and h 0 be a point in Rn. For any h satisfying the inequality I h - h 0 I < 6 the following estimate is valid: T

II •hi llcN(Rn)::::;;; II I llcN(Rn) (1 +I hoI+ 6)N. Consequently, the family of functions (• hf)lh-hol 0 there exists M > 0 such that for all n > 1H the following inequality is valid:

IT ((1- 'l'n) (•d))v I




0 there exists 6

>

0, such that ( 1.10)

From (1.5), (1.9) and (1.10) it follows that for any e > 0 there exists 6 > 0, such that

* f (h) - T * t (h 0 ) I::::;;; e whenever I h - h 0 I < 6. IT

The continuity of the convolution is thereby proved. Moreover, from (1.8) we see that the convolution of any functional from Cf,; (Rn) with functions from C N (Rn) is a continuous linear

CH. II. CALCULUS OF NONOOMMUTATIVE

215

OPER.~TIQRS

mapping from C N (Rn) into C N (Rn), which to a shift, i.e., for any h' ERn we obtain

i~

invariant with respect

-r:h' (T * f (h)) = T * (-r:h' f) (h).

The opposite assertion is also correct: for any continuous linear mapping A of the space C N (R") into C N (Rn), which is invariant with respect to a shift, i.e., forj any h E Rn and f E C N (Rn) the following equality is fulfilled: A-r:hf = T~tAf; (1.11) there exists a functional T E C'N (Rn), such that, for any f E C N (Rn), we obtain Af = T *f.

Prove the assertion. Note that (1.12) Since A is continuous, the mapping f-+ Af (0) yields a continuous linear functional from C';, (Rn) which will be denoted by T. Thus T

(1) = Af (0).

But from (1.12) we obtain T * f (0) = Af (0). By replacing f by -r:hf and taking into account (1.11) we further obtain T * f (h) = Af (h) for any h E Rn. The assertion is proved. Note that T may not belong to C"h (Rn). This assertion, however, allows us to define the operation of the convolution of functionals of the space Ch (Rn). Let Tu T 2 E Ch (R"). Then it is obvious that the mapping Af = T1 * (T2 *f) is continuous and invariant with respect to a shift. Thus, as proved above, there exists T E C'N (Rn) such that for any f E C,y (R11) (1.13) T*f= T 1 *(T2*f). Now we may give the following definition. Definition. The functional T defined by relation (1.13) will be called a convolution of the functionals T 1 , T 2 E C"h (Rn) and denoted by def

T=T 1 *T 2 • By using (1.12) and (1.13) we obtain the relation T*f(O)=T (/) =Tt((T 2 *f)'')

(1.14)

216

OPERATIONAL METHODS

from which it follows that T E C"N (Rn) due to the fact that f-+ -+ (T 2 *f)~ is a continuous linear mapping from C N (Rn) into C N (Rn) and T 1 E Cjy (Rn). Thus, we have proved the theorem that follows.

Theorem. The space C1v (Rn) is invariant relative to the operation of convolution.

Some important properties of the operation of convolution are given by the theorem that follows.

Theorem 1.2. The operation of the convolution of functionals of the space C"N (Rn) is commutative and continuous in factors. Proof. The continuity of the convolution follows directly from (1.14). In fact, IT

(f) ~~II T1ilck II T2 * f lieN~ II T1 lick II T2llck II f lieN.

Hence II T1 * TzllcN(Rn)~ll T1llcN(Rn) II T21!cJV(Rn)

(1.15)

which means the continuity of the convolution in factors. Now show the commutativity of the convolution, i.e., prove that for any f E C N (Rn) ~he equality T 1 * T 2 (!) = T 2 * T 1 (!) (1.16) holds. If we show that (1.16) is fulfilled for any f E C': (Rn), then from Lemma 1.1 it follows that (1.16) is also correct for any f E C N (R"). For this purpose we shall need the lemma that follows.

Lemma 1.2. Let 'Pe E co;; (Rn) and have the following properties: 'Pe (x) ;;?: 0 for any x ERn and ~ 'Pe (x) dx = 1, 'Pe (x) = 0 for I x I ;;?: e > 0; then for any cp E C': (Rn) the following relation is valid:

lim T*('Pe*'P)=T*(jl·

e

~

0

Proof. From the obvious equalities we obtain T * 'Pe * cp -

T * cp = T * ('Pe * cp- cp).

By virtue of the continuity of the convolution we have II T * ('Pe * (jl- 'P) llcN(Rn)~ II T llcf..ro be a sequence of finite functionals convergent to Tlt and {TT}m>o be a similar sequence convergent to T 2 • By considering (1.17) we obtain lim (T~ *ere)* (Tr;' *ere) (/) = Ti * T 2 (/). EtO

But T1 * ere and TT * ere are finite continuous functions, and for this reason we have (T1 *ere)* (TT *ere) (f) = (TT *ere)* (T1 * 0 there exists Jl![ > 0 such that for n > M we obtain

IL

(h* (1 -'\jln) /)

I+

e

II h*f II

~ e

II h*

II f II CN[Rn, B]"

liB•

Any weakly* convergent sequence is bounded. Consequently for any e > 0 there exists M > 0 such that for all n > M the estimate is valid:

I (L, f) (hin)- (L,

'i'nf) (h;';,) I< e.

(1.26)

The function 'i'nf is continuous and of there exists a function fn with a finite such that II fn - f llcN[R", BJ 0 such

for any m > M'. The lemma is thereby proved. Thus the pairing of Cf;; (Rn) with c N B] described by (1.25) is a continuous linear mapping from Cf.r (Rn) X Cn [Rn, B] into B. In complete analogy with the preceding section we may construct the algebra :Iff sr, ... , s ,l (RN), where su ... , sN are integers. For this

mn,

1

purpose denote by c"t· ... , SN (RN) the Banach space of functions continuous on RN and having a finite norm

II g lie

def

8 1'

.. • ,

8N

(RN) =sup s xERn (1+1 Xti) i

I g (x) I · ·.

(1+1 XN I)

s N

The closure of continuous functionals finite in C,~, ... , sN (RN) will be denoted by c;l' ... , "N (RN). Construct $81' .. . , SN (RN)

OPERATIONAL :11ETHODS

224

from C;1, .•.• sN (RN) in a manner similar to the one we have used in constructing !8 N (Rn) from C"N (R"). The isometric isomorphism, which is made use of, will be called the Fourier transform and denoted by ,¥. Sec. 2. The Functions of Two Noncommutative Self-Adjoint Operators Let A and B be self-adjoint operators in a separable Hilbert space H and let T be an operator of the Schmidt class B 2 (H). In Sec. 9 of Chapter I the pair (A, B) of the operators commutative in the everywhere dense set defined on a Hilbert space B 2 (H) corresponded to the pair of operators (A, B). There exists a homomorphism eft->-+-: C (R 2) -+- Op (B 2 (H)), A,B

which translates the function f E C (R 2 ) into the operator f (A, B), ..- induces II f (A~ B) IIB2 (H) ::::;;; II f llc ~ ~II cpj(J) ll$s, s• =II (Pi llll crzii· Thus the norm (5.1) and the product (5.2) induce the structure of the normed algebra into !19 s, s' (R" X R"'). The Banach algebra obtained by the completion of this normed algebra will be denoted by !19 s, s'. (R" X R"'). Let !19 s, s', (R"+"'"-a) be the closure in !19 s, s', (R" X R"') of the subalgebra consisting of finite functions with supports in R" +1" ""'a. The factor-algebra !19 s, s', (R" X R"')/.5W s, s', (R"+"'"- a) will be denoted by !19 s, s'. (a).

Theorem 5.3. The following inequality is ualid: 2

( 1

3 )

II

T'ljJ A, A' ~IJ'IJlll$s,s',· Proof. We have

A') ll=llr'IJl (l, A') (}); 1 (l, A')(Dn (l, A') 11~

llr'IJl(l,

~II T(J),:; (A, A') IIB2-+B3JJ1J:cDn JJ$s, s• 0 (R 2 ) and for any h E B1 , the following formula is valid:

2 (1

lim Tnf An,

3) P (1An, A~ 3) h=Tf2 (1A, A'3) P (1A, A', 3) h.

A~

n-+oo

On the other hand, the sequence of operators

T~ = P (A, A') (jl (A~) Tcp (An)= (jl (A~) T'cp (An)

254

OPERATIONAL l\LETHODS

acting on B 1 to B 2 is bounded in the norm and strongly converges to T'. For this reason

2 (1 3)

2 (1 3)

lim T;,j .An, A;, h = T'f A, A' h. n-+oo The lemma is proved. Now consider the family {T (;)} of regular operators T (;) = A 1 (£) iA 2 (s),

+

where (A 1 (s), A 2 (s)) = A (£) is a generating set of degree s with the defining pair of spaces (B, B). Here the domain D of the operator T (;) does not depend on

s,

lle-iA(~)tll

~~

(1 +It I )8


s,

Q.E.D.

Definition 6.4. Any function :n: (r, s) satisfying the conditions (i) and (ii) of Lemma 6.1 will be called a ff'-function of .II:. The smallest non-negative integer l with the property .:!iOn

(hs/2) = h-lf20n (h

s~l)

will be called the type of .Jf. 26*

OPERATIONAL METHODS

404

Problem 6. i. Let

Then [ 2r;-s

n (r ' s) = {

J+i

for { 2r;-s}

>{

-J for { -

2r-s

2r-s-4 [ -5

2r-s 5- }

1

~5

is a ~-function of .It; here [x] denotes the greatest integer not exceeding x, and {x} = x - [xl. In particular, n (r, 0) ~ 1. Given a Hamiltonian function rJ/8 (p, q, t), a solution s (x, t) of the Hamilton-Jacobi equation with dissipation corresponding to $-8, a smooth function g (p, q, t) and an operator .!! E ~. the following relation will be called the transfer equation with dissipation in the . . d N 2 - or er appronmatwn:

(/!t+(d1'8p(Stx, x, t)+ic!J8pp(Six, x, t)S2x. (/Jx)+ +{-}tr[c!J8pp(Six, x, t)Sxx]+g(S1x,

X,

t)+.it}cp=

=('Js~(hN/2).

(6.1)

We shall consider only those solutions of the transfer equation with dissipation which have the form cp = h-m/ 20s2 (hm12 ), where m is a non-negative integer. The relation (6.1) with the right-hand side replaced by

0s2 (h 112 )

+ (

1.

Proof. We shall construct {ff'~t} and {Qd by induction. Set ff>l (p) = -(1 + Pa) -l p. Then p

+

1

~

.

..e.!_ a' J lg'i (p) =

L.i il I i 1=0

aai

= p-(1 +Pat 1 p-(1 + Po.t 1 PaP+ Q2 (p) = Qz (p), so (6.19) holds for R = 2. Suppose we have constructed ff' 1 , . . . , ff>m and Q2 , • • • , Qm+r so that (6.1 9) holds for R = 2, ... , m + 1, and let ff> m+t be any homogeneous polynomial in p of degree m + 1 with vector coefficients smoothly dependent on ex and t. Then

P+

m+i

m+i

~ .E.!._(_!!___)i ~

LJ jl lii=O

L.i

aa

m+1

It

k=1

li l=m+1 ~

+ I iLJ1=0

LJ jl lii=O

aai

+

~ ~: (a~ )j ~[!>It (p) = Qm+i (p) +

+ Qm+i (p) + m+i

I.

(p)= ~ ..£!._a J!gum+t(P)

@'

k=1

I •1

2.!_ a 3 il

.a?m+i (p) aai '

(6.20)'

+

where iJm+t (P) is a homogeneous polynomial of degree m 1 with smooth coefficient which does not depend on the choice of ff> m+t· Since ali!gum+t(P) (' aai = Pa

a); ff>m+t (p) + Qi. = m+2-i (p),

ap

410

OPERATIONAL METHODS

where 7h,m+ 2 -i (p) is a homogeneous polynomial of degree m + 2 - j with smooth coefficients, (6.20) implies that if ff> m+I satisfies the -equation m+i

~

pi JT

L..J

(t

a )i Pa ap ~m+dP) = -Qm+i (p),

(6.21)

I j J=O

then there is a Qm+ 2 such that (6.19) is satisfied for R = m + 2. Thus, the lemma will be proved if we show that (6.21) has a solution of the form

cfJ'm+i (p) =

Lj

bjp1,

lil=m+i

where bi is a smooth vector function of a and t. Let

Om+i (p) = -

Lj

I i l=m+1

ajp 1•

'Then ff> m+I satisfies (6.21) if b1 ,

~ k!J~k)! [ k,;;j

••. ,

CPa) 1k 1Jbi=ah

bn satisfies the equations

j=1, ... , n.

(6.22)

So it suffices to show that for any j the matrix def ~

j!

t

Aj= L..J k! (j-k)! (~a)

Ik I

k~j

is non-degenerate. But note that ~

~

j!xlkl k! (j-k)! = (1

+x)

Iii*

'

k~j

hence det Ai = det [(1 + t~a) 1 i 1] = [det (1

+ ~a)] 1 i 1:i= 0

.as required. Lemma 6.5. Let {8\} be the sequence of Lemma 6.4, and let N

~N =

Lj ff>d~).

k=i

* This follows from the generalized binomial formula (x

+ yf. =

~

'/ kl u'-k)!

.;ckyi-Jt;

k~j

:here j and k are multi-indices of length n, and x ·• · ., Yn).

=-

(x1 ,

••• ,

xn), !I

=

(y 1 ,

CH. IV. GENERALIZED HALVIILTON-JAGOBI EQUATION'S

411

Then the operator N

R",= ");

(-a )i

_1 p)

i!

~

I\

PN

(6.23)

ar:x

I i 1=0

satisfies the conditions I'

I'

A

I'

I'

A

LNRN= 1 +e 1 ,

(6.24)

RNLN=1-e 2 ,

(6.25)

u•here 2N

I r 1=1 2N

~2 = ~ 1r Gr

br (a, t) (

1=1

(a, t) = OD ( h

a: r '

N+1

-2-) ,

Proof. First we check that Rf.v is a right "almost in verse" of Lf.v, i.e., (6.24) holds. We have N

Lf.vRf.v

=

~

:,

~k

1k 1=0

-

-

a: r ~ i- ~~ (a~ ) N

(

t

=

111=0

S

N

"'

)1

~ -..J I k 1=0 ll 1=0

-lilA!

"'

1

LJ

~

j!(k-j)!Z!

O~j~k

k

~ (-f)-)l+k-j or:x1

ar:x

•

Introducing the index r = l + k- j instead of j, by a change of order of summation we obtain N

N

L"N R"N-- LJ .....::,

"' ~

I k 1=0 111=0 X

fjll+h-r lp~ ~__,.... ....,.k___:_::.:._ ar:x 1 + -r

pk

"'

LJ

(l+k-r)! (r-l)! l! X

l~r~l+k

( f) -:;;::;-)

u~

r

= 1+

2N

"'

(

LJ I r 1=1

ar

f)

0 ~~

)r ,

where f)ll+h-r lp~

"' LJ O~l~rh>r-l

lli>Nihi~N

P" (l+k-r)! (r-l)! l!

f)r:xl+k-r

(6.26)

412

OPERATIONAL METHODS

=

Now introduce the index j becomes N+lll-lrl ar=

~

~

O1 (S) + hkrk (S)},

=0

(2.5)

l=O

where

rk (S) E Hom (Hs+k, H"),

all

s E R,

and

~

U>z (S) =

ala I Qfe ( ----;;-a-

as

ax ,

) 1 x Pa, 13 (S)

all31 axil

(2.13)

p

lz

(S)

+ hkrk (S),

(2.5')

CH. V. CANONICAL OPERAT'OR

ON

443

LAGRANGEAN MANIFOLD

A

(c, A+ B),

but the left-hand member of (2.5') has the form f we use the following generalization of (6.8) of Introduction: n

k-1

n

{ 2

.2i .... ~ 1=1 1 =1

f(A+B)=f(A)+ ~ X

6i 1

+

•••

( 1

{

~

6i 1

•••

(

2

Bi 1

Biz X

••• ,

2!+1)}

Aiz, A;z, ... , An

+

2k

Bik X

•••

1

6ikf A 17

• ••

2!-121+1

3

8izf A1 , ••• , Aj 1 , Aj 1,

1~ji~· . . ~jk~n

X

1

21

Bi 1

11=11_ 1

1

so

1 ••• ,

3

2h-1

2k+1

Ai 1 , Aj 1 , ... , Aik' Aik

+

(2.7) where, as well as in the one-dimensional case, some operators may be additionally included as parameters (cf. the notes following . Theorem 6.4, 6.6 and 6.7 of Introduction, respectively). Thus we obtain (2.5') with Cl>z (S) =

[6j 1

~ · {a~ . .. ·a:. x

1~ji~· . • ~iz~n (

•••

11

1z

1

6}jdf'8) Sx 1,

1 •• • ,

3

Sxji' Sxj 1 ,

21-121+1 •• • ,

2!+1)}

Sxi' Sxjl' ... ,Sxn

(here we have dropped x being the last to operate). In the expression of cD 1, let us change the order of operating so that f)/f)x would be the first to operate. Using the commutation formula of Chapter II, we arrive at CVz =

"" LJ l~la\~2! O"'illl~l

aa61e (~as --;;a 7fX '

X

) Q ( a ) B, a, B 7fX

p

where Q~~ B (S) is an independent of 018 polynomial (with constant 1. coefficients) in the derivatives of S of orders 2, ... , l What remains to be proved now is that Q~: B = P~~ B· It is easily seen that for this it suffices to show that

+

Q~~ B (S) lx=O = p~~ 13 (S) lx=O for all S with Sx(O) = 0. To do this, put dl8 (x, p) = pa. In spite of pa ~ S&oo, both (2.7) with f = J!8, A= Sx, B = p and the commutation formula of Chapter II remain clearly valid in the case, the

444

OPE:RATIONAL ;vrETHO:DS

remainder in (2. 7) vanishing if k is large enough. Hence

;a exp { ! S (~) }

= exp {

~

S (x)} X

1 alo\ X ~ ( - ih) 1~ a. sa-vQ (S) - . .LJ .L.l(a-y)! x y,o axo z v,o

(2.8)

Applying this operator indentity to xB and putting x = 0, we obtain e

_..!__ s (x) h

ih

( -

a a _2._ s fiX) eh

(x)

xi3Jx=0 = ~

( - ih) 1 a!~~

Q~~ 13 (S)

ix=O·

l

Comparison of the last formula with (2.4) produces the desired result, Q.E.D. We now generalize (2.5) to the case where Im S need not vanish. Theorem 2.1. Under the assumptions stated at the outset of the section we have

( 3 2)

( 1)

i

i

N-1

cf/8 x, p eh 8 x =ehS(x) ~ (-ih)k are real. We have shown before that ~}1 > and ~}2 > are symmetric, ~}2 > being non-negative because of the dissipa.tivity condition. Hence we have for any h = h< 1> ih' 2>, h< 1>, h ERn: Im(~rh, h)= -(~h< 1 >, h< 2 >)+(~h< 2 >, h< 1>)+ + (~h(1>, h(i)) + (~(2>h(2), h) = = (~h(i), h(i)) + (~h, h);;::0.

+

Putting h = C rg here, we obtain the required result Im (Erg, Crg) ;;;;::: 0. Now we shall produce some elementary consequences of (C1)-(C3). Lemma. 3.1. If a EM is C-Lagrangean, then the matrix Er (a) is non-degenerate for t > 0.

+ it Cr (a)

+

Proof. Suppose that this is not the case and let g be a. non-zero vector such that (E r it CI) g = 0

+

> 0. Since Erg = -it Crg, it follows that F (g) = -it II Crg 11 2 • Then Crg = 0 by (C3), so B 1 g = 0, contradicting

for some t

=

(C1). The lemma. is proved. Lemma. 3.2. Let an EM be C-Lagrangean with respect to the mapping P = P (a), Q = Q (a), let H (p, q) be a real quadratic form, and let P = P (a, t),

Q = Q (a, t)

(3.5)

be the solution of the Hamiltonian system

P = -Hq (P, Q),

Q = Hp (P, Q),

(3.6)

satisfying the initial condition P (a, 0) = P (a), Q (a, 0) = Q (a). Then an is C-Lagrangean with respect to the mapping (3.5) for any fixed t E R.

Proof. (1) First we show that rank (a.o, = n.

(Pa t)) Qa (a.o, t)

Let P (a, ~), Q (a, ~) be linear in the conditions P(a, O)=P(a), Q(a, O)=Q(a), det a(P, Q) j =f= 0.

a(a,

~) a=cxo

~ERn

functions satisfying

OPE:RATIONAL METHODS

454

Further, let P (a, ~. t), Q (a, ~. t) be the solution of (3.6) with the initial conditions P (a, ~. 0) = P (a, ~), Q (a, ~. 0) = Q (a, ~), and set P. t) = d t a (P (ex, B, t), Q (ex, B. t)) J( • a(ex, B) e a, p, In the same way as in (the proof of) Lemma 2.2. of Chapter IV, we J (a, ~. t) = 0. Thus, J (a 0 , 0, t) =1= 0 which implies conclude that the required assertion. (2) Next we check (C2) for (3.5). We have

:t

Qa (a, t) Pa (a, t)

= =

HppPa (a, t)

+ HpqQa

(a, t),

-HqpPa (a, t) - HqqQa (a, t).

(3.7)

A straightforward calculation yields d

dt{Q, P}i~t=O.

(3) Now we prove that

lm (Pa (a 0 , t) h, Qa (a 0 , t) h)

~

0.

Equation (3.6) implies that d

dt(Pa(a, t)h, Qa(a, t)h)=-(HpqPa(a, t)h, Qa(a, t)h)-(HqqQa(a, t)h, Qa(a, t)h)+ (Pa (a, t) h, H ppPa (a, t) h)+ (Pa (a, t) h, HpqQa(a, t)h) =(P:x.(a, t)h, H 1,pPa (a, t)h)-(Qa(a, t)h, HqqQa(a, t)h).

+

:t

(Pa (a, t) h, Qa (a, t) h) is real, as the sum Thus the derivative of two quadratic forms corresponding to Hermitian bilinear forms. Hence Im (Pa (a, t) h, Qa (a, t) h) does not depend on t, which completes the proof. Now let us consider the Hamiltonian function H = ; (p; + q;). The trajectory of H starting from (Q (a), P (a)) is given by QI (a, t)

=

QI (a) cos t

Q1 (a, t)

=

Q1 (a),

PI (a,

+ PI (a) sin t,

t) = P 7 (a) cos t - QI (a) sin t,

Pr (a, t) = Pr (a).

(3.8)

CH. V. CANONI·CAL OPERAT'OR

=

ON

A

LAGRANGEAN MANIF·OLD

455

Let a 0 E M be C-Lagrangean with respect to the mapping Q = Q (a), P = P (a), and suppose that the conditions O J ( ) def DQ a -na=F' def D

lr(a)=

(Qr, Py) Da

=J=O

are satisfied at a 0 • Let a certain value Arg J (a 0 ) of the phase argument of the Jacobian J (a 0) be chosen. Using the transformation (3.8) we define Arg J 1 (a 0) correlated with Arg J (a 0 ) as follows. First we define the family (Q'(a), P' (a)), 't ~ 0, of mappings M-+ C2n by

Q' (a) = Q (a) - iTP (a), P' (a) = P (a) + iTQ (a).

(3.9)

Next we set DQ' Da

J(a T ) = '

and define Arg J (a 0 , condition Arg J (a 0 , 0)

't)

as a continuous function satisfying the

= Arg J (a 0 ).

The transformation (3.8) induces the following homotopy of the mappings a-+Q'(a) and a--+ (Q 1 (a)- iTP 1 (a),

Q'(a, t)= (Q}(a),

PI (a)+ iTQ1 (a)):

QJ(a) cost+P:f(a)sint),

n

O~t~z-·

For a fixed t-+

T,

DQ~~·

the mapping [ 0, ~ J-+ C defined by t)

la=ao def J ( ao,

't,

t), (3.10)

n

O~t~ 2

is a curve in C starting from J (a 0 ,

and finishing at J 1 (a 0 , 't) der 0 (a 0). Let us show that J (a 0 , T, t) =I= for. 't > 0. In fact, it follows from Lemma 3.2 that a 0 is C-Lagrangean with respect

=

def

DQ} Da

T)

456

OPE:RATIONAL METHOD:S

to the mapping a-+ (Q (a, t), P (a, t)), where Q (a, t) and P (a, t) are defined by (3.8). By Lemma 3.1, for -r > 0 we have de t ( oQ (a, t)

.

-l't

oa

aP (a, t) )

oa

-1-

a=ao

.,.

0.

It remains to be noted that Q' (a, t) = Q (a, t) - i-rP (a, t). Since the curve (3.10) does not pass through the ongm, it defines a certain value Arg / 1 (a 0 , -r) of the argument of the Jacobian J 1 (a 0 , -r). Finally, we defme Arg / 1 (a 0 ) by Arg J r (a 0 ) = lim Arg lr (aa, -r). (3.11) "t-++0

The limit in (3.11) must exist as is seen from the following lemma. Lemma 3.3. If J 1 (a 0 , 0) =I= 0, then Arg / continuous on (0, -r 0 ) for any -r 0 > 0.

1

(a 0 , -r) is uniformly

Proof. Let 0 < -r1 < -r 2 < -r 0 • Consider the following four paths in C: l1 : 't-+ J (ao, -r), 't1 ~ T ~ 't2,

Let

l2

:

t-+ J (a 0 , -r1 , t),

l3

:

t-+ J (a 0 ,

f.,i

T 2,

:n:

0 ~ t ~2'

t),

be the increment of the phase argument corresponding to li>

j = 1, ... , 4. Since the increment of the phase argument corres-

ponding to a closed path is a homotopic invariant and the path l 1 + l 3 - l4 - l 2 in C {0} is homotopic to a point, we have /5,1 f., a = /). 2 /5, 4 • It is clear that Arg lr (a 0 , -r 2) - Arg lr (a 0 , T1) = /5,1 + /),a - /5,2 = /5,4.

+

+

Let m= min o~'~'o

max

Jl 1 (a 0 ,

I / 1 (a 0 ,

-r)J. If

T 1) - /1

(a 0 , -r) I< m,

't1~'~'2

then a simple calculation yields I sin /5,4 I ~ m - 1 I lr (a 0 , 't1)

-

Thus uniform continuity of Arg /

J 1 (a 0 , -r), and the lemma is proved.

lr (ao, 1

't2)

1.

(a 0 , -r) follows from that of

CH. V. GANONICAL OPERATOR

ON

A

LAGRANGEAN MANIF'OLD

457

Definition 3.2. Arg J 1 (a 0 ) defined by (3.11) will be considered concordant with Arg J (a 0 ). Let ao E Q QI r be a point of a Lagrangean manifold with

n n

a complex germ. For this situation we have introduced in Sec. 1 another rule of correlating the argument Arg J 1 (a 0 ) with Arg J (a 0 ), that rule depending on bypassing focuses operation. Let us show that these two rules are equivalent. In the case where a 0 is in the intersection of non-singular patches of the zones Q and Q 1 , this is an immediate consequence of the following result. Lemma 3.4. Let a 0 EM be C-Lagrangean with respect to the mapping a-+ (Q (a), P (a)), and let Q' (a), P' (a) be given by (3.9). Further letT= {1, .. . , k}, and set ,, 8

Q1

def

,

def

,

,

•

,

•

(a, t) = Q1 (a) cost+ ePy (a) sm t,

,

Qi 8 (a, t)=QI(a), ,

def

8

,

P 1' (a, t) =

Py (a) cost- eQy (a) sm t,

P'· 8 (a, t)

P} (a)

def def

Je(a, 't, t)=

DQ'•

8

(a, t)

(3.12)

Da

for any symmetrical orthogonal k X k matrix e. Define Arg J 8 (a 0 , 't 0 , 0) for 't > 0 as a continuous function independent of e (note that Jf: (a 0 , 't, 0) does not depend on e). Further define Arg J 8 (a 0 , 't, t) as a continuous function of t for a fixed 't. Then

Arg Je ( ao, 't, ; ) + :rtO'e, where a e is the number of negative eigenvalues of e, does not depend on e.

Proof. An argument similar to that used in the proof of Lemma 3.3 shows that Arg J 8 (a 0 , 't, n/2) is continuous in 't for 't > 0. Consider the difference

11e, e' ('t) = [ Arg Je ( ao,

- [ Arg Je' ( ao, 't,

'C,

;

)

+ 'JtO'e

~ ) + 'JtO'e'

J-

J.

Since (-1)cr 8 Je we have 11e,e' ('t)

(a

=

0,

't,

~)=(-1)cre'Je(ao,

0 (mod 2n),

't,

~),

OPERATIONAL METHOJ:J.S

458

which implies by continuity that 11 e, e' ('t) does not depend on ,; . Therefore, it suffices to show that !1 8 , 8 • (1) = 0 for any e and e', i.e., that Arg J e (a 0 , 1, n/2) + na e does not depend on e. We have J e (a 0 ,

,;,

t) = v (t) det (C-r: cost+

eCJ sin t),

where

.0 1 1n-k v(t)=det( 10k (cost+smt)-

)>0 '

D (P~, Qj)

~a

Cj =

(a0),

and 1 i is the j X j identity matrix. In particular, ) ~ a (Pj, Qj) a 0 sint. aa Je(a 0 , 1, t)=v(t) det C1 det ( cost+e

Note that P 1 = iQi, hence a (P-I1' Ori) aQ1

1

( i 0k

o

)

1n-k

'

so we obtain le (a 0 , 1, t) = k

= v (t) det C1 (cost+ sin tt-k

IJ

(cost+ i"As sin t) =

s=1 k

= det C1

IT (cost+ i"As sin t), S=i

where "At. ... , "Ak are the eigenvalues of e. For any z EC ""- {0}, define arg z, the reduced phase argument of z, in such a way that -n 0, one can define Arg J 1 (a 0 , 't, t) as a function continuous in t. Finally, let us define Arg J x (a 0) by ArgJK(a0 )= lim ArgJ 1 (a0 ,

't,

, .... ·1·0

~).

(3.17)

The proof of existence of the last limit is quite similar to that of Lemma 3.3. Definition 3.4. Arg J x (a 0) defined by (3.17) will be said to be concordant with Arg 1 1 (a 0 ). Lemma 3.6. If Arg JK (a 0) is concordant with Arg 1 1 (a 0) and Arg J r ( a 0 ) is concordant with Arg J L ( a 0), then Arg J K ( a 0 ) is concordant with Arg ·h (a 0). Proof. We may assume without loss of generality that J(a, .-, t) = d et

J(a,

T1

t) = det

a(Q'I (a),

L

=

0. Set

Q:. (a) cost+ p~ (a) sin t) I

aa

I

1

a(Q_k(a), Q::..(a)cost+P_k(a)sint) K aa

It suffices to show that

~ (-r)

-f)- Arg J (ao,

Arg J ( ao,

T,

+Argl1 (a0,

't,

~)-ArgJ 1 (a0 ,

-ArgT(a0,

't,

~ )+Arg.T(a0,

def

't,

't, T,

0) + O)0)=0

for every 't > 0; here J 1 (a, 't, t) is defined by (3.16), and all the phase arguments of J acobians are assumed to be continuous functions in 't and t. This, in turn, is equivalent to the equality !1 (1) = 0,

462

OPERATIONAL METHODS

because ~ (1:) is continuous for T > 0 and ~

=

(T)

0 (mod 2n).

Let m1 , m 2 , m 3 be the numbers of elements in 1"'-K, K"'l, l"'K· have already shown (see (3.13)) that

respectively.~We

ArgJ(a 0 , 1, ~ )-ArgJ(a0 , 1, 0)= ~ (n-m 1 -m3 ),

-( a

ArgJ

0,

n) -ArgJ(a -

1, 2

0,

n

1, 0)= 2 (n-m 2 -m 3 ).

(3.18}

It is easy to verify that JI(a, t, T)=(sint+cost)-n+mt+m2 X

[

f)

(Qj (a), fJa

P] (a))

fJ

x

(Q1: (a),

cos t +

P~ (a)) .

fJa

Slll

J

t .

Putting T = 1 here, one obtains -iP (a)) .n-mt-m2 -ma J r ( a, 1 , t) -_ D (Q (a)Da • l X

X (cost+ i sin t)m 1 (i cost+ sin t)m2 ,

hence ArgJ1

(

a 0 , 1,

~) -ArgJ1 (a0 , 1, 0)= ~ (m 1 -m 2 ).

(3.19)

The required result ~ (1) = 0 now follows from (3.18) and (3.19), and the lemma is proved. We can now define the index of a complex germ. To make the treatment more transparent, however, we shall first defme the index for a class of objects which are closely related to Lagr.angean manifolds with complex germs. For example, this class contains all (real) Lagrangean manifolds. Definition 3.5. A C-Lagrangean manifold is a real manifold together with a real-smooth mapping I : M-+ C2n such that each point of M is C-Lagrangean with respect to f. We start with the index of a closed path in a C-Lagrangean manifold. First consider a path l (not necessarily closed) lying in a coordinate neighborhood u EM. Let ~ (l, -r), -r > 0, be the increment of the phase argument of det (C (a) - i-rB (a)) corresponding to l. It is ebvious that ~ (l, -r) is independent of the choice of local coordinates; in fact, on making a change of coordinates in u, det (C (a) - i-rB (a)) acquires a positive or a negative factor. For the case where l is not in any coordinate neighborhood, we define ~ (l, -r) by additivity. It is clear that ~ (Z, -r) is independent of 1:

CH. V.

CANO~liCAL

OPERAT'OR

0)1

A

LAGRANGEA:-1

463

MAJ\IFCJLD

provided that l is closed. In this case we set 1

lndl=2n~(l,

n),

-r>O.

One can rewrite this as follows: i

D(Q-ir:P) Da D (Q-ir:P)

~

Ind l = -2il ';}' d ln 1

1

Da

(3.20)

. 1

It is easily seen that if lis homotopic to a point, then Ind l = 0; hence, some 1-dimensional (characteristic) cohomology class of M is defined. Now, let l be a path joining two·non-singular points (i.e., points where DQ!Da. =1= 0). Set ~l = lim ~ (l, -r); (3.21) 't->+0

~l will be called the rotation of Jacobian along l. Existence of the limit in (3.21) can be proved in the same way as in (3.11). Now suppose that Ind l = 0 for every closed path l in M, in other words, that the cohomology class introduced above is trivial. Then the rotation of Jacobian along a path joining a non-singular point a. 1 to a non-singular point a. 2 depends only on a. 1 and a. 2 (but not on the choice of a path). Let a point a. 0 E M, called initial, be fixed; to be definite, assume ry} to be non-singular. Let us choose a certain value of the argument of det Oa (a. 0 ), which determines, in particular, a local orientation of M at a. 0 • We can uniquely define Arg ~~ for any non-singular point a.1 which belongs to the connected component of Jlil containing a. 0 by the formula

Arg ~~ (a. 1)=Arg ~~ (a. 0 )+~(l[a.o, a. 1]),

(3.22)

where l [a. 0 , a. 1 ] is a path joining a. 0 to a. 1 • In particular, (3.22) determines a local orientation of M at a. 1 . Thus, we have defined a local orientation of M at every point of the non-singular zone of a connected component of 111. We shall show that if M is connected, then the local orientations introduced above determine a global orientation of the zone Q. Let a.\ a. 2 be two points of Q, let u1 3 a.l, u 2 3 a. 2 be two coordinate neighborhoods in Q, and let x, y be coordinates in u1 , u 2 compatible with local orientations at a. 1 and a. 2 , respectively. If a. E u1 u 2 , then both x and y are compatible with the local orientation at a.. To see this it suffices to consider the rotations of Jacobian along the following two paths joining a. 0 to a.: (1) l [a. 0 , a. 1 ] l [a.\ a.],

n

+

OPERATFONAL MElTHODS

464

where the whole of l [a 2 , a] lies in u 2 ; (2)

+ l [a

l [a 0 , a 2 ]

2,

a],

where the whole of l [a 2 , a] lies in u 2 • So ~~ > 0, the global orientation of Q being actually defined. Thus, we have shown that triviality of the characteristic class Ind implies orientability of the non-singular zone. But, in fact, triviality of this class implies orientability of the whole of M. To show this consider the mapping a-+ (Q (a) - i-r;P (a), P (a) i-r;Q (a)) (3.23)

+

for some 1: > 0. With respect to this mapping, M is again C-Lagrangean (this is an easy exercise for the reader), and all points of M are now non-singular as seen from Lemma 3.1 We now define uniquely Arg J 1 (a) for every point a E Q1 provided that the initial point a 0 E M is fixed and a certain value Arg ~~ (a 0 ) of the phase argument of~~ (a 0 ) is chosen, by the following procedure (correct if Ind = 0): (1) For 1:;?;: 0, define Arg J (a0 , 1:), where J(

)-D (Q-ir:P) ( 0) 0 a' 1: Da a '

by continuity. (2) For 1: > 0, define Arg J (a, 1:) by Arg J (a, 1:) = Arg (a 0 , 1:) ~ (l [a 0 , a], 1:),

+

so Arg J (a, 1:) is continuous in a. (3) For 1: > 0, define Arg J 1 (a, 1:), where Jr(a, 1:) =

D (Q 1 -ir:P 1 , Pr+ir:Qr) D rx '

to be concordant with Arg J (a, 1:). (4) Set ArgJ1 (a)= lim ArgJ 1 (a, 1:). 't-++0

Problem 3.1. If a Arg JK (a).

E Qr n QK, then Arg JI (a) is concordant with

Returning to a Lagrangean manifold with a complex germ, we recall that it can be regarded as a manifold with the C-Lagrangean subset r (see Example 3.1). For such a manifold, one could define Indr, the index on r, by considering paths lying in a small neighborhood of r. However, it is more convenient to define Indr in terms of the Cech cohomology theory, where the index of a closed

CH. V. CANONICAL OPERATOR

ON

A

LAGRANGEAN MANIFOLD

465

chain of patches plays the same role as before the index of a closed path did. To clarify the connection between these two approaches, we shall describe Ind as a Cech cohomology class of a C-Lagrangean manifold M. We start with the case where M = Q, the non-singular zone. Let {u 1 } be a covering of M by open subsets u1 such that the increment of Arg J corresponding to any path lying in u1 is less than l't/2; such a covering will be said to be admissible. For every j, fix a point a/ E u1 which will be called the central point of u1. Define the 1-cochain ~ Arg J with real coefficients as follows: for every pair of intersecting sets (uh uk), we let (~ Arg J) 1k be the increment of Arg J according to a path l [ai, ak] which joins ai to ak and consists of two arcs, the first lying in uh the second in uk. Obviously, (~ Arg J) 1k does not depend on the choice of paths l [ai, ak], since I ~ Arg J lik < 1t because of the admissibility property of the covering {u1 }. It is easy to check that ~ Arg J is a cocycle. Definition 3.6. We define Ind to be the cohomology class induced I by the 1-cocycle 2n ~ Arg J. Let us show that Ind is independent of the choice of central points. In fact, let [{Xi] be another family of central points, and let 61 be the increment of Arg J corresponding to a path joining ai to [;i and lying in u1. Then the coboundary of the 0-cochain [6 1] is the difference of the 1-cocycles ~ Arg J corresponding to the families {a/} and {a;}, so these two 1-cocycles are cohomologic. Problem 3.2. The class Ind is independent of the choice of an admissible covering {u1 }. I~emma 3.7. The class Ind of a C-Lagrangean manifold M c: M is trivial if, and only if, there exists a continuous branch of the phase argument of the Jacobian J.

Proof. 1°. Sufficiency. Let Arg J be a continuous branch of the phase argument of J. Then {Arg J (ai)} is a 0-cochain with the coboundary ~ Arg J. 2°. Necessity. Let Ind be trivial. Then for some admissible covering {u 1 } (with any fixed family of central points), there is such a family {a 1 } of real numbers that (~ Arg J) 1k = ak - ab where (~ Arg J) 1k is the d.-cochain corresponding to {u 1 }. Without loss of generality · we may assume that M is connected and that aio is one of the possible values of Arg J (aio) for ~orne j 0 • Then a1 is obviously one of the possible values of Arg J (a 3) for any j (to see this it suffices to consider a chain of elements of the covering joining UJ 0 to u1). This value of Arg J (a 1) gives rise to a continuous branch of Arg J in ub so Arg J 30-01225

466

OPERATIONAL 1\IETHOtD.S

is defined as a 0-cochain with coefficients in the sheaf of germs of smooth functions. To prove necessity it remains to show that this cochain is a cocycle. But this is an obvious consequence of the fact that I aj - all. I < JI: if Uj ull. =f=. 0.

n

We shall now remove the assumption made above that the whole of M is in the zone Q. Let a-+- (P (a), Q (a)) be, as formerly, the mapping that determines on M the structure of a C-Lagrangean manifold. Recall that M is in the non-singular zone with respect to the mapping (3.9) with 't > 0. So we obtain the family {Ind~} of cohomology classes of M. But, in fact, Ind~ is independent of ·r. To see this, let 't'1 and 1: 2 be positive numbers, 't'1 < 't' 2 , and let {ui} be a covering of M by such open subsets that for every j, the increment of Arg J (a, 1:) corresponding to any paLh lying in ui is less than n/2 if 0 < 1:1 ~ 't' ~ 1: 2 • Further, let {6~} and {6IU be the cocycles ,1 Arg J corresponding to the mapping (3.9) with 't' = 't'1 and 1: = 't' 2 , respectively. Then we have J:'t2 Ujll. -

J:'tl

Ujk

=

Ail. Ll·q1: 2 -

A} Ll'tft2 1

where ,1~n2 is the increment of Arg J (ai, 1:) corresponding to the change of 1: from 't'1 to 1: 2 • So the cocycles {6Jn and {6Ik} are cohomologic, We may now define the index for a C-Lagrangean manifold by def

Ind

= Ind\

't

>

0.

Lemma 3.7 leads to the following criterion for triviality of the class Ind: Proposition 3.1. The class Ind of a C-Lagrangean manifold M is trivial if, and only if, for each zone Q 1 c M, there exists a continuous branch Arg J 1 of the phase argument of the Jacobian Jr, these branches being correlated in the intersections of zones.

Now let M be a smooth real manifold with a given mapping r c M be C-Lagrangean with respect to this mapping. ,., Consider the following "almost constant" pre-sheaf II on M:

M-+- C2 n, and let

_ { R if n [II.(u) = 0 if u

n r =I= 0 n f=F 0

with the naturally defined restriction homomorphism. We will define Indr as a Cech cohomology class with coefficients in II. First consider the special case where the Jacobian J does not vanish on r. Let {ui} be a covering of M by open subsets with the following admissibility property: if Uj f '=J=. 0, then J does not vanish On Uj

n

CH. V. CANONICAL OPERATOR ON A LAGRANGEAN MANIFOLD

467

and, for any path lying in uh the increment of Arg J corresponding to this path is less than n/2. For any j, fix a central point a/ E uh and for j' k so that Uj uk r =I= 0' define {L1 Arg J} jk as in the case of a C-Lagrangean manifold. Thus a 1-cocycle with coefficients in II is defined, which will be denoted by Llr Arg J. We now define Indr to be the cohomology class induced by 2~ Llr Arg J.

n n

Problem 3.3. Indr is independent of the choice of a covering {ui} and a family of central points {a/}. Lemma 3.8. Let J (a) =I= 0 on r. Then Indr = 0 if, and only if, there exists a continuous on r branch of Arg J. Proof. 1o. Sufficiency. Let Arg J be a continuous on r branch of the phase argument of Arg J. Choose an admissible covering {ui} of M so that

-

=

IArg J (a) -ArgJ (a) I < 2

:rt

a a

whenever there is a j such that both and are in Uj 1\ r. Further, choose a family of central points {aj} so that aj E r Uj if r Uj =I= =I= 0. Thus, a 0-cochain {Arg J (ai)} is defined, whose boundary is L1r Arg J. 2°. Necessity. Let Indr = 0. Then there are an admissible covering {uj} of M and real numbers ai (dependent on the choice of central points ai E ui) so that

n

n

(.11 Arg J)ik = ah - ai

n n

whenever Uj uk r =I= 0. Moreover, for some fixed jo such that Uj 0 n f =f= 0, we can let a; 0 be one of the possible values Qf Arg J (ai~). As before, we choose the family of central points {a1 } so that a 1 E r if Uj n r =I= 0. Assume that any two points a' and a" of r can be joined by a chain (ui 1 , • • • , uim) of elements of the covering {ui} in such a way that a'Euil a"Eum,

Ujs

n

Ujs+t

n r=t=O,

S=1, ... , m-1

(which does not affect the generality). Putting a' = ai?, a" = ai, we see that ai is one of the possible values of Arg J (a 1) whenever Uj n r =I= 0. Define the 0-cochain with coefficients in the sheaf of germs of smooth functions on f by assigning to each j with Uj f =f= 0 the continuous on ui branch of Arg J satisfying the condition Arg J (ai) = ai. The same argument as in the case of a C-Lagrangean manifold shows that this cochain is a cocycle, and the proof is complete.

n

30*

468

OPERATIONAL METHODS

We now remove the assumption that r c: Q. Again using the family of the mappings (3.9), we obtain the family of cohomology classes Ind~, 1: > 0. It is easy to verify that Ind~ is independent of 1:. So we define the index on r by Indr

=

Ind~,

't

>

0.

In particular, Indr is now defined for any Lagrangean manifold with a complex germ. Lemma 3.8 implies the following criterion for triviality of Indr. Proposition 3.2. The class Indr of M is trivial if, and only if, for r branch each I c: {1, ... , n}, there exists a continuous on QI Arg J I of the phase argument; of J I, Arg J I being concordant with Arg JK on QI QK r.

n

n

n

The construction of the canonical operator in the next section will depend on the existence of correlated continuous branches of the roots of J acobians corresponding to various zones (but not of such branches of the phase arguments of J acobians). Since VJ = ..!:..Arg J

IJ Ie2

is defined uniquely, even if Arg J is defined only modulo 4n, the condition of triviality of Indr is, in general, too restrictive. Therefore, it is natural to introduce lndr modulo 2. =

a

=

Given a E R, set Ra =Ria, where (x'""' y) ~ (x- y 0 (mod a)). Obviously, Ra is a module over Z. In the definition of lndr, let us change the pre-sheaf II by using R 2 instead of R. Suppose that J =I= 0 on r, which is essentially the general case as we have already seen. Let indik be the element of R 2 generated by the number{~ Arg JLk· Then {indik} is a cocycle. The cohomology class induced by this cocycle will be denoted by lndr (mod 2). The criterion for triviality {)f Indr (mod 2) coincides word for word with that for lndr (see Proposition 3.2) if Arg J 1 is regarded as a function with values in R 4 n, Arg JI (a 0 ) and Arg J K (a 0 ) being considered concordant if they have concordal!l.t representatives in R. For some applications of the canonical operator (but not for the proof of the Main Theorem) it is not sufficient to consider the class Indr (mod 2) only, and the notion of index needs to be generalized as follows. Let a/al = (a/al1 , • • • , a/aln) be ann-tuple of commuting linearly independent at every point complex vector fields on M. Then on replacing J (a) by det ~i in the above construction of the index, we define a cohomology class of M which will be denoted by Indr, OfOl (mod 2). It is easy to see that this new class differs from Indr (mod 2), in general. The triviality criterion for Indr, O/Ol (mod 2) is quite similar to that for Indr (mod 2).

CH. V. CANONICAL OPERATOR ON A LAGRANGEAN MANIFOLD

469

Sec. 4. Canonical Operator In this section we assume all the phase arguments of J acobians to be defined modulo 4rc. 1. Definition of local canonical operators. We need some notation and terminology. Let us denote by .:ll (An, rn) the set of all equivalency classes of D-asymptotic series on the Lagrangean manifold An with the complex germ rn . .:11: (An, rn) is obviously a left module over the ring of D-asymptotic operators. A point a E An will be called essential for cp E .:11: (An, rn) if cp has a representative non-equivalent to zero at a 0 • It is clear that all the representatives of cp are non-equivalent to zero at a 0 provided that a 0 is essential for cp. The set of all points essential for cp will be called the support of this element, supp q> in symbol. For any cp E ./1; (An, rn), the support of cp is a compact subset of r. Let u be a subset of An. We denote by .Jl (u) the set of all elements of ./I; (An, rn) supported in u. We shall use the notation .:11: (Rn) for the set of all equivalency classes of h-asymptotic series in Rn. Let us say that a y-patch of a Lagrangean manifold with a complex germ is admissible if a dissipativity inequality with e = 0:

c (a)

cvi (a)

~ D (a)

holds in this patch. Let (u, rc~) be an admissible y-patch of (An, rn).

a:n)

Fix an n-tuple :Z = ( 8~ 1 , • • • , of commuting linearly independent at every point vector fields (in general, complex) on u. a(qi + zr, P-+w-) Set J 1 = det az 1 1 and fix a continuous on u branch of the phase argument of J 1 • Then we define the local canonical operator by { ia 1

0. In the case, where s-action does not exist in the vicinity of r, the condition _!__h .t, p dq = 0 ~

rt

r

determines a certain set M of (permissible) values of h. If M has 0 as a limit point, then we can define the canonical operator by allowing h to take on permissible values only. Note 4.2. The case where (An, rn) depends on parameters can be considered in the same way as in Chapter III. If both the classes

~h ~ p

dq an d Indr, a;az (mod 2) are non-trivial, then, to define

""Jr

the canonical operator and to satisfy the quantization condition, it is required that nih

pp dq = Indr, a;az (mod 2), r

(4.8)

CH. V. CANONICAL OPERATOR ON A LAGRANGEAN MANIFOLD

475

which generalizes (4.7) and determines the permissible values of h and the other parameters. For example, if (An, rn) depends on a parameter ~, then the condition (4.8) may be satisfied by a substitution of the form p = p (e, h), where e is a new parameter. To substitute operators for the parameters e and h, one can use the regularized canonical operator in the same way as in Chapter III. 4. Commutation of a canonical operator with a Hamiltonian. Proposition 4.5. Any canonical operator is a monomorphism. Proof. Let X (x) be an infinitely smooth function which equals 1 for I x I < 1 and vanishes for I x I > 2. Set

Xe (x) = X (e-1x), e 8 (ex) = Xe (q (ex)), f 8 (ex) = Xe (p (ex)). Let

;-~)~ O'd ( -'l>lo (N)) > ~ '

so ord (V;p- ~) ~ N/2. Hence ord (V;p- ~) = oo, and the proof is complete. Note. It is easy to see that if V is a quasi-identity then the same is true of V-1 • . Proof of Theorem 4.1. We begin by commuting J/8 (;, ~) with a local canonical operator. Let (u, n~) be an admissible y-patch and let