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 <
in the latter case, once the asymptotics is constructed, the original equation Lu - v
(7)
is reduced to the equation
(1 + QN)O - v,
(8)
or
0(x) - v(x) - f QN(x,y)m(y)dy,
(9)
where QN(x,y) is the kernel of the operator QN. Equation (9) is an integral equation of the second kind with the kernel, which is not only smooth but also small, as the parameter tends to zero. This fact enables existence theorems to be proved for the equation (7) and to construct estimates of the solution, which are uniform with respect to the parameter.
The mixed asymptotics can be most naturally interpreted as the construction of the asymptotics in the scale,* generated by a set of commuting operators Ao - A (multiplication by the large parameter A - 1/h) and Aj 1,...,n. More precisely, define on Co(&n+1) the norm
1012 - fm(z,A)(1-a+A2)s0(x,a)dzdX,
(10)
and denote by X. the closure of Co(R"+l) with respect to this norm (in particular, Xo - L2(Rn+1)).
Then the mixed asymptotica may be interpreted as the construction of an "almost inverse" operator RN in the scale (Xs), such that (3) holds. and QN!Xs+Xs+N ` m, a E R.
(11)
Of course, the Fourier transform with respect to A transfers this scale into the usual Sobolev scale. C.
A Little More on Characteristic Equations
In item A we have obtained some characteristic equations for the KleinGordon equation, corresponding to different kinds of asymptotics. Here we outline briefly the well-known method of solving characteristic equations (detailed considerations and proofs are contained in Chapter 3).
The characteristic equation is a nonlinear first-order partial differential equation. We assume it may be resolved for the t-derivative of the unknown function and written in the form
* By "scale" we mean here nothing more than a family of embedded linear Banach spaces (see Chapter 2 for further information on scales). 6
as + H(c,
at
t) = 0.
(1)
The function H(x,p,t) is called a Hamiltonian function. of item A it has the form
In problems
3
H(x,P,t)
(
°
(2)
p? + m2c4(x,t)w2)
for various values of parameter w E 10,1].
Let the initial condition S(x,0) = So;x)
(3)
for equation (1) be given. The method of solution goes as follows. sider the Hamiltonian system
Con-
is - 2p (x,p,t), (4)
2z (x,p,t).
Equation (4) is a system of 2n ordinary differential equations of the first order. Its solutions are usually called the bicharacteristics of the Hamiltonian function H(x,p,t) (and of the asymptotic problem considered). Prescribe the initial conditions
aSo x(0) - x" P(0) - ax (xo),
(5)
and denote the corresponding bicharacteristic by (x(xo,t),p(xo,t)). The solution S(x,t) may be obtained in the following way. n t H)Ix=x(xoT)dT. W(xo>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)<... p1(Al).
(4)
The described properties ate basic for Feynman ordering, and many useful algebraic identities could be derived starting from these properties. (b) There is another way of ordering the operators, which is called the Weyl ordering. We set intc correspondence to polynomial p(yl,...,yn) the operator
Pw (A1 ,...,An) = aoI+
%
0<W <M a
--,-gin (£A 1
a J(1)"
.
A
a(;al) )
(5)
31
{1,..., where the sum in round brackets is taken over all the mappings a Weyl lal} + {1,...,n} taking each value j E {l,...,n} exactly aj times. ordering possesses the properties: (i) for a single operator A (n = 1) we have :
pw(A) - P(A). 4n, Mz = Bz + b is an affine (ii) (Affine covariance). If M : ¢n transformation, then for any polynomial p(yl, .. yn) ,
(M*p)w(Al,...,An) - Pw((MA)i,...,(MA)n),
where (M*p)(y) = p(My), ...,yn) ' q(y1 +
(MA)i = j BijAj + bi.
(6)
In particular, if p(yl,
"' + Yd' then pw(Ai,...,An) = q(A1 + -..+ An).
The polynomials do not form the class of symbols large enough to be appliThere are various possibilities of cable in the theory of asymptotics. broadening this class of symbols for which the quantization is defined; for example, holomorphic functional calculus based on Taylor expansions or on Cauchy-type integrals, etc. It turns out that the most convenient for the needs of asymptotic theory is the approach to construction of functional This approach calculus based on exploitation of Fourier transformation. enables us to consider the non-analytic symbols and to obtain substantial results for a wide class of operators. Assume that operators A1,...,An generate strongly continuous oneparametric groups in H, i.e., Cauchy problems -i
axtt)
- Ajx(t), x(O) = x o E DAj
(7)
have the unique solutions x(t) = where Uj(t) is a strongly continuous family of continuous operators in H defined for - - 6 t 4 m (in this chapter we give a comprehensive analysis of properties of such operators which will be called generator-3).
Given a function f(yl,....yn), we define the operator UA(f) by virtue of the formula
Al,...,Ad -
(2n)-n/2
r nf(t)Un(tn)x...xUl(tl)dt;
(8)
the integral in (8) being understood in the sense of, say, strong or weak convergence in H. As for Weyl quantization, it may be defined by (2n)-n/2 f
fw(Al,...,An)
where for each w E Sn-1, U(wt), generated by
T E I1,
f(t)U(t)dt,
(9)
is a strongly continuous group
n E
j'1
wjAj.
Of course, we should require in this case
that w E Sn-l. The question is whether these definitions make sense and whether the mentioned properties of the quantizations remain valid for such definition. B.
Functional Spaces and Symbol Classes
In applications the functional space H in which the operators A act turns out to be a Banach space, or, more generally, a total space of Banach spaces, 32
enumerated by elements of the bidirected* set A and connected by continuous The important example of the scale appears 6'. embeddings 116 c H6, for d
in the problem of constructing asymptotics with respect to powers of given For X E X we unbounded operators B1,...,Bn acting in a Banach space X. is the norm function in X and we set set 11x110 = !1x11, where n 11XIIk+ i j=1
11XIlk+I
!IBjxllk, k = 0,1,2,...
.
(10)
Under certain assumptions (see Section 4:C) the subspace Xk C X of such x E X, Hxllk is finite, is a dense subspace which is a Banach space and these spaces form a Banach scale. If with respect to the norm X is a Hilbert space, the spaces Xk may be defined for negative k as dual spaces to X_k. The construction of the scale (Xk) given above admits generalizations which are also described in Section 4. We also discuss there the conditions under which the operator A in X gives rise to a getierator in the scale {Xk}, 1
n
To define f(A1,...,An) by means of (8) we make the assumption that the groups U3(t) = generated by Aj have tempered growth as t space, it means simply that (in the case when H is a
11 exp (iAj t) 11 < c(1+ jt)p, t EIR
(11)
If H is a Banach scale we have the collection of for some c > 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<-isfied (see Section 3:B for description of these conditions and also for example of what pathologies may appear when these conditions aro not satisfied).
,,! (b) are satisfied, the regular representation exists. In this ,e give the proof of existence of regular representation in the By the operators A1,...,An generate a nilpotent Lie algebra'r. the way, this case gives an explanation of the terminology: under the Fourier transform the operators Lj are mapped onto the generators of the regular r associated with r (recall that this repcesentat:2n acts in the space of functions on G by virtue of left shifts). The case of :core complicated commutation relations and the related geometric constructions are considercJ in Chapter 4. If (a) .:cur,
D.
Bibliographical Notes
The main textbooks on functional analysis which we recommend to the reader studying this chapter ire [13, 28, 79]. The presentation of the semigroup theory in this chapter is based to a great extent on these sources and also on original papers [77,78]. The material of Section 2:B,$ and Section 3:A,B,D develops the ideas of functional calculus constructions stated in [50,52,54]. Example 1 of Section 4:B is taken from the paper [627, and Theorem 2 of Section 4:B is an improved version of the KreinShikvatov theorem [47]. The material of Section 4:E is based on the paper [39].
35
2.
FUNCTIONS OF A SINGLE OPERATOR
A.
Semigroups and Generators Let X be a Banach space.
Definition 1. A semigroup (of bounded linear operators) in X is a X, defined for t E, CO, +-), such that family of operators U(t) : X t) (i) for each t, U(t) is bounded; (ii) U(t)U(r) - U(t + T) for all t, T; (iii) U(0) = I (the identity operator in X); and (iv) U(t) is strongly continuous in t (this means that for each x E X, U(t)x is a continuous Xvalued function of the variable t E CO,+ -)).
More precisely, U(t) defined above is a strongly continuous Remark. It was omitted in our definition because semigroup with the Co-condition. semigroups without Co condition are of no interest to us. We deal further only with semigroups, which are in fact groups, i.e., which are defined for all t E kt and satisfy (i) - (iv). Nevertheless, most propositions are formulated in this item for semigroups. Proposition 1.
Then
Let {U(t)} be a semigroup in X.
[U(t)II 6 Me61t
(1)
for some M,w. Proof. (a) JIU(t)N is bounded for t E-:10,11. Indeed, otherwise we should have U(tn) I -+ m for some sequence {tn}, to ei C0,1], kim to to E ED, A.
nBy the resonance theorem [79},l'l(tn)x" is unbounded for some x dicting the strong continuity.
(b) Nov U(t) - U(t - Ct3)U(Ct]) -
It]
6 1 U(t - Ct])A I US 1) II M -
sup
JU(T)0
w
,
U(t - Ct])I! kn
U(t)(U(1))Ct].
II
U(1,) II
X, contra-
Therefore hU(t)II <
t and it suffices to set
U(1)1
TECO,1] Definition 2. The type of semigroup U(t) is the greatest lower bound of the set of w, satisfying (1) for some M. U(t) satisfies the norm condition (for given w), if (1) holds with M = 1.
Now we define the generator for the given semigroup U(t Let DA C X be the set of all x E X, such that the strong limit kim(it)- (U(t)x - x) t-0 exists. Clearly DA is a linear subset of X. Consider an operator A in X with the domain DA, defined by the formula A
kim
It (U(t)x-x), xEDA. 1
(2)
t-.+0 Definition 3. the generator. Theorem 1.
A is called the generator of the semigroup or simply
(a) The subset DA is dense in X and A is a closed operator.
(b) DA - X and A is a bounded operator iff the semigroup U(t) is uniformly* continuous in t. This means that U(t) - U(T) I - 0 as t - T; in other words, U(t)X - U(T)X uniformly in x, IXP <. 1. T11e reader should avoid confusion with uniformity in t.
36
(c) U(t)DA C. DA for all t, and A commutes with U(t); AU(t)X = U(t)AX, For X E p(A), (A - A)-1 also commutes with U(t). x E DA.
(d) For each z E DA the X-valued function x(t) = U(t)xo is differenIt may he defined as the unique solution of the Cauchy problem
tiable.
i !3tt) +Ax(t) - 0, x(0) - xo.
(3)
satisfies (1), each A E t such that -Ima > w (e) If the norm I1U(t)!I The resolvent R1(A) is given by the belongs to the resolvent set p(A). formula def (Al - A)-lx = if (4) m e-iXtU(t)xdt, x E X, -Ima > w, Rx(A)x 0
and satisfies the following estimates
1,2,...;
II (RA(A))m! < M(-ImA-w)-m, -Ima > w, m
(5)
M,w being the same as in (1). (f) Conversely, let A be a closed densely defined operator in X, such that the resolvent R (A) exists for -Ima > w and satisfies the estimates (5).* Then A is a generator of the semigroup U(t), satisfying (1). U(t) is given by the formulae U(t)x = lim (-iUR_iU(A))[Ut]x, x E X;
'(6)
U(t)x = fim eitAUx, x e X;
(7)
u.«
where
All - -iiAR_iU(A)
is a bounded operator in X. with inspect to t. .roof. indur
(8)
Convergence in (6) and (7) is locally uniform
Let DAn denote the domain of the operator An (it is defined X E DAnn if x E DA and Ax E DAn-1). First, we prove that tte
inteia'ection D -
(1
Dan (and hence DA) is dense in X.
We assert that
n=1
the element
x
q(t)(!(t)xdt (Bochner integral)
lies in D fur ar1,it.rary
it
ECo(TR+ \ {0)).
(U(t)>: -x ) 0 0
f.,. O
::
1
it
f
0
Indeed,
¢(t')CU(t+t')x-U(t')xldt'
' ( t ' - t) - f (t' ) U (t' )xdt' . it
so the limit (2) exists and Axe - xio.. By induction, x E D and (An)x x(in,$(n)).The set (x0)x E X,0 E Co is dense in X (if tte sequence +n converges to the Dirac 6-function, xfn converges to x), thus D is also dense. Now let x E DA, x(t) = U(t)xo. Then o
* It is sufficient if this requirement holds for pure imaginary A, Ima > w.
37
I (U(T)X(t)-X(t))
(U(T+t)x0-U(t)x0) _
3
U(t) T (U(T)xO-XO). Since U(t) is bounded, x(t) E DA and Ax(t) = U(t)Ax0, thus (c) is proved. (The cemmutability of U(t) and (a - A)-1 is an easy consequence of the fact 1
x fim T±+O IT [x(t + T) -x(t)3 and, since Ax(t) is continuous in t, x(t) is differentiable in t, then (3) holds.
Moreover, we see that Ax(t) -
that AU(t)x - U(t)Ax, x (=- DA.)
To verify (e) we show at first that the right-hand side of (4) is well defined. Indeed, the estimate
Ire- ixtU(t)xJ
5 me
(Ima+w)t
implies the convergence of the integral for -ImA > w.
Further
iAfine-"tU(t)xdt - fim 1 (U(T) - I) fine-1AtU(t)xdt Ta+O
O
O
fim 1 fine-I t[U(T + t) - U(t)3Xdt - fim i T a0 0
Ifine-iA(t -T)U(t)xdt -
T++0
- fine-ixtU(t)xdt] - fim T-O 0
- fim
c 1.
e
iAT 1T
- 1
0
fTe-11(t - T)U(t)xdt. 0
Tao
Both limits exist obviously when -ImA > 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 <
Ckll k
ITI)k-1:
(ItI+ITI)k+LAI
Since the sum in the parentheses is finite, we conclude that U(t) is uniformly continuous. Vice versa, let U(t) be uniformly continuous. We assert that
Lim -iuR(A) - I = 0.
(14) 39
If it were already pruved, we may conclude that the operator -iuR_iU(A) has a bounded inverse for sufficiently large u and therefore A = -iU By (4) - (R-iu(A))-1 is bounded. The proof of (14) is as follows. -ivR
-iu
(A)x = Jmue-utU(t)xdt. 0
Choose an arbitrary r > 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 <
fue-utdt
f6Ue-utdt 5 z HxII
2llxll
211 X1
Now we estimate the second integral fFue-ut(U(t) - I)xdtll <
ue -udllxll
f u oe-ut(1 +MeW(t+d))dt < Z Ilxl
for sufficiently large u. Thus (14) is proved. Now only item (f) of the theorem remains unproved. Let the assumptions of this item be satisfied. Set Tu - -iiR-1U(A). We notice at first that Eim Tux - x for every x E X. Indeed, i` x E DA,
TUAx - 0
Tux - x
as u + ', s;nce (5) yields that OTJ remains bounded while u + m Now DA is (tense in X, and it follows from above that TUX -. x everywhere in X. Now set
UN(t) - eitAU - eitATU. (Note that TUX (Z DA, so that ATU is everywhere defined. Further AT, = = iU(I - TO) is bounded.) We estimate now the norm of UU(t)y:
PU (t) R= a P
e-tu (I-Tu)
= e-tub etuTu J
ET e-tu k=0 k.
(tu)
II Tkll
By (5) for large u Tu
(R-iu(A))ki '
= u4.
hence
M'(uuW )k. 2
flUU(t)1 < Me-t" E k=0
k (uk
= Me
tu -u) u-W
(15)
t (u' ) = Me
u-"
-W)
in particular, U (t) is bounded uniformly as u + m and t lies in a fixed compact set K C [0,m). Let x E DA. We have
QUU(t)x-Uv(t)xI - N ft -al No(t_4)Uu(4)x)dbsj _
t
0
_ I fo v(t-6)Uu (6)(ATu-ATv)xd6
- I f0Uv(t-4)UU(6)(Tu-Tv)Axd6% ( 0
40
p
where y - Ax; the constant C depends on t but does not depend on u and v
Since Tuy + y, the right-hand side of the above inequality for large u,v. tends to zero as u,v - 0D and we obtain that there exists the limit (16)
U(t)x def Lim Uu(t)x,
the convergence in (16) being locally uniform with respect to the variable The uniform boundedness of the family {Uu(t)} yields that the limit t. (16) exists for all x E X, U(t) is strongly continuous,
IIU(t)II 6 :2im Uu(t)n 4 kim Me (tuw/u-w)
Mew t
Next we have to show that U(t) is a semigroup with the generator A. have, for an arbitrary x(=- X,
We
IU(t+r)x-U(t)U(r)XII +IIU(t)U(r)x-Uu(t)U(T)x +I]Uu(t)U(r)x-Uu(t)Uu(r)xI1, since the semigroup property is valid for UV(t). The left-hand side of this inequality does not depend on u, while its right-hand side tends to zero as It follows that U(t + r)x - U(t)U(r)x, i.e., U(t) is a semigroup. u - W. Let x E DA, then
d[ Uu(t)x - iAUUu(t)x = iUu(t)TAx Thus we may then
converges to iU(t)Ax locally uniformly with respect to t. conclude that
dt
U(t)x - iU(t)Ax and, particularly,
dt
U(t)xlt=0 - Ax, x E DA
For µ Hence the generator of U(t), say A, is a closed extension of A. large enough A + iuI is the one-to-one mapping of DA onto X. By assumption
A has the same property. to prove (6).
Hence DA - DA and A - A.
Now we are in position
For the sake of simplicity we rewrite (6) in the form
uu
i U(t)x - Lim (T )tut]x - tim (I (17) u u We recall that for real p, Cpl denotes the integer part of p, i.e., the greatest of the integer not exceeding p. By (15),
(T )Cut]f P
M(u
uw
A)-Cut]X.
)Cut]
as u - -; in particular fl (T )
<
M(u
)ut - M(1 - W)ut +
uw
Mewt,
(18l
Cut]
is uniformly bounded (u -* -, t belongs to a fixed compact subset oV CO,-)). So it is sufficient to prove (17) for x belonging to some dense subset of X, say to DA2. Let x E DA2. We have then
IlU(t)x- (I - u
I]
A)-Eut]XII
4
(U(t)x-U(t')x
+ IIU(t')x- (I - u A) -ut'xI -If U(t)x-U(t')x
+ IU(t')x- (I where t' - Cut]/u, n - ut' - Cut]. hand side of (19). We have
n
J
+
+
(19)
A)' xj
Now we estimate both terms on the right-
41
t 1U(t)x-U(t')XO= ft Td U(r}xdr wt ewTdT < M Axj. eu
t
< M AXII j
t
jt, gU(r)Ax dr <
since t ' < t ,
,
t - t' < 1/u.
t'
Further,
ll U(t' )x -(I - in' A) -nxj
Jt'll dda 0
[(I
i (t' - s) A)-nU(s)x3Ida.
We obtain from (3) and (9) that
d
ds
C (I - i ( tn ' - s) A)-nU(s)x]
- i (I - i W-n s) A) -n
U(s)(Ax - (I - i(tn s) A)-'Ax) (t' - s) A)-nU(s) t n
(t' - s) A)t-'A2x
n
= t .. s (I - i (t
n
n
=
s) A)-nU(s)A2x, x E DA2.
(The accurate proof uses expansions of the type (12), and we leave it to the reader.) It follows immediately that
11 U(t' )x - (I -
rn' A)-nxll < C C. to 2 = Cl
Thus both where CI is bounded as u -+ m and t belongs to a compact set. terms in (19) tend to zero locally uniformly with respect to t as u So we obtain (6), and the proof of the theorem is complete. Notation. From now on we denote the semigroup U(t) with the generator A by exp(iAt) or e-At whether A is bounded or not. If the reader is attentive, this will not lead to any confusion.
We formulated Theorem 1 for semigroups. some of its items gives us the following: Theorem 2.
The simple reformulation of
Let A be a closed densely defined operator in a Banach
spaceTThen A is the generator of a strongly continuous group U(t) 1ff there exist M and w, such that 1 estimates are valid:
n(A) for ;Imal
> w and the following
nl (AI-A)'i < M(1Im1 -w) m, IImA > w, m = 1,2,...
(20)
These estimates yieldlieiAtjl 5 Mew ltl, and vice versa.
U,(t)
Proof. def
If (1(t) is a strongly continuous group in X, then the formula
t > 0 defines a pair of strongly continuous semigroups. The generators of U+(t) and U_(t) are A and -A, respectively ; hence we obtain (20). Conversely, let A satisfy (20). It follows that both A and -A are the generators, and we denote by U+(t) and U_(t) the corresponding semigroups. The identity U+(t)U_(t) - I holds; the proof of it is quite similar 4o that of uniqueness in the item (d) of Theorem 1. Thus we can define elAt by the formula (1(±t),
e
iAt
=
U+(t), t>0 U_(-t), t< 0,
and all the desired properties are clearly satisfied. fore proved. 42
(21)
Theorem 2 is there-
Given a semigroup U(t) it is often difficult or undesirable to descri"e The following theorem gives precisely the domain DA of its generator A. us a convenient tool to avoid such difficulties (the effectiveness of it will be seen in sections devoted to functions of several operators). Definition 4. Let A he a closed linear operator in a Banach space X (or, more generally, an operator between the Banach spaces X and V) with The linear subset D .C: DA is called a core of A the dense domain DA C X. if A coincices with the closure of its restriction on D. Theorem 3. Let U(t) be a semigroup in the Banach space X with the generator A. Suppose that D C DA is a dense linear subset of X, such that U(t)D C D for all t e CO,'). Then D is a core of A.
We establish first that (A + ip)D is dense in X for sufficiently Proof. Sr.ce large u, namely for u > w, where w is the type of the semigroup U(t). D is dense, it suffices to prove that for an arbitrary x E D there exists Let v > w a sequence xn E D, n = 1,2,..., such that Aim (Axn + iyxn) = x. be fixed.
Then the integral
x = (-A- iu)R_ 41 (A)x - -if e-ut(A+ iu)U(t)xdt 0
(22)
converges and has the continuous integrand. It follows that t'l..:re exists a collection of numbers cns E 4> 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 <
(ImX)-1
Applying Theorem I of item A, we obtain that A is a generator of the group U(t), satisfyingIIU(t)II 5 1. Since U(t)U(-t) = 1, U(t) appear to be unitary operators, as desired. Theorem 2 is proved. Note. One may define the self-adjoi.nt operator in the Banach space x as the generator of a strongly continuous group satisfying II0(t)II = 1. Stone's theorem shows that in Hilbert spaces this definition coincides with the usual one.
46
Convergence of Semigroups
C.
In this item we establish some general results concerning the relatitn between convergence of semigroups and convergence of their generators. These results play an important role in our development of operator calcul:s We apply them, in particula especially of functions of several operators. to the investigation of continuity properties of dependence of functions of
generators on generators themselves. We have already proven one of such results in item A. Namely, it was proved that, A being the generator, exp{itA}= s -lim exp{itAU}, t > 0, where A. = -ijAR_iu(A) is a family of bounded operators. Much more complicated situations are discussed hare. We consider general family of generators, continuous in some special senef and prove the continuity of the corresponding family of semigroups. It appears later that the type of continuity introduced is quite natural in problems connected with functions of non-commutating operators calculus. Here and further, we formulate our statements for the case of sequences of semigroups (resp. generators). One must have in mind that all these statements are valid (proofs being just the same) for families depending We avoid on parameter, lying in a topological space (instead of sequences). such considerations, our aim being to clarify the exposition. X denotes a given Banach space throughout the item, all the operators in question actin; in X.
Now we come on to precise definitions and statements. we introduce a convenient notation:
Following 1401,
Notation. Let A be a closed operator in a Banach space X. We write M,w), if A is a generator of a strongly continuous semigroup which satisfies the estimate
A E
eiAt1
Mews for all t ) 0.
,I
(1)
We shall also make use of the following: I
Definition 1. n
=
1,
2,
... of
Let D be a subset of Rn(t ). Given a sequence {Br(a)}a ED, families of bounded linear operators in X, we say
that it strongly converges to the family (B(a)) a ED, locally uniformly with respect to a, if for each x E X, B (a)x -.- B(a)x and this convergence is uniform when a lies in an arbitrary fixed compact subset of D. s
Notation: Bn(a)
B(a),
ac- D.
Further, the families B (a) will be either semigroups or resolvent families. To begin with, we establish simple sufficient and necessary conditions for locally uniform strong convergence of semigroups. Next we study them in detail and derive sufficient conditions which are much easier to check (we make no investigation of their necessity). At the end of the item these results are applied to strongly continuous groups, in particular to groups of unitary operators in a Hilbert space. Theorem 1.
Let An E G(M,w), n = 1,2,... exp(itAn)
U(t),
t
In order that
.
0
(2)
as n w m for some family U(t) of operators in X, it is necessary that s
R1(An) : R1(A),
IrnA
< -w
(3)
47
for some densely defined operator A in X, and sufficient that* Rao(An) I Rao(A)
(4)
for some A , Im)< -a.
In this case Ac- G(M,w) (in particular, A is then closed) 5J U(t)o- exp{itA}. Proof.
We may assume that w - 0, considering An + iw instead of A.
If (2) holds, d(t) is a strongly continuous semigroup, (a) Necessity. satisfying AU(t)l G M. Indeed, continuity and the estimate for the norm immediately follow from (2); the semigroup property is proved in complete analogy with that in Theorem 1, (f) of item A. Thus U(t) - exp{itA}, A E G(M,O). To prove (3), we note that for arbitrary x E X, Ra(An)x - Ra(A)x
. i(f C0+f
C
if
a- ixt Ce it Anx_Q((t)xldt =
)e-tatEelt 0x - U(t)xldt E II +1 21
Im).
< 0, C > 0
(see (4), item A). If A lies in a compact subset of the lower half-plane, Im. G -6 < 0. It follows that, given c > 0, we can find such C that III2211 < < E/2 for all n (the norm of the integrand does not exceed 2M 11xll exp(-it)). exp(itA.)x converges to U(t)x uniformly for t C CO, C], so ,Illi< E/2 for n large enough, and (3) is therefore proved. Sufficiency. Let (4) be valid. Then we claim that (3) is valid (b) also. To prove this, we need the following two lemmas which will be used later as well:
Let An, n - 1,2,... be a sequence of closed operators in X. Lenin. 1. Define Ab E t as a set of all X E t such that X E p(A.) for n > no = no(X) and llRx(An)ll M(A) < W, n > no(A). Define also As C t as a set of all A EE t such that A E p(An) for n > no(A) and Ra = s - tim R).(An) exists. Then: ^ n''°°
(b)
Ab is an open subset of d; A. C Ab and is relatively opened and closed in Ab (i.e., As is a union of connected components of Ab);
(c) (d)
Rx(An) ' RI, A E As; RX satisfies the resolvent equation;
(a)
s
Ra - Ry = (X - v)RXR,,
X,u E As.
(6)
Lemma 2. Let Rl, X E A C t be a family of bounded operators in X, satisfyi- g 6). Then N = Ker RX and R = ImRl do not depend on A. RX
coincides with the resolvent family of some closed operator Aif£ N = (0). In this case DA - R and A = XI- (Rx)-1 for each X E A.
(7)
Under the conditions of Lemma 1, if Rx = Rx(A), we have A. = Ab() P(A). Proof of Lemma 1. (a) Let A E A . Using the Neumann series ((10), item A), we easily obtain that 11Rx(%n)I M(a )(1 - M(ao)lx for IX - aof < M(ao)-1. n>- 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 <
1J RX(An)CeitA _ e1tAn]RX(A)xll 5
t
(11)
Q (R1(A)-RX(An))eisAxtl ds.
MI
0 The integrand in (11) is uniformly bounded as n
and tends to zero pointwise; by the Dominated Convergence theorem the integral tends to zero as
n +
Thus for n -
RX(An)IeitA - eitAnlu
-0
(12)
for u E Im(RX(A)) - DA and hence for all u E X. Our argument shows that this convergence is locally uniform with respect to t. Further, RX(An)IeitA _
ei.tAn]u -CeitA _ eitAn]RX(A)u
-
(13)
CRX(An) -RX(A)]eitAU+eitAnCRx(A) -R%(An)3u - 0 as n - - locally uniformly with respect to t (the uniformity is evident for the second addend; for the first addend it is a consequence of the fact that e1tAu is a continuous function of t and hence for t varying in a bounded subset of 10,") the values of e1tA lie in a compact subset of X). It follows from (12) and (13) that exp(itAn)v exp(itA)v locally uniformly with respect to t for v lying in Im(RX(A)) = DA, hence for all v E X. Thus the theorem is proved.
Our next theorems establish some conditions under which (4) is valid and therefore the semigroups exp(itAn) converge. At first we concentrate our attention on the behavior of resolvents. Theorem 2.
limit
Let An E G(M,w), n - 1,2,...
.
Suppose also that there
exists as
RX = for some X,ImX < -m, s
's
- l
RX(An),
(14)
Then RX = RX(A) for some AE--- G(M,w) and exp(itAn)
It
exp(itA), t 3 0, provided that at least one of the following conditions is satisfied: (a)
(b)
ImRX is dense in X. -iuR_iu(An) I I uniformly with respect to n, when p
Proof. In view of Theorem 1, we need only to prove that RX - RX(A) for some densely defined operator A. We prove first that (b) implies (a). Applying Lemma 1, just as in the proof of Theorem 1, we obtain that. the limit (14) exists for all A such' that ImX < -w and satisfies (6). (b) means that for arbitrary x E X
50
-iiR- ill (A0)x
+ X,
u - -
(15)
uniformly with respect to ii. Hence we may pass to the limit as n -* m in (15),, obtaining
x = X. kim - iuR -111 I4
(16)
.
Since R = ImRA does not depend on A (Lemma 2), (16) yields x e R (R denotes It remains to show that (a) implies the closure of R). Thus we obtain (a). N = Ker RA - (0). Just another application of Lemma 2 completes the proof We may assume that w = 0, as in the proof of Theorem 1. Then after this. M/IImAt for A lying in the lower half-plane. Thus the family of uRAu operators -iPR_ip, u E (0,'), is uniformly bounded. Now let x E N, then -iuR_iux - 0. We shall prove, however, that
uim - iuRy - y for all y c- X
(17)
In particular, x = Aim 0 = 0 and we obtain the statement of the theorem. Since R is dense in X. it is enough to prove (17) for y E R. obtain
From (6) we
-ipR-iuRA = ARAR-iu + RA - R_iu (18)
- u (-ipR-iu)RA + R1 Fix some A, ImA < 0.
i
(-iuR-
If y e R, y e RAz for some z E X, so that
-iPR_iuy = u! (-itR_iu)y + y -
i
(-iuR-iu)z.
(19)
11
uniformly bounded family, we may pass to the limit as in (19), obtaining (17). Theorem 2 is proved.
Since -ipR_1YY is a y
The following theorem is often rather convenient because no information about resolvents is required; the condition is formulated in terms of strong convergence of generators. Theorem 3. Let An E G(M,w), n - 1,2,... Suppose that for elements of some dense linear subset D C X there exist the limits .
kim Anx
def
Ax, x E D.
(20)
n-
Suppose also that the range
R(A-A)
is dense in X for some A, ImA < - w.
Then A has a closure A E G(M,w) and exp(iAnt) ! exp(iAt). Proof.
We claim that RA(An)
s
RA as n
(21)
where R1 has a dense range and satisfies the conditions,
-A) = liD.
(A
-A)
(22)
Then taking into account Theorem 2, we obtain the required statement. Indeed, the only statement which needs proof is that A has a closure and this closure coincides with the generator of s - kim(exp(itAn )), i.e., with Al - RA-1. But this is evident since (i) (22) implies that (XI - RI-1)ID - A; and (ii) R(A-A) is dense, therefore D . RA(R(A_A)) is a core of AI - RA-1.
51
Moving on to the proof of (21) - (22) first we prove that A - a is M-1 (IImal - w)!Ix 1, x E DAn. G(M,w),fl(All - a)x Since An invertible. Passing to the limit for x E D gives the inequalityll(A - a)xll _> M-11.a l - w)ll xll, x E D, so we may define an operator (a - A)-1 with the A)-11 < M/ domain R(X_A) and the range D, satisfying the estimate ll(A RX is a bounded everywhere defined /(IImAI - w). Denote its closure by Rj. operator with dense range (the latter contains D). It remains to prove Then Rlx E D and Let x E R(?,-A). (21).
Ra (An) x - RZ x = RA (Ail) (An - A)Rax - 0 as n a Since R(A-A) = X, R,,(Arl)x -. RAX
since the sequence IIRA(An)Il is bounded.
for arbitrary x E X and the theorem is proved.
The theorems proved may be reformulated for the case of strongly continuous groups, in particular for self-adjoint generators. Some of the corresponding results are given below, others are left to the reader. Let exp(itAn), n - 1,2,... be a se uence of strongly Corollary 1. continuous groups, satisfying the estimate IIexp(itAn)f 6 M exp(wItI). Suppose that the generators An strongly_converge to an operator A on a dense linear subset D of K and besides R(Al-A) ' R(X2-A) = X for some a1, A2, IW 1 > w, Ima2 < -w,
Then A has the closure A, which is a generator s
of astrongly continuous group exp(itA) and exp(itAn)
exp(itA), t e tit as
n
Corollary 2. Let A. be a sequence of self-ad joint operators in the Hilbert spaceX and let Anx Ax for x E DA, where A is an essentially self-
n
s
adjoint operator in X. Then exp(itAn) immediately follow from Theorem 3. D.
exp(itA), t e R.
Both corollaries
Semigroups of Polynomial Growth
The notion of semigroup type, introduced in item A is too rough from the viewpoint of our aim - construction of ordered operator calculus. The inconveniency of this notion is in particular related to the fact that the semigroup of type wo may not satisfy IjU(t)ll < Mewt with w - woo since the constant M may grow as w 1 wo. Wk
be a Sobolev space X WZ(Rt1), k being an integer t mayLetbeXidentified with the completion of Co(IRI) with respect
Exam le 1. (recall
to the norm* (X) (I
I{OIlk ° (J
2
- 22)ko(x)dx)1/2, 0 E Co(R1); a
1C0'
(1)
the norm in W1 may be also described as 110 Ilk
sup
yE Co tp #o
where
* Note that 1 -
a2.
(R21).
(2)
n-k
f
_ -oo
(3)
is an invertible operator in L2(Rt1), so that (1) makes
aX2
sense for negative k as well.
52
II
is the usual L2 scalar product. Obviously WZ(R') - L2(IR1).) operator A in X with the domain D = C-0 V). defined by
Consider the
(4)
1A¢7(x) = x0(x), 0 E Co(IR1).
We claim that A has a closure, which will be denoted again by A, and this closure is a generator of a strongly continuous group exp(itA), t E whose action on ¢ E Ca(R1) is given by the formula lexp(itA)¢7(x) .
eitx4(x),
t E IR.
(5)
Itl)IkI.
(6)
This group satisfies an estimate
Ilexp(itA) Nk <
Indeed, denote the right-hand side of (5) by U(t)$(x). We have for non-negative k the estimate (6).
NU(04k < C
EO N
s=
as axs
C sI0 I0 I tlfllro
U(t)allo
Q=
=
First we prove
II
s`k
C(I + Itl)kIIm1Ik. For negative k, as (2) shows, W? may be identified with (W2k)* under pairing (3), and we obtain
IIU(t)hk =I1U(t)
(7)
-k =IIU(-t)11-k
Thus we have proved (6), and it is obvious that the exponent Iki in (6) cannot be diminished. The semigroup property and continuity follow immediately from (5), and so does the identity 1 at U(t)o+ AU(t)c - 0,
@ E Co.
(8)
It remains to apply Theorem 3 of item A. Example 2.
Let X = d2, A be a (2 x 2)-matrix
A -
(9)
Then e
ixt
ite
exp(itA) 0
e
iat
iat
,
t
0,
(10)
is a semigroup of the type w - -Ima, but Qexp(itA)II < C(l + t)elWt.
(11)
It is easily seen that the factor (1 + t) appears in (11) because A has a non-trivial Jordan block of order 2. In the general case there is a deep intrinsic connection between these facts which we have no place to In operator calculus we are interested only in strongly regard (see 1521). continuous groups of type 0 (such as in Example 1), so we give the following: Definition 1. A closed operator A in a Banach space X is called a generator of degree k, if it is a generator of astrongly continuous group U(t) - exp(itA), satisfying the estimate
Qexp(itA)Q 6 C(1+;tI)k,
tElt.
(12) 53
The family {AVl of generators of degree k is called uniform if the corresponding groups satisfy (12) with the constant C independent of W. The following theorem gives necessary and sufficient conditions for the given operator A to be a generator of degree k: In order that a given closed densely defined operator A Theorem 1. be a generator of degree k, satisfying (12), it is necessary that A 6 p(A) for Im% # 0 and k IJRI(A)mJ1 6 C j10
k.
7---T JImaj
and sufficient that . E :(A) and (1 constant C is the same as in (12)).
)
-m-k
ImX
1,2,...,
0, m
(13)
is valid for ImA large enough (the
A and -A are clearly the generators of semi(a) Necessity. Prooi. groups of type 0, thus Theorem 1 of item A implies that A E p(A) for Ima We prove (13) for Ima < 0, the proof for Ima > 0 being completely # 0. Using the equality (11) of item A and the estimate (12), we analogous. obtain
c_
(m-1)
1IRa(A)m!I k
C
r"e-tImaI tm-1(l+t)kdt o
k:
(14)
fir,:- ljs i:0 J.. OT
o
et' Ima
'tm + j - I
Since Joe-tatsdt = s/as + I (a > 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 <
Theorem 2 of item A, A is a generator of a strongly continuous group Now we are in position to prove the estimate (12). Let t > 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 <
C.
The lemma is proved.. Our next proposition is, in a sense, the main tool in construction of the mapping f - f(A). Let B be a Banach space. Then there exists an element
Let $ E Ck(i,B), T E Ck'(R).
Theorem 1 .
b e B such that*
T(
) -
(2)
IIbh
(3)
for each h E B* and
f,1$! V Ck
Ck Proof:
Note at first that the left-hand side of (2) is defined cor-
rectly. Indeed, (t) _ clearly is an element of
Ck(ht),
so -that T is applicable. Further, the left-hand side of (2) is linear in R and satisfies the estimate
<MTII f, 110 Il f IFhB. Ck
(4)
Ck It follows that there exists an (unique) element b C- 8** such that (2) and It remains to show that in fact b belongs to 8 (which is (3) are valid. consiftred a subspace in $ ** by means of the standard isometric embedding
Since B is closed in 8**, it suffices to prove the latter propo). sition for finite T.
8 -+ 8
Lemma 2.
The functional T E Ck (IR) may be represented in the form:'
T(f) where T(j) E C k
* (IR).
f E
T
j-0 (J) at3
(5)
f ECk(1R),
also may be chosen
If T is finite, T(j),
finite.
Proof. The latter statement-is obvious, since we can multiply (5) by the cut-off function $ C- Co such that yT - T. To prove that there exists a representation of the form (5), we consider an isometric injection
Ck(IR)
- CO(R) m ... a Ck(IR)
(f + 1) susmands (6)
f(t)
(f(t), si(t),
,
Wi(t))
and apply the Hahn-Banach theorem.
Lemma 2 gives a reduction to the case when T is a finite functional on CO(R). Let 0 CIR be a support of T (support is well-defined for finite functionals). It is well-known that T may be represented in the form T(f) - f0fdu,
where v - NT is a Rhadon measure on Q.
Hence
* Brackets < ,> 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)
<
(h(t) - 11h] (0).
(18)
that
1.It eT-t(I+ITI)kdt
I[Ih](t)I <`h0 o
1(1+ ItI)k f o
jh
1
< (19)
IAI)kdA -. const(1+ ItI)k.
I
Using (18), we estimate recursively all the derivatives up to the order
t and obtain that In E Ck(It) and 11h I L < C j hi 1-1. The investigation Ck Ck of (15) is rather simple and therefore we leave it to the reader.
returl to,the proof of Lemma 4. t E C OR) by the equality
We then
For given T E Ck+ at) define the functional
t(f) - T((i+t)kIIf), fE C(a).
(20)
We claim that in fact T E C+(R). To prove this, consider the sequence In - *nT, Wn being the same as in Lemma 1. We have
58
in (f)- ($nT)(f) - T(Pnf) -T((i+t)kIuIynf) f -1
(*,I)((i+t)kIff) +
(21)
T((i+t)kIjli'Vn]If-1-jf)
E
j-0
(here CI,*n] - I*n - y*,I denotes the commutator of I and the operator of multiplication by yin). The first term on the right of (21) converges to as an operator from T(f), and it suffices to show that the norm of for j - 0,1,..., f - 1.
Co(&t) to Co+l OR) tends to zero as n
[I,y0n]f(t) - Jo eA4n(a+t)-9n(t)]f(a+tjda -
We have
(22)
- a 5 XelXn(t,a)f(a+t)da, where Xn(t,a) is bounded with all Recursive estimates, analogous to in question tends to zero as 1/n. known formulae for derivatives of
derivatives uniformly with respect to n. that of [If](t), show us that the norm We obtain from (20), using the wellFourier transform, thgt
F-16)(y) - ik+P(1 - aq)k(i + y)-fF-1(T)(y),
(23)
and ip view of Lemma 5 it suffices to demonstrate that F-1(T) E C(tit),and Since ii=- C+(tt), its inverse Fourier transform is
IF`1(T)lC G MjTBC*.
given by (10), so the latter inequality is the consequence of (12). (Recall that t - 0.) Further F-1(T)(y) - nim F-1(rynT)(y) and this converIF-1(T)(y)
gence is uniform, since F-1(*nT) - F-1(*nT)(y) < MIT - ( T 1. is a continuous function, since F-1(IPnT) t and Thus F-1(T)(y) is a continuous family of elements of Ctst). is continuous, and Lemma 4 is proved. Definition 2.
We denote by Bk OR) the space of functions of the form
F-1(+), E). We supply Bk(It) with the norm
Ifj
f*
I
(24)
C
k
Clearly Bk(kt), suppli ed with the norm (24) is a Banach space.
It is also
clear that Bk C Bk for V )P f, and the embedding is continuous. One may ascertain from the proof of Lemmas 4 and 5 that multiplication by (i + Y) is an isomorphism between Bk and B,,, while (1 + a_)k realizes an isomorphism
between Bk and B. Now we introduce the union Q
87k ('R) -
(J
8,,
t),
(25)
which will be the symbol space in functional calculus for a single generator of degree k: Some properties of the symbol space 8k Qt) are described in the following: Theorem 2.
have
coT
(a) In order that f(y) E Bk(>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<2) be a function such that x(y) = sign y for IyI > 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
fl
(38)
E1+E2.
Ck
It suffices to show that the latter proposition is true in the case when Indeed, in the general case T2 is a limit of the sequence T2 ie finite. (38) implies then, that the T2n, n . 1,2,..., of finite functionals. sequence T2nT(O(t + -r)) is fundamental in Ckl(Btt) and henceforth convergent
In particular, T2nt(O(t + T))
to some element f r= CklGtt).
f(t) for
Thus f(t) - T2r(O(t + T)). Consider now the case arbitrary fixed t E lt. of finite T2. Let (t) - 0(t/n) be the same function as in Lemma 1, satisfying the subsi2iary conditions 0 G O(r) < 1 for all T. We have ,
T2T0(t+T)) - T2r4(t+t)44n(T)). n > 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 <
j-0 a-0
(1+ItI)k{!0
3TJats Il+E2+0(1/n)):
(41)
thus F 6 CL2(j,Ck1(t)) and the required estimate (40) is valid. the inequality
1+ ItI > 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
A .x
- j f(A)(RX(A))fxni
(61)
I xn0 + 0
I(RX(A))Ll X+ HA
H
as n - m. Further, yn - (X - A)Lzn n- y. y - 0 and (a) is proved.
Since (X - A) L is closed, then
Let D C DAL be a dense linear subset, invariant under exp(iAt), t e tR, f E Bk.
To prove that D is a core of f(A) we need to show that the domain of closure of restriction of f(A) on D contains DAL. Let x E DAL. By Theorem
5 (c) of item A. D is a core of At so there exists a sequence xn E D, n - 1,2,..., such that xn n- x, AZxn Ax. By (48) n
If(A)xn-f(A)xj <
constdxn-xJ +IAfxn-ALxmI ),
(62)
so that the sequence f(A)xn is fundamental and therefore converges to some y E X. It follows that x lies in the domain of closure of f(A) restricted on D.
(c) is proved.
Set D - In, DAL.
It is invariant under exp(iAt),
t E tt, and we obtain that to prove (49), it is enough to verify that,
afo(A)x+ Sgo(A)x - (af + Sg)o(A)x, X E D,
(63)
f0(A)go(A)x - (fg)o(A)x, x E D.
(64)
(63) is obvious. As for (64), we have go(A)D C D (see (b)), and we may write for arbitrary X E X*:
66
<X, (fg)o(A)X> - 2* ?n
(F-1(f)*F-1(g))
(<X.eiAtX>)
-
F(f)t(F(g)t(<X.eiAteiAtx>)) -
2e) F-1(f)t( (21x ?e
F-1(g)T <
(65);-
F(f)t(< (eiAt)*X,go(A)x>) 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 $(A) give identical results.
(20)
We intend
(21)
(21) means that on DAf both definitions
Since DAf is a core of t(A) (it is postu-
lated in the former definition and it will be proved below for the latter), this implies the statement of the Theorem. We have
fQ(a)(dEXx,Y) - f(
((exp(itA)x,y))
2n) 6(eit")](dEax,y); 1
- V(2
m(feitA(dEax,y)),
(22)
(23)
it
(23) being valid by part (a) of the proof. Thus we have to verify the possibility of the application of 0 under the integral sign. Let 0 be a finite functional. Then applying Lesma 2 of item E, we obtain f
0(J" e itl (dElx,y)) -
E
j-O
a? r
jR
3
-R
at3
(24)
Since x e DAf, all the
where duj are the Rhadon measures on E-R,R]. integrals f
eita(dExx,y)}.
1
converge absolutely and we can apply
eltla3(dElx,y),
the operator aj/atj under the integral sign. Further, the integrals 1RRr-ljeitaduj(t)(dElx,y) converge absolutely and by the Fubini theorem
We obtain that (21) is one may change the order of integration in (24). valid for finite 0. Now if the sequence in of finite functionals converges' to 0 in Cf*(1
8o(kt) C CO(lt), see Lemma 6 of item E). By the Dominated Convergence theorem, Thus we obtain (21) for
I 0 (a)(dE),x,y) + r 0(a)(dEax,y) as n + ^^. 0 E 80(It).
is obvious.
It remains to prove that DAf is a core of f Indeed, let x E D0.
o(X)dEx.
But this
Set fn
(25)
dExx.
x.n -
-n
n-'
Then to E DAf, xn
x, and
IO(A)x-$(A)xn12 ' JIII , n10(a)12(dElx,x) n*0, since f
I0(A)I2(dEax,x) is convergent.
(26)
The theorem is proved.
In what follows we shall not be interested at all in arbitrary Borel It follows functions as symbols. Only continuous symbols will be used. from Theorem 2 that if 0 E CO(1t), then DO(A) DAf and 0(A) is an Al-bounded
operator (i.e., J 0(A)xl < C(u x1+ JAtXA)).
In particular, if 0 E C°(IR),
0(A) is a bounded operator. As Theorem 2 (d) of item E implies, 80%i) is not dense in C9(k). Nevertheless, the mapping 0 - 0(A) defined here is in a sense the unique extension of that given by Definition 4 of item E.
Namely, BO(tt) is dense in Ca(R) with respect to the locally uniform convergence of uniformly bounded sequences,* and the following theorem is valid: Theorem 4. for all n.
to y as n +
Let On(y), $(y) E Co(Yt), n - 1,2,..., and 10n(y)lCpo G M
Let also On(y) converge to O(y) locally uniformly with respect Then On(A) n-+ O(A) strongly.
Proof. By Theorem 2 (a) the operator sequence {On(A)} is uniformly bounded. Therefore it is sufficient to prove the strong convergence On(A)x - O(A)x for x E D, where Dc= H is a dense subset defined by
D . {x E Hex - JRRdEIx, R - R(x) + m}. If x E D, On(y) n
O(y) uniformly on C-R(x),R(x)].
G JRR(x)Itn(a) - O(a)12(dExx,x) + 0 as n + m.
(27)
Thus NOn(A)x - O(A)XI24
The theorem is proved.
* Really, Bg(fft) D Co(11R), and Ca(st) is dense in Ca(R) in the sense that is indicated above. 72
G.
Functions of the Operator, Depending on Parameters
The results of item C are applied here to obtain some theorems on continuity of f.(A(c)) with respect to c, f being a given symbols, A(c) being a family'of generators depending on parameter c and satisfying some continuity conditions. We assume everywhere below that c is a variable lying in some open subset E of the space titre, endowed with the induced topology. Theorem 1.
Let {A(E)}EE
F
be a uniform family of generators of degree
k in a Banach space X. Assume that there exists a dense linear subset D C X such that D is a core of A(c) for all c C- E, and A(t) is strongly continuous on D (i.e., A(c)x is a continuous function of c for any x E D). Then for any f C- 61(1R) the operator family f(A(c)) is strongly continuous with respect to c on the linear subset
Xiff-0
D if I- 1
D A
x C- DIAk(e)x is defined and continuous 1 l if k) 2. t with respect to E for k < f
Remark 1.
In general, Df, I ) 2, may be empty.
(1)
However, Df Z) D, if
D is invariant under A(E) for all c and A(c) will be of particular interest to us.
This special case
Proof of the theorem. Set U(E,t) - exp(iA(c)t). We assert that U(c,t) is strongly continuous with respect to (e,t) C- Extit. Indeed, U(c,t)x is continuous in t, and U(E,t)x - U(EO,t)x as c -* Eo, locally uniformly with respect to t by Theorem 3 of item C.
Next, if x E Df,(-is/at)kU(c,t)x-
- U(c,t)(A(c))kx is continuous in (c,t) for k S 1?.
Now let fe q(k), x E=- Df, and let f = F(f) be a finite functional. We obtain, using Lemma 2 of item E: f(A(c))x
P ak E J at U(t,c)x)duk(t), k k-O supp f
(2)
duk.(t) being the Rhadon measures on supp f. The above consideration yields now that f(A(c))x continuously depends on t. If f is not finite, there exists a sequence fn C- 8k (tit) of symbols with finite Fourier transforms fn,
convergent to f in Bj (tit).
By Theorem 3 (b) of item E:
f(A)(c))x-fn(A(E))xj < C
0'n11 * I (A(.))x1 ) n
II f-1011
(3)
Bk
Since (A(c)) Ex is continuous, (3) means that fn(A(E))X converges then to f(A(c))x locally uniformly with respect to c, so that f(A(c))x continuously depends on t. The theorem is proved. Theorem 2.
Let {A(c))EE
E
be a family of self-adjoint operators in
the Hilbert space H , which are strongly continuous with respect to and essentially self-adjoint on the dense linear subset Dc H. Then for any f E Co(lt) the operator family f(A(t)) is strongly continuous in H. with respect to E. Proof.
Since f(A(c)) is uniformly bounded (the bound islIfICO -
= sup !f(a)I), it is sufficient to prove that f(A(E))X is a
continu0ous
aEkt
73
function of c for x E D. Our considerations are of course local and we Since A(e)X is conn.ay assume that a lies in a compact subset E0 C E. tinuous, we have C -
(4)
<
lA(e)'
sup
C EE0 We wish to obtain the uniform-in-c approximation of f(A(c))x by f (A(c))X, of the where fn E Co(R). Thus we have to improve the result of Theorem preceding item. Let f,g E CO(R).
Letmua 1.
M(R) -
Denote for any R > 0: sup
If(a) - g(a)I,
IXI3R m(R) -
sup
IaI
(5)
If(a)-g(a)I Then for any R > 0 the following
Let A be a self-adjoint operator, x E DA. estimate is valid:
f(A)x-g(A)xg2.¢m2(R)II x112M2(R) IIAx112 Let [EA) be the spectral family of A. Proof. non-decreasing function, we have 12
(dEXX,x) <
f
X2> R2
f
R
Since (Eax,x) is a
5 12 fw a2(dElx,x)
X2 >p2
(6)
R
Axe
2 '(7)
R
Thus
IIf(A)x-g(A)X12 - fIf(A)-g(a)I2(dElx,x)
>RIf(l)-g(A)I2(dEax,x) 6
IXI
(dExx,x) + M2(R)f I
aI>R
(dEAX,x) <
6 m2 (R) II x l 2 + M2 (R) I Ax l 2.
(8)
R
The lemma is proved. Now we choose the function f (A) E C0O2) such that and Ifn(a) - f(A)I s (6/2 l x") for 0IaI 6 R, where Ilfn - fNCO 6 21f I CO 0
0
/6, 6 - 1/n.
R = 2/2-C Rfli
Applying Lemma 1, we easily obtain
CO 0
fn(A(e))x - f(A(e))XI 6 n
,
so we have reduced the theorem to the case f r=- Co(IR).
we may apply Theorem 1.
(9)
But Co($i) C 80, so
The theorem is proved.
Remark 2. The reader can easily formulate this theorem for symbols lly growing at infinity (similar to the way it was done in Theorem 1). polynomi aT
Remark 3. We also may prove the theorems on differentiability with respect to parameter under some stronger assumptions. However, the expression for the derivative contains functions of several operators, so we delay the proof of these theorems until we develop the suitable techniques.
74
H.
Semigroups in the Pair of Banach Spaces
Dealing with Banach scales we shall sometin,es use a generalization of the notion of the generator in a Banach space, which is described below. Definition 1. A pair of Banach spaces is a collection of two Banach spaces j, and the continuous linear mapping: j : XI . X2. Definition 2.
in the pair
A strongly continuous tempered group of linear operators
X2 is a strongly continuous family {U(t))tEtit of bounded linear operators U(t) : X1 -* X2 such that U(0)
x11, xEX1
(1)
for some C, k independent of x (k is called the degree of the grou7); s
U(t+Tf)xf = 0
E
(2)
f=1
for any t E 1t, provided that s, T g E 1t, xf E X1, f - 1, ... , s satisfy s
U(Tf)xf = 0.
E
(3)
I
Pro osition 1. For any fixed k1 > k there exists an "intermediate" Banach space id, a generator A of degree k1 in Xm.d,and the pairs jl X2 of Banach spaces such t?at X1 -' Raid; l2 Xmid
-
= j2 o ill U(t) - j2 o exp(iAt) o jl, t E R.
(4)
Proof. Denote by Xo C X2 the linear core (i.e., the set of all finite linear combinations) of the elements of the form U(t)x, x E X1, t E 12. For any r E 1t we define the operator U0(t) : Xo - Xo by s E
Uo(t)xo
U(t +Tj)xf, xo E Xo,
(5)
f=1 where xr. = EL 1U(T2)xf, Tf
12, xf E X1, f - 1,..., s.
The definition (5)
is correct since, if x9 has two different representations of the form given above, (5) gives identical results for both of them (see (2) and (3)). The operators Uo(t) form a one-parameter group. Indeed, Uo(0) = I (identical operator in Xo), U0(t)Uo(T)x0 - U0(t)U0(T)
s
s
E
U(Tf)xf = Uo(t) E
L.1 s
f=1
f=1
U(T +Tf)xf = .
a
U(t+T+(f)xf = (J0(t+T) E U(Tf)xf - U0(t+T)x0, x0EX0. f=1
We have the commutative triangle of linear mappings: Xo
J1 1
/--#
xl
12
(6)
x2
where j2 is the restriction on Xo of the identity operator in X2 and jl is defined since Im(j) C Xo. Besides,
U(t) = j2 o Uo(t) o ji.
(7)
75
Then set N X. a
We drop that I'
- sup
mid
tEft
(1 +
i t i) -kl N j2Uo(t)xo 121 ' xo E Xo.
2 below again considering Xo as a subset in X2. is a norm function on X0 and
(8)
The assertion is
mid
jlx1mid 4 CN X1 11 xEX1.
(9)
N32 X0 12 G Nxo lmid' x EXo.
(10)
To prove this we note first that l- Nmid obviously is positive homogeneous It remains to prove that (9) and and satisfies the triangle inequality. (10) are valid and that N' Nmi does not take the value +- (the implication Nxo Nmid - 0 x xo - 0 follows from (10)). We have E
f-1
U(TI)xf'mid - sup i(1+
N
tEft E
f-1
NU(t+Tf)xf
E
f=1
Nxf Nl sup
(1+It+TkI)k1G
tEit
E U(t+T1)xp N 2) G f=1
sup {(1+ Itl)-k1 GC
ItI)-k1
tE&t
(1 + t I)
s C
E
f-1
1
,
2
G
(11)
Nxf N1(1+ITfI)k
since kl > k. Thus I. Nmid takes only finite values; setting a - 1, Tl -0 in (11), we obtain the inequality (9). (10) is obvious since the expression in curly brackets on the right of (8) is equal to Nj2xo N2 for t - 0.
We assert that Uo(t)xo is continuous in the norm I Nmid for any
xo E Xo and that U0(t)x0 Nmid G (1+ ItI)k Nxo Nmid'
(12)
Indeed, it suffices to prove that U(t)x is continuous in the norm N' Nmid for any x E X1. We have
IU(t+t)x-U(t)xNmid - suP tElt = max{au ITRj
(1+
(1+ITI)-k1
ITI)-k1
GR
NU(t+T+e)x-U(t+T)xN2 -
NU(t+T+e)x-U(t+T)-c N 2 +
(I+ITI)-kl NU(t+T+E)x-U(t+T)xl2} -
+ sup
ITI ;R = max(I1(R,e,x) + I2(R,e,x)).
We estisete first I2(R,E,x):
12(R,c,x) G C ! x j
(1+ It+T+EI)k+ (1+ It+TI)k
l(1+R)k-klsup
TEII
(1+ ITI) k
Cl
G
(1+R)kl-k '
where C11 is independent of e, IEI G 1 (but depends on x,t). Thus, given a 6 > 0, I2(R,t,x) < 6 for R large enough, Ici < 1. Fix such R. Then U(A)x is continuous in the norm N N22 and henceforth uniformly continuous when A lies in a bounded subset of R. Thus I1(R,E,x < 6 for a small enough, and the strong continuity of Uo(t) in the norm The I mid is proved. proof of (12) is reduced to direct calculation: 76
U0(t)x0 11 mid ° sup
{ (1 + I T
TEkt
I)-kl
I Uo(t + T)xo II 2)
- sup { (1+It+TI) kl (1+It+TI)-k1 [I Uo(t+T)x0 fl2) < TEkt
(1 + IT I )kl
< (1+ ItI)klsup {(1+ It+TI)-kl
tl
U0(t+T)xo 1121 E (1+ ItI)k1 IIX0'mid'
Tf--- kt
Denote now by Xmid the completion of X. in the norm I' Ilmid: The closures Uo(t) of the operators Uo(t) in Xmid form the strongly continuous group of linear operators in Xmid, 1Up (t) = exp(iAt), A being a generator of degree The diagram (6) gives rise to the diagram kl in Xmid.
Xmid
(13)
and (4) is valid.
The proposition is proved.
The proof of the above proposition fails for kl = k. One can Note. prove all the statements except the strong continuity of Uo(t) (and henceforth U )). There is a simple counterexample: Counterexample 1. Let X1 = t1 (the one-dimensional Banach space), X2 . C (R) (the apace of continuous functions on kt with the usual sup-norm).
Let y be a coordinate in kt.
For any tE It, xe X1, we set
U(t)x - x.ft(y); ]x - U(O)x, where
ft(y) '
0,
y ) t,
1,
y < t-1/(1+ ICI),
(14)
(15)
(1+ Itl)(t-Y), 1/(1+ ItI) < y 5 t. We have IIU(t)II < 1 for all t, and obviously U(t) is strongly continuous; next, if (3) is valid, we may assume (reducing similar terms if necessary) that Ti > T2> ...>TS. All the terms in (3), except the first, vanish for y E IT2,T11, thus the first one also vanishes identically in this interval and we conclude from (14) and (15) that xl - 0. By induction, xp - 0 for f - 1,..., a and (2) is valid. Thus U(t) satisfies all the conditions of Definition 2 for k s 0. On the other hand, direct computation shows that
sup 1U(t+E)x-U(t)x12 =1x11 tEkt
(16)
for any E f 0,'so that U0(t) is not continuous in the norm 11' Imid for kl . 0.
The operator`A will be called a generator of the group U(t) (one should bear i$ mind, however, that the construction of Xs d is not unique). For any f E Bk(kt) we define the operator f*(A) : X1 + X2 y means of the relation
f*(A)x -
Zn
jf(t)U(t)xdt, xE X
(17)
77
(cf. Definition 4 of item E; the integral in (17) denotes the pairing of
f E Ck+(1t) and U(t)X E Ck(1t,X2)). The following proposition is the immediate consequence of definitions and results obtained in item E. Proposition 2.
(a) f*(A) is a bounded operator from X1 to X2 and
ll f* (A)
<
2n) I1 f
fl
(18)
gk(JR)
(b) If f r= Bk0l(IR), there is a decomposition
f*(A) - j2 o f(A) o jl,
(19)
where f(A) is a function of the generator A of degree k in the Banach space Xmid (cf. item E). The proof is obvious. There is an important particular case of the general situation thus described above. Let E be a vector space endowed with the pair of norms
6- I i, i- 1,2 such that fl x fl 1 a p x 1 2, x E E. Let also A: E- E be a
linear operator such that for any xo E E, there exists an unique solution of the Cauchy problem i aatt)
+ Ax - 0, x(O) - xo
(20)
(the derivative in (20) is with respect to q A1), satisfying
Ix(t) 12GC(l+Itl )kgxo11.
(21)
Denoting by Xi, i - 1,2 the closure of E with respect to the norm and by U(t) X1 + X2 the closure of the operator xo - x(t), we obtain lthe strongly continuous tempered group in the pair (X1,X2) (the mapping j is simply the closure of the identity operator on E). Proposition 1 is now valid for k- k, since x(t) is differentiable and therefore continuous in the norm 1*) which properly enables us to prove the strong continuity in Xmid for kl - k. A slight generalization may be obtained if we consider some wider space instead of X2 (so that the image j(X1) will not be dense :
in X2)-
3.
FUNCTIONS OF SEVERAL OPERATORS
A.
Symbol Spaces Given a n-tuple A - (All ...,An ) of generators of degrees ki, i - 1, n in a Banach space X, we intend then to define the operator f(A)
1 n = f(Al,...,An) by means of the Fourier transform 1 n
f(A)x =_ f(A1,...,An)x F(f)(tl,...,tn)eitnAn
In 2 ',in
...
eitlAlxdt,
(1)
(27r)
in complete analogy with the definition, given in the preceding section for functions of a single operator. Thus we have to define the symbol spaces, for which the integral on the right of (1) makes sense under suitable assumptions.
78
Let a - (a
8 = (31,...,8n) be multi-indices, t = (tl,.... tn e
,an),
EItn. C. ven a fun lion +
:
sup
tEttn
(a
t, we set
stn
def (1 + where (1 + It l)-a at-on for (y1 < 81,.. Yn
(1+ tI)t I - . y<8 aty
(t)
(2:
.
;t1I)-a
8 is the abbrevt... (1 + Itn1z en, Y We denote by Ca(IR°) the Banach space of functions +(t), t E mtn, continuous with all the derivatives, which appear on the right of (2) f r which the norm (2) is finite. Be).
non-negative integer, .e may also define
if N is
On the other
the space C600), using the norm II+
' sup
I7
tE1tn
CN
(: + ItI;-N E Y<8
n
ItI)-N
(3)
2 -N
v (1 + ( E
Here ;1 +
I a2(t)I ty
j=1
o.lowing the ideas of Se tion trespect ve y,
tE, we define the space Ca+(Itn)
CN+(Itn)' as the s.rong closure of the set of finite con-
CN(kn)).
tinuous fun tionals over Ca(lt )
Next the spaces
and BN(Rn) are defined as the spaces of inverse Fourier transforms
Ba(tt
of elemen s of Ca+(kn) or CN(6tn transf r t on.
,
supplied with the norms induced by Fourier
Bo h d functions given above a e pa.t'cu!.ar cases of the more general of which will also be useful. Let (1,...,n} I1U . . U Ik -e a partition of the et {1,.. n} into the sum of non-empty sec in ubstts I1,...,Ik or. let = (vl,...,vk) be ulti- ndices o- the length k, R ...,8n) be multi-indices of the length n Then we -et
1,...,Ik,vl,...,vk sup
(1 +
t
whey.
t
I)-vk
(1+ It
II
tEEtn
Ik
E
Y< 8
I
ate. (t)j,
(4)
aty
and denote by C8
iEI
Il,...,Ik,vl, .. ,vk (IRn)
1
-e spa e of fun tions +(t) continuous with the derivatives of uch t.a th. ex,-ressior on the right of (4) is finite. 8
come back to C. . .
n) or 4, (stn) respectively in the cases k = n and
1
be
nach
We
e
e
he
n
8+ I E CI, M )
h
- CO v(R ) the c
Th, fa ti n ,.iyt
in C v(R° (reca:l
pace,
one is obviou
X-v.. i led
definition of the space CI v(gtn,)() of
sure. of the subspace of finite functionals
ei (yltl + ... +Yntn) belongs to CI.v(Ytn)
I i-.nde; components are non-negative), so for a given
t
t-
roe
ransform may be defined as
79
F-1(T)(Y) - (Zn)n/2T(elYt).
(5)
y being regarded asa parameter on the right of (5). We prove, just as in Lemma 4 of Section 2:E, that F-1 (T) is in fact the inverse Fourier transform of T, considered as a tempered distribution, and in particular F-1 has no kernel.
We denote by BI'v(Itn) the space of functions of the form f - FI(T),
T E Cs+v (stn) and set Of 1I
B
81,ve)
F(f)
II
(6)
B*'v($t n. )
CI,
where F = (F-1)-1 coincides with the (direct) Fourier transform of tempered
(Sometimes we shall use the notation 6v, CB, ... instead
distributions.
of B1.v, CI v, ... if there will be no confusion; also if multi-indices v.., etc. have all their components equal to zero, the corresponding
subscript might be dripped.)
The spaces
and 8N(IRn) are special-
izations of BI v(Rn). The definition of the latter being somewhat cumbersome, we concentrate our main attention on these specializations. This enables us to avoid obscure calculations with no essential loss of generality.
The elementary properties of the spaces introduced are collected in the following: (a) The mapping
Lemma 1.
f (Y)
-
(1 -- -)v1/2' ...
where
d2/ay2
(1 -
aY11
-
az
Z
;
(i+y)-B
32
+ y)-Bf (Y)
aylk
(i+yl)-Blx...x(i+yn)-Bn
i EI ayi
BI is the isomorphism of BB
'v onto B (IR n )
'
n
BO'" "
(7)
(8)
The mapping
f (y) -' (1 - 3y)a(i + Y)f (Y)
(9)
is the isomorphism of Ba(Rn) onto B(4tn).
(b) There are continuous embeddings Ba C B61
B
BI, a. 3 al; 8N C BNi, B S Bl, N a N1;
Ha+l/2+c ORn ) C B
8
n
a
n
a
C CBS ) C HB + 1/2 + C
( n).
(10)
(11)
where a+1/2+c d`f (aI+1/2+c,...,an+1/2+c), and 8+1/2+C is defined analogously; Hs(ln) is the completion of C0(ltn) with respect to the norm y2)-B/2(1 - ay2)a/2 L2 - II (1 + N
A
(here a,8 may have non-integer components). (c) BN(Rn) may be represented as the intersection:
80
(12)
(13)
Ba(1Rn),
Iai
al < N are ..rt,nuous
(a) We assume k = I for the sake of s:mpl .,it Proof. to the Fourier transforms, we reduce the desired staLemen whether the mapping
.
Passing to erification of
f (t) . (1 + t2)N/2(1 - at)-Bf itI The proof o a' ' + It' CN+(1Rn) onto C+(iRn) is the isomorphism of the latter statement does not diffe f,om t:ie ne dim r`onal case (sec s proved ana. g us Y. Lemma 5 of item E, Section 2). (9) .
e p ss to Fourier trats (b) The embeddings (10) are obvi:us these embeddings oincides word for word As for (11), the proof with the proof of Lemma 6 in Se.tio 2:E forms.
enough to conside
It i
(c) Aga n we pass to Fourier tran., orms.
(0,...,0), since it olloi,s from ia,
t'.e case
at multiplic.t on by
,i + Y) -6 isomorphically maps BNOR ) onto BN(IRn)
and '.lso
Thus we ne d t pr..ve that C5(Rn) =
Ba(lm) onto Ba(IRn).
n
Ca(Rn) and
1.1
L
the corresponding embeddings a e c.ntinuot.s. prove that
CN(ttn) =
n
t suffices _o
is turn,
C*(iRn)
(14)
IaI
and the embeddings are continuo s, ec.use on both sides of the latter iden itv :oinc
e
subsets of f mite fu c ionai: On,e this statement c
proved, we obtain by the closed g aph thco em r, t tht norm I T h C* is eq iva ent N
to the norm nI T III
=
a
T II Ca.
<
Thus if Tn
T in
N (R°) , Til
N
C* a(IRn),
so C+(Iln)C N
(i
Tn = TU(ltl/n)
C+(lRn).
I
I
Conve sely, let T e
T ina all C0(R°), where
E C0(P ), j(0
of Section 2'E), so that Tn + , in CN(R') and conversel
T in all
C+Then
()
IaI
cf
Lemma
T E CN(IRn)
Let us prove ( 4). If f E CN(Iln) then f (I+t7)N/2 EC*(Itn) and * (i+t) henceforth f E Ca(IRn) si:ice (i + t)a/ 1+ t2)N/2 is bounded for Ia < N. It follows also thtI'f IIC* < const{If IIC*. Or the op os te, if f e
E
*
a
n
Ca (It )
(I+I I
t
2 ) N/2 <
* f E CN(Itn).
then ,_ons
!
tI a f E E
N
C
*
n
(It )
tla,
for la! < n (here tIa = n1ti ai
(1 + t )N/2f
(a
E
(,n)
and
S.nce
:o ecu ntly
The lemma is proved.
We now continue the invcstigat.on of the properties of the symbol spaces introduced
Lemma 2.
(.) The pointwise multiplication is a continuous bilinear
map from BB1(,Rn) x Ba 2 (ttn) to BB' na a + %81
62 (>Etn
and from BN (IRn) x BNB (1Rn) to
+B2 (IR ) for any N,a,B1162 for which the spaces mentioned have been
defined. 81
(b) The function f(y) - e ytyy a y;1... yYne1Ylt1
...
elyntn belongs
B
to Ba(kn) and to 8(stn) provided that y < 8 (a and N may be arbitrary). The proof of this lemma coincides exactly with that of Theorem 2 (b) and `(d) of Section 2:E and is therefore omitted. (al.....an_1), d - (a,an)> 8 Let a Lemma 3. (B,Bn be multi-indices. The mapping f(yl,...,yn_1) i'n-1) induces the continuous embedding Ba
f(yl.. Proof.
and
n-1) C Ba(Itn)
Passing to Fourier transforms, we find it necessary to prove
T(tl,...,tn-1) x b(tn) E Ca+(kn), where T(t1,...,tn-1) E C64(ttn).
the latter statement is obvious, since the mapping f(tl,...,tn) + f(t1,...,t,0) is continuous.
But the
Ca (IR
Ca(,tn)
n-1
Our next lemur is concerned with the mapping which identifies the pair of arguments of the symbol. Lemma 4. (YIP ... >Yn-1)
f(;}1,.... y0) being the function of n arguments, set 1jf] x Thus, jf is a function of (n - 1)
arguments.
(a) j defines a continuous 8 ' (81,...,Bn-2'8n-1 + Bn)
mapping from BN(ltn) to BN(IRf-1), where 8an-1)
(b) j defines a continuous mapping from B0(IRn) to the same as above, while a - (all ...an_2,min(an-l'an)) Proof.
relatTon
Consider first the case B - 0.
where B is
We define the operator Y by the
CT-S1(tl,...,tn) - (t1.....tn-2'tn-1+ tn), where 4 ie a function depending on (n - 1) arguments.
(15)
Since
I to-1 + to 16 I to-l I+ I to I.
T is a continuous operator from CN(tn-1) to CN (It") and from C60 (tR -l ) to Ca(lm), d being defined in the statement of the lemma. n (or Ba(6t)). " of B(It)
Let f be an element
Denote by T - F(f) its Fourier transform and by Ti 0
the following functional on CN(IR argument:
n-I
a )
n-1
(or Ca(ht
)), continuous by the above
T1(4) - T(T(4))
(16)
Calculating its Fourier transform, we obtain F1(T1)
((T-)n-1T(eiyltix...xeiYn-2tn-_e1Yn-,(tn_, + td) (17)
- V2Tijf(yl....,yn-1)'
so the lemma is proved for 8 - 0.
Let now 8 # 0.
Representing f(y) in
the form f(y) - (i + y1)81... (i + yn)Bnfl(yl.... ,yn), we then obtain the equality ljf](y1,....Yn-1)= (18) (i + yl)B1
82
...
(i + Yn-2) Bn-2(i+
Yn-1)8n-1 + 8njf1(yl,...,yn_1),
which immediately implies, in view of Lemma 1 (a), the desired statement. The proof is now complete. Next we introduce the difference derivation of symbols, which has applications in numerous formulae of the functional calculus for noncommutative operators. First, let f be a smooth function of a single variable y c- R. We define the difference derivative of f as a function of two variables yl, Y2, given by the formula
f(y1) - f(y2) _y2 6f(Yl,Y2) = dy (Yl,y2)
Yl
Y2 (19)
yl
Yl = Y2
f'(y2),
Difference derivatives of
Clearly 6f(y1,y2) is again a smooth function.
higher order are defined inductively; to obtain 6kf/6yk one should fix all the arguments of 6k-lf/6yk-l, except one, and apply the operator 6/6y with
Thus 6kf/6yk is the function
respect to this argument, according to (19).
of k + 1 arguments. It will be shown below that 6kf/6yk is a symmetric function of its arguments so that it makes no difference what arguments are fixed in our inductive definition.
Analogously, for a given smooth function f(y), y a (y1,...,Yn) a Qtn and multi-indices a - (al,...,an), we define the difference derivative of 6Ialf/6ya - (6/6y1)al ... (6/6yn)anf which is the function the order a of n + laI variables, naturally partitioned into n groups, namely, (Y11.Y12....,yll + al;...;ynl,yn2,....Ynl+ an) by suc(Y(1),....Y(n) :
cessive application of the operators (616yi)ai, i = 1,...,n.under fixed values of other variables. In what follows we sometimes use different notation for variables; for the sake of convenience we separate the groups of variables by semicolons and the variables inside the groups by commas. We formulate the following lemma only for n 1 in order to clarify the notation. The proof for n # 1 is identical (up to notation) to that given below.
6kf/6yk is a symmetrical function of its arguments (i.e., Lemma 5. no rearrangement of the arguments affects its value). The following identities are valid:
6kf
1
1-µl
0
0
yk (Yl.....Yk+1) - J dull
f
dug ...
1 u1
3
duk
f(k)(idyl+.+ukyk+(1-u1-...-uk)yk+l)
X
(20)
(here f(k) is the k-th derivative of f); 6
k
6y
f
k
(Yl.... ,yk+l) =
k+l I f(yj)
k yi)-1. II
i=1
(21)
l
i#j
ifyi0yj fori#j: skyk
(Y'...,Y) ' k f(k)(Y);
(22)
6k f
byk (Y,x,x,...,x) _
(23)
83
(a )al ... (ay3 )ak+l6kf(YIP ...'Yk+1)]I
k+l
yl=...-Yk 1-
a.l ... ak+l'
(k+ a ) 'h
(24) f(k + jaI
(Y)
The symmetry of 6kf/iyk is a simple consequen a of (21). P oof. w n cd only. to prove (20) - (24). First we remark hat 1
6f (Yl,Y2) -
.
' (tyl +(l- t)y2)dt.
(25)
0
Inc eo
f(y1)y2)
d
f
=1
dt Cf (Y2 + t (yl - y2)) Jdr-
0
(26)
I
(Y1 - 52)J f' ;tyl + (1 - t)y2)dt, 0
and we obtain (25) applying the definition (19). Thus (20) is proved for We proceed now by induction on k. One has ayk (yl,...'Yk+1) d
I
J
6k-1f
k-1 (Y,Y39Y4,...,Yk+1)) od ( dy dY
J1
Jld- Jlu2du2f 11243 0
0
IY=TYli(1-T)Y2 uk-,dukf (k)
u2
0
0
iu2Yl+(I-t)u2y2'+U3y3+.+(I-1'2-...-vk)yk+1 The fol
change of variables in the integrand:
8(vl,. ,vk) V1 - r112
results is 6kf
2
u3, ,vk - uk: det
- (1- t)u2,v3
u2
a(t,u2,.,v,
the expression yl ...,yk+1)
,dvlfl-v1dv2
J1
...
110 - v2 -... -"k-I dvk
0
e y,
f (k) (v 1Y1 +42Y2 + ... + vkyk + (1 - vl - .. - vk)Yk+l) , whichco n ides with (20) up to the notation of variables proved.
dk: 6y
T'-u' (20) is
N xt,
y,a,x,...,x) =
J
1
1-U1
dyl Jo
o
dy2
.. J1 -1 i - .
- uk-1
o
k
x
x f(k)(x(ul+...+yk)+y'l-ul-...-uk)) After t' + uk'
84
.
'nimodular change of variables vi - ul+ . +1:k, v2 - 2+ ... + k - uk, the latter expression tak,s the form
yxx
6kf
x)
.
6y 2jv2 v3 ... jvk-ldvk} _ 0
Jldv1if(k)(vlx + (1 - vI)Y)jvl 0 0
= j1 f (k) (v1x + (1 - vl
) (k
0
vi-1/(k We a multiple integral in curly r k s eq als S t ing x - y in (23), we come directly 23) denoting r - (1 - vl)
ce
s
ta.n to
2
(
dv1,
1)
ince f,(1 -T)k-ldr
)
f k+I l)(Y)fldu rl-v1d'J2 1
0
0
fl
'
20
y
l/
egas
1
xpression on the left of
th
- 1 - .. - k-1du ual k 1
µak x k
(27)
1-u1-...-uk)ak+1} = f(k+la )( )+(k, l,...,ak+1)1 thee I
notes the multiple integral. Af er e change of variab'es . + yk. we ob ain ul + y21 ...,vk = ul+
v2
,
ak
m k
kavk- { vk - vk-1)ak x
I$...,ak+l) - fldvk{(1 - 0k 0
(28) a-
x jvk-ldvk-2{ ... jv dv1 0 0 Comput
he inner integral.
,2
2- v.)a'valdvl -
0 Thu
We
L1
hen have
vl+al+aljlta
a
-
(
1
2'
1
01'02.
1+a +02
dt
(1 +a, +
al?
0
a, ..,ak+l ) = f(k-l,l+a +
(k
M.
2 -
(
(1+a1+a2)._
k1
-
t er ha d, (O,a) - 1 since (24) '.s .1 ar y -al d for k - 0. app yi g th obtained relation r cursively we g t th
a1 a
(k
(
and wit
-hen
a2)'a3
+
1
1
x $(k-2,2+a1+a2+a3,(1 4,.
On
+1
a2 +a 3)'
k!
24) is p oved It remains to pr ve (21 . F r k - 1, (21) coincides de nits n (19). By induc 'on o k we have k
6
(Y1" '
k I yj - yi)-1 -
I f(
'yk+1) _ (yil Y2)
.-1
j
j
] 2
3
i 2
k+l
- Ef j-2
k 1
, )
1
(Y
i2 ]
Y2- YI
k
(
k+l
E (Y -Y f
j-
]
2
14]
k1 k+1 ]=lf(Yj)(Y5 -y1)1n1(
i#j
-Yi)
k+ }
E
j=1
)
i
] i=1
]
k1
u (y. -y-1 i
ij
] i -1
]
85
In particular, using the difference The proof of the lemma is now complete. derivative, we obtain a simple expression for the remainder in the Taylor expansion, as shown in the following:
There is a representation of the form
Lemma 6.
(y-xrk
f(y) = NEl k-p
k7-
f(k)(x)+(y - x)N 6NN (Y.x,...x)
(29)
6x
for any N and any smooth function f(x), x E hl.
Proof is carried by induction on N. for N - 1 and we have 6N
(Y,x....,x) -
6x =
ff p
N 6N
We come back to the definition
(x,x,...,x) + (y - x)
6x
N+1 N+1 (Y,x,...,x)
6x
N+1
(N)(x)+ (y -x) 6
6
(Y,x....,x)
N+1
6x
for any N ((22) has been used), so that the induction step works and the We give also the n-dimensional formula for N - 1: lemma is proved. We have
Lemma 7.
f(x1,...,xn) -f(y1,...,Yn) _
6f
n
6y =1 (x.J -y.) J
(x 2' "
x.,y.;y. ,...,yn ) "x j,-1'J J J+1
(30)
The proof is obvious. The analogue of the Leibniz formula is valid for the difference derivative.
Lem..e 8.
e have
6y (fg) (y1,Y2) = f(Y1) I. (Y1,Y2) +6Y (Y1,Y2)g(Y2).
(31)
proof.
f(7r)g(Yl) -f(y2)g(Y2)
f(Y1)(g(Y1) -g(Y2)) +g(Y2)(f(Y1) -f(y2)) Yl - Y2
7l - Y2
Next we show that difference derivation acts in symbol spaces defined above and is continuous there. Lemma 9.
Let t
f(tl,....tn-1) -
Than 01 E CN+l
(hn-1)
Consider the function
ftn-1@(tl,...,tn-2.E,tn-1 - E)dE.
and lI PI M CN+l < C
and n.
CN(Rn).
(t1....,tn) E fin,
(32)
CN , where C depends only on N
(roof. Clearly $1(t1....,tn-1) is continuous and, moreover, the inequality J
141(tl....Itn-1)I -
Iftn-10(tl,....
tn-2,E,tn-1 - E)dEI <
< const A f ! CN I f to-1 (I + t2 + ... + to-2 + E2 + (t n-1 - E) 2 )N/2dE i < 0
86
const
to-11(1 + t2 + ... + to-2 + to-1(g2 + (1 - E)2) )N/2dC I CNJoI
S const
C (1 + t2 + ... + tn-1)N+1/2
m
N
holds, which implies the statement of the lemma. Lemma 10.
BN+l
a-1)
Difference derivation is a continuous operator from
to BN (fit°) .
Proof. Denote by I CN+1(ttn-1) the continuous operator such that I - $1, where $1 is given by (32). It follows that the func:
tional T1
:
Now
T(I$) belongs to CN(d.a) provided that T Q CN+1Ot°-1).
-
$N+1($tn-1)
let f E
Set T - F(f).
Then
F-1(T1)(Y1,...,yn) s F-1(T o I)(Yi....,yn) 1
T(eit1Y1.
1
. eitn-2Yn-2 ftn-1eiltyn-: + (tn-1 - )Yn3dt)
-
o
T(eit1Y1.....
-i
1
e itn-2yn-2eitn-,yn-, _
(,/(21r) )° Yn-1 - Yn 6f
i
-
dyn-1
(Y1'...' n-2'yn-1,yn)'
and we obtain the statement of the lemma immediately. Corollary 1.
For an arbitrary multi-index 8 - (81,...'8n-1), the
operator 6/6ypp_1 is continuous from p+l(t°-1) to BN(ltn), where
(81,
Proof. Let f E 4+1 *n-'). By Lemma 1 (a), f - (i + y)dg, where g i_, 81,1+1, 1 8 pbRR cons t If I 8 . By Lemma 8 8N+1 BN+1
6f Y1)81
yn-2)dn-;
x
6Yn-1
x {g(Y1,....Yn-1)
(i+yn-1 )Sn-1-(i+yn)8°-1 +
Yn-1-Yn
y
d
+
(Y1,...,Yn-2;yn-1,yn)(i + Yn)8n-'} -
n-1
g(Y1.-...Yn-1)(i+yl)d1 +
S& 6y n-l
...
(i+yn-2)Sn-=
(y1,...,Yn-2'Yn-l,yn)(i+
8j .l E
(i+Yn-1)3(i+yn)dn-1-3
+
j-1 Yn)dn-,(i +
We obtain by virtue of Lemmas 2 and 10 that g E B0ORn)
yn-2)Sn-2.
6g 6 BN 0 R°) and 6yn-1
87
(i+yl)d1
(i+yn-1)3(1+yn)-n Ff
w(IRn) .
IB(R,3z...3n-2J3n-l
E E
that
t
B OR
the swa A ve t)r spaces, which is nol neces mor , that f: 6yn_1 I < const 11 f d BN; BN+i
mma i
being rather complicated otherwise).
eed, B
I
th', embedding is continuous (Lemma 1
).
(
(IR)
8N'NS)(IR2)
t
tuns
th no
se
B 3
OR
t r:ma
s
d.
p
6i6y is a continuous o ra r from B Corollary 2. (we form-.late the statement only for the one-drmeosior
wha
nd
lv d ii ct)
e
i
den tes
he
:
'J)
n-236n
13
N
j=1
n-1
1
OR
and
t app y th ab.;ve
r
Corollary 1.
To finish with detailed descrip n the spare of symbols which plays in se functional calculus for several n n comm S'(lRn) the following space
spa
n r d
e
n a
a
n
o
We de o e
rs
Bs(&tn)
S°°(Rn
The embeddings established in
symbc the f 'ng o
emma
a)
e
he y
(33)
.ea
s
mete
he
1.
i
fo lowing:
Theorem 1. The functio ) belongs smooth and for some integer m = m aid we have
S
'-If Iy,
ma, be re
n
h spa e F
e
-
an
b
'i
f(A
=
it is
" pan)
35)
v
We begin with def nit on con Let X be a Banach sp c A ..An the degree of gener-to h ng (k1,...,kn). For a - y f 6 b (ERn ,
if
the form
n
uen razor
n ,An)
v
(a1,
t d a
s
for any partition IJU .. U tk of th
f(A1..
o
(34
S OR) -
Functions of Sev ra
i
i
''n);
(y
S°°(Rn) = (y + +Y2).i2. n
B.
f
(1R)
)
man o dering of opera uple of generators x 1, n. Set k = p rator .
in
e
f
Ul 1'
= VA(F)
An
in X by means of the rela -ion < h,f(A,,...,nn)x >
def
_
n, n
2 F f) <
.
-e1ClAlx>
(2
(2
for any x E X, the elements of X' r
-
X Here th and X, whil .
tional F(f) a _' ki n ) on 1
Theorem 1.
X and
88
e s
b
.Q t)
-
he el
,> d note the pairing be ween enote the value of he En. n
n
f(A1,...,An
a co
ec
def ned bounded oper,t-r n
s
f(A1,...,An)
11
(3)
s C II f I Bk0)
for some constant C not depending on f. Proof. for any x e X.
is a continuous X-valued function of t 6 Indeed, 0) A1x =
e1tn0) An. n =
Ee
it
Vn
J nAn ...'e it J +1A'+1
(a itJAJ-
J J)eit j-. (0)A j e itS0)A
e it,(0)A, x-+0
j=1 itAj is strongly continuous by definition of a as t + t(0) since each e generator and all the factors are bounded when t lies in a compact set. Since Ai is a generator of degree ki, i = 1,...,n, one has the estimate
11 eitnAn,...'e1t1A1xq a C(l+I tnI)k'... (1+It1l)kl I x H is an for some constant C. Thus we have proved that a ltn!'n ,,.eit1AIX It follows tthat element of Ck(lRn,X) and its norm does not exceed C Ix I.
the expression on the right of (2) is defined correctly and f(Al,...,An) norm not exceeding is in any case a bounded operator from X to X** with the 1 n
const Qf B . It appears however that the range of f(A1,...,An) is contained in X C X** (see Lemma 1 below); the latter consideration makes the proof complete. Lemma 1. defined by
T C- Ca+(htn).
Let 0 C-
The functional x on X
'. T(), hEX* is determined by a (uniquely defined) element x C- X. to other words, T(O) a fT(t)$(t)dt (the latter notation sometimes will be used if it is convenient) is a correctly defined element of X. The proof of Lemma 1 does not differ from that in the particular case n - I (Theorem 1 of Section 2:E).
Unlike the one-dimensional case, one has to make supplementary assump1
tions in order to define f(A1,...,A1) for growing symbols. The underlying reason for the distinction between one- and multi-dimensional cases is that, generally speaking, in the latter case the analogue of the scale {Hf} (see Definition 5 of Section 2:E) cannot be defined in such a way that the operators Ai; i - 1,...,n, prove to be generators in all of its spaces. In the following items we deal with the case when such a scale may be constructed or is given a priori; here we give a sketch of another approach which leads us to the consideration of f(A) as an unbounded operator in X. Both approaches are of course equivalent for functions of a single operator. We introduce the following: Condition A.
There exists a dense linear subset D C X such that
(a) D is invariant under U(t) -
for any t E ,n.
(b) U(t)x, x E D is infinitely differentiable with respect to t in the strong sense.
89
(c) All the derivatives U(a (t x
a
(.-) (U(t)x) have the polynom;al
-
rate of gr.wth-.
< C 1+ tI)s, xE D,
I!U(a)(t)x
(4)
where s - s(a), C = C(x,a). Theorem 2.
_et
Condition A be sa
For every
sf'.ed.
f E S (ktn)
th, equality
f(A1, ..,An)x (n JF(f)(t)
( )xdt, x E D
(5)
n
defines f(A1,...,An) as a Tiaei
For some B = (f'....,Bn), f E
Proof.
f E BNo(ttn), where N
n
9
ECNo(Eti).
operator w th the domain ').
= max s(a).
On the ofi'r hand, F(f) ECh (Rt
n
n
Applying Lemma l,weobtain
)
I Went f X The linearity of m s proved.
a correctly define h is ob ious, -hu, he i
the operator (5
ees that U(t)x E
C:)nditioi A
a4B
that
n particular,
SN(,n)
3urther, we might inves igate, ie cl bil t, of the obtained operators under suit bl assumptions Howe e , we I ave this subject and go on with our brief re ew of the presente approach -
Assume that some of the oper.tors A1,.. not necessarily the ge era ors pk (y , y E tt l , the operato
An, namely Ajl,.... AJk, are
n X, but for g
en
olynomials pl(y),...,
11
UP1,...,Pk(t:k
..
tJr) - pl( AJ
...
pk( J
jk) (6)
xexp (i Ak
exp(i k jntjn)
k+ tJ +) ..
is defined on the dense linear subs t D and Condition A is s tisfied for instead of t). Then for a y f E S,(R"), which 'Pk(tJk+l'"
Upl'' "
has the form pk(yJk)g(yJk+l ...yJ,,),
F(yc,...,yn) - Pl(Yjl)P2Y 2)
g E S n-k),
we may define the o
i
ator
(A1,..
n An) by means of the
formula f(A1,... An)
`
7) 1
n_k J
(t)(tjkyl... ,tjl) x (7)
X UPi,...,-jk(tjk+l,...,tjn)dt k+,
In particular, if the above c.rndition is Pl,...,pk and D does not depend on the cho the mapping of the set of func ions f E into the set of linear operato a in X wi h
.
dtin
x r= D.
t1sf ed for any olynomials cc of polynomials (7) provides (stn) p lynomial in (ydl' t e omaiT D
'''ik)
* Here is the substitution of t e a (1 . n}; the numbers over the operators on t e right of-(6) have a le.r sense (see ntroduction o this chapter). Namely, they def e the o acr of he factor; the gre. er the number over the factor, the rt er le t is its position in
the prc dt.ct.
90
If some of Ajl,...,Ank are in fact generators we obtain then two different
definitions of f(A1, ..., p )for symbols described above. However, it may be shown that both definitions lead to the same result; this statement will be proved (for operators in scales) in the following items.
We finish this item with the definition of Weyl ordering of generators. We introduce the following: Condition W. The operators A1,.... An are defined on the dense linear + tnAn tlA1 + subset D C X. For any t E stn the linear combination is a closurable operator and its closure (which will be also denoted by Besides, is a generator of degree N in X. I
< C(1+ ItI)N,
(8)
where C does not depend on t.
This condition being satisfied, we are able to define the Weyl-ordered function of A1,...,An for any f e BN(Gtn). Namely, we set fW(A) = fW(Al,...,An) =
nF(f)(t)eit Adt.
In 2 1 9t
(9)
(2n)
The definition (9) is correct and Ilfw(A)
11
6 const if IR .
To approve it
we make use of Corollary 1 of Section 2:C. Since D is a core of t'A, the range of l - to restricted on D is dense in X for all non-real A and we obtain, by virtue of the mentioned corollary, that e1tA is strongly continuous with respect to Now (8) shows that eitAx E CN(6tn,x) for any If also ConX E=- X and the application of Lemma 1 completes the proof. dition A is satisfied for el , we may define fW(A1,...,An) for symbols
f E S').
Linear Operators in Banach Scales
C.
Let A be a bidirected set, i.e., a partially ordered set,_such that for any 61,62 E A there exist the elements 6,6E A, such that 6> 6i, 6 = i = 1,2 (in other words both A and A' are directed sets, where A' coincides with A as a set and is endowed with the inverse ordering relation). Definition 1. A Banach scale (over A) is a dollection of Banach spaces together with linear operators i66, : X6 -+ X6, defined for 6
any pair
i6d
:
6 E A
'
x A satisfying 6 > 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, <
Thus IxY + xy - x - x 161 < c for these
y, y, and this means that xy + z_ l x + x.
The continuity of multiplication
by scalars is proved analogously.
Now if aY + x1, xy - x2, then xy
xY
x1, x1
x1 - x2.
x2 for 6 < 61, 6 < 62. The proposition is proved.
bl
x1, xY
62
x2 for some 61,62, and-
It follows that 1x1 - x2 16 - 0, i.e.,
We now expose some results concerned with integration theory to be used in the sequel. Let 0 ktn - X be a continuous mapping. (This means that 0 transforms convergent generalized sequences into convergent ones. It is easy to see that is continuous if and only if for any to E ktn there exists 60 = 60(to) E A such that y'(t) E X60 for t lying in some vicinity Uo 5 to and :
the mapping
:
Uo
X60 is continuous.)
Let also the distribution T E
E D'(kn) be given. We intend to define, under some conditions, the element T[*] C- X. Let 0 E Co(ktn), f n$(t)dt - 1. Set _,(t) - e"n4(t/e). e E kt, kt
It is well known that
TE(t) - (T*0E)(t) = fT(r)oC(t-r)dt is a smooth function, and T. * T in D'(ktn) as a - 0. Moreover, if T is E.0 the distribution of the order k, TE[f] T[f] for any finite function f with k continuous derivatives. First we define the integral
TECy]
f
n Te (t)*(t)dt.
ks
Let r denote the set of finite unions of compact polyhedra in kt°, ordered by the inclusion relation (i.e., -y< ;if f y c d). Clearly r is a directed set. We set TC14#1 - Aim
fYTe(t)u(t)dt,
y E'r
93
(The definition of fyTE(t) x provided that the limit on the right exists. x p(t)dt is obvious since,y being compact, a standard argument shows that y - X6.) for some 6 E A, 0 is a continuous mapping 0 :
We say that TCy] is defined (and equals x E X) for
Definition 3.
given T E V' (htn) and continuous, 0
X, if for any 0 E Co(L2n),
n
:
fktn0(t)dt = 1, the limit
?im TECO] E-'O
exists and equals x. If W lies in the subspace Ca(ttn,X6) of the space of all continuous
rtn -+ X, and T lies in Ca (stn), we can prove the existence
mappings theorem.
Theorem 1. Proof. Cm(titn).
Then TCy,] is defined,
Let 0 E Ca (Qtn, X6) . T E
described in Lemma 1 of Item B.
TL*] E X6, and equals the element
Let 0 E Co, f0(t)dt - 1.
We assert that TE - T* 0E lies in
Indeed, the Fourier transform F-1(TE) =
F-1(T) E 8a(,n) and F-1(0E) E Ba(lm) (we have made use of Lemmas 1
and
n
B+
2 of item A), so that TEE Ca
* f) for any f E S(tn) and henceforth
Next, we have TE(f) = T($
(multi-dimensional analogue of Lemma 3, Section 2:E) for any f E where
n
B
(0E * f) (t) = fknOE(t)f(t +T)dT.
We have
B< C fl f
M 0E * f
II
Ca
0E * f
B'
CIeI-IBI II
f
B
Ca
Ca
with the constant C independent of t,
(5)
CO a
lei < 1.
Let T9 - yqT, where 4,t E C,(In) be a sequence of functionals convergent to T in Ca+.
We have for lei 5 1
ITE(f) - T(f)I < ITQ(f) -T(f)I + IT¢(0E * f) -T('E * f)I + ITR(f - OE * f)I
(1+C) IITt - T II
11f
Ca
II
+ B
Ca
9. (f - 0E * f)
+ II T I CB* II
CB
a
CL
We assert that ITE(fh) Set here f(t) - fh(t) _ , where h e X6. - T(fh)I tends to zero uniformly with respect to h,Ilh h = 1. Indeed, u > 0 being given, the first summand is less than u/2 if we fix t = to such
that liT9o - T I Ca+ S l
2
(1 + C)-1 {
11y
11
d
)-1.
Besides, the function
0(t) is uniformly continuous together with its derivatives up to the order B on any compact subset of Itn, so it turns out that J,to(t)(fh(tN-11(0E * fh)x x
(t)) tends to zero uniformly with respect to tE Itn, hE
together with all the derivatives up to the order B.
II yp(fh - 0E * fh)
II
B
Ca 94
-' 0 as c
-
1,
In other words,
0 uniformly with respect to h,
II
in
II
= 1,
and our assertion is proved once we choose a small enough.
It follows
that I T,(4,) - T(,y) R 6 a 0 as E -' 0. It remains to prove that for any fixed E, TEC0] exists and coincides with T (0). Let yR E C0(Rn), 0 6 *t(t) 4 1, < m; then for any two and WA,(t) - O for l t l ,' E. Assume that f ynT - T C
a
Y,y C r such that Y r y and Y 73{t1Iti 4£+ e}, we have
TE(t)I(1 +Iti)odt <
f_
I
Y \ Y ITETe((tt))1
Indeed, the function
(1 + ItI)
a
, defined to be zero at the points
where T(t) - 0, may be approximated by smooth functions in L1(y \y), and we obtain that f_
ITE(t)I(1 +Itl)adt 6 sup
Y\Y
TE(O)
P Co(Y\Y) (6)
CO
suPIT(+E*0)I =supl(4gT-T)(OE*$)I CIEI-1"
s* c
hnT - T I
Ca
by virtue of the second of the inequalities (5). Since JpeT - T H a* -. 0 as n + (6) implies that the integral C8 fnTE(t)(1 + ItI)adt
converges-absolutely, so that the limit (4) necessarily exists.
(7)
Moreover,
the, convergence of (7) yielji& that also TclV] ` nim Since the latter expression is simply TE(ty), the theorem is proved. From now on we do not distinguish between parentheses and square brackets in the expression of Tt(y).
We come now to the definition of linear operators in Banach scales. Let X {X6}6E e, y = {Y(T}oE£.be Banach scales, X and Y being their total spaces. A linear operator A : X -+ y is by definition a linear mapping A : X Y of some linear subset Dp C X (the domain of A) into Y. If X = y,
we call A the operator in the scale X. To any linear operator A X ; y the collection {A5e)(6,o)E A X X6 -. Y0'may be of linear operators A6o :
:
L
set into correspondence.
Namely, we set
DA6o = {x
DAIx G X6 and Ax E Yo},
A6ox = Ax, x rc DA6o (8)
Sometimes we drop the subscripts and write A instead of A6o. We denote by L(X6,Ya) = L6a(X,y) = L6o the space of linear operators A : X + y such that A60 is a bounded-everywhere defined operator and by C(X6,Yo) = Cda(X,y) = C6u the space of linear operators A : X -* y such that A6o is a closed densely defined operator. We call the operator A bounded to the right if
95
def
U
L(X.ij)
A
right
L(X5,Y0
6cG uEE
bounded to the left if
ACLleft(X,t/)
def
ll
(lo)
L(XS,Y )-
One The operators, closed to the right (left), are defined analogously. can easily see that A (right(X,11) if and only if A is everywhere defined
and continuous with respect to convergence introduced in Definition 2.
Thus A
:
X + Y is bounded to the right (to the left) if and only if there exists
a mapping o X6
A - E.,
S
a(S) (6
-
E - A, a
:
5(o)) such that A.J(6)
r Y (6) is bounded and everywhere defined (respectively, A6(a)7
X6(0) -' Yo is bounded and everywhere defined).
If A is bounded to the right (to the left) we denote the corresponding Of course, this mapping is not uniquely defined. snapping by OA (respectively, 6A).
In the particular case when X = y and A is a subset of an Abel tan group A, the operator A X - X will be called a right (left) translator with the step s E A, if A is bounded to the right (to the left) and one may choose oA(6) = 5 + s (respectively, 6A(a) = a - s). (We assume that 6 + s F A for any 6 F A in the first case and that - s E A for any a E= A in the second one; of course if A = A the notions of right and left translator with the step s coincide.) :
We return to the general case. If A we define the following sets of indices:
X - 9 is a given linear operator
XA = {16,0) e A xF!A E L(X6,Y0)1 Xr
A
{(6,a)
A x £I A E C(X6' ya)}
AA, AA - projections of XA and XA onto A
EA,EA - projections of XA and XA onto T. T',ese sets will be useful below.
Proposition 3. a' < a.
(a) If (6,0) F XA, then (6',a') E XA for any 6' % 6,
(b) If (6,a) E XA, then (6'.c
E XA for any S' > 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 <
[Tn+l - 2nTn + n2tn-l]e-T (1 -1/n) 0
-
(n+1): (1 -
1/n)n+2
2n!n
(1 -
+ 1/n)'+1
C2t2 (1 - n)-n{ (1 + n) (1 - n)-2 + 1 - 2(1
n2(n-1). (1 - I/n)n
}
.
n)-I 6 C3t2/n.
T.m
It follows that (I - inA)-nx that x E DA2.
U(t)x locally uniformly in t, provided
Since DA2 is dense in X and direct estimate implies that
(I - inA)-nx are bounded in X uniformly with respect to n,* we conclude that (f) is valid for any x E X. " This means that for any 6 E A (a E A) there exists o E A (6 E 4) such that the norms of (I - itA/n) -n as operators from X6 to X0 have a common bound. 100
We introduce, for the sake of convenience, the following notation. ,n is an operator-valued function, we write U(t) E Ca(tRn,°.6,Xo) If U(t), t E if for any x G X6, U(t)x E Cc(IR ,X0) and
R U(t)x II
B
Ca
(tRn.
< C II x II
(17)
6
Xo)
with constant C independent of x E X6. The minimal constant C, for which (17) is satisfied will be denoted by IIU(t) Also Lf LI(t), Ca (htn X6 , X0 ) t E IRn, is an operator-valued function and also TAE D(tRRn), we write .
II
B - TCU7 a T(U(t)] a f nT(t)U(t)dt
.18)
for an operator B in the scale X defined by Bx = TCU(t)x7
(19)
(the expression on the right of (19) should be treated in the sense of Definition 3 of item C, and the domain DB consists of all AE X such tha', this expression makes sense). Proposition 1.
Let U(t) E Ca(btn,X6,Xo), f E Ba(titn).
Then
B = (2ir)-nff(t)U(t)dt e L(X6,Xo)
(here f = F(f) is the Fourier transform of f.. Proof. Let x E Then U(t)x E Ca(gtn,X0). item C, we obtain x E DB and
)-n/2F(f)(U(t)x)
Bx =
(21T
Applying Theorem 1 of
E Xa,
the latter pairing being defined in Lemma 1 of item B. easily obtain
I1Bx,constpfII BBiiUI. a
Proposition 2.
C(kn
a
From thi
11x116'
.Je can
(20)
,X6,Xo)
Let f(y,z) = f(y1,...,yn,zl,..
zm) - gl(yl,.... Yn) g2(zl,...,z), where gl E 8a(1i ), g2 E 8O(tm). Let also U(tl,...,ts), V(ts+l ...`rnT' w(Ti- ',Tm) be the operator-valued functions in a Banach '
scale X such that U E Cal.....fs(tRs,X61,x5 ),
,B
V E CBs+1,
n
as+l,...,an
n-sIX 03 ,X6.
m
wEC 4
(gt .X r:
62
,X F3
Then for anyf x c- X61
Jim+nf(t + (21)
fIR ngl;t)v(tsA1,....tn){ftRmg2(r)w(T)dr)U(tl,...It s)xdt.
101
Proof.
The preceding proposition yields that fmg2(T)w(T)dT c
E L(X62,X63) so that the right-hand side of (21) is a correctly defined element of X64 as well as its left-hand side.
Taking an arbitrary element
h E Xa4 and applying it to both sides of (21), we reduce the problem to the verification of the equality
[gl(t)g2(T)3(<
h'V(is+l'....tn)w(T)U(tl,...,ta)x>) -
- iiI < h,V(ta+1,...It n)(f62(T)w(T)dT)U(t...... ts)x > 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 - . But (22) is an immediate consequence of definitions. proved.
(22)
The proposition is
Now we give the definitions of functions of several operators in a Banach scale X - {Xa)6E A. Let A1,...,An be given operators in X. Let {jl....,in) be a substitution of the set (1,2,...,n). We assume that Aj1,...,Ajkk G n, are generators in X. Let f(yl,...,yn) be a function of the form
un
1'k+l
f(yl,...,yn) - g(Yj1.....Yjk) k+l +.E.+ un 4
Ny3k+l ... Yjn,
(23)
where g(yjl,...,yjk) is a continuous function of tempered growth at infinity (i.e., f is a product of a tempered continuous function in (yjl,...,yjk) and
a polynomial in (Yjk+l
" " ,yjn))
U(tjl""'tjk) -
Consider the expression:
v._.. +.E. +u_auAjk+l' ... Ajn
x (24)
x
which is an operator in X; the number over the operators in (24) denotes the place on which it acts in the product; the operators with lower numbers act before (i.e., they stand closer to the right in the product than the operators with greater numbers) . For example,
13
2
51
O 4
a A B ezp(iTC) + bB2A2exp(irV) (25)
- aB exp(i-C)A + bB2exp(iTD)A2
102
(here a,b E C, A,B,C,D are operators in X). The circles around these numbers will be omitted often. 1 n Definition 3. f(A1,...,An) is an operator in X defined by
f (Al.... ,An) - (21r)-k/2r
dtjk (26) ice it is of
(note that the Fourier transform g(yjl....,yjk) exists tempered growth). We shall also denote 1 n f(A1,...,An) - f(A) - 4A(f) - u
(27)
n (f)
i
All...'An as it will be more convenient.
^-r
Note. The definition is correct only if we fix the substitution For if some of Ajk+1'" ''Ajn were also {jl,.... jn} and the number k.
generators, we might increase k and obtain a different definition of 1 n f(k1,...,An) for the same f. However, it will be proved below that in cases which are of interest to us, all the variants of definition lead to the same result.
Note.
If k - n we obtain the simplest version of the definition: n f(A1.....An) (2n)-n/2
2
(28)
dtn.
Let A,C be generators in X, f(yl,y2,y3)
Example. 1
i 2
3
1
B2g(A,C) 1 x
f(A,B,C)
fg(t,t)exp(iCT)B2
Then
^ y2g(yl'y3).
azp(iAt)dtdi.
Letlg(yjl,n..,y7k) E Ba(gtk) and U(tjl,...tjl,) E
Proposition 3.
E
3
Then f(A1,..... AR) E i(X6,Xo).
Proof.
See Proposition 1. 81
Proposition 4.
Let A1,...,An be generators, f(yl,...,yn) - yl
...
y 8n, n
6 Ca%n,Xd,Xa) for some a.
(29)
Then for any x E X6
f(A1,...,An)x - A_a.. Ailx.
(30)
Proof. The relation (29) means that for x E X p(t) - exp(' tn) ... exp(it1A1 x is a3-times continuously differentiable &walued function. It is easy to see that its B-th derivative equals
0(9)(t)
-
The Fourier transform of f equals Applying h E Xo to
f - il8ld(Sn)(tn) and
.
6(S1)(t ). f on the result,
103
we obtain, by virtue of the preceding formula, exactly . The proposition is proved. The above statement is a step to prove the independence of Definition The next step is (in certain cases). 3 of the choice of k and First given by the following theorem, which is also important by itself. we formulate it in the simplest case: Let f(yl,...,yn)
Let A1,...,An be generators in X.
Theorem 2.
"yn)g2(Ys+1""'Yr)' wherer % s + 1, and also gl E gl S1, (y]ys'yr+l r-s , Bs, or+t, ,dn(!R n ), 92 E B 's+:... .fr Assume that for a (IR E$ al,...,ns,aL+l, ..,an
.,s+1'
r
some 6,61,62,. we have
E CP1'...,as(Rs,X6,X61), s
1
E Cas+1>':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)(<
s+l
)A,)-...-exp(it A )x>) 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
IBv
s
1
xI, xEE.
(2)
We denote by HB,P the completion of E with respect to the norm (1) (the notation it' will be used in the particular case of the norm (2)).
It is
Next, we have the inequality
clear that HR P = X.
nx
P, B,k
< !I
x I P, B, s'
k<s
so that the identical operator on E is extended to continuous operators iak
HB,P -+ Hg ,P for a a k, and we have isk 0 irs - irk, r
s
k.
'
(b) For an arbitrary multi-index a = (al,...,am) with non-negative integer components, we define the norm
IxIBa=
(3)
E
a(v)
1
s
where a(v) _ (a1()),...,am(v)), ai(v) denotes the number of entries of Bi in the product By The completion of E with respect to the norm (3) is denoted by 10B. Again we have continuous operators iaB H$ -+ Hsg defined for B < a, which are the closures of the identical operator defined :
on E.
Theorem 1.
HB,P
{HB P} and HB - (HBO) are Banach scales.
The
operators Bi, i = 1,...,m, are left translators in these scales and moreover 6Bi(k) - k + pi for the scale HB,P, 6Bi(a) +
an) for the scale HB: In other words, Bi is the left translator in the scales defined, and its step is -pi for the scale HB,P and (0,...,0,-l,0, ...,0) for the scale V B.
117
i (a) Consider first the spaces Hasp.
Proof.
The estimate I Bix lk-Pi<
G C Ix Ik follows immediately from the defifiition.
We may write
"1k'llxJk-l+iE: ABix1k-p' xEE, k> 1
(4)
L
pi
We prove by induction on
(here % denotes the equivalence of the normaj.
k that ik,k-1 : HB0 P I HB,p has no kernel and that Bj are cloaurable from It is valid for k - 0, since Bj are closed in X E in the space HB,P' HB P by assumption. Assume that the statement is proved for all k < ko. We prove first that 'kk-1 has no kernel. Let xn e E be a k x for short) sequence convergent in HB,P to some element x (we write xn Set ft - ko.
and let xn
k-.l
0.
k;-Pi
Then xn
Next, from (4) 1, so Bixn k-pi yi E
0, since pi )' 1 for all i.
we obtain that the sequence Bixn is fundamental in
P
HB,-
P
E HB P1.
Since by the induction assumption Bi is closurable in HB P1 P
,P
yi - 0 for all i. Using (4) again, we obtain Ixn fk - 0, i.e., x - 0. It follows that Ker(iks) Thus the triviality of Ker(ik,k-1) is shown.
j.
{0} for all s < k. Next, we show that Bj is cloaurable in Hk.P for any k k It follows that xn k+ 0, and Let xn E E, x y E HBP. 0, Bjxn
Bjxn k+l 1k,k-ly'
Since Bj is closurable in Hg Q, we obtain ik,k-ly - 0, ,
therefore y . 0 since ik,k-1 has the trivial kernel. a
(b) The case of apaces K8 is considered in a similar way. only that instead of (4) we have
x '
B,a
ti
E
i:a. >
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
(7)
Then
'Ax 'k < const g x
(k), x 6 E,
(8)
so that A extends to a bounded linear operator from HB(p) Proof.
to C
We have
IA. N k.
N Bv1Bvz...B,s Ax 6 O, x 6 E.
E
OC + - .
Once it
has been proved, we might estimate these terms, using (6)
J Ku(A)BC1...Bttx 10< C N BE1...BEtX N
0NP ,u>)
C Ix N+() - C Ix N+m(r)-r < CNX
k+ 0 (r) - r S C N x
(k )
(here r - , C denotes different constants in different places). This estimate yields (8) immediately. We prove the required representation of Bvl...BvsAx by induction on s. Assume that Bv2...BvsAx is a linear com-
with + - -
bination of the terms We have
pvl.
Since both terms on the right have the length , we obtain the required representation. The theorem is thereby proved.
Now we turn our attention to the case when the given operators satisfy Lie commutation relations. Let A,,...,A, be operators, defined on a dense invariant subset E of a Banach apace X and closurable in X (we use the same notation for their closures). Assume that A1,...,An satisfy Lie commutation relations on E: n [A., ]x - -i E X! k x, x 6 E, j,k - 1,...,n, (9) J s-1 j k s where
E tt It are the structure constants.
and consider the scale H
-
(A1,...,Am)
We choose some number m < n
k (Al,...,Am)
generated by the
tuple (Aj,...,Am). Let L denote the Lie algebra consisting of all linear combinations of (A1,...,An), and L1 denote the subspace of L generated by (Al,...,Am). We set * It is not difficult to construct a non-decreasing function p*(k) defined for
sufficiently large k, such that A extends to a bounded operator from
Namely, for k > *(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<- r
In particular, LN - L. Theorem 4. Let L be a nilpotent algebra and let A1,...,An be the generators of tempered semigroups in X, satisfying the conditions of Theorem 2 of item B. Then A1,...,An generate tempered semigroups in the scales HA,p for any p - 01,...,Pn. Proof. We consider the case pl ... - pn - 1; the general case is considered in an analogous way (the simple generalization of Theorem 3 is required). Applying Theorem 3, we obtain the desired result. Indeed, all the matrices Aj are nilpotent, and so exp(-At) I has not more than polynomial growth at infinity.
121
N
Theorem 5.
Let L be h stratified Lie algebra L -
E?Lr, satisfying
the conditions of Theorem 2 of item B. Let A1,... , be the basis in L constructed above, and assume that exp(itAl),...,expp(itAn) are semigroups Then A1,...,An generate tempered semigroups in of tempered growth in X. the scale H Proof.
This immediately follows from Theorem 3.
Note. The statement of Theorem 4 is valid for any choice of the basis in L. The special choice of basis given will be used in'the following items to construct a regular representation. D.
Regular Representation of a Nilpotent Lie Algebra
In this item we give the explicit expression for regular representation of the nilpotent Lie algebra and prove that these operators are generators in the scale of symbol spaces. Let I' be a real n-dimensional Lie algebra with the basis a1,...,an and commutation relations n Ca.ba ] - E A.. i,j , 1,...,n. (1) k=1 ij ak 1 j We suppose that I' is nilpotent and that all the matrices Ai, (Aj)kj
-
A1., i,k,j = 1,...,n,
(2)
of the associated representation are strictly lower-triangular, i.e.,
X. - 0 for j) k.
Equivalently, if we denote by rt c r the linear sub-
space with the basis af,af+1'
..
.
an, then
Cr,,r] c r2+l, I - 1,...,n
(3)
(rn+1 = 0 by convention).
Let X be a Banach space, A1, ..,An be the generators of tempered semigroups in X, satisfying the conditions of Theorem 2, item B. Thus, Ai are the generators of the strongly continuous representation of the Lie group G correspondent to r. By Theorems 4 and 5, item C, these operators are generators of tempered groups in the scales HA,p and in H (the (Al,...,Am) latter assertion is valid if r is a stratified Lie algebra generated by (Al,....Am)) The total space of these scales is X, endowed with the convergence, induced by the norm I IX; thus ifor any f E B (the total space of the scale (BS o(tin)}) the operator f(A1,...,An) n is defined as
f(A1,...*An)x =
x E X,
(4)
the expression on the right is understood in the sense of regularization given in Definition 3 of Section 3:C, where the convergence is that in the Banach space X. In order to obtain the operators L1 ...,Ln of left regular representation for the operators A1,...,An, we make some preliminary calculations.
122
Lemma 1.
For any x E D, we have (5)
j = 1,...,n,
L J
where -i
Lj
E Pjk (tj+,,...,tk_1) atk,
(6)
k=j
where Pjk are polynomials with real coefficients,
Pkk = 1 for all k, and
Pjj+: are constants. Proof.
By Lemma 2 of the item B we ha* for any x E D -i atJ U(t)x a
where U(t) = (g )..
Bpq(t) The lower-triangularity of the matrices Aj and the antisvmmetry of true.
structure constants A. with res
ct to the lower indices yields that °in
fact the matrix Aj has the form: 0
0
(9)
A. 0
0
where only the elements marked by * in the lower right block of the size (n - j) x (n - j) may be non-zero. Consequently, the matrix exp(tAj) has the form
exp(tAj)
(10)
(Ej is the unit matrix of the size j x j, Mn_j(t) is the polynomial of the order n - j - 1 with coefficients which are the real matri' the size (n - j) x (n - j): Mn_.(t) = En_j +
where all
+M
Af
nn'j-1)tn-j-i (11)
aJ are strictly lower-triangular matrices.
It fellows Oct
B(t) _ (Bpq(t))p,q-1 is a lower-triangular matrix with ones on the main diagonal,
B(t) _ E+R(t),
(12)
where R(t) is strictly lower-triangular, and Rpq(t) is a polynomial in (tq+1, ..., tp_1) (constant if p - q + 1). Thus the matrix B(t)-1 exists and
B(t)-1 - E - R(t) + R(t)2 - ...+
(-1)n-1R(t)n-1
(13)
123
B(t)-1
is a polynomial in is lower-triangular and the'element tp_1), constant for p = q + 1 (the latterpStatement is easily proved by we obtain the statement of the lemma.
Setting Pjk -
induction).
Then
Let x c- D, F(f) E S(IRn).
Lemma 2.
n
1
n
1
AJQ f(A1,...'An)Ilx - (Li f)(A1,...,An)x.
,.,here
n L. J
a
y P. (-i
1:
k Jk
k-,j
,-i
ayj+i
n
a
ayk-1
(14)
-
(15)
a ....,-iaYI,
= Yj + J
Under assumptions of the lemma, we have
Proof.
1
n
Alt f (A1, ... ,A,)Il x 11rF(f)(t)e]tA
(2n)-n/2
= Ai
(2a)-n/2 JJrF(f)A.elt
(16) (2s)-n/2 1Jr(tLj
(2n)-n/2JF(Li
1 n (Li f)(A1,...,An)x.
Here tLJ - -Lj is the transpose (in L2) of Lj.
The lemma is proved.
Theorem 1. The operators L1 riven by (15) give the left regular representation of the tuple A - (A1,...,An). Proof. Passing to the limit in Lemma 2, we conclude that its statement n n D is valid for any x E X. Now let f E S. x E D 1 1 n f(A1,...,An) (Ljf)(A1,....An)
Then it follows from Definition 3 of Section 3:C that there exists a n
1
sequence fm convergent to f in S'Okn) such that fm(Al,...,An)x converges 1
n
1
n
1
n
to f(Al,,...,An)x, (Ljfm)(Al,...,An)x converges to (Ljf)(A,,...,An)x. and F(fm) E Sokn), m - 1,2,3.... . We have 1 n 1 n (Li fm)(A1,...,An)x - AJQ fm(Al,...,An)]lx, m - 1.2.3,... .
Since Ai is a closed operator, we may pass to the limit and obtain 1 n n Aj Q f(A1,...,An)]lx - (Li f)(A1,...,An)x. I
(17)
The theorem is proved.
X124
Now we give the proof of the analogous theorem in the arbitrary scale X6)6(=-
6'
Theorem 2. Let A1,...,An be generators and right translators in the scale X. Assume that the operators (iA1),...,(iAn) form a representation of the Lie algebra r. Then the operators (15) form the left regular representation of the tuple (A1,...,An). Proof. We need only to show that the statement of Lemma 1 is valid for any x E X under the conditions of the theorem. Then the same takes place for Lemma 2, and the proof goes as in Theorem 1. Next, it is sufficient to prove that Lemma I of item B takes place for any h rc X in the situation considered.
Set Y - X Q ... (t X (the direct sum of n copies of the space X endowed with evident convergence). Set also
B - Alx B : X -
Y is a continuous operator.
...
(18)
F' Anx.
Besides, we have
BAj - (Aj Q I - it T Aj)B, j - 1,...,n.
(19)
Since B is continuous, it follows that
B exp(iAjt) - exp{i(Aj e l- iI 9 Aj)t}B (cf. the analogous argument in the proof of Theorem 1 of item A). the desired statement is proved, and Theorem 2 follows.
(20)
Thus
We establish now the important property of the operators L - (L1,..., Ln) of the left regular representation of the nilpotent Lie algebras. Theorem 3.
The operators Lj, j - 1,...,n given by (15) are generators
of tempered groups in the scale {Ba(t°)}, Proof. In view of embeddinga established in Lemma 1 (b) of Section 3:A, i't s enough to prove that Lj are the generators of tempered groups in the scale {Na(,tn)); Passing to Fourier transforms, we come to the problem of establishing that L5, j - 1,...,n given by (6) are generators in the scale (HU(Itn)}.
Lemma 3.
The operators exp(itLj) may be calculated explicitly. For any function (t) E S'(lln), we have exp(itLj)¢(t) - O(T(.)(t,t)),
(21)
where T(.)(t,t) is defined as the solution of the system of ordinary differ-
ential equations
k - 1, ..., n T(j)
(here we set Pjk
(22)
0 for k < j).
Proof. Indeed, Lj is a vector field, and F(t,t) satisfies the linear partial differential equation of the first order
i ai F+LjF * 0.
(23)
Solving (23) by the characteristics method, we come to the system (22).
Next, since the coefficients of the vector field L3 have the special form (6), the solution of (22) may be given more explicitly. Namely,
125
Tk
= tk, k < (24)
1(i '(t,:)
where Qjk are polynomials.
= tk + Qjk (tj+:,...,tk__T
Thus, the Jacobian (t,T)
det 3L
=
(25)
1
for all T,t, so that exp(i-:Lj) is a unitary group in L2(!R ), and all the
derivatives of T( j) have the polynomial estimate of growth with respect to all the variables. These properties yield the statement of the theorem immediately. Theorem 3 is proved. E.
Pseudo-Differential Operators in Spaces of Smooth Functions
In the sequel (namely, in the chapters concerned with construction of asymptotics), we shall need the developed calculus of pseudo-differential operators defined in spaces of smooth functions (which will be symbol spaces). The results presented below contain somewhat more information than we could obtain directly from general theorems proved in this chapter and: are derived in a slightly different way. Our aim is to expose results which arise in asymptotics theory. The space Hj(tR n ) is the completion
We introduce the scale of S(htn) with respQect to the ncrm
2
hunk
=
QJa-6) k/ _u(x)l2dx)1/2.
(1)
HE
Consider the algebra of smooth functions of tempered growth
H(Rn) = U n Hk(Rn)
(2)
fk f
the field C. When it will not cause confusion, the argument (Etn) will be omitted from the notation.
This is an algebra with unity over
It is clear that H(Qtn) is nothing else but the space S'(tRn) introduced
We also could make use of the spaces Ha(3tn) instead of in Section 3:A. Hk n f($t ), but the latter turn out to be more convenient.
We give the definition of convergence in H.
of functions (fm} C H converges to zero, if a Vk :
A generalized sequence N fm ([
k - 0.
The con-
Hf
vergence introduced is compatible with algebraic operations in H and also separable (i.e., the generalized sequence may have at most one limit). The proof is almost the same as in Proposition 2 of Section 3:C.
Nn)
LH Consider the algebra LH of all continuous linear operators = We let L(B11 - B2y) denote the Banach space of all continuous linear mappings of a Banacfi space B1 into a Banach space B2. in H(ltn).
Lemma 1.
The algebra LH is of the form
LH = nU OU L(Hk - HT).
frsk
(3)
The equality (3) should be considered in the sense that each operator T E
ELH has the following property: 'R3r Vs3k such that T can be extended to 126
a continuous linear mappint, from Hk. into Hr.
the spaces of the scale {HP no confusion. Definition 1. (in the space H).
Since S(ltn) is dense in all
tfii.d extension is unique, act there will be
The elements of the algebra LH will be.calledoperators
Proof of Lemma 1.
Assume that T does not belong to the T may
Let T E LH.
right-hand side of (3). This means that for some fixed V r $s=s
(r)yk,
to Hr(r) HQ not be extended to a continuous linear operator from Consider the directed set A of triplets (r,m,c), where r E IEt, m(-= a + and) E E lt+ \ {0} with the ordering relation defined by (r,m,c) G i(r',m',e') k k' m < m' and t > c'. Since Ht C Hp for k > 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
f rME
n
(4)
k < CS Hi
for e S 6, m > 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 I
k
Hf
s
Hr
4 C(1 + Iti)P, then the operator de
A will be called a generator in H. Theorem 1.
(5)
-i dtt It-O We denote et - et(A).
The operators of multiplication by xj and differentiation
pj - -ih(a/2xj) are generators in H(IRn).*
Let
m be a smooth real function
on Etn, all of whose derivatives are bounded. Then for an arbitrary j - 1, ...,n the operator pj + $(x) is a generator in H(Qtn). Proof. The groups generated by the operators enumerated in the theorem have the form
* Here h # 0 is a real parameter.
127
et(xj)f(x) =
eitxjf(x),
f E H,
(6)
(the multiplication operator) et(pj)f(x) - f(xl,...,xj-1'x. + ht,x3+l,...,xn), f e H,
(7)
(the "shift" operator)
(8)
e H. 0
The required estimates follow immediately from the,.e explicit formulae. The theorem is proved.
Let"Ai,...,Am be generators in H(R ), and let nl,...,nm be real numbers such that it. # nk for j # k. The numbers can be written in increasing order: ... < njm. We consider the product of the groups corresponding njl
to Al,...,Am arranged in the same order, and apply the Fourier transform of the function f to this product: f(A1,...,Am) _
Lemma 2.
(9)
The mapping f -* (Al ....,Am) is continuous and linear from
In other words, Vq VI 3r Vs 3k 3p 3C HdRm) into LH(Qin). we have the estimate nl
0, Vf E
Qtm)
nm
II f(A 1,...,Am )
II
k n a n '4 C h f H, (ht )+Hr(R )
(10) HPP(gym)
As in Lemma 1, we show that the system of estimates (10) is nl nm equivalent to the continuity of the mapping f - f(A1,...,Am). Next, we prove that Vq VP 3r Vs 3k 3p: Proof.
H9Rn)))
(11)
where B is a Banach space, the space of 8-valued
(we denote by
functions on stn with finite norm Ii
jI
Once (11) is proved, the state-
q).
ment of the lemma is obtained immedia?ely; indeed, it is an easy exercise to verify that the Fourier transform maps HQ(&tm) onto H p(Rm) continuously
and that the continuous pairing H P(Iltm) x Hp(!
m
etj1(Ajl) =
(12)
where r - m + Ial, and Bi(t),1. =1,...,r are operators satisfying the property
Vf 3r Vs 3k 3p 3C
128
B(t)
k
s
< C(1+ Itl)p,
(13)
Thus all we need is to prove that the product of the operators satisfying (13) also satisfies (13); then (11) will follow immediately. Let Bi(t), This may be rewritten in the more compact form i = 1,2, sat isfv (13).
Bt(t )
,t)P1(f,s).
Ci(f,s)(1+
ki(2 ,s)_
I
Then
Set B(t) = B1(t)B2(t). iB(t)
(14)
Hs ri(Q)
Hf
C(f,s)(1 + it,)P(1's),
s
k(f s) Hf
(15)
Hr(f)
where r(f) = rI(r2(f)), k(f,s) = k,,(f,kl(r2(f),s)), C(f,s)
C1(r2(f),s)
x C2(f,kl(r2(f),s)), p(f,s) = rl(r2(f),s) + P2(f,kl(r2(f),s)).
The lemma
is proved. m Definition 3. The mapping f - f(A1,...,Am) will be called the ordered quantization corresponding to the ordered set of operators which are
m
r1
Al,...,Am. The numbers -1,.. ,-m indicate the order of the action of the The operators in this set; an operator with a lower index acts first. rm function f is called the symbol of the operator f(Al,...,Am).
\uw we come
in H(1Rn).
to
the
Let f E H(IRn
Definition 4. 2
x
inoestigat n of pseudo-differential operators
ttn).
An operator Si
2
f(x,-ih -)
f(xl,...,xn,-ih
ez ) n
(16)
is called a 1/h pseudo-differential operator. (Here we put coinciding numbers over the operators which compute with each other; it is easy to see that the notation is correct.)
We consider operators depending on the parameter h since this will be useful in the asymptotic theory and we should like to present the dependence on h more explicitly. Lemma 3.
The operator (16) acts according to the formula 1'
Fl/hf(x,p)F1/hu(y),
f(x,-ih aX)u(x) = P_x
u e SORn),
(17)
Y-P
where Fy /hu h =
(2-h)-n/2re-(i/h)u(y)dy,
(18)
fl/h
v = (27h)-n/2re(i/h)<x,p>v(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
t
lal (k+l
II
II
(15)
C Il v 11 ,
which immediately yields (12). Lemma 3.
Under conditions of the theorem, the formula holds: n MAl,...'An)u,v) f k(exp(itkAk)x...xexp(it1A1)u, 1
(16)
mv(t1,...Itk))dt1...dtk. (Here the brackets Proof.
denote the scalar product in H.)
First let f E S_(IRn).
We have
n
1
(f(Al,.... An)u,v) = f uf(tl,. .,tn)(exp(itnAn) x...x exp(it1Al) x x u,v)dt1.- .dtn
x
X u,exp(itk+lA.+l)x...xexp(itnAn)v)dtl...dtn (17) = ftR k(exp(itk.
)x...xexp(it1A1)u,fIR n-kf(t1,...,tn) x eXp(itk+l[tk+l)x...xexp(itnAn)vdtk+l...dtn)dt1...dtk -
x
= fIR k(exp(itkAk)x...xexp(it1A1)u,Pv(tl,...,tk))dt1...dtk, Q.E.D. Sm(3t n
Now if f
) but the estimates (7) hold, we can approximate f by
means of a sequence f E S°°(Qln),
s = 1,2,..., so that the difference f - fs
satisfies estimates 17) with the constant C. tending to zero. Passing to the limit as s - m, we obtain the desired statement. Then combining with (16), we obtain the statement of the theorem.
135
One could prove this theorem in a completely different way, based directly upon the representation (6). Let now the tuple A1, ..,Ap of self-adjoint operators in a Hilbert apace H be given, satisfying the conditions formulated at the beginning of this item, and let, moreover, the following conditions be satisfied: (a) (b)
operators Ak+1,...,An commute with each other;
n 1 the left regular representation L11...,Ln of the tuple (A11...,An)
exists in S' stn), the operators L1,.... Ln being generators in S"(kn).
In other words, according to definitions of item A we have for any
two symbols f,g s S°°(kn): n
1
1
n
1
n
Q f(A1....,An)31II g(A1,...,An)D - h(A1,...,An)
(18)
on V, where 1 n h(y) - f(L1,.... Ln)(g(y)),
(19)
m n and L1,...,Ln are given linear operators in S (1t ), the operator f(L1,..., a
Ln) is defined in S by the formula n
1
f(L1,...,Ln) - ff(tl,...,tn)exp(iLtn)...exp(it1t1).
(20)
Here the convergence is understood in the weak sense. Condition (b) is satisfied, for example, if the operators A1,...,An form a nilpotent Lie algebra and satisfy the agreement conditions (see item B and item D of this section). Analogous Let y - (Y',Y"), Y' ` Y'! - (yk+l .... yn)' m k n-k ) notations will be used for multi-indices a - (a1,...,an). By Sp(lt kk 3 Spm, where m E lt, p > 0, we denote the space of functions f(y) E C (hn) such that I
a1la
I
-4 Ca(l +IyIJ)m.PI«'I, Ia1 ' 0,1,2....,
(21)
By*
n 1 and by Gp we denote the set of operators f(A1,...,An) with symbols f E Sp.
We introduce the following condition on representation operators: Condition (p).
For any m E tt, f E Sp and any interchange it - (w1,...,
of the numbers (1...... n), the operator f(L},...,Ln) is a pseudo-differ-
A
ential operator in S(Itn), 2
f(LI,....Ln) - H(y,-i ay),
(22)
with symbol H(y,E) satisfying the following conditions:
H(y,0) - f(y) E S a
1a+al
0,
a gH (Y.E)I G Ca8(m(Y.E))mp(I«'I ay aE
(23)
(24)
where (y,E) is a function in It2n, such that for some No > 0 the following inequalities hold:
136
1
(25)
l/@(Y,C) < C(1 + IY'I)-1(1 +
(26)
with constant C independent of Theorem 2. (a)
E Sp , i
Under conditions mentioned above:
U Gm is an algebra with filtration. mGAR p - 1,2, then
n
1
n
1
II fI(Al0...'An)]I Q f2(AlI...'An)]I where g E Sp 1
+ m2
.
More precisely, if fi e
1
n
- g(A1,...,An),
(27)
Moreover,
fl(Y)f2(Y) - g(y) E Sp1 +m2 -p.
(28)
(b) For an non-negative integer m, denote by Hm the completion of D , m, which is given by the formula in the norm
I - Io - flu
(29)
M,
where I- l is the norm in H; k
u
in
- IIu IIm-1i
IA.u
E j-1
m - 1,2,...
(30)
(see also item C of this section). n Then each operator f(A ,...,A 1
n E GmP can be extended to a continuous
1
Hs+[-m]
operator from Ha into -t$) for s > 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
alsl
(37)
Fa.N.j(Y,{)I
Estimate (37) yields that the Fourier transform Fa,N,j(y,n) with respect to variables E is summable and flia#Npj(Y,n)Idn < C(1+ lY'I)m
Pej - Pla'I.
(38)
We have N
2 C LO!
G(ao) (Y) +
E
E J
.1 s+ Y+a o
-r FB.N, j (Y,-i ay) x (1 - e)Nf (y) (Y) 3TY (39)
(2x)
E
n
N
a
E
sv-v f nFBN,j(Y.Y - N)(1 -
s+Y'ao j'1
Nf2y)(n)dn.
J
tt
From (38) and the fact that f2 E Sp2, it follows that each term in the sum (39) can be estimated by
C(1+Iy'I)m1+m2-p-PIs'I-PIY'i +C(1+1y,{)m1+m2-p-Plaol Thus the inclusion G(y) E ml
+ m2 - P
(40)
and the Proposition 1 are proved.
Now we prove the validity of statement (b) of Theorem 2. Fist of all w establish the boundedness of 8perators with symbols from S,. Let f E Sr,. We construct a symbol g s SP that
(f(A))*f(A) + (g(A))*g(A) - M2 +*(A), where K + 1 + sup
Yee
if(y)l, *(y) E
SP-k- 1;
(41)
the asterisk denotes the con-
jugation in H. (41) immediately implies that the operator f(A)is bounded in H, since *(A) is bounded by Theorem 1. In the case of pseudo-differential 138
operators the idea to use identities of the form (41) to prove the boundedness was proposed by Kumano-Go. Since operators A1,...,A0 are self-adjoint, the operator conjugate to f(A) is given on V by the equality n 1 (42)
(f(A)) * - f(A1,...,An),
where f is a complex-conjugate symbol, an$ the numbers over operators in f are set in the inverse order. If g E SP, then by Proposition 1 we have
(f(A))*f(A)+(g(A))*g(A) - (If12+Is12)(A)+*(A), 9YESP.
(43)
We solve equation (41) by the method of successive approximations. Set
go(Y) - N`-If(Y)I.
(44)
we It is evident that go e S0P and we have (for the sake of brevity, denote by capital letters the operators, which correspond to symbols that are denoted by small letters): F
*
F + G*Go - N2 + To, 9o C- SP-P.
(45)
0
Now we set for j - 1,2,...,
gj(y) - - 2 ReYp j-1(Y)/8j-1(Y)+8j_1(y) = rj(y)+gj_1(Y),
(46)
where 9j_l should satisfy the conditions:
NZ + Y j-19
F F + G1-1Gj-1
yJ 6 sPU+l) , I1 ]E S P
(47)
P(3+2)
(48)
P
Really, (47)
It turns out that such a choice of 9. is always possible,
yields that Yj - 9j(A) is symmetric In T3. Setting fl - *j(yl,".Oyn), w, - n + 1 - 1, f - 1,...,n, f2 - 1, we obtain by Proposition 1, that
n
1
n
1
n
1
V+j(A) - 9.(Al....,An) - * j(Al,...,An) + X(Al,.... An
where X E SP(3+2), if 9j e so P 0+1)
Set
0! (Y) = 2 lipj(y)+0 (y) +X(y)1;
(49)
then 9.(A) and P! satisfies both conditions in (48). Thus it is sufficient to onstruct symbols 9. satisfying the first condition in (48), and the validity of the second condition may then be provided.
We proceed by induction on j.
Let symbols gj_l, +yj_l satisfying the
induction proposition be already constructed. rj are real ones. Using (46), we obtain:
F*F + GjGj
-
F*F
- N2+'L
Note that the symbols
+ Oj-1Gj_l + G* 1R. + R *Gj'l + R*R.
]-
j -1
+G
R.+R*G.
j-1 3
J
3 3-1
j
J
+R* R.,
J
1
(50)
3 3
139
where rj G SP-pi, gj_1 E SP, gj_1rj + rjgj-1 - 2gj_1rj = -i+j_1, and hence
by Proposition 1, we have
P*F+C.G. = M2+4., yj
(51)
ESPP(J+l)
and the induction step is carried out. Thus we have shown already that the operators with symbols from SP are bounded.
Now we prove that if F E Gp (m being integer), then F
Hs m is a continuous operator.
Aj E G1, j = 1,...,k;
(a)
(b)
A.
F E GP
Hs ,
(52)
Hs-1
Hs
:
(53)
are continuous operators for j = 1,...,k. (c)
:
We have:
Then the implication holds:
CAj,FJ F A.F -FA. E Gp -P, j = 1.....k;
(54)
(d) if F E GP, then F is bounded in H0, (e)
GmGp C Gm+m'. P
(55)
'
The boundedneas of an operator F E Gk from 11s into Hs-k follows from (a) - (e) by standard argument analogous to that carried out in item C, and hence omitted here. The theorem is proved.
It should be noted that the verification of Condition (p) in the general case may be rather difficult. This condition, however, holds in a
the case of usual pseudo-differential operators, n = 2k, Aj+k = xj,
in L.z('Rk ):
Lj = Yj - i
ay k+j
' Lk+j
axj
J
yk+j J = 1,...,k;
H(y,E) - f(yl +Fl,...,yk +Fk,yk+l'...,yn).
One may set (y,&) - I + jy'+ F'J, N = 1. In this case estimates (25) and (26) with No - 1 follow from the PeeIre inequality. Also the following assertion, to be used in Section 1 of Chapter 4, is evident. Corollary 1. Let function f(y(1) satisfy the estimates
ay(1)say(2)yay()a
y(2),y(3)),y(i)
CSYd(1 + (Y
(1) z + )
C-10, i = 1,2,3,
(2))2) (Y
-lal -1Y1 ,
161 + IYI + 161 = 0,1,2,... . Then the operator f(-i
a a
2 2 n ,, E) is bounded in L2(RF) and the upper estimate
of its norm is completely defined by the constants CRY6.
140
III
Asymptotic solutions for pseudo-differential equations THE CANONICAL OPERATOR ON A LAGRANGIAN MANIFOLD IN IR2n
1.
WKB-Approximations
A.
We introduce first some basic notions related to the problems conand let Let x (x1,...,xn) be coordinates in IR
sidered.
,
2
1
2
2
H(x,-ih ax
I
i
h) - H(x1,....xn..ih axl .....-ih as
h) 2 H
(1)
be a 1/h pseudo-differential operator (1/h-PDO in th.. sequel) in R. The definition of a 1/h pseudo-differential operator was gi n in Section 4:E of Chapter 2. We require throughout the chapter thL- the symLjl H(q,p,h), (q,p) a (ql' ....gn'pl*.. "Pn) E tit2n of the 1/h-PDO be a smooth function of the parameter h e 10,13 with values in S-(Rn), i.e., H(q,p,h)
alol +
CaBk(1 + Iq! + IpI)m. h s 10,13, p.q s
181
tRn (2)
aq ap ah
The function Ho(q,p,h) def H(q,p,O) is called the principal symbol (or the Hamiltonian) of the 1/h-PDO. A 1/h pseudo-differential equation (1/h-PDE in the sequel) is the equation of the form 2
1
H(x,-ih ax ,h)u(x,h) - v(x,h), where v(x,h) is a given function. 2
1
2
(3)
If u,v are vector functions and also
1
H(x,-ih ax h) - IIHij(x,-ih ix
h)
fl
is a matrix of I/h-PDO's, then (3) is
a system of 1/h-PDE's.
The typical example of the l/h-PDE is the well-known quantur-mechanical Schrbdinger equation 2
1
-ih at + H(x,-ih aX ,h)4+ - 0,
141
(4)
z
wherd'H(x,-ih
2x
h) -- Zm A + V(x) is the energy operator of the
quantum system under consideration, and
_ O(x,t,h) is its wave function.
We introduce the problem of seeking the approximate solution of the general the form (4) as h - 0; such approximate solutions are known in quantum mechanics as quasi-classical or WKB-approximations (named after Wentzel, Kramers and Brillouin). If the operator H were an ordinary differential operator with constant
coefficients, H - H(-ih aa), H(p) -
HE Ckpk being a polynomial in the vari-
able p 6 IL, the equation (4) would have special solutions of the form a(i/h)S(x,t),
*(x,t,h) -
(5)
where S(x,t) - ax + bt is a linear function in x and t, and besides b + N
+E Ckak - 0, i.e., S(x,t) satisfies the equation k-0 as 0.
(6)
+ H(a) -
As an analogy to this case we seek the approximate ablutions of (4) in the form (7)
*(x,t,h) -
where (x,t,h) depends on h smoothly; for instance, $(x,t,h) -
N E (-ih)k0k(x,t)
(8)
k-0 is a polynomial in h, S(x,t) is a real-valued function. After substitution of the expression (7) into equation (4), we come to the expansion of the form: 2
1
-ih 21t + H(x,-ih ax
+ (-ih)[
T a
+
a_ (x, ap
as ax
(i/h)s(x,t) h)tV
a
a:
aat S
{C
as
+ 1 tr(j-(x, apz ax
N+1(-ih)1Pj(x,-i
as
+ Ho(x, ax)1f(x,t,h) +
xe a axe
+ i aH(x. as .0)1 x ax
a
(9)
i
ax ,t)f(x.t,h)) + hN+2e(i/h)s(x,t)f(x,t,h),
x $(x.t,h) +
j-2 where P3(x,-i '- ,t) are differential operators with smooth coefficients, independent of h; f(x,t,h) is smooth with respect to all the variables. The reader can easily verify the validity of expansion (9) for the case 2
when H(x,-ih ax) is a differential operator*; this will be enough to grasp the idea, see item B for exact formulations in the general case. * Hint: prove the identity 1
e(-i/h)S(z,t)H(x,-ih
,h)e(i/h)S(x,t)
ax
- H(x,11
as
-ih ax11,h),
and obtain the expansion of its right-hand side in powers of h.
142
To obtain the approximate solution we require that the coefficients This requireof the powers of h within the curly brackets be equal to zero. went yields the system of equations:
a4ate
+ Hopx,
as
am
o + F(x,t)Oo - 0,
ax aX k-1
+ Hop(x, at
ax
2
1
Pk-j+1as '00j, k - 1,...,N,
axk + F(x,t)Qk
(12)
where F(x,t)
(10)
0,
at + H0 (x, ax)
32 F>fl
CH
01 ,0) - 2 tr(aP2 i 27h-(X, 2x
00
aza
(x, ax) ax2)'
(13)
Once the system (10) - (12) is satisfied, we have
a
2 _ N+2 (i/h)S(x,t) a g(x,t,h), e ,h)y ' rN(x,t,h) = h -ih 2t + H(x,-ih 2x
(14)
where g(x,t,h) is a smooth function. Assume that the Hamiltonian Ho(q,p) is real-valued. Then we are able to construct the solution of (10) - (12). Prescribe for Lhe sake of definiteness the Cauchy data y(x,0,h) - e(i/h)So(x)@o(x,h), 4o(x,h) -
k-
(-ih)kmok(x)
(15)
-0 First for the unknown function 5 in the equation (4) (we assume ImSQ = 0). of all, we find the solution of the equation (10) with initial data (10) is a well-known, in classical mechanics, HamiltonS(x,0) - So(x). Jacobi equation (it is often called the characteristic equation for (4)), and it may be solved by the method of bicharacteristics. Namely, consider the system of ordinary differential equations (the Hamiltonian system generated by Ho):
q - ap (q,p),
(16)
P - - aq° (q,p).
(The solutions of the system (16) are known as the bicharacteriatics of the equation (4).) Denote by (q(go,t),p(go,t)) the solution of (16), corresponding to the initial data g1t+0 " qo,
as
(17)
P1t-0 ' 2x (qo),
and consider the function t
W(go,t) - So(4o)
+fo tp(go,t)
ago
ap (q(g0,T),p(go,t)').
(18)
- H0(q(go,T),p(go,T))]dt. Consider also the system of equations
q (go, t) - x E
[ltn.
(19)
Since 21- (qo,0) - E, (19) has a smooth solution q0 - go(x,t) defined for aqo
Rn+l,
containing the hyperplane (x,t) belonging to some open set Uo C It - 0) and satisfying the property: if (x,t) E Uo, then (x,t) 6 Uq for any t between 0 and t. For a moment we restrict ourselves to consideration of the points (x,t) E Uo only. We set
143
S!x,t) ° ''I(go(x, t), t).
(20)
The function (20) satisfies (10) and the p-escribed initial condition (see 12]).
Next we turn to equations (11) and (12). functions
Introducing the new unknown
(21)
Xk(go,t) - $k(q(go,t),t), we may rewrite (11) and (12) in the form aX
at k
k-1 2 + F(q(go,t),t)Xk - - E Pk j+1o,-i j.0
a
>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 <
(x,w,h) E K.
(9)
We say that f = 0(h) if f = 0(hN) for any N. Proposition 1.
(a) The following implications are valid:
f - O(hN) *
= 6(h
6(hN) * f = 0(hN-n/2).
N);
(b) For any compact set KC
f = bin
(10)
x n, we have
CK,k11fIIIk< flfIk
k- 0,1,2,..., f E Hh,loc.aupp f C-K, where CK k,Ck > 0, Ck does not depend on K; the right-hand inequality remains valid fof any function f with finite IAflik'
"flak = sup
(w,h)E5
I8 f' k h = sup
'
f of (x,w,h)
(w,h)En t
(12) x
(1 +x2- h2Ax)kf (x,w,h)dx.
The norm (12) is often more convenient than (8), since (12) is invariant under 1/h-Fourier transformation in any group of variables. We introduce the notion of support modulo 6(h-) of the function f E E Hh foc(1tn x n), in a way analogous to the notion of singular support in the theory of distributions. Definition 2.
Let f(x,w,h) E Hh,tocl*n x n).
The support modulo
6(h') of f is the minimal closed subset K C ptn x n, such that for any function $ E Cx'0 (Ytn x n) satisfying supp f (1 K = 0, we have Of = 0(h ).
notation:
The
sing supp f.
We do not consider the singular support in the usual sense here, so Lagrangian manifolds
there should be no confusion while using this notation for 2n in IR
and canonical coverings.
In the space lt2n = (9.p) consider the differential 2-form: w2 -
n E dpj A dqj - dp A dq. j=1
(13)
Obviously, w2 is a closed non-degenerate 2-form, i.e., a symplectic structure in k2n (see 121). Definition 3. Let L tie a smooth manifold, and let i.: L -fR2n be an The pair (L,i) is called a Lagrangian manifold if for any a E L,
immersion n.
i*(TaL) is a Lagrangian subspace in Ti(o)&t2n, i.e., the restriction of w2 on i*(TaL) is a zero 2-form.
150
Proposition 2.
(L,i) is a Lagrangian manifold if and only if i*w2 = 0.
(14)
If (L,i) is a Lagrangian manifold, then dim L 4 n. The proof of (14) is obvious. As for thelatter statement, take any a E L and set A - i*(TaL). The condition w`IA - 0 is equivalent to saying A is
orthogonal to JA, where J is the matrix of the form w2; thus dim A + dim JA G 2n; since J is non-degenerate, dim JA - dim A and thus dim A G n. The proposition is proved. Thus (L and i(L) may be identified locally and even globally if i is a proper embedding), the notion of a Lagrangian manifold is synonymous to that of a manifold isotropic with respect to the symplectic structure W2 ; the term "Lagrangian manifold" was proposed by one of the authors in the book 1507. In the sequel we deal only with Lagrangian manifolds of maximal dimension n. Such manifolds exist, as is shown by the following: Example 1.
Let S - S(x) be a smooth real-valued function defined on
the open subset L C [tx.
We define the embedding i : L +
I2n
by
9,p
q(x) - x, p(x) - 2x (x), x E L. *
*
(15)
di*pdq
Since i pdq - dS, we have i w2 -
- ddS - 0, so that (15) defines
a Lagrangian submanifold in li2n. The definition of Lagrangian manifold, "depending on parameters w E St, may be given in a natural way:
Definition 4.
The pair (L x D,i: L x a + tt2n) is a Lagrangian manifold o
(depending on parameters), if i E C (L x 0) and for any w E 0,(L,iw) is ::w a Lagrangian manifold. We write i(a,w) - (q(a,w),p(a,w)). Our next aim is to cover L x S2 by open sets, such that the restriction of i on each of these sets could be described by equations of the type (4).
For each I C In], we define the mapping nI : L x 0 + ltn
x it,
(xl.&i)
nI
(a,w) + (xl,E-,w) = (gI(a,w),p-(a(w),w))
I
I
(16)
Recall that we have adopted the convention: if I - In] the subscript I may be omitted; in particular, nIn] is denoted by n. Definition 5.
The canonical covering of the Lagrangian manifold
(L x R,i) is the set {(U;I)) - {(Uj,Ij))j
of pairs (U,I), where U E L x 0 E J is an open subset and I C Cn] is some subset, such that: (a) (Uj)j
EJ
is a locally finite covering of L a Q. and Uj are shrink-
able sets.
(b) The projections Uj of Uj onto L and all finite intersections UJ.I
... Js
- UJ.1 n ... n U
are shrinkable sets. is
(c) The restriction of the mapping
on Uj is a homeomorphism of Uj
151
x g,
onto some open subset Vj C ttnX I
1 = Ij, for any j E J, and the inverse
1
mapping iri; belongs to I
The canonical atlas corresponding to a canonical covering {(Ujlj))jE J (here the restriction of the mapping iI. on Uj J.eJ J is denoted by the same letter).
is the set {(Uj,a1 )} j
The pair (Uj,wrl) is called a canonical chart on L x S2, non-singular or singular, depending on whether Ij - [n] or not; (x1J,t:_I
the canonical coordinates in Up q
q(x
IJ.E-
Ij
w) are called
j
We write
w) _ (xIj.q(x Ij
,w),
Ij,E-
Ij
p - p(xIJ,EIJ.w) = (p1J(xI!EIJ.w).EIJ).
(xI,EIJ,w) V.
for the expression io ail of the functions i(a,w) = (q(a,w),p(a,w)) in the J
canonical coordinates of the chart (Uj,a1 ). J
(L x
Theorem 2.
The canonical covering exists for any Lagrangian manifold
U,i).
We make use of the following:
Proof.
Lemma 1.
(lemma on local coordinates)
there exists a subset I C [n)
For any (a
0),w(0))
E L x R,
and a neighborhood U C L x U of the point
(a(0),w(0)) such that (a,w) + a1(a,w) is a coordinate system in U.* Indeed, the Lagrangian subspace iu,*(T(1(O)L)
is projected isomor-
phically on the coordinate Lagrangian subspace (p1 - O,q _ = 0)
for
I
some I C [n] (see [2), [52) for detailed proof). -
det
Thus, the Jacobian
(gI,P-) .
2a
nates on L).
(a,w) f 0 (here a
(al,...,an) is any system of local coordi-
It remains to apply Theorem 1.
By Lemma 1 we may find the covering of L x R by open sets U. in which the coordinate systems of the prescribed type are valid. It may be assumed that all Uo have the form U. - U, X U11, U.' CL, Ua C :Q, where the sets Ua and all their intersections are shrinkable, and also U" are shrinkable (it is enough to choose these sets as small enough open balls in some Riemannian metric). Passing to a locally finite subcovering, we obtain the desired results. The theorem is proved.
The point (a(0),w(0))E L x n will be called non-singular if we may choose I - (n] (in Lemma 1), and singular in the opposite case. The caustic E C n x U is by definition the set of projections a(a,w) of singular points (a,w) E L x p (recall that we drop the subscript I when I = 0).
* We do not require the differentiability with respect to w in the definition of the coordinate system.
1i2
The canonical operator: preliminary definition and discussion.
Let
a Lagrangian manifold (L x 9,i: L x fl + It2n) be given. We define some functional spaces on L x Q. Recall that the continuous mapping r : A + B of topological spaces is called proper if the pre-image r- (K) is a compact subset in A for any compact subset K CB. Cm o
A(L x f2,i) is a subspace of
Definition 6.
a'(L x 0) consisting of
all functions 4 such that the restriction rIsupp 4 of the mapping n on the support of 4 is a proper mapping. AEh](L x 0,i) is a space of polynomials in the parameter h E 10,1] with coefficients from A(L x fl,i). AN[h](L xfl,i) is a subspace of ACh](L x 0,i) consisting of polynomials of the order < N. Proposition 3.
All the spaces described in Definition 6 are Cm's x
aw
x (L x )-modules, invariant under differential (in a) operators on L x n with Ca'0-coefficients.
The space AN[h](L x fl,i) is naturally isomorphic
to the quotient space A[h](L x'fl,i)/IN+lCh](L x fl,i), where IN+l[h](L x A,i)
-
hN+1ACh3(L
x fl,i).
The proof of this statement is obvious.
Let U C L x 0 be an open
We denote by A(U,i), ACh](U,i), ANCh](U,i), IN4hCh](U,i) the subspaces of the spaces enumerated, consisting of functions with supports lying in U. For the sake of brevity, we use in the sequel a shortened subset.
notation for defined spaces: A, ACh], ANCh], IN+11h], AU, AUCh], AU[h], IU
Chl.
The canonical operator K
:
A[h] + tih, foc(ltn x 6)
(17)
is defined in the following way. The set II depends on some cohomology classes on L x 0 and will be defined later. We choose a canonical atlas
{(U,nl ))jEJ of L x 0 and C_:W -partition of unity* {ej}j J
ordinate to the covering CK4](x,w,h)
- JE
J
on L x A sub-
We set
{Uj E J,
JCK.(e.$)](x,w,h), 4 E AN], (x,w,h) E I
x a,
(18)
where K.
:
AU Ch] - Nh foc n x R x (0,1]) J
(19)
'
is an elementary operator corresponding to the chart (Uj,wl) to be defined j later in this item, and the sum (18) is locally finite.
It turns out that for any j1,j2, U71 n Uj2
0, we have #
*This means that ej - ej(a,w) E Ca:W(L x 0), supp ej C Uj for any j, and I e. = 1 on L x n (note that the sum is locally finite).
jEJ J
Since by our
assumptions L x 0 is a manifold, the existence of unity partitions is then guaranteed by standard theorems.
153
Kjl¢ - Kj20 +0(h), 0 E AUjl() Uj2Ch]
(20)
R x (0,1]).
(but (20) is not valid in general for
The property (20) enables us when seeking the solution of the 1/h-PDE Hy - 0 to glue the transport operators modulo I-1[h]into a global operator on The subsequent L x n having the form of a vector field plus a function. terms of the expansion may be taken into account by means of the regular theory of perturbations.
Also (20) implies that the operator
K(") : A°[h] a A - Hh,Roc(fltn x fi)/mod 0(h)
(21)
in the quotient spaces, induced by K, does not depend on the choice of the canonical atlas and the subordinate partition of unity. Elementary canonical operators. Two difficulties might arise if we made an attempt to define the action of thtr,operator Kj on the function 0 by formula (2), where I - Ij, S1 corresponds' to our Lagrangian manifold, and Primarily, the integral 01 is the expression of 0 in canonical coordinates. Secondly, one may pick out might diverge if the set U. is not precompact. from the stationary phase iethod (Theorem 1, item A), that (20) could be held only if 0 were multiplied by the square root of the Jacobian of the canonical coordinates changed under the transition from the chart (Ujl,nl ), i.e., if 0 were not a function but a section of a
to the chart (U. 2,nlJ, 2
certain line bundle over L x R. The first difficulty is eliminated by the virtue of suitable cut-off functions. As for the second one, there are different ways to avoid this difficulty. One may either consider the space of line bundle sections on L x R instead of a function space (just as L. Hbrmander did in his definition of Fourier integral operators 129]) or identify these two spaces by means of some simplification of the arising line bundles. We have chosen the second way here; the simplification is "inserted" into the definition of the elementary canonical operator, so that there is nb need to describe the bundle at all. The adopted variant of exposition is a traditional one in the canonical operator theory (150], 152], [54], etc.). In order to fulfill our intention, we should fix some non-degenerate smooth measure u on L, dependent on parameters w E R. Definition 7. A measure on the Lagrangian manifold (L x 11,i) is a smooth non-vanishing differential n-form on L, dependent on parameters w E R such that its density in any system of local coordinates* (al,..., an,w) on L x R belongs to C-10.
If u is a measure on (L x Q,i) and (al,...,an,w) is a system of local coordinates, we denote the density of u in these coordinates by Du/Da: u - Du (a,w)da Do
1
A do
2
A ... n da n
.
(22)
If (ais another coordinate system, then obviously 1 n Du
Do'
-
Du
Do
Da
Da,
(23)
* (all ...,o ) may depend on w in such a way that a1(a,w)...an(a,w) E CaW; recall that w are parameters so that, for instance, the exterior differentiation d acts only with respect to the variables a, parameters w being fixed. 154
Cm,
o
Thus, the requirement of Definition 7 is that Dp/Da E a w and does not take zero values; by (23) these conditions do not depend on the choice of the coordinate system. In the particular case, when (xl,E_,w) is a canonical coordinate system I
on L x Q. we denote the density of the measure V by VI: D U
_ )
D(xI> & I
r
1)(x1, _,w)
(24)
I
I
Measures on the Lagrangian manifold (L x S2,i) always exist, as is shown by the following:
The differential form
Proposition 4.
o = i (d(q1 - ipl) A ... A d(qn- 'pn))
(25)
is a measure on the Lagrangian manifold (L x Q,i). 0),w(0)) Proof. Let (a C- L x 9 be an arbitrary point. By Lemma 1, for some I C [n], (gl(a,w),pI_(a,w),w) is a system of local coordinates in
the vicinity of (a(0),W(0)).
Consider the following linear transformation
of the space R2n (q,p) -* (q',p')
p' = (Pl; -qi).
q' _ (ql.p-);
SI(q,p);
SI is a canonical transformation, i.e., SIw2 = a2.
(26)
Moreover,
d (q1 - ipl) A ... A d (qn - ipn) (-i)IIISI*{d(q'- ip1) A ... A d(qn' - ipn')}.
-
Set i' - Slo i.
(27)
Since SI preserves the symplectic structure, (L x R,i')
is a Lagrangian manifold, the point (a(0),w(0) we have (-i)III
oL(a,w)
is now non-singular, and Do' Dx1
1
(28)
of = i'*{d(4i- ipi) A ... A d(qn -ipn)}.
Thus, we have reduced the situation to the non-singular case (from this point we drop the primes). We have Do
Dx
a (qI - ipl, ... ,qn - ipn) = det
=
a(xl,...,xn)
a(xI - ipl(x,w),...,xn - ipn(x,w))
det
d e t( E
a(xl,...,xn)
-''
(29)
ap(x w) ax - )
'
In the canonical coordinates (x,m) we have *
0 - i wz - REr thus apax
ap (x,w)
ax r
dxr n dxf k
ap
axr - axP }dxr n dxE;
(30)
is a symmetric matrix with real entries.
155
Let C be a symmetric (n x n)-matrix, C - C} r iC2,.where Lemma 2. C1 and C2 are matrices with real entries, and C2 is positive semi-definite. Then the spectrum of C lies in the upper half-plane. The proof of this lemma is carried out in [52). (which, however, has a zero
Applying this lemma to the matrix
imaginary part), we obtain that the spectrum of E - i half-plane {Rea > 1).
a
w) lies in the
ax Thus Do 0 0 and Proposition 4 is proved.
The measure o is uniquely determined by the immersion i and will be called the canonical measure on (L x n,i). This measure is complex-valued; obviously the real-valued measure on (L x Q,i) exists if and only if L is an orientable manifold. If u is any measure on (L x 0,i), it differs from Cm 0 (L x n). This factor o by a factor, which is a non-vanishing element of a,w will be denoted by '
fu
ip)
def u/o.
(31)
We come to the definition of phase functions, analogous to S,S1 in Let I C Cnl. Consider the differential (1) and (2) in the parametric case. 1-form:
in1R2n.
(32)
p1dgl - gidpi
wi
Obviously dwi - w2, hence the form iwi is a closed 1-form on L
(dependent on parameters w).
Consider the Pfaff equation
dW1 - iwi.
(33)
For any connected simply connected open subset U C L, we may thus obtain Cma' o
a solution of (33), belonging to
w(U x n) and a unique modulo additive term
which is dependent only on w. If I, I C(n) and W1, WJ are the solutions of the corresponding Pfaff equations in U x 0, direct computation yields that
W (a,w) + p(a,w)q_(a.w) - WJ (a.
I
I
I
w) + p -(a,e)q -(a,w)
J
J
+ a(w).
(34)
where a(w) is some continuous function. Let now (U,sI
:
V) be a canonical chart on L.
U -
Definition S. An action, or a phase function in the chart (U,w1), is the function defined in V
S(x l ,t.-,w) - (W I o nil) (x1,1_,w) ,
I
(35)
I
where WI is a solution of (33) in U x n (recall that U is the projection of U onto Q. Thus, action in the chart is defined modulo a continuous function of w. From (33) immediately follows: Proposition 5.
If S(x1,L_,w) is the action in the chart (U,xI), then I
the restriction iIU is given in canonical coordinates by the system of equations
q(xl,tw) - (xI,
as
I 156
(36)
aE(xl.E_.w)); P(x1,C.,w)
I
At Last, we define the notion of the cut-off function in the chart (U,111), which enables us to define the elementary canonical operator. A cut-off function in the canonical chart (U,n1:U + V)
Definition 9 .
ITIx V and satisfying X(xl,xl,Ei,w) defined in tx_
is a function x (x,EI (x,&_ the conditions:
I
),wgR111 ,V) and X(x,E_,w) s 1 in the neighborhood
(a) X(x,E_,w) E C(XO
I
I
of the set VIx_ - q_I (xI ,E_,w)}.
i
{(x,E_,w) E tR'I
x
I
I
(37)
I
(b) For any compact set K C "x x A and any function $ E A(tCh], there exists p number P. such that the conditions (x,w) E K, (xI,E_,w) E supp x
x (a o writ) - n1(supp 0),
(
x ,Ei,w ) E supp x yield that (EI < R.
The cut-off function X exists.
Proposition 6.
The proof will be given below, in Proposition 8, where the stronger statement is formulated. Let the following objects be fixed: a measure u on the Lagrangian manifold (L x f1,i); a canonical chart (U,n1 : U + V) on L x 9; an action S(x1,E_ w) in the chart (U,iI); a continuous branch arg pI x I
x (x1,E_w) of the argument of pI(xI,E
w) in V (such a branch exists since
V is simply connected and pI does not vanish, and since V is connected any two branches differ by 2ak, k being an integer); and a cut-off function
X(x,E-,w) I
An elementary canonical operator in the chart (U,a
Definition 10. is an operator
foc(,n
KeP : Au[h] + Nh
defined by the formula:*
x R x (0,11)
,
1/h
EKef$](x,w,h) - rE-+x_(x(x,E_,w)e
I
)0 o nil) x
uI(x I
I I
(38)
I
x (x,,E-,w,tr)) I
(2nh)III/2 J -
d&i{X(x,E.,w)
x
(39)
(i/h)[S(xl,EW) +x-&-] x e
I
I I
p2(x
I
)($ o ni1)(xI>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
<9(w,h),[Y1> = 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 <
Ca-N(l+Ixl)m-1
lu2(x,a)'
Taking ihto account the above estimate for ju21, derive
175
CNX-N(I+ lxi)-N, x E Rn, for arbitrary N _' 0, and find that the same estimates are true for all derivatives of u2 with respect to x. Further,
ul(x,i) -
(6)
y(x,y,P) _ <x-y,p> +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<:
(R2n)),
X JRn k (B)
(a)
(f "
(11)
(44(P(a)x(B))exptiXS(P(a)'x(g))7)
x (a)
(a) +!t(a)
m i and for in ar:)itrary integer N '. 0. Here R is a linear o;;z,.atct '>1 C ,e order e j with C'-coefficients (depending on S),
Proof.
Let
V(p(a),x(B))
with crmpact (6) we have
be infinitely differenti$ble and
wit, respect to all its arguments.
L(x,;:'1Dx;(i;.) (a)
Then because of
(a) V(P(a)'x(8))
(13)
=
Let V -
A1,P(a)
ei)S4,
+ x
2
LV(P (a)
(a)
,x
)
L(-A-1D
(B)
P(a)
,x
,A-1D (B)
.
x(B)
,P
(a)
)
We obtain the expansions
jV =
v}.1(ix)-jR. +M (p
(a)'x(B);(')-1)
J= 177
The statement of the theorem concerning the remainder follows from (3). We write the first two terms of expansion (9): Rod - LO¢; aL2
R 4 _ < 1
1 CSp( 2
90
(14)
> - < aL°
ax(a) r ap(a)
> -
a0
aP(8)
ax(B)
a2L0
a210
a2S } + SP( (ax(a))2 (ap(a))2
a2S
(15) (ax(B))2
(aP(B)2 axa2ao
(CO
P(a)
P(a)
j=1
(B)
al,
10 + (
2 E
aj pB.
) a(ia)-1
°
0.
Here
Lo - L(- aS(p(a)'x(B))
as(p(a),x(B)) 0 ).
(16)
(aL/a(ia)-1)o
The derivative of x,p as in to.
is taken for (ia)-1 = 0 and for the same values
Next we pass to global commutation formulas. operator on a Lagrangian manifold An. Theorem 4.
;
ax(8)
aP(a)
Let KAn be a canonical
The following commutation formula holds: 2
1
L(x,a-1Dx;(ia)-1)KAn+ a KAn(L(x,p;0)$) + 0(X-1). Proof.
(17)
Let st C An be a canonical chart, KAn(R) the pre-canonical
operator, and 0 E Co(st).
Let st be a non-singular chart.
x (iAS(x))Idon/dxl1/20.
We have
LKAn(n)O - exp(iXS(x))I dzn I1/2L(x,
Then KAnO = exp x
ax ;0)0 + O(a-1) _ (18)
= KAn(n)(L(x,p;O)$) + 0(a-1),
since p - aS(x)/ax on An. Let P be a singular chart with the focal coordinates (p(a),x(B)). Then
LKAn(00
L(-X-'D
Fa1P (a)+xNo I
n apdo a(r) (a)
p
;(ia)-1)
.x(B),P(a),Dx (a)
(B)
I1/2exp(iAS(p(a),x(B)))P(r),
(B)
where r = r(p(a),x(B)).
Function S has the form S(P(a)'x(B)) - Irg < p,dx > - < p(a)'x(a)(P(a)'x(S))> so that as aP(a)
as
x(a)(p(a)'x(B))'
By applying Theorem 3, we obtain
178
,x (a)
(a)
x
LKAn(S2)
_d n(ri
= FN'p(a);x(a)(I dp(a)dx(R)
'/2 exp (iA S (p
(a )
,x
(8)
x
)0 (r)
(19)
C0-(An).
Then (17) follows from
Consequently, formula (18) holds. Let 0 C (18) and (19). Thus, the theorem is proved.
The commutation formula in Theorem 4 was established under the assumption that the differential operator acts first and the operators of multiExactly the same formula plication by the independent variables act second. is valid even if the mentioned operators act in the reverse order. Next, using the terms of the first order in the expansions (1) and (12), we come, after simple but clumsy calculation, to: Let operator 1. and Lagrangian manifold An fulfill the Theorem 5. assumptions of Theorem 4 and moreover the conditions:
(1) Function L(x,p;0) is real-valued, and the equation L(x,p;O) = 0
(20)
determines a (2n - 1)-dimensional Cm-manifold M2n-1(L) in the phase space. (2) An C
M2n-1(L).
(3) Manifold An and volume element don are invariant with respect to the dynamical system dx = aL(x,p;0) dt
ap
'
dP = - aL(x,p;0) ax dt
(21)
(4) There exists a solution of system (21) for all t e k; it is unique 2n-1
and infinitely differentiable for arbitrary initial data (x,p) E= M
(L).
Then the commutation formula SP(22L2x2p;0))
L(x,a-1Dx;(ia)-1)(KAn+(r))(x) = is KAn(dt + aL(x,p;t) 2e
+
2
)0 +0( -2)
(22)
1E=0
Here d/dt is the derivative with respect to the Hamiltonian systen is true. (21), i.e.,
d0(r) dt
<3L(x,p;0) ax(a)
ai(r)> + ,
' ap()
'p (13)
' 2x(8)
(23)
provided that a neighborhood of the point re An admits a diffeomorphic projection on the plane (p(a),x(8)). The generalization of results presented here in the general case, described in items A, B, is obvious, and we leave this reformulation to the reader.
.
179
2.
THE CANONICAL OPERATOR ON A LAGRANGIAN SUBMANIFOLD OF A SYMPLECTIC MANIFOLD
In the previous section we gave the construction which enables Here we intend to generalize this construction, us to solve h-1-PDE's in Rn. thus covering, by the way, the asymptotic solution of h 1-PDE's on manifolds and, more generally, h-1-PDE's in the space of sections of special sheaves on symplectic manifolds (at this stage, our construction appears to be somewhat like the one used in the orbit method and geometric quantization (see [46,42,433 and other papers)). Thus we define a special sheaf over a symplectic manifold M, and the canonical operator should take values in the space of sections of this sheaf. In the case M = T*N the sections of the sheaf reduce to functions on M. In order to make the presentation more smooth, we deal throughout the Section with the spaces of finite or rapidly decaying at infinity functions. Obvious modifications should be made to make it possible to consider the (quasi) homogeneous case. A.
1/h-PDO and Wave Front Sets
Let f(q,h) be a function of variables q c- Rn, h E (0,1). for any s E R
IIfIIHs = sup
hE(O,1)
We denote
"I (1+q2-h20 q )s/2f L2(Rn) ,
(1)
II
where az
q
2 2
aqn
aq1
is the Laplace operator with respect to the variables gl,...,gn, and
IIuIIL2(Rn) : [fJu(q)I2dg31/2 is the I.2-norm in Rn.
Set H(Rn)
n
Hs(Rn).
By a Hamiltonian function we mean any function H(q,p,h) of the variables q E--- Rn, p c- Rn, h E [0,1) (the point h = 0 is not excluded), satisfying the estimates
Col + IRE +YH
aqaap a3h y
(q,p,h) I
< Cc,
(1+ IqE + IpI)
m
(2)
for some m E R independent of a, R, y. We denote by Sm(Rn) the space of functions, satisfying (2), and by S_(R") the union
S.(Rn) =U Sm(Rn). m
Proposition 1.
If H(q,p,h) E Sm, then (1 + q2)-N(1 + p2)-NH(q,p,h) E
Proof.
(3)
It is enough to use the Leibniz formula.
Definition 1. H(Rn) given by:
180
Sm-N
Let H E S. A 1/h-PDO in H(Rn) is a linear operator in
1
fHul(q,h) a H(q2,-ih 3q ,h)u(q,h) (4) H(q,P,hl?(i/h)Pqu(P,h)dp.
ei("/4)(2nh)-n/2J f
=
Here pq = pIgl + " ' + pngn and u(p,h) = e-i(n/4)(2Rh)-n/2J nu(q,h)e-(i/h)Pgdq = [Fl/hu3(P,h)
(5)
R
Note that the inverse operator is given by ei(°/4)(2nh)-n/2J nu(p,h)e(i/h)Pgdp
u(q,h) = Lemma 1.
- CFl/hu](q.h).
(6)
(a) Definition I is correct.
(b) 1/h-PDO form an algebra, namely, if H1,H2 E S', then
H1 o H2 = H3 , where H3 E Sm and possesses the asymptotic expansion N m alai+kH E (-ih) E 1 (q,p,O) m=0 jal+k+2 - m apaahk
H3(q,p,h) =
(7)
alai+eH x
aq
a
P
(q,p,O) + (-ih) N+1 [t
N+l(q,p,h)
for any natural N; the remainder in (7) belongs to S°°. symbols H1,H2 lies in
If at least one of the
s-- n Sm, then so does the remainder in (7). (c) If H is a 1/h-PDO with symbol H E S then f( (formally adjoint operator in L2(Rn)) is also a 1/h-PDO. Its symbol admits an expansion * m Smb(H )(q,p,h) N= E(-ih)E M7-0
a2lal+kR
lal+k=m aqaapa2h
+
k (q,p,0)
(8)
+ (-ih)N+1RN+1(q,P,h)
(here the line over the symbol denotes the complex conjugation; the remainder
'T+1 lies in S' (respectively, S--, if H E S-)).
'
Proof. (a) Since u belongs to a Schwartz space, while H(q,p,h) grows not faster than a polynomial, the integral (4) converges. Prove that Hu lies in H(Rn). To do this make use of Proposition 1. Fix N large enough and set
(1 + q2)-N(1 + P`)-NH(q,p,h) = HI(q,p,h)
(9)
We obtain
Hu = (1 + 42)NHI (1 - A)Nu.
(10)
Since 1 + q2, 1 - :1 are the operators which transform H(Rn) into H(Rn), it is enough to prove the required statement when H CH S 1°(Rn) with m sufficiently large. In this case
181
ei(n/4)n(2xh)-n/2(2nh)-nr
CNu)(q,h) -
x
J R3n
(11)
0
2nH(v,w,h)elgvu(q + h,w,h)dvdw,
x u(E,h)dEdvdw = (21t) ° f
R
where 0
H(v,w,h) _
(12)
is the usual Fourier transform of H. By (12), H(v,w,h) is a continuous rapidly decaying function. Therefore it is easy to see that the function elgvu(q + h,w,h) is, for any natural S, a continuous function of the parameters (v,w) in the norm (1), and besides there is an estimate l
eixvo(x+h,w,h)l k 6 C(1+ Ivl + Iwlk) I 0(-,h )11 k.
(13)
It follows that (a) is valid and that for any natural S (cf. Chapter 2:4): I Hu 1 8
G const n u ll s.
(14)
To prove (b) we make use of the composition formula (cf. Chapter 2:4): 2
1
H1(q,p-ih
Smb(Hlo H2)
aq
, h)H2(q,p,h).
(13)
Expand HI(q,p + hE,h), using the Taylor formula with the remainder: N
m
DIak+kH
H (q,p+hE,h) = Z h '(
apa2hk
k+Ial= m
m=O
1
E
(q,p,O)Ea)+
1 (16)
N (N) + h H (q,p,E,h).
It follows from Chapter 2:4 that the operator 2
H(N) acts in S -, S .
1
(q,P - i aq h)
Thus (7) is an immediate consequence of (16).
The proof of (c) is quite analogous to that given above, and so we omit all details. The lemma is proved. Definitior 2. some real r. denoted by Ir .
Let f(q,h) be a function such that h-rf(q,h) E H for
Then we write f - o(hr).
The space of such functions will be
From Lemma 1 it follows that HIr C Ir for any 1/h-PDO H
and any rE R. Let H(p,q,h) be a Hamiltonian function. We say it is finite and write H E So, if for any fixed h the support supp H lies in the ball lgzl + lp2l < R, where R does not depend on h. It is easy to see that So C S. Definition 3.
Let f - o(hr) for some real r.
The wave front of order
k C- R U {+ m} of the function f is the smallest of the closed subsets
K C R2n = Rq x RP, having the property: for any H E So, such that H - 0 in the vicinity of K, the relation Hf - o (hk+1)
(here m + 1 = m) is satisfied.
182
(17)
The wave front of order k of the function f will be denoted by WFhk)(f) if k = W).
(WFh(f) also,
(a) Definition 3 is correct.
Lemma 2.
o(h
(b) For any 1/h-PDO H and f
r),
WFhk)(Hf) C WFhk)(f) C WFhk)(Hf) U U{(q,p)/H(q,p,0)
01.
(18)
U WFhk)(f).
(19)
(c) If f = o(hk+1), then WFhk)(f) _ 0.
(d) There is a relation WFh(f) =
kER
(a) We should show that the intersection Ko of sets K posProof. sessing the described property possesses this property as well. Let H E so, Since supp H is a compact set, one can find the sets K1, supp Hfl K,.= 0. ...,Kr with the mentioned property, such that
supp H fl Kl in ... fl Kr = o.
(20)
.By means of unity partition we represent H in the form of the sum H = H1 +
+ ... + Hr, where supp Hr fl Kj = 0. H;f
We have
o(hk+1),
J
1,...,r
and, consequently,
Hlf+...*Hr f = o(hk+l)
Hf
(21)
(a) is proved.
(b) The left inclusion (18) follows directly from the definition and To prove the right inclusion it is enough to mention that if from (7). H(go,p ,0) f 0, then using (7) we easily construct the symbol G(q,p,h)
such tgiat 2
H o G - F(q,-ih 2q ,h),
(22)
where F(p,q,h) - 1 in the vicinity of (go,po). The statements (c) and (d) follow directly from the definition. The lemma is proved. it set introduced here is quite analogous to The notion of a wave f HBrmander's one. We note that in (36) the wave front was defined in
another way (namely, as a support of the limit in D'(R2n) of the corresponding density function e(i/h)pgf(q)f(p)(2nh)-n/2
o(q,p,h) -
).
RIn
Let U C be some subset. We u iote by Hk(U) the subspace in H(R2n), consisting of functions f(q,h) such that WFhk)(f) C U,
and by Hk(U) the factor-space Hk(U) - Hk(Rn)/Hk(R2n \ U).
(23)
183
Lemma 3.
There are natural mappings an! commutation diagrams: Hkz (U)
(a)
k, for
ki
(24)
jU3U,
(25)
ki 2 k2 z ks; (b )
}tk (U., )
jU2Ul 3U2U3
Hk(U1)
__- + H(U3)
for U1 > LI2 D U3; Hki
(C)
(U1) -- Hk, (U2) (26)
Hk2(U)
for k,
_
0
Hk'(U)
k2, U_ , 1f2
the existence of commutation diagrams (24) - (26) follows from Proof. the natural embeddings H
k(U)'r Hk (V), U CV;
H
k{il) - Hf (if), K
(27)
which, in their turn, ;immediately follow from the definition of the space
I;k (U)
.
Lemma',.
The space Hk(U) is invariant with respect to 1/h-PDO (and
thus 1;h-PDO are correctly defined in Hk(U)). 'hen the commutative diagram
if H E SW, U C V, K
H
Hk (V)
jUV
-+- k HM
I
1
I
Hl
Hk(U) -+-hi +Q if
juV
H
(U) -- .
(28)
(U)
cE
H (U)
;he l/h-PDOin the space Hk(U) is completely determined by restriction of its symbol on U. ,exists.
the ,roof is obvious. Thus, _:f for any domain U C R2n, we set S (LI)
= +.H(gl,p,h) E C'(U x (0, 1))
,
(29)
3H1(9,p,h) - Sam, H1
1{i
(Q, P)EU
then to any symbol H E S (U) there corresponds an operator
184
H
:
Hk(U) -- Hk(U),
(30)
which we call the h-PD) in Hk:.) with the symbol H E S(U). By Lemma 4 the operatuts (30) are consistent with the restriction homomorphisms, introduced in Lemma 3. 1
B.
quantization of Canonical transformations of Bounded Domains in R2n
Let U,V be bounded simply connected domains in R2n, and let g be a mapping, given by relations (q`,P`)
(q(q`.P'),P(q',P')) = E(4'>P').
:
U w V
(31)
and which is a canonical transformation of U into V, i.e., a smooth mapping such that
gF (Lip n d(j)
= dp' n dq' .
(32)
Here
dpndq = dpi ,dgl+... +dpnndgn.
(33)
Assume that g(U) = V; we will ren;;re That the mapping might he continued into some neighborhood Uo D U of the closure of U. Consider in tR(q, p,)
R24,P) the graph F(g) of the transformation g
continued into Uo:
(a'.p')t-ilo, lq.p) = g(q',p')}. Lemma J..
(34)
The graph ','g) i_ a Lagr,.r.oian sub-manifold in IR4n with
respect tr the '-form
- dp
dc, -
dp' ? dq'
(35)
(k.e., the _astriction of the form t; on '(g) is equal to zero). Proof.
the statement of the lemr:.i fellows from the relation
(32).
Let x g be t canonical operator on the manifold r(g), taking 1/h-
oscillatins; functions (2n into. correspondence with smooth finite q,q')of E (<. functions an r(g), polvnomial.iv dependent on h. To define the canonical operator w, must Fix a non-degenerate measure u on r(g); we shall make use of p = (dp n dc} n ... A (dp n dq).
(36)
n times
The unity partition, the initial point and the coincident choice of measure density arguments will be fixed in an arbitrary way. Example 1. If the manifold r(g) may be defined by virtue of the unique generating function (El]) S(q',q), i.e., may be described by the equations p = 2q (q',q),
P'
'I,l'
(q',q),
(37)
then the result of action of the canonical operator Kg on the function $ r_ C_(r(g), may be given by the formula
185
-S (q',q))
1Kgt1(q',q,h) -
(38)
aq-zaqr
{det
(q',q))-1/2
(here and in the sequel we consider (q',p') as a coordinate system on r(g); in particular, the function is written in terms of these coordinates). In (38) we have fixed an arbitrary, continuous inUo branch of the argumen, of the Jacobian 2
det aq- aqr (q',q(q',p')).
We construct now a linear operator T(g)
:
Hk(U)
Hk(V).
In the following way: let 0 E C0(U0),
(39) ,
1 in U.
We set
e-i(n/4)(2nh)-n/2r n iKgo3(q',q,h)f(q',h)dq'.
CT(g)fl(q,h) -
(40)
Lemma 2. (a) The operator T(g) Hk(U) - Hk(V) is defined by (40) correctly, i.e., it does not depend on the choice of function 0, satisfying the mentioned condtions, and maps :
Hk(U)
Hk(V).
(b) The mapping T(g) defined in (40) commutes with the restriction mappings, introduced in Lemma 7, i.e., the following diagram is commutative
T (g) 0 Hk(V)
Hk(U) i
i g(W)V
WV
T (g) Hk(W)
(41)
11
Hk(g(W))
Proof. Let H be a 1/h-PDO with the symbol H E S'°° n S(g(U)). follow ni g commutation formula holds
Ho T(g) - T(g) o H1
The
(42)
where d=
H1(q',p',O) = (g*H)(q',p'.0)
H(q(q',p'),O),
supp H1 C.g-1(supp H).
(43)
The proof of (42) follows immediately from she stationary phase method. It follows from (42) that the operator T(g) maps Hk(U) into Hk(g(U)). The remaining assertions of Lemma 2 are proved quite analogously to the corresponding assertions of Lemma 4 in Section 2:A, The lemma is proved. Note 1. It should be noted, however, that the operator T(g) is defined not uniquely but only up to a factor of the form
exp(iC(1)/h+ iuC(2)), C(1) a IR, C(2) E 7l
(44)
dependent on the choice of action and the measure density argument on r(g).
186
Study now the algebraic properties of the correspondence g -+ T(g). 2n
Let U,V be the bounded simply connected domains in tit the set of canonical transformations g : U + V, g(UZ continued in some simply connected neighborhood of U.
Denote by Sp(U,V) V, which might be
if U,V,W C tt2n are bounded simply connected domains, the composition law is defined (45)
SP(U,V) " SP(V,W) , SP(U,W), (gl,g2) + g2o gl.
Theorem 1. 2n in Gt
Let U,V,W be bounded connected simply connected domains Then
g1 E Sp(U,V), g2 E Sp(V,W).
,
T(g2)T(gl)u = exp{ h Cg1g2 + inCg2g2}T(g2o gl)Vglg2u
In (46) C(1) is a real number and C62) 9192
for any u E Hk(U).
(46)
is an integer,
depending on the choice of action and the measure density arguments on the is a 1/h-PDO of the form graphs f(gl), r(g2), r(glg2); V9192 2
Vg1g2 = 1- ihRg1g2 (q,-ih aq ,h)
(47)
E SW(U).
where R glg2
We use the proof scheme used in [39] and 161].
Proof.
Lemma 3.
2n
Let g
:
U -+ IR
be a canonical transformation. There exists
a smooth family {gt}, t e [0,11, gt such that
:
U ,R 2n of canonical transformations,
go - 1, gl - g.
(48)
Set for Z - (q,p) E U
Proof.
V t(z) + 2tg'(O), g(z,t) 1
t -i
2
t e [0, Z ],
(49)
[g1((t2)z>-g1(0)1+g1(0), tC- [2
where Pt is a smooth family of linear canonical transformations,
o = id, l - dgt1z-0 (such a family exists since the group Sp(n,&1) is connected).' function f[0,11 + [0,3/2] be strictly monotone,
f(0) - 0, f(l) - 2 , t
f(2)
,
2
f(k)(2) - 0, for k
Let the
1.
(50)
d=f
Then g j(-,f(t)) is a smooth family of canonical transformations, satisfying the required conditions. The lemma is proved. Consider now the vector field t
Vt(z) -
((gt)-lz)
(51)
It is a Hamiltonian one; thus there exists a function H0(z,t) such that
187
vt n r aH0 az Thus gt
0
E
-E
0
T=
(52)
Outside the bounded domain
g4 0
U =
n
(53)
gt(U).
tGCO, it roe continue the function H0(z,t) in such a way that the inclusion can be satisfied: akHO(z,t)/atk E S'(tR2n), k = 0,1,2,..., t E 10,11. Set now H(q,p,h,t) = H0(q,P,t) - ihH1(q,p,t),
(54)
S'(IR2n),
where H1(q,p,t) E
22H0
H1(q,P,t) = Set Vt = gt(U).
tr
agap
(q ,p, t)
(55)
.
Consider the operator family + (ia/2)b(t)T(gt),
T(t) = e(i/h)a(t)
(56)
T(t)
Hk(U)
:
=
Hk(Vt),
where a(t) and b(t) are chosen in such a way that the operator (56) continuously depends on t.
Lemma 4.
The operator (56) satisfies the equation
-ih aT(t) u+ H(t) o T(t)u = 0(h2), u E H(lRn).
(57)
The proof is obvious. Perform now the above construction for g = g,.
We obtain that the
operators T(gt)o T(gl) and T(gtgl) both satisfy (57) and have coinciding initial data. Since H(t) is formally self-adjoint modulo 0(h2), the formula (46) is valid. The proof of the Theorem 1 is complete. C.
The Sheaves of Rapidly Oscillating Functions on a Symplectic Manifolc'
Let M be a smooth manifold of even dimension 2n, supplied with the symplectic form w2. Let
{lfa,ya
:
2n Va C IR(q,p)'aET
Ua
be a canonical atlas on M, i.e., an ,.alas such that
ya (dp n dq) = w for all a E T.
Let a,6,... E T.
We denote
had... VC,
(6... )
(58)
(1 U3 (1...,
(J
_
a
(Lfa6... ) C Va, (59)
Ha(s...) = U(Ua
188
(and, consequently, their co-
It is assumed that all the sets Ua,Uos
ordinate images) are bounded, connected, and simply connected. The canonical coordinate mappings are defined 1
gas
V
Ya 0 Ys
(a)} Va (8)
and, consequently, we may obtain the operators Tab a T(g,B)
Hs
:
(CO
HO(B)
(60)
From Theorem 1 of the previous item we obtain, that if Ua6Y #
'
(1)
EaBy
Taso TsY =
+ ine(2Y}
(61)
In (61) e(1) EIIt, E(2) E a 2. Thus, two-dimensional aBy aly 0a) (1) and E (2) of the covering {Uo} with values in tR and a 2, cochains E respectively, are defined. It turns out that the following lemma is valid.
in the space HO
The cochains E(1) and E(2) are cocycles.
Lemma I. Proof.
By Theorem 1 of previous item,
A # 0,
Tab o Tsa = so that Tas is an invertible operator.
We have
Let UasY6 # 0.
Tas a TsY o TYd = TaY o Ty6exp{i hs1+ iirc(2Y)
= Tad-exp(i
(62)
(2)
(1)
Ehs1 +
i
Ehy6 + ine(1) aBy
ine(26 Y
'
on the other hand,
Tas o Tay o TY3 - Tas o isdexp{i 6+ inE(Y6} c(0) hyd
= Tad'exp{i
E(1) + i hs5 + ines26 + ineas6
(63)
Y
From (62) - (63) we obtain, multiplying by
Ede(1)/h+nde(2))
-
= (Easy-ea66+Eay6 -
E(1))/h+n(E(2)-E(2)+e(2)-E(2))
asY
sy6
for every h E 10,11.
ayl
sY6
_ 0(mod 2n)
It follows that de(1)
The lemma is proved.
all
(64)
= 0, dE(2) = 0(mod 2).
(65)
Thus, we have defined the cohomology classes
f(1)EH2(M,JR), e(2) E H2(M,71 2) on the symplectic manifold M.
189
^x(!. It
Definition 1. The pair (w,h) satisfies the quantization condition (two-dimensional)-if for the given values of w and h, we have fih
i.e., E(i)/nh +
E(2)
E(')+E(Z) = 0 (mod 2),
(66)
defines a zero cohomology class in H2(M,IR.,).
Let the quantization conditions be satisfied.
Then there exists a
cochain 1r - {pas) of the covering (Ua) with values in tR2 such that for
Ua8y 0 0. _ ay +
(du)asy
U
By
u
nh 1
(1) (2) EaLY + Ea6Y (mod 2).
(67)
Set
Tai = Ta3exp(-in}ia6), (68)
Tao = Taeexp(-cpa6). From (67) it follows that the operators Ta6 satisfy the agreement conditions
Tab o TBy ' Tay
(69)
and, consequently, define some sheaf F° of linear spaces on M. Definition 2. The sheaf FO is called the sheaf of rapidly oscillating functions on M. Give an explicit description of the sheaf F°. Recall, first of all, the definition of the sheaf (cf. (201).
The sheaf of linear spaces over manifold M is a collection of linear spaces F(U) where U varies in the set of all open subsets of M and linear operators iUU,
:
F(U') ; F(U),
defined for U C U' such that the iollowing conditions are satisfied: (a) For U C U' C U", we have
iUU, o iU,U" ' iUU (b) If U -
U U
(70)
and f E F(u) is such that
jET j lU.Uf
0
(71)
j
for all j, then f - 0. (c) If U
jU
TUj and the tuple of elements fj E U, j'E T satisfies
the agreement conditions
'Ujn Uk,Ujfj - 'UjnUk,Ukfk 190
(72)
for all j,k E T, then there exists a unique element such that (73)
iU.Uf = f for all j,k E T.
F(U) is called the space of sections over U, while the mappings are called the restriction homomorphisms. The sheaf F° is constructed in the following way. For each open subset U C M denote by FO(U) the subspace in the direct product
II H0(Yn(U fl Ua)),
aeT
consisting of tuples {ua}a E (such that (74)
ya in the space Ho(yaUfl Uns) for all pairs (a,R) such that Uas # 0.
If
U C U', then by
iUll,
: f' (U' )
3 ° (U)
we denote the mapping induced by natural projections
HO (Y(, (U' n Ua)) - H°(',a(U fl uo)). It is not difficult to verify the correctness of the given construction, while the validity of sheaf axioms (a) - (c) follows readily from the definitions. We see that the construction of the sheaf F° depends on the choice of the one-dimensional cochain u of the covering {U.} such that
du = e(1)tth+E(2) (mod 2).
(75)
Let u,u' be the two cochains, satisfying (75); then d(u - u') = 0 and thus u - u' is a one-dimensional cocycle. Let T'S = Tasexp(-inuas),
(16)
and let 30 be a sheaf constructed by the means described ah,,v-, using the transition operators T'5. Assume that u - u' is a cobo,irdary modulo 2. Then
ua8 - uaf3 = AC, -
6
(mod 2)
(77)
for U06 # 0; and the mapping {W'}'ET -, {exp(inaa)*a}aET induces the sheaf isomorphism F - F°'.
(78)
Thus we have proved the following:
Theorem 1. The sheaf of rapidly oscillating functions on a symplectic manifold M exists if and only if the cohomology class
`(1) tth
t(2) +
a H2(M,1t 2)
equals zero. If this class vanishes, the sheaf of rapidly ,scillating functions is fixed (up to an isomorphism) by the choice of some (also dependent on parameters) cohomology class u C H1(Ut2).
191
E F°(U).
Now let U C M be an open subset, Definition 3.
*Ch-
wave front of * is the following subset of M:
WFi(.i) _
ya1. (u"))C:U.
(WF
(79)
It is easy to see that the set (79) depends only on i, not on a canonical atlas and the representatives WC'. Let H be a smooth function on M, growing slowly in all the charts of some canonical atlas.
Definition 4.
The operator H in F"(U) given by 1
H(Wa) = ((H0yai)(q,-ih q)V'a)
(80)
is called a 1/h-PDO in the space of sections of the sheaf F°.
Lemma 2. The operator (80) is a sheaf homomorphism F° - F°. The statement of Lemma 2 of item A (k = 0 only) remains valid for l/h-PDO in the space of sections of the sheaf of rapidly oscillating functions. The proof is obvious. U.
(rantized Lagrangian Sub-Manifolds and the Canonical Operator
Now let M be a quantized symplectic manifold with the form w2, and A be a connected Lagrangian sub-manifold in M (i.e., the restriction w2ITEA is equal to zero for all z E A). Let also do be a non-degenerative measure on A. We construct here an operator K
: Ca(A) .O(U)
(81)
for any open subset U C M, which will be called a canonical operator. Let again
2n
(ya be a canonical atlas on M.
Ua - VaC L(q,p) :
T
For any a
Aa = ya(A n u,
(82)
is a Lagrangian sub-ma.ifold in IR2n D Vo; we assume that the covering (Ua) is su;h that A. is connected and simply connected. Under these conditions on each A. there exists a canonical operator, amd we may consider it acting in he spaces Ka
:
Co(A (1 Uo) - Co(Aa) -+ Hu.
(83)
Lemma 1. Let the intersection ;ifl tin fUS be non-empty. Then for any function E C0 '_(A U Ua U U8) there is an equality in the space HO (8): Kam - K8
exp(iirba8),
(84)
where ba8 is the cocycle on A with the values in the group of real numbers modulo 2. Proof.
This Is an obvious consequence of the stationary phase method.
Let now the cohomology class
b E H1(A,tR2) 192
(85)
be trivial. We say in this case that the quantization condition on A is Multiplying K. by the factor Sa satisfied. R,Jj = 1, we can reach the identity Kam = KQ for all a,8 such that A fl U, fl Ua is non-empty and ¢ EE Co(A n Ua fl UB).
It follows that the global operator K
:
-- F(u)
C0'(:1)
(86)
is defined for any open subset U C M, coinciding with Ko on Co(A fl Ua). Definition 1. The operator (86) is called a canonical operator on the quantized Lagrangian submanifold of quantized symplectic manifold. I.
3.
THE CANONICAL SHEAF ON A SYMPLECTIC IR+-MANIFOLD
A.
Some Function Spaces
From now on we consider the R+-homogeneous version of asymptotic constructions which, in particular, enables us to prove the main quasi-invertibility theorem (this is performed in Chapter 4). This section presents a necessary preliminary to the subsequent considerations. We list a number of function spaces to serve further as local models of the Poisson algebras, spaces of canonical sheaf sections over a R+-homogeneous symplectit manifold, etc. We define also pseudo-differential operators in the introduced function spaces and give some supplementary information and notations. Let Rn be an arithmetic n-dimensional space with elements denoted by (x1,...,xn) and let 1 = (A1,...,an) be a fixed n-tuple of non-negative real numbers. Set x
Io = (j c- In]laj = 01,
(1)
I+ = In3 \ Io = Io. Here En] is the set of integers from 1 The equation
We assume I+ to be non-empty.
to n.
c E R+
TX -
(2)
defines free smooth action of the group R+ (multiplicative group of positive real numbers) on the space
Ip
x (R I+l \{O,; )def = R* i
(3)
For any x E R*, set A(x) _ 1-1,
(4)
where T = T(X) E R+ satisfies the property 21
2
E
(TX). =
jEI+
E
jEI+
T
2
jx.
1.
(5)
3
It is evident that A(x) is a smooth non-vanishing function on R* satisfying the properties: A(TX) = TA(x),
cA(x) <
E
Ixj1l"j S CA(x)
(6) (7)
JEI+ 193
Table 1.
Functional Spaces
Conditions for f(x) to belong to this space
Notation of the functional space
(xl,...,xn)I5 CaA(x)
f
S(al,...,Xn)(U)
f(`A)(xl,...,xn)j<
m
S(a1,...,An)()
J
J
CaA(x)m
Ej(1-
0
Yaj
f(a)(xl,...,xn)1,< CaA(x)m +
L(a1,.... X )(U)
m
SO1
)(U); in addition, n f(ix) = imf(x) for T : 1, A(x) ) R where
same as for
(U) X
R depends on f
'
(U); in addition, A) n
same as for
(U) 1
n
1
N1 (N)
for any N ? 0, f (x) =
fj (x) + RN (x) ,
E
j=0 )(U), mj 5 m;
where fj(x) E 073
j
= 0;
n
N1(N) % W Sm-N ('l,
1
n
)(U)
We introduce the real-valued function A(x) on with some positive c and C. Rn, setting A(x) = A(x) for A(x) ) 1 and regarding that A be a smooth For I fl i+ i 0, we set function depending on x and ) 1/2 everywhere. AI(x)
For example, if n = 2,
>.1
=
AI(xI) = A(x) 1 YI+\
(8)
I -0'
= 12 = 1 and therefore A(x) _ (x
large x, then A(11(x1,x2) = t(x1,0) = ;xlj for large x1.
+ x2)1/2 for
Obviously, the
following inequalities are A1(x)
cA(x) 5 1 +
i A(x),
Ixj11/,J F.
(9)
CA(x)
(10)
jEI+ with positive constants c and C in (10) different, in general, from those
in (7).
We have gathered the functional spaces most frequently used in the sequel in Table 1 given above. Some elucidation is necessary. U The denotes an arbi ary domain in Rn; the case U = Rn is also included. elements of a space are smooth functions in (I up to the boundary of U. In the second column of our table conditions for f(x) to belong to a specified functional space are presented; they must he satisfied by in U. The lowest possible values of constants in the estimates form the tuple of seminorms, defining the topology in the described functional space. m denotes any real number. The notation of the functional space may include an additional subscript 0 (which makes sense for U z Rn and indicates that supp f U for any f from the space) or c (which means that the elements of the spare have compact supports with respect to x1). C
U
194
The elements of the space 0 (U) are called (A1,...,A )-quasi(All n)
homogeneous functions of degree m for large x P ° 1
(All ...IAn)
those of the space
;
(U) are called asymptotically (x ,...,x )-quasi-homogeneous n
1
functions of degree < m.
Except for 0m1
)(u),
we allow m to take
n values +- and -, denoting, respectively, the union and the intersection 1
of the corresponding spaces for all finite m. Obviously, the inclusions between these spaces are valid: (a) Except for 0m1
)(U) the space with the lower value of index n
m is contained in the space with the greater one; (b) 0(a
... ,1 1,
`m
)(u) C Pm
(
n
n
1
S (alV :.,An) x (u) C L(A 1
u)
C
Sm
(u) C
n
1
,...,a )(u)' n
Also it is evident that A(x) E
.An) (U), We present the useful
property of quasi-homogeneous functions Proposition 1 (the Euler identity).
If f E=- 0m
)(U), then
All it
for x E u, A(x) large enough, we have mf(x)
E ajaf(x)/axe.
(11)
j=1
Differentiate Tmf(x)
Proof.
The space Sm
f(TX) with respect to T and set
1.
i
) (U) will be a local model for the space of the Poisson
1
'
n)
algebra, and the space Lin
1 )(U) will be a local model for the space
n
1
of can'-nical sheaf sections. We need some considerations of pseudo-differential operators in the latter space. Let m
E L(A
(R u). 1
l
...
We intend to define the action of pseudo-
A) n
differential operators by means of the common formula (cf. Chapter 2): n)jf(x,p)ei(x-y)P$(y)dydp.
(12)
f(x, -i a- )O(x') = (2n)
The integral on the right in (12) is, generally speaking, divergent. However, under certain conditions it may be regularized and give rise 'to a continuous operator in Lea n l )(Rn)11
Definition 1.
n (R2n)
We denote by 1.
the space of functions
f(x,p), x,p a Rn satisfying the following conditions:
alnl+Islf(x,P)
I
C
axoap0
A
x)m-£3X303+£3(13
-1) 3
aB (
(13) A
X (1+E.jpjA(x)i
-1
1)
m1 +
Ia.l
,
loll + iBI
195
Set also
where ml does not depend on a,B (but depends on f).
R2n
T(al.....xn)(R2n) - m T(A1
T(Al,...,An)(R2a
T(al,...,an)(R2n) -
m Theorem 1.
For f E T
(A1,...Xn)
(R2n) the integral (12) admits a
regularization (described later in the proof of this theorem); once f belongs to TEA
)(R2n), the operator (12) is continuous in the spaces
A
n
1 1
f(x, -i aax)
:
(R2n) ; La+m+d
Ls
(R2n)
(14)
(Al,..., m)
(Al,....An
for any s E R and any 6 > 0. (211)-n
Proof. Denote the integral on the right-hand side of (12) by I(x). We perform some variable changes and transformations over I(x) (our calculations to this end being purely formal). We introduce the new variables of integration (j, nj, j - 1,...,n by the formulas: A(x)1-A j.
yj - xj +(JA(x)AJ; pj -
(15)
Substitution of (15) into I(x) yields
1(x) - C1feiA(x)Lnf(x,nA(x)1-AWx +CA(x)A)d(dn]A(x)n. nA(x)1-A
In (16) and subsequent formulas
(16)
stands (for short) for (n1A(x)1-11,
...,nnA(x)1-An)
etc.
Then (16) may be written in the form:
1(x) - A(x)nf5eiA(x)EnF(x,n)4(x,E)dEdn,
where F(x,n) - f(x,nA(x)1-A), (x,() - $(x + (A(x)A). The conditions that f(x,p) E TEA
1"
(17) '
)(R2n) and $ E LEA
A
n
1"
)(Rn)
A
n
turn into the following estimates on F and 0 are: a
IaI+IBI a ax an
a(z,n) 14
m- E.A.a.
](l+ Inl)ml + lal
C.8A(x)
(18)
1-1 + IBI - 0,1,2,...; xg)
(A(x)A)8 +Ej(1-Aj)YjA(x)£]A]Yj,
1 < CYA(x +
aYa(Y
(19)
IYI - 0,1,2,....
We consider first the case when m1 is negative and sufficiently large, namely, ml < -n and then we point out how the general case may be reduced to this one.
Let e1(z) be a smooth finite function in in equal to I in the vicinity of the origin. For any e > 0, we have the partition of the unity
1a
eI(A(x)1-C
(
e1(A(x)1-C
196
)+(l-eI(A(x)l-C
f)+
e2(A(x)1-C
(
)
and, respectively, l(x) = II(x) + 12(x), where I.(x) is obtained from I(x)
via the multiplication of the integrand by ei(A(x)1-E E), i - 1,2. The By (18), taking into account that m1< - n, I1(x) converges. regularization of I2(x) goes as follows: consider the differential operator 1
L-
E C.
3
(20)
.
iA(x)EZ j-1 J anj
Obviously, the coefficients of L are smooth on supp e2 and LelA(x)En - eiA(x)En.
(21)
Therefore we obtain by formally integrating by parts N times: 12(x) - 1N A(x)n-NrreiA(x)Ene2(A(x)1-E E)O(x,E) 1
1
(22) (E21
a2nj)NF(x,n)]dEdn.
C( E
x
)N
Estimate the integrand in (22).
Denoting it by JN(x,E,n), we have Inl)m1IEl-N
IJN(x,E,n)I 4 x A (x + EA (x)
a a
x
(23)
6 cons t A(x)m+8(1
+InlmII&I-N(1
+
E
j6I+
leJI1/xj)lsl ,
We see that for N large enough the integral on the right-hand side of (22) is absolutely convergent (note that ICI >_ CA(x)e-1 ). 0 on supp JN)
We take the formula (22) as the definition of the regularization of the integral I2(x) and therefore we have defined the regularization for I(x) as well, since I1(x) needs no regularization. It is obvious that the regularization does not depend on the numbers N,E and on the choice of el(E). We also see from (22) that II2(x)I 5
(24)
.for any Ni.
Indeed, since JEI > CA(x)e-', we have
(1+ IEI)CA(x)E-1 where co = sup CA(x)E-' <
CA
(X)E-1 +CC
Therefore
a
(1 Cco)NA(W)N-NE
1
ICI on supp
N
1
(1+ ICI)
N
e2(IEIA(x)1-E).
Substituting this into (22) and (23), we obtain II2(x)I 6
which immediately yields (24).
Consider now the case m 2 -n. We introduce the C°° partition of unity in the space Rz with the properties:
i)
suppPoc(Izl <1}, supp p.c[Izl>Z, IzjI
)
Izkl,k - 1,.,.,n}; 2
197
IaI m 0,1,2,..., j e O,...,n
C C
ii)
and represent I(x) in the form:
I M
I(x)
= J15ff(x,p)pj(pA(x)a
E
j-0
-0
(25) fff(j)(x,p)ei(x-y)P4(y)dydp
E
j-0
(note that f(j)(x,p) E
T(J11,...,an)(Rn),
j - O,...,n; the reader can easily
verify it).
As for
Each of the integrals I(J)(x) may be considered separately. 0)(x), I
nothing has to be. proved, for the resulting integral in (17) is
taken over (InI < 1). In the expression (25) for I(J)(x) we perform (formal) integration by parts over y., which yields the formula (12) with (y) J
replaced by aM(y)/aye and f(y,p) by f(i),(y,p)(ipj) -M
After that we go
.
along the familiar ways and-obtain
IM(x) x
A(x)ffeiA(x)Enf(J)(x.nA(x)1-a)
X
-
M (i n. A(x)I-1J)-M M (x +EA(x)A)dgdn M
A(x)-M(1-aj)rreiA(x)EnFJM(x,n)OM(x,
(26)-
)dtdn,
where
alai+Is F.IM (x..n)
.m-E A a C
as
a:aan8 (we have taken
A(x )
k k k(1
-mi-M+IaI
+I n I)
(27)
nto account that InI < CIr1.I on supp pi(n)) and
ay4b (x,E) aE Y
EAkyk
s+M(1-aj)+Ek(1-ak)Yk
1
< C A (x + EA(x) a
I
A (x)
)
(28)
.
Choosing M so that m1 - M - n, we can perform the regularization just as above. The correctness of this procedure can be easily verifica. Again we obtain II2J
x)I <
CN1A(x)-N1
for any N1.
Estimate now T1(x) (resp.
IiJ)(x)), which according to (24) is the "essential" part of I(x).
We have
I1(x) °
(29)
where F(x,n) satisfies (18), while (19) on supp e1(A(x)1-eE) reduces to
8y(e1UMx,O) I G CYA(x)
s+IYI
(30)
at Y
Combining these estimates, we obtain, taking into consideration that the
voltme of integration over E in (29) does not exceed II1(x)I <
198
CA(x)ID+e+nc,
(31)
and exactly the same is valid for IQ) (x).
Thus we have estimated I(x)
alai itself and we need to estimate the derivatives
a
I(x), Jul = 0,1,2,...,
ax
It is an easy computational exercise to verify that the derivatives may be obtained as follows: one should apply the derivatives under the integral sign in (12) and then follow the regularization procedure described above. The calculation omitted here leads to the desired result: as well.
alalI(x)
C.(x)
,
lal
0,1,2,...,
(32)
axa
where the constants Ca depend linearly on the constants in the estimates It remains to set e - d/n. Theorem 1 is thereby proved.
.for (x).
We need also to establish an (approximate) composition law for pseudo-differential operators. This is the matter of the following theorem. 2
Set Tm
,n
1"
J1
Theorem 2.
T
T(All
(R2n)}'
"
T(a ,...,a ) .
,
An)
n
1
(All...,an)'
in
(x
1
) _ { f ( x , -i ax )If E
, . . .
1
x ' n
The set T
(al,...,an) is an operator algebra in the space
More precisely, the operator multiplication maps
)(Rn)
Tf into Ol.... ,an)
(Al,..-,an) -
x (R2n) and
Tm+f
(al,.. ,an) and for given f E Tm(al,...,ln) X
2In
the symbol h E T(A1,
E TjA1'"
.a n
(R2n) of the
product Cf(x , -i ax)]1g(a , -i a x)3 has an asymptotic expansion:
al
h(x,P)
E
(
a--i-,
a
lal=0
al
lal
of (x,P) ap
ax
(x,P).
(33)
The equality (33) means, by definition, that the difference h RE.. (x,P) 101 a lal (-i) 2n -r- a belongs to T(1 . . . (R ) for any natural R.
a f am+f-R-1
10,1-0
ap
ax
An)
1l
Proof.' We sketch the critical points of the argument. Let f and g be as in the formulation of the theorem. Using (12) twice and performing the simple variable change, we come to the following statement: for any 0 E 1(All ...an)(Rn)
0-9 )
h¢,
(34)
where* h(x,p) _ (2w )-nffei(x-y)Pf(x,P + P)g(y,P)dydp
(35)
and the integral (35)'is defined via regularization as in Theorem 1. Now estimate the integral (35). Perform the change of variables (15) and set also
pj = nj A
(x)1-a
(36)
* We denote Ei (xj- yj )pj - (x - y)p for the sake dt brevity.
199
We cbtain:
h:x,
t
iA(x)1-a
x
x g(x + CA(x)A,nA(x)1-A)dCdn =
(37)
A(n)nffein(x)CnF(x,n + n)G (x, ,n)d,dn.
where the functions F(x,z) - f(x,zA(x)1-') and G(x,{,n) = g(x + CA(x)
IA(x)1-N)) satisfy the estimates IaI+IBI FF(x,z) a
m-EX.a. I
< CaBA(x)
a
m1+Ial
1 1(1 + IzI)
(38)
:
ax az )1-a)m2
A(x)
(39)
IG(x,C,n)I < CA(x + CA(x)a)t(1 + InI
A(x+CA(x)
)
By the Pcetre inequality [45}, we have
Le.t ml < - n.
(1+In+nl)m1 < (1+IjJ)mi(1+IT)))Imii.
(40)
Following the lines of the previous proof, we obtain ih(x,nA(x)1-a)I
< CA(x)m+L+6(1 + InI)m2+Im1I
(41)
(the technical details are left to the reader), or IPA(x)A-ll)m2+Im1I.
CA(x)m+,Q+6(1+
h(x,p) <
(42)
Assume now that mi a - n. We use again the partition of unity p(z) introduced in the proof of the previous theorem and represent the j-th term of the sum in the form h
n
(j )
(x,P)
(2n)
f(])(x,P+P)
i(x-y) P
l1e
(P] + p] )M
M (P.+ p.) g(Y,P)dYdP
J
(43) (2s)-nrrei(x-Y)P 1111
f(.i)(x,p+p)
6j + Pj)M
8
E(P-i )Mg(Y,P)3dydp. ]
ayj
a
The symbol gJM(y,p) _ (pj-i j )tig(y,p) satisfies the estimates: i a
al+IalgjM(Y,
t+M(1-a.)-Ea..+E(a.-1)8. a P)I < C
ayaaps
as
J
i
A(Y)
]
x
(44) (Y)X-lI)m2+M+IaI
(1+ IPA
after changing the variables as in (26), we obtain hQ) (x.p) _ where F.
(2a)-nA(x)n-M(1-aj)JJeiA(x)EnF]M(x,')+ n)GjM(x,C,,)&.drl,
satisfies the estimates (27).
Ih(x,P)I <
CA(x)m+Q+6
Choosing M > n + ml, We obtain
(1+
In any case we may take M - ml + n + 1, '_Lm= ohtgtn.ng
200
(45)
CA(x)m+f+6i1
h(x,p)I <
+
lpAA-il)ma+ml+2(n+1)
(47).
Performing the Next we should estimate the derivatives of h(x,p). derivation in (35) and using the elementary properties of Fourier transformation, we obtain immediately: ala+Blh(x
_
)
(2n)-n1Jfei(x-y)P
E
alyl-lelf (x,P + P) x ax Y ap 6
0*k Y6a
ax a
.
04068
(48)
ale-Yl+ls-elg 01-Y
x
ax
ap
a-0
Each term in the sum may be estimated exactly in the same way as h(x,p) itself. We obtain finally x,P )
E Cae
I
ax a aP
B x
J
m+R+6-E.aa+E(a-1)B
ala+Blh( I
(1+
i
J
J
J
J
A(x)
(49)
IPA(x)a-1l)ml+m2+lal+2n+1. Tm+A+6
Thus we have proved that h(x,p) E
Ul,...,1.n
)
(R2n)
for any 6 > 0.
It
remains to prove the expansion (33) which shows, in particular, that we may set 6 - 0 in the above inclusion. To perform this we express f(x,p + p) in (35) in the form of the Taylor's expansion with the remainder. To avoid extended calculations we derive the principal term of the expansion (33) only; we hope that the reader has already grasped the idea of our estimates and can proceed with the general case without assistance. We have 1
f(x,P+P) " f(x,P) + f
n
z Pj
o j-1
ag app
(x,P+6P)d6.
(50)
Substituting (50) into (35) yields h(x,p)
f(x,p)g(x,p) 51)
- i
E
ffei(x-y)P(fl of
(x,p + eP)de)
app
j-1
a
(y,P)dydP axe
Denote the expression in parentheses by fj (x,p,P).
following estimates are valid for f(x ,p, p) and 2lal+l
It is obvious that the - (x,p): 3
l+
Ylf(( x,P,P)
m+X.-1-EX kak+E(Xk 1)(0 k+Yk) i
a ax cap 4Y
G CasA(x) Y
(52)
S
P)A(x)1-11)max(ml,0)+lal
x
a
(1 + I(P +
l a i + l s l
ax aap6
k - aj -EX kak+E(ak l)Bk
ag axe
x
CaBA(x) l
(53)
x (1+ IpA(x)a-ll)m2+1a1 Therefore, applying the estimating techniques developed previously, we obtain
201
L_1+6'1
(54)
(R2n)
h(x,p) - f(x',P)g(x,p) E
n
1
2n).
+f 011... .1
Thus, the )(R n principal term of the expansion (33) is obtained; all the subsequent terms Theorem 2 is proved. are derived in a completely analogous way. for any 6 > 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 <- M},
(7)
where ml,m2,and M are positive numbers. (f) The projection a(r(g)) of the graph r(g) C T*Rn x T*Rn of transformation g on the space Rx x Ry is a uniformly proper set in the following sense: if we set 10 - {j1A. = 0}, Jo - {jluj = 0}, I+ - In] \ I0, J+ - In] \ JO,
then for any compact set Kl E RIIO1 the set K2 - {yjo1(x,yJ,y3 ) E tr(r(g)) I
for some y
and x with x
E K3}
o
0
+
is bounded and the diameter of K2 depends
J+ Io only on the diameter of Kl, and the same property holds if we also exchange x and y.
We should also be concerned with canonical transformations defined on Defisome R+ invariant connected simply connected open subset U C T*Rn. nition 2 applies to this case as well except that, instead of (3), we have g
Proposition 1.
:
U - T*Rn.
(3')
If g is a proper R+-invariant canonical transformation*
then:
(a) g takes proper Lagrangian manifolds into proper ones. (b) The graph r(g) C T*K x T*Rn of the transformation g is a Lagrangian maniford with respect to the form 02 - dp A dx - dq A dy.
(8)
Proof. (a) Let L be a proper Lagrangian manifold.
To prove that g(L) 1 -1J
is proper we need only to establish the inequalities lpjI < CA(x) on. g(L) or, equivalently, that IpI is bounded on g(L) for h(x) - 1. From (6) the property A(x) - 1 yields ml < A(y) < m2 with positive m1,m2, and thus 1q( < M since L is proper. By item (c) of Definition 2 Ipl is then bounded. Q.E.D.(b) is obvious. The operator T(g) will be defined in terms of its kernel, T9(x,y), on the space Rn x R°, this kernel being given by the canonical operator on the graph r(g). Thus we need to define the corresponding auxiliary objects. The 2n-tuple (A
defines the action of R+ on R(X
1,.
)
.Y
- Rx x R° and the corresponding function A(x,y).
The Lagrangian manifold
r(g) C T R2n - T Rn x T*Rn is not proper since the basic inequalities hpjI < CA(x,y)I-Xj. 1gjI < CA(x,y)
1-1j,
(9) j - 1, .... n
* We drop the words "R+-invariant" in the sequel since non-R+-invariant objects are not the matter of our consideration. 216
However, for any constant C the intersection rC(g) of r(g) with the set defined by the inequalities (9) is a proper Lagrangianmanifold, and therefore the canonical operator may be defined on it (provided that the measure is chosen). do not hold on 1'(g).
There is a special distinguished measure homogeneous of degree n defined on r(g), namely. the pullback of the Euclidean volume measure on T R n
)t (dql A ... A dqn A dyI A ... A dyn) .
u
(10)
r(g) Here
= T*Rnx T*Rn
IT
T*Rn, i = 1,2
(11)
Since being canonical is the natural projection on the i-th factor. preserves the Euclidean volume as well, and u can also be defined as
u = (n 21
r(g)
)*(dpI^...^dpnAdx;^...- dxn).
g
(12)
For the sake of convenience we make use of a special canonical atlas on rC(g).
Proposition 2. There exists a proper canonical atlas on rC(g) such that the canonical coordinates in any chart of this atlas have the form (x,yl,q_) for some I C In]. I
Proof.
manifold in
For any fixed x the set {(y,q)Jx(y,q) = x) is a Lagrangian R2n
Thus, by the lemma on local coordinates, we may choose
(y+q)
canonical coordinates of the form (x,yl,q_) in R+-invariant neighborhood I
of any point'of rC(g). the inequalities
We need only to prove that, in obvious notations,
J E J,
JqJ.J 4
(13)
hold in such neighborhoods; other properties of the proper canonical atlas m.:y be satisfied by suitable choice of the covering. But (13) immediately follows from (9) and (6) since
S
A(x,y) S
(14)
Thus Propositioi 2 is proved.
We assume in addition chat there is a finite number of fixed canonical charts
on r(g) so that a proper canonical atlas on any rC(g) may be
given by (U. n r
and that the density of the measure u in these:'
charts satisfies the conditions imposed in item B.
We note that r(g) is a simply connected manifold; therefore the quantization conditions are necessarily satisfied on r(g). Denote by 1$ (g) the
D-m(g)- {1Dn(g). union of spaces Dm(rC(g)) for all C > 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 <
3
C (here the constants c and C are the same as in (6)) and define
finally:
X(x,Y) - Xl(xlo.ylo)X2(xl+,yJ+ Clearly X(x,y) equals
the n x R y),
in
I
n
0
S(A1,...,Xn,Vl,...,un)(Rx
(19)
neighborhood of r(g) and belongs also to (R4n)
therefore to Tm
Now it follows from Definition 1 and Lemma 1, that for any $ E D" (g) X(x,Y)[K9$I(x,y) -[Kg$](x,y)
E L(Xi,...,Xn,ul,...,un) (Rnx x Rn)y ,
, (All...,Xn,ul,...Ad (Rnx
and consequently modulo L-
(20)
x Rn) XCK $7 does not g y
depend on the freedom of the choice of X(x,y). Definition 3.
By T(g,$) we denote the operator (as usual, arg i = n/4)
1T(g,$)f3(x) -
(2n)n/2fX(x,y)[K9O1(x,y)f(y)dy.
(21)
* The reader should take into account that the phase functions, etc. in canonical charts for Ky are constructed in concordance with the choice of signs in (8). 218
Lemma 2.
E 17m(g) then the operator T(g,+) acts in the spaces
If
T(g,o) = LS
(ull.... un)
for any S E R.
Modulo L7.
(Rn) Y LS+m+n + !(IuI-IAI)(Rn) (Al,...,an) x y
(22)
n
1'...,an)
(R ), the result of action of T(g,o), x
does not depend on the choice of the cut-off function X(x,y). Proof.
By Proposition 3, the kernel K(x,y) of the integral operator
m+n- WX1+IuI)
(21) belongs to L
(al,...Xn,ul"' .,un)
n n x R).
(R
x
In the expression
y (23)
[T(g,0)f](x) - fK(x,y)f(y)dy we perform the change of variables yj = A(x)ujnj, j = 1,...,n.
(24)
JK(x,y)f(y)dy = A(x)IuIfK(x.nA(x)u)f(nA(x)u)dn,
(25)
Then we obtain
where the volume of integration is bounded uniformly with respect to x by construction of the cut-off function X(x,y). Estimating the integral on the right-hand side of (25), we obtain that it does not exceed }(IAI+IuI) =
coos
t-A(x)S-+n + }(IuI-I"I)
The x-derivatives are estimated in an analogous way and thus we obtain the proof of Lemma 2 (the second assertion of the lemma is self-evident). Our next aim is to establish the composition formulas for operators We begin with formulas of composition with pseudo-differential T(g,O). operators. In establishing and writing the composition formulas, it Remark 1. is convenient to slightly modify our notations in the following way. Thus consider the projection 7T1
T*Rn x T*Rn -+ T*Rn (26)
(y,q,x,p) -
(y,q)
(cf. (11)) and its restriction on r(g), which will be denoted by the same T*Ry letter.
We consider now functions on
rather than on r(g) (or, in
other words, consider (y,q) as standard coordinates on r(g)) and denote by T(g,Q) the operator, which in the notations adopted above should read
T(g,m o nl) . In these new notations our results take the simple and readable form: Theorem 1.
(a) Let H(x,p) E
,xn)(R2n).
There exists differ-
ential operators Pj on r(g) with the properties: (i) Pj is a differential
operator of order 4 j with'coeffickents which are linear combinations of the derivatives of the function (g H)(y,q) of order 6 2j with smooth real coefficients. The. operator Pj acts in the spaces Pi
: Vm(g) - Dm+S 3(g), j = 0,1,2....
(27)
219
(ii) In particular,
for any m.
Po - (g H) (Y,q)
(28)
(iii) For any N, N-1
T(g, I (-i)3P0,
H(x,-i
(23)
j=0
}(IPI-IHj)(Rn) (Rn) to LS+m+n-N + (X1'" ''Xn)
modulo operators acting from Ls for any S. (b) Let H(y,q) E TSu
'
)(R2n).
u
Then there exist differential
n
operators P. on r(g) such that (i) and (ii) of item (a) are valid with H substituted3for g*H and N-1
T(g, I (-i)3P0) ayj=0 1
2
(30)
j
to within the same modulus as in (29). (a) Theorem 2 of item B would give the desired result but it
Proof.
cannot be applied directly since H(x,p)
TAX
)(R4n).
u
'
1
In
n
n,v 1
order to avoid this difficulty, we represent the kernel of the operator (29) in the form 2
K1(x,y) = X2(A(Y)/A(x))H(x,-i aX)K(x,y) + (1- X2(A(Y)/A(x))) x 1
(31)
1
1
2
X H(x,-i ax)K(x,Y)
H1(x,Y,-i ax)K(x,y) + H2(x.Y,-i
aX)K(x>Y)
Here K(x,y) is the kernel of the operator T(g,$) and the function X2 is the same as in (18).
The function H1(x,y,p) belongs to TW
(al,.... Xn,
1
,
)(On) and it is easy to see that application of Theorem 2 of item n
B yields the result (computational details are left to the reader) once we prove that the second term in (31) is inessential. To do this we write it in the integral form: 2
H2(x,Y,-i -)K(x,y) _ (32) -nJelp(x-')H(x,p)[1-X2(A(Y)/A(x)))K(C,y)d&dp.
- (21)
Perform the variable change xj = A(x)Aiwj,
gj = (33)
yj , A(x)u3&j,
pj - A(x)1-Xje3,
j - 1,...,n;
then (32) takes the form 2
A(x) n
2
H2(x,Y,-i aX)K(x,Y)
(
2n
) !e
iA(x)B(w-n)
where w -n # 0 on the support of the integrand.
H11- X2]Kdnde,
(34)
Integrating by parts over
(A(x))-N
P, we obtain the factor
220
with N arbitrarily large; the case when
the integral over p diverges is considered in just the same way a.s in items We do not go into further detail. A and B.
(b) We have
1
(T(g,$)H(Y,-i 35 (211)-n
1IfK(x,y)H(y,E)eiE(y-n)*(n)dndEdy
=
- fKI(x,y)ip(y)dy,
where K1(x,y) =
(27T)-n fK(x,e)H(e,E)ededC
2
= 11(y,
i ay)K(x,y)
(36)
Although we did not consider such operators in previous items, their 2
1
theory is just parallel to that of operators of the form H(y,-i
2y).
Con-
siderations analogous to those performed in the proof of (a) lead to the We have no space to present a more comprehensive study of identity (30). The proof of the theorem is now'complete. the question at this point. Our next theorem establishes the relations between the:composition of canonical transformations and composition of the correspondent operators T(g,O). In addition to previous considerations, fix a non-zero tuple (e1,...len) of non-negative numbers and the corresponding action of the n
group R+ on the space Rn and the cotangent space T Rv = Rn T-Rn. T1
Consider the proper canonical transformations *
gI
T R
n
. T*Rn y
(37)
g2 : T*Ry I T*Rn and their composition
g2 gl = g3
* n
:
n
*
T Rx + Tv. R
(38)
Obviously, glis a proper canonical transformation. We assume that the canonical transformations j = 1,2,3, satisfy all the additional conditions imposed in this item. Under these assumptions and some further restrictions on canonical atlases, the following theorem is valid: Theorem 2.
E
Dml+m2-k
(g3), k
Let 0. E Dm3(gj),j = 1,2.
There exist functions 4k E
0,1,2,..., such that the following properties are
satisfied:
(i) The function m is a bilinear form of the values of functions ¢1 and g1*$2 and their derivatives up to the order 2k. The coefficients of these forms are smooth real functions. (ii) The funct.ton to has the form
m o a $1 (iii) For any natural N the equality
* 0 2.
(39)
91 .
N-1
T(g202)T(g1,41)
T(93,
£
(-i)mJ .)
(40)
0 j
221
holds modulo operators, acting from LAX
(Rn) to LS-N+ml+m2+n +
j X
n
(X - 9 )(Rn), for any S.
The sign + or - in (40) depends on the choice of the branch of the argument of the density of canonical measure on graphs r(gl), r(g2), and r(g) (recall that for a connected simply connected Lagrangian manifold there exist exactly two ways of choosing the concordant family of arguments of the measure of density). Before proving the theorem, we wish to concretize the further Remark 2. The point is that without restrict'--ions on canonical atlases mentioned above. additional assumptions even the function (39) may not belong to toDm1+m2(g3). However, such a situation is rather pathological; in practically interesting
cases the space pm(g) merely coincides* with the set Tm, functions f e Tm (R2n) such that (Xl,....Xn)
IpAX-1I
,
(R2n) of.
is bounded on supp f
(here we assume in our notation that g acts on elements of T RX, where TX - (TXlx
TXnx
T,Xl,
).
n IXn)(R2n) into 1
....
Since conditions of Definition 2 imply that g takes kXII....X we see that additional conditions
mentioned are necessarily satisfied in these cases. coincides with T (Xml, ....
To ensure that Vm(g)
(R2n) it suffices to require that:
Xn'
)
(1) As before, the number of charts in a proper canonical atlas is finite; (2) in any chart the Jacobian of canonical coordinates with respect to "standard" coordinates (y,g) is greater than some positive constant on any set of the form (7). Thus, (2) is an example of additional conditions under which Theorem 2 is necessarily true. Proof of Theorem 2. Using the partition of unity and transition operators Vjkk (see Theorem 1 of item B), we reduce the theorem to the case when only one canonical chart is employed on each of manifolds r(gi), i.' 1, The proof is based on the stationary phase method; as before, the critical 2,3. point is to verify the applicability of this method, since the formula (40) In itself is not new and unknown from the purely computational viewpoint. fact it was established by Hdrmander (293 (in other notations) for Fourier integral operators; irt the 1/h-case the method of the Cauchy problem was used
in [39] to prove this result - see also Section 2 of the present chapter.
Thus, taking into account that there is lack of space to repeat the known arguments, we restrict ourselves only to demonstrating the applicability of the stationary phase method. To establish (40), we need to calculate the asymptotic expansion of the kernel of the product of operators on the left-hand side of (40). According to previous considerations, we assume that the kernel of T(gl,ol) has, in obvious notations, the form Xl(Y.x)feSl(Y.xl.P iC 7 i )-x i p ial(x,Yl.pi)dpi.
K,(Y.x) a
(41)
and the kernel of T(g2,02) has the form X2(v,y)feiCS2(v.YT.gT)-YTgTIa2(9,v.YTgT)dqT.
K2(v,y) -
* See Remark 1.
222
(42)
Here al,a2 are the total amplitude functions (including square roots of Thus, the the measure density), and Xl,X2 are the cut-off functions. has the form: kernel of the product K3(v,x) = JK2(v,Y)K1(Y,x)dY = fffeitS1(Y,xi,Pi)-xIp_+S2(v,yT,gT)-y_gr]X1(Y,x) (43)
x
=
x X2(v,Y)a1(x,Yi,Pi)a2(v,YT,gT)dPldgTdy (note that (43) is an integral over a finite domain due to our construction of the cut-off functions). To establish (40) we need to consider the partial Fourier transformation Fx
-
p-(K3(v,x))
k
ffflei[S1(Y,xi,ni)+S2(v,YT,gj)-xini yjgj+kpkIxl(Y,x) x
(44)
x X2(v,y)al(x,yi,ni)a2(v,yT,g3)dpidg3dydxk, where as above the integral is taken over the finite domain. (44) the variable change: v
.
Av ( )eJz.1)
Perform in
,
j E K,
xj = A(v)pj4j(1)
yj =
pJ = nj =
qj '
A(v)1-Xjz;2),
j E K,
(45)
A(v)1-1jzj3)
A(v)1-uj4;4)
j E
xj =
After obvious transformations, we obtain Fxk
Pk(K3(v,x)) _ (46)
where z = (z(1),z(2) ), 4 = (4(1),...,4(5)). the functions (D(z,4) and A(z,4,A) are smooth and bounded uniformly with respect to A with all the (z,4)-derivatives. Also $ is real-valued and the integration in (46) is performed over a bounded domain whose size does not depend on z and A. Application of the stationary phase method to (46) is therefore valid and after prolonged computations, we come eventually to (40).
Then, as an easy consequence of Theorem 1, we obtain the following:
223
The operators T(g,$) satisfy the property: if K E=
Proposition 4. Ess (w), then
g(K (1 supp $) E Ess(T(g,$)0)
(47)
If $ Proposition 5. This proposition is somewhat more complicated. is a CRF associated with a proper Lagrangian manifold L, then T(g,$)lp is a CRF associated with the proper Lagrangian manifold g(L). The proof is based on routine application of the stationary phase method.
Now we may define the operator T(g) corresponding to a proper R+-invariant canonical transformation satisfying the additional conditions described previously.
The operator
Definition 4.
0-
0-
T(g)
:
n L(ui,.... un)(Ry
(48)
L(al,...,an)(Rx
Om
is defined as follows: for any $ E L g(y,q)
6 T(U
(R2n)
(Rn)
(U1....,Un)
choose the function
Y
such that $(y,q) - 1 in the neighborhood of some
1, ..., 11n)
set K C Ess(p) of the form K = {(y,q)I!
(49)
S CA(y)1-Uj, j - 1,...,n}, I
$ may be interpreted as an element of 00(g) (see Remark 1).
T(g)4' Proposition 6.
Modulo L-"
deaf
Set
T(g,$)' .
(al,...n
(50)
)(Rn), the operator T(g) is a linear x
one and the result of action T(g) on y in (49) does not depend on the choice of the function $, satisfying the conditions of Definition 4. Proof.
Then
Let $1,$2 be two functions satisfying these conditions.
we haye 1
$ _
(51)
3(y,-i ay),y(mod L(Ul,...,pn)(R1)) -Un ) with $j a 1 on supp $3, j - 1,2.
for suitable $3 E TIu
Using
1
Theorem 1, we obtain immediately def
T(g,$1)$
T(g,$)'$3* =T(g,$2)$3V = T(g,$2)*
(all the equalities are valid modulo L-" 6 is proved.
(Rn)).
(52)
Thus Proposition
Thus we have defined the operator T(g), which is the "quantization" It should be emphasized that T(g) of the canonical transformation g. depends essentially (except for the "principal term") on the choice of the partition of unity on the graph r(g). In the sequel we sometimes work with the families of canonical transformations, families of functions $ to which the operators T(g) should be applied, etc. In all these cases we assume that the representatives of our operators may be chosen and that they are chosen in such a way that the function $ in (51) and the elements of the partition of unity depend smoothly on the parameters involved.
224
Before formulating the theorem on composition of "quantized" canonical transformations, we enlarge the class of admissible symbols of pseudo-
0
differential operators acting in the space L(X
'n
1
0,
(al,...,an)
)(Rn).
Denote by
(R2n) the space of smooth functions t(x,p) satisfying estimates:
a+glm-E ?. .a.+E.(a.-1)B. axaa(3.P) 1
J
5 Ca6MA()
J
J
J
J
J
(53)
P
for
is >Isi = 0,1,2,....
M;
Ipj;l(x)XJ-lI
(54)
Thus we do not impose any restrictions on the rate of growth of f as 0 j Ip.A(x)Aj
lI
e L (al.
M
(X1,...An)
0 Xn)
(R2n).
= U Tm
We set T( X
,
(All ....An)
Let
For some C > 0, the set
(R2n).
K = ((x,p)IIPjI 5 CA(x)1-XJ, j = I,..-,n}
0
,Xn)(R2n)
If f E T(al
belongs to Ess k,.
(55)
we clearly may choose the
An)(Rn) such that f - fo = 0 on K (make use of the
function f 6 T(All ....
cut-off function of the form X(p1A(x)XI-1,...,pnA(x)Xn 1)). 1
def
2
4w = f(x,-i
We set
1
2
fo(x,-i a-YP x).
(56)
)(R2n) does not depend oD"the choice n of function fo. It is also clear that the composition theorem, such as Theorem 2 of item B and Theorem 1of the present item, remain valid for this extended set of pseudo-differential operators if we consider all the Clearly fy(x) modulo L(711
operators to act only on
E L(
(R2n 1
n
0
The symbol space T
(al,...,an)
it admits "asymptotic summation." is valid:
(R2n) possesses the useful property:
More precisely, the following proposition
"mj
Let fj(x,p) E T (A
Lemma J.
.
as j There exist = max in., such that j J N f(x,p) -
(R2n), j = 0,1,2,..., where m. + 2n X )(Rwhere m
f(x,p) E Tm (X 1' "' '
=
n
max m.
E f.(x.P) E T(x'3l j=0 1
.
.,A
)(R2n)
(57)
n
for any natural N. Proof. This is a variant of the famous Borel lemma, proved for the classical PDO in 130]. The function f(x,p) may be taken in the form
f(x,p) = J!O(l-X(cjA())fj(x,P),
(58)
where X E Co(R1) equals 1 in the vicinity of zero, and the positive numbers cj tend to zero rapidly enough to ensure the validity of estimates
225
(53) and inclusions (57) (note that the sum (58) is finite for any fixed The details of the choice of tj are completely analogous to those in 1301 and are omitted. x).
The property established in Lemma 3 enables us to obtain the asymptotic composition formulas to within operators whose image lies in the space Xn)(R2n)
rather than in L(Nl1
L(X1,
fixed previously.
Xn)(R2n) for N arbitrary, but
For example, the following theorem holds: *
T*Rn and g
*
n
n
a T R be canonical v transformations satisfying all the above conditions together with their composition g2o gl. Then Theorem 3.
Let g
T*Rn
:
x
1
y
:
2
T R
y
T(g2) o T(gl) _ +_ (1 + R)T(g2 o gl) ,
0 modulo operators acting from L (X1,...,Xn)
(Rn) to L x
(59)
(ell ...8n)
(Rn), where v
the sign + or - in (59) depends on the choice of the branch of the square root of the measure density on Lagrangian manifolds r(g1), F(g2), andF(glg2) 2
v) is a pseudo-differential operator with the symbol
and R = R((v,-i 0
R(v,{) E T-1
(O1, .O n) (R2n)
which has the following asymptotic expansion:
R(v,{) =
£ (-i)3R.(v,{); j=1
R13)
Tj
,...,0
(60)
3
)(R2n) is a real symbol satisfying n
Rj(TOv,t1-O{) = T 3Rj(v,{) for large A(v),
(61)
and the sign = in (60) is used to denote that the difference between R and the sum of N terms of the series belongs to Proof.
T-N-1
(R2n
(81'...,en)
Using Theorem 2, we come to the problem of solving the equation
(for any 'y E L(X
(Rn
X
n
1
(1 +R)T(92 0 gl)V) = T(g2 0
(62) 0 X
with respect to k, where ,+ is a certain element of T(X 1
pendent of iy,
and X is a cut-off function with (X - 11 E Ess(ip).
more, 0 is asymptotically quasi-homogeneous, _
E T(a1,...Xn)(R2n)' Set H = 1 + R. the form
)(R2n) inde-
' n
4j(TXx,Tl-Xp)
Further-
£ ., where 0j E
_ j=0 = T 3$.(x,p) for large A(x), and00 - 1.
Using Theorem t, we obtain the system of equations on R. of
0(g7o g1)*(Rj) = Fj1q,R1,.... Rj-13,
(63)
where F. is a given differential expression. This system may be solved recurreftly since 40 = 1. Applying Lemma 3, we obtain (62) and henceforth (59). Theorem 3 is proved.
226
We intend to finish this item with consideration of the case when the canonical transformation is defined on some R+ invariant open subset U C C T*Rn. x
Let g
:
(64)
U -,, T*Rn
be such a transformation.
set U
We assume that g is defined on (or may be extended to) some large open U such that there exists a function 0UU with the properties*: I
UU E
T(
ODU
1 in U; suPP UU C U.
(65)
The formula (51) is then modified and reads
T(g),P where
dif T(g.m o fUU)*,
is chosen as in Definition 4.
Om
2n
l )(R
p E L(Nil
. n
(66)
The formula (66) makes sense for any
); however, the result is independent of the choice of
Om
)(U) where
0U6 only for y 6 L(1 n
1
01
L(Nil ...,X) (U) -
(here m E R U {m}).
(y, E Om L(Nil ...INn)(Rn)13K C U: K 6 Ess(V+)}
(67)
Let now gl - U1 - T*Rn, U1 C T*Rn and g2 : U2 - T*Rn,
U2 C T*Ry be the canonical transformations satisfying the above conditions together with T*RV.
g2o B1
:
(68)
U1 n g1-l(U2) ~
The statement of Theorem 3 remains valid if both sides of (59) are con1
sidered as operators acting on elements of the space L
(Nil ...,an)
(U 1
fl g x 1
x (U2)) D.
The Canonical Sheaf F+
In Section 2 of the present chapter we defined the sheaf of rapidly oscillating functions on the symplectic manifold M in terms of factorIt seems to be spaces, their elements being the equivalence classes, etc. the most convenient way in view to clarify the exposition as far as possible. Here we choose another approach. We work in terms of local representatives rather than their equivalence classes and construct only the space of sections of the sheaf over the whole symplectic manifold M. This is because we have in mind our general aim, to apply the described results to the theory of operator equations. There is no other way (known to the authors) to do this except working with representatives, since we shall have to substitute in the latter the tuples of non-commuting operators instead of their arguments.
On the other hand, the experience induced by the material of Section 2 enables the reader to think over the homogeneous situation from a different viewpoint which, however, appears to be ill-adapted to the mentioned applications. We begin with the notion of a proper symplectic R4-manifold. * In this case (U,U) will be called a proper pair.
227
M i= a proper symplectic R+-manifold if and only if
Definition 1.
the following co::li:.ior-s are satisfied:
(a) The fret infinitely differenti:lble action of the group R+ on M is given such tha. Lie set of orbits M/R, admits the structure of a smooth MIR+ is a smooth mapping. manifold for which the projection M (bi T.,
hrmogenecu
closed non-degi:nerate dilforential 2-form w2 on M is given, of degree one:
Tu:1 = ts,z, : E R+.
(1)
(c' The smooth non-vanishing function AM, homogeneous of degree one, is given:
AM(TZ) = TAM(z), t E R+, z E M.
(2)
AM will he called the weight function on M. (d) The finite atlas M
N Uo ua of connected shrinkable coordinate a=L
charts (ua,u,ti
:
ua >Da C R2n}al is given* with the properties:
(d1) For each a the non-zero n-tuple (A1"
" 'An) _ (A1(a)'"
n(n))
is a homogeneous canonical transa " on R2n, defined by this tuple: formation with respect to action of R+
of non-negative numbers is given and
Ua(dp(a) A dx(a)) = w2,
(3)
(TZ) = (4)
T E R+, a E ua.
T
Here (xln''"'x
(a)'pl'>"'pn
)) are the coordinates in RZn : Da, (xla)(z),
xna)(z),p(a)(z),...,pna)(z)) are, the components of the mapping aG. *
(d2) For each a the function AM(z) and uat:a(z), where Aa(x(r)) is
constructed in correpondence-with the tuple, are equivalent on a in the sense that there exist positive constants c and C such that for foAM(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
<
E IYjXJ - Ej e
tcv)
i(1-aj)Yj
We choose arbitrarily and fix the argument arg VIxia),I p))
for
any
chart of the canonical atlas and define the elementary canonical operator, Ka
242
V (Va) -
F+(M)
(21)
by means of the formula is (x(a),p(a))
Kom - jo(a)
(a)fe
p_
.,x_
I
I
I
I
I
(Aa1(x(a))) x (22)
X
I
V0D(VO).
I
Here va is the coordinate. mapping in the canonical chart Va; the expression in outer braces is a usual elementary canonical operator (see Definition 4 of Section 3:B), 0-
ja :L a , F+(M),
(23)
Om
where a - a(o) is a natural mapping, taking $L
into the equivalence class of too}, where :ys - 0 for S f a, 4a - 4y. The application of the stationary phase method immediately yields the following: Theorem 1.
For any non-empty intersection Vafl Va
there exists an
integer Haa, and a sequence of differential operators voo,j, j - 0,1,2, in Vafl Vo, with the following properties: The operator is a differential op4Fator of order 4 2j with the vaa,j real coefficients acting in the spaces i)
voo,j
Vm(Vo fl Vo,) + vm'i(Vofl V0,)
:
(24)
for any m and homogeneous of degree -j (i.e., decreasing the degree of homogeneity by j). ii)
If we denote by voo, the asymptotic sum* voo, =
E (-i)3vao,., j.0
(25)
then for any $EV (Vafl Va,) Ka,# - exp(iiHaa,)Ka(vaa,4).
(26)
iii) The operators vaa, satisfy the cocyclicity condition vGo'va,oo = °ao" in Va fl V0, fl y0 (27)
if vafl va, flva 0 0 and for the principal term vaa,o we have
voo'o
0 in V1. flVo,
(28)
for any a,o' with Va fl Va, iv)
Hoo, is a cocycle modulo 2 on L.
Proof of Theorem 1. This is clear from our previous discussions (see Theorem 1 of Section 3:B; Proposition 5 of Section 3:C).
Or The asymptotic summation-follows along the lines of Lemma 3 of Section 3:C.
243
Definition'4.
L is called quantized if
is a coboundary modulo
2 an L (and henceforth the branches of arg u1(xia),p(a)) in different I
charts may be chosen concordant so that factor (irHoo,) in (25) is eliminated). Remark 1. Whether L will be quantized or not depends on two factors: first, on the choice of the measureu on L (more precisely on the equivalence class of u with respect to the equivalence relation: ul ti u2 if and only if uI/u2 : L -
t\ {0) is a mapping, homotopic to a constant one), and
second, on the choice of concordant branches of the argument when defining We do not go the transition operators TnS of the canonical sheaf F+(M). into further detail here; see Section 2 and also 138], 155]. Consider a quantized proper Lagrangian manifold L C M (we assume, consequently, that the atlas {Vo} and the measure V are given and fixed). - 0) and We assume that the concordant branches of arg ul are chosen (H cat
that the-partition of unity (eo) is chosen subordinate to the covering {Vo} such that eo E DO(L) for all a. We define the canonical operator on L, K
.
D-(L) + F+(M)
(29)
by the equality K$
d`f
LoK0(eo+). 4 E V%).
(30)
Theorem 2. For any "classical" pseudo-differential operator H E Ts(H) the commutation formula is valid
HK# - KPH,
(31)
where the operator P is the asymptotic sum P =
E (-i)JP., j-0 J
(32)
and the operators Pj possess the following properties: i)
Pj is a differential operator of order 42j on L, acting in the spaces P.
Dm(L) - Vm+s-j(L)
:
(33)
for any S, homogeneous of order m - j.
The coefficients of the operator P7 are linear forms with the smooth real coefficients of terms of the asymptotic expansion of the symbol Ha and its derivatives up to order 2j.* ii)
iii) The operator P. coincides with the restriction of the principal symbol of the operator H on L. iv)
If the pair (L,u) is associated with H in the sense that
hoIL - 0
(34)
and L
V(ho)u - 0,
(35)
* This is valid for any coordinate chart Ua, containing the considered point of L. 244
where LV(h0) is a Lie derivative along the trajectory of the Hamiltonian vector field* V(ho) J w - -dho,
(36)
then Po - 0 and P.
= V(h0) + F[ho,hsub]
(37)
where FCho,haubI is a linear form with the smooth real coefficients of haub and the derivatives of ho of order < 2, restricted on L. Proof.
reduces the the "local" here is the argument is
Employing the unity partition {eo} and the operators voaf
problem by virtue of our remarks about essential subsets to case considered in Theorem 2 of Section 3:B. The only novelty asymptotic sum (32). This (purely technical) point of the treated similarly to Lemma 3 of Section 3:C.
Corollary. If * K, where K is a canonical operator on a proper quantized Lagrangian manifold L C M, then L C Ess(4+).
The elements y E F+(M) of the form q - K@, where K is a canonical operator on L, will be called the canonically representable functions (CRP's), associated with the proper quantized Lagrangian manifold L. Solution of the Cauchy Problem for a Pseudo-Differential Equation in +(M) D.
In F+(M) we consider the Cauchy problem (-i 2t + H(t))iy - 0,
(1)
(0) _ 'o e F+(M)
(2)
on the segment [0,T3, where H(t) E T1(M) is a "classical" pseudo-differential operator in F+(M), (+o e F+(M) is a CRF associated with a proper quantized Lagrangian manifold L. Let ho(z,t), where z is a point of M, be the principal symbol of A (homogeneous of degree 1) and hsub(z,t) be the subprincipal symbol of R (homogeneous of degree 0). Consider first the case when ho(z,t) is real-valued. Then the procedure of finding a solution to (1) - (2) is very familiar We seek the solution to (1) -
(from experience in various other situations). (2) in the form
(3)
V+ = KO,
where K is a canonical operator on the Lagrangian manifold L C :R = Mx T*Rt, which is constructed in the following way. Let 'Yo = Kooo,
where Ko is a canonical operator on the Lagrangian manifold Lo C M. consider the Hamiltonian system
(4)
Now
(2ho(z,t)/2t) V(ho)(z,t)
(5)
* (35) makes sense since under the condition (34) the field is tangent to L.
245
(here E is the momentum dual to t) on M with the initial data (to,Eo,zo) - (0,h0(zo,0),zo), zo E Lo.
(6)
The set of functions of system (5) with the initial data (6) for various zo a Lo forms a Lagrangian manifold L C M. We define the measure on L, setting p - yo A dt
(7)
in coordinates (zo,t) on L, where uo is the given measure on Lo. easy to verify that
(8)
(ho+E)IL - 0' V(ho+E)P - 0; that is, L is associated with the operator E + H - -i
It is
(9) at
+ H.
We impose the requirement that L be a proper Lagrangian manifold (generally this may not be the case since the Hamiltonian system (5) does not necessarily preserve the inequality (in local coordinates)) IPjI < CA(x)
1-Xj
(10)
Since L is shrinkable to Lo, L is necessarily quantized if Lo is. Thus the form (3) may be employed, and we may choose in such a way that (K4)(0) - Koo(0)
(11)
for any 0 E U°'(L). Substituting (3) into (1) and using Theorem 2 of item C, we obtain the following asymptotic problem for :
PO - 0, 4(0) - Ao
(12) (13)
This system is easily aplved since the principal term of the operator P has the form (in coordinates (zo,t) on L): P1 - 2t + F(zo,t),
(14)
where F(so,t) is a given function.
We require that the solution of (12) - (13) belong to Om(L) (again this may not be the case since in pathological cases for some values of t, infinitely many canonical charts on L, intersecting with the trajectories of the Hamiltonian vector field V(ho + E) coming from supp o, may occur). If our requirements are satisfied, we readily obtain the solution to the problem (1) - (2) on the segment (0,T]. Consider now the case when the principal symbol is essentially complexvalued. The solution then would be given by a canonical operator on a Lagrangian manifold with the complex germ. In this book we.have no space to present its construction in R+quasi-homogeneous case. Thi/ construction is a result of complicated synthesis of the ideas, used here in the case of a real Lagrangian manifold and the theory of complex germ (527 (see also 154,59,613). Fortunately, the existence theorem for this case, which will .be used in Chapter 4, may be formulated in terms of real geometrical objects and afew additional notions should be introduced to give the formulation.
246
We assume as before that the initial data have the form (4), i.e., o (Hoever, this
is a CRF associated with the realLagrangian manifold Lo C M. restriction is not essential.) Denote
H(z,t) - Reho(z,t), (15)
H(z,t) - Imho(z,t)o H(z,t) and H(z,t) are homogeneous functions of degree 1.
For any K C M and e > 0, denote by Ut(K) the subset in M consisting of all points z such that there is zo - zo(z) E K so that: z,z r= U. for some a - a(z), and the points (x(°) ,p(°`)) - %(z) and (xoa),p0 satisfy
) - Vc(zo)
xoj)I ( rAa(xoa)Aja) (16)
( tAa(xoa))1+A3(a), j - 1....,n.
IPJa) - Poj
Condition 1.
For some t > 0 the inequality is valid: H(z,t) (0 for z C- Ut(L0), t C -10,C].
Condition 2.
(17)
Set Ar - Ut({z r= MI z e Lo,H(z,0) - 0}).
(16)
Trajectories z(zo,t) of a (non-autonomous) Hamiltonian system z(zo,t) - V(H(z(zo,t),t)) exist
for t E [0,T3, zo E Ot.
(19)
There is a constant C such that CAa(x(a)(z(zo,t))1-aj(a)r
IP)(z(zo,t))I (
j - 1,...,n
(20)
on these trajectories for any a, such that z(zo,t) E U. Also H(z(zo,t),t) ( 0
(21)
on the trajectories. Condition 3.
For any t E [0,T] Ot E D_(L(t)),
(22)
where mt is the solution of the transport equation, corresponding to H(z,t)- Reh9(z,t), and L(t) is the shift of Lo along the trajectories (19) during the time t. Condition 4.
For t - T, we have H(z(zo,T),T) -c -E
(23)
for z0E Rt. Theorem 1. Let Conditions 1 - 3 be satisfied. Then there exists a solution' y+ of the problem (1) - (2). If in addition Condition 4 is satis-
fied, this solution satisfies the condition 4(T) - 0.
(24)
247
Remark 1. The solution is given by a canonical operator on a Lagrangian manifold with a complex germ, obtained by the construction somewhat
similar-to that in the case of a real-valued symbol. See [52] for the theory of the canonical operator on a Lagrangian manifold with a complex germ.
Remark 2. Conditions 1 - 4 are called the absorption conditions (for the init manifold L. and given e > 0). liar
248
Iv
Quasi-inversion theorem for functions of a tuple of non-commuting operators 1.
EQUATIONS WITH COEFFICIENTS GROWING AT INFINITY We consider in this section the quasi-inversion theorem (presented in
general form in the subsequent sections) in relation to a particular problem,
namely, to partial differential equations in Pn, whose coefficients may have polynomial growth as jxj -* m. Numerous papers were devoted to the "elliptic" case, when the principal symbol of the operator in question is a non-vanishing function of (x,p),,e.g., homogeneous in (x,p) of some The degree. However, the non-elliptic case remained uninvestigated. quasi-inversion theorem is the very tool that enables us to consider it and to construct asymptotic solutions of the differential equation such that the consequent terms of asymptotics (and, respectively, error terms on the right-hand side of the equation)become more and more smooth and decay more rapidly at infinity.
A single example is presented in item A, while the general equation with coefficients growing at infinity is considered in item B. A.
Model Example
Consider the following example. of the form: [Lu](x,t) =
a2u - as2u.+ at2
Let anfequation be given in tt2 =3(x,t)
c(x,t)x2mu(x,t) - 0,
(1)
- ul(x),
(2)
ax2
with initial data ult-O - uo(x),
au/stir,,
where m is a positive integer, c(x,t) is a smooth function, bounded with all its derivatives and satisfying c(x,t) > c > 0, and the initial data uo(x) and ul(x) are tempered distributions in the space W.
We seek the solution of
(1) - (2), asymptotic in the following
sense:
Definition 1. The functional sequence uN(x,t), N - 1,2,..., is an solution of (1) - (2) on the interval 10,T) if, for any N, (2) is satisfied and asymptotiT
249
sup
<
llxk(-i l )rLuN II
f
tE(O,T) k+r4N
(3)
L2(I11)
More precisely, in the situation described we shall speak of {uN) as an asymptotic solution with respect to powers of operators x and (-i ax)'
For m - 1, the asymptotics in the above sense were constructed for (1) - (2) in 1521 by reduction to a system of first order. In what follows, we denote
c(x,t).
C0 = inf
(4)
(x, t)EIR x CO,T] By assumption co is strictly positive. To solve (1) - (2) asymptotically, we set 1
3
2
1
uN(x,t) = GNO (A1,A2,B,t)uo(x) +G
N,l
3
2
(A1'A2'B,t)u1 W,
(5)
where Ai, i = 1,2. and B are the self-adjoint operators in L2(R1):
Al = -i(a/ax), A2 = x, B = x,
(6)
satisfying the commutation relations CA1,A21 - CA1,B] - -i; CA2,B] - 0, and GN,i(yl'y2'y3't), i - 0,1, are symbols to be determined.
(7)
The equation
in question may be written in the form -(-i 2t)2u +{A1+ c(B,C)A2m}u - 0.
(8)
The major sense of introducing the operator B A is that we can explicitly separate the polynomial growth at infinity powers of A2), while the remaining coefficients are required to be bounded functions of the operator B. Then it is sufficient to construct an asymptotics with respect to powers of Al and A2, uniform with respect to B. The left regular 2
1
3
representation for the tuple (A1,A2,B) has the form aaye
L1 =
yl - i
- i aay3
L2 = LA2 - y2' L3 = LB - y3
,
(9)
LA1
(see Chapter 2, Section 4 for the definition and the method of evaluation of regular representations. In our particular case the calculation is rather simple and therefore left to the reader.) Theorem 1. For any natural N there exists a natdral N - N (N) such 1 1 that the the estimates a°ay
N1,
C(1+ ly11 + Iy2I)-N1+I°l, I., -
)
(10)
valid for the symbol T(yl,y2,y3), imply the operator norm estimates E
k+r 4N
250
k
a
r
1
2
3
x (-i aX) T(A1,A2,B)
<
L2-L2
(11)
Proof.
Then, once k + r C N,
N being fixed, let N1 be large enough.
n the function k2(yl-
(12)
i ay2)rT(Yl,Y2.Y3)
fkr(Yl,Y2,Y3) ' satisfies as1
3f
a2r
ala
a yl
By Theorem I of Chapter 2, Section 4:F, the operator
for al + o2 S 3. 1
(13)
(yl,y.2,y3 )1 S c(l + lyll + IY21)-3
y2
2
3
1
2
3
k
fkr(A1,A2,B) _ (L_LrT)(A1,A2,B)
r
a
2
1
3
ax) T(A1,A2.B)
(14)
is bounded in L2(R), and the theorem is proved.
Next we derive the equations to determine symbol: GN i(y,t). We obtain through substituting (5) into (8) and using the left regular representation operators (9): -(-i at)2GN
ay2 - i ay 3)2 +c(y3,t)yZm)GN,i ' RN,i(Y,t),
+ {(Yl - i
(15)
with initial conditions GN,i(Y,0)
6io,
aaN,i
(y:0) = 6i1
(16)
(here 6ij is theKronecker delta). Symbols RN i(y,t) in (15) are the arbi-
trary ones, satisfying the estimate (11) uniformly with respect to tE[O,T]. Next we perform the change of variables in (15), depending on the parameter A > 0. Namely, yl
- Ax1, Y2
A
1/m
(17)
x2, y3 ' x3
Dividing by A2, we obtain {(xl+X-1/m(-iA-1
-(-i1-1 at)2gNi(x,A,t)+
ax2
- ix-1 2 )2 + c3(x,t)x2m}g.
(18)
(x,A,t) ' A-zrN .
3
where small letters denote expressions of functions, denoted by capitals, in the variables (x,A,t). We construct below an asymptotic solution for A + - of (18) with initial data (16) and next show that this asymptotic leads to functions GN,i(y,t) satisfying the desired estimates. Thus, consider th8 equation -
t )2y(x,A,t) + {(x1 + -(-iA-1 a
A-1/m(-iA-1
aX ) 2
- is-1
(we write (x,A)
ax
)2 + c(x3,
(19)
0(A 8)
3
\ {0) and any
0(A s) iff for any compact set K C :k2 x
1x2)
multi-index a - (a1,02,a3) there exists a constant Cindependent of x3, such that
251
8+
a
axnlax2axa3
(20)
K'°
for (xi,x2) E K, a > 1). The solution of (19) is sought in the form of alinear combination of functions s-1
iAS(x,a-1/m,t)
P(x.a,t) - e
(-ix)
E
-k -1/m 4k(x,x ,t),
(21)
k-0 where S(x,E,t) and +k(x,E,t) are smooth functions in all their arguments, and S is a real function. Substituting the function (21) into equation (19), we obtain
{-Cat - i1-1 2t]2 + (xl + r ax + axS 2 3 + c(x3,t)x2
}s-1.s
iEa-1 ax
-
is-1 X )2 +
2
(-il)-k$k(x.E,t)
3
-
(22)
0(a-3),
E
k-0
where e - a-1/m, or after opening brackets and collecting the terms with equal powers of a and E,
{{C-(a!)2+(x +as )2+c(x t)x2m]+2E aS (x1 +as) + at i ax3 3' 2 .:'2 ax3 + C2(2S )2}-is-1{[-2 aS 2-+ 2(x +as) ax2 at at ax3 1 as ax3
+ 2c[ (x + 1
+
222
ax2
+ as
a
a
ax2 ax3
ax2
]}-a2{[- a2 +a2 at2
+
]+2E
ax3
E
-
a2S+a2s at2 ax2
]+
a2s as a + ] + E212 ax2ax3 ax2 ax2
(-ix)-k *k(x,c,t)
x
a
ax3
a2 ax2ax3
+E2
2
(23)
}} x
ax2
- 00'-s).
k-0
The notation f - 0(A 8) means that the function f is locally uniformly adecreasing as a + m together with all its derivatives as .
The equations for S and k follow from (23). These equations are somewhat different in cases m - l,m -2, and m 3, and we present all these cases below: (a) m - 1 (i.e., c - a-1); then we have
at)z
LS_)
- (xl + ax 3
+ c(x3't)x2
(24)
(Hamilton-Jacobi equation);
as {-2 aS 2-+ at a2(x1+ax3)
a a2s as as )} aX3 -as te+aa2+2i aY (x1+X
2
3
X Qk (x, t) + {-
252
a
ate
+
a2
ax3
+ 2i[ (x1 + ax
3
3
ax2 + ax2 ax3 +
x
as a2s 3-(ax2 + ax2ax3
a a2s ax2 ax2 + axe 3 +
aS
{-C2
)y
}+k-1(x' t) +
(25)
a2+-3 2
+ 2i axaax 2
}+k-2(x,t)
- 0,.k - 0,1,...,s-1
2
-
3
ax2
(transport equation; for the sake of convenience the notation +p(x,t) = 0 is used for f < 0). a-1'2), then we have
(b) m - 2 (i.e., c
as
e2A(x,e,t)
+ (xl + ax3)2 + c(x3.Ox2 +2c ax2 (xl + ax3)
(26)
(Hamilton-Jacobi equation); {C-2
,IS
at at+2(x1+ax3)
ax
t2
3
2
+2+iA(x,e,t) 3x3
i(ax2)23+2eC(x1+ax3) ax2+3x2
ax3 (27)
a
2
+a
2t.
2
ax3
-2i
as
-i22S 3+2c
a
ax2 ax2
a
2
ax2ax3
ax2
}+k-l(x,e,t) -
a2+
- it2
k2-
(x,c,t) +iak-l(x,t,t) ° '-2akkx,e,t), k - 0,1,...,s-1
ax2
!C
(transport equation).
3, then we have
(c) m
-(3t)2 + (xl
+2-S 3
)2 +c(x3,t)x4m+ 2e as2 (xl +
as
)+ 3
(28)
+ C2(-BS )2 - CM 2
(Hamilton-Jacobi equation);
([-2 at at+2(x1+az) 3
32S ax
at
3
2
a
a as ax3 + ax2ax3 + 2e { (xl+ 'S ax3) ax2 +ax2
3 + c212
+
a2
3.2
IA(x,e,t)3 +
3
'S ax axa + 2
2
a?2 7}+k(x,e,t) + a x2
(29)
+ {[-
a2
a2
3 + 2e
at2 + ax23
a2
ax ax 2
+ e2 3
32
+iak-1(x'e't)
ax2 }+k-l(x'c't) 2
emak(x,c,t), k - 0,...,s-1 (transport equation).
In (26) - (29) A(x,c,t) and ak(x,c,t) are arbitrary
smooth functions, a_1(x,e,t)
def 0.
First we construct the solution in case (a). into two equations
Equation (24) splits
at + /(x1 + (aS/ax3)) + c(x3.t) 2 - 0
(30)
253
and
at -
(x1+ (as/ax3))2 + c(x3,t)x2 - 0.
(31)
We construct the solution S+ of equation (30) and the solution S_ of equation (31), satisfying the following initial conditions S+It-0 - S-It-0 -
0.
(32)
(As it will be shown below, these initial conditions agree with initial conditions (16) for functions gN i.) Equations (30) and (31) are Hamilton.
Jacobi equations with the Hamiltonian function H+(x,p,t) -
p3)2 + c(x3,t)x2.
(33)
Their solutions have the form [2): t
S+(x,t) - (Jo[p+(xo,T)H+p(x+(xo,T)P,(xo ,T'),T (34)
H+(x+(xo,T),P+(xo,T),T)dT}Ix 0
- x
0±
(x,T)'
In formula (34) x+(xO,T),p+(xO,T) are solutions of the Hamiltonian system, corresponding to the Hamiltonian function H+withinitial conditions x11. x0, PIT-0 ' 0 and x0 - xo+(x,t) is the solution of equation x - x+(xO,T). (We assume that the segment [0,T] is sufficiently small, so that then the Jacobian det(ax+/3x0) does not vanish, and this solution exists; the fact
that such T > 0 may be chosen is the consequence of homogeneity of functions H+ in (xl,x2) and of uniform boundedness of derivatives of c(x3,T) in x3.)
Write the Hamiltonian system corresponding to the Hamiltonian function H+. The variables x1 and x2 are parameters in equations (30) - (31), hence we
have non-trivial equations only for x3 and p3.
(Here and below we drop
indices ± and arguments x1 and x2 of x3 and p3.) These equations have the form x2(ac/ax3)(x3,T) xI+ P3 x3
2H+(x,P,T)
H+(x,P,t) ' P3
(35)
x3(0) - x3o, P3(0) - 0.
It is easily seen that the solution (x3,p3) of system (35) is a pair of homogeneous functions of degree O..end 1 respectively in variables xI and x2. Hence S+ is a homogeneous function of degree l in the same variables.
Now we turn to transport equation (25). It can be rewritten using equation (35) and the fact that H+ # 0 for x2 + x2 > 0 as follows:
2H+(x,P,t)(C8t+x3
ax
I+ft(x.t))$k+(x,t) + 3
+ RI±Ok-1± + R2±0k-2± + R3t0k-3+-
(36)
0, k - 0,1....,
where f+(x,t), Ri+, i - 1,2,3, are functions and differential operators, which can easily be calculated once the function S ±
254
is obtained.
Let
(37)
A+(x,t) = exp(-fof+di)
(the integral in (37) and below is taken along the trajectory of system Then the recurrent formulas (35) which meets the point x at time t). to-1H-1 I R (38) 0 d-0 0k+(x,t) = A+(X,t)(0ko+(xo(X,t)) 2 f ± + k-'+ jo
-
j=1
define the solution of transport equation (25) (here 0ko+(x) are initial data for functions 0k+(x,t)). , we see that the functions
Constructing the functions S+, 0k+
iXS+(x,t) S-1(
0+(x,X,t) = e
-iX)
£
-k
0k+(x,t)
(39)
k=0
are smooth for x2 + x2 0 0 and satisfy equation (19).
We set
c(x 3 ,o)x2)-1;
0(1)(x) =
(40)
0(O W 0, k = 1,2,...,s-1, kof
(41)
y(1)(x,X,t)
= 01)(x,X,t) + 4(1)(x,X,t).
The upper index in brackets means that the functions, which we construct, Then correspond to gN
'
1.
(1)
a4(1) It=o = °'
at
It-0 =
X.
(42)
We set (the number s = s(N) will be chosen below) gN 1(x,X,t) = (4)(1)(x,X,t)/X)X(x1X,x2X).
(43)
where X(x1,x2) = 0 is a neighborhood of zero in R2',
1-XECo(It2). By setting oo+(x) = 2 , 0ko)(x) = 0, k - 1,2,...,s-1,
(44)
we obtain for 0(0) - 0+0) + 4.0), 0
It=0
1'
at
to
We set gN,o(x'X't) = 1P
(°)(x,X,t)X(x1X,x2X)
(46)
It is easily seen that equations (24) - (25) are homogeneous with respect to (x1,x2), as well as the initial conditions (32), (40), (41), and (44) so that the obtained solutions can be written in the form gH,i(X,X,t) - GN,i(X1X,x2X.x3't), i - 0,1,
(47)
GN,i(y,t) = gN,i(Y,l,t).
(48)
where
255
The functions GN,i(y,t) satisfy initial conditions (16) up to smooth functions finite with respect to (yl,y2) and independent of Y3. If we subtract
these functions from CN i, we obtain on the right-hand side of (15) and additional term which is finite with respect to (y1,y2) and, consequently, satisfies the estimate (11). Thus we may assume the initial conditions to be satisfied precisely. Substituting functions GN i into equation (15), we obtain R
C
a
't)y2)rl N,i
ac)2GN,i +{(yl- i ay2 - i ay 32 +
(-i at)2 + (y1- i 0101
y2
- i as )2 + c(Y3,t)y2,X(Yl,Y2)7y(1)(Y,l,t) + y3
(49)
+ X(Y1,Y2){(--i tt)2 + (Y1- i aye - i y3+ c(Y3,t)y )V(i)(Y,1,t)) (in formula (49) the square brackets denote a commutator). The properties of function X yield that the first term on the right-hand side of (49) is a smooth function finite with respect to (yl,y2). Using (47), we rewrite the expression in curly brackets on the right-hand side of (49) as follows:
F(y)
def {i2(-(-i}
I
3C)2 + (x1
- ia' I ax )2+c(x3,t)x2) x
is-2 x 2
X V,(1)(x,a,t)}
3
x = 1
l
/k
=
x = y 1
l
=
x
1
Ix3=Y3
1= yl/a
y2/a x3=y3
x2 = Y2 /a
Ix 2 = Y2/\
(50)
0(X-s+2)
X20(X-1)
x2 =
x3-y3
since the functions (i) (x,a,t), satisfy the equation (19).
i 1,2, which have been constructed, For (yl,y2) C supp X, y2 + y2 > r2 > 0 holds,
so setting a = 4 (y2 + y2) 3 1, xi = yi/A, i - 1,2, x3 - y3, we obtain by (20) (choosing K = ((x1,x2)jxi + x2 - r} C t12 \{0}) that 2
Fa(y)
y2)-8+013+2,
(y2 + y2 3 1)
al 013 16 Ck.a(yl + ay1 y2 ay3
(51)
Hence, RN. also satisfies the estimate (51), probably with other constants C
k,a.
Choosing s - N1(N) + 2, we obtain by Theorem 1 that the function (5) with symbols GNO(y,t), 6N.1(y,t) constructed above solves the problem (1) - (2) in case (a).
Now consider the case (b) (the case (c) is quite analogous). Hasilton-Jacobi equation in case (b) has the form Ti- +
H±(x,p,c,t) - O(cm)
The e
(52)
rich the Hamiltonian
H+(x,p,c,t) (53)
H,(x,p,0,t)
256
Hot -
p3)2 + c(x3,t)x2
.
Let
Equatior. (52) can be solved by the successive approximations k
Sr = S.(x,t,e); 5,
(?k
' (x,t)
S,(x.t,c))ir=0;
at
then we obte:it: the following systota of equations, which allows us to find SW,
the functions
(x.t);
SW
0) + H
c+
nt
as
(1)
ax
+-off (x ' ap
it ±
aH
as+k')
as
(x,
(0)
as
t)
±
ax
ax
t) = 0,
-
(1)
ax
as+k)
as(O)
(0)
as+
aH
-+ac± (x
+
ap
(x'
ax
+ Fklx,t,
ax
't)
0,
(54)
as(k-1)
as(0)
o, _
at
' 0' t)
ax
..'
ax
ax
0,
]
where Lhe functions Fk can be easily obtained in a recurrent way. The solution of the first equation in (54) coincides with the above--constructed solution of equations (30) - (31) (one should only replace x2 by x2m). All subsequent equations in (54) are ordinary linear differential equations along the trajectories of the Hamiltonian system, corresponding to the Hamiltonian function H f; hence they can be solved by ordinary integration. Quite analogously the 'ransport equations can be solved by the methods of perturbation theory with respect to the small parameter r. Consider now some special cases. Let m - 1. If the function c(x3,t)= c(x3), i.e., it is independent of t, equations (35) can be integrated in a more ";,l;cit way. Namely, the Hamiltonian function H. is independent of t in this case and therefore it is constant along the trajectories of system (15) 12]. Differentiating the first equation with respect to t, we
obtain
--
x
-
-; 2(x2 + c(xao)x2) X2 c' (x )
--
2
±
xl
(0) = x ao ; x3(0) x3 3
(x2 + c(xao)x2)
;
(55)
hence
6, )2
x
x2c'(x3)dx3
x.+o xl
x2
c(xeo)x2
X2 + c(xao)X2
':'c(x ) = 1 - -- 2
(56)
3
,
x3(0)
xap.
XZ +c(XSO)y.
The sign of x3 coincides with the sign of x when t belongs to a neighborhood of zero, which depends on (x1,x2), but as xl - 0 the sign of x3 is opposite to the sign of c'(x,o). Equation (56) is an equation with separable variables and thus *an be integrated: dx3 (57)
(1 -
x2c (x ) 2 3
)
xl +c(Xao)x2
257
(the signs ± in (57) are, generally speaking, alternative on different seseents of trajectories and, to be more precise, one should know the concrete form of function c(x3); an example will be studied below). It is easily seen that c(x 3 )x 2
(pH, p - H+)(x.P.t) - xlx3 +
V'(x1
(58)
+ c(xse)x2)
hence St(x,t) - xI(x3 - xfe(x,t)) (59)
x2
t
J0c(x3(x'e(x.t).T))dr.
t
(xi + c(xsa(x,t))x2)
In particular, if c(x3) - cog x3 + 2, then x3 satisfies the equation of pendulum oscillations:
x3 +
X2 2
2(x2+x2(cos xfe+2))
sin x3 - 0, (60)
x3(0) - t x1/r'xi + x2(2 + cos xse)
x3(0) - x50,
with the parameter depending on x1, x2, and initial conditions.. a 1. Now consider the case c(x constructed asymptotic. with the usual ness. Note that the equation obtained by other methods and is used here only
In this case we shall compare the asymptotics with respect to smoothin this case can be solved precisely as an example.
The Hamilton-Jacobi equation has the form a38 act ± ((xl + x )2 + x2) - 0, S±It-0 - 0.
(61)
3
It has the solution
S±(x.t) - + t/xi+x2.
(62)
The transport equation (the solution of which is evidently independent of z3, so we omit the terms including derivatives in x3) has the form:
242
f
ato
258
Hence
x2 +x z
k-1 ±
)
k-2
a24 k-3
(63)
- O, k - 0.1....,
3x2
xxt 12
X12 +X2
0
0
0. (64)
xl x 2t
zi and so on.
-
(xi +X2)3/2
a _i
2ix1 2x - t2 2
1
1
or
atl
ate
+
tX2
+
2
(z a x)
Ok + {
a 22 ) (z 1 x
tz t (2
2
2
t 2N+sZ tt + 21
-+
4
2
1
1
2/(x2 + x2)
22
at2
+ 2ix
2
- t2 1 ax2
X22
xl +X2
)f 0 ,
X x
o(x,t)
1 2
o(x,0)exp{ 2 t2 X2
+ x2
1
t
m (x,t) - + f (
C-
1
at2
x2)
0
1
a? + 2ix
1 axe
xl+x2
exp{
So taking initial
0).
x 1
()(x,t)
(65)
c`)idi:,
(t`
and so on (here we used the fact that m1(x,0) conditions into consideration, we have 0±
]x
2
2
x2 + x2
x1x2
i
oo(x,7))eW:
x2
a
i t2 2
1X2
xl+x2 (0)(x,t) -
(66)
x x
1 2
2 eXp{ i t2 2
xi+x2 exp{ 2 t2
GN,1(Y,t) - X(Y1,y2)
21Y22 }
2
2
2
x(Y1.Y2)
x
x
expi
(y2+y2)
it y1y2
}x
(67)
2(y2+y2)
+y2) (+ lower-order terms);
x
it2yly2
GN,o(y,t) - x(yl,y2)exp(
y2) (+ lower-order terms).
2
2
2(Y1 +y2)
(68)
The usual asymptotics with respect to smoothness of the problem (1) - (2) can be constructed as follows (m - 1, ,.(x,t) = 1). We shall seek the asymptotic solution in the following form: I
u(x,t) - Go(-i
2
2
as
,x,t)uo(x) + G1(-i
2x
,x,t)u1(x).
(69)
The function Gi(yl,y2,t) satisfies the equation
-(-i ac)2Gi + { (y1 - i
'Y 2
);+ y2}Gi
R. >
(70)
where R is the symbol of the smoothing operator, or after the change of variables y1 - ax1, -(-ia-1
+i(x
is-1
Y2 = x2,
(71)
)i +a-2x2}gi - 0(a-s)
Ix
(72)
2
Seeking gi in the form of a linear combination of the functions ,P(x,a,t) - eiaS(x,t).
s-1 (-ia)-k.
L
k(x,t),
(73)
k-0
we obtain the Hamilton-Jacobi equation for S:
259
as
' (xl + az2)2 ,
c)2
and the transport equation for C_2
+ C-
at at at
a2S+ 2-S
-
a
ax2
2x2
1
at2
+
k
ax2
2
L2_ at a 2
k(x,t):
+ 2(x + aS )
aS a
(74)
(75)
22
+ - - x2]@k-1(x,t) = 0, k = 0,1,2,... 2
(recall that k - 0 for k - -1 as it was assumed before). equation (74) with zero initial condition, we obtain
By solving
S+(x,t) - ±tx1.
(76)
Then the transport equation becomes
+ aik +aX2 + X 1
C-
a20Z-1
2-
+
aZ-1 aX +
at
2
x20 k-1] ' 0
(77)
or amo
ado
+ax2
ak at
aok
(78)
- 0,
az k-l + a20 k-l +x
+
+ ax2 ' - x1
at2
2
-
2 k-1
ax2
].
(79)
2
1
By setting -o(lo*(x) - {
io;)(x,t) - + 2x
:
, we obtain
;o,)(x,t) - 2
(80)
.
1
Introducing new variables E - (x2 + t)/2, n - (x2 - t)/2, we rewrite equation (79) (for the upper sign) as follows: aan+
- - xl C (E + n)2 + -87C -an ]+k-l+
(81)
with the initial conditions ®k+lE-n - 0, k
1,2,...
.
Hence 0k+(E,n)
z 1
fn[(E + n) 2 - aEan E
Cx2 2x2
1+
3
k-l+
- (x2+L)3 3
k = 1,2,..., (82)
1
and so on.
Analogously,
$il) - - i2 C.3- (x2+t)3], 6x1 (83)
(o) j1+
260
--
x3
C 3 - (x2±t)3 3 2x1 1
Finally, we obtain
G1(Y1,Y2.t)'=
(e-ity,-e'1") + 1
Z
y Go(Y1,y2,t) =
Cy2-(Y2+t)3)(eityl-a-ityl)...,
6y1
2(eityl - e-1tY1) -
Cy2- (y2 + t)3](eityt - e-ityl)..., 6y (84)
1
G1 = sin(ty1)/yl + (lower-order terms), Go = cos,(tyl) + (lower-order terms).
If concrete initial conditions u0, ul are given, then the asymptotics with respect to smoothness and the mixed asymptotics with respect to smoothness and growth at infinity of the solution of problem (1) - (2) are given by formulas (69) and (5) respectively. These formulas may be rewritten more explicitly: raipx{Go
uam(x,t)
(
(85)
(p.x.t)uo(p) + G1(p,x,t)ul(p)}dp
and
Je'px{G o(p,x,x,t)uo(P) +G 1(p,x,x,t)ul(p)}dp,
U mix(x,t) -
(86)
where uo(p) and ul(p) are Fourier transforms of initial data fe ipxui(x)dx, i - 0,1,
ui(P) _
and the functions Gi, Gi are given by formulas (67)r
(87)
(68), and (84).
Formulas (82) and (83) show that the asymptotics us (x,t) (see (85)) is not uniform in x. One can show that the more terms o? asymptotics taken, the stronger this non-uniformity becomes. Since the functions Ok± behavf as positive powers of'x2, the mixed asymptotics umix(x,t) gives not only uniform smoothness of the remainder on the whole axis, but also its de-
crease as 1xI
-* m.
If we restrict our consideration to bounded domain D e Itt only, then yields the asymptotics usm' Here we show for x C D the asymptotics u mix this only for the principal term of asymptotics. Consider the behavior of function (67) for y2 e D and lyll + m. We have
Y 2 2)
(y 2 1
exp{
Y Y 2 2l22
+ a2)
=
+y22)
=
r1+Y2 exp(
it 2
a
(88)
sin(tyl) }
+
1+a2
yl
+ 0(a)
as a -+ 0 (here a = y2/y1 w 0 as 1y1I -. m uniformly in y2 E D).
Thus
-N,1
differs from G1 by a symbol of the smoothing operator, which is uniformly smoothing in any bounded domain. Analogously, one can show that GN,o turns into Go in any bounded domain. In conclusion we present the principal term of asymptotics for special initial data. Let initial data have the form: uo(x) = 0,
ul(x) = eix/h,
(89)
261
where h + 0 is a small paramEter. Then the principal term of asymptotics at infinity, which is uniformly smooth with respect to h, has the form u
mix
(x, t) -
Y-
(h
eix/he(it2/2)(xh/l+ x2h2)sin(h 1 + -.2h2). (90)
(I + x2h2)
The asymptotics constructed above in the form given here are Note. valid only on segment t E [0,T] such that there are no focal points when In the the solution of the Hamilton-Jacobi equation is being constructed. general case the solution is given by means of a canonical operator (see Chapter 3), but we do not give these formulas to avoid cumbersome analytic constructions. B.
.Theorem on Asymptotic Solutions
In this item we formulate in general form the theorem on quasi-invertibility of (pseudo) differential operators with growing coefficients and give the scheme of its proof. The details of the proof are omitted since this theorem is a special-case of the general theorem on quasi-invertibility First of all which will be formulated and proved in subsequent sections. we give necessary definitions. The function f(z) - f(zl,...,zn) is called (ml
Definition 1.
,. .. 'mn)
quasi-homogeneous of degree r (here a1,...,:rn,r are real numbers, nj > 0,
j - 1,...,n), if for any A > 0 the following equality holds: f(avlzl,...,amnzn) ' Xrf(z1,...,zn). Definition 2.
(1)
The function $(z1,...,zn) is called (a1,...,nn)-small
of order s, if the estimates 2az2 z I
G
wi3-s-kal, Io,
' 0,1,2,...
(2)
hold for-EIzil2/ni > 1. Definition 3. (m1'" -'mn)
Smooth function f(z1,...kzn) is called asymptotically
quasi-homogeneous of degree r, if for arbitrarily large a the
following representation is valid: N(s) fk(zl,-...,zn) + +9(z1,...,zn), f(zl,...,zn) E k-0
(3)
where s(zl,...,zn) is a (a1.... ,trn)-small function of the order a, and fk(zip .... Zn) is a (ir1,...,an)-quasi-homogeneous function of the degree rk, o r - r> r
1
> r 2>
...
.
If f also depends on additional variables w, then it is called asymptotically (a1....,arn)-quasi-homogeneous of degree r with respect to (z1, zn) uniformly in w, if the following conditions hold: (a) The functions fk from (3) are bounded together with all their
derivatives on the quasi-sphere EIz,12/m1
1 uniformly in w.
(b) The constants from (2) in the estimate of the "remainder" 48 in (3) may be chosen independent of w.
262
Theorem 1.
Let F(xl,...'xn,zl,...'zn,Ell...'En) be an asymptotically (p1,...,p2n)-quasi-homogeneous function of degree r with respect to (z1,...,
zn,gl,.
,En) uniformly in xi,...,xn and min{Pn+l'." 'p2n)
1.
Let the
Hamiltonian function H(w,h,q,p) 2 H(wl,...,wn,h,gl,...,g2n'pl,...'p2n) +hPn+1-lpl
def hrF(gl,...,gn,h PIgn+l,...,h-Png2n,h Pn+l(wl
+
hPl+Pn+1-lpn+l,...,h P2n(wn
2
2
(4)
hP2n-lpn + hPn+P2n-1p2n
+
Then the operator
satisfy absorption conditions (see below). 2
+
1
2
F = F(xi.... ,xn,xl,...,xn,-i
z
1
,...,-i
x
)
(5)
n
in L2(ttn) has the right quasi-inverse in the following sense: there exists the sequence {G.} of operetoys in L2(l1tn) such that
FoGH - 1+RN,
(6)
where for u + v < N the operator
TuvN - x9(-i 2x)v N
(7)
is bounded in L2(ltn).
Formulate now the absorption conditions. The quasi-homogeneity of the function F and formula (4) imply that, for arbitrary N, function H(w,h,q,p) can be expanded in a series in terms of (fractional) powers of h modulo
0, C.
H(w,h,q,p) N
E h JHj(w,q,p); j-0
here 0 - co < El < ..., Lim E. - + J-'m J
(8)
We set E.
Heas(w,h,q,p) '
h JHj(w,q,p)
E
(9)
and call function (9) the essential Hamiltonian of the operator (5) (see
Thus the essential Hamiltonian contains together with the zero term of the expansion of function H(w,h,q,p) in powers of h, all the subsequent terms of this expansion with Ej < 1. Denote by 0 the set of points (w,q,p) such that p - 0,
n 2/P 2/P. E (1-ii n+1 + Jqn+ij 1) - 1, H0(w,q,p) - 0
(10)
and by 0E, E > 0, the domain RE ' {(w,q,p)jdist((w,q,p),H) < E}. (cf. [521). We shall say that the considered problem Definition 4. satisfies s the absorption conditions, if there exist c > 0, T > 0, and a continuous function
263
T - T(w,gip), 0 < Tw,q,p) C T. (w,q,P) E Oe, such that:
(a) The trajectories (q(go,po,w,t),p(go,po,w,t)) of the Hamiltonian system aReH 0
ap
q
(w,q,p),
aReH
o
P'
(w>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)
<021do*g®do*h> (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
{a{ = 0,1,2,...
(9)
.
la the inequality (9), a - (a(1),...,a(n)) _ (a a
nru
), and
=
n E A J {a (J) j=1
Lemma 1.
r
E
EJ A j a j k.
(10)
j-1 k-1
{
)(Rn) if r1 = ... - rn - 1, no
It is obvious that rm (Rr) = SMX n
1
matter what ml is equal to.
n
rrml lj(R ); The operator 6/6x
acts in the spaces
if
d/6xjf Proof.
m
(rl,...,r,...,r )
(R
J
°
(r1 ....,r+l,...,r n J
).
(11)
For the sake of convenience, we denote xjf by y, xj,r.+1 by
z, and omit the other arguments under the function sign.
ajf -
f (y) - f (z)
df
y-
1
fp
2f 3y
We have
(tY+ (1-t)z)dt..
(12)
The equality (12) makes it evident that (11) holds if Aj - 0 (the integrand in (12) satisfies the estimates (9) uniformly with respect to r 6 10,1]). Let now Aj > 0.
The space Rr. where
divided into two cones Rr
(r1,...,r3+1,...,rn), may be
Dl U D2, where A.
Dl - (x E Rr (13)
D
2
- {i ERr
a'
and where e > 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 <{ (1+IYI1' 3+P))XJ. where P - ElxkokllAk > 0.
(15)
(15) yields
IY - zI where ¢ is continuous, m(0)
(16)
(c)(1+ IYI + Ip1Al), 0.
We obtain
I+ Ty+ (1-T)zI +
<1+IYI+IPI J+(I-T)$(E)(1+IYI+IPI J)
(17)
thus
A0(x(T)) <
(18)
On the other hand, we have
<1+ITY+(1-T)zI+IP1 3+(1-t)0(E)(1+IYI+IPIXJ); (19) thus A0(x(T)) 3
(x),
(20)
provided that (p(E) < 1.
Thus we obtain the desired estimates in DI by differentiating under the integral sign in (12). To obtain the estimate in D2 we make use of the identity
alai 3xa
(
6f)
axle
a6l+82
(
f(a)(Y)-f(a)(z)
(21)
Y-z
aysl + azs2
where a includes the derivatives with respect to all the variables except y and z. We have a
I6I
s ax
(6xf) JA
si
82
E
E
C 8Y(y _z)
- I-Y1-Y2
x
YI=o y2-o
(22)
01-Y1 + 82-Y2 ) - f(a)( z )] ay8 1-Y1az62-Y2 Cf(a)(Y
where CBy are constants. ferent from zero.
,
Only the terms with y1 - 81 or y2 - 82 are dif-
Estimating the typical terms, we obtain
z)-1-Y1-82.(!)81-Ylf(a)(Y)I I(Y -
<
x
X Ar(x)m-m1- < a,a >--(81--1)x] S
d,A> (23) r
r
273
or
(y _ z)-1-1'2-51 8)02-Y2f (a)
J (1+91+Y2)Ar(i)IPl
(z) <
X
(24)
Ar(')m-ml-<X,a>-(92-Y2)Xj <
X
(here I is obtained from x by replacing y on z). holds in D2 and the lemma is proved. Lemma 2.
Thus the required estimate
The multiplication operation (fg(x,y) - f(x)g(y)) maps rmm+f l+fl(Rr+r').
(25)
r(Rr)r(Rr') m1 fl ;
Lemma 3. The operation of identifying the pair of arguments within f(x(1);...;xkl,...,xk.,rk-l; the k-th "block" (f(x(1).....x(k).... ,x(n)) +
;x(n)) c f(x(l);...,xk1" ..'xk,rk (r1,.. ,rk,.. ,rn)
m r
m1
) - r
(R
lxk,rk-1'...;x(n))) acts in the spaces:
m
' (R
(rl,...,rk 1,....rn) ).
(26)
The proof of both lemmas is quite evident.' Combining these two assertions we may obtain various statements about the products, for which the arguments in the function-factors (partially) coincide. To end this item, we introduce the function space rm (Rr).
By the
definition f(x) E rm (Rn) (m1 > 0, m < ml) if and only iflthere ex,}sts a } j EI ,
function Af(x), depending only on {x,.
satisfying the following
properties: (i)
Afx) <
Af(x) > 1, x E Rr.
(27)
(ii) For some e > 0 in the domain defined by inequalities EAr(x)a1,
j c- I+, f,r e (1,...,r.),
Ixjf - xjkj <
(28)
the inverse inequality Ar(x) < holds.
(29)
(iii) The function f(x) and its derivatives satisfy the estimates CaAr(x)mlAf(x)m-ml+Ej(1-Xj)aj
(30)
If(a)(x)I <
for a such that
E(1-A.)aj+m-ml <0,
(31)
and If(a)(x)I
<
CaAr(x)m+Ej(1-Aj)aj,
(32)
provided that
E(1-A+m-ml > 0. The space rml(Rr) is a generalization of L(.
is a generalizationof Sm
01.... An) (Rn).
(33) n)(R°) just as rmml(gr)
The statements of Lemmas 1-3 extend,
with corresponding modifications, to the case of spaces rml
274
(Rr).
Conditions on Operators and Definition of the p-Mapping
B.
Let a Banach (in particular, Hilbert scale) overR or Z be given (see Definition I of Chapter 2, Section 3:C), i.e., the collection of the Banach (resp. Hilbert) spaces X - {X6}, 6 E R or Z, together with, the continuous embeddings X6 C X. for 6 > e. We assume that there is a subset
given so that any X6 is the completion of D in the norm 6.
DC fl X6 = X, 6
We slightly restrict here the general definitions of Chapter 2, Section 3:D X6 and assume everywhere that tempered generators A in the sp--e X, =
6
satisfy additional conditions: D C DA, AD C D and exp(iAt)D C D for any t E R. Such operators A will be called merely "tempered generators (in the scale x)." We assume that the n-tuple X _ (Nil X ) (see item A) is .... fixed throughout the exposition. Let (All ...,An) be the n-tuple of tempered generators in the scale X = {X6}.
We introduce some "good-behavior" conditions on this tuple. The first of these conditions was primarily used in the paper [351 in somewhat different form. Condition 1.
For any
Let j1,. ..,jm E {l,...,n) be a finite sequence.
u E D consider the D-valued function u(tl,...,tm) - exp(iAjmtm)x ... x exp(iAj1t1)u, t E Rm.
(1)
We require that u(t1,...,tm) be infinitely differentiable in any space X6, and the derivatives satisfy the estimates 3au(t
1
,
2ta
..,t.) m
116 S Ca6(1+11 tl)ma6,
(2)
where Ca6 depends on u E D, and m06 depends only on 6 and al, and does not 0
(here I, = {jI)k
depend on a-
I
= 01, l+ _ [m]\ Io). J ]
Proposition 1.
m
Under Condition 1, the operator f(Ajl ...,A.
any f E L, (Ni., . (Rm)
)
for
at least on D and acts in the spaces
Jm in
i
A.
f(A
Proof.
)
Let 6 and u E D be fixed.
1 mAjm)u = (2n) f(Aj......
:
D
f) X6.
Jm
J:
(3)
6
We have,
f_v mally,
-m/2
ff(tl,...,tm)u(tl,...,tm)dt1,...,dtm,
(4)
where u(t1,.,.,tm) is defined by (l.), and f is the Fourier transform of f.
What one needs to show is that the integral (4) may be regularized. represent f(x1,...,xIs ) in the form f(41,...,xm
fl(xl,...,xm)(l + x2 )k(1 + x2 )f 1o
with some natural k and Q. estimates f(Y)(xl,...,xm) 1
Since f e L(a,
)(Rm),f1 satisfies the
Jm
C(1 + x? )-k(l +X2
wh.-re M and c ace some positive constants.
(5)
I+
.... A,
Ji
10
So
)M-f+olYl,
I+
(6)
(6) yields, by well-known
properties of the Fourier tra.sf.,tr;, that the following estimate is valid: 275
C,
1,1 +Itl)rfl(tl,.... tm)I
nd M - f + or < - 2 II+I.
provided that k > 2 1Io1 (4) as follows:
f(A....,A j1
(2r)
--M im )u
"2 rf 1 (tl,...,tm){(1 -
(7)
Using (5), we rewrite
2)k (1
a-.)f
X u(t ,...,t ))dt ,...,dt
m
1
1
m
X
1+
Io
(2n)'/2JVfi(tl>...>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.
<
(9)
Under this choice of k and f, the integral (81
converges absolutely 2 (in the strong sense) and gives the desired regularization. Proposition 1 is proved.
Our next condition deals with the estimates of the norm of product of semigroups, generated by Al,...,An in pairs of spaces (X6,Xa). Let jl,.. ,jm E {l,...,n}.
Condition 2.
U(tl,...,tm)
Ujl,...,jm(tl,...,tm)
`
For any N > 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
j
(10)
xk )
(lower-order terms in wjk contribute to B.Ef,g) with 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 <
Xk(x,tl,...,tk-1'to, (63) X 0,...,O)_,xk+l,...,xn) < c2A(x),
;,covided that (X1(x,tl,...,tk-l'to,0,...,0),...,Xk(x,tl,...,tk-l' 0
E (U,tk1.
Then, making use of Condition
3 of item B and the fact that under (63) the symbols occurring in the remainder belong to r
r
mm-N
($t
) with various ml and r, we come to the following
result: given 6 and Nlwe l always can choose N large enough, so that u`u(3t)B(aC )'K(`'`o) p6 0
6+N
< C for Iel,161,IYI 4 N 1 (64)
IsI,I - O,y = 0 if k E I. Establish thus the conditions under which (63) holds. the argument of A in (63) is merely
Note first that
X(x,tl,.... tk-l'to,0,...,0).
Also it follows from Lemma 1 that the mapping X(x,t) defined by (53) satisfies the identity X(T1x,Tl-lt) - T1X(x,t)
(65)
1-11
in any Y set of the form { t. I < CA ( x) j = 1 ....,n} for T ;? 1, A(x) large enough. In order to3establish the desired conditions, perform in X(x.t) the change of variables I
tj - A(x)1-1j6j, j : 1,...,n
(66)
Z(x,e) - X(x,A(x)1-Ae).
(67)
and
Clearly on any set of the form {Iejl < C, j - 1,...,n}, Z(x,e) satisfies the condition Z(T1x,0) - T1Z(x,e)
for T > 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)
<< CA(x) j- k, j,k = 1,....n,
det IIa2/axk 1j,k=1
TX-
(71)
C-'
Then A(x) and A(Z(x,O)) are equivalent on K; more precisely, cIA(x) < A(Z(x,e)) < c2A(x),
(72)
where cl and c2 depend only on C. Proof. Z(x,e) satisfies (68) for A(x) 3 c3, where c3 depends only on (72) is evident for A'(x) < c3, so it suffices to consider the domain where (68) holds. The Euler identity (see Proposition I of Chapter 3, Section 3:A) holds: az.(x,e) n (73) A.Z.(x,9) - E Akxk, j = I......n. a J J xk k=1 C.
To *The right inequality in (72) follows from (70) and (73) immediately. is by (71) invertible prove the left one, we note that the matrix and the dlements of the inverse matrix B satisfy, in view of (70) and (71), the inequalitigs A._A (74) IBjk(x,e)I < c4A(x) J k II
where c4 depends only on C. We have n Ak xJ = E Bjk(x'e) X Zk(x,e). j E I+, k=1
(75)
J
and therefore Ix .I < c
J
A._A A(x) J kIZ (x,6)I, j e I.
E
Sk
(76)
k
EI+
It follows from (76) that for any j E I+, there exists k + k(j) E I+ such that
A-A IxJI < cSII+IA(x) j
-X
i e 1+,
(77)
j E I+.
(78)
or IxjIA(x)Ak(j)-Aj
<
Raising (78) to the power 1/Ak(j) and summing over j E I+, we obtain (Ix.Ip.(x)-Aj)1/Ak(J)7A(x)
C
< c6A(Z(x,e)).
E
(79)
3EI+ The expression in the square brackets in (79) is, however, greater than some positive constant on the set A(x) > C3; therefore, we come to the left inequality in (72), and Lemma 2 is proved. Assume from now on that all the systems (29) for k - 1,...,n satisfy the property: the solution (31) is defined for any initial data and any to and the following estimates are valid: alai+f
(k) (x,to),- xj)I < C,MA(x)
AJ-+I-Ak for It01 6
(X3
I
a
x ato (80)
< MA(x)Ak, IaI,M = 0,1.2....
.
289
We refer to commutation relations with the principal part satisfying this condition as normal commutation relations. Definition 1.
U C li.2°
)
is a left normal domain if:
(a) The system of equations (81)
X(Y,P) = x,
where X is given by (53), has for (x,p) E U a unique differentiable solution (82)
y - L(x,p)
0; in particular, the Jacobian
turning into y = x for p
y
aX Y
= det
ly-L(x,p) #
0 for (x,p) E U.
(83)
(b) For any R > 0 the intersection RA(x)1-13,
UR - U (){(x,P)Hlp,jl <
j - 1,...,n}
(84)
satisfies the properties: (bl) the functions (82) satisfy the estimates
a.-+
SC
1
2xaap
aBR
A(x) 3
(85)
for (x,p) E UR and lal,161 = 1,2,...; (b2) denote by UR
CIR2n
UR -
(Y. P) E tR
p(1)
p
1
the set
2n13P(1) E
p(1) -p -
1'
"' k 1
k 1'Pk
(x,P(1))
E UR,y - L(x,P(1)),
Up(1) p k+1 k
-p n=0 for some
_
(86)
k = (1,...,n) and y E C0,1]}. The conditions of Lemma 2 are satisfied for (x,t) _
(x,A(x)X-1e)
E UR
In what follows we usually call a left normal domain simply a "normal domain" omitting the word "left." Lemma 3.
For C > 0 sufficiently small, the domain CA(x)l-aJ,
U - {(x,P)EGt2nIjpjj <
.1
=
1,...,n)
(87)
is a normal domain.
Estimates (69) and (70) are always valid in the domain of the Proof. form (87) (t)je second of these possibly with another constant C). Both DZ/Dx and DX/Dy are equal to unity for t = 0 (respectively, p - 0); thus for C small enough these Jacobians will be greater than some positive constant. The lemma is thereby proved except for some tech-:al details which we leave to the reader. Let now the support of f(x,p) E TM
(al,...,an)
domain U, supp fC U.
290
It follows then that
OR2n) lie in a normal
(Paf)(X(x,P),P) E
TMs,...,a (R
n
1
(88)
)
for any s and therefore we may obtain the asymptotic sum of the series (60) -m
(we denote it also by F(x,t) E T
(al, ... , an)
).
We claim that (4) is valid.
Indeed, using the asymptotic expansions for F(x,t) (and, respectively, for Wk(x,y,t,to)) up to an arbitrary order, we obtain the estimates (64) for arbitrary b and N1 since estimates (63) take place via Lemma 2 and condition (b2) of Definition 1. The estimates .(64), in turn, yield after simple calculations, the estimates (5) for the remainder in (4). The only thing remaining to obtain (6) is to solve the "inverse problem," i.e., to find f(x,t), F(x,t) being given. From (60), and (81) and (82), it is easy to see that the described solution has the form E (PSF)(L(x,p),P),
f(x,p) = F(L(x,P),P) +
(89)
s=1
where PS are differential operators of order 5 s + 1 in x and the s-th term (IR 2n) provided that also f(x,p)
of the series belongs to Ts
(al,...,Xn)
E Tm
A )OR ).
E
Thus we have proved the following theorem:
n
Theorem 1.
Assume that 4(x) E Lst (A
'
(IRn) and that for some
A
n)
1
K G Ess 0, K CA where U C 1Z2n is some normal domain.
For any F(x) C
n
n)(IR ) the composition formula
E S(X1,
2
F(A)o I(A) _ (H(x,-i ax)(P)(A) (mod Op(x))
is valid, where H(x,p) E T n)
(90)
(IR2n) is any symbol such that for
(x,p) in some neighborhood of K H(x,p) = F(L(x,p)) +
E (PSF)(L(x,p)),
(91)
$=1
where P are given differential operators of order fi s + 1, and L(x,p) is a solution of the system of equations (81). In a completely analogous way the following problem may be solved:
given the symbols F(x) E S(1l,
n and l(x) E Ls t,(Al, an)(1R) ....
an)(JR °
....
0
construct a symbol 1)1(x) E L( m
al,.. ,an)
OR") such that
(D(A)o F(A) = 01(A) (mod 0p (x)).
(92)
We do not perform the calculations again but merely formulate the final result. (29).
Again, let X(k)(x,t) = X(k't)(x) denote the solution of the system We set
Y(x,t) - X(n,-tn) o ... o
X(1,-t;
(x),
(93)
and define the mapping Y by Yi (x,t) = Yi (x,t1,...,ti ,0'...,0), j - 1,...,n.
(94)
Set :i-so 291
2(x,8) - Y(x,A(x)l-19)
(95)
y - R(x,p)
(96)
Denote by the solution of the system of equations (97)
Y(y,p) - X.
The domain U C t12n.P) will be called right normal if it satisfies the con-
ditions of Definition 1 with X replaced by Y, and L(x,p) by R(x,p), and UR replaced by
UR - {(y,p)E tR 2n12,(1)
(x.p(1)) E UR,y - R(x,p(1)),
pn1) -pn'"''pk+l-pk+1'pk lipkl)'pk-1-
(98)
pl - 0 for some k E {1,...,n}
and µ C-[0,131. The analogue of Lemma 3 is obviously valid; the following theorem takes place: OW
'(x) E Lst A
)(R°) and that for some n For any F(x) E K E E8s @, K CU, where U C Et2n is a right normal domain. Theorem 2.
Assume that
(
E S(1
)(,1n) the composition formula
1 1
1
1
n
(mod Op (x))
$(A)o F(A) _ (G(x,-i
is valid, where G(x,p) E T
(tt2n) is
(99)
any symbol such that for
(x,p) in some neighborhood of K G(x,p) = F(R(x,p)) +
E (QsF)(R(x.p)),
(100)
s-1
where Qa are given differential operators of order < s + 1. Definition 2.
The mappings
F ; L(F) - U(x,-i aX), F E S(al,...,an)G(m) and
2
F + R(F) - G(x,-i ax),
FE
(101)
(102)
constructed in Theorems 1 and 2, respectively, are called the left and right I n (local) regular representation for the tuple (A1,...,An
It should be emphasized that the operators L(F) (R(F)) of the left (right) regular representation are therefore correctly defined (up to L
(al, ..., a n)
(t2n)) operators on the subspace in Lmst,(al,
..., A n )
tn) con-
sisting of functions (x) such that some K E Ess @ lies in the left (right) normal domain.
292
4.
SYMPLECTIC MANIFOLD OF A POISSON ALGEBRA AND PROOF OF THE MAIN THEOREM
A.
Effects of Variable Changes
Let a V-structure on the Poisson algebra in Rn be given in the sense of Definition 2, Section 3:B, i.e., a mapping n
1
f
+ f(A1,.... An), f E L(al,...,Xn)(Rn),
(1)
where (A1,...,An) is a proper tuple of operators in a Banach scale X satis-
fying the commutation relations n
1
[Ai 'A k] -
(2)
where the principal part of wjk(x) equal to O.k(x)/Q is the tensor defining the Poisson algebra structure.
Our final aim is to construct asymptotic (in the scale X) solutions of the equations of the type i n f(A1,...,An)u - v, where f 6 (3) st.(a ,...,) )(R2n 1 n this, in turn, requires the solution of an auxiliary Cauchy problem with subsequent integration over t (we have this construction in item B):
-i t +F(A1,...,An)u (here we assume that F E P1
st'(Xi.... 'An
0, uit-O - v
)(R2n)).
(4)
We seek the asymptotic
resolving operator for the problem (4) in the form 1 n U(t) - §(A1,...'An.t),
0
where O(x,t) E LsL'(i
i
)(R2n).
1'"'' n
(5)
Using Theorem 1 of Section 3:D, we
obtain for 4(x,t) the equation 1
-i 2t + H(x,-i 2-),P E L(al,...,an)(R2n)
(6)
with the initial condition ,It-p
1,
(7)
where H(x,p) is described in the cited theorem, provided that some Kt E-= E Ess((P(t)) lies in a normal domain for the considered values of t. Now assume for a moment that f(x) is real-valued. The solution to (6) and (7) may be found with the help of the canonical operator (see Chapter 3, Section 3:B) by a method quite similar to that given in Chapter 3, Section 4:D,, Thus, m(x,t) - CK$](x,t)
(8)
is a CRF associated with the proper Lagrangian manifold L(t) C R2n which is nothing other than a shift of the manifold L(0) ° Lo ` {(x,P)Ix 6 Rn.p - 0)
(9)
293
during the time t along the trajectories of the Hamiltonian vector field corresponding to the Hamiltonian function H1(x,P) .. F1(L(x,P)),
(10)
(R2n) is the principal homogeneous part of F. where F E O1 1 st,(a1,...An)
For Iti small enough everything is alright since L(t) lies inside the normal domain. This may not be the case for greater values of t, and the However, method fails once L(t) attempts to "leave" the normal domain. this does not necessarily mean that there is no solution for these values of t (the situation is analogous to focal points in the WKB method). This means only that the solution no longer has the form (5) with @(x,t) given by (8). Introducing another family of operators (B.,....Bn). functionally dependent on Al,...,An, and considering functions of these operators, gives the possibility of obtaining the solution for these values of t as well. It turns out that the development of this idea leads us directly to the consideration of the symbol of the resolving operator as a section of the canonical sheaf on a special symplectic manifold, which we call the symplectic manifold of a Poisson algebra. We come to detailed considerations. To simplify the subsequent arguments, we assume that the following condition is satisfied for the operators: If f(xl,...,xn) E L(All
Condition 1.
n 1 n A )(R ) and f(A1,...'An) E n
L(All
0--(x), then f(x,...,x p
n
1
.... an)
(Rn) (= S
(Rn)).
This
(al,.... xn)
Condition is not necessary for the validity of the subsequent theorems; however, the requirement that the u-mapping be "asymptotically monomorphic" enables us to present much more obvious proofs and to avoid cumbersome calculations.
Let U C R2n be a normal domain (see Definition 1 of Section The mapping
Lemma 1. 3:D).
L
:
U
Rn,
(x,p) -; y = L(x,p),
(11)
where L is defined in the cited definition, preserves the Poisson brackets, i.e., for any f,g E C-(Rn), *
{L f,L g} = L {f,g) in U for A(x) large enough
(12)
d of (in (12)'(L f)(x,p) f*L(x p)); {f,g} is a Poisson bracket of f*and g with respect to R, and {L f,L g) is a Poisson bracket of L *f and L *g generated by the symplectic structure w2 - dp A dx
(13)
in U, i.e., {L*f,L*g) =
E
{
j-1
Proof.
a(Lg) *
apj
axj
- 8(L*f axj
L* ap.
}
(14)
Obviously (12) is valid for any f,g E C_(Rn) if and only if
aL.(x,p) al. (x,p) k g{ axs aps
294
a(Lef
aL (x,p) aL.(x.p) -
k ap5
axa
}_ Rjk(L(x.P)), j,k = 1,...,n. (15)
1 )(R
To prove (15) consider any symbol (x) E Lst (Z
'
2n
) and calculate
n
1
the symbol of composition (equal to zero by commutation relations)
n
1
1
Q (Ai oAk - AkoAi + imjk(Al,...,An)Ilo@(Al,...,An
(16)
0
using Theorem 1 of Section 3:D (we assume that R C U for some K(=- Ess m). We obtain that the symbol of operator (16) equals 2
(17)
H(x,-i a.)0(x), A.+A -1 where H(x,p) E T(A ,k
)(R2n), and where n
1
X .+A -2
aL.(x,p) aL (x,p) k ax
Ho(x,p) - -iE{
s
a
ps
k-
H1(x,p) E
H(r,P) = Ho(x,p) + H1(x,P).
(
(18)
AR)(R2n),
DL (x,p) 3L.(x,p) k
a
s
+
ps
ax s
s
(19)
+ ittjk(L(x,p)) in U (we used the composition formulas from Theotem 2 of Chapter 3, Section 3:A). Substituting for m various CRF's associated with Lagrangian manifolds 1-a
a
L C U we deduce, using Theorem 2 of Chapter 3, Section 3:B, that H(T x,T p) rapidly decays for T In particular, P) Ho(x,p) - 0 for A(x) large enough -
since for these A(x) we have Ho(T1x,Tl-1 Lemma 1 is proved.
13+?tik-1Ho(x,p), T
> 1.
Thus
We denote by Rn,
U
f
(20)
the R+-homogeneous mapping coinciding with L for large A(x). analogous way we may prove the following: Lemma 2.
Let U C R2n be a right normal domain. R
U + R
n
(x,p) '
,
In a quite
The mapping
y - R(x,p),
"anti-preserves" Poisson brack^_ts, i.e., satisfies the property
{R f,R g} _ -R {f,g) for A(x) large enough, for any f,g e C'(Rn).
Also, if U is both right and left normal, then *
*
{R f,L g} = 0 for any f,g E C (Rn). Proof.
We leave it to the reader as an easy exercise.
As above, we
denote by
r
:
U -*Rn
421)
where U is a right normal domain, the R+ homogeneous mapping coincident with R for large A(x).
195
Next we study the transformation of symbols which accompanies passage from one proper set of generators (A1.....,An) to another one, say, (Bl'...,Bn). Let u - (ui,..
Nn) be the n-tuple of non-negative numbers We denote To - (jluj-01),
satisfying the same conditions as A - (A1,...,xn). T+ - (il
We assume that (Bi,...,Bn) is a proper tuple
> 01} - [n] \ To.
of generators in the scale X, corresponding to the tuple u - (ui,...,un) and that the following conditions are satisfied: For j - 1,...,n
(Change of variables).
Condition 2.
1
B. - Y.(Alt ...'An);
(22)
the symbols Y.(xl,...,xn) satisfy the following conditions: uj
Y.(x ,...,x ) E P n
1
Y(x ,...,x )- £ d
3
kE I0
l
a
stn) x
jkxk
(0) for j E T+,
E P
st,(Al,...,An)
(23)
(e) for j E To, x
(24)
where ajk are some given constant.coefficieAts; symbols Yj(xi,...,xn) have the asymptotic expansions of the form Y.(xl,...,xn)
Y.
k-0
where
J,k
(xl.... ,xn),
(25)
u -k Yjk(r11xi,...,TAnxn) - T 1
(26)
Yj,k(xl,...,x0)
for large A(x), and the difference between Yj and the partial sum of N terms of the series on the right-hand efde of (25) belongs to v -N-1 at
x (Rn).
n
We denote yj(xl,...,xn) - Yj,o(xi,...,xn).
(27)
and require that the coordinate change y - y(x) be strongly non-degenerate in the sense that
Idet 2 (x)l 3 CA(x)
(28)
I
with some positive constant C. Remark 1. Under these conditions A(y(x)) is equivalent to A(x) (where of course A(y) is constructed according to (u1,...,un)). The proof is
quite analogous to that of Lemma 2 of Section 3:D. Condition 3.
(Commutation relations).
For j,k n [Bj,Bk] - -iwjk(B1,...,Bn), 1
where
(29)
-1
k Wjk E Pst,(ui,...,un)(R
ilk -
We denote
296
E Ost,(ui,....
(30)
Vn)(Rn).
(31)
)x
0Jk(Y) ° Condition 4.
(32)
The 2n-tuple of operators (All .... An,B1,...,Bn) is a
proper tuple (with respect to weight tuple (X1..... an,ul,...,On)); moreover,
the item (b) of Condition 3 of Section 3:B remains valid for this 2n-tuple if we replace the space !'m
by the space rm (R) in the definition of which ml
we require only (cf. Section 3:A) that
A(x) < CAf(x,y) in the R+-invariant neighborhood of the set
A - i(x,Y)jxJI=...°xjr'JE I+,yjl`...-Yjr
.J EJ+' n +J
7
yJ ' YJ(x). J E T+). Lemma 3.
For A(x) large enough, we have ay. ay
Jk(Y(x)) =
(33)
L nfm(x) ax axk
f,m
m
A
Proof.
We calculate [BJ,BkI from (22). Modulo 0p (x), we have n LBJ,Bk] - -i{yJ'yk)(A1,...,An)(+ lower-order terms). (34)
On the other hand, we have by (29) 1BJ,BkI - -imJk(11 Y1(All ...'An)II,...,Q Yn(All ...'An)11)
(35)
We claim that modulo 0-(x) the right-hand side of (35) may he represented 1 P n as the function of (A1,...,An): 1
n
1
n
n
I
w.k(Q Y1(A1,...*An)]J ,...,11 Yn(A1,...,An)]1) n 1 S2Jk o y(Al,,..,An) (+ lower-order terms).
(36)
Indeed, (36) follows by successive application of the K-formula ([52;) and then by using Condition 4. Lemma 3 is proved. Let the symbol d(y) 1= L intend to solve now is such that
(R°) be given.
The problem we
om t:
1 n 0(B1,...,Bn)
find
a
symbol
0 (x) E L 1
1
_
n
(R')
st'(al' .. 'A n)
- m1(A1,...'An) (mod 0p (x))
(37)
(in particular, we obtain the tool to verify the identity (36)). To do 1 n this, we turn to the definition of (B1,...,Bn) via the Fourier transformation:
n 1 (B1,.. ,Bn) - (2r)-n/2 r4(nl,...,nn)e1nnBn
...
einlBldrl ... drn. (38)
Our first step will be to find the asymptotics for the product of einnBn ... eif1B1 of the form
297
elnnBn
...
1 n e1n1B1 - E(n,A1,.... An) + R(n)
(39)
with the appropriate estimates for the remainder R(n). In some analogy with procedures performed in Section 3:D, we successively solve the definitive equations for ein1B1,...,einnBn
More precisely, we consider the sequence
of functions Ej(n,t,x) of (n,x) E R2n, t E [0,l], j - 1,...,n, satisfying the conditions:
E1(n,0,x) - X(n,x), (40)
E3(n,O,x) - Ej-1(n,l,x), j - 2,...,n;
E3,is an asymptotic solution of the problem* -i 2t E.(n,t,A) - n3Yj(A)0 Ej(n,t,A) +
(n, t), t E 10,11, (41)
Rj(n,t) E Op (x), j - 1,...,n. In (40) x(n,x) is a cut-off function,
X(n,x) - 1 if In.I <
CA(x)1-uJ,
j - 1,...,n; (42)
C1A(x)1-u j
X(n,x) - 0 if In jI > I.I.IeI a
nx
a
j<
for at least one j;
C.SA(x)-+,
-0,1,2,....(43)
ax an
For example, we may set X(n,x) - c(nA(x)u-1) where { is an appropriate finite function. 0m
Assuming that E .(n,t,x) E J
(a1,..., n)
(Rn) for any n and t under con-
sideration and that some Kj E lies in the normal subset U, we obtain, via Theorem 1 of Section 3:1, the following equation for Ej(n,t,x):
-i at-Ej(n,t,x) - n3H3(x,-i ax)E.(nt,x) (mod i(a1,...,A )(5 )),
(44)
2
where the principal symbol of H.(x J ,-i az) in the neighborhood of Kj equals yj(L(x,p)).
Rigorously speaking, theorem 1 of Section 3:D was estab-
lished only for symbols belonging to the space S(0 1,...,an)(Rn) and for does not belong to this space. However, using the representation (24) it is easy to show that this theorem remains valid for Yj(x), j c- To;
j E To.
To produce the asymptotic solution to (44) we make use of the theory of canonical operators on the proper Lagrangian manifold developed in Chapter 3, Section 3:8. However, here we need Lagrangian manifolds depending on the parameter n, also involved in the R+ action, so we indicate the main modifications which have to be made in definitions and estimates of Chapter 3,. Section 3:B.
1
n
* Here we again use the shortened notation f(A) - f(A1,...,An). 298
First of all, Definition 1 of that item reads now: L C
ie is
a proper R+-invariant family of Lagrangian manifolds if: (a) L(n0) - L f) (n - no) is a Lagrangian manifold in R(x p) for any
no c- R; (b) L is R+ invariant (the action of.R+ is. defined by) (45)
t(x,p,n) -'(-r x,tl-lp,tl-un),
in obvious abbreviations; (c) the inequalities CA(x)1-xj,
CA(x)1-uj
(46)
Injl <
Ipjl G
hold on L with some constant C.
Next, the formula of Definition 2
S1 - a1x1p1 + (a_ - 1)p-x_
(47)
II
I
for the action in the canonical chart with the coordinates (x1,pis now valid only for n = 0; generally S1 must be defined as the unique homogeneous solution of the Pfaff equation dS1 - pldxi- xidpi.
(48)
0m
1 )(R
As for estimates, the conditions on ! to belong to L
now read
n
a1°I+I8'P 1 +for 1.1,101 1
ax an
Iar
1
- 0,1,2,...,
- 0.
(49)
0
We do not indicate here numerous minor modifications still necessary; all the theorems of Chapter 3, Section 3:B remain valid in the new context. Let us solve (44) successively for j - 1,2,...,n. of Lagrangian manifolds L0, given by the relations
Consider the family
C1A(x)i-ui,
p - 0,
lnjl
<
j - 1,...,n,
(50)
and let Lt, t E 10,1] be the shift of L0 along the trajectories of the Hamiltonian system x - -n1 ap (y1(L(x,p))),
+nl as (Y1(L(x,p))),
(51)
corresponding to the Hamiltonian function -n1Y1(L(x,p)). Note that for Cl small enough the trajectories of (51) are necessarily defined for t 6 10,1], and Lt is a proper family of Lagrangian manifolds lying completely in U. Note also that since L. obviously satisfies the quantization conditions, so does Lt for any t. Thus we may find E1(n,t,x) in the form (cf. Chapter 3, Section 4:D) E1(n,t,x) - [Kt#](n,t,x),
(52)
299
where K is a canonical operator on Lt, and Q satisfies the transport equation and the initial data (53)
+Jt=0 - X(n,x)
(we assume that the measure chosen on Lo has the form dx1A ...A dxn). E1(n,t,x) thus constructed satisfies the equation 2
Then
1
-i a[ E1(n,t.x) - n1H1(x,-i aX)E1(n,t,x) + R1(n,t,x),
(54)
where R1 satisfies the estimates (49) uniformly in t e 10,13. We proceed by induction on j.
Once (44) is solved for E3_1, we impose
for Ej the initial condition (40) and solve (44) for Ej, using the Hamiltonian function -n.Y.(L(x,p)). We obtain 2
1
-i at E.(n,t,x) = njHj(x_i ax)E.(n,t,x)+R.(n,t,x),
(55)
where Rj satisfies the estimates (49) uniformly in t e (0,13. From (55) and Theorem 1 of Chapter 4, Section 3:D it follows easily that E.(n,t,A) - njY.(A)0 E.(n,t,A) + R.(n,t),
-i
(56)
2t
where uniformly in R. n y anal A
f or any 6 ,N and 6+ 6+N 6 C adyN
(57)
Ial,IYI - 0,1,2,..., jaj01 - 0.
Taking into account (22), we obtain -i
at
E.(n,t,A) - njB.o E.(n,t,A) + Q.(n,t),
where Q.(n,t) satisfies the same estimates (57). the fact that Bj is a tempered generator, .n
E.(n,1,A) - e
n.
.
3
Ej(n,O,A) - ie
.
(58)
Next we obtain, using n.B.t
1
> if e
Qj (n,t)dt;
(59)
0
thus
Ej 0, 1,A) - e
iB.n.
where Q.(n) again satisfies (57). setting
> 3 o Ej (n,O,A) + Qj (n) ,
(60)
Combining these estimates, we obtain,
E(n,x) - En(n,l,x),
(61)
.that einnBnx
... xein1B1
- E(n,A) +R(n)+einnBnx
... (62)
where R (n) satisfies the estimates (57). Make further transformations on the identity (62). x(n,A) - X1(n,B) + X2(n),
where x2(n) satisfies the estimates (57),
300
Then we have (63)
),
xi(n,Y) E T(u 1
> . . . > u )
(64)
k
n
and may be expanded into the asymptotic series xl(n,Y) = jZ0x1j(n,Y), xl1- (n,Y) E T(ul,...,ynQR2n
(F,)
where (66)
x(n,x), X1j (n,Y(x)) = (ny)(n,x), j - 1,2,..., where ;j are differential operators.
To prove.(63) it suffices to use the K-formula (cf. proof of Lemma 3) recursively. Thus
e1nnBnx
... x e1nIB1 - E(n,A) +einnBnx ... x e1nhB1o(l_X(n,B)) +R(n), (67)
where R(n) satisfies (57) (but is different from that in (62)). (67) into (38) we obtain, using Theorem 2 of Section 3:D,
(B)
(2,r)-n/2Je(n)E(n.A)dn+(2n)-n/2JC(1-X2(Y2 ,-i
Substituting
ay ))$](n) x (68)
x e1nnBn x ... x elnhB1dn+
(2,r)-n/2fR(n)kn)dn,
)QR2n) has the asymptotic expansion
where x2(y,n) 6 Tou 1
n
X2(Y,n) = X1(n,RB(Y,n)) + (QlXl)(n,RB(Y,n)) + ...,
(69)
where Q1,..., are-differential operators and RB(y,n) is the mapping constructed in Theorem 2 of Section 3:D (according to the tuple of generators 1 n (B1.. ,Bn)). The third term in the sum (68) belongs to the space Op (x). to the following conclusion:
e(B) - 01(A) (mod Op x),
We come
(70)
where $1(x) - (2,r)-n/2Ji(n)E(n,x)dn, (71)
provided that there is some subset K E Ess(e) such that K (lsupp(1 - x2 x
x (y, n)) - 0. Evidently, this is always the fact if the set { I ni I < CAW 1-oi } belongs to Ess(e), where C is small enough. Study now more thoroughly the transformation (71). has locally the form
The kernel E(n,x)
(2n)111/2JeiCS1(n,xl,Pi)+x1pi]a(n,x1,P1)dpl,
(72)
E(n,x)
where S1 is quasi-homogeneous of degree 1, and a is asymptotically quasihomogeneous. The function S1 is equal to
S1 ` S- IP_,
(73)
where all the functions are considered as functions on L(n), and dS - pdx
(74)
301
on L(n) (n being fixed!). It we consider S as a function on L, n no longer being fixed, we have obviously
dS - pdx+Adn, where A - (Al'...,An) is a certain set of functions on L.
immersion j = L
-
,,,n (x
x :
Consider the
given by (recall that LC It 2 nx
,
(Y
, P) j
It2n
(75)
(
(x,P,n) -
2
x 6l
n
P)
n) :
(76)
(x,P,A,n).
In view of (75) this immersion defines a Lagrangian-manifold in II2X,P)xlt2Y,n) (with respect to the form Sit = dp A dx - dq A dy), which we
again denote by L.
Consider the subset of L given by the inequalities 1-u
(77)
We claim that for t > 0 small enough this subset is the graph of the proper Indeed, it is It,-invariant canonical transformation g (y,n) not hard to calculate yj - Aj in (75) at the point (n - 0). :
To perform this, fix nl = ... - n.
J-1
- nJ.+l - ... - nn
0, and let n . vary J
in the vicinity of zero. Then, from (50), p varies in the vicinity of zero and the coordinate system (n,x) may be chosen on L in this neighborhood. The Hamilton-Jacobi equation for S(0,...,n.,...,0,x) reads an. - Yj(LA(x,P)).
(78)
J
A. - Yj(LA(x,P));
Ajln- 0 - Yj(LA(x,0)) - yj(x)
a
azs
In-0 = ax
and thus det L23nax f 0 for n - 0.
(79)
(80)
(81)
It follows that L is the graph of canonical
transformation in some neighborhood (77) and, obviously, this canonical transformation is a proper one (see (28) and Remark 1). Considering all that is presented above, we come to the following theorem. Theorem 1. Assume that a u-structure on a Poisson algebra in Itn is given and that the variable change and operators B1,..., Bn satisfying the Conditions 2 - 4 are given.
There exists for a R4-invariant two-sided normal
domain U C It2Y,n) necessarily containing the set (nj < cA(y)1-uj) for c > 0 small enough, a properlR+-invariant canonical transformation + IR
g
and an asymptotically (modulo
ential operator K e T
: U
(III'
ul,...,un )
Lst, (ul 302
(x,P)'
(82)
)(Il2n)) invertible pseudo-differn
such that for any function @(y) e,
'on) such that K E U for some K EssG (m) , the formula is v slid
n
1
n
1
O(Bl,...,Bn) = CT(g)K0](A1,...,An) (mod 0p(x)),
(83)
where the operator T(g) is defined in Section 3:C. The canonical transformation g possesses the properties: g(y,O) - (x(Y),O).
(84)
where x - x(y) is the solution of the equation y - y(x), and x(IB(Y.n))
EA(g(Y.n)). (85)
x(rB(y,n)) ° rA(g(Y,n)). (Y.n) E U. Consider any symbol Only the proof of identities (85) needs a hint. 0(y) satisfying the condition of the theorem and also any symbol H(y) E
E S8t,(ul,.... un)n).
The composition H(B)o m(B) may be calculated in
two different ways, using Theorem 1 of Section 3:D and the present theorem. Substituting various CRF's for @(y), we easily obtain that the principal homogeneous part of H(LB(y,n)) and H(y(LA(g(y,n)))) are equal, from where the first of identities (85) immediately follows. identities is proved in an analogous way. B.
The second of the
Globalization and Proof of the Main Theorem
Let a manifold N. diffeomorphic to btn, be given with the smooth action of the group 1t+ and the Poisson algebra structure homogeneous of degree -1 (i.e., the Poisson bracket (f,g) is homogeneous of degree m + A - 1 if f and g are homogeneous of degrees m and f respectively). Let also a smooth function
A=AN : N-+El/2,W)
(1)
be given, homogeneous of degree 1 for A 3 1. We assume that the following conditions are satisfied: (i) A finite set of coordinate systems x(1) . (x(1),... x(1)). n
1
...................... x(k)
. (x(k),....xnk)
(2)
...................... x(N) m (x(N) 1
is given on N.
..
x(N) n
Each of these coordinate systems covers N, and the action
of the group lt+ is given by TX
where
(TA(k)x(k)
(k)
_
TX(k)x(k))
TX(k)x(k) (3)
1
(k)
(a(k)...... (k)) 1
n
(4)
is a tuple of non-negative numbers such that
303
{j
1.k)
[.i]Iask)>0)
(5)
is a non-empty set (we denote also 0}).
Ink) - In] - I(k) - (j E
(6)
The inequalities cA(x(k)) G A 6 CA(x(k))
(7)
hold with positive constants c and C (here A(x(k)) is defined according to the tuple a(k)).
(ii) To any coordinate system x(k), k - 1,...,N, the proper tuple of generators A(k) - (Ak)
l,....Ank)
(8)
is taken into correspondence, so that the local u-structure
f(Aik),...,Ank)), f e L' (ktn) (A(k) ,...,An(k) )
f
(9)
in the sense of Section 3:B is defined, where the operators A(k),...,Ank) satisfy the commutation relations
CAk) ,ALk)]
-
(Aik),....A n k)
(10)
of wjt)(x(k)) equals the corresponding com-
and the principal part
ponent of the tensor, defining the Poisson algebra structure in the coordiWe assume also that Condition 1 of the item A is valid for any
nates x(k).
of the tuples A.
(k)
(f)
,x (iii) For any pair x of coordinate systems, the corresponding operator tuples are related as described in Conditions 2 - 4 of item A. In more detail, the notations will. be
A(f) 3
X(fk)(A(k) .:.A(k) 1 n
where
j - 1,...,n,
(11)
(f)
X,fk) E P 3
O R
St, (A (k)
X(fk)
a.fk)x(k) E 6
-E s
i
El(k) ]s
with given constants ask)
(k)
j E If)
po
(fin
St.(a(k),...,a(k))
symbols
X(fk)
(12)
j E I(f) o
(13)
have asymptotic expansions in the
homogeneous part for large A(x(k)) functions with integer step; the prinX;fk) cipal homogeneous part of xjl) o (x(k))-i (note that this, equals together with conditions (12) - (13), implies serious restrictions on the possible coordinate changes). Let for any k E {1,...,N}, a two-sided ht,-invariant normal domain
vk C R2n(k) (x
304
be fixed.
(k)
>p
)
By Theorem
1
of
item A,
for any pair
k,E E {1,...,N} the proper Qt+-invariant canonical transformation gkk : Vki ~ VEk'
(14)
where vkE C vk, vik C vE, is defined so that I
O(A(k),...,A(k)) - (T
n
1
Ek
4)(A(E)
..
1
A(E)) (mod
n
p
(15)
if K C vkf for some K E Ess m;
TEk - T(gEk)oKfk , where KEk E TW (k) (Al
operator.
is a given invertible pseudo-differential
(k)
,...ran
(16)
)
Standard argument treating Condition 1 of item A shows that (Tfk o Tka)4 - TEam (mod L(a(E),...,a(f))n)).
(17)
n
1
once K C vak () gk,(vkf) for some K E Ess(m) and, in particular,
8Ek ° 8ks - gEa
(18)
on the common domain of definition of the right-and left-hand sides of (18).
The equality (18) and equalities (84) - (85) of item A lead to the There is a symplectic kt+-manifold of dimension 2n following conclusion. and the mappings
Eµ+N,
(19)
r :H
N,
(20)
N +M.
(21)
j
:
such that: (a) these mappings are R+ invariant and smooth except, possibly, the set (0) C M of fixed points of R+ action; (b) E preserves the Poisson bracket while r anti-preserves it, and t and r are in involution, i.e., {f*f,f*g) - E*{f,g),
(22)
{r*f,r*g} - r*{g,f},
(23)
{r*f,E*g} - 0
(24)
for f,g r=- C (N) (here on the left we have the Poisson bracket on the symplectic manifold, and on the right, on the manifold N).
The mapping j is given in local coordinates by j(x(k)) - (x(k),0).
(25)
to j - r o j - idN
(26)
and s atisfice
(the identity mapping of N).
The manifold M is obtained from vk - a via transition functions gEk Thus, there is a covering N
EU 1UE
M
(27)
305
and coordinate canonical tit+-mappings
Uf . vf,
(28)
gfk = vfo vkf,
(29)
of
:
such that
and the mappings (19) - (20) in local coordinates are defined via Theorems I and 2 of Section 3:D. We may define the function
AM : M + tl/2,o),
(30)
AM ° fAN.
(31)
setting
Then we require that
(iv) vkf C vk are chosen "properly," i.e., in such a way that Mis a separable manifold (roughly speaking, this means that vkf are maximal possible ones). (v) M \{0) is a proper symplectic IR+-manifold (in parti.'ular, the atlas (UL) may be inscribed into appropriate atlas {Uf}, so _hat (Uf,Uf) will be a proper pair for any I). [0,T] - N of the (vi) For any function f E C '(N) and any trajectory y Eulerian vector field corresponding to f, there exists the trajectory :
(0,T1 4 M of the Hamiltonianvector field of f*f, such that Y (0) = J (y (0) )
(32)
(and, consequently, f(y(t)) v y(t), t e [0,T]). We say Definition I. Let the Conditions (i) - (vi) be satisfied. .then that a global u-structure on the Poisson algebra on N is given, and that M is a symplectic manifold of this Poisson algebra.
Condition (vi) is the crucial one, since all other conRemark 1. ditions are always satisfied in the local version (one should merely set N - 1). Denote A. =
x. =
1;1),
j a 1,2,...,n,
(33)
Pst,(al,...,an)n),
and let f l-= 1 n f = f(A1,...,An).
Definition 2.
(34)
The function H - f*(A(x)-m+lf) (35)
on M, where the subscript "1" denotes the principal homogeneous part of the function in parentheses, is called a left Hamiltonian function, or simply a left Hamiltonian, of the operator f. Now we are able to formulate and prove the main theorem of this book, namely, the quasi-invertibility theorem.
306
Theorem 1 (Main Theorem). Let the HamiLtonian function H of the n 1 operator f = f(A1,...,An) satisfy the absorption conditions of Chapter 3,
Section 4 for some e > 0, T > 0 and the initial manifold given by the immersion
j . No. P, neighborhood of the set of zeros of the
where N C N is an k+ principal part of f having
lie form
No a {xENjjxj -x .'
01.
quasi-inverse operator G, i.e., an
Then the operator f has a operator such that
fo G = 1 + R,
(36)
where R E Op-w. Remark 2. The theorem on the left quasi-inverse operator is formulated and proved in a quite analogcus way.
The most esserrial work was performed already in Chapters 3 Proof. and 4, and it remains to ga.ier all the results and to complete the argument. The operators TEk given by (16) satisfy the cocyclicity condition modulo L-" and therefore define a sheaf, F+, over M (this sheaf slightly differs from that considered in Chapter 3; the operator Kfk is present in
the definition of transition operators, but since KEk is almost an invertible pseudo-differential operator, all the results hold in this case automatically). Using the condition (15), we may define the mapping
#
Fst,+(M) -' 0p(x)/Op W.
(37)
'
where F
(M) is the set of sections with representatives "stabilizing at st,+ infinity," by means of the formula
E mE(Ai,...,An) E) E=1
the condition (15) guarantees that the mapping (38) admits If
F(x)
and
[{,pN=l7 a Fst,+(M). 1 n then the composition F(A1,....An)oi
# (0)
(40)
(x) calculated
may be modulo 0 p
in the following way.
We have (all calculations modulo Op (x)):
307
i
F(Al,...,An)o
E
@f(AAnf))
R-1
=
E
f-1
IFf(Aif), ..,Anf))7111't (AAnf))II
N
2
i
(41)
n
1
E
ax
k=1
where the principal symbol of H(x(f),equals ax
C(f*f) o v17(x(f),p(f)), where f is the principal homogeneous part of F.
(0) -
F(A) o
(42)
Thus,
(H(O)).
(M) with the principal
where H is a pseudo-differential operator in f symbol f f.
(43)
at,
Turn now to the quasi-inversion problem posed in Theorem 1.
Multi-
plying f(A) by ALA) l from the right, one reduces the problem to the case m - 1. We seek G in the form
G= 4 (A)
_if T
q (t)dt,
(44)
0
where 4(A) satisfies modulo 0P (x) the problem
f (A) o O (A) - 1- p (A).
(45)
and modulo Op (x)
-i
aatt) + f(A) o q'(t) - 0,
i (O) - p(A).
(46) (47)
Here the symbol p(x) is chosen in such a way that
p (x) - 1
(48)
in the IR*-invariant neighborhood of the set of zeros of the principal part of the function f and supp p(x) C No. The equation (45) is easily solved by means of the composition theorems presented in Section 3:B. We have
*(x) - (1- p(x))/fl(x)(+ lower-order term).
(49)
where f1(x) is the principal homogeneous part of f.
Further, we seek the solution of (46) - (47) in the form
((t) -+y (t) ,
(50)
or, more precisely, as some representative of pp(y,(t)), and obtain for y(t) E Fet +(M) the Cauchy problem
-i 308
+ Hp - 0,
(51)
*Y(0) = Koo,
(52)
No -' M where K. is the canonical operator on the Lagrangian manifold j and H is*a pseudo-differential operator in Fst +(M) with the principal symbol E fl. :
'
Theor:m 1 of Section 3:D guarantees that the problem (51) - (52) has a solution and that (T) z 0. Therefore, substituting the operator (44) into (36), we obtain
foG = 1-p
T
1-p(A) +p(A) -p(T)
T d
(t)dt 1 (mod Op (x)).
The theorem is therefore proved.
309
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