Asymptotics of Operator and Pseudo-Differential Equations (Monographs in Contemporary Mathematics)

Asymptotics of Operator and Pseudo- Differential Equations V. P. Maslov and

V. E. Nazaikinskii

Asymptotics of

Operator and

Pseudo-Differential Equations V. P. Maslov and

V. E. rlazaikinskii Moscow Institute of Electronic Engineering Moscow, USSR

CONSULTANTS BUREAU 9 NEW YORK AND LONDON

Library of Congress Cataloging in Publication Data Maslov, V. P.

(Asimptotcheskie metody reshen:ia psevdodifferentsial'nykh uravnenii. English]

Asymptotics of operator and pseudo-differential equations / V. P. Maslov and V. E. Nazatkinskii.

cm.-(Contemporary Soviet mathematics) p. Translation of Asimptoticheskie melody resheniia psevdodifferentsial'nykh uravnenil. Bibliography: p. InLiudes index.

ISBN 0-306-11014- 8 1. Operator equations-Asymptotic theory. 2. Differential equations-Asymptotic theory. I Nazaikinskii, V. E. II. Title. III Series QA329.M3613 1988 88-3984 515 7'24-dcl9 CIP

This translation is published under an agreement with the Copyright Agency of the USSR (VAAP)

© 1988 Consultants Bureau, New York A Division of Plenum Publishing Corporal on 233 Spring Street, New York, h Y. 10013

._Al rights eserved

No part of this book may be pprodu-ed, stored in etrieval system, or transmitted in any form or by any mean$;ielectronk mechanical, photocopying, microfilming, recording, or otherwise, without writer permission from the Publisher Printed in the United States of America

CONTEMPORARY SOVIET MATHEMATICS Series Editor: Revaz Gamkrelldze, $teklou Institute, Moscow, USSR ASYMPTOTICS OF OPERATOR AND PSEUDO-DIFFERENTIAL EQUATIONS V. P. Maslov and V. E. Nazaikinskii

COHOMOLO(iY OF.INFINITE-DIMENSIONAL LIE ALGEBRAS D. B. Fuks

DIFFERENTIAL GEOMETRY AND TOPOLOGY A. T. Fomenko

LINEAR DIFFERENTIAL EQUATIONS OF PRINCIPAL TYPE Yu. V. Egorov

OPTIMAL CONTROL V. M. Alekseev, V. M. Tikhomirov, and S. V. Fomin

THEORY OF SOLITON8: The Inverse Scattering Method S. Novikov, S. V. Manakov, L. P. Pitaevskii, and V. E. Zakharov

TOPICS IN MODERN MATHEMATICS: Petrovakil Seminar No. 5 Edited by O. A. Oleinik

Contents

CHAPTER I INTRODUCTION 1.

2.

3.

4.

Examples and general statement of asymptotic problems for linear equations. What do we call characteristics? Relationships between characteristics and asymptotics

1

Quantization procedure and quantization conditions for general symplectic manifolds

13

Feynman approach to operator calculus: its properties and advantages

17

The outline of the book. Preliminary knowledge necessary to read this book. The section dependence scheme

27

CHAPTER II FUNCTIONAL CALCULUS 1.

Introduction

30

2.

Functions of a single operator

36

3.

Functions of several operators

78

4.

Regular representations

106

CHAPTER III ASYMPTOTIC SOLUTIONS FOR PSEUDO-DIFFERENTIAL EQUATIONS 1.

The canonical operator on a Lagrangian manifold in Q12n

141

2.

The canonical operator on a Lagrangian submanifold of a symplectic manifold

180

The canonical sheaf on a symplectic 1R+-manifold

193

3.

v

Pseudo-differential operators and the Cauchy problem in the space F+(M)

4.

234

CHAPTER IV QUASI-INVERSION THEOREM FOR FUNCTIONS OF A TUPLE OF NON-COMMUTING OPERATORS .

Equations with coefficients growing at infinity

49

2.

Poisson algebras and nonlinear commutation relations

266

3.

Poisson algebra with u-structure.

270

4.

Symplectic manifold of a Poisson algebra and proof of the main theorem

References

vi

Local considerations

293

310

I

Introduction

1.

EXAMPLES AND GENERAL STATEMENT OF ASYMPTOTIC PROBLEMS FOR LINEAR EQUATIONS. WHAT DO WE CALL CHARACTERISTICS? RELATIONSHIPS BETWEEN CHARACTERISTICS AND ASYMPTOTICS

A.

The Klein-Gordon Equation Consider the equation ou_m`c'

32- 2c4 u. -a2u+ E u=0 at2

h2

j=l 3x

(1)

h2

(the Klein-Gordon equation for a scalar classical field with the nonzero mass m*), and pose the question: what is the characteristic equation that naturally corresponds to the equation (1)? One may obtain, at least, two different answers to this question.

From the physical viewpoint, one should write the Hamilton-Jacobi equation describing the free motion of a relativistic particle with the rest mass m, namely, 2

-

3 E

j=1

2

(2Xj )

-m2c4 = 0.

(2)

On the other hand, from the viewpoint of mathematicians specializing in the theory of hyperbolic equations, the characteristic equation for (1) would read 3

(3 i)2 -

E j=1

(2z j

L = 0.

(3)

The question is: which of equations (2) and (3) is the right characteristic equation for the equation (1) in the standard sense? We give the following answer: neither of (2) and (3) is, or, more carefully, both of them are right characteristic equations for (1). The nature of discrepancy between

* c is the velocity of light, h is Planck's constant divided by 2ir. In order not to restrict ourselves to equations with constant coefficients, we assume, although there may be no evident physical interpretation, that c may depend on x,t: c = c(x,t).

1

It is due to the fact that two different (2) and (3) is easy to see. asymptotic problems for equation (1) are considered. Treat first the problem of the so-called quasi-classical approximation Quasi-classical approximation (originating from to the solution of (1). quantum mechanics) is none other than the asymptotic expansion of the solution in the'case when h may be regarded as a small parameter, i.e., it is the asymptotics of the solution for h - 0. One usually seeks the quasi-classical asymptotics in the form u(x,t,h)

e(i/h)S(x,t)[0o(x,t)+ hOl(x,t) +...1

(4)

j : 0, (or as linear combination of such functions), where ... are smooth functions; S(x,t) is real-valued. We see that S(x,t)/h is a rapidly varying phase, and $(x,t,h) - mo(x,t) + hml(x,t) + ... is a slowly varying amplitude of the "wave" u(x,t,h). Thus, u(x,t,h) is rapidly oscillating as h - 0 (the wavelength is of order grad SI/h). 1,

The characteristic equation is the equation which determines the evolution of the phase S(x,t). To obtain it, substitute the function u - exp(iS/h)$ into the equation (1). This substitution yields

ou- m2c4 u . h- 2a (i/h)S(x,t)

as

h2

m2c4)

- ih[acZ+2c

2 3 as a a 2 [(2t-ih ar) -3Y1(ax3-ih aX,)

- h 2e(i/h)S(x,t){[(as)2 at

ac

E jeI axj

(ate+X 3

aX 3

-

E (as )2 - m2c4]o j-l axj 3

a[

(5)

a=]0) - 0.

j=1 axj

Equating to zero the coefficient of h_2 in (5) readily gives the HamiltonJacobi equation (2).* The equation (2) is a nonlinear partial differential equation of the-first order. The method to solve such equations is well known; we recall it in item C and now we turn the reader's'attention to other problems for equation (1).

Namely, consider the problem on propagation of singularities for the equation (1): given the initial data with prescribed discontinuities, we intend to find the solution to within smooth functions (asymptotics with respect to smoothness) and thus to determine the evolution of initial data discontinuity. We assume that the initial data are smooth except for the surface too(x) - 0) and have a jump discontinuity on this surface (we assume that

0o EC°°( R3)and grad-P, x 0). Any such function, up to a smooth function, may be written in the form uo(x) - t'(lo(x))w(x),

(6)

where O(x) is smooth, 6(z) is the Heaviside 0-function,

* The solution of (2) being found, the equations for l, 2' 31 be obtained by equating to zero the coefficients of h-1, h0, h, do not discuss this matter in the Introduction.

2

...

may We

(

9(z)

0, z(0, (7)

1,

0.

z

We seek the solution as the linear combination of functions of the form

u(x,t) = 'R(i x,t))m(x,t) = 9(m(x,t))4o(x,t) + (8)

(m(x,t))29(m(x,t))42(x,t) + ...

+

.

It is useful to note that each term in the series (8) is generally more smooth than the previous one; thus, the first term 0(9)Q has a jump discontinuity at the surface {O(x,t) = 01, the second term me? )Q1 is continuous but its derivatives of the first order have jump discontinuities, Substitute the function (8) into the equation (1). We have etc. a2m(x,t)

a2

6(4,(x,t))

+

It2

at2

+ 26(m(x,t))

am at

(x,t)

at

a20 at2

(x, t) + 6(m(x,t))

(x,t)

(9)

x m(x,t) + 5,(m(x,t)at (x,t))2m(x,t) where 6(z) is the Dirac 6-function, 6'(z) is its derivative. the x-derivatives in a similar way we obtain 2

3

4

ou- mm c

u - 6'(m)C-(at)2 +

+ 6((D)1-2

am a

at t+

3

2jE1

3

am ad a2m axj axj

32f

+ e(m)C-

2

30

t (ax) 10 + jal

h

ate

az0 +

ate

Calculating

3

a2m

01 +

(10)

+jEl ax2

m2`4

s j=1 axe h

03.

Equating to zlero the factor at the main singularity 6'(m) in (10), we obtain for m(x,t) the characteristic equation (3), determining the evolution of the discontinuity surface for solutions of equation (1), with initial data m(x,0) = mo(x).

Assume now that we intend o construct the asymptotics with respect to smoothnes9 for arbitrary initial data and show that this problem all the same leads to characteristi_ equation (3). Note first of all that the Cauchy problem with ar',*- ary initial data for equation (1) reduces to tre following one:

G- m2c4 2h2 G 0' atlt=0

n(x-y)

0, (11)

6(x-yl)6(x2-v2_lx3_Y3);

the function G(x,y,t) is call"J the fundamental solution of the Cauchy problem for (1). Thus, it suff'c.es to construct the asymptotics with respect to smoothness of the fundamental solution, i.e., the so-called parametrix. To do this, we make use of the Fourier transform and represent 6(x -y) as the superposition of plane waves:

6(x-y) =

1

leip(x-Y)d3p

(2,')3

3

Thus, we should solve the Klein-Gordon equation with the initial condition e1P(x-Y) and then integrate over p; the properties of the Fourier transform make it clear that IpI - pl + Pz + p3 should be regarded as a large parameter, with respect to which the asymptotics should be constructed. Then set 3

w J

-

j - 1,2,3;

J

E w? - 1. j=1 J

(13)

We seek the solution in the form of a linear combination of functions G(x.y.P.t) -

eiIPIO(x.Y.w.t)(,o(x.y.w.t) + IPI-1+1(x.y.w,t) + ...).

(14)

Substituting (14) into (1), we obtain (o - mhc4)G(x,y,p,t) - IPI2eiIPI$C(C)2

E (2Xj )2I + j l (15)

+ terms of lover order with respect to IpI.

Formula (15) leads immediately to (3) with initial data @It-0 Thus, there exist two types of characteristics for equations (x1 - yj). with a small parameter at senior derivatives which describe, respectively, asymptotic expansions with respect to powers of the parameter (rapidly oscillating asymptotics) and with respect to smoothness (parametrices). In fact there is a close connection between them, as one sees when comparing (4) and (14), and the Fourier transform helps to establish this connection. It is noteworthy that despite such evident correspondence between these two kinds of asymptotics they were studied independently for a long period of time. It is quite natural to pose the problem of constructing a mixed asymptotic expansion for this class of equations, namely, the expansions which give solutions modulo functions which are smooth and simultaneously decaying as the parameter tends to zero. It turns out that for the Klein-Gordon equation the characteristics corresponding to this latter problem are defined by the family of Hamilton-Jacobi equations, depending on the parameter w 6 (0,13: (at)2

j

E1 (2X.)2 _ w2m2c4 - 0. J

(16)

For w - 0 these characteristics coincide with those for the problem of constructing asymptotics with respect to smoothness and for w - 1 they coincide with those for rapidly oscillating asymptotics. To demonstrate this, consider the solution of the Klein-Gordon equation (1) 4s a function u - u(x,t,X) of the variables x,t and A - 1/h, and perform the Fourier transformation with respect to A, i.e., set v(x

,

t

.

x 4 ) - (2rr)-1/2I ..m

e-iAx4u(x ,

t

,

A)dX.

(17)

We see that, u(x,t) being the solution of (1), v(x,t,x4) satisfies the hyperbolic equation

a2v+ at2

3

a2v+m2e4 a2v

j-1 axe

7

(18)

and that the mixed asymptotic expansion of the solution u(x,t,A) of the Klein-Gordon equation is equivalent to the expansion with respect to smoothness of solution v(x,t,x4) of the equation (18).

4

Similar to the above, we seek the parametrix for (18) as,an integral over p = (pl, P2. P31 p.) of a linear combination of the functions of the form

IPI-1O1(x,x4,y,y4,w,t) + ...),

where w = (w1, w2,w3,w4), and obtain the equation for 4(x,x4,y,y4,w,t)

(aat)2 -

E

J=1

(aaxe

with initial conditions

)2 -m2c4(aax4 )2 = 0,

(19)

4

01t=O

(20)

E

j-1

Set

'P(x,x4,y,y4,w,t) ` S(x,y,w,t) + w4(x4 _yd'

(21)

where S does not depend on x4,y4 (such substitution is possible since The equation for coefficients of the equation (19) do not depend on x4). S reads:

(LS )2 _

at

j

E (aS )2 -m2c"w2 a 0, 4

3x

(22)

i.e., we obtain (16) (the condition w4 E 10,11 follows from the fact that 1).

The mixed asymptotics and corresponding characteristics play an essential role in applications. For example, the mixed asymptotics is well adapted to the mathematical description of effects of Cherenkov type in which the tail of rapid oscillations (light beaming) is observed behind the particle (described by the 5-function) moving at the velocity exceeding the light velocity in the medium. The domain of rapid oscillations is defined by a corresponding family of characteristics 152]. B. General Statement of the Problem of Finding Asymptotics with Respect to Smoothness and Parameter

Now we may give the more formal and general definition for asymptotics of the type described above. Let H. denote the Sobolev space of index s:

{J nu(x)(1-6)su(x)dx}t/2,

(1)

where t is the Laplacian in Qtn. The problem of construction of asymptotics with respect to smoothness or to the parameter may he considered as follows: given a (differential) operator L,

L : Hs - H5--m for s E Qt,

(2)

we seek "almost inverse" operator RN such that

LRN - 1+QN,

(3)

where QN is either a smoothing operator II QNIIHsIHs+N

` m, s E fft

(4)

5

(this corresponds to the case of asymptotics with respect to smoothness), or a "small" operator II

QNHHs-Hs 6

s E 6t,

(5)

in the case of asymptotics with respect to a small parameter h.

For mixed aaymptotics we require that the operator QN be simultaneously a "small" operator and a smoothing one: (6)

IIQNflHs-H5+N
t) = So(xo) +J0(Jal pi apj

Set (6)

p'P(xo,T)

Solve the equation x = x(xo,t)

(7)

for xo: xo = xo(x,t) (by the implicit function theorem it is always possible for sufficiently small t, since the Jacobian detlax(xo,t)/axo] is equal to

:unity for t - 0).

The function S(x,t) = W(xo(x,t),t)

satisfies the equation (1) with initial data (3).

The characteristics appear to be connected with rich and complicated geometric structures in the phase space in which the bicharacteristics are defined. The questions of the type "what is the form of the solution to the asymptotic problem if (7) cannot be solved for xo (det(ax/axo) vanishes) and therefore S(x,t) does not exist" lead eventually to quantization conditions in which he topology of Lagrangian manifolds (i.e., surfaces, formed by bicharacteristics) plays an essential role. We cannot detail this matter further in the Introduction and refer the reader to Chapter 3.

7

D.

System of Equations of Cryst$

Lattice Oscillations

Analogous mixed asymptotics and corresponding characteristics may be constructed for difference and difference-differential equations (see 152]). Consider here a simple example of a differential-difference equation, namely, a system of equations describing the atom oscillations in a one-dimensional crystal lattice with the step h on a circle of the length 2xrN = Nh (here N is the number of atoms). This system has the form d2un c2 (un+1- tun + un-1); n = 1,...,N; (1) dt2

h2

in (1) we set uo = uN, ul +1; un denotes the displacement of the n-th atom (with respect to its equilibrium position), and the constant c is the velocity of sound in the crystal.

We assume that the radius rN of the circle remains finite, while N ti 1/h is a very large number, and look for an asymptotic solution under such assumptions. Consider a smooth 2nrN-periodic function u(x,t) taking the values uin the lattice points. Equation (1) may then be rewritten in the form

a2un

at2

c` = h2

Cu(x+h,t)-2u(x,t)+u(x-h,t)]

(2)

or, using the identity e i[-ih (a/ax)]

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

(3)

which may be verified either hya Taylor expansion, or by means of the Fourier transform, in the form 2

4c2sin2 (- Zh ax)u = 0.

(4)

h2 at2 + The family of characteristic equations corresponding to the posed problem even in this simple case is not standard and follows from the first author's general concept 150]. It has the form (at )2

4

c22 sin2(2 W as

3x) = 0.

w

(5)

The parameter w varies in the interval 10,2irrN] (it is connected with the fact that the lattice characters (2nk/N)rN vary on this interval). In the three-dimensional case the parameters in the characteristic equation vary in the Brillouin zone provided the lattice corresponds to an Abelian discrete group. If the lattice corresponds to a non-Abelian group, the parameters in the characteristic equations vary in some extraordinary domains. E.

An Example of a Difference Scheme

Closely related with those presented in the previous item are the results for a simple finite-difference equation, namely, the difference scheme approximating the.wave equation a2u(x,t) _

2 a2u(x,t) (1)

at2

ax2

Using the grid with the step h both in x and t axes and denoting un d'f

u(nh,mh),

(2)

we come to the following difference scheme m+l

un 8

- 2un + unm-1 /h 2 = c 2(un+l - 2un + un-1/h ) . m

(3)

The family of characteristic equations for the scheme (3) has the form

aS + 2aresin [c2 sin2 j 2(`ax as)] = 0, at

(4)

w

These characteristics where the parameter w varies on the interval 10,2n]. define the spread zone for the so-called unity error, i.e., for the solution which at the initial moment equals unity at some grid point and equals zero at all other points. Such characteristics for different schemes were first introduced in 151]. The Mixed Asymptotics with Respect to Smoothness and Decreasing at Infinity F.

In item B the statement of a mixed asymptotics problem was outlined for the case of commuting operators A. = X. Aj = 1,...,n with respect to powers of which the asymptotics is constructed. A somewhat more complicated situation arises when we make an attempt to construct asymptotics with respect to a tuple of non-commuting operators.

As an example, consider here differential equations with coefficients growing at infinity, i.e., the equations containing both the powers of the differentiation operator Al -i(a/3x) and of the operator A2 = x (multiplication by x). We consider the following example:

Lu = - 2-2u +.L2-u - x2(1+b(X))u, b EC-(91). o atz axz

(1)

It is natural to pose the following problem for the equation (1): an "almost inverse" operator RN should be constructed, such that LRN = 1 + QN, where the remainder QN has a kernel QN(x, y) not only smooth, but also vanishing as x * ... This requirement may be more precisely formulated as follows: consider the scale (Xs), where Xs is the closure of C0°°(R) with respect to the norm 11u11s

z

=

u(x) (1 +xz

ax2)su(x)dx.

(2)

In the scale (Xs} the remainder QN satisfies the estimates (11) of item B, thus the "almost inverse" operator RN gives, for non-smooth and nonvanishing at infinity initial data and right-hand sides, the solution to within functions that are smooth enough and decaying as !X1

-_

The characteristics for the problem (1) are defined by a tuple of noncommuting operators A, and A2. The following Hamilton-Jacobi equation occurs as the characteristic equation (cf. [52]):

(Dt)2 = (ax)z+w(1+b(x)

(3)

where w is a parameter varying in the interval 10,1]. It is no wonder that both terms in the right-hand side of (') contribute to the Hamilton-Jacobi equation. Indeed, assume for a moment that b(x) = 0 in (1). After the Fourier transform with respect to x both ..he equation (1) and the system of norms (2) remain unaffected. Thus, the first and the second terms on the right-hand side of (2) are equipollent; the asymmetry in (3) is due to the fact that Fourier transform does affect the model solutions such as (4) or (8) of item A; this symmetry d'sappears if we consider the general form of solution given by the canonical operator 150,52] and Chapter 3 of the present book.

9

Most noteworthy is the fact that the characteristic equat*ou (3) looks as if a/ax and x (but not a/ax and b(x)!) in (1} were commutative. In fact the impact of non-coimnutativity displays itself in lower-order terms only, and this is why the asymptotics may be constructed. For general equations with growing coefficients, analogous characteristics may be constructed and global asymptotics of the solution with respect to smoothness and to the growth at infinity is obtained (i.e., the almost inverse operator in the scale {XS} defined by the norms (2) is constructed (see 152,541). Examples and general techniques of constructing asymptotics in this case are briefly considered also in Section 1 of Chapter 4 of the present book. Example of Constructing Asymptotics for Equations with Degenerate Characteristics C.

If we consider the equation Lu - f with singularities, or an equation with degenerate standard characteristics, it is sometimes possible to find the appropriate operators A11...,A, such that the non-standard characteristics of the equation with respect to these operators are non-singular. In this case it would be possible to construct an almost inverse operator in the scale generated by operators A1,...,An. This scale ie defined by the sequence of norms

I -Is - R (1 +A1+ ... +A2)8/2uI

(1)

(we restrict ourselves to the case when Aj-s are self-adjoint operators in some Hilbert space).

The almost inverse operator RN of the operator L satisfies the equation

LRN - 1 + QN, B QN 1s-.s+N < m,

(2)

and can be expressed explicitly in terms of functions of non-commuting model operators A,,...,A. provided that they form a representation of a nilpotent Lie algebra or nonlinear Poisson algebra ((52,53,36,35] and Chapters 2 and 4 of this book). Even in the case of geometrically trivial characteristics, namely. for some degenerate elliptic equations, this idea has led to profound and unexpected results (see [17,70,68,26,69] and other papers where the almost inverse operator in this case is constructed in the scale, generated by the tuple of vector fields At,...,An which form a nilpotent Lie algebra).

Here we show by the simplest example how the ideas discussed above enable us to solve the problem of oscillating solutions for a hyperbolic equation with degeneracy in characteristics. Consider. the following problem: a 2u(x,t,h)

_ c2(x)Au(x,t,h)

f(z ,

t,

h),

(3)

ate

°It=o

(4)

°tIt=o ' 0,

where c(x) E C"QRn), c(x) > 6 > 0 and state the question of finding asymp-

totic, for the solution of (3) and (4) as h -+ 0 for the right-hand side f(x,t,h) of the form f(x,t,h) - etS0 x) h+o(x), So(x) - x2= E

where o E C 10

(tn), 0(0) a 0.

j1

xj ,

(5)

By means of 0uhamel's principle

the solution of the problem

(3)

and (4) may be expressed by the formula

t

u(x,t,h) = j v(x,t - i,h)dt,

(6)

0

where v(x,t,h) satisfies the Cauchy problem a2v - c2(x) v - 0,

(7)

a t2

vii=0 - 0, vtit-0

a

iSo(x)/h

(8)

The standard scheme of constructing the asymptotic solution to the latter problem is as follows (cf. item A): the asymptotic solution is represented in the form of a linear combination of functions of the form ais(x,t)/hb(x,t,h). Evidently, the result of action of the wave operator on such a trial function may be written as follows: cz(x)o7eiS(",t)/hb(x,t,h)

-

[

asz - - h2{C(at-)

-

c2(x)(s)2] x

ate

as a0 - 2c (x) as 30 + x b - ih[2 at ax ax at

(a's _ c2os)b] at2

ax

h2[320 - c2 (x)o$]}. at2

(9)

The characteristic equation ta'.es the form

(at)2 - c2 (x) (2x)2 - 0,

(10)

with the initial data Sit=0 - S0(x).

(11)

We have (2S0/ax)Ix-0 = 0, therefore the solution= of the h racteristic equation (10) with initial data (11) at not smooth fun ticns at h a point. Thus the standard scheme of constructing r pidly oscillating asymptotics fails in the considered case.

It is evident, however, that the characteristics of the problem (7) - (8) corresponding to the asymptotics with respect to smoothness have no Indeed, the wave 6perator is homogeneous with respect to degeneracy. hence the bicharacteristics start from the sphere operators ipj - I so that tae degeneracy point ipi - 0 falls out (cf. items A and Q. Therefore we are able to construct a parametrix for the problem (7) and finally, solving the integral equation of the second kind, represent the solution of the problem (3) - (4) in the form u(x,t,h) - (RNf)(x,t,h) + (rNf)(x,t,h)

(12)

In (12) the operator RN may be calculated explicitly and all that we know about rN is that rN has a sufficiently smooth kernel, so that rN is a smoothing operator (RN : He + He+} and rN :Hs Hs+N are continuous operIn our example we have in the space HN ators). u(x,t,h) - (RNf)(x,t,h) - 0(hn/2), that is, the leading term of the solution as h y 0 is given by RNf.

(13)

Indeed,

the difference (13) equals r f and is a convolution of the exponent exp(iS0(x)/h)b0(x) with the (sufficiently smooth) kernel of the operator rN, and the desired estimate follows from the stationary phase method.

In order to obtain the next terms of the expan ion of rNf (and therefore the solution u(x,t,Y,)) w'th respect to powers of h, one should substitute RNf into the quation. This procedure yields a wave equation with zero Cauchy II

The latter problem differs from the initial data and smooth right-hand side. one in that the wave operator needs to be inverted on functions, smooth (non-oscillating) with respect to the small parameter. This task can be easily performed by a computer. However, it should be mentioned that the accuracy of approximation of the solution by the leading term RNf depends on the type of initial conIxI4, then the accuracy is 0(hn/4). ditions. For example, if S0(x) Thus, the estimates of the leading term of asymptotica are connected with individual conditions. The leading term has a rather complicated form, namely, it may be asymptotically represented as an intejral with integrand having a singularity and simultaneously oscillating with respect to the parameter h.

For example, in the three-dimensional case and for c : 1, it has the form iI l2/h

if - 1 f 6n

e l-x-E

ix-EI6t

In the case of variable coefficients and arbitrary dimension the leading term has the same properties. The outlined scheme is applicable as well in the case of general operator equations which have degenerate characteristics with respect to one tuple of operators and non-degenerate characteristics with respect to another tuple. In particular cases this scheme was magnificently realized in the papers (18,11,58,24,25,21,51. H.

General Statement of the Problem of Constructing Asymptotics

The examples considered above evidently lead us to the general definition of asymptotics and consequently to the general statement of the asymptotic problems. From the abstract operator point of view, all the variety of asymptotics considered above can be treated by a single approach as the asymptotics in some Banach scale (Hs}. Specifically, by an asymptotic solution of equation Lu - v E Hs,

(1)

we shall mean a sequence of elements uN E Hal for some sl, N- 1,2,..., such that

LuN - v E Hs+N

(2)

for any N. The operator RN, which takes v into uN will be called an "almost inverse" operator; thus, for an almost inverse operator

LRN - ItQN;

QN

: HsiHs+N

(3)

is continuous for any s, and the problem of constructing asymptotics is the problem of finding the almost inverse operator in the scale. Usually the Banach scale (Hs} is generated by some operator A or, more generally, by a set of operators Al,...,A. defined in the space Ho in such a way that u E He if and only if u s H,_1 and Aju E Hs-l,j ' 1,...,n.

(4)

Sometimes, for example, if A7 are the self-adjoint operators In a Hilbert space Ho, the case of n operators may be reduced to the case of a single operator by setting

12

A - (1 +A2 + ... +A2)1/2.

(5)

then is none other than the domain D(As) of the operator As (cf. 150,521).

The space H

Two main conclusions follow from the above considerations of this section. First of all, characteristics play the most essential role in the process of constructing the asymptotics for given problems. They define the geometry of the problem and lead eventually to explicit expressions for the terms of asymptotic expansion of the solution (or at least for the Moreover, the role of characteristics is not restricted leading term). only to asymptotic expansions and related topics. The characteristics say to specialists something else and contribute essentially to their intuitive understanding of the problem, e.g., enabling them to say what kind of problems (Cauchy problem, boundary-value problem, etc.) it is natural to impose for given equations. A number of simple concrete examples considered above show that it is necessary to give the general definition of characteristics. This necessity is also emphasized by close connection of physical and mathematical problems concerning the notion of characteristics. In the second place, the characteristics themselves depend on what asymptotic problem is considered for the given equation, and this in turn is determined by the choice of tuple of operators, with respect to which the asymptotic expansion is constructed. In other words, in order-to define characteristics we need a tuple of operators (the model operators) with respect to which the asymptotics is constructed. Further, the initial equation should be expressed in terms of these model operators. This procedure itself demanded considerable preliminary investigation, namely, construction of the calculus of non-commuting operators 152]. Then the definition of characteristics is given in terms of the symplectic structure of a phase manifold corresponding to the commutation relations for the model operators (see 136]). It may happen that "standard" (i.e., commonly used) characteristics have degeneracies, while "non-standard" characteristics, obtained by appropriate choice of the operator tuple, are well behaved. For individual initial data the asymptotics in the new sense may produce the old-fashioned one.

Thus, the general theory of asymptotics deals with two principal ranges of ideas: a) b)

The geometric and analytic concepts connected with the notion of characteristics. Functions of commuting and non-commuting operators in functional spaces and related topics.

They are preliminarily considered in Sections 2 and 3 of the present chapter respectively.

2.

QUANTIZATION PROCEDURE AND QUANTIZATION CONDITIONS FOR GENERAL SYMPLECTIC MANIFOLDS

The following difficulty arises both for asymptotic problems in terms and for mixed asymptotics in terms of a tuple of operators. The phase space for these problems is not necessarily a cotangent bundle. Thus, the construction of asymptotics demands the application of general phase manifolds and, in particular, the construction of the appropriate calculus of pseudo-differential operators. Here we consider this problem for h-pseudo-differential operators (and, respectively, asymptotics with respect to a small parameter).

of a parameter or smoothness

13

If the configuration space U of the mechanical system is compact, then the phase space of the corresponding quantum system possesses a discrete structure.

Let, for example, U - S1 be a circumference of unit radius. The corre* sponding phase space is not a cylinder T U - S1 x tR1 but rather its submanifold, diffeomorphic to S' x a and consisting of circumferences, lying on this cylinder with the distance h (the Planck constant) between them. Indeed, the *-function *(x) defined for x E S1 may be expanded in the Fourier series

y,neinx. n--m

*(x)

(1)

rather than the Fourier integral and therefore the possible values of momenta, namely, points of the spectrum of the operator $ - -ih(a/ax), form a

discrete set of the form k - 0,+1,±2,...

p - kh,

.

(2)

Note that the values (2) are the very values satisfying the well-known Bohr-Sommerfeld quantization condition

(3)

2nh f Ypdq E7L (here y is any closed 1 cycle in the phase space).

Similarly, if some classical phase space X is compact with respect to momenta, the coordinates become discrete after quantization and vice versa. One may assert that if in some field theory analogous to the Einstein theory the classical phase space is curvilinear and compact with respect to momenta then its quantization will necessarily lead to discretization Non-trivial examples arise, for of coordinates (the fundamental length) instance, when the phase space is non-compact both in coordinates and in momenta. Consider again the system of equations for crystal lattice oscillations (see Section 1). The phase space even in this simplest case is not a cotangent bundle. Indeed, the configuration space for this problem is a two-parametric family of circumferences depending on a continuous parameter

h > 0 and dis rete parameter NC-Z. (the length of the circumference equal to 2wrN - Nh). The coordinate x being discrete-valued, the momenta space is also a circumference, the length of which equals 2w. The phase space is therefore a two-dimensional torus T2, for which the equality holds 2nh f

(the number of Atoms).

dpAdx - N

(4)

T2

It turns out that the condition that the left-hand aide of (4) be an integer is necessary and sufficient for the existence of the correspondence which takes any symbol f, i.e., any smooth function on the torus, into the operator £, acting on functions defined on the circumference 0 < x < 2rrrN, so that the following properties are valid: (a) commutation formula

Ii,g] for any symbols f and sponding to the form dpadx),

fg-gf - - ih{f g}+0(h2)

is a Poisson bracket on the torus, corre-

{f,g} - (af ax

and (b) formula of operator action

14

(5)

2f) ap

ax ap

(6) '

rapidly oscillating exponential:

e-(i/h)S(x)f(e(i/h)S(x),)

iLCop (x'

3

ax

= f(x,

S)P(x)

-

2 ¢ )x (ap (x, TX)

(7)

E

2 )3x3p (x' cx)] ; 0(hz). Here fi(x) and S(x) are arbitrary smooth functions on the circumference, S(x) is a real-valued function, and the estimate 0(h2) is in the norm The operators f will be called, in analogy with the case of Ll(-mrN,nrN). Euclidean space, the h-pseudo-differential operators (with symbols defined on the torus).

The construction of calculus of h-pseudo-differential operators may be performed on anarbitrary phasemanifold (thiswill he made in Chapter 3:2). It leads to the analogue of condition (4),and this "two-dimensional quantization condition" is what we intend to discuss below. Let X be an arbitrary symplectic manifold, i.e., a manifold on which a closed non-degenerate 2-form w2 is defined. As in the case of the torus, consider, instead of a single manifold, the family of diffeomorphic manifolds, depending on the parameter u, varying in the compact set. It is more convenient to assume that the manifold X is fixed, and the symplectic form w2 depends on the parameter u, w2 - w(u), or equivalently, the Poisson Recall that the Poisson bracket f. , }(v) depends on the parameter v. bracket may be defined in the following way: by Darboux theorem [2] the

form w2 has, in the appropriate coordinate system, the form w2 - dp A dx, and then

the Poisson bracket in this coordinate system is given by (6).

The condition in question reads: for p - p(h), for any two-dimensional cycle I on X the integral of the symplectic form divided by 2nh should coincide modulo integer numbers with half of the value of the second Stiefel-Whitney class, i.e..

I Cw(v(h))7 2nh

W2(moda).

(8)

2 The condition (8) being satisfied, it is possible to establish the corref, where f is a smooth function on X, and f is an operator spondence f } in the space of sections of some sheaf, so that (5) is valid with {. , _ ( . }v(h) and (7) holds locally (see [36,37,38] and detailed reproduction in 1551, and the Spanish translation of [52]).

Call the equation (8) the quantization condition for coordinateIn analogy with the Bohr-Sommerfeld quantization condition momenta. 1

2nh

pdq u/2 E71,

the class 2 W2 in (8) will be called the vacuum correction.

(9)

Recall that

the vacuum correction in the Bohr-Sommerfeld quantization condition, which at first seemed to contradict the evident expectations based on the foundations of quantum mechanics, further obtained its interpretation as the energy of vacuum and led to explanation of such delicate phenomena as Lamb's displacement. The physical sense of the vacuum correction term in quantization condition (8) is not yet clear, but one may expect that It is also connected with some delicate intrinsic properties of the field theory models with a non-plane phase space.

It is remarkable that condition (8) (with zero vacuum correction W2 = 0) already arose when co-.structing a rather narrow class of pseudo-

15

differential operators with symbols, locally linear in momenta, namely, differential operators of the first order within the framework of the geometric quantization [46,73,74,75].

For the nonzero vacuum correction the condition (8) within the context of the first order operators was obtained for Kaehler manifolds [10] and in the case of general real manifolds [273; in both cases the existence of complex polarization was required; this requirement is absent in our approach.

When constructing the calculus of pseudo-differential operators on the orbits of a compact Lie group, the condition (8) numerates the irreducible representations of the group. Thus, the coordinate-momenta quantization coincides with the Weyl rule of integrity for the major weights of irreducible representations [7,421. The vacuum correction in this case equals zero. It should be noted that for general phase manifolds the coordinatemomenta quantization condition is a sufficient condition for the existence of the canonical operator on real and complex characteristics. This fact enables one to apply on general phase manifolds the theory of global asymptotic solutions for h-pseudo-differential equations which is constructed in detail for the case of phase space 8t2n in [50].

The global calculus of h-pseudo-differential operators can be defined also on symplectic V-manifolds [373. We should note that when constructing the calculus of common pseudodifferential operators on homogeneous symplectic manifolds (see 18,23]), the quantization conditions (8) disappear.* In Chapter 3:2 we give the coustruction of the calculus of h-pseudodifferential operators on a symplectic manifold X, which, in particular, yields the quantization conditions (8). We wish to mention here that this

construction elucidates the relations between the vacuum correction

W2

2 conin (8) and the index class [50,3] in the one-dimensional quantization dition: the relation between these two classes is just the same as between the classes of dp A dq in(8) and pdq in the one-dimensional case. More precisely, the situation is as follows. The one-dimensional classes correspond to the Lagrangian manifolds and the two-dimensional ones correspond to symplectic manifolds. The graphs of canonical transformations are considered, which define the transition diffeomorphiams from one canonical coordinate system to another system. If the intersection of three coordinate charts is considered, we may construct a closed one-parametric family of canonical transformations which passes successively through diffeomorphisms corresponding to coordinate changes. The graph of a canonical diffeomorphism may be regarded both as a Lagrangian manifold as well as a symplectic manifold; the correspondence between two-dimensional classes

and one-dimensional classes is established then via the familiar Stokes formula:

f any' - jndw.

(10)

This relation might probably give anew outlook on the integer two-dimensional

class occurring in the quantization condition for coordinate-momenta.

* However, these conditions take place if the phase space is homogeneous only with respect to some of the variables. These conditions guarantee the existence of regularized canonical operators [523. 16

FEYNMAN APPROACH TO OPERATOR CALCI'i..US: ITS PROPERTIES AND ADVANTAGES

To this end, the aim of this book has become clear: we intend to study various kinds of asympto,ics for linear equations within the limits of the unified approach bound up with characteristics in the general operator context.

discussion of the main tool that we use to

Here we make a solve this problem. A.

Feynman Ordering of Operators

In quantum mechanics one deals with operators and commutation relations between them from the very beginning - these are essential basic notions of the theory. In general problems of constructing asymptotics we also come to the necessity of dealing with systems of non-commuting operators and functions of them. The ordered operator calculus proved to be a very convenient technique to treat the functions of non-commuting operators. It bases on Feynman's observation 1151 that once we rupply the noncommuting operators with numbers which indicate the order in which they act, we may treat them as if they were commuting ones. Later this idea was thoroughly developed in 1521. Illustrate this by a simple example. We write indicating numbers over the operators, the operators with smaller numbers act before the operators with greater ones; coinciding numbers over operators, which do not commute, are not allowed. Thus, for example, if

f(x,y,z) - E ckfmxk)fzm' then a substitution of the operators 1,B,C into f yields the operator 1

2

3

f (A, B, C) - E ck fmCmB Ak

i one lJcr the arobien: [151: expand the 'product of exponents eBeA into the homogeneous in B and A. In our notations we have

sum rf

1

eBeA

2

2

1

eA + B =

n=0 (A +n.B)n E

we see rh,..at the Fevgvn.'a notations. llowed combinatorial computations (though simple in this roonp:e) to be ,.voided and to obtain a solution readily. The same is the si.Luiticn in the complicated problems. There is a list of simple rules of calculations a:7d transformations for functions of Feynmanordered operators which simpiii greatly the computations and arguments in numerous problems ccncerned with non-commuting operators. These rules will be derived and proved ccmrletaly in Chapter 2; all these rules are evident for polynomials and used in examples below without rigorous proof

We now show how the method works for differential equations. In the theory of linear ordinary differential equations the Heaviside method is well known. For the equation

Y" - y = sin h

(1)

,

this method goes as follows. Denote the differentiation operator by D, D - d/dx; the equation reads now:

(I)3 - 1)y = sin ti

.

17

Then the solution is D21

y

D21

=

eix/h

_

(eix/h - e-ix/h)

x

(sin

1

I

(D+i/h)2-1

21

1

2i

1- a-ix/h

1

1

(D-i/h)2-1

(here we used the "permutation with exponent" rule

f(D)elx

=

eXxf(D+\),

The latter expression up which is easily verified for polynomials f(z)). to solutions of the homogeneous equation y" - y = 0 reads {eix/h

y a 1

e-ix/h

1

2i

1

-1 - h-2

-1 - h-2 (eix/h _ e-ix/h)

1

h2sin(x/h)

2i(1+h 2)

1+h2

It is easy to see that we have obtained the precise solution of equation

M. Consider now the equation with variable coefficients

y - x2y = sin h ,

(2)

or, in above notations,

x' )Y = sin hx .

(D'

The Heaviside method fails now; however, we may write the numbers over operators and try the solution of the form l

1

2,

(sin h

D2 - x`

Using the "permutation with exponent" rule

f(x,D)e1 x a eiaxf(x,D+1),

(3)

and the identity 2

1

f(x,D)l - f(x,0) since 1 is an eigenfunction of D with the eigenvalue 0, we obtain

y

1

1

2 _X2 2 2l.

e

(e

-ix/h

ix/h

-ix/h

1

(D-i(D-i/

1)

ix/h

1

) - 2i

-e

1

/-T`

(D + i+

eix/h

2t (x2+h-2

_ h2sin(x/h)

I + h2x2 Substitution of this inro (2) yields

18

1

{e

a-ix/h

x2+h-2

1

-

Y.,

- x2y = -

+ sin

h

i

h I+h2x'

x

- h-1 cos

+

+h2x2sin(x/h)

}

(1 + h2x2)4

h

2h2x

h (1+h2x2)2

2h2 (1 + h2x2)2

x

= sin

sin x +

h2{-h-'sin x

1 + h2x2

4 2sin(x/h) _ 2h2sin(x/h) = x +2h3xcos(x/h) +2h x

h

I

(-

2xh

cos

x2+h-' I+(xh)`

(1 +h2x2)2

(1 + h2x2)3

(1 + h2x2)2

x+ h

2(xh)'

sin

(1+(xh)2)2

x _ 2sin(x/h)

h

1+(xh)2

The quantity in the curly brackets is bounded uniformly with respect to Thus the function (3) is not a precise solution,, but an x E=IR, h E [0,1]. "almost solution," which satisfies the equation up to the "disturbance" 0 and decays as x tends to infinity, as 1/(x2 + which tends to zero as h The ordered operators method enables the construction of all the + h-2). subsequent terms of such mixed asymptotic expansion with. respect to parameter b -+ 0 and growth at infinity, but this technique is beyond the scope of the Introduction. The method of ordered operators has a widespread area of application, from obtaining of the so-called regular representations (see Chapter'2) and defining the characteristics in asymptotic problems (cf. B and C of this section where two examples are considered) to deduction of certain

identities in the theory of nonlinear equations such as the KdV equation, etc . Of course, the ordered operators method is only a technique and often the results obtained may be proved without it; but it is a very convenient. technique, which provides great computational economy and lucidity of In these problems, discussion in problems that this book is concerned with. to give up the use of the operator method would be just the same as if one were dealing with the limits of increment ratios instead of derivatives throughout the theory of differential equations.

The operator technique is extremely fruitful in various concrete In general situations, it is also much more convenient to examples. formulate and to prove the results in these terms. Feynman himself was dealing with continuous families of ordered operators rather than finite ones (T-products, the corresponding continual integrals in quantum field theory, etc.). This matter is not considered in the book; however, it is quite natural that consideration of operator calculus for a finite number of ordered operators, to which Chapter 2 of this book is devoted, may serve as a useful preliminary step in thus becoming acquainted with these questions and as an introduction to Feynman operator It is interesting to note that the relations between operator calcalculus. culus for afinite number of operators and Feynman's investigations are not confined to that mentioned above. For the equations of quantum mechanics and quantum field theory the method of ordered operators gives asymptotic approximations to precise solutions (formally) expressed via Feynman continual integration technique. The definition (2) of,the function of ordered operators is purely . illustrative; it is valid only for polynomials or convergent power series.

The definition in terms of Fourier transform f(A1,...,An) _

(27r)-n/2r

f(t1,...,tn)el

to

...

elAltldtl...dtn

(4)

19

t

iA1 j

is a group of linear operators generated by the operator Aj, where e is most appropriate for our aims, permitting the consideration of nonIt is thoroughly studied in analytic symbols f and unbounded operators. Chapter 2.

Consider now the Feynman approach in a very brief comparison with some other existing methods of constructing the operator calculus. We begin with the simple observation that, in fact, in the theory of differential (and later pseudo-differential) operators the ordered form of notation was always used. Indeed, if aX)a

L

F

(5)

aa(x)(-i m

is a differential operator, then its symbol is usually considered to be I

jaj <m

(6)

aa(x)

and this means that in our notations 2

L = L(x,-i ax).

(7)

The same form of notations is often used in quantum mechanics, when some given operator P is expressed in terms of creation and annihilation operators a* and a. The "normal form" reads

m

P

F

m,n

Pmn(a*)na 1

(8)

,

2

P = P(a,a*),

where P(zl,z,) =

F

m,n

(9)

is a Wick symbol of the operator P 161.

As for pseudo-differential operators, they are often written in the form (see 130]): Pu(x) =

(2n)In

2

(10)

f n lR 2

where p(x,£) is the symbol of P, i.e.,

1

p(x,-i(a/ax)) (see Chapter 2 1

2

for detailed explanations); sometimes the opposite order x, -ia/ax is used (cf. 145]).

However, there exist different approaches to construction of operator calculus. We do not mention here those which are based on the theory of analytic functions and the Cauchy integral formula since the class of analytic functions is too narrow for asymptotic problems to be considered and also these approaches are mainly oriented for functions of bounded operators.

Among the methods which could be used also for our aims, the Weyl approach to functional calculus should be mentioned (see 11]). The characteristic feature of this approach is that if f(xl,...,xn) = g(alxl+ ...+ anxn),

(11)

fW(Al,...A,) = g(aIAI+ ...+anAn),

(12)

then

20

where ;r. the right-hand side of (12) we have the function of a single operator alA1 + ... + anAn. Compare the Weyl and Feynman ap,orc,"ch on an example of polynomial functions of two operators A and B:

Function

Weyl operator

Feynman-ordered operator 2

1

f(A,B)

f(x,y)

(x+ y) 2

fW(A,B)

2

1

(A+B)W=(A+B)2 = A2+AB+BA+B2

(A+ B)2 = B2+ + 2BA+A2 12

AB = BA

xy

(AB )W = 4 (A + B)2 -

-(A-B)2] = 2 (AB+BA) 1

x2y

2

A2B = BA2

x2y`

(A2B)W

A2 B2 = B2A2

1 (A2B+ABA+BA2) (A2B` + B2A2 1

(A2B2)W = 6

+ AB'A+BA2B+BABA +ABAF) (in the last columns we used the property (11) - (12) for computations, representing the polynomial in question in the form of the sum of powers of linear functions; for example,

x`y = 6 I(x+y)3-(x-y)3-2v31),

(1'a)

Assume that the operators A and B satisfy the commutation relation

(L:)

AB - BA = 1,

[A, B3

and compare Feynman and Weyl symbols for some polynomial operators. present also the Jordan symbol defined by

W

Cf(A1,B)

fJ(A,B) = 2

0

+f(A,B)].

0,wrator

Feynman symbol

Weyl symbol

Jordan symbol

I

f(x,y)

g(x,y)

h(x,y)

1

(P

AB BA2

B2A2

2

f(A,B)

=

gW(A,B)

=

hJ(A,B))

xy+1

xy+2

xy+i

x2y

x2y -x

x2y -x

x2y2

x2y2 - 2xy +

x2y2 - 2xy 2

The comparison. yields the conclusion that the Weyl calculus, being more "symmetrical," is more complinated from the computational viewpoint.

The symmetry of Weyl calculus is a useful property: If all the operators A1,...,An are self-adjoint and f(x1.... ,xn) is real-valued, then fW(A1,...,An) is also a self-adjoint operator. The same property is valid for Jordan calculus.

21

However, we have chosen the Feynman ordering of operators for the following reasons: a)

b)

All the computational formulae are essentially simpler for the case of Feynman ordering. In the case of unbounded operators the Weyl calculus imposes much more rigid functional-analytic requirements on the operators A1,...,An which do not seem to be natural for applications except for the case when all the operators A1,...,An are self-adjoint.

In the subsequent two items we present two examples in which the introduction of operator notation simplifies greatly all the exposition. B. Exam le I. Commutation Formula for the Schrbdinger Operator and Rapid y Oscillating Exponential

In the problem of quasi-classical approximation to the solution of quantum-mechanical equations the main role belongs to the characteristic equation, which was derived in Section 1:A for the particular case of the Klein-Gordon equation. Here we consider the Schrbdinger equation: 1

Hu = H(x,-ih

-)u(x,h) a 0

with the arbitrary Schrbdinger operator H.

(1)

We seek u(x,h) in the form

u(x,h) - e(i/h)S(x)O(x,h),

(2)

where ¢(x,h) may be expanded in a series with respect to powers of h, and the problem is to obtain the result of application of the operator H to the function (2). We do not give rigorous proofs here, but only demonstrate that the operator approach gives the result readily after simple calculations. We intend to obtain the operator P such that e(i/h)S(x)pm(x,h);

(3)

in other words

p = e-(i/h)S(x)He(i/h)S(x). Proposition 1.

For an invertible operator U, 1

n

n

1

f(U-IAIU,...,U-IAnU).

U-lo f(A1,...,An)o U Sketch of the proof.

U-10f(A1,...,An)o x

(4)

We have (2n)-n/2

U

eiAltlUdtl...dtn

x

(5)

f

(tl,...,tn)U-leiAntn

,,, x

(2n)-n/2 ff(t1,...,tn)U_1eiAntnU

x

(6)

U-1e iAltlUdtl...dtn,

so it suffices to demonstrate that U-leiAtU = eiU-IAUt (7)

This is true, since U-leiAtU1

22

(U_1eiAtU)

t=O = I,

dt

= iU-1AeiAtU

= -iU-IAU. U-leiAtU,

(8)

thus the validity of Proposition 1 is demonstrated.

Since

e(-i/h)S(x)xe(i/h)S(x) = x (9)

e(-i/h)S(x)(-ih 3 )e(i/h)S(x)

_ -ih

x

+JS(x) 3x

2x

we obtain 1

2

a

(i/h)S(x)

H(x,-ih TX-) (e

y(x,ti)) = e

(i/h)S(x)

H(x,

3S 3x

1

- ih

8

-X

(x, h) ;

(10)

this formula is the generalization of the permutation formula (3) of item Thus the equation for Q (x,h) reads

A.

1

H(x,

3x

- ih -x)n(x>h) = 0;

in particular, in the principal term we obtain the characteristic equation (12)

0.

H(x,

It is also easy to obtain the subsequent terms of expansion of t

H(x,

as

- ih ax

Note that the formula of commuwith respect to powers of h (see below). tation of Hamiltonian and exponential (10), which is readily obtained in ordered operators notions, was derived in a very complicated way (see 1501, for Fourier integral operators see 1291) by means of the stationary phase method.

Next, we obtain the second term of asymptotic expansion of 1

H(x,

3X -

ih 3x)

with respect to powers of h. By the way, we introduce some simple rules for treating a function of non-commuting operators. Proposition 2.

The formula is valid 1

a

f(A+B) = f(A)+g 6X (Al,A+B),

(13)

where

7x

(x,Y) = f (x) _ fY (y)

(14)

Proof. We use the following rules of the ordered operator calculus (they are proved in Chapter 2): sl

a)

b)

sn

The operator f(A1,...,An) does not change, if we replace the numbers sl,...,sn by other ones so that the mutual position of on the real axis remains unaffected for any pair (j,k) such that [Ai,Ak7 # 0. ("shifting indices") m m m f(.... A,.. ,A,...) =

,A_.),

(15)

where

g( ,x,

..j = f(..

,x,..

(16)

23

...,rsiif [a,

If ai,...,sm E [a,bI, segment, then

c)

b], where Ca,b] C L4 is some given

s1 rl rs am f(Al,...,Am)g(AID+1,...,Am+s)

A

ri

rs

Cg(Am+1,...,Am+s

(17)

where a E [a,b] is arbitrary, am

61

C = f(Al,...,Am)

(ls)

All these rules are evident (the reader can easily verify them for polynomial symbols). We also use a convenient notation. If we need to use some function am f(Al,...,Am) as a single operator in some more complicated operator expression, we put it into the so-called "autonoit us brackets" 152]: 81

Q f(A1,...,An)D ; then the numbers si,...,sm are "valid" only inside the brackets and do not "interact" with numbers outside the brackets; the number, under which the whole operator acts, is written over the left bracket. In this notation, the equality (17) may be written in the form rs r1 rs am rl al am a 51 f(Al,....Am)g(Am+1,.. ,Am+s) - Q f(Al,...,Am)])g(Am+1... .Am+s).

Now we prove (13).

(19)

We have 3

1

f(A+ B) - f(A) + f (A + B) - f (A) - f(A)+f(A+B)-f(A)

/b/

3

6f

1

f(A)+(A+B-A)

4 3 /a/ 6t p - f(A)+(A+B-A) bx (A,A+B) -

3

1

1

dx (A,A+B)

/c/

04

3 4 2 3 6f (A,A+B) = f(A) +QA+B-All 6x - f(A)+B2 ofx (A,A+B) 1

2 6f

/a/

f (A) + B

3

1

(A,A+ B),

bx

(20)

Q.E.D.

Obviously the proposition remains valid if f depends on additional operator arguments. Now we have 1

2

H(x,

as

2

a

1

2

aS

a

3

3x - ih x) - H(x, ax) + (-ih ax) L

a)

= H(x,

4

6H

4

aS _

aS

(x; ax ' ax

6p

>.haax) .

4

- ih

2

p (x, ax

8x

3

1

(21)

'ax) S

-h2 a a 62H 6 as as as a ih ax)' ax az 6p2 (x' ax' ax 'ax

Thus the second term of asymptotic expansion of 1

2

H(x,

ax

- ih ax) 2

1

4

with respect to powers of h has the form: - ih

24

ax 6 H P (x'

3

aisas x'ax)'

2

1

Using the rules a) - c), we may commute 3X and 3S 3S 611 = ax 5p (x; 3x .z1 3

ih

ih

3H

3s

3x

a

ap (x, 3x) 3x

,

obtaining

2s) a2s ih 32H apt (x' ax 3x2 2

(we leave the calculation as an exercise to the reader). obtained the commutation formula

e(i/h)S(x)(H(r

3S)4,

ax

3H-(x,S)

(x, h) - ihE

ap

ax

3x

(23)

32

3X5S

3pA + 2

Thus we have

X)e(j/h)S(x)s(x,h)

H(x,-i =

(22)

.O(x,h)] + O(h2)}, (x'

)

3x2

from which the characteristic equation and equation form o(x) = 2(x,0) easily follow. Example 2. Characteristic Equation for an Equation with Growing Coefficients C.

In the previous item we used the operator method to derive the characteristic equation in the familiar case, namely, for quasi-classical asymptotics of the Schr6dinger equation. Here we show that in an analogous way the characteristic equation for the non-standard asymptotics (the mixed asymptotics with respect to smoothness and growth at infinity) may be obtained. We consider the equation (cf. Section 1:P): a2u

a2u

ax2+ (1 +b(x))a2u = 0, ult=0 s 0, utjt'0 - uo(x),

(1)

at2

where b(x) E Co(al).

Intrcduce the operators

Al =- iX A2=x, B=x.

(2)

We seek the asymptotic solution of (1) in the form 1

2

2

u(x,t) = G(t,A1,A2,B)uo (x),

(3)

where G(t,xl,x2,a) is a linear combination of functions of the form x

(t,x1,x2,a) = e

iA(x)S(wl,w2,a,t)

C1po(wl,w2,a,t) + (4)

+ A(x)-10 1(w1,w2,a,t) + ...3,

where A(x) ix1 + x2}1/2, wl = xl/A(x) w2 - x2/A(x). Thus, the symbol G(t,xl,x2'a) of the resolving operator for the Cauchy problem (1) is sought in the form of an asymptotic expansion with respect to powers of A(xl,x2) {xl + xJy}'". The fact that such expansion gives the mixed smoothnessgrowth-at-infinity asymptotics follows from the estimates proved in Chapter 2.

The equation takes the form (where c(x) = 1 + b(x)):

LT Dt2+Ai+c(B)A2711t f(t,A1,A2,B)I) = 0,

(5)

25

where = denotes the equality modulo operators with kernel sufficiently smooth and decaying at infinity. Proposition 1. 2

1

2

2

2

1

1T 12 +Aj+c(B)A2 ]1QT(t,A1,A2,B)11 = fl(t,A1,A2,B), at2

(6)

where z

fI(txl,x2,a) = Proof.

-i

2

ff + (xl- i

at2

a x2

aa)2f+c(a)x2f.

(7)

Using rules a) - b) from item B, we obtain 2

2

1

3

2

1

2

= A1F(A1,A2,B) _

AIQ F(AL,A2,B)]1

32

24

1 4 2 1 2 4) SF 3 = A1F(A1,A2,B) + AI(B - B) 7C, (Al;A2;B,B) _

3

33

4

2

1

2

4

3

2

1

4

IF

(A1;A2;B,B) _

a

(8)

2

2

1

24

2

1

6F

- AIF(Al,A2,B) + Q AI(B - B)])

- A1F(A1,A2,B) - i as (A1,A2,B)

IF (AI,A2,B) (12 2

1 z 2 1 = AIF(A1,A2,B) ->_

as

IF

-i

2

1

a x2

2

(A1,A2,B).

Thus the multiplication by AI from the left is equivalent to action by the

operator (xl- i a - i

Since A2 and B commute, we have

on on the symbol.

ax2

1

2

2

2

2

2

2

2

1

A21I F(A1,A2,B)]1 = A2F(A1,A2,B); (9) 1

2

c(B)Q F(A1,A2,B)11

Therefore, we come to (7). We have by (4).

[

a2

are

xx

2

2

Calculate now the function (7) if f is given

2_) 1+

+ (xI

1

c(B)F(A1,A2,B).

aa)

2 ca)x27e

iA(x)S(wl,w2,a,t) x

X [¢o(w1,w2,a,t) +...] - e1A(x)S(wl,w2,a,t) x (10)

x [-(A .

-i

at)2 + (xl +

3

ax2

(AS) + A as - i a - i L)2 + ax2

+ c(a)x2]4o(wl,w2,a,t) +...7. We extract from (10) the leading term (with respect to powers of A), which It has the form determines the characteristic equation.

-A2 (as)2 + (xi + A aa)2 + c(a)x2 = A2[-(a t)2 + (wI

IS +

+ c (a)w27.

(11)

Thus after the change S. m - aw the characteristic equation takes the form (recall that c(a) - 1 + b(a}):

(at )2

- (as)2 _w2(l+b(a)) = 0, w2 E [0,1].

(12)

The detailed consideration of mixed asymptotics for equations with growing coefficients is given in Chapter 4, Section 1.

26

4.

THE OUTLINE OF THE BOOK. PRELIMINARY KNOWLEDGE NECESSARY THE SECTION DEPENDENCE SCHEME TO READ THE BOOK.

The main part of the book consists of three chapters: Chapter 2 through to and including Chapter 4. Chapter 2, entitled "Functional calculus" is devoted to thorough development of functional-analytic baseground for ordered operator calculus and construction of asymptotic solutions. No special preliminary knowledge on these subjects is required; the reader should only be acquainted with basic notions of functional analysis such as Banach and Hilbert spaces, The exposition is detailed and as elementary as linear operators, etc. possible. We begin with the theory of semigroups and functions of a single operator. These questions are considered in Section 2, including functions of self-adjoint operators and functions of operators depending on parameters. Various symbol Section 3 is concerned with functions of several operators. spaces are introduced and investigated, the functions of several operators are considered in Banach spaces and in Banach scales as well (the later notion is introduced and discussed in detail). The main results are in Section 3:D where the mapping n u

:

f(xl,...,xn)

f(A1,...,An)

(1)

is introduced and the main rules of operator calculus are proved then for operators in Banach scales. In Section 4 regular presentations of tuples of non-commuting operators are introduced and investigated. By definition, the (left) regular representation of the system of operators A1,...,An is the system of operators L1,...,Ln, acting on functions in such a way that

n

n

(Ljf)(Ai,...,An) = Aj o f(A1,...,An).

(2)

The regular representation enables the composition law in the set of func1

n

tions of A1,...,An to be established and thus to reduce general asymptotic problems to pseudo-differential ones. The functional-calculus conditions of existence of regular representations are studied, the case of Lie algebras is considered as an example; some boundedness theorems are proved which result from the existence of regular representations.. In Chapter 3 the analytic apparatus necessary for construction of almost inverse operators is treated. Two cases are considered: the case of small parameter asymptotics (which stands somewhat separately) and the correspondent technique in the homogeneous case when the action of the group ht+ of positive number is given.

Only elementary knowledge of mathematical analysis is necessary to understand the material of this Chapter. All the notions used which are out of this framework are introduced and explained in the text. The analytic apparatus developed includes the canonical sheaves on symplectic manifolds (both with Ul+-action and without it), the calculus of pseudo-differential operators, and the canonical operator in general symplectic manifolds. In Chapter 4 the results of previous chapters are applied to the general problem of construction of almost inverse operators. The operator approach to this problem is illustrated in Section 1 by an example of constructing asymptotic solution to equations with growing coefficients. In Section 2 the notion of Puisson algebra is introduced and discussed (for

27

the case of the small parameter theory).

In Section 3 the same notion is

used.in the JR+-homogeneous case and the central object of the theory, namely, the u-structure on the Poisson algebra, is defined in its local version. In Section 4 the global version of the introduced notions and the symplectic manifold of the Poisson algebra are defined and representations of the canonical sheaf on this latter manifold are used to prove the main theorem, which establishes the relationships between the geometric properties of the n

1

symbol f(xl,...,xn) of the operator f(A1,...,An) and the existence of the The proof of this quasi-inverse operator (and its explicit expression). theorem ends the book. The exposition is self-closed; the only exception is the theorem on solution of pseudo-differential equation (Chapter 3:4), the proof of which is based on the complex germ techniques 1523 not considered in the book. But we present the material in such a way that if one merely takes this theorem as given, then none of the other statements will require the information not contained in the book. It should be noted that geometric and analytic concepts of Chapters 3 and 4 (symplectic manifold of the Poisson algebra. canonical sheaves and the canonical operator on general symplectic manifolds, etc.) were not developed on the blank space. We try here to review the history of the question briefly. The sheaves on symplectic manifolds analogous to those used by us are well known in literature. They primarily occur in geometric quantization and in the method of orbits in the theory of representations of Lie groups (see [41,43,73,4,741, etc.). In all of these papers the existence of polarization was required or, more generally, the condition that the Chern class e E E H2(M,ZL ) be even was required, i.e., instead of the quantization condition

Za Cw] - 2 E H` (M,7L ) ,

(3)

where Cw7 is the class of the symplectic form, it was separately required that

2rh

[w] E H2(M,7L );

c

- 0 (mod 2).

(4)

The condition (3) was first written for odd c on Koehler manifolds with the complex polarization in 110]. In 127] the same procedure was held for arbitrary real manifolds (with complex polarization). It was discovered that in general situations the second Stiefel-Whitney class W2 plays the role of a in (3). The analogue of condition (3) was obtained independently in 136] for the procedure of construction of pseudo-differential operators calculus mod 0(h) on general symplectic manifolds. The condition reads 2-rh

[w] - 4 E HZ (M, 71 4) ,

(5)

where the class v E H2(M,7L 4) is calculated via indices of paths on Lagrangian manifolds 150,3]. No polarization was required. The class in (5) is in fact even [55]; however, it may be odd for manifolds with conic singularities 137].

The canonical operator on general symplectic manifolds was constructed primarily in [381 (where also the pseudo-differential operators calculus was constructed modulo 0(h2)). The main novelty is that the classes [pdq] and one-dimensional index are not defined separately. The application of this developed apparatus to asymptotic problems begins from the papers 135] and 133].

28

The notion of the sympiectic manifold of the Poisson algebra together with the notion of theasymptotic group algebra was introduced in the paper (36). Also the formulae for symbols of regular representation operators was announced in this paper (the complete proof is contained in 1341).

Thus all the main topologic and geometric notions used in Chapters 3 and 4 were recently developed in the cited papers for the case of small parameter asymptotics. However, the algebraic structure of considered objects being unchanged, the technique of proofs and especially of estimating the remainder terms is completely different (and much more complicated) for ti., case of homogeneous manifolds and asymptotics with respect to general type of operators. These estimates are the main results of Chapters 2 and 3, since they lead to proc:f of the quasi-invertibility theorem. Interesting independent results in the homogeneous case may be found in the papers 18,233. Reproduced here is a scheme of dependence of chapters and sections (dotted lines mean "weak" dependence).

Chapter !

Chapter

Chapter

Chapter

Chapter 4:2

LHJ _[ Scheme 1.

Chapter 4:3 I

Ch3p3er_

Chapter 3:4

Chapter 4:4

Dependence of the sections.

29

II

Functional calculus

1.

INTRODUCTION

In this chapter we develop the apparatus which is used in this book for obtaining asymptotics in functional spaces. This apparatus is the he functional calculus for several non-commuting operators acting in a Banach space, or, more generally, in a Banach scale. The aim of the introduction is to give in short the exposition of the main results presented in the chapter and to clarify the main ideas. The bibliographical notes are gathered at the end of the Introduction. The reader, who is not interested in precise formulations and detailed proofs, may restrict himself to reading only the Introduction of this chapter and then go to the subsequent chapters. We assume that the reader has become acquainted with the basic notions of functional analysis such as normed spaces, closed operators, resolvents, spectral expansion of a self-adjoint operator, etc. No information is assumed to be known about semigroups in. functional spaces. We assume that the foundations of theory of Lie groups are known to the reader. Sometimes we use the results of distribution theory. A.

Functions of Several Operators.

Feynman and Weyl Orderings

Let A1,...,An be linear operators in some functional space H. also p(yl,...,yn) be a polynomial

on

al

P(Y1,...,yn) ` Iaj E6 M a yl

Let

... yn .

Generally speaking, the result of substitution of A1,...,An instead of Y10...syn into the polynomial p is not uniquely defined if the operators Al,.... A. commute. The ordering of operators-is the method by which to avoid such indeterminacy. (a) Let al,... sn be a set of real numbers, such thatai f aj if Ai does not commute with These numbers may be rearranged in the non-decreasing order: 63... 6 njn. We set °il < *j2

p(A1....,A,)

aj x a An 1a1 GM a in

30

.:.

A

ii

'.

(1)

Since [Ai,Ail - AiAj - AjAi = 0 for "i = 'r j, the definition does not depend The ordering (1) was first introduced on the choice of the rearrangement. by K. Feynman. The numbers Ill ...... n indicate the order of action of the operators A1,...,An; an operator with the lower index acts first. The mapping nn IT

;:A

... An

A1.

defined by (1)

p '' p(A1,...,An)

:

n

1

is called the (Feynman) ordered quantization corresponding

Ill

an

to the ordered set of operators A = (A1,...,An), and the polynomial p is Feynman ordering possesses simple called the symbol of the operator (1). properties: (i) the quantization depends only Tillon the real axis; in particular,

nn then p(A1,...,An)

1

(ii) If p(vr. (YJS+r'

,

,

n q(AJ1,...,Ajn).

I

..

the order of numbers nl, if p(y;, .. Yn) = 4(Yji, , Yjn),

on

yjr.),

El

..

v0) = p1(Yj;,

.

VJ:+

..

Yjn)p!'

then

s 1

r+l

r+2 n Jr+11A J r + 2 " -. A in) x

s

°n s+l

11,

x

jr+2.

(2)

r

4' p2(Ajs+l,...,Ajr)l'

where ^ E ks + 1, rJ is arbitrary.

We used a convenient notation in (2): The expression contained in the "autonomous brackets" Q D is considered as a single operator, and the place on which it acts is indicated by the number over the left bracket. Thus, there is no correspondence between the numbers over operators inside and outside the brackets, and the notation " s+l

It

is equivalent to c, where

r

s+l

p2(AJs+l,...,Ajr)I1

r

c = p2(Ajs+l' 'AJr)'

(iii) If Ajk = AJk+:' then k k+2 n "1 'n p(A1,...,A.) - q(Ajl,...,Ajk,Ajk+2,...,AJn), 1

(3)

where

q(Yjl,...,yjk,YJk«2....,Yjn) = P(YI....,yn)lyjk+l=yjk. The properties (i) - (iii) yield that the mapping p p(A) is the homomorphism of the algebra of polynomials of one variable into the algebra of operators in H and that if p(yl ... yn) = P1(y1) " " Pn(On), then 1

n

p.(A1,...,An) = Pn(An)^Pn-I(An-1) 0, p > 0. estimates of the type (11).) If the operators Aj satisfy the desct'ibed I

n

property, formula (8) defines f(A1,...,An) for any f whose Fourier transform t is a continuous functional on the corespondent space of functions slowly growing at Infinity (details in Sections 3 and 4). In particular, the l

expression f(A1,...,An) makes sense for f E S`°(tRn) (the latter inclusion means that for some m E 92 and any multi-index a the function (1 + lyl)-m' aaf/ayo is bounded in IV. The properties (i) - (iii) of the Feynman ordering remain valid in the case considered (however, the precise formulations become more complicated). Under similar assumptions the formula (9) defines correctly the Weyl-ordered operator fw(A1,...,An). The novelty is the requirement that U(t) be strongly continuous in t GIRT, (this property

holds automatically for U0(t)

... , U 1 (t), i.e., that the group exp(twA)

converge to the group exp(re'A) as w a'. The conditions under which this convergence takes place are investigated in Section 2:G.

The important particular case of the general constructions is the calculus of pseudo-differential operators which may be regarded as functions of operators x of multiplication by independent variables in 1Rn and also operators of differentiation: -i(a/)x.). As will be shown below, most of the problems concerned with solution of equations, including functions of several operators, may be reduced to pseudo-differential equations for symbols (or systems of such equations). Thus we give in Section 4:E the construction and the analysis of pseudo-differential operators almost independent of the other material of the chapter. To finish with, we emphasize that the problem of estimates is the crucial one in functional calculus, and purely algebraic consideration is not sufficient for constructing the asymptotics.

* The set A is called bidirected (or directed in both sides) if it is partially ordered and any pair (61,62) of its elements has a majorant

and a minorant.

33

C.

Regular Representations n

1

Fix the ordered tuple A = (A1,...,An) of generators in a functional

n

1

n

1

g(A1,..,An)Il of two In general, the product 11f(A1...... Feynman-quantized operators cannot be written in the form space H.

1 n n Q f(A1,...,An)iI Q g(A1,...,An)]

n

1

1

- k(A1,...An)

(12)

But if the tuple A satisfies some additional ). conditions, the representation (12) exists and for some symbol k(y1,...,y,

k(Yl,...,yn) = f(Yl,...,yn)* g(y1,...,Yn),

(13)

If (13) holds, we

where * is a bilinear operation in the symbol space. may reduce the equation 1 n f(A1,...,An)u - v E H

(14)

Namely, we seek u in the form

to the equation in the space of symbols. n

1

u = g(A1,...,An)v,

and we obtain the equation for g: f(y11....Yn) * g(Y1,...,Yn) = 1.

(15)

If we solve the equation (15) not exactly, but modulo some "good" class of symbols, we obtain the asymptotics of the initial equation (for example, in classical PDE theory "good" symbols decay at infinity and correspond to smoothing operators). The structure of operation f* g turns out to be rather simple.

Set

(Ljg)(Y1,...,yn) = yj * g, (16)

) - 1,...,n.

(Rjg)(Y1,...,Yn) = g* yj,

Then n

1

(f * g)(Y1.....Yn) ' Cf(L1,....Ln)gl(yl,...,yn) (17)

n

1

= Cg(Rl.... ,Rn)f](yl....,yn).

(Note that (17) obviously follows from (16) if we consider only polynomial symbols.)

The n-tuple L = (L1,...,Ln) is called the left regular representation of the tuple A - (A1,...,An), and the n-tuple R - (R1,...,Rp) is called the right regular representation of A. We give here a simp a example of Let Al - -i(3/2x), A2 - x be opercalculation of regular representation. ators acting in the space of functions of one variable x. Using the properties (i) - (iii), we may calculate the left regular representation for 1

2

the tuple (A1,A2) as follows. 1

2

Take an arbitrary symbol g; we have 3

1

2

2

1

2

A2Q g(A1,A2)I1 = A2g(A1.A2) - A2g(A1,A2), so L2 = y2 (the multiplication operator).

34

Next,

2

1

3

2

1

3

4

1

Alt[ g(Af A,)]l = Alg(A1,A2) - Alg(A1,A2) + 4

2

-4

2

1

2

4

1

4

-g(A1,A2))/(A2-A2)]

+

(note that the symbol [g(yl,y2) -g((yl,y2)/y2 - y'2] has no singularities). We may write then

Al A2A2)[(g(Al,A2)-g(A1,A2))/(A2-A2)] +

A1QgiA1,A2i]] 3

4

1

1

+ A1g(Al,A2) 2

1

2

1

3

4

2

3

AIg(A1,A2)+[TA1(A2-A2)]1X 4

1

4

2

1

1

2

- I(g(A1,A2)-g(A1,A2))/(A2-A2)]=AIg(A1,A2) 1

2

2

4

2

4

1

2

1

1

2

-i[(g(A1,A2) - g(A1,A2)/A2- A2)]= AIg(A1,A2) - i(ag/3y2)(A1,A2).

Thus L1 - yi - i0/3y2).

Similarly R1 - yl, R2 - Y2 - i(a/ayl).

The existence of regular representation in this example-is connected with the fact that there is a commutation relation [Al,A2]

iI

(I denotes the identity operator). The general situation is quite similar. We require that (a) (b)

A1....,A1 satisfy the "rich enough" system of commutation relations; some agreement conditions of the functional-analytic nature are then s w.

Then we obtain

A(ifine-iatU(t)xdt) = A.if'.-iatU(t)xdt- x

0

0

for an arbitrary x E X.

The operator x a

is bounded for

-ImA > w (its norm does not exceed therefore Mfoe(w +

Hence for

x E DA ifine-1ltU(t)Axdt - fim 1-t fine-i't[U(T + t) - U(t)]xdt = 0 o Ta0 ifine-iatU(t)Axdt

-x,

0

just as above. Thus (4) is proved. RA(A) is bounded and defined everywhere, therefore closed. So A - A is closed as well and the proof of (a) is complete. To prove (5) we observe that n A E p(A), x E X dan 1RA(A)x - (-1)nn(RA(A))n+ix, (9) (RX(A))-1

(indeed, the resolvent of a closed operator A is a holomorphic function on It may be represented by the Neumann series

p(A).

RA(A) -

E

(A))n+l,

(Ao -A)n(R,

Xo E p(A),

(10)

n-0

convergent in the circle !A-A01 < HRx (A)II -1; see [40]). 0

38

Thus

(_ml)ml)- 1

(Rl(A))mx !

im

dm-

dam

1

foe-lI[U(t)zdt

-1

=

fine-'X ttm-lU(t)xdt.

(m-1

o

(The differentiation of the integrand is correct, since convergence of the resulting integral is locally uniform with respect to A in the domain under consideration.) Since

r

fe(w+Iml)ttm-1dt =

(m-1)! (-Ima - w)m

0

, w+lma < 0,

we obtain (5), and (e) is proved. Now we move on to the proof of uniqueness Let x - 0 and x(t) satisfy (3). We assert that x(t) vanishes in (d). identically. o prove it, consider an auxiliary X-valued function

y(t,T) = U(t-T)X(r) defined for 0 < T G t. We have y(t, 0) = 0 and y(t, t) = x(t). It turns out that dy/dz exists and vanishes identically. This immediately yields x(t) __ y(t,t) - y(t,0) - 0. Let T E CO,t] be fixed, E E C-T,t- TI (so that T + E G 10,t]). By (3) we have x(T) E DA and ,

x(T + E) - x(T) + iEAX(T) + er(E), where Nr(E)l + 0 as c - 0.

(12)

Thus we obtain

Y(t,T + E)- y(t,T) = 1U(t- T- E) - U(t - T)]X(t) + + U(t - i - E)I X(T + E) - X(T)] _ -icAU(t - T)X(T) + + Erl(E) + U(t - T - e)CiEAX(T) + er(e)], where 11rl(E)I - 0 as c -+ 0 (the expansion of the first term analogous to (12) is valid since x(T) E DA, U(t - T - E)X(T) being therefore differen-

tiable in e, as proved above). rewrite the formula obtained:

Since (c) has already been proved, we may

y(t,T+E) - y(t, T) = EIi(U(t -T - E) - U(t - T))Ax(T) +

+ rl(E) +U(t - T-c)r(E)] = ER(E). Since U(t) is strongly continuous and bounded locally uniformly in t, then IR(E)N + 0 as c i 0. Hence y(t,T) is differentiable in T and dy/dT = 0,

as .we desired.

If A is a bounded operator, the aemigroup U(t) has the form U(t) - exp(itA), where exp(B) is defined as the sum of the convergent series: exp(B) = eB =

t

1. Bk.

(13)

k-0 k

Hence we. obtain, using the inequality Itk - TkI G It -

exp(itA)-exp(iTA)I
0. Since U(t) is uniformly continuous, there exists 0 such thatlkU(t) - II < e/2, when 0 5 t < d. Since dt = 1,

-iuR_iu(A)x- x = f ue "t(U(t) - I)xdt = 0

d

e-ut(U(t) - 1)xdt+ f°° tie-ut(U(t) - 1)xdt. lie-lit

The estimate of the first integral is:

ll Jove -lit (U(t)-I)xdt
tns F to,W), n = 1,2,..., s = 1, ,Sn such that Sn

x = f'm ( I cns(A + iu)U(tns)x) _ :£im (A + "i)xn, n-- s=l n-

(23)

Sn

where xn -

I

cnsU(tns)x l D by our assumptions.

Thus (A + ip)D = D is

s=1

dense.

R_iu(A) being bounded, D is a core of R_,u(A). result is an immediate consequence of the following:

Now the desired

Lemma 1. If B is a closed operator in the Banach space X and B 1 exists, D C X is a core of B iff BD is a core of B-1. Indeed, the closure of the operator is equivalent to the closure of its graph in X x X, and the graph of B is transformed into the graph of B-1 after the interchange of the factors. We obtain that D is a core of A + iu, hence of A. Thus the theorem is proved. In the sequel we sometimes make use of the following evident result:

Lemma 2. If B is a closed operator in the Banach space X and a dens. linear subset D C DB is invariant under RX(B) for some X'E p(B), then D is a core of B. The proof is obvious.

Next we formulate a rather useful criterion, which enables us to con elude that the closure of a given operator is the generator of a strongly continuous semigroup. Theorem 4.

Let D be a dense linear subset of the Banach space X, Suppose that for each xo r= D there exists a unique function x CO,=) + D, t a x(t) such that* A : D + D be a given linear operator. :

x(0) = xo' i

2-x (t)

+Ax(t)

0, t > 0.

(24)

* Derivation in (24) is in the sense of topology of X.

43

(t) 11 < C(t)I;::0I!.

(25)

where C(t) does not depend on xo and remains bounded when t belongs to a compact set. Then A is closurable, and its closure is the generator of a strongly continuous semigroup U(t), and besides the solution of (24) has the form x(t) - U(t)xo. Proof. Let U(t) be the unique bounded operator, coinciding with the operator xo -, x(t) on D. By (25) I!U(t)jl < C(t), and since U(t) is strongly

continuous on D and bounded locally uniformly, U(t) is strongly continuous on X. Since the solution of (24) is unique, we easily obtain that U(t)U(t)= = U(t + t). Thus U(t) is a strongly continuous semigroup and A is the

restriction of its generator on the subset D. By Theorem 3, the closure of A is the generator itself, and the proof is complete.

Our last theorem is a collection of almost obvious properties of the powers of a generator. Theorem 5. Let A be a generator of the semigroup U(t) in the Banach spare X. The following statements are valid: U(t)x is f-times differentiable iff x e DAf. In this case the f-th (a) derivative is continuous and may be given by the formula d

1

f (U(t)x) = i ' U(t)A R x, x E DAf.

(26)

f

Let p(y)

(b)

£

piyR, pf

0, be a polynomial of degree E.

Then the

j= 0

operator

p(A) = ;EO pjA1> DP(A)

(27)

DAf

is closed.

Let D C DAs be a subset of X, satisfying the conditions of Theorem 3.

(c)

Then D is a core of p(A) for each polynomial p of degree < s. Proof. (a) For k = 1 this statement follows from Theorem 1, (d) and the definition of the generator. The induction on k gives the general result.

(b) This is a general property of closed operators with the non-empty resolvent set. Considering A + X instead of A, if necessary, we may assume that 0 E a(A), i.e., A-1 exists and is bounded. We proceed by induction on f (fog f I the proposition is valid, see Theorem 1, (a)). Let xn1= E DAf, n x, p(A)xn 1,2,..., xn z. We show that x E DAf. Indeed, nn+ set P1(y)

jzpjyi = P(y) - P 0 . q(y) = p(y)/y = j=1 E 1 1

Then pl(A)xn n+ z - pox

zl and A-lpl(A)xn - q(A)xn

p.yl-(28) A Izl.

Next

q(A) is closed by the induction assumption, so x E Dq(A) = DAf-1 and q(A)x= A _1z1 E DA.

It follows that x E DAf and p(A)x = Aq(A)x + pox = zi +

+pox= Z. (c) Let Ima < -w, where u is the type of the semigroup U(t). Since every polynomial p(y) of degree < s may be represented in the form p(y) _ s

£j=0Cj(y -x)1, it suffices to prove that D is a core of (A - A)1 for

44

Just as in the proof of Theorem 3, we establish that (A - 1)3D is j 5 S. Namely, we have instead of (22) dense in X. lj

(A-A)J(Rx(A))Jx ° (J for x E D. B.

):

Jme-iattj-1U(t)(A-A)Jxdt

ff

(29)

0

Further, the proof is identical to that of Theorem 3.

Examples of Semigroups

The abstract notion of semigroups, introduced above, is supported with some examples in this item. In particular, we give the proof of Stone's theorem, having in mind that functions of self-adjoint operators are of special interest to us . The theorem mentioned asserts that the notion of a self-adjoint operator is identical with that of unitary strongly continuous group generator. We begin with the following: Let X = L2(btl) be a space of measurable square-summable Example I. Define the complex-valued functions on the real line with the usual norm. group U(t) in X by

U(t)f(x) = f(x+t), t EtR, f GL2(IRl).

(1)

The operators U(t) are unitary in X and clearly satisfy the semigroup U(t) is strongly continuous in t on C°°(ltl) and therefore on X, property. due to the uniform boundedness. The generator o? the semigroup (1) is the operator A = -i(8/3x) with the domain DA, consisting of all absolutely continuous functions f e L2(R1) such that f/2x E L2(tI). The operator S so defined is self-adjoint. Indeed, C'0(al) is invariant under the action of the semigroup (1), and (

i dt 1U(t)f(x)1)1t=0 = -i

3ff(x)

,

f GCo(Itl),

(2)

so that the restriction of A on Co(htl) is -i(a/ax). It follows from Theorem 3 of the preceding item, that A is the closure in LZ((tl) of -i(2/Bx), defined on Co(al). As for the proof of the given description of the domain DA of this closure, we refer the reader to standard textbooks on functional analysis. The fact that A is self-adjoint is a consequence of Stone's theorem to be proved later in this item. Example 2.

(Dissipative operators)

We suppose that X is a Hilbert

space.

Definition 1. A contraction semigroup is a strongly continuous semigroup satisfying the condition IIll(tA < 1 for all t > 0. A dissipative operator in X is a closed densely defined operator A such that Re(Ax,x) < 0 for arbitrary xfimmOG DA. Theorem 1. A is a generator of the Contraction semigroup iff iA is a dissipative operator and R(I-iA) - X (here RB denotes the range of B, i.e.,

the set of all y G X such that y - Bx for some x E DB). Proof. Let A be a generator of the contraction semigroup U(t). The theorem yields that A is closed and the point A _ -i belongs to p(A), thus R = X. If x E DA, we have (I-iA)

Re(iAx,x) =

t Re(U(t)x - x,x) a fim t (Re(U(t)x,x) - (x,x)} < +0

< 0, since Re(U(t)x,x) < I(U(t)x,x)1 < 1!xJ12 - (x,x).

45

- X. Conversely, let iA be a dissipative operator with R (I-iA) > 0, we have

If ReX >

X12 4 Re1XII x12 - (iAx,x) 1 E 1 Xx - iAx!I. hxI, xE DA or II x

so that if for some X 0o RX0(iA)I

1/ReX0.

(Al - iA)x1,

ReX

, x E DA,

(3)

X, then Xo E o(iA) and 1X01-W this estimate for the norm yields (taking

ReX 0 > 0, R

Moreover,

(10) of item A into consideration) that p(A) contains the open disk with We start with Xo = 1 and consequently the radius Re\o and the center Xo. apply the above arguments to points lying in disks constructed before. One can easily see that we are able to construct the set of such disks covering the right half-plane of the complex variable X. Thus the right for ReX > 0. half-plane is contained in p(iA) and lRX(iA)N S Applying Theorem 1 of item A, we obtain the desired result. The theorem is proved. The analogous theorem for Banach spaces may be found in more special literature (see bibliographic remarks in Section 1). (ReX)-1

Example 3. Hilbert space.

(Self-adjoint operators)

Theorem 2. operators in X. Conversely, each continuous group

(Stone) Let U(t) be a strongly continuous group of unitary Then its generator A is a self-adjoint operator in X. self-adjoint operator in X is a generator of a strongly of unitary operators in X.

Proof. we have

We assume again that X is a

We prove at first that A is symmetric.

Let x,"c 0A.

Then

(U(t)x,y) = (x,U*(t)y) _ (x,U(-t)y), hence (Ax,y)

dt U(t)x,y)lt=0 = -i dt (U(t)x,Y)It=O =

-i dt (x,('(-t)Y)It=0 = (x,i dt U(-t)y)It=0 - (X,Ay), as it was claimed. Further, Theorem 1 of item A implies that A is closed and all non-real X belong to p(A). Thus A has defect indices (0,0) and is therefore self-adjoint 1401. Conversely, if A is self-adjoint, all nonreal X belong to p(A) and

I((A-X)x,x)I2 = I(A-ReX)x,x)I2+ (ImX)2;IxII' > (ImX)2IIxII4, x E DA.

It follows that II Ax- XxII > ImX IIx II or II RX (A) II
- no(a), i.e..Ab contains the disk lA - aoj < aol)-1

M(ao)-1,

(b) The inclusion As C Ab is an immediate consequence of the resonance theorem 179]. To prove that As 3s open we note that if Ao E As, then

* Bn s B denotes the usual strong convergence. 48

there exists a - Rim (RAo(An))k =

(RAo)k, k - 1,2,...

The Neumann series

.

for RA(A0) is dominated by the convergent series Ek=OM (A )k+1IA. - Aok, when

Thus s - Rim RA(An) exists for these A.

Ix - Ao1 < M(ao)-1.

n-

Note that we

have also proved (c). To prove that As is closed, consider A E Ab such It Am, Am E As. Choose m such that 1Am - AI < 1/2M(A). that A = R

follows from above that one may assume M(Am) < M(A)(1 - M(A)IA - Aml)-1 < < 2M(X). Therefore Ix - AmI < 1/M(Am) and the above considerations imply (d) is valid, since RA is a strong limit that A C-,A,. Thus (b) is proved. of operator families, satisfying (6). First we note that (6) implies commutability of

Proof of Lemma 2. the family R1. Further

RAu = (I+ (A -p)RA)R.u, so that Ker Ru C Ker RA, and

Ruu - RA(I + (u - A)RV)u, so that ImR,, C ImRA. if Interchanging X and u, we obtain that Ker RA = Ker R,, and ImRA - ImR.,. Conversely RI - (A - A)-1, then clearly Ker RA - (0}, since (A - A)R1 - I.

let N = M. Define the operator Al with the domain R by the formula (7). Then clearly AA is closed (since RA is), (AI - AA)RA = 1, RA(AI - AA) - IIR (the restriction on R of identity operator). It remains to prove that AA We have does not depend on A. Let x E R, then x = RAv for some v E X. AAx - AARAv - (XI- (RA)-1)RAv - ARAv -v;

AUX - ApRAv = (iI- (RU)-1)RAv - yRAv- (1\)-'%v - (RU)-1(A-u)RAR,v - uRAv - v - (A - lu)RAv - ARAv - v - AAx. Only the last assertion of the lemma remains unproved. If A,u C-Ab n p(A), we have for n large enough (such that resolvents are defined)

%(An)-RU(A) - 11+(p-A)R,(An)]

(8)

(this equation is an easy consequence of resolvent identities). If we choose A E A. (which is non-empty by assumption), we obtain that (An) + Rp(A) (since R1(An) I RA(A), and RU(An) is uniformly bounded). hus

s

A. - Ab n p (A) . We return to the proof of the theorem. Since A 6 G(M,w), estimates (5) of Theorem 1, item A imply that Ab contains the lower half-plane. (4) means that A E La. Applying Lemma 1, we see that A. also contains the lower half-plane. Now we can apply Lemma 2 and thus we obtain (3). It follows from (3) that RA(A) satisfies estimates (5) of item A. Hence A E G(M,O). We set U(t) - exp(iAt) and prove that (2) is valid. Establish first that

RA (An)CeltA- eltA11]RA (A) (9) iftei(t

- s)An(RA(A)

- R1(An))eisAds,

IaA < 0

0

(integral on the right in the sense of strong convergence).

Indeed,

49

CRX(An)el(t - s)AleisARX(A)]

d

ds

)eisARX(A) +

-iel(t - s)AnAnRX(

iel(t-s)AnRX(An)eisAARX(A) (10)

iei(t- s)AnC(I- XRX(An))RX(A)

=

iei(t

- RX(An)(I- XRX(A))]eisA =

s)An(R1(A)-RX(An))eisA

Integrating with respect to s from 0 to t yields (9). < 0, fixed. (9) implies

We now put X, ImA
0), we obtain (13).

Suppose that (13) holds for IImAI > R. Then (b) Sufficiency. 1R1(A)mp t M/(JImal - R)m, IImAI > R for some M. Indeed, as we have just seen, the right-hand side of (13) equals the right-hand side of (14), so

T Je-t I I m a cm- 1(1

C

+ t)kdt 6 (m- I. 0 C T f"-tIma -R)cm-Ile-tR(1+t)k]dt 5 m-1) o (IImaI-R)m I R (A)mll

(15)

M

where

M-

le-tR(1

sup

(.

tE-lo'o )

+ t)kl
0 (the case t < 0 is considered analogously). By Theorem 1 (f) of item A

Thus ',,'

exp(iA;:).

n

(-iuR_iu(A))Ct,t7

exp(iAt) = s - kim

= s - kim (-i t R

u

n(A) )n;

-it

thus

n n II exp(iAt) I E ni m ((t)

C

(

Joe

(T/On Tn

(1 +.

T)kdT)

n

C fim IIr -; Joe-TnTn- 1(I + rr)kdT1

nz (n-1) n

C kim l (n n -

54

0

£ r :k

,

tj J

e-TnTn + j - 1dT1

(16)

k

aCBimC E nj''0

tj(n+'-1).)]-C(l+t)k.

k!

1

r

(n-1):n}

since the expression in parentheses tends to unity as n i m. theorem is proved.

Thus the

The groups of polynomial growth being particular type of strongly continuous groups, all the results concerning the latter are valid for the former (one should only remember that dealing with groups it is necessary In theorems of item C to study resolvent behavior in both half-planes). we must substitute "An is a uniform sequence of generators of degree k" instead of "An 6 G(M,w)." The attentive reader has undoubtedly noticed that, after such substitution, it does not follow directly from these However, it is theorems that the limit group satisfies the estimate (12). the immediate consequence of the inequality Is - fim Un n+

Cim

1

Unj.

n+

Here we finish our excursion into the semigroup theory and move on to the calculus itself. Functions of a Generator in Banach Space

E.

We deLet X be a Banach space, A be a generator of degree k in X. of symbols f (y) -functions of a real variable y, for which an operator f(A) in X is well defined. After this we prove that the mapping f -+ f(A) has quite natural algebraic properties.

scribe here the class

8 being a Banach space, we denote by Ck(R,B), (f,k 3 0-integers) the space of all i-smooth (i.e., having A continuous derivatives) mappings $: R - B with the finite norm 1tI)-k

sup Ck If

tER

(1+

E

'a- - (t)ll

(1)

j-0 3t

k - 0, this space will be denoted by C(R,B), while if B - t, it will

f.

be denoted by*k). Clearly Ck(Bt,B) supplied with the norm (1) is a Banach apace.

Let Ck (it) denote the space of continuous linear forms on Cleat) with

the usual norm. The definition of the support of the functional T E Ck (R) is not obvious, since one might prove, using the Hahn-Banach theorem, that

Ck*(1t) contains functionals which vanish on all finite elements of Ck*(Rt). Nevertheless, we define the notion of a finite functional.

T E Ck (R) is

finite if it takes zero value on elements $ E Ck at), vanishing for ttl < R. R - R(T). Now we define Ck+(iR) as the minimal closed subspace of Ck* at), containing all finite functionals.

Then let 0 E Co(R), y,(t) = 1 for Iti t 1, * (t) - *(t/n). Lemma 1.

A functional T on Ck(R) belongs to Ck+(Y() iff the sequence

Tn - yinT converges to T as n a

in the norm of Ckd(R).

Proof.

Since Tn are finite functionals, one of the statements of the Ck+ lemma is evident. To prove the other, suppose that T 6 This means (it). there exists a sequence T(m) of finite functionals convergent to T. The sequence of operators of multiplication by (1 - *n(t)) is uniformly bounded in

35

Cf(R); denote its bound by. M. Given an c > 0, choosem such that IIT- T(m) II< r/M. Then for n large enough (1 - WT(m) = 0, hence ITn - TI - 11 '(1-Wn)TQ - H (1-Jn)(T-T(m))I
denote the pairing between the elements of 8* and 8.

56

(8)

b -

is a well defined element of B (see [711, Chapter 3), and clearly b satisThe theorem is proved. fies (2). Notation.

In the situation of the theorem we shall write

j T(t)4(t)dt.

b-

(9)

It is clear from the proof of the theorem that if 8 is a Remark. B), the statement is valid for arbireflective -Banach space (i.e., B**

trary T E Ck (1R) .

As we are going to define f(A) by the formula f(A) - (1/T)f f(t) X

X exp(itA)dt, we study next the inverse Fourier transform of the elements of Ck 00

Note that there is a continuous embedding SIR) C Ck(R), where S(ki) is the apace of smooth functions, rapidly decaying at infinity with all the derivatives. Thus each element T E Ck (1R) defines-an element of S'(I&), i.e., a so-called tempered distribution. Lemma 3. If T G Ck+(1t) and the corresponding tempered distribution equals zero, then T - 0. Proof.

that Tn = 0.

C- S(ft), m

By Lemma 1, T - niimm Tn, where Tn - *,T. Indeed,

let f E CkOR).

1,2,..., such that fm

We are going to prove

There exists a sequence fm E-

ynf in Ck(OR).

X (ynf)- Tn(f). So Tn(f) - 0, as we claimed.

Thus 0 - T(fm) AT X

Consequently, T - 0.

The

tempered distribution, corresponding to T E Ck+(R), will also be denoted by T. The Fourier transform of tempered distribution is a well-known construction (see, for example, (71]) and we give the following: Definition 1. The inverse Fourier transform of the element T E C1+(1R) is the inverse) Fourier transform of the corresponding tempered distribution. Lemma 4. Let T E Ck+(t). function given by*

Then its inverse Fourier transform is a

7-2

1-1(1)(Y)

T(eity)

(10)

(

Further, F-1(T) E CQ(IR) and

I F-i (T) !j( k Ir

MIT qCL*,

(11) ;

k

where the constant M does not depend on T. Proof. Let at first T be a finite functional. Then (10) is valid ((791.- eorem 4, Section 3 of Chapter 6). Moreover,

* On the right-hand side of (10),eity is the function of the variable t, depending on the parameter y.

Obviously elty E Ck(1R).

57

C IF-1(T)(y)I < C MTIt*(1+ lyl) f

(12)

k Ck+ (this inequality immediately follows from (10)). Now let T E Olt), Tn be the sequence of finite functional converging to T. Then IjTn1 is uniformly bounded, and F-1(Tn) + (1//2n)T(elty) pointwise. Pairing with arbitrary elements of S(R) and applying the Dominated Convergence theorem, we obtain (10). To prove (11), we need the following:

The mapping

Lemma 5.

f(t)

(1 +

(13)

at)f(i+t)-kf(z)

is an isomorphism of CfOR) onto C(R). This statement remains valid if we change some of the pluses in parentheses into minuses. Proof.

It is enough to show that the mappings

Clot) -Ck 1(!t),

f(t)+ (1 + at)f(t), k - 0,1,2,..., f = 1,2,...,

1 it Ck at) i Ck-1OR), f(t) + (i+ t)-1f(z)> k

are isomorphisms.

We begin with (14).

1,2,..., 1

(14)

0,1,2,...,

(15)

Direct computation gives

(1 + at)f(t) 1-1 < 2I f(t)B f. Ck

(16)

Ck

Further, consider the operator

[Ih](t) - 1

t

th(T)dT, hE CkQR).

eT

(17)

A-1(R) We claim that I maps Ck k(R) and is the two-sided inverse for onto CL (I + ac).

Indeed, one can easily verify that (1 + at)

identity operator. that

To prove that Ih E Ck(R), if h C1), we note

aatJr (Ih](t) - atj We obtain directly from (17)


x) it is necessary for f(y) to derivatives, satisfying for some N the estimates

aJ (y)I < aye

C(1 + I iI)N,

(26)

j = 0,1,...,k, and it is sufficient that it has k + l continuous derivatives satisfying (26), j = 0,1,...,k + 1. (b) BkOR) is an algebra with respect to pointwise multiplication of More precisely, if fl C- 911 (N), f2 a Bk2(IR), then fl,f2 E

the functions.

_EBk1+12(R), and II flf2ll fl+f2

II

Bk

fl11Bkfl

II f2l1 f2.

(27)

Bk

In particular, Bk(1R) is a Banach algeora.

(c) The function f(y) = yse1Ty belongs to

Bk(pt)

for

i e ht,

s - 0,1,2,

S.

(d) Let X E C-(t R. Then X does not belong to Bk(iR). Theorem 2:

We need the following lemma to prove

Denote by Ha(TR) the completion of C0(IR) with respect to the

Lemma 6.

norm

fI00Ha '

(1+y2)-B/2(1

-

II

a? )°/2@,IL2

(28)

aye

(here II f

ICI f (y) I2dy is the usual L2-norm; clearly Ho(1R)

II L2

WZ($t),

see example 1 of item D); then for each c > 0 there are the continuous embeddings

k HQ+t+l%2(R) CBkf

CHk+e+1/2(18)

(R) C

(29)

The embedding 8fk(R) C CLk M) was already proved in Lemma 4.

Proof.

For a S 0 the elements of Hc' are measurable functions (we do not reproduce a rather standard proof of this fact). So it makes sense to speak about embeddings on the right and on the left of (29). We prove first the right one.

Let f E Ck(R).

The norm in H

II

k

s{

j=0

a

is equivalent to the norm

(1 -22 7) k/2 (1+y2) -(f + e/2) - 1/4 ay

ET,

HE +E+1/2

f+ e + 1/2

L

4

(30)

a k! 2 -(f +c/2) - 1/4PI2dy) 1/2 T'-(k -)) _mI ayj (1+y ) m

Since f E Cp(ht) (1 +

y2)-(f + e/2) - 1/4fil

y2)-1/2 - e

(1 +

6 CII f 112

The right-hand side of (31) being summable in iI, f G Hk

f+c+1/2

II f II

6 coast Il f

k

f+c+1/2 Now let f E H + e + 1/2(ltt), f0 Fourier transforms

60

(31)

Cl

aye

C" fn

and

k. C1

n

f in

f +E+1 / 2 ( 1 ).

Then the

fn(t)

F(fn)(t)

=_

It suffices to show that the sequence fn(t)

lie in S(it), hence in Ck+(1R).

Denote Fn(y) - fm(y) - fn(y), Fmn (t) = fm(t) -

is fundamental in Ck (fit). *

Then we obtain by Lemma 5

Let h c; Ck(tt).

fn(t)

h(t)

h(t)Fmn(t)dt

C(1+t2)k/2+e/2+1/4Fffi(t)3dt

2 +1

(l+t2)k

f

h(t)

-

(32)

Je lytfn(y)dy

(

4

f f

a

(l+t2) k 2+e 2+1 4 { (1 + i) I (I + t2

k/2+E/2+1/4-

Fmn(t) }dt

}{Ik(1 + t2)k/2+E/2+1/4F

kh2;e

( (1 - at)R

(1+t2)/2+1/4

1

t2)k/2+e/2+1/4F 1/4+c/2 L21 (If'(1 +

< constghj 11 Ck

=

mn(t))dt 6 (t)I L2 =

(1+t2 )

const t h I

(i + Y)-f (l -

ay2)k/2+e/2+1/4Fmn(y)

fl L2'(

Ck

< const j hj f J tmnI k+c+1/2'

(33)

Hf

Ck

Since AFmnl+e+1/2 - 0 as m,n

we obtain that fn(t) is a fundamental

Y

sequence in Ck (Qt).

Thus the lemma is proved.

Proof of Theorem 2.

(a) If f E Bk(tt), then f E BkGt) for some N and,

by (29), f e CN(6t). Conversely, let f 6 write the chain of inclusions (we set a 1/2):

C %+l (fit) C B` *t) C Sk(kt)

f E CN+l thus (a) is proved. Definition 3.

Twice using (29), we

(34)

To prove (b) we introduce the following:

Let Ti G CIJ+(Qt)

and T2 is an element of

j - 1,2.

The convolution T of T1

Ck(f1 kf2)+(tt), defined as

T(4) _ (Ti * T2)(0) " T1tET2r(m(t +t))3,

(35)

where the subscript t (resp. r) denotes that the corresponding functional acts on functions of a variable t (resp. t). We claim that the definition is correct and the following inequality holds

IT I * T2

6 IT,

12'

(36)

F-1(T1 * T2) _ 1F-1(T1)F-1(T2)

(37)

1

Ckl+f2

I T21

Further, we assert that

(pointwise multiplication on the right-hand side of (37)).

$ince the norm

in Bk is given by (24), (b) is clearly a consequence of (36) and (37). To prove the correctness of definition (35), we show that f(t) - T2r($(t+r)) is the element of Ckl(Rt) and

61

If R

CkE1

no. where no is large enough.

(39)

We show that the mapping

Fn : t -* Fn(t), Fn(t)(T) - O(t+T)l'n(T) is an element of Ckl(kt,Ck2(,LT)), and

(40)

I F"IC11 < n T21Ck2* I ONCk1+E2 +0(1/n), as n k

this immediately implies our proposition. a"+B

Since supp yn is compact, all the derivatives

-o -a (# (t +T)0n(t)) at at

are uniformly continuous for a < E1, B < E2, when t lies in a bounded set. It follows that these derivatives may be interpreted as continuous mappings

R-00, and thus t- F(t) is a mapping together with E2 derivatives.

at

E £1

- sup (1 +ITI)-k

arFrt E Ckl

from 1. to CklOR), continuous

Further,

'Sup

a

j+r

j-0 at3 atr

f+S

It+rl)-k

< (1+ ItI)ksup (1+

£2 I a C0(t+T)*n(t)31

£1

tart

[O(t+t)9Vn(T)]I
1+ + tt here.)

(We used

(42)

Now we have for Ti E Cki G), i - 1,2,

F-1(Tl * T2) 2n) Tlt(

2a

T1t(eiyt

7,1

Zn) (Tl * T2) (elyt) 2e

T2T(el(t+t))) -

T2t(eIYT)) - i1

(T1)F(T2).

(43)

So (37) is valid. Thus (b) is proved. We have shown simultaneously that convolution is an associative and commutative operation (since the same

62

Further, the Fourier transform of yseiTY up to

holds for multiplication).

the factor coincides with 6(s)(t - T) (s-th derivative of Dirac 6-function (c) is and thus defines the finite continuous functional on Bk for V. > s. proved. To prove (d) we note that if x(t) were a continuous linear form (Here Do(lt) is on Ck(1R), it would be a continuous linear form on Do(a). a space of continuous functions on It with compact support. The sequence f E Vo(lt), n = 1,2,..., converges to zero if supp fn c K for some fixed compact K not depending on n and fn(t) converges to zero uniformly on K.) We have

x(y) - sign y+xl(y), where xl(y) F(x11) is a Do(r). The v.p.(l/t).

(44)

Then is a finite function with a jump discontinuity at y - 0. continuous function and thus defines a continuous functional on Fourier transform of sign y is, up to a factor, the distributio (Recall that v.p.(1/t)(0) = C

`t) Jltl - t

dt.

E

(45)

Let now iP E Co(lt), y(t) > 0, y(t) = I for Iti S 1 be fixed. Let also 4(t) be a smooth function p(t) % 0, 4(t) = 0 for t 6 0, $(t) = 1 for t ' 1.

Set ¢n(t) - y(t)0(nt)(tn.n)-'I P. Then we have On (t)E Co(pt), On no 0 in Do(lt) . On the other hand v.p.(1/00 n

3(kn n)''/' f

foO(t)

(nt) dt(kn t

n)_'/2

dt 0(t) O(nt) dt a ($n n)-i/' 1/n t t 1/n 1

1

(tn n)-'/21n n = (In n)'/'

=

as n - m

Thus v.p.(1/t) does not define a continuous functional on Do(lt). that x * Bk and the proof of the theorem is complete.

(46)

It follows

Now we have come near to the basic definition of this item. Let A be a generator of degree k in a Banach space X. If X E X lies in the domain DAt, then eiAtx belongs to Cf($t,x) (see Theorem 5 (a) of item A, Eq. (26)).

Thus, if f E Bk(It) , the expression fo(A)x

(

j_F(f)(t)e Atxdt,xEDAf

(47)

(see notation (9)) is, by Theorem 1, a correctly defined element of X. Definition 4. Let A be a generator of degree k in X, f c- B Gt). We denote by f(A) = 11A(f) the closure of the operator f9(A), given y equality (47). The operator f(A) is called a function of A with symbol f. Theorem 3. (a) Definition 4 is correct (i.e., fo(A) is a closurable linear operator).

(b) If f E Bk(IR), f(A)DAs C DAs_f for s > f, and the following estimate is valid:

!As-ff(A)xJ+If(A)xp 6 MsfIffl I(I

xEDAt;

(48)

Bk in particular f(A) is a bounded operator in X for f E Bk(lt). 63

P

(c) If f 6 S k(Gt), each dense linear subset of DAP, invariant under

exp(iAt), t E R is a core of f(A). (d) The mapping UA algebraic properties:

:

f(A), f e 8k fit) possesses the following

f

UAf)uA(g) - UA(fg), or f(A)g(A) - (fg)(A) (49)

PA Of + Bg) - aUA f + BUA(9). or of (A) + Bg(A) - (af + Bg) (A)

(the line above the operator denotes its closure), UA(eiyt) - eiAt, t E 1R. UA(Y) - A (50)

UA((a - y)-1) - R1(A), Ima 0 0.

Before proving this theorem, we carry out some discussion of the result. The theorem asserts that the image of UA is a set in C(X), which forms a commutative and associative algebra with respect to the operations

aB, a 1

0

0, a - 0,

B+ C - B + C, B o C - BC.

(51)

Further, UA is the homomorphism of algebras continuous in the sense (48) 8k(t) - B(X), where B(X) is the set of continuous (in particular UAIRp in1cX,

is a Banach algebras' homomorphism). We pay more Introduce the following norm on assertion.

linear operators

attention to the latter DAP:

(52)

QxIIP - 1-1 + JAfx1, xe DAP,

and denote by HA the space DAP supplied with norm (52) (for.! - 0, we set

xg - X. gxlo - qxl) Lemma 7.

HA is a Banach space.

are continuous and dense. !

s + is.

Embeddings HA -+ HA

The operator (RA(A))s, A E p(A)

for P 3 m - a.

,

f - 1,2,...,

The operator As is continuous from HA to IF, for

is continuous from HA to f

A is a generator of degree k in Hj for each L.

Proof. first we prove that HA is a Banach space (i.e., is complete). Recall that Al is closed (Theorem 5 (b), item A). Let the sequence xn be This means exactly that x fundamental in the norm is fundamental in X (and hence convergent to some x 6 X) and Afxn is f ndamental in X (and hence convergent to some y E X). Thus X E DAP and AAx -Py; IX - xnit

x - xnI + AA xn - y

-

Next we prove that As

0 as n

i.e., xn n+ x E HA in the norm

. HA - HA, t > a + m is continuous (in particular

for s - 0 we obtain that the embeddings HA - H Clearly ASHA'C lFA for ! 3 a + in.

,

!

m are continuous).

Let now xn E HA, n - 1,2,..., and

IIxnHI£ - 0, IlAsxn - YIIm - Oasn i m. Itfollowsthat IIxnHI -+ 0, IIAsxn - YII 0. Since Asis closed in x (Theorem 5(b), item A),y - 0,and we obtain that Aa

64

:

HA + HA is closed and everywhere defined. By the closed graph theorem,

"'

Now we are able to show that (R1(A))s is a bounded operator

As is bounded. HA+s

(the case m < f + s follows automatically, since the em-

from HA to

s

beddings HA - HA -1 are continuous as proved already).

Indeed, (A -x) D f+a. Hj,

By Banach theorem (on = D I for a E p(A) and this operator is continuous. Hp the open mapping) (Rx(A))s = ((X- A)s)-1 is also continuous. To prove that HA is dense in HAf -1 (with respect to the norm 11'14-1), we make use of the following commutative diagram:

A

-

A

(X -A)

Hf-1

embedding

HI

.(R:(A))f-1

f-1 H1

A

'

embeddin

DA

?1 (X -A)

f-l

(a E P(A)),

HO . X HA

(53)

which gives reduction to the fact that DA is dense in X (Theorem 1 (a) of Further, item A).

Aexp(itA)xp+ N exp(itA)AfXI 4

(54)

C(l+ ItI)k,xIf

C(1+

so A is a generator of degree k, and the lemma is proved. Definition 5.

The sequence of dense embeddings

X . HAS HAD HAD ... O HHD ...

(55)

(48) means that µA realizes a

is called a Banach scale* generated by A.

continuous mapping from Bk(lt) to B(Hs,HA f) (here 8(X,Y) is the space of In other words, p is a continuous bounded linear operators from X to Y). homomorphism of the algebra B8(It) into the algebra of continuous operators in the Banach scale (55) (we do not give the detailed definitions here; these will be given in Section 3).

First of all we notice that if x E

Proof of Theorem 3. U(t)x E Ck(Gt,HA

Hss, eiAtx

) for a 3 f, and

U (t) x1Ck(Et,HA f) 4 const M x

HA.

(56)

Indeed, this is an easy consequence of Lemma 7 and equality (26) of item A. Thus, if f e Bk(tt), T -

eleiaent of HA - f (Theorem 3), and

h - T(U(t)x) is a correctly defined

s-f 4 consti f

hj

HA

fq xO Bk

(57) HA

On the other hand, we have f(A)x - h1 - T(U(t)x),

(58)

*See Section 3 for generalizations.

65

where the pairing of Ck(t,X) and Ck+(tt) stands on the right. f

h coincides with hl (under the embedding HA

To prove that

C X), it suffices to prove

that <X,h> - <X,hl> for each X C X* (recall that X* C (HA R)* and is dense since HA

f

is dense in X).

But the latter assertion is evident since

<x.h> - <X,hl> - T(<X,U(t)x>) with usual pairing of

Ck+(dt) and Ck(t) on the right.

Next we show that RX(A)f(A)x = f(A)RX(A)x, f e for any X E X

Thus (b)) is proved. Bk(pt), x E DAR.

Indeed,

<X,RX(A)f(A)x> -

27 F(f)()

-

27ir

F(f)(<X,RX(A)(t)x>)-

(27 F(f)(<X,U(t)R),(A) x>) -

- <X,f(A)RX(A)x>

(59)

(we used Theorem 1 (c) of item A).

Now let f E 4(k)' xn - DAL, n - 1,2,..., xn i, 0 in X, Yn - f(A)xn + n n+m yin X. Set zn - (RX(A))IYn

for some X E p(A).

(60)

Then

zn^ - fl (RX(A))Lf(A)xn

)

-

F(f)t(F(g)t(<X.eiAteiAtx>)) -

2e) F-1(f)t( (21x ?e

F-1(g)T
) F(f)(<X,eiAtgo(A)x>)

2rt)

- <X,fo(A)go(A)x>,

and (64) is proved. (50) is now obvious, since F(eiyt) - 21 6(t - t), F(y) - i/61(t) (d denotes the Dirac delta-function). As for the last of identities (50), y)-1 The ' 1. it follows from (49) and from the identity (A - y)(A theorem is proved.

At the end of this item we intend to show that, although rather complicated, the symbol space 8 is nevertheless quite natural for one-operator functional calculus in genera Banach spaces. Namely, we present a simple example in which all bounded functions of a given generator A of the degree 0 are of the

form f(A), f e 8:%). Being quite similar to the case k - 0, the construction of examples for k 0 0 is omitted here. First we give some extension of the notion of a function of a given operator A. describing the latter in "external" terms.

Let Definition 6. Let B be a bounded operator in a Banach space X. also C be a closed operator in X. We say that B and C commute, if for every x E DC, Bx E DC, and BCx - CBx. Example 1.

(66)

A being a generator, A commutes with eiAt.

Definition 7. Let A be a closed operator in a Banach space X. The closed operator C in X is called a function of A, if every bounded operator B, commuting with A, commutes also with C. Lemma S.

E 4(k).

Let A be a generator of degree k in a Banach space X, f E

Then f(A) is a function of A in the sense of Definition 7.

Proof. Let B be a bounded operator in X commuting with A. Then B commutes with exp(iAt). Indeed, for x E DA exp(iAt)Bxand B exp (iAt)x both satisfy the same Cauchy problem ((3), item A) with xo - Bx, and therefore coincide (item A, Theorem 1 (d)). Next we note that, commuting with A, B commutes with At as well. Let x E DAf. We have

<x,Bfo(A)x> - - 7 S F(f)() -

-

2n

F(f) (<X,exp(iAt)Bx>) - <X,fo(A)Bx> ,

(67) X C_ X*,

so Bf0(A) - f0(A)B. Passing to the closure f(A) of fo(A), we obtain the desired result. Now we have all we need to construct the next example. Example 2. Let X be a space of complex-valued continuous functions f(t) of the variable t c- IR such that fin f(t) - 0, equipped with the usual C-norm: ITI-

67

MIX = sup If(T)I.

(68)

TE6t

X is a Banach space, and each f r=- X is a uniformly continuous function. f(T)I < E/3 for ItI > R. Then f(T) is uniformly continuous on [-R,R] so there exists 6 > 0 such that If(T1) - f(12)I < t/3 for Tl,T2 E C-R,R7, IT1 - T2I < 6. It is easy to verify, however, that If(T1) - f(-2)I < c for any T1, T2 such that then L 1 - T2I < 6 under these conditions.

Indeed, given an E > 0, choose R such that

Consider now a one-parametric group U(t) in X defined by

(U(t)f) (T) = f(t+T), fEX,tett.

(69)

Clearly JU(t)j 1 for all t, U(t) is strongly continuous since the relation IU(t)f - U(tl)fAX 0 is equivalent to uniform continuity of f(T). t y tl

The generator of the group U(t) is the operator A its domain DA consists of continuously differentiable functions (r), T E g such that 4 E X and 0' E X. Thus A is a generator of degree zero.

7.

Proposition 1. Let C be a bounded function of A in the sense of Definition Then C - #(A) for some $ E Bo(R).

Proof. C being a function of A, it commutes in particular with U(t). Let now f E X. We have

[Cf](t) - CU(t)Cf](0) - ECU(t)f](0) = XC(U(t)f),

(70)

where XC E X' is defined by

XCU) _ <XC,f> = [Cf](0). Let now X G X'.

(71)

We have

<X,Cf> _ <Xt'<XCTU(t)f>> _ <Xt'<XCT'f(t+T)>>

(72)

(the subscripts t and T mean the same as in Definition 3). By the Hahn-Banach

theorem, X and XC may be extended to linear continuous functions on Co(R), i.e., elements C0of'C0*(hR).

We assert that these extensions may be chosen

belonging to O+(1R). The convolution of T1 and Tz (Definition 3) is then well defined and commutative (see proof of Theorem 2), so that*

<X,Cf> = XCt[XT(f(t+T))] = XC(<X.U(t)f>), and, consequently, C - ¢(A) where prove the following: Lemma 9.

_

'F-1(X C)

E BOO(R).

(73)

It remains to

Every functional X E X' is the restrictiod of some element

of Co+(IR) Proof. Consider the mapping : h - S1, (T) - 2tan " T (here S' is the unit circle with the angular coordinate e (-rr,n]). Under this mapping X is transferred into the subspace X of C(S1) (space of continuous functions on S1), consisting of functions vanishing at - r. Co(lt) is transferred '

into the space X of bounded functions on S1, defined and continuous where except the point * - a. Thus we have embeddings

* We denote the extensions of X and XC by the same symbols. 68

every-

C C(S') C X.

(74)

The norm in each of the three spaces is given by the same formula II

sup

fiI =

(75)

I f (,v) I .

E(-rr,n)

Finite functionals on C0 (R) are clearly transferred into functionals on

whose support does not contain the point - n. Now let E X*; by the HahnBy the Riesz theorem Banach theorem there exists an extension h E C*(S')of h is a Rhadon measure on S1. Let a = h({s}). Then h = h - ad( - a) is also an extension of x and

h(Q) = tin hn(Q) = nim h(Q n L-rr + n , a - n ])

(76)

Obviously measure hn(Q) induces finite

for each Borel subset Q of S1.

functionals hn on X, h = Lim hn exists in X*, and restriction of h onto coincides with X. F.

Lemma 9 is proved.

Functions of a Self-Adjoint Operator in a Hilbert Space

Example 2 of tie preceding item shows us that we may scarcely hope to enlarge the class 8k of symbols of bounded functions for general generators of degree k in general Banach spaces. It turns out, however, that in the special case (self-adjoint operators in Hilbert spaces) the class of of symbols, which give bounded operators, may be essentially enlarged. In the first section we This item may be divided into two sections. recall some well-known facts about self-adjoint operators, while the second one is devoted to relations between two approaches to the construction of functional calculus for self-adjoint operators. To begin with we study in short the spectral measure of a self-adjoint operator A in a Hilbert space H. No proofs will be given since we assume that the reader has already become acquainted with this material (see 113], 1711 or any other textbook on functional analysis).

The decomposition of unity in a Hilbert spice H is of bounded operators in If, possessing the following

Definition 1. the family EX XE IR

properties: (a)

For any A, EX is an orthogonal projector in H, i.e., EX = EX,

(b)

p

(1)

X

= E11 for P

X;

(2)

EX is strongly continuous on the left, i.e., s - Lim Eu = EX (or

u+X-0 (d)

EX;

{EX} is a non-decreasing family, i.e., EXEV = E E

(c)

EX

EX has strong limits as A

fin Eux = EXx for all x C- H); ± m, and these limits are

s - Pim EX = I (identity operator), a - fin EX = 0. X-

(3)

y-X-O

(4)

X- -

69

It follows from the definition that for any x,y E H, the function

Ex.y(A) - (EAX,y),',

(5)

is continuous on the left and dEx (A) is a correctly defined complex-valued Borel measure on at. We shall usehe notation (dEAx,Y) for this measure. Let now $(A) be a Borel function of A E ht. Clearly 0 is measurable with respect to all the measures (dEAx,y), x,y E H. Set Dt - {x e HII0(A)IZ)is integrable over IR with respect to measure (dEAx, x).

(6)

Set also (p(4)x,y)

f (a)(dElx,y), X E Do. y 6 H.

(7)

Proposition 1. The equation (7) correctly defines a closed operator u(e) with domain Dm; the following estimate is valid:

IIv(m)xll2 ° f- Im(A)I2(dEAx,x).

(8)

-

The following notation will be used: 0(A)dEA.

f

(9)

Now we may formulate the basic theorem on the structure of self-adjoint operators.

Theorem 1. For any self-adjoint operator A in the Hilbert space H there exists a unique decomposition of unity E1 - Ex(A) called the spectral family of A, such that A - f

AdEA.

(10)

This spectral theorem enables us to construct the functional calculus for a given self-adjoint operator A. Given a Borel function D(A), A E R, we set

m(A) - f

i(A)dEA(A).

(11)

-ou

The properties of the introduced correspondence $ - (A) are summarized in the following: Theorem 2. bounded and

(a) If @(A) is a bounded function, the operator @(A) is

sup

D(A)H

(12)

AE a (A) if I,(A)I < C(1 + IAIk), A E a(A), then DAk C DO(A) and

(b) Moreover,

X E DAk.

O(A)xj 4

(13)

(c) The following relations are valid: (m(A))* - 4(A), 0(A)

01(A)D2(A) if

We have the equality sign in (15)

'(A) = 01(A)t2(A).

(14)

(15)

if both (A) and m2(A) are bounded on

o(A).

4(A) Z) a1m1(A)+ a2D2(A) if m(X) ° a1f1()L) + a2m2(A)

(16)

(we use the notation C1 C C2 if DC1 C DC2 and Clx - C2x, X E DC1).

A)-', then 4(A) - Ru(A). Imp f 0; and (A) - Ak,

(d) If '(A) if @(X) - ak.

Proof. The assertions (a), (c), and (d) are standard, and their proofs may be found, e.g. in [13]. Let (A) be a Borel function, satisfying A E o(A). Then we have for x E DAk: E C(1 +

fIO(A)j2(dExx,x) < C(f(dEAx,x) +f2k(dEax,x)) (17)

E C(II xII 2 + Akx 11 2) < m, and this implies (b) immediately.

Thus the theorem is proved.

Next we investigate the relations between the definition of 4(A) introduced above and that given in the preceding item (Definition 4 of item D).

Both definitions coincide for 0 E 80(!R).

Theorem 3.

Proof. (a) We show first that mt(A) - exp(itA), where ot(A) - exp(ita). Indeed, mt(A)j 6 1 for all t E ht. Next, let x E DA2. We have

V t+E(X) -

eia(t+e)

eiat(1+iAE)+E2),2FE(a)

=

(18)

- mt(a) + icot(A)X + E2a2FE'tU), where Fe't(A) -

e

iXe

- 1 - iXC eiat

is a bounded function.

e2A2

By Theorem 2 (c) we conclude that 0

t + e (A) x- 0 t (A)x - ieA((t t(A)x)+E2FE t(A)A2x

(19)

and besides F.,t(A) is bounded uniformly with respect to e. Consequently, mt(A)x is differentiable with respect to t and satisfies the Cauchy problem i 3i (0 (A)x) + A(tt(A)x) - 0, o(A)x - X. By Theorem 1 (d) of item A, 0t(A)x - exp(itA)x. and 4t(A) is bounded, mt(A) - exp(itA). (b) Let now 0 E 80'(tR) be arbitrary.

Since DA2 is dense in H

The Fourier transform of 0, which

we denote by 0(t) = F(Q)(t), lies in Ca+ (1R) for some natural L. to prove that

f-@(a)(dEax,y)

2n) m((exp(itA)x,y)),

x E DAf, y E H (here EX - Ex(A)). of $(A) give identical results.

(20)

We intend

(21)

(21) means that on DAf both definitions

Since DAf is a core of t(A) (it is postu-

lated in the former definition and it will be proved below for the latter), this implies the statement of the Theorem. We have

fQ(a)(dEXx,Y) - f(

((exp(itA)x,y))

2n) 6(eit")](dEax,y); 1

- V(2

m(feitA(dEax,y)),

(22)

(23)

it

(23) being valid by part (a) of the proof. Thus we have to verify the possibility of the application of 0 under the integral sign. Let 0 be a finite functional. Then applying Lesma 2 of item E, we obtain f

0(J" e itl (dElx,y)) -

E

j-O

a? r

jR

3

-R

at3

(24)

Since x e DAf, all the

where duj are the Rhadon measures on E-R,R]. integrals f

eita(dExx,y)}.

1

converge absolutely and we can apply

eltla3(dElx,y),

the operator aj/atj under the integral sign. Further, the integrals 1RRr-ljeitaduj(t)(dElx,y) converge absolutely and by the Fubini theorem

We obtain that (21) is one may change the order of integration in (24). valid for finite 0. Now if the sequence in of finite functionals converges' to 0 in Cf*(1R

(dEAX,x)
6', such that:

(a) All the operators i6d, are continuous embeddings, and besides X6 ± X6 are identical operators. (b) For any 6,6',6" E A such that 6 > 6' > 6", the diagram

(1)

of Banach spaces and homomorphisms is commutative (i.e., i66" o i66,).

- i6'6" o A Banach scale is called a dense one if all i66, are dense em-

beddings.

91

{W2(ltn))kElt, where WZ(ltn) is the space of Fourier trans-

Example I.

forms of measurable functions, for which the norm Q4 N

2`(tt°)

-f

10(x)12(1+x2)kdx, ($ - f(m))

is finite, is the well-known Sobolev scale.

(2)

It is a dense scale.

Example 2. Let A be a set of pairs of multi-indices & - (ct,B) of the length n with natural components with the ordering relation 6 3 6' Z (a ) a', The collection of Bn). a ( B') _ (al > al'" ''an BL" .. 'Bn an'B1 spaces {Sa ORn)}(a,8)E

A

(these spaces were defined in item A) with natural

embeddings form a Banar_h scale over A, as follows from Lemma 1 of item A. A. Also {HaB (as)}(a ,

B)E A

is a Banach scale; here the set A may be extended to

co1sist of all the pairs of multi-indices with arbitrary real components. The latter scale is dense. The next proposition follows immediately from the definition. Proposition I. Let E be a vector space. Assume that for any element 6 of the bidfrected set A the norm P6 is defined on E and satisfies the

condition I x A

dd

) Ix 1 6, for any x e E, 6' < 6. Denote by X6 the com-

pletion E6 of E in the norm p q6 and by i66, the closure of the identity operator on E in the corresponding pair of spaces. The collection of these objects forms a dense Banach scale provided that the following compatibility condition is satisfied: if the sequence {x0} of the elements of E converges

to 0 in the norm 1 N 6 and is fundamental in the norm I

I a. for some 6' >

> 6, then it converges to 0 in the norm A A6, as well. Notation.

The Banach scale will be denoted by X - (X6)6E A; X6 will

be identified (under the embedding i66 ') with the subset of X6, for 6 > 6'. Consider the union

U

X -

Ea

X6.

(3)

X possesses the structure of a vector space. Indeed, if xl x7 E X, then xi e X¢ for some 6i, i - 1,2. Let 6 ( 61, 6 < 6; (such 6 exists, since A is Then xl,x2 E X6 and we may define their linear combination + bx2 E X6 C X. The vector space axioms are obviously valid for all such a definition. In applications the vector space X usually has a natural interpretation. For example, for the scale {Ha(lts)} the union (3) is nothing else than the space S'(ltn) of tempered distributions on V. We refer to X as the total space of the scale. If some linear subspace E C :X is dense in any X6 C X, 6 E A (and, consequently, we are in the situation of Proposition 1), it will be referred to as the scale base.

We introduce the convergence in the total space X of the Banach scale Let x r - X, y -* xY be a generalized sequence in X (this means

{X6}6C-A'

:

nothing more than that r is a directed set.)* Definition 2.

{xY}

Er converges to x SEX if and only if there exist

60 E A, yo E r such that zYE X6o for all y * We shall also speak of (XY)Y E r

92

as

yo, x e X60, and I xy - x ! 6o

a r-sequence for short. 4

converges to zero (i.e., for any e > 0 there is y - y(e) E r such that Ix In particular, the sequence +cn converges to - x 16o< a for y' ) y). x If and only if xn E X6, x E X6 for some 6 t and all n> no, and IIxn -

xll6*0asn-

We write xY -. x or Lim xY - x if the r-sequence xY converges

Notation.

yer

6

to x. We write xY + x if ^ xy - x P 6 -- 0. We do not introduce any topology associated with this conver-

Note. gence.

Proposition 2. The convergence introduced above is compatible with linear structure on X (i.e., linear operations are continuous with respect to this convergence). The generalized sequence may have at most one limit.

Proof. Let xy - x, X. -* x, y E r, y E r. sequence xY + xy converges to x + x. 61 < 6, 61 4 6.

61x

Then xY ylx, xy

< e/2 for y) y(e/2), Y ) y(r/2).

We prove that (r x r)-

Indeed, let x., + x, xy

so hxY - x 161 < c/2,

8

x; let also

xy - x 16,
6, o' % a.

Proof.

is b:,m,d d

(a) Ao, - raa,A60i

tnd everywhere defined if

the same holds for A60, (b) Let A E C(X6,Yo).

If xn G DA, n - 1,2,..., xn

then xn a is and Axn Q y, thus is E DA and y = Ax.

The proposition is proved.

C

Proposition 4. X6

Let (B8o}

(6 ,a )

ix

b'

x and Axn

This means that A C-

be a family of linear operators,

Ya defined for (5,o) lying in some subset X E A- F. lowing conditions are equivalent: B6o

96

'

Cr

The fol-

Y,

(a) There exists a linear operator A : X + y such that DBdo C DA for X and B6ox = AX for x G DB6a. any (6,0)

M

= 0 and xj E DBf o

E xj j=1 we have:

of el:ments of X such that

(b) For any finite set

j = 1,...,m where (Si, j) E X, j

.

m

E B6 0x = 0 j=: J J J Proof.

(b) * (a).

It is evident.

(b).

(a)

We denote by DA the

linear core of all the sets DBdo, (6,0) E X, an: set for xE DA k

k Ax =

E

j=1

B6,0,xj if x J J

E

xj, xj E D0

j=l

(12)

J i

The definition (12) is correct since the condition (b) yields that that

k

k

m

in the case when the equalities x = JEl x. =

B6jajx

jEl B6j0jx.

j=k+1

m E xj are two different representations of the element x E DA. Thus, j=k+1 The proposition obviously, the operator A defined by (12) is a linear c ie. is pr,'ed The operator constructed in the proof of this proposition is

called the (minimal) operator generated by the family Proposition 5. B6a

X6

*

Let (Bda)(d,^)

{B6a}.

E X be a family of linear operators

Assume that:

- Yo.

i)

for any (61,01) E X, (62,02) E X such that 61 3 62, 01 and B61alx = B62a2x, x E D.Bdl01

i) that

f

ol,

a2, we have

(61,01) F X, (62,02) E X, there exists a pair (6,0) E X such a., i = 1,2. Then the conditions (a) and (b) of Propo

;,uun '. are satisfied.

m

!.et vj E Dx6

)of.

j

J

= 1,...,m,

j

E xj = 0. j=1

By induction on m

nroee. u:::nx (ii), that there exists a pair (6,0) E X such that 6 < 6j, i,. .,r:. Using (i), we obtain .3j.

E 46.o.xj = E B6axj = BdaExj = 0, J

since $

o

i

is a linear ope-tor.

The proposition is proved.

Proposition ',. satisfy the conditions Let the family {B60)(6,0) E X of Proposition 5 If all B60 are closurable operators, then the family also satisfies these conditions. (B6o)(d,a) E X

Proof.

Only (ii) should be proved.

Let x E DB6101' Bd1alx = y. This

means that there exists a sequence x E DBd, n = 1,2,..., such that n 101 Xn

.61

x, B6i71xr, 02

3d

101

y.

y.

But then, according to (i), xn

Thus X E DB

and Bd232x = y.

62

x and B6202xn

The proof is complete.

6)02

97

runctions of Several Operators in a Banach Scale Definition 1. A tempered (strongly continuous) group of linear operators in a Banach scale X - {X6}6E is a family {U(t)} telt of linear A

operators U(t) in X such that:

(a) The domain DU(t) - X for all t c- &t and

U(t)U('r)x - U(t+ r)x, U(0)x - x, xEX, t,t EQt.

(1)

(b) For any 6 E A there is a a E A and conversely for any a GA there is a 6 E A such that: (i) U(t)i=- L(X6,X,) for any t C- IR; (ii) U(t) is strongly continuous in t as an operator from X6 to Xa; and (iii) for some c - c(6,a) and k - k(6,a) the estimate

(2)

IU(t)xs c(l +ItI)k Ix16 is valid for any x E X6.

Note. One obtains the definition of a strongly continuous group of linear operators in X omitting the requirement (iii). We denote by AU C C A x A the set of pairs (6,a) for which (i) - (iii) are satisfied.

Definition 2. The generator of a strongly continuous group {U(t)) of linear operators in X is the operator A in X defined by

Ax - Aim(ic)-1(U(c)x -x) a _it (U(t)x)]I dt E+0 t-0

(3)

(the domain DA C X is the set of all elements x E X for which the limit (3) exists). We write U(t) - exp(iAt). Theorem 1. and besides

(a) The domain DA is dense in X and invariant under U(t)

AU(t)x - U(t)Ax - -i d[ U(t)x, x E DA.

(4)

(b) The operator A is closed (i.e., if r-sequence (x Y} converges to x and (AxY) converges to y, then x E DA and Ax - y). (c) The Cauchy problem.

ddtt) + Ax(t) - 0, xlt-0 - x0 E DA

1

(5)

has a unique solution x(t) - U(t)xo (thus, there is a one-to-one correspondence between groups and their generators). (d) For Imo f 0 the resolvent RX(A) - (XI - A)-1, e-ixtU(t)xdt, x E X,

RA(A)x - if

(6)

0

is defined (the sign in (6) coincides with the sign of (-Imo)) and moreover RX(A) E Lright (X,X) n Lleft(X,X), XRX(A) D AU, (e) RX(A)m E L(X6,Xa) for any is,

Rl (A)' x a

c

k

,-O

(6,a) EAU and

k!(m j-1):

I

Im'X

-m-k

x6 (7)

Imx 0 0, m - 1,2,...

98

k = k(6,o) are the same as in (2)).

are valid for ,6,:-) &.',U (c =

(f) There is

t

representation

exp(iAt)X = Fim(I - nt A)nx, x G X,

(8)

n -

the convergence in (8) being locally uniform with respect to t.

Proof.

W. gives here averv brief proof of this theorem, since it repeats

with ,light modifications the proof of Theorem 1 of Section 2:A and of Theorem 1 of Section 2:1). (a) For x E X and q E Co(R). set

x0 = f_µ,.(t)1i(t)xdt.

(9)

and consequently The integral (9) makes sense since x G X6 fo: some The set of e'ements U(t)x is a continuous Xc-valued function for some a. of the form (9) is dense in X; on the other hand the limit fim(ic)-1 x (U(E)x4 - x,) exists and equals x. have

,.

f+D Next, if x lies in DA, then we

it iU(E)U(t)X-U(t)X) - U(t) 11- (U(E)X-x)J.

(10)

U(t) E L(X6,Xa) expression in square brackets tends to Ax in some X6 for some c, so the right-hand side of (10) tends to U(t)Ax in X0, therefore in X. We obtain U(t)x E DA, AU(t)x - U(t)Ax. (d) and (e)

Set

m

R(a,A)x = `in _7 m

x EX. Ima < 0

10

(11)

(the case Ima > 0 is considered in a similar way). The integral (11) is convergent for any x E X, since the estimate (2) is valid. Further, the calculation identical to (14) of Se_tion 2:D shows that Rm(X,A) satisfies the estimates (7). We show next that Rm(X,A) = (XI - A)m. Indeed, :m-1

m-1e -iX(t-E) U(t)xdt

1

(U (E)-I)Rm(X'A)X iE

e(m-1): im-1

m-1 -iXt U(t)xdtJ

f0t

(m

e

E f (t-

-

E

m

(t- E)m-leiXE - tm-1

1), C!0

U(t)xdt

C

E)m-le-iX(t -E)U(t)xdt7.

(12)

0

The limit on the right as e

-

0 exists and equals

XRm(X,A)x - Rm-1(X,A)x for m > 1,

XRI(X,A)x- x for m - 1. Thus, RM(X,A)x E DA and (XI- A)Rm(X,A)X = Rm_1(X,A)x (here we denote def R0(X,A) I). Similarly, R.(X,A)(XI - A)x = Rm-1(X,A)x for x E DA, so Rm(X,A) _ (XI- A)m. Thus (d) and (e. are proved.

(X -

(b) A = X - ((X .- A)-1)-1 (ImX # 0) and is th,refore closed (since A)-1

is; we leave details to the reader).

99

(c) Let x(t) be a solution of (5), and set (13)

Y(t) - U(-t)x(t).

It is easy to verify that y(t) is differentiable and dy/dt - 0, thus y(t) E xo and henceforth x(t) - U(t)xo. (f) Using (11), we may write for t > 0

(I - in ) -n x - ( t)n(lt - A) -nx ([)n

11). loin-ie-nt/tU(T)xdr

(14)

n mtn-ie TU(tT)xdr.

1

n-1).

J

n

o

Then

Let x E DA2.

(1

U(nT)x . U(t)x+t(Tn nn)U(t)Ax+t2(Tn n)21 (1-u) x 0

(15)

- t7U(t)Ax+t2/n2 (t-n)ZY(t,T) x U(t+Nt(Ln. ))A2xdu a U(t)x+ it-r n For some 6 E A we have

and 6 of item A).

(cf. Lemmas

Y(t,r)

16 4 Cl(l +n)k < C2eT/n,

(16)

where the constants are independent of t if t remains in a compact subset of Substituting (15) into (14), we obtain

It.

(I - itA)-nx + t2

In_lU(t)x + t(In - In-1 rn-1 (T

f

n

Jtke-Tdt

where Ik - I

1.

We have

-n)2e-Ty(t,T)dT

t n f-Tn-l(T 0 C2 t 2 n C2 t2 -l nnr-(

- n)2e-Ty(t,T)dT,

0

5

fl 6
3.

Since the action of the tensor product of distributions may be obtained by successive application of the factors, it is enough to prove that

g2C< h.V(ts+l"..tn)w(T)U(tl....,ta)x > 3 < h,V(ta+1,....tn)(fw(T)g2(T)dT)U(tl,...,ta)x > for any fixed t E ttn.

Setting hl - V"(Cs+1' .... tn)h s X63, xl - U(tl,...,

ta)x E X62, we may rewrite the equality in question in the form

g2C 3 - ':armor-s.X61,X62),

E

exp(itnAn)

Then for any

(31)

r

s+c

E X: n

s+l

f(Al,...,A,)x = C

s n+s+l-r s+2 gl(A1,...,As,Ar+1,....An )x, I

(32)

where r-s

1

C = g2(As+l,...,Ar ). Proof.

(33)

This is an immediate consequence of Proposition 2.

Corollary 1.

Under obvious assumptions, we have k

k+l

k+l

n

n

f(A1,..,Ak)g(1,...,AR;x = g(Ak+1,...,An)Y,

(34)

where k

1

y = f(A1,...,Ak)x.

(35)

This is a particular case of the above t.eorem for r = n. The identities (32) and (34) may be written in a more compact form,

i E we introduce the u eful notation- the so-c ailed "autonomou, bra. kets" 11

11. By convention, the expression in these brackets is regarded as a single operator, i.e., the numbers over the operators inside these bra kets do The place not "interacr" with the numbera yet the operators outside them. on which the operator marked bylonumousbrackets acts is denoted by the number over the left bracket 11 or over the line over the whole operator. Using this notation, we may rewrite (32) and (34) in the following form s

r+l

n

s+l

r

81(Al,...,AsAr+11 ..,A,)g2(As+1,....Ar)x = r-s s s+2 n+s+.-r 1 Q g2(As+l,...,Ar )11 g1(Al,...,As'Ar+1,....An )x, s+l

104

(36)

k+l

k

I

n

f(A1,...'Ak)g(Ak+1,...,An)x =

k

1

1

n-k

1

2

k+l'...,An )II x

= Lt f(A1,...,Ak) li It n-k

1

(37)

k

Q g(Ak+1,...'An )D Q f(A,,...,Ak)I1 x. he me theorem is valid if some of the operators A1,...,An are not the g.nera-or:., but the symbols in question are polynomial in variables, corresp ._'ng tc those operators. We do not write the analogue of the condition 31) for this case since it would take too much space. On tie other h..n "iese conditions are obvious and the reader may write them himsel:, desired. Combining the proved theorem with the preceding pro;-osit we conclude that under obvious conditions the definition of

f(A1,...An( does not depend on k and {j1...... in)Our n', theorem is concerned with the case when two of the operators A1,....An co side and no other operators act between them. Again we give the form 1,tion only for the case when all the operators are generators in X.

("sh'fting indices" theorem). = As+1,...,An be generators in X,

The-:-em 3

Al,...,A

Let f(yl,...,yn) E 88(Iltn

(38)

x exp(itIA1) E C6

a

Then f-r any r E X. we have n

s

n-1

s+l

f(A1,... An)x = g(A1,...,As,As+2,...,An )x ,

(39)

g('1....,ys,ys+2'....Yn) = f(y1,...,Ys'Ys,Ys+2'...,yn). Note.

We adopt the convention that the right-hand side of (39) will s

i

s

2rcof.

n-1

s+l

be written in the form f(A1,...'As,As,As+2'

" ''An )x.

By Lemma 4 of item A, g E 8(htn-1)' where a

d = (a .. ,a s-i 2 (

1'

nin(a s.:c s+1 ,a

s+2 ''

..,an

'3 s-1,Bs+ s+l'"s+2'.. ,8n).

We assert that E f8

'

I

A

G

,

so that the right-hand side of (39) is well defined.

a

In eed, it

easy to prove that if a Xo-valued function F(t,T) satisfies . F(t,-), is(Bs'.S,+1)-times continuously differentiable, and satisfies the estimates i

r(t+ X,T - X)

2

1-

v1

et then f(t)

at"

F:t,T) 5C(l+1tI)ss(1+frI)os+1, Y1

S. Y2s+1'

(40)

F(t,t) is (8s +8 s+1)-times continuously differentiable and

105

layf(t)

I < C(1 + It

I)min{as,as+1),

2ty

Y 4 B

s

+ B

s+l'

(41)

Next, taking any h E X6, x E X6, we may write

+t

F(f)(
) 1 I

- F(g)(< (This connection between actions of F(f) and F(g) on test functions may be easily ascertained from the proof of Lemma 4 of item A.)

We now summarize all the theorems proved and give their application to the case when all the operators involved are translators and all the symbols considered belong to S. Theorem 4. Let all the operators A1,...,A.0 be right translators in Then for any symbols f,g,h E S , polynomial in the variables corresponding to that of operators A1,...,An which are not generators, we have: 1 n (a) f(A1,..., ) is the right translator in X independent of the choice of k and {jl,..''k1 in Definition 3.

X.

r r+l n s s+1 1 f(A1,...,As,Ar+1,...,An)g(As+1,...,Ar) -

(b)

a

1

n

r+l

s+1

1

r-s

(42)

- f(A1,...,AsAr+l,...,An) Q g(Ae+1,...,Ar )Il. (c)

1

n

s+1

a

h(A1,...,A

s

A

s+1,...,An) =

a a n-i ,...,A s,As,...,A u)

h(A1, 1

(43)

if As - As+l.

The same statements are valid for left translators with the only difference that they are valid not on all x E X, but for x E X6, where 6 is sufficiently large (6 > 60, where 6o depends on the operators and symbols involved). Proof. Using the definitions of Sm, generators, and right translators, we may always choose for given 6 such a. S, o that the requirements of the above theorems are satisfied. The case is similar for left translators.

4.

REGULAR REPRESENTATIONS

A.

Definition and the Main Property

Let X be a Banach scale, A1,...,An be an n-tuple of operators. For the sake of simplicity we assume throughout this item that all these We consider also the scale B - {Ba(ttn)) There is a mapping PA, described in Definition 3 of Section 3:D, which sets into correspondence an operator f(A1, ...,An) in X to any element f E -B. operators are. generators in X. (see example 2 of Section 3:C).

Definition 1. The left regular representation of the n-tuple A (A1,..., An is the n-tuple L - (L1.... Ln) of the operators in the scale B such that the following identity is valid:

106

AiIA(f) - VA(Lif), i - 1,...,n

(1)

for any f E B. Definition 2. The right regular representation of the n-tuple A 'Alt .... An) is the n-tuple R - (R1,...,Rn) of the operators in the scale 8 such that

VA(f)Ai - "A(Rif), i - 1,...,n

(2)

for any f e B.

Of course, regular representations do not exist, in the general case, The existence of a regular representation yields a system of relations satisfied by this tuple. This system has the form n A A x - fij(Alt .... An)x, x E DAiA.,i,j - 1,...,n, (3) for an arbitrary n-tuple A of generators in X.

i j

where

fij(y) - Li(yj)

(4)

Conversely, if the operators A1...... A satisfy certain algebraic relations and some functional-analytic restrictions, th4 regular representations exist. These questions will be considered in the sequel; here we assume that the regular representation exists and investigate the consequences of this fact. Moreover, we restrict ourselves to consideration of the case when the left regular representation L - (L1,...,Ln) exists. Theorem 1. Let also

Assume that the operators L1,...,Ln are generators in B.

UA(t) a

C-

(5)

S Ca(Re,X6,X61) n Cu(te,X61..Xo) () Cpate,X6,Xo);

E

UL(t) =

(6)

E Cu(ttn,8a(ttn),8p(ttn)).

Then for any xE X6, g E Sa(tin), f E Bu(tte), we have Q f(A1,...,An) 31 Q g(A1,...,An)11x - (f(L1,.... Ln)g)(A1,.... An)x

(7)

or, in another form,

VA(f) o VA(g)x - VA (VL(f)(g))x.

(8)

In other words, the knowledge of regular representation enables us to write 1 n the product of two functions of (A1,...,An) in the form of some new function n 1 of (A An).

Proof.

Fix some x E X6.

The mapping mx : g + VA(g).x is a continuous

linear mapping from Ba(ttn) to X61 and from 8p(tte) to X0. prove that

mxCUL(t)g] - UA(t)mxIg], g e Ba(tte)

It is enough to

(9)

107

indeed, applying to (9- the distribution (2-.T)-'/2 F f), w, obtain the desired result. We prove (9) by induction on k, setting

ULk(t) = UL(tl,...,tk,0,

.0), (10)

UAk(t) = UA(t1'...,tk.0'...'0)

aAii proving that mxEULk(t)g] = UAk(t m

g), k = 0 1

..,n.

(tl)

(11) is evident for k = 0, so we mu t only show how the induction step

works. One should disting fish betw.en two cases; (a) Vk # 0. In this case both sides of (11) are differentiable with respect to tk Xo-valued and atk

m CU

x Lk

(t)g] - m [LkU x

Lk

(t)g] = iAA m CU

k x

Lk

(t)g]

(12)

by (1),

UAk(t)mx[g] _ iAkUAk(t)mx[g].

(13)

atk (11) is valid for tk = 0 by the induction assumption, so we may use Theorem 1 (c) of Section 3:D and obtain (11) for any tk E fit. (b) vk - 0. We use the same idea as above but the technique is now more complicated. Let t1....,tk-I be fixed. We introduce the norm d It on B8

mid

setting

g 11 mid = sup

tkC-k

Cl + It ki )-"k-1 II ULk(t)g

n,

n

8

(TR

(14)

)

and denote by 8mid the completion of Ha with respect to the normI IImidThe results of Section 2:H are applicable to the case; here the mapping j (see Definition I of Section 2:H) is given by j = UL(tl,...Itk-1'0,...,0).

(15)

By Proposition 1 of Section 2:H, ULk(t) gives rse to a strongly continuous group w(tk) in 8mid and ULk.(t) = j7w(t)jI.

where jl

8a(tn) w 8mid and j2

Define the mapping M.

:

8u.id

8"(N") aie contine_)us mappings.

8mid .- Xa by setting

%(g) = mxo j2 g) Clearly M. is a continuous linear mapping. the group w(r). We intend to show that

If only (18) were proved, it fo:lows that Mxw(tk)

= exp(itAk)Mx and consequently,

108

(17)

Denote by L the generator of

MxLg = AkMxg for any g E DL.

(16)

(18)

mxULk(t)g - mj2w(tk)jlg - Mxw(tk)jlg (19)

UAk(t)mxg

- exp(itkA.)mxj2jlg = exp(itk k)mx L(k-1)8

for any g E

as desired.

Let g E DL, h - j2g.

We prove (18) in the following way:

We claim that h E DLk and (20)

Lkh - j2 L9-.

It is enough to show that exp(itkLk)j2(8) - j2(w(tk)g).

The latter identity

is valid on the subset jl(8) dense in Bmid and therefore valid for any

g E Bid (exp(itkLk) is continuous from Bp(ktn) to some Bp,tn) and therefore closed in 8p(itn)). From (20) we obtain

MxLg - mxj 2Lg - m .kj 2g = A.mxj 2g - AkMxg, i.e., (18) is valid. B.

(21)

The theorem is therefore proved.

Agreement Conditions

It turns out that in general even the most simple algebraic properties are not valid for functions of several generators. Thus we have to impose some additional conditions on the mutual behavior of generators (or their First of all groups) in order to obtain substantial functional calculus. we present a striking example of pathological behavior. Example 1. There exist two self-adjoint operators, Al and A2, in the Hilbert space H, such that

(a) For some dense linear subset D C DA1 () DA2 the restrictions A1ID and A21D are essentially self-adjoint. (b) Al commutes with A2 on D,

A1A2x - A2A1x, x E D.

(1)

(c) Nevertheless, the corresponding groups exp(iAlt) and exp(iA2t) do not commute, contradicting, the expectations based on formal algebraic calculations. The construction is rather simple. Consider the Riemannian surface r of the multi;valued analytic function / - 1x x and y may be considered as coordinates on r, each point (x,y), x2 + y2 0 0 corresponding exactly to two points of F. Hence the Lebesgue measure u - dxdy is defined on r, and we may define theHilbertspace L2(r,p) of square-summable functions on r. Next, for each point (x,y) such that y f 0 we may uniquely .

define its shift g,(x,y) _ (x + t,y), t EAR by therequirement thatgt(x,y) be a continuous curve on r. If y - 0 this definition fails when x(x +xt) < 0. However, the subset {y = 0} C r has a zero measure, and we are able to define the following group of unitary operators in L2(r): -

U1(t)f = (gt)*f, f E L2(r), t c- ht_ The generator of the semigroup Ul(t) will be denoted Al. Obviously

A1f - -i f , f EC 0(r)

(2) Let D - Co(r).

(3)

109

Proposition 1. Proof.

C(?') is a core of A1.

Consider the subspace Dl C Co(") defined by

D1 = t 9 E= CO (F)

I

y i 0 on supp c 1.

D1 is dense in L`(C) and invariant under U1(t). by Theorem 3 of Section 2:A. The proposition is

(4)

Thus D1 is a core of 41

All the more, C0(7) D D1 is a core of All

Quite analogously we define the shift gt(x,y) and the corresponding group U2(t) = (iA2t): U2(t)f - (gt)*f;

(5)

A2f = -i ay , fE Co(l).

(b)

Thus, A and A2 are essentially self-adjoint operators defined on C00'). 1 They commute on Co(P), since a f/axay = 32f/ayax. On the other Indeed, let f E Co(F) he a function exp(iAft) and exp(iA2t) do not commute. with support in the small neighborhood of the point (1,1) on one of the sheets of F. Then the function f = U1(-2)U2(-2)U1(2)U2(2)f

(7)

has its support in the neighborhood of the same point on another sheet of r and does not therefore coincide with f. It is easy to see that in the discussed example even the commutability Alexp(iA2t) = exp(iA2t)A1 does not take place. Indeed, the domain DAl is not invariant under exp(iA2t); moreover, for f C- Co(f), exp(iAZt)f may be jump discontinuous in the x-axis direction for suitable values of t.

After this consideration we proceed to the discussion of positive results which guarantee us the required commutability. It appears to he most convenient to represent them in the form given below, since this particular form admits easy applications to commutation relations. Let X,y be Banach spaces, A,C be the generators of strongly continuous X semigroups in X and Y respectively. Let also B Y be a closed densely defined linear operator. We are looking for conditions under which Bexp(iAt) = exp(iCt)A (i.e., DB is invariant under exp(iAt) and :

B (sxp (iAt )x

exp (iCt )bx) . The above counterexample shows that besides the

algebraic condition "BA = CB on a sui able dense subset," we have torequire We shall speak about the s,ime additi,nalfunctional-analytic conditions latter ones as of agreement conditio:s.

Theorem 1. Assume that there exists a core D C DB of the operator B such that for any x E Do a ix E Djx GDA and Ax E D}, we have Bx E -Dc and BAx - CBx. Then exp(iAt)x E DB, B exp(1At)x . exp(iCt)Bx, x e DB,

(8)

that at least one of the following agreement conditions is then satisfied:

a) Dp is also the core of B, D is invariant under exp(lAt), and v(t) exp(iAt)x is a contin.ous Y-valued function of t for x E D.

110

(b) D is invariant under R1(A) for Iml < moo, where w0 is the maximum of the types'of semigroups exp(iAt) and exp(iCt). Then the linear subset D - DB Conversely, let (8) be satisfied. satisfies all the mentioned properties. Since D Proof. Assume first that (b) is satisfied. Let Iml < -wo. is invariant under R1(A), we have R1(A)x E Do for z E D, so that (C - 1) x x BR1(A)x - B(A - 1)R1(A)x - Bx, x e D, or

BR1(A)x - R1(C)Bx, x ED, Iml < -wo.

(9)

Cut]

Since D X. u > wo; x E D and t > 0 being fixed. Set xu is invariant under any power of R1(A) as well, we obtain by successive application of (9) that

yu ° Bxu - (-iuR-iu(C))[ut]Bx.

By Theorem 1 (f) of Section 2:A, y + exp(iAt)x and y + exp(iCt)Bx as Since B is closed, exp(iAt)x E DB and B exp(iAt)x - exp(iCt)Bx. u + m. Since D is Thus (8) is proved for x E D. Now let x E DB be arbitrary. a core'of B, there exists a sequence xn E D B-convergent to x (i.e., xn + Set yn - exp(iAt)xn. Then yq + exp(iAt)x and By, + x Bx). Thus (8) is proved, since B is B exp(iAt)xn - exp(iCt)Bxn + exp(iCt)Bx. a closed operator.

Assume now that (a) is satisfied. It is sufficient to prove (8) for We note first that Dq is x E DV and then to repeat the above argument. invariant under exp(iAt). Indeed, let x E Do. Then exp(iAt)x e D (since D is invariant) and A exp(iAt)x - exp(iAt)Ax E D, since Ax E D, so that exp(iAt)x E Do. To prove (8) for x E Do consider the expression CBU(t + e)x -BU(t)x)/E, where U(t) - exp(iAt). We have

1 CBU(t+E)x-BU(t)x] -

iBf1U(t+AE)Axdl. o

Ax G D, therefore BU(t + Xc)Ax is a continuous function of X. Since B is a closed operator, we may apply B under the integration sign, obtaining

f CBU(t+E)x-BU(t)x] -

if1BU(t+AE)Axdl. O

Using the continuity of the integrand again, we conclude that

e CBU(t + c)x - BU(t)x)

E O

iBU(t)Ax - iBAU(t)x - iCBU(t)x, x E Do

Thus we (the last equality is valid due to invariance of Do under U(t)). have proved that for x E Do, y(t) - BU(t)x is differentiable and satisfies the Cauchy problem

i dy(t)

+ C y (t) - 0 , y (0) - Bx .

(10)

Since C is a generator of a strongly continuous semigroup, the solution of (10) is defined uniquely (Theorem 1 (d) of Section 2:A), so y(t) - exp(iCt)x x Bx, and (8) is proved for x E D0. Assume now that (8) is satisfied and set D - Dg. Then DB is invariant under exp(iAt) by assumption, B exp(iAt)x is obviously continuous for x E DB since it equals exp(iCt)Bx. Next we prove (b).

We have

R1(A)x - if a lltexp(iAt)xdt, Iml < -wo. 0 111

For x E DB, the integral

if Be-lItexn(iAt)xdt = if e-rxtexp(iCt)Bxdt = R1(C)Bx 0 0 converges absolutely and has a continuous integrand; since B is closed we conclude that R1(A)x E DB, so DB is R(A)-invariant. Moreover, (9) is We mention that Do = (x i x = RX(A)y for some y E DB) valid for x E DB. -y + Xx E DB, s(tc E Do. Indeed, if x = R1(A)y, then Ax = (A - X)x + Xx Now we RX(A)y, where y = Ax - Ax E DB. Conversely, if x E Do, then x are able to prove that BAx = (Bx far x E D0. Let x = RA(A)y, y c- DB. Then by (9)

BAx = B(A-),)x+XBx = -By +\BR1(A)y = By +AR((C)By = (.?RX;C) - I) By = CR,,(C)By = CBR1(A)y - C3x,

For x E DB, set It remains to prove that Do is a core of B. = -iuR- (A)x E D0. We claim that X. is B-convergent to x as v + m. Indeed, by heorem 1 (f) of Section 2:A

as required. xFi

(-iuR

(A))

Cut]

x

exp(iAt)x, u

t E CO.T] for any T -

uniformly with respect we obtain

-iuR-iu(A)x - exp(

u.+

x.

.

Thus,

0,

A)x

0,

i exp(u A) x - x so that -ibR_iu(A)x - xu

Set t = 1/

On the other hand

Bxu = -ilBR- iu(A)x = -iuR-iu(C)Bx, yp

and the same argument shows that yu u+ Bx. Thus Do is a core of B and The reformulation of the theorem for the case of the theorem is proved. strongly continuous oups is obvious We employ Theorem 1 to prove the modification of the Krein-Shikhvatov theorem, concerned with strongly continuous representations of Lie groups in Banach spaces. First of ail we recall some notions and facts from the We omit the proofs since theory of Lie groups and their representations. they may be found in standard textbooks. Let r be a n-dimensional Lie algebra over tR with the basis a1,...,an, so that knI

[ai,a.I =

E

aijak. i,j

-

1,...,n,

(11.)

=

k

where X . are the structure constants of r (with respect to the basis al,

...a). Let G be a Lie group with Lie algebra F. We shall use the canonical coordinates of the second genus in the neighborhood of the neutral element e E G: the coordinate tuple x - (x1,..., ) lying in the neighborhood of. zero in &(n corresponds to the element g(x) E G equal to g(x) = gn(xn)g rl(xn-1) ... g1(xl),

(12)

where gi(t) is the one-parametric subgroup of G, corresponding to ai E r (i.e., gi (0) = c, gilt + r) = gi(t)gi(r), and ai is a tangent vector of the

112

curve gi(t) for t = 0). The composition law in the coordinate system (xl,...,xn) has the form

g(x)g(y) = g(r:(x.S)), where q' is a smooth mapping of the neighborhood of thcori,!:,: :n IR" < k" into IRn, y,(y,0) _ ,y(O,y) x

(x,y).

y.

is easy to calculate the derivative (1b/ox) x

It

Since our consideration is local, wemay assume, by the Ado theorem,

that f is realized as a matrix Lie algebra, and the vicinity of e in C as tha in a matrix Lie group. Thereafter g(xl has the form g(x) _

(14)

where exa is the usual matrix exponent. We calculate now the derivative (2 /2y)g(y). Later we substitute x and 41 _ >y(x,y) into the obtained expression. We have 3

ylal

ei ]a

ctPnanc4'n-1an-l.

g(Y) =

a

(15)

Using the fact that for given -lemenus a,b of a matrix Lie algebra e

tb

ae

-ti

= e

tad,.

(a),

where adb is an operator of commutation with b in this algebr = 1b,a], we obtain y,nadan aW

K(','')

(16) :

adb =

*jada.

= lie

(17)

To simplify the expression in square brackets we note that in the basis (ai,. an) the operator adas has the matrix As with the elements (As)pq = _ Xsq and consequently the operator

is represented

by the matrix d

Thus ,s

g($_) =

F.

p=1

(18)

n pL

Bpi (V%)ap)gM'

where B(yh) = B(yl,...,yn) is the matrix with the elements Bpq($') =

(19)

In particular, B(O) = I (identity matrix), su that the inverse matrix

C(,i) = B-' (,y) is defined when y is close to zero.

Calculating Eli., derivative with respect to x on both sides of (13), we obtain n

ax. g(>y(x,y))

37.. t

dX

P,.]=1

ia

t

g(J+(x,y)) = dz. (g(x)g(y)) = t

(,Y)apg(>y(x,y)),

(20)

n L Bpt()aPg(x)g(Y) _

(21)

P n

E Bpi(x)apg(l(x,y)) p=1

The matrices ap, p - 1,...,n are linearly independent and so are a g(i (x,y )) since g(ay) is invertible. Thus the comparison of (20) and (21) yiRlds n

a

E Bpj(y) a.

Bpl(x)

(22)

j1 from where

a

n

a (x,Y) i

E Cjk(I(x.Y))Bki(x) k=1

or simply 2x - C(ip)B(x) = B 1(i)B(x).

(23)

Recall the definition of the repre-

sentation of G in a Banach space X. Definition 1. A (strongly continuous) representation of the Lie group G in the Banach space X is a function g + T(g) on G. whose values are bounded linear operators in X, such that T is continuous in the strong sense, T(0) - I and T(g1)T(g2) - T(9192) for any g1,g2 E G. In particular, for any element a e r of the corresponding Lie algebra, the family Ta(t) - T(ga(t)), t E where ga(t) is a one-parametric subgroup of G, corresponding to a, is a strongly continuous group of bounded linear operators in X (Definition 1 of Section 2:A) and therefore has the form T (t) - exp(itA), where A is its generator. The operator A - A(a) will be called the generator of representation T, corresponding to a e E.

The theorem given below enables us to construct representations of Lie groups starting from representations of their Lie algebras. Theorem 2. Let r be a Lie algebra, given by relations (11), and G be Assume that Al1...,An a corresponding connected simply connected Lie group. are the generators of strongly continuous groups of bounded linear operators in the Banach space X; D C X is a dense linear subset such that

(a) D C DA1f ... n DA and is invariant under operators Aj, their resolvents RA(Aj), and groups ell, j - 1,...,n. (b) The operators iAj form the representation of r in the space D, i.e., n CAj,Ak]h - -i E AskAah, h E D, j,k - 1,...,n. (24) s-1

Then there exists a representation T of the group G in X, such that Aj are its generators, Aj - A(aj). Proof. Since G is connected and simply connected, it is enough to construct T(g) for g lying in the neighborhood of unity, namely in the neighborhood which is covered by canonical system of coordinates. For such g, we set

T(g(x)) a T(x) -

(25)

T(x) is a bounded strongly continuous function, and besides Tai(t) - exp x X (iAit) is a strongly continuous group with the generator Ai, so all we need is to prove that the operators T(x) satisfy the group law in the vicinity of zero, i.e.,

T(x)T(y) -

(26)

for x,y small enough. It is sufficient to prove that (26) holds on the dense subset D C X. Note that D is a core of each Ai by Lemma 2 of Section 2:A. 114

Let h E D, we have

Lemma 1.

n e

iAstAkh =

E lexp(tAs)]pkApelAst h, s,k = 1,...,n.

(27)

p=1 X ID

. m X. The Consider the Banach space Y = X x do = X O n summands operators in Y may be represented as matrices with operators in X as their X Y which is the closure* from We introduce the operator B elements. Y Y, which D of the operator h * (A1h,...,Anh) E Y, and the operator C. has the form (As denotes the transpose of As):

Proof.

:

:

Cs - As 0 1 + iI ® As As

ASll... XnlI

0

............

+i 0

(28)

DCs - D A S

... s)DAs.

1RI ... An i

As

C. is a generator of the strongly continuous group exp(iCst) in the space Y. Indeed, direct computation shows that we sho,ild set

exp(iCst) = exp(iAst) ® exp(-As't). Further, we have by (24), respect to lower indices,

using

(29)

also the antisymmetry of 1jk with

BASh = (AIAsh,...,AnAsh) _ (AsA1h,...,AsAnh) (30) - i (

AisAfh,..., E ansAkh) _ (As ® I)Bh + i ( I x A')Bh = C P-1 B=1 E

h E D.

...e D is invariant under resolvent Rx(A5), we may apply Theorem 1 and obtain that B exp(iAst)h - exp(iCst)Bh, h E DB D D.

(31)

The latter identity may be written in the form n

Apexp(iAst)h - exp(iAst)

E lexp(-Ast)]pkAkh k=1

(32)

n

exp(iAst)

E lexp(-Ast)3kpAkh. k=1

(32) is equivalent to (27) since exp(-Ast) is the inverse matrix to exp(A.st). The lemma is thereby proved. Lemma 2.

For any h E D, T(A)h is differentiable with respect to

AEOr -and T(A)h Proof.

Set h(X) = T(X)h.

B is closurable since hn

E Bpi (1)APT(A)h.

(33)

P=1

]

For any c

(Ell ...,tn) E htn, we have

0, Bhn ' h = (hl,...,hn) implies Aihn

i . 1,...,n, so that hi a 0, since Ai are closed operators.

115

n

h(X+e)-h(A)

E

j-1 +tn)

- h(AI,.... A3,A3+1+

n iAn(An+ i E eje j-1

En)

... x

x eiAj+1(Aj+1 + ej+l)jldTeiAj(aj + 0

n

when It 9

1A1A1h

ie

E1 j e j-

+0( be

N ),

- (ti + ... + en)1/2 + 0; we have used the strong continuity of and the invariance of D under exp(iAit).

Thus

h(A) is differentiabl2' Ind -i

(34)

h(X) -

a.

Successive application of Lemma 1 yields

-i as h(A) -

E

P.1

J

The lemma is proved.

For h E D, set now

h1(x.y) - T(x)T(y)h, h2(x,y) - T(ijr(x,y))h.

We have hl(0,y) - h2(0,y) - T(y)h.

(x,y) - 1 Bp.(x)Aphl(x,Y),

h

n

ah ]

(36)

p-1

J

-i ax2 (x,Y) -

By Lemma 2, n

2hI -i

(35)

ay (x,y)

E

ax

s-1

]

n E Bpe(W(x,Y))Aph2(x,Y) p-1

(37)

n

- pI Bp.(xAph2(x,y), since (22) is valid.

We note that

Bpi(xi,...,xj,0,...,0) - Cexp(x.A.)]pi - dpi, since the j-th column of Aj.consists of zeros

(38)

0), so we have

ahi -i a` (x1,...,xj.0,...,O,Y) - Aihi(xl.....,xj,0,...,O,Y), J

(39)

i - 1,2, j - 1,...,n.

Now we may prove that hI(xl,...,xj,0,...,0,y) - h2(xl,...,xj,0,...,0,y)) j - 0,...,n by induction on j. It is valid for j - 0 and, if it is valid for j - Jog we have that for j - jo + 1, both hl and h2 satisfy identical Cauchy data at x - 0 for equation (39). Application of Theorem 1 (d) of Section 1:A yields the desired result. Thus T(x)T(y)h - T(*(x,y))h. The theorem is proved.

116

Scales Generated by the Tuple of Unbounded Operators

C.

Now we proceed to the investigation of more concrete scales of the type arising when the problems concerned with asymptotics are considered. The general results may be essertially improved for such scales, and many more detailed theorems may be stated. In this item we introduce different types of scales associated with a given collection of closed operators in a Banach space.

Let X be a Banach space with the norm I I, E C X be a dense linear Assume also that closed operators Bl,...,Bm in X are given such that E is the core of each of these operators and E is invariant these operators.

subset.

Consider

(a) Let p = (p1, ..,pm) be a m-tuple of positive integers. the following norm defined on E:

IBv

E

Ixlk=lxlp.B,k'

05 < k

xxEE,k=1,2,....(1) s I .. .'Bv

Here the number of elements s in the sequence (vi} of integers, satisfying 1 < v < m, is not fixed: the empty product (s - 0) is by convention the (the identity operator, is defined by - pv + Pv + ... + pv

number of entries of B in the product is counted lwith the "weight P.); the sum is taken over 11 the sequences (vl,...,v8) satisfying then t e pm = 1, we obtain enumerated conditions. In particular, when pl = ... the definition of the norm E

O<s

x I B,(al,...,ai 1,....an)

We mention

+

r

+ I B.x n B.(al,.. ,ai-1,.. and the proof proceeds by induction on

an)

a > 0,

(5)

I aIinstead of k. The theorem is proved.

Let A : E + E be Now we concentrate our attention on the scale H a given linear operator. We are seeking the conditions under which A gives rise to a translator in the scale HBP. These conditions are rather simple, however. Introduce first a convenient terminology. Given a product Bv1Bvz ... Bvs, we call the number - pvl + ... +pv the length of this product.

Similarly, pv1+ ... +pv a is called the length of the commu-

tator K.(A) - [BvI[Bv2,,,[Bvs,A]...)].

We adopt the convention that A

itself is the commutator of the length zero. Theorem 2. Assume that there exists a function 0 71+U{0}+ 71+U{0} (where 71 + is the set of positive integers), such that for any r c- a + 0{0} and any commutator Kv(A) of the length r, the following estimate is valid: :

xEE

(6)

with the constant C depending only'on r - . Then A induces a left translator. in the scale HB,P' More precisely, set

(k) = 118

max

(k+¢(r) - r), k E 7L+U{O}.

0 *(0) set **(k) max(r1*(r) C Q. Then p clearly ** is non-decreasing and A is bounded in the mentioned pair of iki(t*(k)) *(**(k)) A V*(k) spaces since there is a dec exposition 40p HB

to HB*pk).

%19

-

%.P

119

L2 = L1 +11,111, 11, L3 = L2 + 1L1,L27,..., Ls = Ls-1

+[L1,Ls-17,....(10)

We have the non-decreasing sequence of subspaces L1 C L2 C ... C Ls C ... of L.

This sequence becomes stable for some s = so, i.e., Lso = Lso+l = = El, where L1 is the Lie subalgebra in L generated by (A1,...,Am). Proposition 1.

The elements of L1 are left translators in H n

More Precisely, if B =

(A1,...,Am)

E a.A . EL , then s .1 J

j-1

n

Bx 11 k S Ck II x

k+s( j=1

where Ck depends only on k. Proof.

The elements of Ls are sums of products of the length < s of r

the elements Al,...,Am.

Let Cl,...,Cr be a basis of Ls

lar' < const v=1

E

£ BrCr v=1

n

r

and besides

Then B =

fail,

all the norms in a finite dimen-

j=1

sional vector space L. are equivalent. Thus (11) turns out to be an immediate consequence of the norm definition (2).

Assume now that the operators A1....,An satisfy the conditions of Theorem 2 of the preceding item (E plays the role of the subset D mentioned there). In particular, the operators A1....,An (and all their real linear combinations) are generators in X. We should like to find out whether they are also generators in H For this we shall make use of the (Al, ....

identity

).

Ajexp(iBt)x - exp(iBt) Here B =

n

n of E IR,

n £ lexp(-At)] kjAkx, x E E. k=1

(12)

A = fIafAe l is the corresponding matrix of the

f Il afAf,

associated representation of L. The identity (12) was proved in item B (see (32) of item B) for basis elements of aLie algebra. but it is easy t'o see that a posteriori this proof is valid for arbitrary B of the above form.

Theorem 3. Let LB C L .be a minimal invariant subspace of the matrhC* A, containing Ll. If LB C L, B is a generator in the scale H

(Al-

(in particular, Aj itself is a generator in the scale H

Am)

(Al,...,Am)

If LB C Ls, then the estimates

for j

exp(iBt)x Nk 4 CkPB(t)(Pt(t))k flx IIsk, x C H(A1,...,Am)

(13)

are valid, where PA(t) is the norm of exp(-At), PB(t) is the norm of the operator exp(iBt) in the space X. Proof. We proceed by induction on k. Assume that the statement is proved for k = ko. Set k = ko + 1. It is enough to prove the estimate (13) for x E E

* A,,... An is the basis of L and the action of an (nx n)-matrix A in L is defined as the action on the coordinates with respect to this basis. Thus, AAj 4 EAkjl

fEkafI jAk

matrix of the operator'iadB.

120

iEfatCAt,A.I = i(B,Aj1, i.e., A is the

m

U exp(jBt)x k - IIexp(iBt)x Uko+ E A'exp('Bt)x i)ko

j-1

l exp(iBt)x II k+ E Q exp(iBt) E texp(-At)] k Akx j o j-1 k-i

4 Ck PB(t)(PA(t))o( k IIx1Ik S+ E o

0

j-1

ko

5

(14)

E Cexp(-At)3 k3Akx11 k s), xC- E,

II

0

k-1 n

by the induction assumption. We claim that E [exp(-At)] kJAk E LB for any k-1 Indeed, the latter sum is the result of action of the j E We have operator exp(-At) on the element Aj E L1.

exp(-At)Aj -

E 1/r:(-t)tArAj

r-0

and for any r, ArA E LB since LB ED Ai and LB is invariant under A. Applying Proposition 1 to the right-hand side oP (14), we come to the estimate

Q exp(iBt)x I k< Ck PB(t) (PA(t))ko( x k s+ const p x lI k e+e) 0 0 0 x

lx II

Z )texp(-At)] k.j 4 Z j-1 k-1

x

(k +1)s 0

The theorem is proved.

We consider now the case of nilpotent Lie algebras. Assume that the algebra L is nilpotent. We assume that the basis A1,...,An of L is chosen in such a way that ajk - 0-for s 4 k. In other words, all the matrices Aj of associated representation of L are strictly lower-triangular matrices. A An important example is given by the so-called stratified Lie algebras. N

Lie algebra L is a stratified Lie algebra if L - rC±1Lr (the direct sum of linear subspaces) and

ILr,Ls] C

Lr+s (Lr+s d=f {0),

if r +s >N)

(15)

and besides L1 generates the whole algebra.

If we choose the basis Al,...,Am in Ll and extend it to the basis A1,

r(j)

, where r(j) is a non.,An of the whole algebra, such that Aj E L decreasing function, this basis will satisfy the condition formulated above. We mention that in this case the spaces defined by (10) have the form Lr -

(±

Ls.

(16)

s k' and this embedding is cptri.

tinuous, the above argument yields that for any (r,,e) E A, we may find f - frmc e H with the properties:

fMce Hp

(a)

f

(b)

j

for all intege4s k such that 0< k< m;

c

k

HE

Tfrmell

(c)

s(r)

> 1 or TfrmE

Hr(r)

The generalized sequence {frmr}() E A converges to zero.

Indeed, for

any k e I ko,where ko E a + is any fixed element greater than k, C is the norm of embedding operator Hfo C On the other hand, (c) means that is not convergent to zero so T is not continuous. The obtained Tf t

contradiction proves the lemma since the inclusion of the right-hand side of (3) into LH is evident. There is a natural convergence in LH: the'gener-

alized sequence {Tm} C LH converges to zero if Vt 3r Vs 3k so that Tm -'0 k s in L(Hp - Hr). Since the space H is not covered by theorems of Section 3, we need to construct the functional calculus in H independently. Definition 2. If in the algebra LH we are given a one-parameter multi ltcativegrogrup {et,t E IR} which is differentiable and rows slowly

as It + m, that is, V1 3r Vs 3k 3p > 0 Vt E Iv(p)dp

are the 1/h-Fourier transform and its inverse, respectively. proved by standard calculations using formulae (6) and (7).

Lemma 3 is

Consider the function e (x) expi(i/h)xp}, which for each p E Iitn Let T be Rn arbitrary operator in LHORn). We associate with ita functionof the variables p,x E gn by means of the formula belongs to H(Qtn).

Smb{T}

def

e_p(x)(Tep)(x).

(19)

129

Definition S. The function defined by (19) is called the symbol of the operator T. We note that if 2

T - f (x,-ih .) is a 1/h pseudo-differential operator, then Smb(T) - f (it follows from Lemma 3). It is natural to assume that an analogous assertion holds in the general case: 1

2

T - Smb{T}(x,-ih ax).

(20)

(20) is valid in the case Smb{T} E H(kx x kp)

Indeed, the operator

T - Smb{T)(x,-ih aX) annuls the function ep(x) for any p

kn, and the set

of-linear combinations of functions ep(x) is dense in H(Itn). But Smb{T} We will describe belongs, in getteral, to a broader algebra of functions. it next. Let H9;1'=2 (k° x kn) denote the completion of C-(k, x kp) with respect ,,r2 p to the norm:

If

181,82 R

- (ff(I+Ip12)-r2(l+IxI2)

rlI(1-Ap)62/2(1-AX)81/2

x

rl,r2

(21)

x f(x,p)I2dpdx)1/2.

We set

x kp def

(}

u fl 'U Hr1,r20Etx x ttp).

s2 rl al y2

(22)

2

1

In this algebra of functions we introduce a natural honvergence by analogy with the above constructions. Clearly HORx x kp) C Hn x kp), where the embedding is continuous.

First suppose that T6 LB(kn). It is easy to show that the function Smb(T), defined by (19), belongs to the algebra HII x kp). It turns out that each element in H(k. x kp) can be regarded as the symbol of some 1/h pseudo-differential operator. Theorem 2.

The mapping 2

f - f (x,-ih a;) can be extended to a linear homeomorphism

H(kx x kp) ; LH(Rtn)

(23)

of the algebra of symbols onto the algebra of operators. mapping is given by the formula v-1(T) - Smb{T}, Proof.

Since for f E L2

The inverse

x t) the operator 2

f a f(x,-ih aX)

is a Hilbert-Schmidt operator in L2(kn), we have

4 [Tr(f*f)]1/2 -

Af@ L2(kn)+ L2(kn)

130

I (2xh)°

2

If I

(24) L2(kX x kp)

As a consequence of this estimate in the case of a scale {Hk) we have the inequality fl f r+k(IR n)

Hr,kn)< Cr,t,s,k I f I

_

Hr.fn x fin) s,k

+s

x

(25)

p

which holds for all f E eck. In HORn x tt').we consider the convergence which is induced by H(R

X S) is a dense subset of H(tsx x p), and (25) yields that

H(l

x

Since LHGtn) H + LH(ttn) (defined on this dense subset) is continuous. is a complete space (the verification of this fact is any easy exercise), the mapping u can be extended to a continuous linear mapping betweetl these spaces. y

We will show that N is a homeomorphism; that is, the inverse mapping Smb : LH + H is continuous. To do this we establish the following estimate on the symbol of an operator T in the algebra LH(kn) : yf E 22 + y-i

ZoGIR YrE 2a+ 3k GIt: Smb(T}

Hr,f

s+f,r-k

nx x

Rn)G Cr j,s,k

k+noRn)+Hr(IR n).

Tp

r+n

p

(26)

s

Here 271 + is the set of non-negative even integers. This estimate is equivalent to the assertion of continuity of the mapping Smb : LH H.

We now prove (26).

Since T E LH, we know that

Vf E 2 7 1 + 3s E IR Yr E 2 7 1 + 3k E k t

:

T I Hk+n

r+n

. Hr < s

We set Smb(T} - f; then 2

T - f(x,-ih t7,) - f.

From (24) we find that

f

- (2irh)nTr(f f s+f, r-k

fl Hr

where f1(x,p) - (1+ Ixl2)-(r+s)/2(l +ipl2)(k-r)/2(1 - nx)r/2(1 -4 p

L/2 )

f(x,p).

Consequently

f

rf

Hs+f, r-k

6C(2) k,a

fl o (1 - )n/2 o (1 + IxI2)n/2 B L2

~12

In the scale {Hf) we have to estimate the operator fl in terms of the norm This is easy to do by using the formula which connects the symbol fl to the symbol f (we choose r,f E 2a + so that f1 is expressed in terms of the derivatives of f of integer order). As a result we obtain of f.

bf

r,f

Hs+f,r-k

GC1,T,k,a f

k+n

Hf+n- H.r

The theorem is proved.

131

By virtue of theorem 2 the linear structure and the structure of the convergence in H(Itn x) and in LH(Itn) are isomorphic. But the algebraic By using the homeomorphism u we structures of these spaces are distinct. can transfer from LH to H the non-commutative multiplication *, with respect to which p is an isomorphism of algebras. Definition 6.

The product 1

1

(27)

Siah(Cf (x,-ih aX)] o Cg(x,-ih 2x)3}

f*g

is called the twisted product of the functions f and g.

n) of continuous mappings from H which associLH We introduce the mapping v

We consider the algebra LHn x H(In x Rp) into itself.

:

d-f Q(1), where ates with each continuous operator Q in H the function v(Q) 1 is the identity in H. Thus the mapping v applies the operator to the

idrtity. .)ie note that, analogously to (3),

. LH(dt°x ttp) - n u n u n u n u

s2 k2 fI r1 81 k1 12 r2

(28) 1

2

1

2

Therefore we can define a homeomorphism

j

:

LH(1ln) -+ LH(1Etn x Itp)

(29)

by means of the formula j(T) - Ce-(i/h)xp1T.1e(i/h)xp],

(30)

where T. is the operator T acting with respect to the variable x in H(Itxn x x Itn), and Ceixp/h7 is the operator of multiplication by the exponential.

Frog our definitions, we obtain the following assertion. Lemma 4. tative:

(a) The following diagram of an algebra homeomorphism is commu-

PI

LH(Itn)

P

i

LH(It& x knp)

Here H is regarded as an algebra with the twisted product *. (b) We have the formulas 2

J1(f) - f(x,p - ih ax), (f * g) (x,P) - (ji(f)Ig(x,p).

(31)

These fotmulae give us a method of calculating the symbol of a composition The proof is obvious. of operators in the algebra LH(hn). Some Estimates for Functions of a Tuple of Non-Commuting Self-Adjoint Operators F.

In this item we consider estimates for functions of a n-tuple A1,...$An of self-adjoint operators in a Hilbert space for which a regular represent n tation exists. As it was shown in this chapter, an operator f(A1,...,An) is bounded without any additional assumptions, if f E B0(Itn). However, in applications this requirement is often too r?strictive; in particular, a

132

smooth function f(ylhomogeneous of degree zero for large lyl, does not necessarily belong to

0

It is well known that pseudo-differential operators with symbols from Sm

P.6

(ltn x I ) are bounded in the space LZ(Itn) for m - 0 (see, for example, n

1

The boundedness of f(A1,...,An) for symbols of a functional class 130,45]). wider than B0(IRn) follows in this and many other cases from: (a) (b)

the existence of non-trivial commutation relations; the self-adjointnesa of operators A1,... ,An.

We assume throughout this item that the left regular representation In terms of this representation we formulate a sufficient conexists.

*n-k .,,k)

are dition, under which the operators with symbols from Sm P.0 bounded in the scale of spaces, generated by opera rs (A1,...,Ak).

Being combined with the methods of 152] and 139], the results of this 1 n item yield the proof of boundedness of operators f(A1,...,An) with oscillating symbols of the form f - exp(iS)4. However, we do not dwell on these questions. We establish some estimates for functions of tuples of Feynman-ordered non-commuting self-adjoint operators in a Hilbert space. Further, by saying that self-adjoint operators A and B commute, we mean in fact that their spectral families commute:

(1)

EX (A)EU (B) - E11 (B)Ex (A) - 0, X,i E IR.

Our basic aim is to establish the estimates for functions of tuples, for However, we first prove some prewhich a regular representation exists. liminary results. Let H be a Hilbert space and A1,...,An be (unbounded) self-adjoint operators in H. From now on we assume that the intersection DAlfl ... f1DAn of domains of operators A1.... ,An contains a linear subset D, which is dense in H and invariant under operators Aj and corresponding unitary groups exp(itAj), j - 1,...,n; further, we assume that for any sequence of indices ils...'Jm c- {1,...,n} the product of groups exp(itmAjm x...xexp(itlAjl) is infinitely differentiable with respect to t c- Qtm in the strong sense on the domain D, and all its derivatives grow no faster than the powers of t:

'ata (exp(itmAjm)x...xexp(it1Aj1)u) 1< C(1+ 10)

where N may depend on I a , and C depends on ja I

Under these conditions the formula 1

N,

u E D,

(2)

and on the choice of u E D.

n

f(A) = f(A1,...,An) . +

(3)

where f(t) -

(25)-nff(E)e-itEdE.

t.E E ttn.

(4)

is the Fourier transform of the function f, QA = QA(t) - exp(itnAn)x...xexp(it1A1),

(5)

133

and the brackets < ,> denote the pairing of a distribution with a test

function in the weak topology on D , defines for any f G S (Gan) a linear operator f(A) on a dense linear subset D. D D, invariant under all operators The operator (3) is correctly defined and bounded (for any self-adjoint operators A1,..., An) if f c- B (bin), i.e., f belongs to the space of Fourier transforms of the limits of finite functionals over the space C(Rn) of continuous functions bounded in IRn with the norm if - sup f(y) (see Section 3 of this Chapter). f(A), f E S (btn).

C

yElRn

Under these assumptions operator (3) can be rewritten as a multiple Stiltjes integral 1 n f(Al,...,An) f(al,...,an)dEan(An)x...xdEal(A1), (6) where dEl(Ai) is the spectral measure of operatorAi; the integral is considered in the sense of weak convergence in H. Formula (6) makes sense for a wider class of symbols than formula (3); however, one should remember that for f 0 S (btn), operator (6) need not necessarily be densely defined. Theorem 1. Let operators Ak+l.....An be pairwise commutative, and the function f(yl,...,yn) satisfy the estimates

a1+...+ak IY21)-k -E

f(yl,...,yn)l 6 C(1+ lyll + ayal 1

...

(7)

3yak k

for some e > 0 and all a - nal,...,ak) with lal - al+... + ak

k + 1.-

Then the operator f(A1,...,An) is bounded in H (and hence can be extended by continuity on the whole space H). We divide the proof into several lemmas: Lemma 1. Let self-adjoint operators B11,...,Bm be pairwise commutative in a Hilbert space H, and let the sequence gn(x) of continuous bounded functions in x - (xl,...,xm) E Gtm be uniformly bounded (i.e., Ign(x)l 4 M for all n,x) and converge to the function g(x) as locally uniformly with respect to x E btm. Then the sequence of operators gn(B1,...,Bm) converges to the operator g(B1,...,Bn) as n -+ - in the strong sense. Proof. First of all since spectral measures dEll(B1),...,dEAm(Bm) commute, we have the estimate

lgn(B1,...,Bm)

lg(x)l 4M;

6 sup

(8)

xEbt

hence it is sufficient to verify the convergence on some dense subset of H. We choose this subset to consist of the following vectors u - EA1(B1)EA2(B2) ... EAm(Bm)v, v e H, where A .., A. are compact Borel subsets of the real axis. form (9), then Ign(Bl,.... Bm)u - g(Bi,...,Bm)u A < sup

XEA1x...xAm and, consequently, gn(B1,...,Bm)u + g(B1,...,Bm)u.

134

(9)

If u has the

Ign(x) - g(x)l

(10)

The lemma is proved.

Lemma 2. Under conditions of Theorem 1 the Fourier transform Ov(tl, tk) of the H-valued function v E H,

mv(Yl,...,yk) is a continuous function and

f

k

ll dtl...dtk 6 C Il v

II

(12)

,

where the constant C is independent of v. Proof. Since f(yl,...,yn) is continuous and bounded with respect to is a all the variables, Lemma 1 yields by estimate (7) that continuous summable function; hence the Fourier transform exists:

mv(tl,...,tk)

-k fOv(Y1,...,yk)e-itydyl...dyk,

and it is a bounded continuous H-valued function. tomv(tl,...Itk) _

(13)

(2n)

Further,

(2,r)-kfl(-i ay)aOv(Y1,...,yk)le ltydyl...dyk

(14)

for any multi-index a with lal K, k + 1 (the differentiability of mv(y.,..., Y ) follows from Lemma 1), hence the norm of t(;v is bounded for lal S k + 1 and can be estimated by const Ilv ll. Now we have

cl + tz)

II

H 0 the following inequalities hold:

136

1 max(-1-m],0). S are bounded in H.

(here [-m] is the integer part of the numbe In particular, the operators with symbols from

P

.Proof of Theorem 2.

First we prove the following:

Pro osition 1. Let fi E mi, i - 1,2; (Trl,...Inn) be an interchange of numbers 1,...,n). Then g

def -

rn

fl(L1,...ILn)(f2(y1,...,Yn)) E Sol + m2

(31)

and

g(y) - f1(Y)f2(y)j_- Sp1+m2 - p

(32)

Item (a) of the theorem is a special case of Proposition 1. Proof of Proposition 1. we have by Condition (p)

Obviously, (32) implies (31); to prove (32), 1

2

g(y) - H(Y,-i t: )f2 (y) - H(Y,0)f2(Y) -

- if0dt{ } 1

fl(Y)f2(Y) - if ldt() 0

it fl(Y)f2(Y) - iG (y) +41(y),

.117

denote where *(y) E Sml fi762 -P by virtue of (23); the brackets summing with respect to the coordinate number from 1 to n. We estimate Set ej + 0, j E {1,...,k}, ej the derivatives of the function G(y). j E {k+l,...,n}.

By (24) we obtain

ala+sl I

a'

Y a&

" HCj(Y'T01 < CMY,E))

0

ml -P(I0, I + Is"I

(34)

J

uniformly in T E 10,1] (here and subsequently the letter C denotes different constants). Using (25) and (26) and the inequality 1, which follows from (25), we can obtain the following inequality from (34)

I

aI°+eI ayaa4s

H&j(Y,TO)l < (35)

C(1+lY,I)m1-pej-Pla'I(1+I6I)Nolml-Pej-Pla'II

4 with the constant C independent of x E 10,1]. We set (fl FaN.j(Y,E)

-

a a

lal

j HE.(Y,Tf)dt](1 + E2)-N.

(36)

o ay

For N large enough and all a < ao (where ao is any fixed multi-index) and 1,2,..., we have IBI

t)Xj, k - 0,1,...,N, (22)

as

where Pk-j+l are some new differential operators, and the sum on the right is meant to be zero for k - 0. The equation (22) for Xk is called the transport equation and is an ordinary differential equation along the biWe may successively resolve the system (22) by means of characteristics. elementary integration: -f tF(q(go,z),i)dt o

Xk(go.t) - e

k-1_

t -1TF(q(go.T'),T')dT' ok(go) - S e o

(23)

2

I Pk-j+l(qo,-i aqo ,T)xj(go,T)dT, k

X

X

0,...,N.

j

Now make some conclusions. First of all, the WKB-approximation in the considered situation enables us to obtain for arbitrary N the "approximate solution" of the equation (4), which satisfies the Cauchy data (15) and satisfies the equation up to the remainder rN(x,t,h) (14) which may be estimated in the following way: 1(-ita )a(-ih

Cas(K)hN+2, (24)

lal +a - 0,1,2,...,

(x,t) E K

for any compact subset K C Uo. We write rN(x,t,h) - 0(0+2) in Uo for the function rN(x,t,h) satisfying the estimates (24),* and call any function I)(x,t,h) satisfying (15) and such that

-ih at + H + 0(hN) in the domain U C tn+l,

(25)

the asymptotic solution modulo hN in U of the problem (4), (15). And secondly, the WKB-approximation in the presented form does not provide the asymptotic solution for all (x,t) even if the solution of the Hamiltonian system (16) exists for all t. Indeed, the simple example (one-dimensional harmonic oscillator - H(q,p,h) - 2 (q2 + p2)) shows that the solutions of system (10) - (12) have singularities in the points of the caustic (i.e., in the points where the Jacobian det(aq/aqo) vanishes). On the other hand, it is known that in the same example the exact solution with finite Cauchy data exists for all (x,t) and has no singularities. Thus we should like to improve the m thod in order to obtain global asymptotic solutions, provided that the Hamiltonian flow is globally defined. (We note, however,

* The definition of 0(hN) will be refined in the subsequent item.

144

that additional conditions should be imposed if non-finite Cauchy data are to be considered.) Such improvement becomes possible as soon as we broaden the class of Let I c {1....,n} z [n] be some subset I = (1,...,n} \ I; we consider the WKB-approximation in mixed coordinate-momenta representation: functions within the asymptotic solution constructed.

4_,t)

(i/h)S (x 1

x-le

W(x,t,h) 1

I'

I

(26)

OI(xl,4_,t,h)7: I

.I

N

here SI(xl,&_,t) is a real-valued function, OI(xl,&_,t,h) I

x (xi,{I,t).

k

E (-ih) ON x

I k=0 The representation of the asymptotic solution in the form

Substituting (26) into (7) is the particular case of (26), when I = 0. (4), we obtain the system of equations (see theorem on commutation in item B; however, the result is not unexpected since the 1/h-Fourier transformation transfers differentiation into multiplication by coordinates and vice versa): aSI a`t

asI

asI

ac + Hopl(xI,- a&- , aXT

I

a. Ik

asI I

I

aX1

aOlk

axI k-1

FI(xI,&I,

+

ax

I

(27)

0,

.

a

2OIk

asI

aSI + Ho(XI, -

as1 a - +

- Hox- (x1,

I

2

(28)

I 2

a ax

t)OIk ` - E Pk_j+1xl+E j=0

I

a I

I

Olj,k = 0,...,N,

analogous to (10) - (12). If the support of Ol lies in the set (1E-1 < R) I and the system (27) - (28) is satisfied, we obtain i

-ih at + H(x,-i aX ,h)W - r- (x,t,h) N+2

(i/h)SI(x1,E_,t)

1/h

x_(e I

I

(29)

N+2- (171/2)

gI(xl,s-,t,h)} = O(h

)

i

(the last equality may be established by direct estimate).

The fact that

the accuracy is less than in (14) (the factor h-111/2) is not essential since N may be 2hosen as large as we desire. (Note also that this factor disappears in L2-estimates.) Solving (27) by virtue of the method of b,icharacteristics, we come exactly to Hamiltonian system (16), while the equation (28) for OIk is reduced to an ordinary differential equation along the bicharacteristics. It turns out that the global asymptotic solution of the problem (4) and (15) may be written as the linear combination of functions of the form (26) with every possible I; all these functions correspond to the same family of bicharacteristics, defined by (16) and(17), and are combined together by means of partition of unity. To prove the formulated assertion, one should show that, primarily, the solutions of the form (26) may serve as continuations of each other and, secondly, that for any point lying on the bicharacteristic of our family such as I C {l,...,n} may be found so that theasymptotic solution of the form (26) may be defined in the neighborhood of this point. The latter property is guaranteed by a lemma on local coordinates (see next item). As for the former one, we give here the proof of it

145

Before doing this, we present a theorem on asymptotics in the simplest case. of integrals with rapidly oscillating integrands.

Let I(x,w), x E Gtn, w E 0 be a smooth real-valued function. l

Definition 1. Th: point (xpp,w,,) E n x IRm is a stationary point of 0 (with respect to the variables x), if am

as

(xo,WO) = 0.

(30)

The stationary point (xo,wo) is non-degenerate if the determinant of the matrix

T

a2m a2m .... ax1axn ax1ax!

a20

(X,W) =

(31)

axe

arm

320

axnax1 " " axnaxn is not equal to zero at (xo,wo).

Note that if (xo,wo) is a non-degenerate stationary point of by the implicit function theorem the equation

ax

then

(x,w) - 0 defines in the

vicinity of (xo,w0) the unique smooth function x = x(w). Definition 1 also makes sense if the dependence of on w is only continuous (the function x(w) is continuous in this case). Consider the integral ei(an/4) It4](w,h) -

where m(x,w), O(x,w) are derivatives with respect ball 1xj < R(k) provided $ is called the phase of

its amplitude.

Theorem I.

r

(2sh) n

/2 J

t

(i/h)O(x w) '

ne

(32)

O(x,w)dx,

functions, continuous in w together (x,w) to x, 1 0, O(x,w) vanishes that w belongs to a compact set K. the integral (32), and the function

with all their outside some The function 0 is called

(Stationary phase method).

(a) Assume that the phase @ has no stationary points on the support Then Itm](w,h) decays faster than any power of h as h + 0 locally uniformly with respect to w. If 0,0 depend on w smoothly, the same is valid for all the derivatives of i10] with respect to w. supp 0.

(b) Assume that there exists a non-degenerate stationary point x x(w) of the phase $ and no.other stationary points of $ lie on the support of $. Then the integral (32) has for any N an asymptotic expansion It01(w,h) - e(i/h)O(x(w),w)(dett- a?2 (x(w),w)])-1/2 ax

x

N-1 z {$(x(w),w) +

(33)

E (-ih)(Vkt(P]P)(x(W)'W)) +0%10'01(W)' k-l

where Vkt@] are independent of h differential operators in x of the G 2k with coefficients dependent on the derivatives up to the order of the phase $ with respect to x taken in the stationary point; the of the square root is fixed by the following choice of the argument a

dett- aX2 146

].

order 2k + 2 branch of

arg detl-

2

a

x(w)M] -a x2

n arg Ak, E k-1

(34)

where -3n/2 < arg lk < n/2, A1,...,an are the eigenvaluei of the matrix

a2 2

(x(w),w), counted with their multiplicities.

The remainder RN

ax

RN[.,4](w) has the estimate: for any compact subset X c

em (35)

1(-ih aX)aRNI < Ck,a

((35) is valid for Jn - 0 if ,0 depend on w continuously and for any a if m depend on w smoothly ).

Consider now the WKB-approximation of the form (26) and show that under certain conditions its asymptotic expansion due to Theorem 1 results in the approximation (7). We have f

p(x,t,h) - e (2,rh)

1I1/2

Ile

tt

(i/h)[t-x- + S (x ,t_,t)] #I(xl,t ,t,h)dt_. I I I I I I

I

(36)

The stationary point of the phase of integral (26) is given by the equation

x_+

BSI aE_

(xl,t_,t) - 0,

(37)

while the matrix of second derivatives of the phase with respect to E_ (x,t) of the a2S equation (37) exists and is unique on supp $I, and that det[ at_aE (xl,t_

coincides with that of SI.

Assume that the solution t- I

x (x,t),t)] f 0.

I

Then by Theorem 1, we obtain (i/h)[S (x ,t_(x,t),t) +x

I

r4(x,t,h) - e

I I

II

I

X

I

I

m(x,t,h) +0(hN) (38)

__

e(i/h)S(x,t)4(x,t,h) +0(0),

where S(x,t) - SI(xl,t -(x,t),t) + x-t-(x,t), I I I

(39)

and the explicit form of the function ¢(x,t,h) may be obtained from (33). The equations (37) and (39) show that the functions S and SI are the Legendre transforms [2] of each other. It follows, in particular, that the equations as,

qi - - 2g_ (q1,p-,t),

aSI P1 - axI (g1,PI,t)

(40)

define the same set of points (p,q) E It2n as the equation

as P -

(q,t).

(41)

In other words, we see, taking into consideration the connection between solutions of Hamilton-Jacobi equations and bicharacteristics, that S and S1 correspond to the same family of solutions of (16). Our considerations in this item have a preliminary character, so we do not give further details here; in particular, we pay no attention to

147

is transformed, although this matter is rather how the amplitude We hope that the interesting and leads to some topological conditions. motivations for construction of global asymptotic solutions have become clearer, and thus we move on to detailed formulations IR2n

B. The Canonical Operator on a Lagrangian Manifold in

The canonical operator is a global version of the WKB-approximations .to solutions of 1/h-PDE's, described in the preceding item. As we have already seen, the WKB-approximations of the form O(x,h) = e(1/h)S(x)O(x,h),

(1)

and

(i/h)S1(x1,E

1/h

V1(x,h) = F

X_ I

{e

01(xl,Ch)},

I

(2)

I

where ¢,01 are polynomials in h; S,S1 are real-valued, define functions

that are coincident (up to a unimodular factor) modulo 0(hN) as h - 0, provided that the systems of equations p -

and

as

(3)

(q)

aSI

PI = aX

IS1

(gl.Pi).

4I

(4I,PI),

-

(4)

aI

I

and, q,p E lRn, define the same submanifold in the space tt2n - ktq x besides the functions Q and 01 are properly related. The submanifold

L CIR2n, given by (3) or (4), is isotropic with respect to the differential form wz =

F dpi A dqj

(5)

j-1

H2n

(i.e., the restriction of w2 on the tangent space of L vanishes identically and, as will be shown below, any isotropic submanifold L of dimension n may be locally represented in the form (4) for a suitable I c [n]. These observations lead to the idea of represen-

in

ting the global asymptotic solutions of 1/h-PDE's in the form of linear combinations of functions, having the form (2) with various I C [n] and corresponding to the same isotropic submanifold L. The canonical operator is the realization of this idea; roughly speaking, the canonical operator acts on functions defined on L and transforms them into functions on t, dependent on the parameter h; for functions on L with support small enough, the result has the form (2). - KO (where K is he canonical It turns out that the condition on 0 is reduced operator on L) to be an asymptotic solution to the l/h-PDE

to geometric conditions on L plus the transport equation form (the reader has

become acquainted briefly with in item A). It turns out also the existence of the canonical L C Rn, namely, some element of

a local variant of these latter conditions that, in general, there is a hindrance to operator on a given isotropic submanifold The requirethe cohomology group H'(L,IR).

ment that this cohomology class be zero is known as the "quantization condition"

and leads, in particular, to asymptotics of eigenvalues for 1/h-PDO's. We come now to exact definitions and formulations. Parameters. In the subsequent exposition we deal with various objects, .depending on parameters, such as smooth functions, mappings of manifolds,

148

Usually these parameters will be vector fields, differential forms, etc. (wl,...,wm) and vary in some simply connected open subset denoted by w 0 C Gtm or, more generally, on a simply connected manifold 12 (possibly with a Cm'0(M W 0 x(2) non-empty border M. If M is some manifold, we denote by C ' E

a,w

a,w

the space of (complex-valued) functions f(a,w;, a E M, w E f2, continuously (Equivalently,.this means that f(a,w) dependent on w in C--topology on M. is infinitely smooth in a and continuous in w with all its derivatives with respect to a.) As a rule, we assume that the components of all our mappings, forms, vector fields, etc. in local coordinates belong to the space Ca:W; all the exceptions to this convention are mentioned explicitly..

The convenient notation will often be used: if the function, mapping, def

etc. f(a,w) depends on the arguments a,w, we write fw(a)

f(a,w).

We present the implicit function theorem for C''0-mappings. a,w Theorem I.

,an,wlI...,wm)

Let fi(a1,

f

i(a,w) E C a ', 0 i = 1,...,

k 5 n, and assume that fl(a,w) _ ... = fk(a,w) = 0 for (a,w)

(6)

(a(0),w(0)) and the Jacobian aft

aft

aaI ... aak D(fl,.... fk) D(al,.... CL k)

= d et

(7)

afk

afk

aa1

aak

is not equal to zero at (a(0),w(0)). Then the system of equations (6) has a unique solution (al(ak+l'" .'an'w),...,ak(ak+l,...,an,w)) in some neigh-

borhood of (a(0),w(0)) and ai(ak+1'

The space Hh.eoc(htn x R).

.. ''an'w) E C (a

0

k+l'

..,a ),w' n

First of all, we define the space to which our

approximate solutions should belong. Assume that a set S2 C S2 x (0,1] of pairs (w,h) is given (this set arises naturally from the quantization conditions, as we shall see soon). For given _h c- (0,1], we shall denote by Dh C S2 the set of w E 12 such that (w,h) E Q.

Definition 1.

Hh,Yoc E° x 0) is the space of functions f(x,w,h),

defined for x E tn, (w,h) E S2 smooth in x and satisfying the condition: for any function 0 E Cx'W(titn x P) with compact support supp expression

"Of II k = sup

_

II

II

(w,h)ES2

(Z Etn x 52,

the

k,h = (8)

sup

J

(Tf)(x,w,h)(1- h2A

(w,h)ES? tgn

x n

is finite for any integer k 3 0 (here Ax =

a2

is a Laplacian).

E

It

1 ax3

is obvious that if x

(!Rn x

IIf

Ilk is finite, then

I!m

fIk

s finite for is

any-T4-

x

w

S 2) .

149

For any real N we write f = O(hN) if f/hN E Hh,toc - Hh,toc .n x a). and any Also we write f -_O(hN) if for any multi-index 8 compact K C tt" x n, the estimate is valid: IBI aaxaf

I

CK.BhN-i8l'

(x,w,h)l
E->w.h)}.

I

Note that for (x,w) E K (a compact subset in IRn x ft) and $ E AUCh], the integration in (39) is in fact over the finite region 1E < R - R(K,$).

Also it is clear that Kef is a linear operator.

* In (39) 1 is a continuous branch of the square root defined by I - pll1/2x X exp(2 arg p1), where the branch arg VI was fixed above.

157

(a) Definition (10) is correct (i.e., the operator Ket Theorem 3. really acts in the space (38));

(b) For $ E AUthJ, Kef$ does not depend modulo 0(h) on the choice of the cut-off function x; (c) The support modulo 0(hm) of Kef$ is contained in the set w(supp $). Let X1, X2 be cut-off functions, e - X1 _X2'

Proof. We start from (b). Then the integrand in

,E_,w) + x_EJ (i/h)1S(xII

i(,r/4)III

I(x,w,h) - e

j

(2irh) I 1 IR II

I I

dE_{g(x,E-,w)e

x

I

I

(40)

x ( I(4owI1)](x,w,h))

vanishes in the neighborhood of stationary points of the phase 4(x,Ei,w) S(xl,E_,w) + x_E_ (see Proposition 5 and Definition 9). I

We may rewrite

I

(40) in the formI I{x,w.h)

hN + CI,II+1/2]

(III/2)JIR IIIdLI{a(xl,E_,w,h) x (41)

W) x (0+1111+1/21 (i/h)O(x,E F e

)),

where a(x ,E-,w,h) is a polynomial in h, smooth with respect to (x1,...), a(xl,E.,wih)I- 0 for IEII > R(K) when (x,w) E K, where K is any compact in

e x D, and L - -iI

EI

am

a@

I

_j_

Ls

_

(i/h)0 f h-1e(i/h)0

(42)

I

Inte-

is a differential operator with coefficients non-singular on supp a.

grating by parts N +E 0(hN).

since if

11+1

3 times we obtain immediately that I(x,w,h)

Since N is arbitrary. (b) is proved.

Similarly, (c) is valid

E Cx' DIn x R), supp p n i(supp $) - +, then we may set 9(x,E_,W) I

$(z)X(x,E_,w) in (40) and then proceed as above. (a).

Let *IC Ci': Gn x R), K = supp ry be compact.

It remains to prove

By similar argument we

obtain that modulo 0(h)

iCKefN

1/h E_; (X (E_)e I I I

(i/h)S(zl,E-,w)

1(# o mIl ) x (43)

I

where X(CI) E

X(Et)

0 for IEII > R(K,`). Using Proposition 1,

we obtain that *CKef+] - i8 (h ) - 0(h) C Nh, foce x a x (0,13).

The

theorem is grgwd. The lobal canonical o orator and uantizatio3conditions. Now we may pteHent t compete construction of the canon cal operator, outlined some pa$es earlier. Assume that a Lagrangian manifold (L x sl,i) is given. With

18

no loss of generality, we assume that L is connected (otherwise one should consider connected components of L) and we recall that 11 is assumed to be connected and simply connected. We fix: i)

some measure ;r on (L x L,i);

ii)

some canonical atlas {(U3,n1J

:

Vj)}jc

Uj

Ca'w-partition of the unity, {e.}.

J

of (L x R,i) and some J on L x 0, subordinate to the

covering {Uj}jE J;

iii) the family of cut-off functions {Xj(x,t_ w)}

in the canonical J. EJ

charts, satisfying the condition: for any compact set K C stn x fl and any function 0 C A[h], XJ.(x,E w) ' 0 when (x,w) E K and (xIj w) E I.

Ci

(supp m n U

I.

for almost all (i.e., for all except some finite

subset) j c- J; the existence of such a family is proved in Proposition 8 below and such a family will be called concordant with the canonical atlas; iv) v)

the point a(0) E St which will be called the initial point;

some continuous branch arg Da (a(°),w) of the argument of N/Do for a(o)

fixed, w C n (such a branch exists since n is assumed to be simply connected).

To determine uniquely the elementary canonical operators K. corresponding to the canonical charts (Uj,nl ) according to Definition 10, we

have only to fix the choice of the phase function Sj(xl,r;w) and of a I continuous branch of arg u1 ,(x1J,4 - w) in VJ . for each j E J. We perform I.

J

this in the following way. For each j E J= we choose a point a(J) E U., which will be called the central point of Uj and a differentiable path

yj

:

10,1] -* L, yj(0) - a(O), yj(1) . a(J)

(44)

Definition 11. We set i*w1 + j

Wj(a,w) ' 1

(J)

yJ

arg. Da (0,,w) = arg

i*wl - p- (a,w)q

IJ

IJ

(a,w), a E Uj,

21-1

Do (a(0),w) +1y.d(arg Do) + 1a

a(J)

J

(45)

d(arg Da), a E Uj(46)

a for any w C 0; JO(j) is taken over any path lying in Uj and connecting the

points a(J) and a (note that the form d(arg Do) is defined correctly, since various branches of arg Do differ by a constant) arg vI (xI J

,w) J

C(arg. Do n-1](x J

Ij

D7")

w) +

Ij

'j,

-I

.

J

+ arg J

J

Iij

J

where the branch of arg oI (xl,,&, way:

(47)

W), (x1 ,E- ,w) C V.,

J

Ij

J

w) is chosen in the following special j

159

n

arg al (xI.,4- ,w) '

)

where py

it

< arg ak

Ij

J

2 ,

n

Z arg Ik - 2 IIjI, ksl

(48)

ak are the eigenvalues of the matrix (a(qlJ ,- iplJ.,

+ iqiJ ) )/a(qlj,p_ ), counted with their multiplicities.

We also set

Ij

j

S .(x J

,w) a (W .oir Ij)(xIj ,E- ,W), (xIj

Ij

J

Ij

E V J..

(49)

Ij

Ij

The above definition is correct.

Proposition 7.

Proof. We have to show that (a) W. given by (45) satisfies the Pfaff equat on see Definition 8); (b) lk uses in (48) lie in the right halfplane; and (c) (47) really gives some branch of the argument arg ul,(xl,, E I .W),

*

J

(a) Since Uj is shrinkable and i WI is closed, (45) is a correctly defined expression ((46) is as well). Next, we have

dWj - i*wi- d(i*pgI ] i*(p

i*(pdq - pI dqI - gljdpij)

dq Ij

]

J

J

- q- dp_ ) . i*wl Ij

I I.

I

I. ]

as desired.

(26), the matrix

(b) Under the canonical transformation S1 a(qI J

ipI,p

+ iq Ij

Ij

J

J

)

becomes 3(cl

aqi

and we have shown already in the

a(g1J,pIJ)

proof of Proposition 4 that the spectrum of the latter matrix lies in the right half-plane. Next,

+Iq I- ) al (xl ,E J

J

,W) '

(50)

3(4

Ij

iJ

IJ

so that (48) really gives some branch of the argument arg al

.

J

(c) This follows from the fact that

uI.

Dµ i (DonIJ)ooI.'

(51)

The proposition is proved. We define now the elementary canonical operator K., corresponding to the canonical chart (U ,a ) by virtue of Definition ld, and the precanonical operator Ij o

K

by virtue of the formula

160

A[h] + Hh,Coc n R x (0,17)

(52)

J

K -

F1/h

E K.e.O

jE1 3

E

x

jEJ Ij

3

{X(x,E-

Ij

3

Ij

w) X

(53)

(i,'h)S. (x I. X e

w) uI.(x

Ij

,

j

J

)(e.$ o

Ij

3

(recall once more that the argument of the 'square root in (53) is assumed to be equal to one half of the expression (47)). Definition 12.

The canonical operator K

:

A[h] -

Hh,Poc(n

x R) 0

is a composition of the pre-canonical operator K Hh,l'oc(Rn

tion map lh,PocOkn x R x (0,17) _

(54) with the natural restric-

x R).

Here the set R C R

x (0,11 is selected by quantization conditions as given below: (a) The pair (w,h) E R x (0,11 satisfies the quantiDefinition 13. zation condition if the cohomology class

A (w,h) = h li*wl) --I Ed arg Dv 1 E H1 (L,tR)

(55)

is trivial modulo 2a, i.e., for any 1-cycle y on L we have

w1+2 var arg

= h Y

D-u

d arg D is the variation DO -= 4 Y

with some integer k - k(y) (here var arg Y

of arg Do along y).

(56)

- 2wrk

Y

(b) R C :Q x (0,1] is the set of all pairs (w,h), satisfying the quantization condition. Note. It is clear that it is enough to verify (56) for some collection {ys} of cycles such that {Eys]} is a base of the homology group H1(L,R). The set R is selected by the system of equations:

< A (w,h),1Ya] > = O(mod 2tr)

(57)

which turns out to be a finite system in most of the applications. Also the observation that the second summand in k55) does not depend in fact on (w,h) is useful. Indeed, P is assumed to be connected, Qd arg

Do

1,1yl>

depends on parameters continuously and may take only discrete series of values, multiple to 2z, therefore being a constant. Thus, (57) may be reduced to a system 1

'nh 4Ya i*w(pdq)

k + 1/2 '

(58)

111

depending on whether var arg Do is a multiple of 4n or not.

(58) is the

Ys

well-known quantum-mechanical quantization condition (where parameters w usually include energy and other physical characteristics of the considered system). For further information on quantization conditions see item D of the current section.

161

(a) There exists a family of cut-off functions (Xj x Proposition 8. concordant with the given canonical atlas. jG J Ij ,w (x,E_

x

(b) For the concordant family of cut-off functions the right-hand side of the equality (53) is a locally finite sum (thus, the definition of the pre-canonical operator is correct). Proof.

(a) Let X E Co(EZ) be a function, satisfying the conditions: We define the cut-off

X(z)-1 for IzI G 1/2 and X(z) - 0 for IzI > 1.

l. x Vj by

function Xj(x,t_ ,w) on tlxIJ

Ij

w) - X(Ix I -q _ (x

I j

Ii

J

t- ,w)12), IJ

(59)

Ii

where 1.12 is the square of the usual Euclidean norm in ttIll. We claim that the function (59) satisfies the conditions of Definition is concordant with the canonical atlas. 9 and that the family {Xj).

J

Indeed, let $ E AEh].

If A denotes the support supp m, the restriction

nIA

A+ 0 x 11 is a proper mapping.

set.

We denote by K1 C tin x 0 the compact set, consisting of the points

Now let K C kn x 0 be a fixed compact

(x,w) E ttx x 0 such that dist(x,x') G 1 for some (x',w) C K (the distance is defined in terms of the usual Euclidean norm in ). (x,C_IJ,w) E supp X. and (a,w) - >r 1(x IJ,t _ j,w) E A.

Assume that x(=- K,

If follows from the

I

definition of

Xj

that (x1 ,q_ (xi J

Ij

j

,w),w) = wr(a,w) E K1, so that (a,w) Ii

belongs to a compact set n-1(K1)fl A. Since the canonical covering is locally finite, we conclude that for at most a finite number of elements j 6 ] all the inclusions mentioned above may be valid, i.e.,for x E K only a finite number of terms on the right-hand side of (53) may be npn-zero. Further, from the above arguments it follows that ( w) belongs to a

compact set r

lJ

(x'1(K1) fl A), so that the estimate It

I

< R for some R

large enough is valid, i.e., the conditions of Definition 9 are satisfied. The proposition is thereby proved.

Now we come directly to the comparison of elementary canonical operators on the intersections of the canonical charts which will give us the foundation of the introduced quantization conditions. Theorem 4.

Let Uj,Uk be any pair of elements of the canonical covering

with non-empty intersection Ujk - Uj fl Uk. i)

nb bers on w,

ii)

Jk

There exist:

E Is and C(2) E w2 Z, where CM(w) continuously depend Jk Jk

differential operators V!k, 9-0,1,2,..., on L of the order 4 2s with

the coefficients independent of h defined in the intersection Ujk and belonging to C.-:w(Ujk)'

such that for any $ e AU Ch) and any natural N the equality holds: iJ

162

C(i)(w) + iC(2)}KV(N)0 -6(h N+I), Jk jk J Jk

K m - exp{ k

(60)

where

V(N)[h] -

V(N) Jk

s-0 joc(IRn

0(hN+l) is meant in the space

(61)

(-ih)SVlk;

E

jk

x

St

Hh

C;k) and Vjk are

x (0,1]).

uniquely defined by these conditions, while Cjk) is defined uniquely modulo the multiple of 21. intro-

and C;k) and the operators

The numbers

Theorem 5.

duced in the preceding theorem satisfy the following properties: (a) VfN) is the identity operator, Vf!) - I for any j and N. JJ

For any

JJ

non-empty intersection Ujk, Vjk is the identity operator defined in this intersection, Vjk - 1.

If Ujkf - Uj n uk n uI is non-empty, we have

V

V3k) o Vim) -

(62)

(the latter identity is understood as the equality of formal power aeries in h with the coefficients which are operators in AUjkf)' (b) We have (under suitable choice of C;k))

C(l)+C(1)_C(1) - 0, C(2)+C(2)-C(2) . 0 jP Jk kf Jk kf Jf provided that the intersection Ujkf is non-empty.

CO) k)

-

(63)

Also

- 0

Ck(22)

(64)

for any k s J.

C. Consider any path

(c) There are explicit formulas'for 10,1] -

:

L such that yjk(0) -

(J),

Yjk yjk(10,1/23) C U. and yjk((1/2,1])

yjk(1) -

fy i*wllfy i*wl _f y

k

Jk

21

Car

DN (.(k)'.) DU - argj gk Do

(see Definition 11 for Wk,Wj, argk D

a(k), and besides,

U. Then i*wl,i

(65)

Yjk

j

(j) (a

,w) - var arg Du Do

(a, w)]

(66)

Yjk

, argj

D ).

(d) Formulas (65) and (66) enable us to extend the definition of Cfk) onto the set of all pairs (j,k) such that U. (1 u J

f a, thus

J

preserving the properties (63) and (64). the 1-cocycles of the covering

Therefore Cjk) (w) and

are

of L and therefore define the co-

G J homology classses C(1)(w), C(2) E H (L,tl).

We have

C(1)(w) - [i*w13, C(2) - 2 [d arg

Do

].

(67)

163

Theorem 6.

(a) We have 2wm for all U3k S~ 0

(68)

Ti

for some m E 7l if and only if the quantization condition is satisfied for the pair (w,h) E 0 x (0,1).

(b) The canonical operator K, defined by (54): (i) does not depend on the choice of the central points a(i) and paths y.; (ii) modulo 0(h') does not depend on the choice of the family of the cut-off functions concordant with the canonical covering; and (iii) modulo 0(h) does nit depen., on the choice of the partition of unity and of the canonical covering. (c) Thus, the operator

K(°) : A =_ A°[h] -+

(69)

b'foc(n x 0)/mod 0(h)

in quotient spaces depends only on the Lagrangian manifold (Lx 0. i),measure v, the initial point a(0), and the prescribed value of arg

Do

(a(0),w(0))

for some w(0) E 0. Note. One might introduce on the right-hand side of (45) the auxiliary additive term, depending only on w E 0. Once it has been done, the canonical operator will depend on the choice of this term as well.

Proof of Theorems 4 (60) is valid with C3k

5 and 6.

and C3k

First of all, we prove that the identity

given by (65) and (66), respectively.

It is enough to prove (60) for functions 4 with a compact support. Indeed, if y E C"':OORn x 0) has a compact support, then *Kk4 - 4Kk41, where 41 E AU

[h] has a compact support and iK:Vlk)$1 (these 3k facts are simple consequences of the definition of the cut-off functions).

Next, if the support of 4 is compact, than

K# lc

(i/h)Sk(x

E/hi Ik

(a

x

'k

I1L

X (4oxik)(x

Ik µ(x) lk'

.E- .w)

(70)

,E

.w.h)}+O(hW),

and the analogous formula is valid for the right-hand side of (60).

At last,

for (+ E C:'° stn x 0) with the compact support

*f I s,h C const f I a ,h 4 constli fils.h,

(71)

the latter inequality being valid by Propos'rion 1 (b), the constants in (71) depend only on a and p. Thus, we have gotten rid of cut-off functions of any sort and, using the invariance of the norm under the 1/h-Fourier transform, we may reduce our problem to the following one:

Given a function #c- AU. (h] with a compact support, one should verify that 3k

164

F1/h x_ +{_

F

1/h

{e

E_ +x_

Ii Ik

Ij

(i/h)Sk (xi k,F.. ,w) l1ri

Ik

x (4 o n-) (xlk,f_7 ,w, h)) - exp{ h

x e

,E_ IJ

Ilc I

k,w) x

(w) +

V.

(i/h)Sx

E

I

x (72).

w)

A1) X

(x

VI.

IJ

]

Ij

J

J

J

X (xIj'h) + N+I(xIJ.E-,.w,h), where the remainder RN+I has the estimate t1

s h < Csh"

N+111

,

s - 0,1,2,....

(73)

The validity of expansion (72) will be proved by means of the stationary phase method_(see Theorem 1 of item A). First of all, we make reduction to the case I. - 0. To perform this, consider the canonical transformation SIj defined b3 (26) and set (x',C') - SIJ(x,E) = (XIJrEI,,EI,r- IJ)i (74)

(q',p') - SIJ(q,p) = (glj,p. ,plj,-.gIJ) (a,w) + Thus, we define in fact the new Lagrangian manifold (L x 1,i' - (SI (q(a,w),p(a,w)),w)). In the "primed" variables we have now: :

J

(75)

F1/h

- e-i(e/2)jIj\lkj l/h

r1/h 1j

Ii

Ik

Ik

arg vI (XI ,E_

1k

Ik

w) - arg v'(x',w) -arg a'(x',w) +

Ii

3

j

+ arg aI (XI ,E- ,w) - erg v'(x',w) ]

J

2

iIj1;

(77)

J

arg vlk(xlk,Ei ,w) - erg v' I,k(xl,k.Ellk.w) 1,(x',c

- erg u'

k,E_

k

1

w) +arg aL

k

- erg WI, (x1,

k

k

-Ic

r

,

k

k

2 Ilk1 '12

jlkj

(78)

(we easily obtain (77) - (78) from (47) - (48); cf. the proof of Proposition 7),

IVI (x1 ,EI '01 - j'x',w)j, J_.

J

J

165

w)I

IL Ik (x lk

Iu, (XI, ,E. ,w)I, I k k

Ik

Ik

S (x

k

I

,

z

(79)

,w) = Sk(xIp ,El, ,w),

Ik I

k

k

k

C(2), . C(2) Ik Ik

c(1),(w) ` Ik jk Since

etc.

- 2 III\IkI 1I

4 ITkH+4 Ir1I .

4 1'k

-4 (11

119 \

Ik

4 1k - ICI =

i\zkl+Ilkl-IIk\Ij!) - -

4

(80)

II I.

(72) becomes in new variables (we omit the primes now): (i/h)Sk(x pE/h+

Ik

x_

.E

I,w

k

lk

{e

(x w

u

1k ,Eik.w.h))

Ik

(i/h)S.(x,w)

x

((VjN),)o n-1) (x,w,h) + N+1(x,w,h). The left-hand side of (81) may be

where RN+l is expected to satisfy (73). rewritten in the form

(i(b)ESk(xik,E

e i(n/4)Ilki J

I(x,w,h) `

x

(81)

r,u(x,w) x

I

exp(

(2,rh) Ik

2

w) 4 ;.,

e

k= x

(82)

) (m o it-,) (xi .Eik,w,h)dCIk;

uL (x

the integrand in (82) has a compact support. We apply Theorem 1 of item A. The equations of the stationary point L. ` E-I (x,w) are Ik

'k

a

(xlk,Elk,w) - 0;

-E

Ik +

(83)

I1c

by Proposition 5 they are equivalent to equations of the Lagrangian manifold in the chart (Uk,alk): xIk

- q_ (x Ik

Ik

.E- ,w). Ik

(84)

Since supp 4 C Uk fUj , the equation (84) then has a unique solution on cupp 0 o trik; this solution is given by equations of the Lagrangian manifold in the chart (Uj.1r):

E_ . p- (x,w). Ik Ik

Moreover, on supp

o r-1, we have k

166

.

(85)

det(-

a2S k

aqi

) - det as-k - det a(x a

aE2Ik

Ik

,

Ik

-) t 0.

(86)

Ik

Next, we obtain, using (45) and (65), that

Sk (xL ,p

(x,w).w)+x_ p_ (x,w) - (Wko7r-1)(x,w)+x- p_ (x,w) - (87)

Ik Ik Ik k - ((Wk+ie(p- q- ))oa-1)(x,w) - {C(fY +fa(J))l w11on-1)(x,w) Ik Ik J a lk

a r ((f19+ f1jk-fYk+1Yk+fa(k) )i*wlon-1)(x,w) - C!1)(w)+.(x,w). Si Jk

Thus, we obtain I(x,w,h) - e

(iIII) ICM (w) + S.(x,w)I. Jk j {Cdet a(x3,E_

3-1/2 x

Ik N

x sE0(-ih)sVsU Ulk(x C Xf o A-1)(xlk.Ei

I -p- (x w) +

k +

%+I

k

k

(88)

(x,w,h),

where the remainder satisfies the required estimates, Vs are differential, (in E_) operators of the order 14 2a, and V0 - 1. Denoting -)_1Va

Ik

Vjk -

(we regard (x

V(N) - NE1(-ih)9Va s-0

(

(89)

,E- ,w) as the coordinates on L x A in formula (89)), we nay Ik

rewrite (88) in the form I(x,w,h) - e

(i/h)EC!1)(w) +S (X,W)l Jk

3'1/2 x

{Cdet a(x/,EI k

x

Ui (x

)((Vjk)$) o w=1)(x

+

,EIk,wW}itr

- p..(x,w) +

(90)

%+I(x,w,h).

In (89) and (90) the argument arg det a

E arg ,1m, - 2'r < arg as

-

z

E'

(91)

2 ,

m-0

where am are the eigenvalues of the matrix

We have

a xIk,E 'k

D(x

(-Ik

vlk(xi ,Eik.w)(det a z Ic

lk

D x D,E_ ) Ik Ik

Dz

,EI )

k

-DL

DX

(92)

(our notations are not completely pure, but it seems that confusion is unlikely to occur). Thus, to prove (60), it remains to show that for our choice of the arguments we have 167

k

w) -arg det

(xlk,E.

(mod 4n) (93)

8(x

k

Ik

Ik

To verify (93), we rewrite

21 (93) is obviously valid). do in the form

arg u - arg }ii - arg det a xia .-L

c da

k

k

(.)d(arg ) -5 d(arg Do)

r

g Da) +5

+ arg a- arg a'k- arg det 2(x

f (k)d(arg Da) +

(94)

aq

,- ) f:

k (here a E Ui (1 Uk, (x,f) _ (q(a,w),p(a,w))).

Since (fo(b) -fa(k))d(arg x a a d(arg DP), (93) turns out to be equivalent to

x DQ) - fY jk

arg a- arg a

arg det a(xIk'4-

0 (mod 4n).

(95)

I k-

Ik

To prove that (95) holds, we fix any point (a,w) E and consider the matrix-valued function (we write I instead of Ik themafter): A(t't)

a( - i

-1

a(q -itp ,(q -irp_)cos t + (p +irq_)sin t)

I

I

al,.. ,an

3-1. (96)

a al,...,an)

In (96) (al,...,an) is an arbitrary coordinate system on L in the neighborhood of the point a; A(t,t) does not depend on the choice of this cooordinate system. We set also J(t,T) - det A(t,T).

(97)

Since a is a non-degenerate measure, it is clear that J(t,r) 0 0 for any (t,t) for which it is defined. We assert that A(t,T) is defined for T > 0. Indeed, we have a(q I-

iTp ,(q -iTp_)cos t + (p +itq_)sin t)

I

I

a(al,...,ad

i

I

3(q- itp) a(al,...,on)

(98)

where

q - (g1,gIcos t +p sin t), p

(pl,pIcos t -q sin t).

(99)

I

The transformation (q,p) ; (q,p) of R2n, given by (99) is a canonical one, i.e., it preserves the form w2. Thus, 1 L x ft - 6t2n, i(a,w) - (q(a,w), p(a,w)) is a Lagrangian manifold. It is enough to prove that the form 6 - i*(d(ql- irpl) A ... A d(4n - iTpn)) is non-degenerate. Make a change :

of variables: q - q,

T.

Since then m2 - dp A d4 - t-1dp A dq, the

mapping i (a,w) ; (q(a,w),p(a,o)) is also a Lagrangian manifold and it suffices to apply Proposition 4. Next, we have obviously :

J(0,0) . det a( 8x1

. o,

J(2 ,0) - det a(xi,ip) - all

168

(100)

(101)

y3

ft

n 2

Fig. 1.

J(t,l) = det

3(q-ip) 3(a ,...,an )

Edet

a(qi ipl,(gi ip..)elt) an) 3(a1'

1

e

J

-ilIIt

(102)

1

Consider a connected simply connected domain r C: tt2

t,T

such that r

contains the half-plane i > 0, the points (0,0) and (2 ,0), and J(t,r), is

defined in r (the existence of such a domain follows from the above consideration). The continuous branch arg J(t,i) of the argument of the Jacobian J(t,T) in the domain r exists and may be fixed by fixing its value in any point of r. We'fix the choice of it by setting arg J(0,0) - arg a

(103)

((100) implies that the definition (103) is correct).

We assert that

arg J(Z ,0) - arg al.

(104)

To prove (104), consider in the domain r the contour y . Yl + Y2 + Y3' shown i n Figure 1. Obviously, arg J(Z ,0),= arg J(0,0) +var arg J(t,T) _ Y

(105)

- arg a +var arg J(t,T) +var J(t,T) +var arg J(t,T Y2

Y1

Y3

From (102) we obtain immediately var arg J(t,T)

(106)

2 IIi.

Y2

Next, var arg J(t,T) = var arg .Y(t,T), where Y3 Y3 T) = el("/2)jIji(Z

J(2 (107)

a(qi iTpl,P-+iTq-)]-1

a(q1-ip1,Pg+iq..)

de t

a(al,..

.Ed et

an)

3(al,...'an)

We have also J(0,T) = det

3(g -ip)

3a

3(-jr

Edet

Da

J-1.

(108)

We intend to show that n

var arg J(t,T) 1

var arg J(t,T) = Y3

-

E arg am, m=1

n E arg um. m=1

(109)

169

a (qi-ipl,pi +iq .)

a(2x 1p)

where Am, um are the eigenvalues of the matrices

,

3(x ,L-) I

1

respectively, the values of their arguments being taken in the interval

(-2,2). The canonical transformation S1 (26) reduces the proof of the second of the equalities (26) to the fir at one (the difference in sign is the Thus, it consequence of the fact that y1 and y3 have opposite directions). is enough to prove the first of the equalities (109). To do this, we choose a special coordinate system (al,...,an), namely, the canonical coordinate system (x1,...,xn). In this system we obtain

J(O,T) = where B =

.

det{(E-iB)(E-icB)-1}=_

detM(c)m,

(110)

It was already discovered in the proof of Proposition 4

that B is a symmetric matrix with section) imaginary part.

i

non-negative (in

fact zero in this

We assert that the eigenvalues am(T), m = 1,...,n, of the matrix M(r) _ (E - iB)(E - itB)-1 lie in the open right half-plane for all t a 0. Indeed, we have the representation

Xm(T) _ (1 - 1Km) (1 -

(111)

1TKm)-1 ,

where Km are the eigenvalues of B, which lie in the upper half-plane by Lemma 2. We have thus Re am(t) - (1 + r2IKmI2)-1 Re(1 - iKm)(1 + iTKm) _ (112)

_ (1+r2IxmI2)-1(1+rIKm12+(1+T)ImKm) > 0. We have M(1) = E, M(0) - a(g-1p) 2x

,

thus X m m

arg am(1) - 0 and define arg xm(t) for t above that - 2 < erg xm(0) < 2

.

a If we prescribe m(O) . 0 by continuity, w'_obtainfrom the

We have thus

n

n var arg J(t,T) = yl

where arg X

E (arg am(l) - arg am(0)) m=1

2 ,2). m

m m

E arg am, m=0

(113)

1,...,n, and (109) is proved.

Now (104) follows from (105), (106), (109), and Definition (48) of the measure density arguments. Indeed, combining these identities, we obtain n

erg J(Z ,0) - erg o -

E erg A.-! III + M-0

+

n

n

M-0

m=0

E erg um = E arg um-

(114) IT

2

III = arg aI.

Next we show that arg J(Z ,0) - arg J(0,0)

a(xaqA)

arg det

(115) I

(thus, the left-hand side of (95) is in fact precisely equal to zero). Since a(q - ip)/aa is a constant matrix,

170

var arg J(t,T) (116)

Y

a(q -irp ,(q. itp)coo t + (p +itq-)sin t) - var arg det[

I

Y

].

a(al,...,an

Choosing the canonical coordinates (x1,...,xn) as (al,...,an), we obtain

m var arg J(t,T) - var arg det(C(t,T) - iB(t,T)) Y

Y

I var arg A.(t,T), (117) j-lY

where a.(t,T) are (continuously dependent on t,T) eigenvalues of C(t,T) i8(t,r); arg aj(t,T) are continuously branches of their arguments, a(g1,gTcoo t+ pi sin t)

C(t,T) -

x a(pi,p.coo t- g1sin t)

,

a

Bit T) - T '

(118)

ax

Both C and B are symmetric matrices, and the imaginary part of C- iB is negative semi-definite (recall that t E [0,Z ]).

By Lemma 2, the

spectrum of C- iB lies in the lower half-plane, as (t,T) E y; thus the branches arg aj(t,T) may be chosen satisfying the conditions

- n < a(t,T) < 0.

(119)

Since

C(0,0) - iB(0,0) - E, (120)

a(girp.) aq C(Z ,0) - iB(2 r0) - (a(=Ir_) ax R

x

I

we obtain immediately that (115) is valid. Hence, we have proved that the assertion of Theorem 4 is valid under C(2) described in Theorem 5 (c).

the choice of

The uniqueness of

Vjk and the property (a) of Theorem 5 are the consequences of

Cjk),

c.ie following: Proposition 9. The elementary canonical operator Kef is asymptotically monomorpliic; more precisely, this means that the following conditions are equivalent for any N: (a) $ @ IUCh];

(b) Kel$ - 0(hN).

Also if Kef corresponds to a non-singular canonical chart, condition (a) is equivalent to: (b') Kef+ - O(hN).- See 152] for proof of Proposition 9.

In addition Theorem 5 (b) follows from Proposition 9; however, we give below its direct proof, which also provides the validity of Theorem 5 (d). We observe first that the formulas of Theorem 5 (c) give the correct definition of

Wt C(2) for any non-empty intersection Uj n U. We should

verify that

Ci) +C ) + C M - 0, 1 - 1,2,

(121)

171

Fig. 2.

if Ujkf - Uj (1 Uk n 6f f 0, and that 0, i - 1,2, j

(122)

J.

The latter is evident since the path yjj is homotopic to a constant path a(]), t 1E C0,11. To prove (121), consider the system of paths on yjff.(t) _-

L,}shown in Figure 2.

Here aOkf) Has in Ujkf, each of the small "triangles" lie, in a single chart and is therefore homotopic to zero.

We have

C(i) +C(i) +C(i) - f w(i)g i - 1,2, kf fj t Jk

(123)

where r - yjk + ykf + yfj and w(1) is a closed 1-form, w(1) - i*w1, w(2) - 2 d arg DDM

(124)

.

Thus (123) may be represented as the sum of integrals over small triangles, the latter being equal to zero, so that (121) is proved. It follows that (Cjk)} represent some cohomology classes of the covering (U3), hence co-

homology classes of L. since the covering (U.) satisfies the conditions of Leray's theorem which asserts that cohomologies of the open covering are isomorphic to cohomologies of the manifold itself, provided that the sets forming the covering and all their finite intersections have trivial cohomologies. It is not hard to establish that these classes are exactly C(1) - Ei*w1], C(2) - 2 Ed arg D-o ].

(125)

Indeed,

CJk)

- bkl) -bbl) +CJk) - (ab(i)) jkJk), i - 1,2, 1

Jk

yjk

i*wl, Cf2) - 1 f Jk

are cohomological to

i.e.,

d arg

2 yjk

Dc

,

(126) (127)

given by (127) and hence represent

the necessary classes. Thus, Theorems 4 and 5 are proved, and it remains only to prove Theorem 6. (a) For fixed w E R, consider the subcovering (Uj}jE J (Uj}jEJ,

tains w.

Clearly (Uj}j

of Leray's theorem.

h 172

of the covering

consisting of all Uj such that the projection of wUj onto 11 con-

EJ

is a covering of L, satisfying the conditions

Let now

2irm, m - m(j,k) E 7L

(128)

for any j,k E J, Ujk # 0. E Jw, Ujk # 0.

If y

Then all the more (128) is satisfied for j,k E

10,1] -+ L is any closed path on L, then it is

:

obviously homological to a finite algebraic sum of the closed paths rjk

Yj + Yjk - Yk' j,k e Jw, and therefore 5 (h i*wl + 2 d arg t)

< 8 (w,h),[y]>

2nk(y), k(y) E u .

(129)

Conversely, if the quantization condition is satisfied, then (128) is valid since it is a particular case of (129) for y - r. k* Note that we have proved that it is sufficient to require (128) on'y for j,k E Jw. Next we show that it is also sufficient to require (128) only for j,k such that the This means that, in a sense, the projection of un Uk onto {1 contains w. quantization condition is not only sufficient, but necessary as well for the existence of the canonical operator, mod d(h) independent of the choice of the unity partition and other auxiliary objects (we say "in a sense" Since the structure n outside the subset {h-4 c}, t being any fixed positive, plays no role in the construction). Thus, assume that (128) is valid Ujw n Uk # 0 (here Uhf denotes the intersection of U C L x for that such j,k x A with L x {w}).

Since {Ujw}je J is an open

Now let j,k be such that Uj (} Uk # 0.

covering of L (and obviously UP C Uj), we may cover the path Yjk by a , finite sequence U.w,...,UW of open sets jl - j, jm - k, and U. n UW Ji 3i+l im J1 i - 1,...,m - 1. We obtain thus:

1 C!1) jk

mEl(1

Jk

i-i

ji i+l

+C(2)

jiji+l

2irm(j,k), m(j,k) E 7L .

)

(130)

(b) The independence of the canonical operator of the choice of a As concordant family of cut-off functions is clear from Theorem 3 (b). for its independence of the choice of central points aW and the paths y3, this follows from the fact that if 8 (w,h)/2r is an integer cohomology class on L, we have

h Sj+Z erg NI

(h Wj+2 arg.

)o ail

(f h i w1 +

J (131)

d arg D - pi qi }o ail + arg

+ fY

(01(1)),w) (mod 2n),

J

2

where y - 10,1] - L is any path with y(O) - a(0), Y(1) - a, (a,w) E i.e., the left-hand aide of (131) modulo 2w does not depend on the choice

of a(3) and yM. (c) The independence of

K(O)

: A0[h] = A

lih.foc(Ytn x R)/mod 0(h)

of the choice of the canonical atlas and the unity partition is also clear. Indeed, the union of canonical atlases (subdivided if necessary) is a canonical atlas itself; thus it is enough to prove the invariance under the choice of the unity partition. Let {ej}, {aj} be unity partitions, subordinate to the canonical covering; then since Vjk - 1, we have the

following sequence of identities in the quotient space Nh,Eoce x n)/ /mod 0(h):

173

E K.e.+ =

j

ii

E

K.e.e,+ =

j,j,JJJ

j,j,] J]

K.,e.e:,+

E

and the desired assertion is proved. now complete. C.

E K.,e+ E A, J

(132)

J

The proof of Theorems 4, 5, and 6 is

Commutation of a Pseudo-Differential Operator and a Canonical Operator

In this item we establish the commutation formulas for X-PDO and the canonical operator. Here X - 1/h is a large parameter; the proofs in the form given here were first proposed in [561, and we use here the material of [561 with slight shortenings. Our task is to derive asymptotic solutions for A + m of the equation 2L(x'X_11

- 0, (Dx - -i.3/3x).

If Lu - Ek, ak(A_1Dx)ku, x E R1, and the coefficients ak are constant, then these solutions can be sought in the form exp(iAS(x)). Moreover, whenever the characteristic equation has simple roots, any solution will be a linear combination of the exponential solutions. Thus in accordance with this example, we shall look for a solution in the form of a formal series (iA)-k+k(x),

exp[iAS(x)] E k-0

or, more generally, in the form K+, where K is the canonical operator as described above. It is necessary to specify how a A-PDO acts on a rapidly oscillating exponential, i.e., on a function of the form +(x)expliAS(x)]. Let x 6 R1; then

(a dx)exp[iAS(x)] - S'(x)exp[iAS(x)]; 1-)2expCiAS(x)] - C(S'(x))2 +1

S'(x)S"(x)lexp[iAS(x)];

(ia dx)m(espCiAS(x)])

- eapCiAS(x)]C(S' (x))m+m2l

(S'

(x))Ck--1S"(x)

+0(A-2)].

By using the Leibniz formula, we obtain (a d;)mC+(x)exp(iAS(x))]

- exp[iAS(x)]C(S'(x))m+(x)+1a m2m-1

(S'

(x)+(x) * E (iA) j-2

where Rj is a differential operator of order j.

(R.$) (x)1, Operator ((1/iA)/(d/dx))m

mpM-l, has the symbol L(p) - pm, and since dL/dp d2L/dp2 we get for the operator in question the following expression:

L(A-ID x)C+(x)exp(iAS(x))] - exp(iAS(x))CL(S'(x))+(x) +

+1

174

dL

dp(x

+' (x) + 2l d2L(S' (x)) S"(x)+(x) + ...7. 'LX

dp

m(m-1)pu-2,

2

This formula is true also for differential operators L(x,a-1Dx) with vari able coefficients, since we differentiate first with respect to x, and then multiply the obtained expressions by functions of x. Finally, as the operators a/axj, a/axk commute on smooth functions, we have

m

(1)

(ix)_jRj(x,Dx)O(x),

= exp(iAS(x)) E j=0

0

where are linear differential operators of order j. If in particular L is a differential operator of order m, the coefficients of which are polynomials in (ia)-1, then we obtain (RoO)(x) = L(x,

(R10)(x) _ < + C1

;0)0(x),

ax

aL(x,(aap/ax);0)

Sp(d2L(x,(3S(x)/ax);0) 32S(x)) +a 3p2

2

axe

Let symbol

Theorem 1.

'a2xx) > +

(2)

L(x, !S(x) ;t)

_

t-0

ax

at

10(x)

E C'([0,1],S_(1R2n)) and function

S(x) be real-valued. Then for a '_ 1 and for anarhitrarv integer N

0, we

have

L(2X,a-'Dx,(il)-1)1O(x)exp(ixS(x))J

= (3)

exp(iXS(x)) E (i\)Rj(x,D)O(x)+0_N-l(x,,A). x j-o

Here

is a linear differential operator of order `= j with coef-

ficieni?s from tilt' Class C_(kn).

The estimate of the remainder is given by -r

Cra-N-l+ al(1*

0x0-N-l(x'X)i

with arhitr::ry r > 0, x E Rn. Proof.

(4)

Ro and R1 satisfy the formulas (2).

Let u(x,X) be the left-hand sideof (3).

Then

u(x,A) _ (x)nJexp[ia< x,p ? ]L(x,P;(ia) ')I(p,a)dp,

I(p,)..) = fO(x)exp[i).S(x) - <x,p> ]dx. Let M = {p

:

(5)

p = aS/ax., x G supp ;) and 0(p) C RP be the exterior of

a finite domain, C(p)(1 M = 0. We construct a C -partition of the unity: has the compact support supP %(P) c n1(p) + ')2 (p) = 1, p E Rn, where C G(p), and we correspondingly set u(x,X) = u1(x,X) + u2(x,X). Further,

we obtain

n a.. u2(x,a) _ -

2

1

fexpl:ia < x,p > ] E

ia1xI for x # 0.

j=1

CI(P,a)L(x,p))dp

as

pj

Consequently, for jx+ = 1 and arbitrary N > 0, we have
+s(y), where integration is performed over a finite domain in R1 x R. Let x C K. (as a function of y and p) has a single stationary point The function y = x, p - aS/ax. Let H(x) be the matrix composed of the second Q(x) :

derivatives with respect to u and p of the function y at the point Q(x), i.e., 1,

H

j, k : n.

1

Then det H(x) _ (-1)n, the signature of Further,

H(x) is zero, and the eigenvalues of it are ± 1. 'Y(x,Q(x)) - S(x).

If 1x1 I R, then we obtain (3) and (4).

If IxJ > R, and R> Ois large enough,

then the integral for ul does not contain stationary points, and thus lul(x,A)i

i. CNA-N(l + 1xp-N, (xi

> R,

where N -> 0 is arbitrary; an analogous estimate holds for all derivatives of u with respect to x. Thus (3) is proved. Corollary. The asymptotic expansion (3) can be differentiated with respect to x and a any number of times.

The following theorem can be proved exactly in the same way as for Theorem 1: All statements of Theorem I remain valid for operator

Theorem-2.

L(xDexcept that ao(x) > +

aL(x,aS(x)/ax;O) ap

ax

'

32S(x))

+ [1 Sp(a2L(x,(as(x)/ax);0) 2

ax

+ 2 t L(x, 3SSyx) ;t)I E=0 + Sp

+

(7)

ax

a2L(x,(aS/ax);0) ]$(x). 3x3p

(Note that now L(x,- a Dx;(ix) 1)1 0'(x)exp(iaS(x))]

(2n)nfJL(y,P;(ia)-1)0(y) x (8)

x expli.a(S(y) + )]dpdy.) Now we arrive at establishing the commutation formula in general canonical coordinates. We decompose the set (1,2,...,n) into two disjoint subsets (a),(B) (a) - (all ...,ak), (8) (Bl,...,8f), where k + L n, :

ai f 8j for all i,j (one of the sets (a),(8) can be empty). We set x = (xal,...,xak), and analogously p = (P(a)+P(B)) We - (z(a),x($)), "(a) _ denote k ,p > = E x p , dx < x - dx ...dx j=1 aj aj (a) (a) (a) al ak

176

and analogously < x(6)11(6) > ' dx(B), dP(a), dP(B)

We introduce the A-Fourier transformation over a part of the variables by

(F A,x(a)P(a)u(x))(P(a),x(B)) (9)

fexpt-iA < x(a)'P(a)> ]u(x)dx(a)" e

where, as usual, transformation yield

in/4

Then well-known properties of the Fourier

FA,x(a)+ p(a)L(x,A-1Dx;(iA)-1)F"P(.), x(a)u(P(a)'x(8)) (10)

2

2

L(-A-1DP

1

1

;(iX)-1u(P(a),x

,A-1Dx

,x(6),1

)

(a)

(B)

(6)

2

for any A-PDO, L = L(x,A-1). The above formula allows us to construct a class of f.a. solutions of the form

u(x,A) - FA1P

(a)

Theorem 3. Let symbol L(x,p;(iA)-1) E C0([0,179S S(p(a),x(B) be real-valued

m(P(a)'x(B)) E CoOtp

No

S(P(a)'x(B)) E

ID

A

function

x jn-k

C OR

(B)

(0(P(a)'x(B))exp[iAS(P(a)'x(B))7)

lily;(-iA)-1)F-1

d;.ffe.rerai 0.

In particular, h(x,p) (-= T

To end this item, we introduce the notion of asymptotic expansion in ...*X )(Ra). the homogeneous case, i.e., in the space n 1 .

Definition 2.

Let (x) be a given element of the space Lea 1

'n

(Rn)

E $.(x) gives an asymptotic expansion j'0 ln)(Rn) and ;im mj - - m .(x) if 0j(x) E

We say that the formal series of $ and write $ 'o JI

'" '

and if for any N there exists no such that t

- JI $.(x) E L(al....,a)(Rn),

(55)

n

-0 J

once t z no.

In terms of Definition 2 one maZ say that Theorem 2 gives an asymptotic expansion of (f o g)$ in the space L(1 l.. ox )(Rn). n 1 Stationary Phase Method, the Canonical Operator. and Wave Fronts in the Quasi-Homogeneous Case B.

(Rn) there is a class of

Among the elements of the space L'

special interest for us - namely the class of canonically repreeentable functions. This class will be described and investigated here in some detail as well as some geometric objects connected with these functions. Tte simplest example of the canonically representable function (CRF in the sequel) is fi(x) - eiS(x)J(x),

whe:-e S(x)r= 01(All .... an) (Rn)

'

a(x) E SR

(k I.

(1)

...n)

(Rn)

and ImS(x) 2 0.

It

is not hard to establish that under these .:onditions P(x) E L

)(Rn). n In th. present item we confine ourselves to only real-valued phase functions 1

'A

The experience in 1/h-theory suggests that the consideration of the Lagrangian manifold associated with the phase function S(x) of CRF would be useful. This Lagrangian manifold L is given by the equation S(x).

pj - pj(x)

2x. (x), j - 1,...,n,

(2)

J

and possesses the properties which are formalized in the following definition (here and in the sequel we assume the numbers 11,...,an to be given and fixed). Definition 1.

202

The Lagrangian manifold L is called proper If:

(a) the inequalities CA(x)1-AJ,

j z 1,...,n

IpJ .1 6

(3)

are valid on L with some constant C - CL; (b) for A(x) sufficiently large L is invariant under the aCLion of the group R+ on R2n, defined as follows: (T1tX1,...,TAnXn'T1-A1p1,...,T1->npn),

(T>'x,t1-A p)

T(x,P) _

-

(4)

(x,p) E R2n. T C R. In other words, (b) means that if (x,p) E L and A(x) > R - RL, then for T ) 1 the point t(x,p) necessarily belongs to L. However, in what follows we use a somewhat different version of condition (b), more convenient for our needs: L is supposed to be everywhere invariant under the action of R+, but the equation (2) (or similar equations in the mixed coordinate-momenta representation) defines L only for A(x) large enough. The equation (2) defines not a general proper Lagrangian manifold, To deal with but one which diffeomorphically projects onto the x-plane. the general case one should be concerned with the mixed coordinate-moments representation. To motivate our considerations thoroughly, we first study the partial Fourier transform of the function (1) under some additional assumptions.

Fix some subset I C In] and suppose that the inequalities IX iI C CAI(x)AJ. j E I

(5)

holds on the support supp 0. Adding to ' a suitable function bounded with its derivatives and finite with respect to x_ therefore belonging to L

IUI+

,

(Rn), we may assume that (x) . 0 for A(x) < R and therefore (A1,...,An)

S(x) is quasi-homogeneous on supp 0. I(xl,pr)

I

Consider the partial Fourier transform

i[S(x)-px_]

_i III/2 je (Z-

I

(x)dx-.

(6)

I

The integral (6) converges since, due to (5), it is taken over the domain. The stationary point of its phase is given by

pj - p1(xl,x_) = ai(xl,xI)

(7)

To calculate the asymptotic expansion of the integral (6) perform the change of variables xj n

jtAI(x)Ai, (8)

-A J,

pj - nJAI(XI

J E I.

In the new variables we have I(xl,p-)

where IAII -

i III/2 A(x) (2i)

iAI(x)IS(E1,s-)-n Je I I I

x

x .0(AI(x),EI,EIWIP

(9)

4(A,c) - (&A1)/Am.

10)

E A.

jEI

203

The integral in (9) is taken over a compact set {IEjI 6 C,j E I}. for derivatives of $(A,E) with respect to E_ are

Estimates

I

m-Ex .a.+EX.a.-m

EX.a.-m

as

m(A,E)I - IA

I

>

C0A

+(°)(EAA)I

>

>

>

- ca.

(11)

I

Thus, all the E. derivatives of a(A,E) are bounded uniformly with respect I

to A and we may apply the stationary phase method to the integral (9), considering AZ(x) as a large parameter provided that the stationary point or the phase in (9) is unique and non-degenerate on supp 0. The equations of the stationary point coincide up to notation with (7). Let EI EI(Elini)be the (resolved) equation of the stationary point and assume that det aEaaE_ (E1,E

I I

(EI,n_))x O, provided that(EI,E _(EI,n_)) E supp 4.

I

I

I

The usual stationary phase method (see, for example, [56)) gives now Ix 1-111/2 I(xl,p _) - (Zn)III/2AI(x)

1

a

iAI(x){S(EI),Ei(El.ni)>-T1I - EI(El.ni)1

x

e

{@(A I

1

(EI,E(El,n_)))1/2

(det aEaaE

x),E' ,E(E ,n-)) + I I

I

(12)

+ AI(x)-101(AIWAIST) _) + ...}+ R(AI(x),EI,n-),

I

I

where erg det a22S

is chosen in a special way, and all the derivatives

Z

of the remainder R have the estimates ak+1% Ck,a,Nz_N(1 + Inil)-N,

(z,ET,ni)I a

Raaz aEalI anal

(13)

k,IaI,N - 0,1,2.... Returning back to initial variables, we obtain I(xI,P-) - e I

iS(xl,p_)_ I ¢(xI,p-) + R(xI,p-),

(14)

I

where S is equal to the phase value in the stationary point (7) and the following estimates are valid with m1 - m + IxI_I -I1I/2: IP,I ( CA,(x)

1-x

ml -E

3101;(x TIP ) a a_i

J, j E I, on supp -E _a (1-x ),

a X

I( CQA1(x) JET 9

j

j C-1 1

(15)

IaI = 0,1,2,...;.

axIIapi1

(10

aIaIS(x ,p_) I C9

Ia

1-E

I

a x

CCAI(x)jEI j

-t a (1-x i JEI j

), (17)

on supp 0,

204

Ial = 0,1,2....;

aI°IR(x ,P-) Ia_I a

I

I

< ca Al (x)-N(1+

E

(18) I

IaI,N - 0,1,2,....

The estimates (15) - (18) are just those corresponding to the essence of the matter, as follows from: (a) If the estimates (18) are valid then*

Lemma 1.

xI(R)

pI+

(19)

L(al,...,an)(Rn)'

(b) If the estimates (15) - (17) are valid and Im 5' -

m1+I1I-1A

2 0. then

I

(20)

1(al,...,A )(Rn)' Proof.

pI (a) In the integral

I(x) - (

I 1 /2 fe1 1 1R(x,P-)dP , I I I

pxiR) (x) - (1 2n

(21)

perform the change of variables (8) and obtain

III-IaI

iA (x)

_ n

1

I fe

1(x) - 2n)II/2AI(x)

I IR(AI(x),E1,ni)dni,

(22)

where R(z,t1,n_) satisfies the estimates (13). Integrating by parts then yields the estimates: N1.

I(x) 4 CNAI(x)-N(1+

N,N1 - 0,1.2.....

(23)

I

and the same estimates are valid for derivatives of I.

It remains to note

that

AI(x)-N(1+ It_I)-N1 s Mx

)-No

for large N1.

(24)

(b) Perform the change of variables (8) in the integral (20). I(x) =

We have

(2n)III/2lei( S(xI'p)+xipi);(xl,P)dpi p I

I

I

III-Ix ilfe iAl(x)(S(EIAI (x) A,r) iA_x)1-I)/AI(x) +

(1)III/2A 2n I(:) + Mini)

1-a

a

W 1AI(W )

i

)dnI

nIAi(W )

2n)

ml+lIl-lail Iil/z A(x)

iAI(x)CS(AI(x),tI,*n_)+E n

x fe

I

I

I O(AI(x),tIoni)dni

(25)

where the functions

* T

p-+x-

denotes the inverse Fourier transformation.

205

(26)

S(z" 1,ni)

(27)

satisfy the estimates In

0,1,2,...,

C,

i

(28)

< cc" IcI = 0,1,2,..., on supt. 0.

(29)

alalSa_

an : For

Set M = max sup suPp (25) obtaining the factor J

J

M one may integrate by parts in

-N

for any N; thus we derive that

EII

-N

II(x)j C

const A(x)

(30)

if we choose N sufficiently large. The derivatives of I(x) may be estimated (20) will be called a CRF in a similar way. Lemma 1 is thereby proved. in the mixed coordinate-momenta representation. .

Now we are almost in a position to define the canonical operator since

there are enough observations to be summarized. Let L C R2n be a proper Lagrangian manifold. It is well knows that there exists a canonical atlas on L (see, for instance, Section 2 of the present chapter). However, an arbitrary canonical atlas is not sufficient for our needs; roughly speaking, the validity of the conditions of Lemma I (b) should be guaranteed. Thus we introduce the following: Definition 2 .

Let {(u.,y.

JE J, U u

uj - R(x

:

L be

I(]) a canonical atlas of a proper Lagrangian manifold L. proper if the following conditions are satisfied:

This atlas is called

(a) {ui } is a locally finite R+-invariant covering of L. (b) Let (u,y : u+

be any chart of the atlas.

In " = y(u)

the inequality holds .PJ .

CAI(x)1-4 J, j E I.

(31)

Also the function S (x ,P-) -

I

I

I

E A .p.(x ,p_)x.+ J I I J

JE1 J

E

jEI

J]

(A. - l)p.x.(x,P-), J I

(32)

where P1

PI(xT,PI)

x_ -

xI_(xI,pI_)

(33)

I

are the equations of L in the local coordinates (xl,psatisfies the estimates (17) in u for AI(x)

206

Ro = R0(u).

(c) All the intersections uj fl uk are R+ precompact sets (that is, the sets u. fl uk/R+ are precompact; it is equivalent to the assertion that the set uj fl ukfl {A(x) - 11 is bounded).

(d) There exists a partition of unity {ej}jE J on L. homogeneous of degree zero, such that for any j C- J the function (e 0 Y-1) (x1

Op

satisfies for AIM (x) ) Ro - RO(uj) the estimates (16) with mI . 0. Lemma 2.

The proper canonical atlas always exists on a proper Lagran-

gian manifold L C R2° Proof.

Let ao E L be any point.

Let j E=- I+ be chosen from the con-

dition Ix.(ao)l - max (ao)l. Then, obviously, A(x) s in J kEI+ J some R+-invariant neighborhood of no. Since xj(no) # 0, aj # 0, we have dxjlao

0 on L (use Proposition I of item A and homogeneous.local coordi-

By the lemma on local coordinates (2) in some nates to prove this fact). neighborhood u of a., a canonical system of coordinates (xI,pI) may be chosen with j e I; since I is R+-invariant this neighborhood may also be chosen R+-invariant. Since L is proper, (31) is satisfied. For each ao E L take a R+-precompact coordinate neighborhood of the described type and select a locally finite subcovering of L; it is not hard to see that this canonical covering is a proper one. The lemma is proved. Remark 1. The proper canonical covering constructed in the proo of Lemma 2 co,isists of R+ precompact elements; however, the case of particular interest is the one where the charts are not R.+-precompact (not like their intersections). It is difficult to formulate general existence theorems concerned with this matter; nevertheless practical problems supply a lot of examples with non-R+-compact charts.

Let a proper Lagrangian manifold L with a proper canonical covering

(u.,vj : u. -* Vi C R 2n

,P-I(J))}jEJ be given. I(J)

.

Definition 3. The space e (L) = Dm(L,{u.}) is the space of functions f on L, satisfying the following conditions: J

(a) The support supp f intersects at most,, finite number of uj -s. (b) For any k E =-J the function

fk (x1,PI) - (f ovkl)(xl.Pi), I

I(k)

(34)

satisfies in Vk the estimates al-If

ak (xI ,p-) I _

m-.

S CoAI(W ) JE

-

J JE1

. 1

J*

(35)

axITap We set OW(L) = m D°(L), D W(L) = mn Dm(L).

TF.e canonical operator we

I )x

intend to define acts on Junctions from D -(L), taking them into LEI n x (R ).

n

1

To define it we need to 3icuss the quantization conditions in

207

the }ua.x-hom.3eneouscase. We do not repeat the discussion. of these conditions l1_rtoT*1e.i ;n S.ccion 1 of the present chapter, but merely indicate some c,:untial modi('ications engendered by the additional R+ structure.

First of all, we note that the "first quantization condition" connected with the cohomology class of the form pdxIL disappears in the quasi-homogeneous case, namely, the form pdxIL is an exact one.

To prove this, consider a point (x,p) E L and the trajectory of the group R+ starting at this point:. (tllxl,...,tlnxn,tl-llpl,...,tl-hnpn3..

(36)

(x(t),P(t)) _

The tangent vector of the trajectory (36) at (x,p) has the form V =

E (X.x. ax. + (1- X.)pj p ). 3 J j=1

(37)

V.is tangent to L and therefore L being Lagrangian V J dp n dx vanishes on the tangent space of L: n 0 - V J dp n dxIL - 4:1Epjdxj -X (pjdxj + xjdpj)]IL = (38)

- (pdx-d(Ekjpjxj))IL' Thus we have the equality .dS

pdx - d(EXjpjxj)

(39)

on L, which proves our assertion.

It follows that the function SI(x1,p) given by (32) is a generating I

function of L in the sense that aS1 PI

X,

(x1.P

axI

(40)

as - ap I (x1+PI) I

are the equations of L on the local canonical coordinates (x1,p-).

Indeed,

I

we have (omi-.ting

the sign IL of restriction onto L):

ST - S-p_x-, I I

dSI - pdx - p _dx _ - xI _dp_ - p dx - x_dp_ , I I I I I I

(41)

which immediately proves (40). Also we note that in the intersection uj n uk with I = I(j), T - J = I(k), we have SI - SJ + pJxJ - plxl;

(42)

that is (ronuing ahead); the collection of phases {SI(j))je T on L is concordant with respsct to the stationary phase method. Thus only the second quantization condition remains in the quasi-homogeneous case.

208

Lei. u be a smooth non-degenerate measure on L, homogeneous of degree In canonical local coordir with respect to the action of the group R+. nates* u = uI(yi,p-)dxI A dp_,

(43)

I

I

where the function ii (xi,pwhich is the density of the measure y in local I

I

coordinates (xl,p_), is quasi-homogeneous of the degree r -

x (1-A.): uI(tzxl,tl-a p .)

=t

r - j F I 7-F.

jEI

E

(1-A

F.

jEI

A. -

jeI 1:

x

)

j ul(xI,PI)

(44)

We require that (a) the pair (L.::) satisfy the second quantization condition (i.e., the class of dln(j..Io) be trivial modulo 4n, see Section 1 of the present chapter);(b) for an ciart of the given proper canonical atlas the measure densit, uI(xl,p ) atisfies the estimates I

sau (x ,p-)

r - F

I- I

a

(1+a.)A. - E

E CaA1(x) jCI

J

J jEI

(l+a.)(1-A.)

J

(45)

dxIIap I I

for AI(x) large enough.

We assume that the concordant branches ct arg uI

in canonical charts are chosen and fixed. Definition 4. Let all the above requirements be satisfied canonical operator is the mapping K

Then the

0(L)* L(al,...,an)(Rn)

= K(L,N)

(46)

defined by KA ° ZKj(e10),

(47)

"elementary canonical operator" K. is given by the equality (here

where tb

+E D(L supp d ` u

1)(xl,pi)l(48)

iSitxi,p

CKj+.'(x) _ P-,e

1

1l(+ o ))

i

The functions where R0(uj). E(z) = 0 for z I(j), - E(zIE of the `trm [4,j are called canon:ca:ly representahle functions (CRF). ,ht rrectness of Definition 4 is verified easily. Since q E D-(L), only ifini_e number of non-zero terms occur in the sum (47); we use Lemma.1 to estimate:the elementary canonical operator and obtain the following: :

'

Proposition 1.

The elementary canonical operator acts in the spaces**

m-}E°. A.+}(r+III) ]°lAn) J (Rn). 0,(u.)+L (All ....

(49)

We need several of the canonical operator which are establ+shtd in the subsequent theorems.

In (43) 1x1 (:p.,

dpi denotes the exterior product of all dxj, jEI and

j c_ I rearranged is the order of Increasing index. ind

low we denote Dm(u) _ {m E V (L)Isupp + C ul for

L.

209

Let uj (1 uk # 0.

Theorem 1 (The cocyclicity theorem).

There exists

a sequence of differential operators Vjkt, f - 0,1,2,..., acting in the spaces Vjk!{

:

uk)-.Dm-t(u3

Dm(uj n

(1 uk)

(50)

(m is arbitrary) with the properties: (a) Vjko ' 1,

(b) Vjko is the operator of order 6 22 with real, smooth coefficients. If we denote N -1 E

(-i)IV.kt, N = 0,1,2,...,

(51)

R=0

E Dm(uj n uk): n M_ 1jE=X.+ j(r+j!(j)j)-N

then for any

J)

L

(Rn).

(52)

n Thus, Kk and K. coincide "in the principal term." Let K = K(L,v) be a canonical

Theorem 2 (The commutation theorem).

Let also ri(x,p) E T'

operator described in Definition 4.

There exist operators Pp, 1 P

al....,an)(R2n).

0,1,2,..., acting in the spaces

Dm(L) + D

s+m a tl

(53)

(m is arbitrary) with the properties: Its coefficients are (a) Pf is a differential operator of order s2L. linear combinations (with smooth real coefficients) of restrictions on L of H(x,p) and its derivatives up to the order 21.

(b) The operator P. is an operator of multiplication by H(x,p)IL. (c) If we denote

N-1 P(N) =

E

(54)

(-i)EP1,

P-0

D(L):

then for any

m+s - En l X. +!(r+n-l)-N n

2

a (N) H(x,-i ax)K0 - KP $ E L(a1,...,A )J

(R )

(55)

Also, if H(x,p) is associated with (L,P) in the sense that H(x,p) IL - 0, V(H)ILl - 0,

(56)

where

v(H) '

ap ax

(57)

ax ap

is a Hamiltonian vector field generated by H, then PO - 0, P1 = {v(H) 2

E

j`1

ax.a j

)IL'

(58)

p]

Proof of Theorems 1 and 2. As forthe purely computational aspects, no new ideas in comparison with the 1/h-case are involved. The main problem is

210

to show that the performed expansions are valid in the considered situation, it being proved that the formulas for the terms of asymptotic expansion merely coincide (with obvious renotations) with that for the 1/h-case. We intend to prove the expansions to be valid and thus complete the proof. We begin with Theorem 1.

T

Let

C V (u. fl uk).

m

I(k),

Denote I

I(j), and

I(xl,Pi) _ c(AI(x))(uI(xl,pi))t/2($o and analogously for $T(xT,p ).

(59)

We have

fi

(Zr)lII/2reiCSl(xl,p)+

(K .$)(w ) _

PI]$I(x1,P-)dpI-.

(60)

TPT7OT(xT,pI)dpfi

(61).

J

(Z;)Ifil/2re1fST(xr'pfi)+

(Kko w _

Let x(z-) E C0(RI1I), X(z-) = 1 in the vicinity of the origin. I 0 I claim that for e > 0 small enough

0- x(EEZ(x)))(Kk$)(x) E L(a ,...,A )(R°), 1 n

We

(62)

where A.

Ei(x) - xj/AI(x) 3 .

(63).

Indeed, consider the set M - ir(u.fl uk) C Rx, where w

is a natural projection. henceforth,

:

L - Rn, (x,p) + x

Since u. (1 uk is R+-precompact, so is M and,

.Ixjl G CA(x)Aj, j

with sow constant C for x f-= M. finally

1,...,n

(64)

Since M C lr(L ), A(x) 5 C A_(x) in M, and ] I I

Iti(x)I 6 C

0,

j - 1....,n

(65)

for x E M, where Ei(x) are given by the equality (63).

In terms of function

ST(xr,p7), M may be described in the following way: M = {x E R11 13p.: (xr.PT) E vk(u.fl uk) and xfi + 2p- (xr,pfi) = 0).(66)

r

T'us, necessarily, xj + 3p (XT,pr) f 0 for some 5).

if x'does not satisfy

In (61) perform the change of variables A

.

xj = A(x) ]Yj, j - I,...,n; (61)

P] - A(x)1-Ajnj, i E fi.

The function (61) then has the form (Zx)l7I/2A(xlrl-IaTI

(Kk$)(x) eiA(x)CST(YT,nT)+yfinfi7

x I

x

x

(68)

mT(A(:) Yr. A(x)1-xnfi)dnT.

211

It ahou'd be emphasized that in (68) integration is over a compact set independent of x and, supp $ being R+-precompact,y.f varies also In a compact Consider the set set. Ml - (x E R°{ k. I > Co + 1 for some j E [n]).

(69)

The above considerations make it evident that inf x E MI

IgradnTCST(YT,nT) + yT1I > 0.

(70)

(xT,P T-) E suPP $T

Thus we may set a so small that X(sL_(x)) - 1 for It (x)I < Co + 1, j E I, and integration by parts in (68) yields (62). technical details of this proof.

The reader may easily recover

Now we represent K k $ - X(et_x))f $ + ( 1 -

I

The second

I

term As shown to be inessential; applying the Fourier transformation of Fx-a to the first term, we obtain the integral pI

I

(ITI-III)/21jeiCST(xT,q T)+x7.

T(x_) lop - (1

1

xlpI

x

(71)

x X(eLi(x))$T(xT ,gj)dq.dxi.

It is not hard to see that integral (71) is taken (for fixed (x1,p_)) over a compact set.

We perform the change of variables A.

xj - AI(x) JFj, j - 1,...,n, qj - AI(x)` -Jnj, j 6 To

(72)

pj - AI(x)I-Ijnj, j E I.

The integral (71) takes the form X.

I(xI.P-) (73)

x !1e iA(x)C

(!;T,nr)+Ernr-C Ie_1

I X(et I)1sT(Er,nr)dnrdE z

where the integral is taken over a fixed compact set. The further treatment of this integral is completely analogous to that of (9), the usual phase method thus being applicable to this integral. We obtain (60) as the first term of expansion, and all the subsequent terms, thus proving Theorem 1. Next we give a sketch of the proof of Theorem 2. First of all, it suffices to prove the theorem when the elementary canonical operator K is substituted for K; the general case then follows via successive use unity partition and transition operators V%a) .jk Let H(x,p) E Tel

1r

l

)(R2n), $ E V°1(uj), and assume, to shorten the

P X

the calculations, that 1(j) - 0.

212

We have, by definition:

2

H(x,-i ..)K,0

(2a)-nffH(x,p)ei1(x-y)P+S(Y)3E(A(y))(P(Y))I'2(0ovjl)(Y)dydp

= (74)

(2x)-nrrH(x.P)eil(x-y)P+S(Y)30(y)dydp

Perform in (74) the change of variables a.

A(x) Jo.,

xj

(75)

yj = (ej + 4j)A(x)lj, pj ' nj A(x)1-11

and we obtain the integral x

I(x) ' A(x)nr

H(6A(x)l.nA(x)1-1)

(76) m((e+f)A(x)A)dEdn.

x

a

This integral is regularized as in the proof of Theorem 1 of item A. Applying the stationary phase method with A(x) as a large parameter, we The reader can easily recover the technical details come to Theorem 2. omitted here. To finish with this item, we introduce the notion of the wave front (Rn). for elements of the space Lm (All ...,an

Definition 5.

Let

Lea

In )

The wave front

(Rn).

is the

minimal of all closed R+ -invariant subsets K C R2n, satisfying the property:

if the support of the function H(x,p) E T

(al,...,an)

(R2n) lies in a closed

R+-invariant R+-compact set K1 and K1 f) K = 0, then 1

H(x,-i ax) E L (11,...,A )(R).

(77)

n

It is obvious that Definition 5 is correct and WF(O) may be defined as the intersection of all closed R+-invariant subsets K C R2n, satisfying the mentioned property. Theorem 2 of item A and Theorem 2 of the present item, being combined with Definition 5, show that the following obvious assertion is valid: Theorem 3.

n)(R2n) and

(1) For any H E T(A1

L(al

WF(H4) C WF(y,) C WF(H*) U ((x,p) lfim H(TAx,Tl-Ap) - 0). T

n)(Rn):

(78)

+m

(2) For any canonically representable function 0 the wave front WF(WW) is contained in the corresponding Lagrangian manifold.

213

C.

Quantization of Quasi-Homogeneous Canonical Transformations

In the previous item we constructed a class of canonically representable functions (CRF's) associated with a given proper R+-invariant Lagrangian manifold L C R2n. Here we establish the correspondence between a R+homogeneous canonical transformation g satisfying several additional conditions and linear operators T(g) with the main property: if y is a CRF associated with the Lagrangian manifold L, then T(g)4, is also a CRF, associated with g(L). We establish also the composition formulas and the commutation formulas with pseudo-differential operators for the operators T(g). We proceed to precise definitions and formulations-. First of all, we introduce a notion somewhat different to that of the wave front (defined in the previous item) but much more convenient to our needs. The notion of the wave front is well adapted to the case when one considers pseudo-differential operators with R+ finite symbols and/or R+compact wave fronts, etc. On the other hand, removing the R* finiteness condition for symbols of "test" operators in Definition 5 of item B would yield severe difficulties when proving that WF(*) exists for general (+. Fortunately, to develop the theory of quantization of canonical transformations there is no need to use such precise information as WF(,y) should give; it suffices to employ the following: Definition 1.

Let 4 C

For a given closed R+-invariant

n subset K C R2n we say that K is an essential subset for 4' and write that 1

2

K E Ess(4), if for any operator f

f(x,-i -) E T 2x

that supp f n K - ¢, we have f4 a L-" (Rn). (al,.... %d

of Ess (y)

are collected in

m (al,...,an'

(R2n) such

The main properties

the following proposition:

Lemur 1. (a) If K C Ess(4,) and K1 D K is a closed R+ invariant subset, ther Kl E Esaly).

(b) If K E Ess(4), then WF(4,) C K provided that WF(4,) exists. (c) If 4,

is a CRP associated with the Lagrangian manifold L, then

L C Ess (4+) .

2

(d) For any H = H(x,- a-) C T(x 1,

..., a

)(Rn) the implication is valid:

K C Ess(ip) -;r K n supp+ H E Ess(H4).

where supo+ H is the R+-invariant envelope of supp H.

(e) K1

Ess(ipi), i - 1,2 -i+ K1 U K2 E Ess(4,1 + 2)

(f) 0 CEss(4,) if and only if y C L(

)(R2n). n

1

Proof.

The properties (a), (e),and (f) are evident.

\ K then by definition of WF(4i) there is a symbol f E T

(b) If a E WF(y)\

(al,...ran)

(Rn) with

the support in a small enough R+-invariant neighborhood of a (so that supp f () K = 0) and such that fey

definition of K.

L7711

.1 )(a) , which contradicts the n

(c) This immediately follows from Theorem 2 of item B, and (d) also immediately follows from Theorem 2 of item A. Lemma 1 is proved. 214

if R2n \ U E Ess(ip).

The set U C R2n will be called inessential for 0

1An)(Rn) the subspace of

In)(Rn) C

We denote by L(al,

(xl,

functions y such that Ess(1) contains the set {(x,p)I

Ip.1

5 CA(x)1 A3

j = 1,

A..,n) for some C = Cy. One should note that all CRF's belong to m (Rn) (see Definition 1 of item B). L (al,....an)

The operators T(g), which we are going to define, act on

0

L"

(All... An)

)(Rn). To begin with, describe the

(Rn) rather than on L

class of admissible canonical transformations g. be two non-zero n-tuples of nonLet (All .... A ) and negative numbers. nEach of these tuples defines an action of the group R+ on the space Rn and, consequently, on the cotangent space T Rn - R n. TX - (T ll x ,...,T

A n

xn), x e R

n (1)

(TAlx,...,TlnxnTl-Xlp1,...,T1-lnpn),

T(x.P)

(x,p) E T*Rn -

R2n,

and

TY = (Tuly1,...,TUnyn), y E Rn. (2)

T(Y,4)

un (T Ul yl.... ,TYn,

T1-ul

ql....*T

1-un

* n Qn). (y.4) E T R

R In

Here and below we adopt the convention: if the coordinates are denoted by (x,p), then the action (1) is assumed, while for coordinates (y,q) we Although this notation is not completely rigorous, confusion employ (2). should not occur. Definition 2.

The mapping T*Rn,

g - T

(y,q) - (x,p) = (x(y,q),p(y,q))

-

(3)

is a proper R+-invariant canonical transformation if the following conditions are satisfied: (a) g is smooth outside the set (A(y) = 0}. (b) g is a canonical transformation, i.e., preserves the symplectic 2-form: *

g (dp A dx) - dq A dy.

(4)

(c) g commutes with the action of the group R+: Tg(x,p) = g(T(x,p)), i

X(T y,T

1-p

or

A

q) = T x(y,q),

P(TUy,T1-u q)

(5)

T1-ap(Y,9)

(d) g "preserves the equivalence class of A" in the following sense: there exist positive constants c and C such that cA(x(y,4)) < A(y) < CA(x(y,q))

(6)

for all (y,q). 215

(e) Functions p.(y,q) and all the derivatives of the functions x.(y,q), J pJ.(y,q), 3 = 1,...,n, are bounded on any set of the form: K = {(y,q)lml < A(y) < m2"gl 0, r (g) -U The above considerations yield the validity of the following: Vn(g),

Proposition 3. The canonical operator Kg on the graph r(g) is defined and acts in the spaces:

217

K

:

g

Dm(g) -

Lm+n-}(IXI+JvJ)

(RnxRn) x y

(All ....

(15)

*

for any m.

The above statement immediately follows from Proposition 1

Proof.

of item B. be the projection of the set x Rn (1'(g)) C Rn Let Ko = a Io Jo o yJo in xIo x and define x R2n , on Rn f(g) C R2n (x,P) yJo (y,q) x1o Ko = {(yJ0,xI0 )Idist((yjolx

IQ

),K,) 14 2}

(here dist(z,K0) is the distance between the point z and the set KO the usual Euclidean metrics), and set JX1(x1o -E10,yJ. - nJ.W IQdnJo, (tIo"Jo) E Ro,

X1(xlo,yJo)

(16) in

(17)

where X1(zl ,UJ ) E Co(R11o1+IJol), X(zl , U J ) - 0 for Izl 1 2 + JUJ 12 > 1 0 and

10

'U

q JO

10

m 1.

dUT

It is obvious that x(x I ,yJ ) is a smooth

function equal to one in the neighborhood of the set KO, bounded with all the derivatives. The requirement (e) of Definition 2 guarantees that the diameter of the set {YJoIX(xlo,yJ0 is bounded uniformly with resepct to xl .

f 0)

Then set also

0

X2(xI+.YJ+) = X2(A(y)/A(x)),

(18)

where X2(3) E Co(R1), X2(z) - 0 for z > 2C or z < c/2, and X2 (z) = 1 for

< z
1

ct (z) < l:aAa(z) < CA,d(z), Z E',}

(5)

(d3) All non-empty intersections of charts ua are shrinkable sets (in other words, the covering ua satisfies the conditions of Leray's theorem), and if the intersection ua n u8 is non-empty then the mapping

Y(3a = Usua`

: ua(ua n u4)

Ue(ua n i

)

(6)

is a proper R+-invariant canonical transformation satisfying the conditions of item C. (d4) There exists an inscribed covering M = lJ Ua, Ua C Ua satisfying all the above conditions such that for any a (D,'D,), where Da = Ua(Ua), is a proper pair (see last in item Q.

* Our methods can be extended to consider locally finite atlases.

228

Mo

(d5) There exists a partition of unity

subordinate to the

covering ua, such that Taa is a homogeneous function of degree zero for (R n). large AM(z) and (ual) 0a E TO (al(a),...,Xn(a))

Here saying that {0a)

is a partition of unity we mean that E0a = 1 for AM(z) > const.

This is

a

because we intend to avoid (where possible) explicit use of the numerous cut-off functions.) Example 1.

Let an n-tuple (a1,...,a ) be given.

The space

R2n

with

the action of R+ given by T(x,p) =(TAlxlr,...,TXnxn,Tl X1p1.....T1 anpn the form w2 = dp n dx, and the atlas, consisting of exactly one chart, do However, if we delete the set not satisfy the conditions of Definition 1.

tx1+ = 0}, the space R2n \{(x,p)Ixl+ = 0) gives a simple example illustrating the definition.

Next we should define transition operators. For the sake of convenience and in order to shorten the formulas, we introduce the following notations: Om OM La

L

-

Om

OM

(7)

L0B = L(al(a),...san(a))(Ua(ua n us)), 0, om La 66 = L(al(a),...,A (a))(ua(ua n uB n u6)), etc., where m E R U {W}.

Denote also

La

n (g)

L(A1(a),...,an(a))(Rx(a)

We assume that the partitions of unity subordinate to canonical coverings on graphs r(yas) are chosen and fixed for all a1Bsuchthat ua n up # 0. Set Taa = 1 and for 8 # a for ua n uB # 0

TBa = T(YBa) : L aB

(9)

1 0Bat

where T(YBa) is given by Definition 4 of item C. Lemma 1. The operators TBa of (9) satisfy the conditions: for any non-empty intersection ua n ua n uY there exists an asymptotically homogeneous symbol RaBY(x(a),

0p (a) ) E T(ai l(a),...,an(a)(R2n),

(10)

having expansion of the form ((60) of item C), and integer number such caBY that the equality holds

Taa o P

= (1 +Ra8Y)

modulo operators acting from

0 LYaB

o

Tayexp(inseBY)' .

to La

(11)

(E.By} is a (Cech) 2-cocycle on M. .

Proof.

We readily obtain (11), applying Theorem 3 of item C.

The

cochain EaBY is none other than the cochain e(Z) from Section 2; it is a cocycle according to Lemma 1 of Section 2:C; we denote the corresponding cohomology class by

229

Cs] E H` (M,Z).

(12)

Definition 2. A proper symplectic R+-manifold M is quantized or satisfies the quantization condition if the cohomology class Cc] is trivial modulo 2.

We assume that M is quantized. Than there exists an integer 1-cochain {u7,8) of the covering {u a) such that Ea3Y is a coboundary of ua8' (13)

ca BY - uaB - 1'a1 + ;"BY (mod 2). '

Replacing Tab by Tasexp(-inua3), that is, making a proper choice of arguments of measure density on graphs C(Ya6), we cancel out the factor of exp(ineaiY) in the equality (11).

From now on we assume that M is quantized

and the arguments of the measure are chosen in such a way that (11) holds with eaBy = 0. It appears that the factor (1 + RaBy) also can be eliminated by means of an appropriate adjustment of the operators T. There exist symbols Q

Theorem 1 . x

(R

(x(6),p(B)) E T-1

X

), having the expansion

I Qa6j(x(8),F(3))

_

Qa8

(14)

j=1 2n

0-

is a real-valued symbol satisfying

where QaBj E T(a1(B).

QaBj(T1(8)x(B),T1-1(B)p(6)) =

T-jQa4,(x(B),p(B)).

(15)

such that the "adjusted" operators

Tab = Tas(1+Qa8)

(16)

satisfy the cocyclicity conditions

Tab o TBY

Tay for Ua n UB n UY

&0,

(17)

0m

modulo operators acting from LYaB to La Proof.

(a) Preliminary stage.

Tas )

We first construct the operators

= Ta8(1 + Qas)), Qas) E T(al(d)....,an(B))(R2n

(18)

satisfying the properties:

TMT(0) eB BY

_

(1+R(o))T(O) aay ay

0

R(r) (1 (a),...,an(o)) (R2n(19) a6y GT-1

modulo operators acting from LYas to La ; T(0)T(0) = 1 a6 ' Ba 0

0

modulo operators acting from L

a

to L am a

To do this, set

(0) F 0 for B < a,

Qa 6

230

(20)

(21)

and for a > a consider the equation on Qa6) (22)

TBaTa8(1 + Qa8)) s 1, or

+RBOB) (1+Qaa)) . 1.

(23)

(Here and below we do not write the moduli to within which the equations are valid since they are obvious from above considerations.) We write formally the Neumann series

1- RBaB + (RBaB)2 -

Q(s)

(24)

BOB)3 ...

Using Theorem 2 of item A and'Lemma 3 of item C, we sum this series asympIt follows from composition formulas that (19) totically and come to (20). is also valid. Assume now that for some N we have constructed

(b) Induction process. the operators T (N) aB

(1 + aB

(N)

(N)

QaB )' QaB

0

2n

(25)

E T(aI(B),...,an(B))(R

so that

T(N)T(N) aB

Ba

(26)

1

and

T(N)T(N) aB By

(1+ R(N))T(N) aBY ay

,

(R2n R(N) E T-N-1 aBY (a l(a),...,an(a))

(27)

CEs+1)

The problem is to pass from N to N + 1. (N+1) T018

(N)

(N+1)

m TaB (1+"%S

)'

We seek T(

(N+1)

in the form 2n

O-N-1

%B

28)

so that Q(N+1) aB

(N) E °r-N-1

%

(R2n

(11(B),...,an(B))

B

Recall that all the symbols involved are asymptotically homogeneous, admit expansions of the type (60) of item C.

(29)

i.e.,

Let r(N) denote the principal homogeneous term of R(N), and Aq(N+1) aBY aB *By (N+1) denote that of AQ'B , set also (N) 0a By

o (N+l)

a aBY '(r

(30) u6(Aq(N+1))

aB Calculate the composition T(N+l)o T(N+I) We denote by z with various aB Ba subscripts the terms of lower order in our expansions. We have, using the composition formulas,

231

(N+1) (N+1) Tab o TBY

(N+1)

(N)

Tao (1 + P1.8

(N) (N+1) ) )TBY (1 + 6QBY

+a (N+1))+z )T (N) (1+ (u-1)*(p(N)+v(N+1) ay a aBY a8 BY 1 (1+(u-1)*(p(N)+a (N+l) +Q (N+1)-o(N+l))+z )T(N+l) a

CL $y

a8

o(N+l) _ (N+1) +o(N+l) ay

2

ay

Thus, we need to solve the system of

(we omit the detailed calculations). equations

aB

aY

By

(31)

By

.

P(N) in U (1 U

a

aBY

(1 U B

y

(32)

on M.

P(N) is a cocycle on M, i.e., it is antisymmetric with May respect to its indices and for Uafl UB fl Uy f) U6 Lemma 2.

p(N) - p (N)+0(N)_0(N) a0y a66 ay6 BY6

-0 in ua n u fl uy nU6 .

(33)

B

Proof. For any non-empty intersection ua8y6 - ual uB fl uyi u6, consider the product (N)

(N)

(N)

(34)

TaBY6 - TaB o TBy o T y6

We may calculate this product in two different ways:

TaBY6 = (1 - ii(aBy N)

+

z3)T(N) o T(N) OB Y6

(35)

(N) + z (1 - iiaBy ) (1 - ir(N) - (I_ ir(N) + z5 )T(N) ay6 + 4z )T(N) a6 3 a6 a6y - ir(N) ay6

on the other hand, TaBY6

TaB) (1 - ir6y6 + z6)TSa) (36)

- 11-i(y*Bar(N))+z IT(N)oT(N) - 11-i(y*8ar(N))-ir(N)+z IT8 (N) a6 7 as aB6 8Y6 86 8y6 Multiplying by T(N) from the right, we obtain

r(N)+r(N)-r(N)r(N))+z8 - 0. aBy ay6 a66 (y* Ba BY6

(37)

We claim'that (37) yields (N)

(N)

(N)

*

(N)

r aBY + ray6 - raB6 - y8arBy6 - 0

(38)

in pa (ua f I u8 f) uy f) u6) . Indeed, denote the left-hand side of (38) by F(x(a),p(a)). F(x(a),p(a))E

e 0T(al (a).....an(a)) (R 2n) and is quasi-homogeneous of degree 1 for large A. ,po(a) ) # 0 for some (xo(a) ,po(a) ) E po(ua (1 u6 (1 uy fl u6) a Then if F(x(a) o 1

consider the function

(x) - exp(iS(x(a)))Q(x), where S E 01 (Rn) is a real-valued function satisfying (al(a),...,an(a))

232

(39)

as (x(a)) = P. P(a) az o

(40)

(a)) (Rn) is a cut-off function in the small R+n invariant neighborhood of x(a). Substituting into (37) yields a contra-

and O(x) E So

(AI (a).....

diction in view of Theorem 01 of item A, Theorem 2 of item B, and Lemma 1 (c) of item C. Passing to functions on M, (38) gives (33). The antisymmetry follows from (26). Lemma 2 is proved. It is well known from the theory of sheaves that equations (32) are solvable on M. One may set a(N+1)

= EP(N) 0

6 a66 6

a9

(41)

'

where 86 is the partition of unity described in Definition 1. to establish that the function T(N16)

from

6) 1(aas+1)) may be prolonged to a function

For arbitrary choice of lower-order terms in " '

(N+l) AQa

It is easy

n(6)).

the operators (28) satisfy T

(N+1)

a6

o T (N+1) By

a( I+ B (N+1) a6y

(N+l ) )TaY

(42)

o_

with RaNBYI) E TX X (a))

We have, in particular,

(R2n).

T(N+I)T(N+1) = 1+ R(N+I) aB a8a Ba

Adjusting

(43)

TB(NN+l)

for a < B with the help of the Neumann series analogous to

(24), we obtain T a6 (N+I)T (N+l) Ba

while (42) remains valid. remains to set

= 11

(44)

Thus the induction process is performed.

QaB

Qa6

Qa6

N=I

Qa6

It

(45)

)

(the sum of asymptotic series is obtained via Lemma 3 of item C). 1 is proved.

Theorem

We come to the definition of the sheaf F+; more precisely, the space F+(M) of sections of F+ over M. N

Now consider the set of tuples (4

.

}a01, where CL

0

yo E La

(46)

We say that the tuple (yea} is equivalent to zero, if for any a the sum 0

Fa

ip

ETay Y

C_

n

L(xI(a)....,an(a))(Rx(a)

(47)

satisfies the property: there exists a set Ka c R%(a) \ Dc, such that Ka E Ess(Fa).

(48)

233

Definition 3.

F+(M) is a factor-space 0

F+(M) -

equivalent to 0).

(49)

equivalent to 0}.

(50)

Also by F+(M) we denote the space flL0"/{{,J1)

Fm(M) =

I

4. PSEUDO-DIFFERENTIAL OPERATORS AND THE CAUCHY PROBLEM IN THE SPACE F+(M) In this section, the notions of pseudo-differential equations and the Cauchy problem in the space of sections of the sheaf, constructed in the previous sections of a canonical sheaf on the basis of this discussion, the theorem on sufficient conditions for the asymptotic solvability of the Cauchy problem for a pseudo-differential equation of the first order is established.

Pseudo-Differential Operators in the Space F,(M)

A.

Let M be a proper quantized symplectic R+-manifold, and let F+(M) be the space of sections of a canonical sheaf on M, constructed in the previous section. To define pseudo-differential operators in F+(M), first of all turn to local representation. Let H = {H ) be a tuple of pseudo-differential Om

operators, Ha being an operator in L

1 (1

(

a)) (R

) with the symbol

X

0 Hct(x(a)'p(a))

( 0)

(a) .

(1)

T(al(a).....an(a))(R2n

0m

Let

E F+(M); {y'a} C RL a being a representative of Q1'. a

def

H{yea)

Consider the tuple

(H'}.

(2)

H correctly defines an operator in the space F+(M) if and only if the equivalence class of the tuple (2) depends only on W or, in other words, if {'ya} '\' 0 implies {Ha'o) ti 0. We assume that the operators Ho satisfy the compatibility condition:

for any a,6 with Ua fl u,

0

HaTaB ° Tad Hd , modulo operators acting from LBa to L_. F+(M). We have ETYaHa`V

a

Let {'V} define a zero class in

= H ETaq + E (T Ha - H T a)V+a = 01 + @2. yY aY

a

(3)

Ya

(4)

As for the first summand in ( 4 ) , there exists K C R2n \ V such that K E E Ess ml. Indeed, this is valid for ETyaiya and the pseudo-differential opera!or does not enlarge essential subsets (see Lemma 1 (d) of item C). As for the second summand, the condition (3) yields that the operator T H - H T may be represented in the form Ya

a

Y

ya

- T(YYa.m),

(5)

supp 0 n ua (Ua fl uY ) = 0

(6)

TYuHa -H'a))(R2n)

where 0 E T ....

234

and X

(we omit in (5) the cut-off factor depending on the function to which T(y Ya,0) is applied).

From the statements on the behavior of the essential sets (see Proposition 4 of item C) it follows that there is a E Ess((TyaHa -H T )i ) Y Ya

a

such that Ka C R2n \VSince the sum (4) is finite, we have Ko = K U

U (a Ka)

Ess (W1 + @2) r K. C R2n \ DY.

Thus, the tuple (Ha'1'a) is equivalent to zero, and the mapping (2) gives rise to the mapping of equivalence classes, i.e., of elements of F+(M). We come to the following natural definition:

The pseudo-differential operator in F+(M) is a linear

Definition 1. mapping

H

F+(M) + F+(M)

:

(7)

induced by the mapping (2) of the representatives, where the tuple {H(,,} satisfies the compatibility condition (3). The operator (7) and the tuple {Ha) are denoted by the same letter, but there should be no confusion since everything is clear from the context. We denote by Tm(M) the space of pseudo-differential operators H for which

Ha E

It is obvious that if H E Tm(M), then H acts

..ana))(R2n).

in the spaces F++m+d(M)

H

:

F+ (M) -

(8)

for any s E-: R and any I > 0.

Next we intend to define several global objects associated with pseudodifferential operators. To perform this, we restrict ourselves to the considerati.r; of "classical" pseudo-differential operators. By saying that 2

the pseudo-differential operator H(x,-i aX) is classical of order m, we

mean here that its symbol H(x,p) E Tm )(R2n) has an asymptotic expansion of the form: (1 l'" '' n H(x,p)

where H. (x,p) E

E H.(x,P), j=o

(9)

(R`n) and 1

n H](TAx,ti-1P)

- Tm 1Hj(xrP)

(10)

for T

1, A(x) large enough. The operator H E Tm(M) is classical if each Ha is a classical pseudo-differential operator.

Introduce some function spaces on M. (9) and (10) will be denoted by

(11

0

x (R2n) - U On

(R2n)r

p0-

.,a

The space of functions satisfying )(R2n); set also P(1 ,...,I ) n

n

1

(R2n)

m

n m Pm(J11,...ran

(R2n).

0

We denote by pm(M) the space of functions the function (pa)

such that for any a

(f) defined in Da may be prolonged to a function

235

X

Then by

(al(a),..., 1 n (a))(R2n) (here,lm E R U

belonging to P1° 0

Pm(u), u c H we denote the space of functions on u, which are restrictions 0

Properties of the transition homomorphisms y0a (see of elements of Pn(M). Definition I of Section 3:D) make it evident that Y (ua)

ua(P(al(a)....,an(a))(R2n).

Let H : F+(M) + F+(M) be a classical pseudo-differential operator of order m. Set Pm(ua).

ha s ua(Ha)

(11)

The compatibility condition (3) yields, via composition theorems (Section 3:C), certain conditions on the functions ha. These conditions read

ha = ta8h6 in U an U(12) here tab is an asymptotic differential operator; tab ~

E (-i)jtaBj' J.0

(13)

where taBj

u6) :

Ps-'(ua n u6) for any a.

Pa(uo n

(14)

is a homogeneous of degree -j differential operator of order < rj with real coefficients in ua fl

u6' ta60 ' with respect to degree of homogeneity

1.

Consider the asymptotic expansion

0

ha =

E (-i)jha., -0

haj F Pm J(ua);

(15)

haj is quasi-homogeneous of degree m - j for large AM, and consider also the analogous expansion for h6. Collecting in (12) the terms with equal degree of homogeneity, we obtain the infinite system of equations: h

no

h8o

;

ha

= h l

+ t

al

h 061 Bo'

........... .

(16)

The first of equations (16) leads to the following: Proposi;ion 1.

The principal term hao of ua(Ha) is a globally defined 0

function on M. This function will be denoted by ho E Pm(M) and called the principal symbol of the operator H.

As for lower-order invariants, the situation is rather more compliIt is probably impossible to define the "subprincipal symbol" of the operator invariant to any canonical coordinate changes. However, a substitute may be defined in our case; this substitute is valid only in the canonical charts of the atlas. It follows from the cocyclicity conditions for Tab that the operators tab satisfy the (asymptotic) cocyclicity conditions: cated.

taatBy = toY in ua n U6 n uY' ta6t6a = 1 in Ua n U6.

'(17)

(18)

Since the principal term of the operator tab equals the identiy operator the conditions (17) and (18), in particular, yield the-following conditions for the operators ta61 ta61 - -tSal

in Ua fl U61

ta91 - tayl + t6Y1 = 0 in Ua fl U6 fl UY>

(19)

(20)

that is, ta61 is an operator-valued 1.-cocycle on M.

Again it is not difficult to find the operator-valued 0-cochain as on M such that ta61 is a coboundary of it:

oa - a6 ' ta61 in Ua fl U6.

(21)

We make use of the partition of the unity t9a) and set

as = E9 ay tl in U. a

(22)

Y Y

Clearly as - a6 - 18 (taYl- t8y1) = 10 ta61

ta61 in Ua n U6;

(23)

here wo made use of the cocyclicity condition (20). Proposition 2.

The functions haub,a defined by

hsub,a a boil - oaho coincide on the intersections Ua fl U6.

(24)

Indeed, we have

hsub,a -hsub, 6 = hal - h61 + a8ho - raho = (25),

h81 +ta6lho - h61 - ta6lho = 0 in Ua fl U6. b which coincides in Thus, there is a globally defined function h willube Ua with the function (24). The function hsub called a subprincipal symbol of the operator h. The knowledge of the principal and subprincipal symbol allows the reconstruction of the first two terms of the expansion of the symbol Ha (x(a),p(a)) in each canonical coordinate system (U0,ya : U + Va)). Further "invariants" (we put this word in quotes since these quantities depend, however, on the choice of the partition of unity) may be constructed in a similar way with only technical complications, but we do not go into detail. P"1(M)

Also a question arises: the functions ho C Pm(M) and hsub C-

being given, can we construct a pseudo-differential operator H E Tm(M) having these functions as its principal and subprincipal symbol, res,ectively? The answer to this question is affirmative, as the following proposition shows: Proposition 3. There exists an operator H E Tm(M) such that its principal symbol is equal to ho and the subprincipal symbol is equal to

hsub Proof.

For any a, set I

ha = Etay6yh, where

(26)

Y

237

(27)

h - (1 -ECta61.86))ha+ hsubin Ua. d

Here the square brackets denote the comutator of operators. h does not depend on the choice of a; indeed, 11ta61

E(tadl,e63 - EE(t661,e63

The function

- t061,e63 (28)

ECta611861 - CtaB1,13 - 0 in Ua n u8 6

Thus,

tBaha

Et8atay6 h - EtBy6 h - hB'

(29)

Next, the first two i.e., the compatibility conditions are satisfied. terms of the asymptotic expansion of hn have the form: hao - ho.

hal

(30)

- Itayl8yho+ hsub - ECt061, 8 3h o - h sub + yE8y t aylho . 6

6

y

(31)

Thus the principal symbol of the constructed operator equals ho, and comparison of (31) with (22) and (24) yields that its subprincipal symbol equals hsub Proposition 3 is thereby proved. B.

Pseudo-Differential Equations and Statement of the Cauchy Problem

Let H be a pseudo-differential operator in F+(M), H E Tm(M). equation of the form

The

(1)

H4+ - v,

where v is known and * unknown elements of F+(M), is called a pseudodifferential equation in the space F+(M). Our aim in this book does not include the solution of general pseudo-differential equations in the space F+(M). We consider only one special case which is a necessary stage in the procedure of solving operator equations, considered in Chapter 4. We mean the Cauchy problem. Let M be a proper quantized symplectic R+-manifold. Consider the line R1 with the coordinate xo = 1 and the trivial action of R+. The direct product l4 - M x T R1 is obvioJhly a proper quantized symplectic R+-manifold. Indeed, the canonical charts on M x T * R may be obtained as the direct products of those on M and of T*R1:

ua - Un x

T*R1

,

(2)

and all the conditions of Definition 1 of Section 3:D are easily verified. Lit + E F+(M).

For each fixed t E R1, the mapping i*

+ : F+60 ± F+(M)

(3)

is defined taking the equivalence class of (To) to the equivalence class The mapping acts in the spaces

of (T It-to)'

i+ : F°(M) + F°(M)

for any m.

238

We denote i+(T) - r(t).

(4)

Let H E T1(M) be.a pseudo-differential operator in the space F+(M). def pa + H E T1(H). Then H The Cauchy problem in F+(M) is a problem of solving

Definition 1.

the equates (-i 8t + H)y =HIV - 0 in F+(M)

(5)

with the initial data 0o E F+(M).

(6)

We also consider the non-homogeneous Cauchy problem, which diff%rs from(5) - S6) in that on the right-hand side of (5) we have some given element of FF+(M1. Also the case may be considered when the operator H depends on (t). Finding the solution of the problem (5) - (6) usually fails since the estimates which one manages to obtain are not uniform with respect to t E R1. -

Hence we may consider sore fixed segment K C R1. K S 0; for example, K - (0,T], then set M - M x T K and repeat the above arguments. Thus we The initial obtain the definition of the Cauchy problem on the segment K. data may be imposed for any fixed to E K, not necessary for.to - 0. Proposition 4. Cauchy problem

(Duhamel's principle).

Consider the non-homogenous

(-i at + H(t - v 1

(7)

*(0) - 0

If the solution of the auxiliary homogeneous

on the segment t E [0,T]. Cauchy problem

(-i

I

+ H(t))xt - 0

at

(g)

xt (to) - v(to) o

on the segment Cta,T] exists for any to E (0,T] and depends continuously on to (in the sense that there exists a family of representatives (x to

the elements of which depend on to continuously), then the problem (7) has a solution of the form: t

*(t) - ifo to(t)dto.

(9)

where the integral with respect to the parameter is defined via integrals of the representatives. Proof.

Passing to representatives, we have a

-i

where Ess(ETa AC

) 9 K

xt a at

+ Haxtaa - At,a

: K () P

- ¢.

(10)

Set

*a(t) - if xto t 0 (t)dto. 0

We have then

239

t

-i at (pa(t)) ° J:(-i at Xtoa(t))dto + Xtoa(t0) (12) t

v(to)+iJ (At o(t)-H(t)Xt0 a(t))dto = v(ta)-H(t)i (t)+Ea, 0

0

where t

Ea . J 0 t0

a(t)dta

(13)

and therefore Z. satisfies the same condition as At a.

Proposition 4 is

o

proved.

To obtain solutions of homogeneous Cauchy problems for classical pseudo-differential operators in F+(M) we need to construct the canonical operator acting into the space F+(M). This is performed in our next item. Canonical Operator on a Lagrangian Submanifold of a Proper Quantized Symplectic_R: Manifold M C.

Define first of all essential a.ubsets for elements of the space F+(M).

Let H E T(M) be a classical pseudo-differential operator. point z C- ua. The symbol H. has an asymptotic expansion:

Ha(x(01).p(a))

=

E(-i)jHaj (x(a),p(°))

Consider any

(1)

(cf. (9) of item A). We say that zess supp(H) if and only if there is a neighborhood of the point Ua(z) such that for (x(a),p(a)) belonging to this Ha.(rX(a)x(a),TI-X(a)p(a))

neighborhood the function

vanishes for r large

enough for each j e 0,1,.... 31f z E ua (1 u8, this condition on z does not depend on the choice of the chart in view of the compatibility conditions (12) of item A. It is clear that ess supp R is a closed R-invariant subset

of M. Definition 1. Let 4) E F+(M). The closed R+-invariant subset K Cr M is called an essential subset for y (we write R C Ess(*)), if we have 4 - 0 for any classical pseudo-differential operator H satisfying the condition:

K n ess supp(H) = 0.

(2)

Proposition 1. Let q E F+(M). There exists a representative 4a} of 41 such that K C Ess(y) if and only if the closure p.(K) n Da is an essential subset for ya, a - 1, ..,Mo. Proof.

First of all we construct the family of operators p(a) E T '(M)

such that (i) ess supp(p(a)) C ua and (ii) Ep(a) - 1.

Each operator p(a)

is given by a collection of symbols, whose image on M we denote by p(a)s, B = 1,...,H0. Note that by virtue of (i) the definition defines the operator p(o) completely. We denote p(()a by pa for short. The condition (ii) reads

I

ItaBpB '

1 in Ua for all a.

(3)

tool - 1

(4)

Note that and, consequently,

240

tasjl - 0, j - 1,2,....

We seek ps in the form

(It follows from the fact that

ps -

(5)

E p8., p8.E P M.

(6)

j-O

It suffices to satisfy the system of equations EPso - 1,

EtaY1Pyo in Ua for any a,

EP61

(7)

EP B2 = -ItayiPyl - Ytay2Pyo in Ua for any a,

in such a way that supp pYj C UY.

Set

Pso - 6s,

(8)

where Be is a partition of unity (see Definition 1, Section 3:D). second equation in (7) reads now

The

Sspsl - -EtaY16Y in U.

(9)

The right-hand side of (9) does not depend on a. Indeed, in view of the cocyclicity conditions ((20), item A) it may be rewritten in the form (in ua n us)c

- Et 6 6 6 + Et It Y ayl y Y 6yl y Y a61 y

= YEt6yl B+ y

t a611 - Et 6yl 6y .

(10)

Y

Thus we may set Psl = 6s(EtaY l0 ) in U S n Ua

(11)

.

Y

Repeating the process (the next stages are, however, somewhat more complicated), we construct the desired operators p(a). We obviously have def

` a(a)Y

(12)

a(a) M

For each V+(a) we may choose a representative (0(0)s}0`1 such that only 0(a)a is different from zero.

Indeed, it suffices to set

{(a )a -

It is clear that (V+(a)a}Mo

ETa$(p(a)W)s.

(13)

s

is a representative of P.

ua(K) n Da E Ess(4P (%)a) for any a.

Now let K C M,

Then if K n eas supp(H) - 0 for any a,

We have a ess supp(Ha ) n u a (K) n Da - 0, and it is easy to establish that

a (a)a r 0' 0

Conversely, let K C Ess(*), and let H

be an operator in I (a1(a),...,

a

ln(a))(RX(a)) such that aupp Ha n ua(K) n Da - 0. (a)

Consider the element

E F+(M) induced by the representative

0, 6 {*(a)6}

a B - a.

(14)

241

We have obviously (15)

H(a).p(a)*,

p(a)

where-H(a) is any element of T.(M), prolonging the operator Ha, and thereProposition 1 is proved.

fore q+o . 0 since ess supp (H(a),p(a)) fl K - 0.

We now come to the construction of the canonical operator. Definition 2. Let L C M be a Lagrangian manifold. proper Lagang a manifold if for any a the manifold La

µ,(L n U

C R2n(a) (x

L is called a

(16)

(a)

,p

)

is a proper Lagrangian manifold in the sense of Definition I of Section 3:B. Definition 3. Let L CM be a proper Lagrangian manifold. The proper L of L by open sets together atlas on L is a locally finite covering Y Vo

with the coordinate mapping, defined on these sets, such that the following conditions are satisfied: i)

For any a the corresponding a - a(a) is given such that Va C L fl u..

ii)

The coordinate mappings have the form (x(a)(z).p(a)

V a 9 z + (x(a),P.a)) `

I

I

(z)).

(17)

where a - a(a) and I = I(a).

iii) All the intersections Va f1Va, of the canonical charts on L are also R*-precompact. iv)

For any a the charts {Va}a(a)

form on the Lagrangian manifold

La - ya( U

Va),

(18)

a(a)-o a proper atlas in the sense of Definition 2 of Section 3:B.

on L such that 0 may be

We denote by Dm(L) the space of functions represented in the form

(19)

EOa.

where (ua)-l$a E 1

( L ) , and by D°1(Va) the space Om(Va) - ($ E Ot(L)Isupp x

Va).

Let u be a given measure on L, homogeneous of degree r, and such that

(a)

in any canonical chart Va the density uI(XI I(a)) satisfies the estimates

jaIYIvI(xia),p(a))I Y

Lj

(a) )

$p";

(here a - a(a), I


q>P)>

aq

glt.0 ` q0, pIt_O

(12)

p0, (w,g0,p0) E SIC

are defined for 0 14 t < T(w,go,p0) and the mapping

[O,T] x P, + 10,T] xttgnX[t', (t,w,g0,p0)

-I,

(t,q(g0,P0,w,t),w)

is a proper one.

(b) The inequality

6 0

(13)

(w,h,q(g o,P0,w,T(w,q0,P0),P(q0'P0 ,w,T(w,90 ,P0 ))) < -E.

(14)

ImH

ess

holds on these trajectories; besides ImH

ess

Proof of the theorem.

We shall seek the operator GN in the form 2 2

GN - GN(-i ax ,x,x);

(15)

then we obtain the following equation for the symbol GN({,z,x): 2

2

2

2

F(x ,...,x ,z ,...,z ,t

n

1

n

1

1

1

1

axl

azn

1

n

1

1 aicn -1 azn

x

(16)

X GN(C,z,x) - 1+RN(E,z.x), where one should take GN in a form such that RN satisfy the conditions of Theorem 1 of item A (more precisely, the multi-dimensional version of this theorem is considered). We change the variables: Ei

h p°+lwi, xi

qi, zi - h plgn+i, i - 1,...,n.

(17)

Taking into account Theorem 1 of item A, one may rewrite equation (16) in new variables in the form (here gN(q,w,h) - GN(h pn+1w1,...,h p2nwn,h PI x x qn+1,...,h-Png2n,gl,...,qn)): 1

hH(w,h,q,-ihq)(gN(q,w,h)) = h-tHgN(q,w,h) - 1 + O(hN1).

(18)

Thus we come to the h-l-pseudo-differential equation on the function of

gN(q,w.h) Using the partition of unity, we can rewrite the right-hand side of (18) in the form:

264

(19)

1 - P1(q,w)+P2(q,w). where supp P1(q,w) C x

((q,w)I(wI/RPn+1/2,

..,wn/RP2n/2,g1,...,gn, X

E2/p

(20)

gnat/Rpl/2,...,g2n/RPn/2, 0) co}.

Here

n+j + Iqn+j l 2/p j);

R - E (J w.3

j

Ho(w,q,0) # 0 for

(q.w) E supp P2.

We shall seek gN in the form (21)

gN - hr(gNl + gN2),

where gNi satisfy the following equations:

_

+6(01), i ' 1,2.

HgNi ' Pi(q,w)

(22)

First we solve equation (22) for i - 2. Since Hp(w,q,0) # 0 on supp p2, the solution can be obtained by means of successive approximations: N1

E 8N2)(q,w,h)hk, k-0

(23)

where g(2)(q,w,h) depend continuously on h E 10,1).

We obtain the system

gN1(q,w,h) -

of equations which enables us to find the functions %2)(q,w,h): (w,h,q,0)(-ih 2q)k'jz08N2)(q,w,h)hl ' P2(q.w) + 0(hN)

(24)

JkIE0 2pk

This system can be solved recurrently, since H(w,h,q,0) # 0 on supp P2 for sufficiently small h. We shall seek the function gN1(q.w,h) in the form: T f0 BN19.w.h ' h

(25)

where the function * satisfies the following Cauchy problem:

-ih at+Htp - 0(hN*2).

t-0

-

P1(q,w)+0(bN+1).

(26)

The absorption conditions imply (see 152)) that the solutjonNof Cauchy problem (26) exists on the segment (0,T] and that *(T) - 0(h ) due to (14). Although to be precise this fact is proved in [521 only for finite initial data, nevertheless we use the initial data which are not finite. However, the proof given in 152] is suitable without essential changes if in the definition of absorption conditions one replaces the requirement of finiteness of initial conditions by the requirement that the trajectory tube projection on physical space be proper. This was made in Definition 4. Then we have:

265

1T -

Hgtfl

Ti '

HiydT+0(hN+l) - -rT 0

0 +0(hN+1)

(0)

dT+O(hN+1)

at

(27)

- ol(x,w) +0(hN+1)

Hence the solution of problem (26) is constructed and problem (18) is thus solved. Returning to initial variables, we complete the proof of Theorem 1.

2.

POISSON ALGEBRAS AND NONLINEAR COMMUTATION RELATIONS

A.

Poisson Algebras

Let N be a smooth manifold of dimension n, and let a smooth homomorphsim 0

:

(1)

T N - TN

of vector bundles over N be given.

(Thus n maps linearly the fiber T N y

over the arbitrary point y c- N into the fiber TyN over the same point, see (191.) the induced homomorphism of section spaces we denote by the same letter T°N + TN.

Given a function f E C(N), we consider a vector field Yf on N, given by Yf - n(df),

(2)

and define in C(N) the bilinear operation {f,g} _ {f,g)0 - Yf(g), (3)

f,g E C'(N).

Definition 1. The space C _(N), supported with bilinear operation (3), the P aeon algebra on N if and only if for any functions f,g,h E is called

E C"(N), we have (f,g} - -{g,f}, (4)

({f,g},h) + ({g,h},f) +{{h,f},g} - 0.

We denote the Poisson algebra by P(N) = P(N,0). Let (yl,...,yn) be local coordinates in some coordinate chart U C N.

Using standard coordinates, corresponding to the bases (dyl,...,dyn ) and 2

) in the fibers of T U and TU, respectively, we set a matrixayn valued function Inik(y) Il into correspondence to the mapping n.

'

(.

,...,

Lesms 2. The conditions (4) are equivalent to the following conditions, given in terms of local coordinates:

nik(Y) +nki(Y) - 0, an. (y)

Eln k

n. k (Y) it (Y) -'L-+ ayk

an

(y) 81 ayk

The proof consists of simple calculation. tions (4) of Definition 1 are satisfied.

(S)

an.. + Psk(Y) ayk

U. 3

- 0.

(6)

Further, we assume that condi-

Definition 3. The function (3) is called the Poisson bracket of the functions' f and g, and the vector field Yf, given by (2), is called the Generally, the Eulerian vector field correspondent to the function Yf. field Y will be called Eulerian if it may be locally represented in the form If for some f E C°'(N).

The Poisson bracket and the Eulerian vector field possess Lemma 4. the following properties: Yf{g,h}'- {Yfg,h}+ {g,Yfh},

[Yf,Y9] s YfYg-YgYf =

(7)

Y{f,g),

(8) (9)

LYf (n) - 0.

In the local coHere LY is the Lie derivative along the vector field Y. ordinates the Poisson bracket and Eulerian vector fields are given by the formulas

m

Yf

3f

E

i,k=1

{f,g} =

nik(Y) 2

yk

a

3

(10)

yi

i,k=lnik(Y) aYk ayi

(11)

The equalities (10) - (11) follow directly from definitions; Proof. (7) i valid since (2) and (4) are. To prove (8) consider an arbitrary Applying the commutator [Yf,Y91 to it, we obtain function h E C°°(M). CYf,Y9]h = {f,{g,h)} - {g,{f,h}) _ (12)

-{{g,h),f) -{{h,f),g) _ {{f,g},h) - Y{f,g}h (here we used the Jacobi identity).

Now we prove (9). Let ¢t be a local one-parametric group f diffeomorphisms of N, generated by the field Yf. We will show that 0tn - n. Really, it follows from (7) that 0t preserves the Poisson bracket: mt{g,h} -

(13)

We may interpret St as a section of vector bundle TN ®TN; in this interpretation the formula which defines the Poisson bracket becomes

{g,h} _

(14)

(the brackets denote the pairing of covariant and contravariant tensor fields). From this formula it follows that the invariance of the Poisson bracket implies the invariance of S2. The lemma is proved.

Note 5. Contrary to the case of symplectic manifolds and Hamiltonian vector fields, we cannot consider the equality Ly(n) - 0 as the definition of the Eulerian vector field since even locally there may be no function f E C`°(N) such that Y - If. (A trivial example: n - 0. Then Y is arbitrary although Yf = 0 for any f E C°°(N).)

Thus we have shown that the Eulerian vector fields on N form an algebra Eu(N) - Eu(N,n) and that the correspondence f Yf is the representation of P(N) in Eu(N). Now we prove simple assertions about the homomorphisms of Poisson algebras.

267

The homomorphism of Poisson algebra$ is a Lie algebra

Definition 6. homomorphism

S

:

P(Nl,AI)-P(N2'n2),

(15)

for some smooth mapping 0 : N2 + N1.

such that 0 '

Let P(N) be a Poisson algebra and f E P(N) be an element such Lemma 7. that the vector field Yf generates the global group {fit} of automorphisms of N. Then {4*} is the group of automorphisms of Poisson algebra P(N). The proof follows from (9). Let Lemma 8. point y E N2 we have

be a homomorphism of Poisson algebras.

0*(D2(y)) ' R1 OW).

Then for any (16)

Moreover, for any function f E P(N1,f21), there is a relation

m*(Y0*fW) ° Yfl)4(y)). Proof.

Let g,h E P(N1).

Then

<S21,dgedh> (m(y)) ' *{g,h}(y)

d*h)(y)

_

(18)

(y) - <S22,m(dg ® dh)> (y) and since g and h are arbitrary. we immediately obtain (16). (17). Using (16), we obtain

0*(Y(2)W) `

*(n2(dO*f))(y)

Prove now

° (19)

a(n2($ df))(y) - S21(df)(O(y)) ' Y(1)(m(y)). f

The lemma is proved.

The well-known example of the Poisson algebra is the Poisson algebra The symplectic form w2 obviously defined a linear isomorphism of spaces of vector fields and differential 1-forms on M which sets into correspondence to a linear field Y E TM the differential 1-form of functions on a symplectic manifold M.

a (Y) =Y,lw2

(20)

(the fact that a is an isomorphism follows from the independence of the form w2). *

Denote by 0

:

T M ± T M the inverse mapping 0

a-1.

(21)

Lemma 9. The closure of the form w2 is equivalent to the condition (6) ((5) follows from the fact that w2 is an exterior form). The proof reduces to a straightforward computation.

Thus on any symplectic manifold there is a natural Poisson al ebra with a nondegenerate mapping Q. Vice yersa, let the mapping Q : T"M -+ TM defining the structure of Poisson algebras, be nondegenerate. Then M is even-dimensional, orientable, and we may define the symplectic structure on M, setting

268

(22)

w2(Y,X) = ST-2(Y)(X).

Next we study Eulerian vector fields on M. Let f E C'(M). a-2(df) or field Y = Yf is defined by condition Y y .l w2

Then the

df.

(23)

We have also LYW2

= d(Y J W2)+ y J dw2 = ddf =

(24)

0

since w2 is closed.

On the contrary, let Y be such that Lw2 - 0. Then the form y J w2 is closed and, consequently, there exists always a function f (locally) such that (23) is satisfied and thus Y - Yf. Hence the algebra Eu(M) for the symplectic'manifold M is the Lie algebra of Hamiltonian vector fields. Let y E M be an arbitrary point. v By the Darboux theorem in the vicinity of y there exists a system of local coordinates, in which the form w2 reads E dpj A dqj a dp A dq. j-1

w2

(25)

In the coordinate system (gl,...,gn,Pl,...,pn) the HamilLemma 10. tonian vector field and Poisson bracket are defined by the equalities a of a of a n of a Yf - of ap aq - aq ap = JEL(ap3 aqJ - qJ a -j)

Proof.

(f,g} = Yf(g).

(26)

Calculating Yf J w2, where Yf is defined by (26), we obtain

YfJ w2

E (dgj(Yf) Adpj- dpj(Yf) A dgj) -

J-l E

j-l

(27)

(af dp.+af dq.) j - df, apj aqj J

i.e., we come to (23). The second of the equalities (26) is the immediate consequence of the first one. The lemma is proved. B.

Poisson Algebras and Commutation Relations with Small Parameter

Important examples of Poisson algebras arise in the consideration of nonlinear commutation relations with small parameter h - 0. Let H be a Hilbert space. Assume that an n-tuple Al - A1(h),...,An - Ah(h) of self'13,

adjoint operators, depending on a small parameter h 6 to that the Commutation relations

1Aj,Ak] . ihf2jk(A), j,k - 1,..,n are satisfied.

Here

is given and

(28)

njk(yll.... yn) are the given symbols and we use the

standard notation I n f2jk(A) = Qjk(Al,...,An

Perform the coordinate change

zl - l(y),...,zn = mn(y)

(29)

and introduce the operators

269

(30)

B1 - 1(A),...,Bn - On (A). Thnn a4

(BBk3 s ih E

r,4 - ihrEb(

-

0(h2) (TYU ayk Drb)(A) +

r

6

(31)

ask ayb Orb)(

(B)) + 0(h2)

(to prove (31) it suffices to apply the formula of indexes permutation and It follows that after the coordinate change the K-formula of 152]). collection of functions Qjk(y) transforms as a contravariant tensor of rank 2. Namely,

It turns out that the conditions (5) - (6) are rather natural. [Aj,Ak] +[Ak,Aj] = 0,

so it is natural to require that the identity (32)

njk(y) + nkj (Y) - 0

hold; further, the following statement is valid. Lemma 11.

(135]).

For any symbol f there is a commutation formula

(A.,f(A)] - ih E 3

(Q.

m-1

f)(A)+0(h).

(33)

]m aym

Applying this lemma, we obtain I n

a4

[A6 ,[A ,A k ]] - -h2 E (n bm aLA) (A) + 0(h2). m M-1

(34)

Using the Jacobi identity for commutators, we come directly to (26). The above considerations were not completely rigorous and played essentially the role of a hint, but we have shown the natural role of Poisson algebra in the asymptotic theory. We come now to exact discussions and formulations.

3.

POI88011 ALGEBRA WITH U-STRUCTURE.

LOCAL CONSIDERATIONS

In this section we give the construction of v-structure for a Poisson algebra of fu.,ctions on Rn, with the given fixed coordinate system. Item A is purely technical; we introduce some new symbol spaces which enable us to perform later the asymptotic expansions and to estimate the remainders. In item B the conditions on operators are imposed and the v-mopping is In items C and D we establish the composition formulas for the defined. product of an element of the algebra with, respectively, another element of 0

the algebra and a general operator, whose product lies in L

(Rn

n (in particular, it may be a CRF (see Chapter 3, Section 3)). Almost all the geometric constrictions were previously developed by Karasev (33,34], Karasev and Maslov [38,36] for the case of the small parameter asymptotics; however, the main ic!a being slightly modified, the proof technique is completely different since the methods within the cited papers to estimate the remainder do not work in the quasi-homogeneous situation. 1

Z70

It seems quite probable that the conditions imposed on the operators A1,....An in item B have not taken their final form yet, and the relationships between them still need further investigation. Some Auxiliary Function Spaces

A.

Let A - (A1.... ,An) be a given n-tuple of non-negative numbers such

that the set I. - {JA. > 0} c [n] - {1,...,n} is non-empty.

We assume A

to be fixed throughout the subsequent exposition.

For any n-tuple r - (r1,...,rn) of natural numbers consider the space Rr def Rrl Rr20 ... Rrn with coordinates x (x(1)....,x(n)) divided into "blocks" or "clusters" x(j) - (x.1,...,xjr.), J action of the group R+ on Rr by

and define the

r(x(1),...,x(n)) - (Tllx(1)....,tlnx(n)).

(1)

i.e., the element f=- R+ acts as multiplication by r J within the j-th "block" of variables, j - 1,...,n. r

Denote by Er the set of all the mappings a

:

I+ -

N - (1,2,...},

such that of < rj for all j E I+. function A0(x) - Aa((xjoj)j

j + aJ.,

(2)

For any a E Er we define the smooth

I+), satisfying the conditions: (i) A0(x) > 2

for all x; (ii) A0(x) is quasi-homogeneous of degree 1 for A0(x) > 1, i.e., A0(rx) -

rA0(x) for r > 1, A0(x) > 1;

(3)

(iii) there are the two-aided estimates cAa(x) < 1 +

Ixjai11/J1J < CA0(x)

E

(4)

j e- I+ with positive constants c and C (see Chapter 3, Section 3:A for detailed construction of such functions). We also define the function Ar(x) Ar({x(j)}j I+), satisfying the same conditions except that instead of (4),

e

we have cA (x) < 1 + E r

EJ Ix,kI1/aj < CAr(x). J

(5)

JEI+ k-1

Set

Ar(x) - min A.W.

(6)

aE £r max

The following statements are obvious; (a) Ar(x) is_equivalent to A0(x); (b) if rj - 1 for all j E I+, then Ar(x) - Ar(x).

oEEr

d

The operator of the difference derivation d 6xjf if naturally acts in the spaces

6jf

(r1,...,rJ ,....r C -(R

n)

)+

,

f < r.

Cm(R (rx,...,rj+l+...,rn)

]

),

271

f(x(1),....x(n)) * ax

je ' (xjf -

(7) (j)'....x(n))-f(x(1),..., R(j).... ,x(n))).

xj,r+1)-1(f(x(l)......

where R(j) _ (x(J)'xJ,rj+l)' R(j) - (xj1,...,xj,rj+l,...,xjrj) (xj,rj+l stands in place of xjf).

Our aim is to present function subspaces in C'(Rr), in which the difference derivatives act in a natural way and which coincide also with )(Rn) in the case when all r. are equal to one. n real numbers such that SCI

0, m < ml.

ml

We denote by Fm (R

r

Let m, ml be

(8)

(Rr) the space of functions f E C-(Rr

rn

1

n)ml

1

satisfying the estimates 3

{

a

{

l

f(x)

Ixa

{

< C A (x)

ml

r(x) ml

A (x)

r

m-m 1

0 is chosen small enough (see below). We claim that for e small enough the functions A-(x), Ar(x), and Ar(x(t)), T E (0,1}, are equivalent in Dl uniformly in t,

272

A (x)

(x)

A

Ar(x) < const, Ar(x)

< const,

A(z) r P

r

(14)

Ar(x(T))

< const, x E DI

Ar(x(T)) < const,

Ar(x)

and so are the functions Ai (z),Ar(x), and Ar(x(T)), where x(T) is obtained via replacement of xJ.f in x by Ty + (1 - T)z.

Indeed, it is enough to

prove the equivalence of A0(x) and Ao(x (z)) for any o E Er. We have A (z) ue A,, (x(T)) if o) # and if of = k, then by definition of DI and by (4 3

lY-zl ...>tm) X

=

(8)

ukf(tl,...,tm)dtl,.... dtm.

X

where by (2)

ukf(tl,...>tm) 9, 6 Ckf(l+ ItI)m(k). Now we choose k > + a(m(kj + m)

1Io1 and then define f in such a way that M - f +

2 II+I.


0 there exists N1 (al,...,am) satisfying

Set

- exp(itmAjm) x ... x exp(it1Aj1). (10)

0 such that for any multi-index a -

ak - 0 if Ajk - 0, Eakajk < N, k the following estimates are valid:

3t01

U(1,... , tm)u 16-N1 6 Ca6 (l + I t

(11)

I)m(N,6)

(12)

Bu96

for any 6 (the derivative in (12) is taken in the strong sense in X6-N1).

Next we impose the condition which makes it possible to develop the asymptotic theory. Condition 3.

(the "asymptotically diagonal spectrum" condition):

(a) Let x E Rr,

f (x) E rm (Rr ) m

(13)

1

for some ml.

Then rll

r2r2 rnrn rn1 ;A2 ,...,A2 ;...;An ,...,An. ) E Op (x)

11121 x21

f(Al ,...,A1

(here rij are arbitrary pairwise different real numbers, and

(14)

Om(x) denotes

the set of operators BX_ -X_ such that for any 6, B is a continuous operator from X6 to X6-m). (b) Given N and 6, there exists m such that for any r and ml, the inclusion

276

f(x) a rm (Rr)

(15)

1

implies that 71 1 nnrn f(AI ,...,An

(16)

Xd+N

X6

is a continuous operator.

X = {

Definition 1. A proper tuple (of tempered generators in the scale ) is a tuple of tempered generators satisfying Conditions 1,2 and 3. 6

Some explanations need to be given to make the situation clearer. Conditions 1 - 3 are of complicated functional nature and it is doubtful whether they could be derived from some simple assumptions within the framework of general theory. In practice, these conditions should be verified for concrete operators to which this general theory is to be applied. Condition 3 is of crucial importance for the theory; the reason for its name is that, roughly speaking, it asserts the following: if f(xll,", ..,x nl,...,xnr ) decays as A(x) - W in the R+-invariant vicinity of

xlr1, n the diagonal set A - (x.1 =... a xjr

for all j E I+), then the corresponding

operator is a "smoothing operator" Sin the scale {X6}. It is not difficult to show that the 2n-tuple of operators a

(-i

ax1

a

,...,-i

axn

xl,.. ,xn}

in the Sobolev scale is a proper one (here

A1- ... ° An

l Anti

' ..

2n

° 0).

It is useful to note that Conditions 1 3 and Definition 1 (and therefore all the subsequent arguments) depend on the choice of the tuple (All ...,?n).

Let (A1,...,An) be a proper tuple of generators in the Banach scale X - {X6}, and assume that they satisfy the commutation relations 1 n iwjk(Al,...,An), j,k - 1,...,n,

[A.,AkJ

(17)

where the symbols

n

Aj +Ak 1 w.

Jk

(x) r-z P (All

(18)

...,), n)( R )

have the asymptotic expansions

(S) J.k (x) =

(S) (x) E 0

x j+ak1-S

F. 0w ].k (x), w.

S=

Jk

(R-),

(19)

(als ... I an)

and the functions '.+a -1 njk(x)

E 0(Al,k..'An)(Rn

(20)

define the Poisson algebra structure

(f,g) =

i 0. of j,k=1 ]k axj axk

(21)

on C (Rn), i.e., the functions (20) satisfy the relations (5) and (6) of Section 2:A of the present chapter.

277

Definition 2. The above conditions being satisfied, we say that a The up-structure is defined over the Poisson algebra given by (21). mapping is a mapping n)

Op(x),

L(Nil ...In)(R

(22)

def

f(xl,...,xn)

u(f)

n f(Al,.... An), 1

where Op(x) is an algebra of operators in the scale x, defined at least on D. In the sequel we are particularly interested in the action of the u-mapping on a certain subspace of L(A L

A

)(Rn), namely, on the. subspace

By definition,

functions "stabilizing" at infinity.

of

at,(A 1,...,An'

1'"'' n Lam

(23)

t,(A1,...,An)(B.)

Lst,(A1,...,An)(Rn

m and f(x) E Lst'(A1 ,.

n)(Rn), if and only if f(x) E Lm

and there exists the function fo(x1 ) E L(A +

(a)

(a)

fo

.,A 1

(x1 )I < Ca N(1 + I xl I) + , o

_N

(kn),

Nit-. Ad

)(Rn) such that

n m+

)

A(x) (24)

N,IaI = 0,1,2,...,

i.e., f(x) stabilizes rapidly together with the derivatives as Ix1 14-

o

m

Thhe spaces Sst,(A1,...Xn)' Pst,(al,...,an), etc.are defined in a similar Y.

In what follows, we require that

A.+A-1 wjk e Pst,(a1,.

(25)

,An)(R°

Next we come to establishing the composition formulas for operators lying in the image of the p-mapping (22). C.

Composition Formulas for Elements of an Algebra In this item we establish the asymptotic formulas for the composition i

n

If(A1'...,An) in the case when f,g E S(A1l

1

n

g(A1,.... An)

= f(A)o g(A).

(1)

.An)(Rn); in particular, when f and g are

asymptotically quasi-homogeneous functions. Theorem 1.

Let f E

g E 5121,...,An)(Rn).

Then the

composition (1) satisfies

f(A) o g(A) -h(A) E 0 (x) = fl ON(x), N

where the symbol h(x1,...,xn) E Sl+m2 pansion

P

(2)

An) (Rn) has the asymptotic ex'

h(x) = F(x)g(x) +

E Bj(f, g](x); j=1

(3)

here B.1f,g] E S'3> )(Rn) is a bilinear form 3

n

a E` j b. 3a 6 (x)f(a)()g(B)(x),

B.Ef,g](x) =

+Enk

(4)

fBl'j

1'

where bja6() E p(

=l k jak+Bk)

(R n);

in particular,

n (5)

B1Cf,g] - B1Cg,f] _ -i{f,g},

where {f,g} is the Poisson bracket of f and g. it is easy to see that (5) is a principal term of the symbol Remark I. of the operator lf(A),g(A)] (the product cancels out). Thus, the theorem, f in particular, asserts that the mapping u v(f) = f(A) is an "almost :

(Rn) with the Poisson bracket

representation" of Poisson algebra Sm given by (21) of item B.

We make use of the permutation formula

Proof.

1

2

2

620

2

1

(A,B)- $(A,B)

CB,A]

1

524 (6)

(A,A;B,B).

bxldx2

This identity may be easily obtained using Theorems 2 and 3 of Chapter 2, Section 3:D. We have 2

2

1

2

1

1

2

3

3

AC

5

1

60 dz

4

60 dx

(A, A; B) 1

1

3

2

(A-A) dx (A,A;B) 1

34

52

1

3d

1

¢(A,B)-0(A,B) _ (A,B)-Q(A,B)

2)

620 16X2

(A, A; B)] = 1(B - B)

dx

1 12 4 (7)

(A, A; B, B) _

1 3

6

(AlAS'B2B4

CB,A] dx26x 1

2

Q.E.D.

In this formula + may depend also on several other operators none of which, however, acts "between" A and B. Apply the identity (6) to the in f with An,...,A2 in g, then product (1), permitting successfully A 1

coming forth to A2 in f and so on.

n+l

2n

n

I

We obtain in this first stage:

n

1

n

1

f( A1,...,An)g(A1,...,An) = f(A1,...,An)g(All .... An) k+2

j Ekw3k (A 1

- i

k+n+1 ) df . An

dxj

d

6xk

I j-1 k+l k+n+2 2n+4 (A ;...;A. A A. ;A.

l

3-1

3n43-j J1''.

.

J

]

1 k-1 k k+n+3 k+n+4 (A 1; ..;Ak-1;AkAk ;Ak+l ,

'An

)

x(8)

2n+3 '

,An

)

to thespace fm1+m2-1 (R(rl,.'''r1) where rj E {1,2;3,4} (S = max{ml +m2 - 1, 0, Xj + Ak - 11),

The typical termof the sum in (7)

(8).

Next we apply.the permutation procedure described above to the sum in As a the difference derivatives transform'into usual ones and

279

SI+m2-2(Rk.) for some r and S, depending

the remainder appears belonging to

on the difference derivatives of f, g (and wjk). repeated N times, we-obtain

This procedure being

N-1

f(A) o g(A) = (fg) (A) + E BEf,g](A) +RN,

(9)

j=1 j

where the symbol RN of the operator RN belongs to the space

fm:+m2.`N(RrN)

SN Note that at any step of this process RN is a bilinear form of the difference derivatives of f and g, and that the operators under f and g function signs always act in the proper order. This yields that when passing from RN to RN+i the order of the difference derivatives of f and g involved increases exactly by 1 and subsequently depends on the derivatives of f and g of order 4 j. Summing the asymptotic series, we obtain the function h(x) satisfying (3).

Equations (3) and (9) togetherwith Conditipn3 of item B show that f (A)o,g(A) - h(A) E O-N(x) for any N; therefore (2) holds.

It remains to verify (5).

We have BlCf,g] _ -i E

njk(x) af(x) aaa(

j 1).

Since njj

0

and njk + 0kj = 0, we have

- of ate) B1 Cf,g] - B1 Eg,f] - -i jE< k0 j k(af 2xj axk xk xj

_ (11)

of

-i E ci jk axj axk j,k

-i(f,g), Q.E.D.

Theorem I is thereby proved. D.

Composition of Elements of an Algebra with Operators

Whose Symbols Lie in L(A

1 )(R ) n

1

This item is devoted to the solution of the problem: given the symbols F(x) E Ss

t,

(al,...,an)(Rn) and O(x) E Lst,(al,...,an(Rn). 0

construct a symbol l(x) E La

t,(11, .an) (Rn) such that

F(A) o 0(A) __ m1(A) (mod OP (x)).

(1)

A brief analysis shows that the argument of item C fails in the considered case and that some new techniques are necessary to provide the construction of A1(x). However, the idea is quite simple (although the estimates appear to be cumbersome). We have* ,P(A) - (27r)

2 -n/2 6 (t)eiAtdt a (2n)-n/2JFx i tEX(x,_i 1 1 2

x)'(x)I x (2)

11

x eiAtdt (mod 0p (x))

_

(2-ir)-n/2J@(t)CX(-i a[ ,t)eiAt]dt (mod Op (x}).

* Here and below we write eiAt

280

def eiAntn, eiAltl

for short.

)(R2n), X(x,p) = 1 in the neighborhood of

where X(x,p) E Tst (X

n

1

some set K E Ess(t). (R

Assume that for any f(x,p) E TW ( st' X1" ' find a function

2n

), we have mansgeu to

Xn)

def L(f)(x,p) F(x,p)

E Tst,(X1,...,Xn)(R2n

such that

2

1

f(-i

(3)

,t)eiAt fl F(A,t) o eiAt+g(t),

(4)

2t

where R(t) is a smoothing operator in the scale X, satisfying the estimates

to (at)8R(t) 116 - 6+N for any a, 6, 6, N with Isl

Then we obtain, modulo 0p (x)

- 0.

I

(5)

Cn86N

0

(2n)-nl2r'(t)F(A) o f(X) (A, t) o

F(A) o (P (A)

eiAtdt.

The composition F(A) o f(X)(A,t) may be calculated using the results of item C and we obtain F(A) o @(A) _ (2n)-n/2J;(t)H(A,t)o eiAtdt,

where H(x,p) E Tst

(R2n).

X

(X l,,

.

If L-1(H) is defined, then we obtain 1

n)

2

,t)eiAt}dt;

F(A) o 0(A) _

at

that is, (1) is valid with 2

1

0(x) = L-1(H)(x,-i at)l(x).

(6)

Our argument, carrying out the above program, consists of two stages: (1') we perform some formal calculations and expansions to obtain (4); and (2') the estimates of the remainders are derived, thus validating the calculations made in the first stage. We have

We come to the first stage. 1

n

1

,t)(eiAt)

f(_1

n

f(A1,.... An,t)eiAntnx ... x eiAltl.

3t

(7)

We intend to find a function F(x,t) such that

n

n

n

2n-1

2n-1

f(A1.. A n ' t)el(A1 -A1)tlx... x ei(An 1

1

- An)tn

n

(8)

F(A1,...'An,t) (mod Op (x}).

The problem of solving (8) for F(x,t) still cannot be treated by straightforward expansions like that in item C, since we are a posteriori interested in values of t of order t. ti A(x)l-X]; therefore the derivative ax x

i(x - y)t

1-X

-X

(e > 3 ) has the order A(x) 3 rather than A(x) 3 necessary for these expansions to be applied. However, it appears that the problem may be reduced to solving a number of differential equations of the first order x

in t.

281

To perform this we introduce the system .of unknown functions

Fk(x1,.. ,xn,yk+l,...,yn,t1,...,tn),k = 1,...,n (9)

Gk(x1,...,xn,yk'yk+l'...,yn,tl,...,tn),k . 2,...,n+1 (Gn and Gn+1 do not depend on y) and require that the following conditions be satisfied: F,(x1,...,xn,y2.....yn,t1,...,tn) - f(xl,y2....1yn,t11 ....tn); Gn+l(xl,...,xn'til .... tn) = F(x1,...,xn,t1,...,tn);

2n -n 2n n n+k+l Fk(Al,...'An.Ak+1 ...,A n,tl,...,tn)e ( A n -A

n)tnx...xe

n+k ( Ak

(11)

-k

Ak )

2n n n+k+l = Gk+l(All ...'AnAk+l ....,An),tl,...,tn)) x

2n x

ei(An-An)tnx n

Gk(A1.

n+k+l

-n

k. (12)

-k-1

xei(Ak+l -Ak+l)tk+l-Rk(t), k - 1,...,n; 2n

2n

n+}r

,An,Ak ,...,An'til .. n

i

(10)

-n

n+k

.t)ei(An- An)tnx

n+k+l

2n

2n

-k

...x ei(Ak - Ak)tk -n (13)

i(An - An) tn x

= Fk(Al, " 'An'Ak+l r...,AnIt1,...,tn)e n+k x

k

... x al(Ak -Ak)tk+Rk(t), k -

where Rk(t), Rk(t) satisfy the estimates (5).

Our scheme is as follows: we start from k . 1 and define Fl by the equality (10). Next we solve (12) for G2. After this successively for k = 2,...,n we solve (13) for Fk and then (12) for Gk+l. Finally, F is defined by (11). The crucial point of our analysis is the solution of (12). We introduce the function Wk(xl'-""n'yk+l'yn't1,...,tn,to) depending on the additional parameter to E 10,tk] such that n

dto

2n n+k+l {Wk(A1,...,An,Ak+1 ,...,An,t1....,tn,to)el

2n

-n

An Ad tnx

... X

n+k x el(Ak+l - Ak+l)tk+lei(Ak - Ak)(tk - to)) . R (t,t ) k o n+k+l

-k-1

(14)

where Rk(t,to) satisfy estimates (5) uniformly in to e 10,tk7 and

Wklto=o . Fk;

(15)

so clearly we may set Gk+l . W klto.tk

(16)

Rk(t) - -ft k kk(t,to)dto EOp (x).

(17)

which yields

282

To solve (14) we calculate the derivative on the left-hand side of this equality and obtain aWk

d

dto

{...} _

n (All ...,A,,,yk+l,...,yn,t,to) +

(

ato

i1'' ky.t. 0 n n+1 1 J= J JI + i(Ak - Ak )Wk(Al,...,An,Yk+1'...,yn,t,t0))e

n+k x Yk

(18)

....fin.

x e

yn-An

e-iAntn

i

k(tk to)e-1 'k+ltk+l

.. x

x

We seek the formal solution for Wk vanishing the expression in-the curjly 2n n+k instead of Ak ,...,An and omitting temporarily brackets. (Writing

the product of exponents occurring in (18) is a convenient tool to thus We have shorten the notation.) n+l

0

n

1

(Ak - Ak

dWk

n j+l Aj,Ak7

E 1 j:l

j

1

n+2

j+2

(AI;...;A.,Aj ;...;An ,Y,t,to) _

dx.J

(19)

n n j j+n+l j+l j+n 6Wk 1 (AI;...;A.,A. _ -iJI1wjk(A1 ,...,An 1 ; ;An,Y,t,to) dxJ

Continuing this process, as in item C, we obtain 0 n+l 1 n (Ak - Ak )Wk(Al,...,AnY,t,to)

=

(20)

1 N-1 * n E (LksWk)(A1,...,An,Y,t,to) + QkN(Y,t,to)

s4 for any natural N, where

a +E.a.a.-s Lks ks

a 4s E

C ksa

(

)

a

ax

a

' Cksa

E P(a

(21)

1,...,A )(R°).

n

QkN is a sum of terms, each of which has the symbol of the form daWk Q(x,Y,t,to) - CkNa(z)

(22)

a

dx with jal -4 N, x E R(r1'

'rn)'

A +Ea.a.-N -1)(Rr); CkNa(x) E rN(2xma 6J x

here Amax = max a.. j

J

In particular, n Lkl = -i E 1l.k(x) as

j1 J

(23)

.

(24)

xc

We seek Wk in the form of the formal series Wk(x,Y,t,to) ti

E Wkf(x,Y,t,to), f-0

(25)

283

satisfying Taw

o

+ i E L.k Wk = 0. j=1

(26)

In term-by-term form (26) reads aw

Ti. aWkf

E R.k(x) aXkO = 0, j=1 3

E=1 aWkf E R.k(x) _ -i E ax. j=1 3 3

(27)

n

ac + o

aw

n ko +

W s=OLklf-s+1

P

ks'

1,2,...

(28)

Consider the system of ordinary differential equations of the first order (29)

xj = 41 kj(X), j = 1,...,n, X E Rn.

Denote the solution of this system with the initial data Xj(0) - xj

(30)

X - X(k)(x,to)

(31)

by

(here to is the "time variable"). Lemma 1. The solution (31) of the system (29) with initial data (30) has the property

X(k)(TAx,Tl-Akto)

- TAX(k)(x,to)

(32)

for T > 1, A(x) is large enough, 'tot 6 Proof. Differentiating both sides of (32) with respect to to yields (assuming that A(x) is large enough):

(X(k)(TAx,TI-akto))j dto

dt

=

T1-Aknk3(X(k)(TAx,Tl-Akt0

)),

(33)

TAjnk3(X(k)(x,to)) = Tl Aknk3(TAX(k)(x,to)) (34)

(TXX(k)(x,to))3 0

Thus both sides of (32) satisfy the same system of equations and (32) holds by the uniqueness theorem for ordinary differential equations (the fact that initial data coincide is trivial). Lemma 1 is proved. Calculate the solution of the system of equations (27) - (28). Using the solution (30) of the system (29) and taking into account the antisymmetry of nkj' these equations may be written in the form de

dt

0

0

Wk o (X(k)(x

o ,-t o ),Y,t,to

)

0,

(35)

A-1 x s Wk (X(k)(xo,-t0).Y,t,t0)= -iEELk,k-s+1Wks3 x

(X(k)(xo,-to),y,t,to),

( 36 )

A - 1,2,3,...

.

From (35) we ,btain Wko(X(k)(xo,-to),Y,t,to)

284

a Wko(xo,y,t,0)

(37)

or, resolving for x0 the system of equations X(k)(x0,-to) = x, (38)

Wko(x,Y,t,to) = Wko(X(k)(x,to),Y,t,O) Also we have for f = 1,2,..., Wkf(X(k)(xo,-to),Y,t,to)

- if

. Wkf(xo,Y,t,O) (39)

to L-1 E CLk,-s+1WksJ(X(k)(xo,-t'o),Y,t.to)dt'o,

0

S -O

or

Wkp(x,Y,t,to) - Wkk(X(k)(x,to),Y,t,O) (40)

t o C-1

- if

ta).Y,t,t'')dt'o.

E

o

s.0

However, the latter expression admits further simplification; to carry this out, we begin from f - 1, using the expression (21) for Lk aIaIW

to

Wkl(x,Y,t,to) -

C E

Wkl(X(k)(x,to),Y,t,O)- if

0

Ck2a

X

ak0

ax

IC-142

X (X(k)(x,to - t''),Y,t,to)dt'' - Wkl(X(k)(x,to),Y,t,O) -

- if taI 0

C

E

IaI 42

(X (k) (x, to - to)) ((t k2u

aXxk)

(41)

(x, to - to))-1 aX)a]

X 14ko(X(k)(x,to - to),Y,t,t'')dt''.

From (38) we have Wko(X(k)(x,to),Y,t,O)

Wko(X(k)(x,to - to),Y,t,to)

Wko(x,Y,t,to); (42)

thus (41) takes the form Wkl(x,Y,t,to) .

Wkl(X(k)(x,to),Y,t,o) + Lk2(to)Wko(x,Y,t,to),

(43)

where Lk2(to) is a differential operator of second order, to

Lk2(t0) _ -if

E

Ck2a(X(k)(x,to - t'')) x

0 10142 X ((

t ax (k) ax

(44)

(x,t0 - to))-1 ax )adto.

Similarly we have t

Wk2(x,Y,t.to) = Wk2(X(k)(x,to),Y,t,O)- if oELk3Wko) x 0 (X (k)

(x,to

to - t'),Y,t,to)dto - ij0

Ik2(Wkl(x,to),Y,t,O) + Lk2(t0 ) (45)

dt' - W

(k)

X Wko(x,Y,t,to)7I x=X

(x,to-to)

o

k2

(X(k)(x.t

0 ).Y,t,O) +

+ Lk3(to)Wko(x,Y,t,to) + Lk2(to)Wkl(x,Y.t,to), and generally,

285

f=1 Wkf(x,Y,t,to)

Wkf(X(k)(x,to),Y,t,O) +

sl1Lk,f-s+l(to)Wks(x,Y,t,t0)(46)

where Lk5(to) is a differential operator of order < s, which may be expressed in an obvious way through Lks, with s' < s. Next we intend to find the formal solutions for equations (13) in order to obtain the initial conditions for Wk+l from the end point values of Wk, n

1

It is quite simple since no exponents act between A1....'An k = 1,...,n-L. n+k n+k and Ak on the left-hand side of (13). 'We commute Ak in the k-th place, using the permutation formula from item C; we obtain the asymptotic expansion Wk+l(x,yk+2'...,yn,t.0)

= Wk(x,xk+l'yk+2,...,yn,t,tk) + (47)

+

(Lks)Wk)(x,xk+l'yk+2'....yn,t,tk),

7

s=1

where Lks) are differential operators in x,yk+l of the form similar to (21). Of course, (47) may be rewritten in the term-by-term form just as (26) was. .Now we are ready to obtain the formal solution for the problem (8). We begin with the principal term. Using (15), (16), (38), and (47) and taking into account that Lk$) in (47) contribute only to lower-order terms, we obtain successively: Flo(x,y2,...,yn,t) - f(xl,y2,...,Yn,t); F,o(x,y3,...,yn,t) = Wio(x,y2,...,yn,t,t1)1 y2

-x

2

= f(Xi1)(x,t1),x2,y3,...,yn,t);

(48)

Fso(x,y4,...,yn,t) = W-o(x,y3,...,yn't't2)ly3=x3 =

=

f(X(1)(X(2)(x,t2),tl),X(2)(x,t2),x3,y4,...,yn,t);

and, finally, the principal term of F(x,t) occurring in (8) takes the form F(o)(x,t) =

Gn+t,o(x,t)

- f(X(x,t),t),

(49)

where the j-th component of the mapping X is defined by X.(x,t) =

(50)

In a more' convenient form the mapping (50) may be defined also in the following way: for any j E (1,...,n} and tj E ht, set

X(j,tj)(x) daf X(j)(x'tj). j - 1,...,n.

(51)

X(x,t) - X(1,tl) 0 X(2,t2) 0 ... o X(n'tn)(x).

(52)

Define

In other words, the point X(x,t) may be obtained if we move, beginning from x, along the trajectories of system (29) subsequently for k = n,n-1,..., 2,1 during the time intervals tn,....tl, respectively. X(x,t) may now be defined by 286

1,...,n.

X.(x,t) - Xj(x,O,...,O,tj,tj+l,...,tn) j

(53)

Thus, seeking F(x,t) as the formal series (54)

F(x,t) = sZOF(s)(x,t),

we have found the principal term of this series; it is given by (49). F(1)(x,t), we have, calculate the lower-order terms in (54). As for (15). (16), (38), and (47):

Now using

Fll(x,y2,...,yn,t) - 0,

F21(x,y3,...,yn,t) - Wl,(x'y2....,yn,t,t1)ly2.x2 + + CL11 Wl,(x,y2,...,yn,t,tl)]ly (EL22(tI) + Lll ]f(XI

(55)

-

x 2

2

(x,tl),y2,...,Yn,t)}ly2-x2.

where 'L12(tl) is a differential operator of order C 2 in x while L11) is an operator in x and y2.

The expression in curly brackets may be rewritten is some new differential

as (L12(tl)f)(X(1)(x,tl),y2,...,yn,t),

Xil)(x,tl).

operator acting on f before the substitution xl obtain F21(x,y3,.... Yn't) -

Thus we can

(112(tl)f)(Xil)(x.t1),x2,Y3.....yn,t).

(56)

and further application of this technique yields

- (Plf)(X(x,t).t),

F(1)(x.t)

(57)

and, generally, (58).

F(s)(x,t) - (Psf)(X(x,t).t),

where P. is a differential operator in x of order 4 s + 1, and, besides, F(5)(TXx,Tl-at)

a

Tm-sF(a)(x.t) (59),

for T > 1, A(x) large enough, provided that f(TAx,tl-At) . Tmf(x,t) for T _> I and large A(x) (we omit routine calculations leading to this result). If we set P. = 1, we may write the formal series solution of (8) in the form

F(x,t) =

E

s=0

(P f)(X(x,t),t) - f(X(x,t),t) + E (P s f)(X(x,t),t). s s-l

(60)

Our next task is to validate, under certain conditions, the expansion (60); that is, to "sum" the asymptotic series occurring throughout the calculations and to estimate the appearing remainders. Also we need the conditions under which (60) is solvable for f(x,t).

Let f (x,p) E Tmt (A . 1 I-a 3 for if lpjl > CA(x)

0 a

l a i + l

l f (x , p )

axa ap

a

a A

.l ) (t2n) (recall that this means that f(x,p) n some 3,.where C - C(f), and that

,

1a1,181 - 0,1,2,...,

plus stabilization conditions at infinity with respect

(61)

to XI.).

we

287

1,...,n, the partial sum of N - i terms of the series (25) take for Wk, k where the Wkf's are obtained from f recursively via (46), and substitute this expression into (14). The expression for Wk reads

Wk(x.Yk+l,...,yn,t,to) _ (Lkf)(X1(x,tl,...Itk-l'to,0'...,0), X2(x,tl,...,tk-l'to,0,...,0),...,Xk(x,tl,.... tk-l'to,0,...10), X

(62)

'k+l'...,yn,t,to); to a 10,tk1;

he Lk'is some differential operator whose explicit form is of no interest tb us. The remainder Rk(t,to) in (14) after our substitution would be a sum o' a number of operators with symbols of the type (22) time, the product

of

Assume that (for some given positive cl and c2) cl'(>.l
1, A(x) large enough. Lemma 2.

Assume that a subset K C x,e) is given such that the

following inequalities are valid for (x,8) e K with some positive C: 288

(68)

(69)

1631 < C, j = 1,...,n, A

Jaz. axkl def

(70)