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M. A. Shubin
Pseudodifferential Operators and Spectral Theory
M. A. Shubin
Pseudodifferential Operators and Spectral Theory
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M. A. Shubin
Pseudo differential Operators and Spectral Theory Second Edition Translated from the Russian by Stig I. Andersson
Springer
Mikhail A. Shubin
Department of Mathematics Northeastern University Boston, MA 02115, USA
email: [email protected] Stig 1. Andersson
Institute of Theoretical Physics University of GSteborg 41296 Goteborg, Sweden
Title of the Russian original edition: Psevdodifferentsialnye operatory i spektralnaya teoria Publisher Nauka, Moscow 1978 The first edition was published in 1987 as part of the Springer Series in Soviet Mathematics Advisers: L. D. Faddeev (Leningrad), R. V. Gamkrelidze (Moscow) Mathematics Subject Classification (2000):35S05,35S30,35P20,47G30,58)40
Library of Congers CaulogiagmPublicaum Data Shubin. M. A. (Mikhail Aleksardmvich). 1944. rPaevdodiffemrsiarnye operatory i spektral'nais teoriia. English) Pseudodifteremia) operators and spectral theory / M.A. Shubin : translated fio® the Russian by Stig 1. Anderson. 2nd ad. p. rxn. ISBN 354041195X (softcover : alk. paper)
1. Pseudodiffe emial operators. 2. Spectral theory (Mathematia) QA381 .54813 2001 515'.7242dc21 2001020695
ISBN 354o41195X SpringerVerlag Berlin Heidelberg New York ISBN 3540136215 1st edition SpringerVerlag New York Berlin Heidelberg
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from SpringerVerlag. Violations are liable for prosecution under the German Copyright Law. SpringerVerlag Berlin Heidelberg New York a member of BertelsmannSpringer Science+Business Media GmbH httpzf/www.springer.de 0 SpringerVerlag Berlin Heidelberg 1987, 2001 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Daten and LichtsatzService, Wilrzburg Cover design: design 6 production GmbH, Heidelberg 44/3142LK  5 4 3 21 0 SPIN 10786064 Printed on acidfree paper
Preface to the Second Edition I had mixed feelings when I thought how I should prepare the book for the second edition. It was clear to me that I had to correct all mistakes and misprints that were found in the book during the life of the first edition. This was easy to do because the mistakes were mostly minor and easy to correct, and the misprints were not many. It was more difficult to decide whether I should update the book (or at least its bibliography) somehow. I decided that it did not need much of an updating. The main value of any good mathematical book is that it teaches its reader some language and some skills. It can not exhaust any substantial topic no matter how hard the author tried. Pseudodifferential operators became a language and a tool of analysis of partial differential equations long ago. Therefore it is meaningless to try to
exhaust this topic. Here is an easy proof. As of July 3, 2000, MathSciNet (the database of the American Mathematical Society) in a few seconds found 3695 sources, among them 363 books, during its search for "pseudodifferential operator". (The search also led to finding 963 sources for "pseudodifferential operator" but I was unable to check how much the results of these two searches intersected). This means that the corresponding words appear either in the title or in the review published in Mathematical Reviews. On the other major topics of the book the results were as follows: Fourier Integral operator: 1022 hits (105 books), Microlocal analysis: 500 hits (82 books), Spectral asymptotic: 367 hits (56 books), Eigenvalue asymptotic: 127 hits (21 books), Pseudodifferential operator AND spectral theory: 142 hits (36 books). Similar results were obtained by searching the Zentralblatt database. And there were only 132 references (total) in the original book. So I decided to quote here additionally only three books which I can not resist quoting (in chronological order): 1. J. Bruning, V. Guillemin (eds.), Mathematics Past and Present. Fourier
Integral Operators. Selected Classical Articles by J.J. Duistermaat, V. W. Guillemin and L. Hormander., SpringerVerlag, 1994. 2. Yu. Safarov, D. Vassiliev, The Asymptotic Distribution of Eigenvalues of Partial Differential Operators, Amer. Math. Soc., 1997. 3. V. Ivrii, Microlocal Analysis and Precise Spectral Asymptotics. SpringerVerlag, 1998.
Preface to the Second Edition
VI
These books fill what I felt was missing already in the first edition: treatment of more advanced spectral asymptotics by more advanced microlocal analysis (in particular, by Fourier Integral operators). By the reasons quoted above I did not add anything to the old bibliography at the end of the book, but I made the references more precise whenever this was possible. In case of books I added some references to English translations and also switched the references to the newest editions when I was aware of the existence of such editions. I made some clarifying changes to the text in some places where I felt these
changes to be warranted. I am very grateful to the readers of the book who informed me about the places which need clarifying. Unfortunately, I did not make the list of those readers and I beg forgiveness of those whom I do not mention. However, I decided to mention Pablo Ramacher who was among the most recent and most thorough readers. His comments helped me a lot. I am also very grateful to Eugenia Soboleva for her selfless work which she generously put in helping me with the proofreading of the second edition. I hope that my book still has a chance to perform its main function: to teach its readers beautiful and important mathematics. March 21, 2001
Mikhail Shubin
Preface to the Russian Edition The theory of pseudodifferential operators (abbreviated `PDO) is comparatively young; in its modem form it was created in the midsixties. The progress
achieved with its help, however, has been so essential that without `PDO it would indeed be difficult to picture modem analysis and mathematical physics. `PDO are of particular importance in the study of elliptic equations. Even the simplest operations on elliptic operators (e.g. taking the inverse or the square root) lead out of the class of differential operators but will, under reasonable assumptions, preserve the class of'FDO. A significant role is played by 'I'DO in the index theory for elliptic operators, where `PDO are needed to extend the class of possible deformations of an operator. Y'DO appear naturally in the reduction to the boundary for any elliptic boundary problem. In this way, `I'DO arise not as an endinthemselves, but as a powerful and natural tool for the study of partial differential operators (first and foremost elliptic and hypoelliptic ones). In many cases, `PDO allow us not only to establish new theorems but also to have a fresh look at old ones and thereby obtain simpler and more transparent formulations of already known facts. This is, for instance, the case in the theory of Sobolev spaces. A natural generalization of `PDO are the Fourier integral operators (abbreviated FIO), the first version of which was the Maslov canonical operator. The solution operator to the Cauchy problem for a hyperbolic operator provides
an example of a FIO. In this way, FIO play the same role in the theory of hyperbolic equations as'I'DO play in the theory of elliptic equations. One of the most significant areas for applications of `PDO and FIO is the spectral theory of elliptic operators. The possibility of describing the structure of various nontrivial functions of an operator (resolvents, complex powers, exponents, approximate spectral projection) is of importance here. By means of `PDO and FIO one gets the theorem on analytic continuation of the function of an operator and a number of essential theorems on the asymptotic behaviour of the eigenvalues. This book contains a slightly elaborated and extended version of a course on `PDO and spectral theory which I gave at the Department of Mechanics and Mathematics of Moscow State University. The aim of the course was a complete presentation of the theory of'PDO and FIO in connection with the spectral theory of elliptic and hypoelliptic operators. I have therefore sought to make the presentation accessible to students familiar with the standard Analysis course (including the elementary theory of distributions) and, at the same
VIII
Preface to the Russian Edition
time, tried to lead the reader to the level of modern journal articles. All this has required a fairly restrictive selection of the material, which was naturally influenced by my personal interests. The most essential material of an instructional educational nature is in Chapter I and Appendix 1, which also uses theorems from §17 and §18 of Chapter III (note that § 17 is not based at all on any foregoing material and § 18 is based only on Chapter I). We unite all of this conventionally as the first theme, which constitutes a selfcontained introduction to the theory of `I'DO and wave fronts of distributions. In my opinion, this theme is useful to all mathematicians specializing in functional analysis and partial differential equations. Let me emphasize once more that the first theme can be studied independently of the rest. Chapters II and III constitute the second and third themes, respectively. From Chapter II the reader will learn about the theory of complex powers and the function of an elliptic operator. Apparently the theorem on the poles of the i; function is one of the most remarkable applications of `PDO. The derivation of a rough form of the asymptotic behaviour of the eigenvalues is also shown in this chapter. In Chapter III there is a more precise form of the theorem on the asymptotic behaviour of the eigenvalues. This theorem makes use of a number of essential facts from the theory of FIO, also presented here. Let us note that it is in exactly this way that further essential progress in spectral theory was achieved, using, however, a more complete theory of FIO which falls outside the framework of this book (see the section "Short Guide to the Literature"). Finally, Chapter IV together with Appendices 2 and 3 constitute the final (fourth) theme. (Appendix 3 contains auxiliary material from functional anal
ysis which is used in Chapter IV and is singled out in an appendix only for convenience. Advanced readers or those familiar with the material need not look at Appendix 3 or may use it only for reference, whereas it is advisable for a beginner to read it through.) Here we present the theory of `I'DO in IR° which arises in connection with some mathematical questions in quantum mechanics. It is necessary to say a few words about the exercises and problems in this book. The exercises, inserted into the text, are closely connected with it and are an integral part of the text. As a rule the results in these exercises are used in what follows. All these results are readily verified and are not proved in the text only because it is easier to understand them by yourself than to simply read them through. The problems are usually more difficult than the exercises and are not used in the text although they develop the basic material in useful directions. The problems can be used to check your understanding of what you have read and solving them is useful for a better assimilation of concepts and methods. It is, however, hardly worthwhile solving all the problems one after another, since this might strongly slow down the reading of the book. At a first reading the reader should probably solve those problems which seem of most
IX
Preface to the Russian Edition
interest to him. In the problems, as well as in the basic text, apart from the already presented material, we do not use any information falling outside the framework of an ordinary university course. I hope that this book will be useful for beginners as well as for the more experienced mathematicians who wish to familiarize themselves quickly with 'PDO and their important applications and also to all who use or take an interest in spectral theory. In conclusion, I would like to thank V.I. Bezyaev, T.E. Bogorodskaya, T.I. Girya, A.I. Gusev, V.Yu. Kiselev, S.M. Kozlov, M.D. Missarov and A.G. Sergeev who helped to record and perfect the lectures; V.N. Tulovskij who communicated to me his proof of the theorem on propagation of singularities and allowed me to include it in this book; V.L. Roitburd who on my request has written Appendix 2; V.Ya. Ivrii and V.P. Palamodov who have read the manuscript through and made a number of useful comments and also all those who have in any way helped me in the work. M.A. Shubin
Interdependence of the parts of the book Chapter I
I §18
i
Chapter 11
Chapter IV
Chapter III
Appendix 2
I §17
H Appendix I
I
I
Appendix 3
Preface to the English Edition There are so many books on pseudodifferential operators (which was not the case when the Russian edition of this book appeared) that one naturally questions the need for one more. I hope, nevertheless, that this book can be useful because of its selfcontained approach aimed directly at the spectral theory applications. In addition it contains some ideas which have not been described in any other monograph in English. (I should mention, for instance, the approximate spectral projection method which is a universal method of investigating the asymptotic behaviour of the spectrum  see Chapter IV and also a review paper of Levendorskii in Acta Applicandae Mathematicae'.) Certainly many new developments have taken place since the Russian edition of the book appeared. The most important ones can be found in the monographs listed below. September 3, 1985
M. A. Shubin
References
Egorov, Yu. V.: Linear partial differential equations of principal type. Mir Publishers, Moscow 1984 (in Russian). Also: Consultants Bureau, New York, 1986. Hi rmander, L.: The analysis of linear partial differential operators, v. IIV. SpringerVerlag, Berlin e.a. 19831985 Heifer, B.: Theorie spectrale pour des operateurs globalement elliptiques. Societe Math. de France, Asterisque 112, 1984 Ivrii, V.: Precise spectral asymptotics for elliptic operators. Lecture Notes in Math. 1100; SpringerVerlag, Berlin e.a. 1984 Kumanogo,H.: Pseudodifferential operators. MIT Press, Cambridge Mass. 1981 Reed, M., Simon, B.: Methods of modern mathematical physics, v. IIV. Academic Press, New York e. a. 19721979 Rempel, S., Schulze, B.W.: Index theory of elliptic boundary problems. AkademieVerlag, Berlin, 1982 Taylor, M.: Pseudodifferential operators. Princeton Univ. Press, Princeton, N.J. 1981 Treves, F.: Introduction to pseudodifferential and Fourier integral operators, v. I, II. Plenum Press, New York e. a. 1980
See also: Levendorskii, S.: Asymptotic distribution of eigenvalues of differential operators. Kluwer Academic Publishers, 1990.
Table of Contents
..... .......... .. ...... ..... .. .. ........ ......... . ...................... ............. . .......... ..... .... .........
Chapter I. Foundations of `PDO Theory §
1.
§ 2. § 3. § 4.
§ 5. § 6. § 7.
§ 8.
I
Oscillatory Integrals . . . . . . . . . . . . . . . . . . . . . . . . Fourier Integral Operators (Preliminaries) 10 The Algebra of Pseudodifferential Operators and Their Symbols . 16 Change of Variables and Pseudodifferential Operators on Manifolds 31 Hypoellipticity and Ellipticity 38 Theorems on Boundedness and Compactness of Pseudodifferential Operators 46 The Sobolev Spaces 52 The Fredholm Property, Index and Spectrum 65
Chapter II. Complex Powers of Elliptic Operators
....... .... ..
§ 9. Pseudodifferential Operators with Parameter. The Resolvent . § 10. Definition and Basic Properties of the Complex Powers of an Elliptic Operator §11. The Structure of the Complex Powers of an Elliptic Operator § 12. Analytic Continuation of the Kernels of Complex Powers . . . . § 13. The CFunction of an Elliptic Operator and Formal Asymptotic Behaviour of the Spectrum § 14. The Tauberian Theorem of Ikehara § 15. Asymptotic Behaviour of the Spectral Function and the Eigenvalues (Rough Theorem) .
..................... .....
1
77 77 87 94 102
...... . .... ... .. ..... 112 ..
.. ....... ..... .......... .
Chapter III. Asymptotic Behaviour of the Spectral Function
.. .. ..
120 128 133
§ 16. Formulation of the Hormander Theorem and Comments . . . . . 133 § 17. Nonlinear First Order Equations 134 § 18. The Action of a Pseudodifferential Operator on an Exponent . . . 141 § 19. Phase Functions Defining the Class of Pseudodifferential Operators 147
...... .. .... ... ..
§20. The Operator exp(itA)
........ ....... ......... ..
§21. Precise Formulation and Proof of the Hormander Theorem §22. The Laplace Operator on the Sphere
150 156
........... ..... 164
XII
Table of Contents
Chapter IV. Pseudodifferential Operators in IR"
............ 175
§23. An Algebra of Pseudodifferential Operators in IR. §24. The AntiWick Symbol. Theorems on Boundedness
and Compactness
........ 175
......................... .
186
§25. Hypoellipticity and Parametrix. Sobolev Spaces. The Fredholm §26. §27. §28. §29. §30.
......... ................... ........193 197 .................... 202 ..................... 215
Property Essential SelfAdjointness. Discreteness of the Spectrum Trace and Trace Class Norm The Approximate Spectral Projection . . . . . . . . . . Operators with Parameter Asymptotic Behaviour of the Eigenvalues . . . . . . . .
Appendix 1. Wave Fronts and Propagation of Singularities
.
.
.
.
.
206
.
.
.
.
.
223
....... 229
........ ........ ..................... 269 ............................... 275 ............................. 285 ..... ..........................
Appendix 2. Quasiclassical Asymptotics of Eigenvalues
240
Appendix 3. HilbertSchmidt and Trace Class Operators
257
A Short Guide to the Literature Bibliography
Index of Notation Subject Index
287
Chapter I Foundations of `I 'DO Theory §1. Oscillatory Integrals 1.1 The Fourier transformation. The simplest example of an oscillatory integral is provided by the Fourier transform of a function (or distribution) of tempered growth. Let S (IR") be the Schwartz space of functions u (x) c C' (R') all derivatives of which decrease faster than any power of I x I as I x
oo, i. e. for
arbitrary a, fi sup I X, (DO U) (x) I < + oo .
(1.1)
XER'
As usual x here stands for (x, , ... , x"); a and fi are multiindices, so for example a= (a...... a") and a , is a nonnegative integer; xe = x,  . . . a = (a, , ... , a") a a 181
where a, =
a xj
; as = a; ... a.0 =
axwith If I
M"...
+ $,. The
"
left hand sides of (1.1) define a collection of seminorms in S(IR") which turn S(IR") into a Frechet space. The Fourier transform of a function u (x) e S (IR") is given by the formula
f e' u(x)dx,
(1.2)
i = 1/1 and dx=dx,...dx" is
where HEIR", x
Lebesgue measure on IR". The integral in (1.2) is taken over the whole of 1K",
which will always be the case unless a domain of integration is explicitly indicated. It is well known that the operator F defines a linear topological isomorphism
F:S(IR")+S(IR") and that the inverse operator (the inverse Fourier transformation) is given by the inversion formula (F'4) (x) = u (x) = ! e,x ' e u
where d = (2n)"dd, ... dd..
d,
(1.3)
Chapter 1. Foundations of I'DO Theory
2
Now we are going to show how to extend the Fourier transformation (1.2) to
continuous functions u(x) satisfying the following condition: there exist constants C > 0 and N > 0 such that I U (x) 15 C',
(1.4)
where (x) stands for (1 + IxI2)112 and Ix12 = x2 + ... + x.. We will define u(l;) E S' (IR"), the dual space of S(IR"), i.e. the space of all continuous linear functionals on S(IR"). So we want to regularize the integral
= If e` "
(1.5)
eS(lR"), an integral which we will also regard as the value of the functional u at the element If u(x)eS(1R") it is obvious that with
since in this case (1.5) converges absolutely. We give two equivalent means of regularizing (1.5) both differing from the wellknown method, based on the Parseval identity, and both extendable to considerably more general situations.
First method. Put D; =
a
i1 d xj
, D = (DJ , ... , D") and _ (1 + D; + .. .
+ D,,) "2 (usually we will make use of k with k a non negative even number so
that k becomes a differential operator). The vector D will also be used to indicate differentiation in the variable. To avoid confusion we then denote by D. the just described vector D and by D, the same vector but acting on the variable. We have e"x.4 = kkelx4.
(1.6)
To begin with, suppose that u (x) e S (IR"). Then inserting this expression for e4'4 in (1.5) and integrating by parts, we obtain
= Ile
,x*cu(x)kkV/
(S)dxdd.
(1.7)
This integral is now defined not just for u (x) e S (R") or for absolutely integrable u (x). Indeed, if u (x) satisfies (1.4) and k > N + n, then (1.7) converges absolutely
and we can consider it as the required regularization of (1.5). Exercise 1.1. Verify that formula (1.7) defines a continuous linear functional u e S' (]R") fork > N + n.
Second method. Suppose that qp (x) a Co (IR") (the space of compactly supported infinitely differentiable functions on IR") and that V (0) = 1. Put
§1. Oscillatory Integrals
3
E>0.
1
(1.8)
This integral converges absolutely. It turns out that there is a limit I= lim I, »o
independent of the choice of W (x). Indeed, carrying out in (1.8) the same integration by parts as before, we get
IL= J$eix r (ex) u(x)rt and if k > N + n, by the Lebesgue dominated convergence theorem, the limit as E0 exist, and equals as defined by formula (1.7).
Exercise 1.2. Verify that for different values of k formula (1.7) leads to the same functional u. 1.2 Definition of the oscillatory integral and its regularization. Now consider an integral more general than (1.5) 1° (au) = J J
e'°(x.6)
a (x, 0) u (x) dx dO.
(1.9)
Here 0 E IR", x E X, where X is an open set in IR" and u (x) a Co (X), i.e. u (x) a Co" (X) and there is a compact set K c X such that u 1 x\ K = 0. To describe a (x, 0) and 4P (x, 0) we introduce a number of definitions.
Definition 1.1. Let m, a and b be real numbers; 0< b 5 1, 0:5 Q 5 1. The class S,',6 (X x IR") consists of functions a (x, 0) a Cm (X x IRN) such that for any multiindices a, fi and any compact set K c X a constant Ca, a. K exists for which (1.10)
10,'00a(x,0) 1:_5 C..B.K 0k+1, ..., 0,, 101)dt, j=1
0
where au, = a d eSQ a °(B x 1R+). It remains to carry out the inverse
substitution.
10
Chapter 1. Foundations of `PDO Theory
Proof of Theorem 1.3. Assume that Q + b = 1. Then, if a I c. = 0, by lemma 1.3 with 0; = a8 , we may represent a (x, 0) in the form
a= Y a; ;_,
a
a;eSQ a"(U).
a0;
(1.27)
ao
However, taking into account that e'm= i a e'm, we obtain, on inteao; ao; grating by parts, that
But a©; e SQ d a  °(U), demonstrating the second statement of the theorem. From
this proof it is obvious, that if a (x, 0) had a zero of infinite order in C, then b (x, 0) could also be chosen to possess this property. So in proving the first statement we can assume a (x, 0) eSQ ' (X x R"), M as large as desired. But then the integral (1.21) converges absolutely and uniformly in x as do the integrals obtained from it by differentiation of degree S 1 (M), where I (M) + + oo as
M  + oo, and hence the smoothness of A (x) follows. D Exercise 1.10. Prove theorem 1.3 when the second of the assumptions (1.22) is fulfilled (0 (x, 0) linear in 0).
Hint. It amounts to applying of part c) of Lemma 1.2.
§2. Fourier Integral Operators (Preliminaries) 2.1 Definition of the Fourier Integral operator and its kernel. Let X, Y be open sets in IR'x and IR"r. Consider the expression
Au(x) =
a(x, y,0) u(y) dydO,
(2.1)
where u (y) E Co (Y), x e X, 0 (x, y, 0) is a phase function on X X Y X IR"' and
a(x,y,0)eSm(Xx Yx1R') with Q>O and b
with u e 8'(Y), v e Co (X) we are done.
Exercise 2.2. Verify that the operator A, defined in this way, is indeed an extension by continuity of the map (2.3). So an F10 A with operator phase function dy maps Co (Y) into C'(X) and 1'(Y) into 2'(X). We now study the change in the singular support under the action of A. Let us settle a notation. If X and Y are two sets, S a subset of X x Y and K a subset of Y, then S o K is the subset of X consisting of the points x E X, for which there exists a y e K with (x, y) c S. Theorem 2.1. The following inclusion holds:
sing supp Au c So ° sing supp u
(2.10)
where Sm = (X x Y)\ Rm consists of those pairs (x, y) for which there exists a 0 eIRN\0 with 0; (X, y,©) = 0.
§2. Fourier Integral Operators (Preliminaries)
13
Proof Splitting U E 9'(Y) into a sum of a function in C, '(Y) and a distribution with the support in a neighbourhood of sing supp a we see that it suffices to demonstrate that sing supp (Au) c S. o supp u. Let K = supp u and K' an arbitrary compact set in X not intersecting So o K and so that K' x K C Rm. Since R. is open, there are open neighbourhoods S2 and ST of the compact sets K and K' respectively such that ST x S2 C R,. So it suffices to verify that Au E C°°(Sl'). But this is evident, since KA(x, y) E C°°(R,) and in particular, KA (X, y) E Coo (ST X S2).
Exercise 2.3. Verify the statement used above that if K4 E C (Q' x (2) then A maps d'(Q) into C°°(Q').
2.3 Example 1: The Cauchy problem for the wave equation. Consider the Cauchy problem (2.11)
f I(=o = 0
f' 1,=0 = u(x)
(2.12)
where x e IR", f = f (t, x), d is the Laplacian in x and  to begin with u (x) a Ca (1R"). We solve (2.11)(2.12) with the help of the Fourier transformation in x, putting je,,..cf(t,y)dy.
In this way, we have a
are
_
.JIt=o = 0,
(2.13)
.1j" It=o = uO
(2.14)
where u(i;) is the Fourier transform of u(x). From (2.13) and (2.14) we easily obtain that
f (t,
)=
(,),in t I1
Therefore by the Fourier inversion formula
f(t,x) = f f =ff
sin t14 lu(y)dyd e`1f1)u(y)dyd(;.
We would like to split the last integral into two parts separating the exponents ei'111 and el"41. However this would lead to a singularity at t = 0. To avoid E Co (lit"), such this singularity, let us again use a cutoff function X = x that x (!) = I near 0 and split the integral into three parts:
f (t, x) = f+(t, x)  f_(r, x) + r(t, x)
14
Chapter 1. Foundations of 4'1)0 Theory
f+(t,x) = f Q. x) = f
r(t,x) =
ffe'lc.,3'>t+rIt11(1
 x(4))(2i X (.))(2il I)dyd4, I1
sintI dyd4.
It is clear e.g. that f+ = Au where A is a FIO with the phase function
) (t, x, y, ) = (x  y) . This is an operator phase function. Since 45{ = x
+ t 1C.
y+
we have
Cm = (Q. X, y, ) : y x = tIII), sm=((t,x,y):Ixy12=t2).
The second term f_(t, x) can be similarly presented as f_ = Au, where A is a FIO with the phase function
1,
(3.14)
where C. does not depend on t, that is for cp (8/t) and t z I we have uniform in t estimates of class S10.0 . In fact aaq (0/t) _ (aa(p) (©/t)  t1a1
and
101 5 t5 2 101
for 0 esuppaegp(9/t), from which we obtain (3.14). Further from (3.14) follows that m'e 1a1 +6161
l aeax [1P (0/t) aj(x, e)J I < Cj
if x e K,
, t? I and 1 a I+ 1#1+1:5j. Let us observe now that 1 /21 f' (0) I for Ai 1 t l 5 1 /2 I f' (0) I, I t 151. Denoting
d = min
1 }we have I f'(t) I >1/21 f'(0) I for t e [ d, A]. We have
2A2
J
2A0? If(A) f(A)I ? 2A If'(O)I 2
and consequently, 11(0)1 5 2 e ° = 2Aomax
IA2
IT(O)i,
i}.
or If'(0)I 5 2A,, i.e. either If'(0)12 II (o)I 5 4AOA2 or If, (0) 12 S 4A2 and thus (3.17). O
This implies that either If'(0) 1
54ApA2.
Lemma 3.2. Let K, and K2 be two compact sets in lR° so that K, C Int K2 (the set of interior points in K2). Then there exists a constant C> 0, such that for any smooth function f on a neighborhood of K2, the following estimate holds
(sup
jaj=1
IDaf(x)I
sup If(x)I +ja1=2 sup Z I Daf(x)I(3.18)
Csup If(x)I (xc.Kz xeX2
Proof. Immediate from Lemma 3.1.
0
§3. The Algebra of Pseudodifferential Operators and Their Symbols
23
Pro of of Proposition 3.6. Let b  Y a, (such a function exists by Proposition j=3
3.5). Putting d (x, 0) = a (x, 0)  b (x, 0), we have for every compact set K c X the estimate
xeK,
Iaea,d(x,0)I < C',
(3.19)
where C and p depend on a, fi, K, and additionally
Id(x,0)I 5 C,', where C,= C,(K). Set do(x, ) = d(x,
xeK,
(3.20)
Then a{a° ds(x, x) 14 =0 = aBax d(x, 0),
and applying Lemma 3.2 with K, = K x 0, K2 = k x { I 151 }, where k is a compact set in X such that Int k =>K, we obtain from (3.20) d(x,0)I\2 ( sup
I0B0P
xQK iai+ipisl
C 0 and by C for m  (Q  b) N + by < 0 (in both cases C is independent of S, q and t). Taking into account the factor we obtain from (3.31) that for sufficiently estimated in absolute value by Cm (Q
8)N+av for
large v I
SC
f
P(1a1.
,
Iql>Ifl/2
where p = max { m  (Q  6)N, 0}. If p  (1 O v + n + 1 < 0, it follows that IRa.O.t(x,S)I S
CP(1a)v+n+1 jn1 dq S
C doesn't depend on x, and t (for x e K, t e (0,1 ]). Selecting a large enough v we can make the exponent in (3.32) p  (1 b) v + n + I negative and as large as we like in absolute value. Taking (3.29) and (3.30) into account, we obtain for R,,, the estimate C«)'"(Qd/N+.,
xeK,
te(0,1],
which ensures the applicability of Proposition 3.6 and so finishes the proof. U Remark. The method of proof of Theorem 3.1 is very typical for the theory of `PDO and the corresponding arguments are to be found in all versions of this theory independently of the mode of presentation. We therefore strongly urge the reader to carefully study the proof of this key theorem.
26
Chapter I. Foundations of'I'DO Theory
3.5 The symbol of the transposed operator and the dual symbol. The transposed operator 'A is defined by = (x, y, 0) and a(x, y, 0) E Saa (X x X x IR") (cf. formula (2.1)), where
1QSS 5'(x),)
(4.21)
A
where 'P (x, n) is a polynomial inn of degree no higher than 1#1/2 (with C°° (X)coefficients) independent of A and where 4P0  1. It remains to show that these polynomials are given by (4.13).
We will compute the polynomials P (x, q) with the help of differential operators. For the differential operator A we have iy' q
ly=x(x) CA,(Y,q)ly=x(x)e = eix(=)'" aA(z D:) eix(x) "I 1:=x A)ely."
(4.22)
(here a, (z, D:) denotes the operator A, acting on the variable z). We write now x (Z) = X (x) + X' (x) (z  x) + Xi (Z),
from which
(z) qx'`X'(x)n.
X(z)
Putting this into formula (4.22), we obtain 0 A, (Y, i) l y=x(x) = e' ',(x), {aA (Z, Dz) eis 'x'(x)q eix;(Z) q,l I. x
(4.23)
Now use the Leibniz rule (3.46) (Exercise 3.6) to differentiate the product of two exponents in (4.23). We then obtain clearly
aA,(X(x),n)=Y
1
aA(a)
(x,fX'(x)n)'Da
a)
:eix:(:)
I==x
(4.24)
(we have used here yet another obvious formula for differentiating a linear exponent : aA (z, D:) a i=  c =
e114
aA (z, c))
Formula (4.24) signifies the validity for differential operators of (4.13) for
the polynomials P] (x, q) in (4.21). But in view of the universality of the polynomials 0a(x,n), then (4.13) is valid also in the general case. Examples. 00 = 1, 4i,=O for I P 1= 1, bg (x, )7) = Dr (i x (x) n) for I Ii I = 2. Corollary 4.1.
0A,(Y,n)  aA(XI(Y), (N
i(Y))ln) ESQ a
z(o)lz)(X1 xlR").
(4.25)
This statement shows, that modulo symbols of lower order, the symbols of all operators obtained from A by a change of variables form a welldefined function on the cotangent bundle TX.
36
Chapter I. Foundations of `l'DO Theory
Corollary 4.2. If A E CL"(X ), then A, e CL'" (X, ).
Proof. Obvious from formula (4.11).
4.3 Pseudodifferential operators on a manifold. Let M be a smooth ndimensional manifold (of class C"'). We will denote by C' (M) and Co (M) the space of all smooth complexvalued functions on M and the subspace of all functions with compact support respectively. Assume that we are given a linear operator
A: C(M)C(M). If X is some chart in M (not necessarily connected) and x: X . X, its diffeomorphism onto an open set X, c 1R", then let A 1 be defined by the diagram
C( (X)
'' i C°°(X)
CC (X,)
"' C"(X1)
(note, in the upper row is the operator ra.o A o ia., where ix. is the natural embedding iX: Co (X)  Co (M) and rx is the natural restriction rX: C"(M)  C' (X); for brevity we denote this operator by the same letter A as the original operator).
Definition 4.1. An operator A: C o (M) . C°` (M) is called a pseudodifferential operator on M if for any chart diffeomorphism x: X  X, the operator A I defined above is a `'DO on X 1.
Theorem 4.1 shows that the `PDO on an open set X C IR" for 1Q 3 < Q are `PDO on the manifold X. Furthermore, from Lemma 1.2, we see that the class of symbols SQ a(T*M), as well as the class of operators LQ,,,(M), are welldefined for I  o 0) in the parametrix of the classical elliptic operator A in the scalar case can be expressed via am ,t (x, 4) by 2j+1
,
b_mj(x, ) = Y c,(x, ) (am(x,
(5.19)
1=2
where c, (x, ) is a function positive homogenous of degree m (11) j in , polynomial in the functions am , a.,, ..., am _; and their derivatives of order 0
(6.5)
iti W xeK
for an arbitrary compact set K c X. Therefore we derive Theorem 6.2 from the following proposition. Proposition 6.1. Let C e LQ, a (X) and be properly supported, 0 5 6< Q 5 I and let C* = C and assume lim Re ac(x, ) > 0 fEK
for arbitrary compact sets K c X. Then there exists a properly supported operator
B E LQ4(X) such that R = B*B  C has a C°° kernel. Lemma 6.1. Let a (x, ) a SQ a (X x IR") and let a (x, c), for arbitrary (x, ) e X x IR", take values in a compact set K c C. Let a complexvalued junction
f(z) be defined on a neighbourhood of K and be infinitely differentiable as a function of two real variables Re z and Im z. Then
f (a (x, )) E SQS(X X M')
(6.7)
Proof.a Denote u = Rez and v = Im z. Then we evidently have vyJ (a(y)) =
Y_
x Op (Rea) ... ay.(Rea) from which (6.7) follows, since I (a;pvg1 f) (a (y))1:5 Cpq .
(a(y)) x
a;.(Ima),
(6.8)
1j
Proof of Proposition 6.1. It follows from Lemma 6.1 that j/Re ac (x, ) belongs to SQ,,, for large . Therefore there exists a properly supported WM Bo E L°q a(X), such that if bo (x, ) is its symbol then
From this it follows that C  Bo B0 a L; (61? 6)(X).
(6.9)
The operator Bo serves as the "zero order approximation" to B. We will seek a first order approximation in the form Bo + B, , where B, e L.1 °"(X).
§6. Theorems on Boundedness and Compactness of Pseudodifferential Operators 49
We have
C (Bo + Bi) (Bo + B,) = (C Ba Bo)  (Bo B, + B* Bo)  Bi B,
.
(6.10)
The point is to reduce the order of the operator on the lefthand side, taking B1 to be properly supported with symbol b1 (x, ), such that for large
2b, (x, ) bo(x,) = ac_(x,),
(6.11)
which is obviously possible, since by Lemma 6.1 bo '(x, ) E S.O. a for large . It follows from (6.10) and (6.11), that
C(Bo+B,)* (Bo+B,)EL..(6.12) Arguing by induction, we may in exactly the same way construct properly
supported `PDO B; E L(X), j = 0, 1, 2, ..., such that C  (Bo+ ... +B,)* (Bo+ ... +B;) ELQ aQs'(X).
(6.13)
Now let b1(x, ) be the symbol of B3 . It only remains to construct a properly
supported operator B, such that Y b, (x, ) i=o
It follows easily from (6.13), that this operator will be the one we are looking for.
Thus Proposition 6.1 is proved and together with it Theorems 6.1 and 6.2. D
6.3 The compactness theorem. We will derive the compactness theorem from the following much more general statement. Theorem 6.3. Let A E LQ,a(IR"), 0:!9 6 < q:5 1, let the kernel K,, have compact
support in IV x IR" and let the symbol a,, (x, ) satisfy I'M I a,, (x, c) I < M'
(6.14)
Then there exists an operator A, such that A  A, E L (IR"), the kernel K,, has compact support and
IIA,uII S MIIuII,
ueCO (R").
(6.15)
P r o o f. Let X e C o (IR) be such that X (x) ? 0, f X (x) dx = 1, 0 < X 1. Such a function can be found. Indeed, to begin with let the function Xo (x) be such that Xo (x) a C o (IR"), Xo (x) z 0 and J Xo (x) d x = 1. Then obviously I Xo 1. Put now X (x) = f Xo (x+ y) Xo (y) dy. In view of the fact that j (e) = 110 ( ) I Z the I
function X (x) fulfills all the requirements.
50
Chapter I. Foundations of `YDO Theory
Now put xL(x) = E"x(x/E) and define the operator AL by (6.16)
AL u = Au  A (xL * u),
where (xL * u) (x), the convolution of xL and u, is defined by
(XL*u)(x)= 1x.(xy) u(y)dy= ju(xy) X.(y)dy Now, in view of Theorem 6.2
IIALuII2 s M2IIuxL*u112+ (R(uxL*u), uxL*u),
(6.17)
where R is an operator with kernel R (x, y) e Co (IR" x Ilk"). Note that the Fourier transform of u  xL * u is (1 2 (En)) u ( ) and from the condition 0 < X < 1, it follows that
IIuxL*ull < (lull.
(6.18)
Further, denote by RL the operator which maps u into R (u  xL * u), then its kernel is given by the formula
RL(x,y)= R(x,y)  IR(x,z)
E)
or
RL (x, y) = R (x, y)  I R (x, y + t z) x (z) dz,
from which it is obvious, that supp RL (x, y) lies in some fixed compact set K (independent of z for 0 < &:5 1) and, in addition, for
sup X. Y
It follows that II RL II i 0 for a  0. We now obtain from (6.17) and (6.18) that IIALu112 < M2IIu112+
IIRLUII (lull.
(6.19)
From the conditions of the theorem it is evident that we may replace M by M  6, where b is sufficiently small. But then it follows from (6.19), that for sufficiently small z > 0 11ALuII2 s M211u112
Put A, = AL . Since the symbol of the convolution operator with xL is (Ed) e S  (1R" x IR"), it is evident that A  A, a L W (IR"). It is also easily
verified, that the kernel KA of A, has compact support.
§6. Theorems on Boundedness and Compactness of Pseudodifferential Operators 51
Theorem 6.4. Let A e LQ (IR"), 0 s' and K a compact set in M. Then the embedding operator
is": Hs(K) Hs'(K) is a compact operator.
Proof. By the equality (7.8), we obtain
§7. The Sobolev Spaces
61
(A: A_,) (A. u)  (A: R,) u _ (As A,) (A. u)  (A.,, A
(A, R. u) + (As R,) (R,u)
(3'1(M) (hence from Corollary 6.1 for any compact set K, Since A; A_,eL1.0 one can find a compact set K2, such that A,. A _, is a compact operator from L2(K1) to L2(K2)) it is clear that if u runs through a bounded set in H'(K) (and, consequently A,u and R,u run through a bounded set in L2(K,)) then As u runs through a precompact set in L2(K2). Similarly one shows that in this case QR,,,u runs through a precompact set in L2(K2) for any differential operator Q. But this, in view of the equality (7.8) and the definition of the norm, implies the compactness of the corresponding set in H' (K2), hence in H"(K), since, actually, it belongs to H" (K) and the topology in H"(K) is induced by the
one in H(K2) provided K C K2. A generalization of Theorem 7.4 is Theorem 7.5. Let A e L"Q, a (M), A properly supported, either I  Q < b < Q or b < Q and M = X, an open set in IR'. Let the numbers s, s' e IR be such that s' < s  m. Let K be a compact set in M and k a compact set in M (depending on K) such
that A 4f '(K) a r'(R). Then the operator A: H5(K)  Hs'(k) is compact.
Proof. Theorem 7.5 is a consequence of Proposition 7.5 and Theorem 7.4, since
the operator A: H'(K) , H" (k) can be viewed as a composition
H'(K)
A H'M(R)
H''(R).
Cl
Denote by CP(M) the space of functions on M having continuous derivatives of order Sp in any local coordinates. The topology in CP(M) is defined by the seminorms IIu1IA,K= sup I Au(x)
(7.16)
xeX
where A is any differential operator of order Sp. We denote by Cg(K) the subspace of the functions u e CP(M) with supp u c K. It is clear that the topology of CP(M) induces a topology on Cg(K), which can be given by a Banach norm.
Theorem 7.6. If s > n/2 + p, then H;a(M) c CP(M) is a continuous embedding. If K is a compact set in M, then the embedding H'(K) c Cg (K) is a compact operator under the same assumption s > n/2 + p.
Proof. Since differential operators of order p are continuous maps H'(K)  H'P(K), it is obvious that it suffices to consider the case p = 0. Further, it suffices to verify that for s > n/2, we have a continuous embedding
Chapter I. Foundations of PDO Theory
62
Hs(K) c Co(K), since the compactness of this embedding is obtained by writing it as a composition
H'(K) c H'`(K) c Co (K) (e > 0 such that s  e > n/2) and using Theorem 7.4. Finally, it is clear that it suffices to consider the case of K lying inside a chart, i.e. the question reduces to the case M = IR". Thus, let K be a compact set in IR" and s > n12. It follows from Lemma 7.3,
that it suffices to prove the estimate sup Iu(x)I < C11u115,
u c Co (K),
(7.17)
xCR'
where C does not depend on u. We will prove this estimate with C even independent of K. We have
lu(x)I= I je1z fu( dal [jlu( )12< >z'd where C =
+oo, as required.
Corollary 7.4. f Hja(M) = C(M). This corollary is obvious. Let us also note the dual fact: U Hs(K) = f'(K) for any compact set K c M. This fact follows from the wellknown statement of distribution theory, that if u e d'(K), then u can be written as u = Y_ Q;v;,
where v; E L2 (k), k is compact and Q J are differential operators. if If m is the
greatest order of the Q;, then uEH'(K). 7.7. Duality. Let there be given a smooth positive density du on M. This defines a bilinear form = j u (x) v (x) du (x),
(7.18)
for instance, if u r= Co (M) and v E C°° (M).
Theorem 7.7. The bilinear form (7.18) extends for any s e IR to a pairing (separately continuous bilinear mapping)
H.'.P(M) x H,; (M)  C
(7.19)
which we will denote as before by < , >. The spaces Ho,,,P and HL are dual to each
other with respect to this pairing, i.e. any continuous linear functional 1(u) on can be written in the form for some v e H1« '(M), and any
§7. The Sobolev Spaces
63
continuous linear functional 1(v) on H,« (M) can be written as , where If the manifold M is closed, then the transformation which attaches
u
to any v eH'(M) the linear functional is an invertible linear continuous operator from H'(M) into (H'(M))* (where the latter space is endowed with the natural Banach space topology).
Proof. 1. First let us verify that the form (7.18) extends to the pairing (7.19).
Note that the operator A appearing in the definition of the Sobolev space can be chosen symmetric with respect to the given density, i.e. such that = s').
Exercise 7.3. Let T' = R"/2,r71" be the ndimensional torus (7Z" is the lattice of points with integer coordinates in ]R"). If f E C°°(lr) (= Co (1r)), then f decomposes into a Fourier series fk e",
f(x) =
(7.22)
keZ*
where fk are the Fourier coefficients, given by the formula
f = (2n)" $ f(x) e"dx.
(7.23)
The same formula also applies to f e 9'(1'") if as the integral in (7.23) we take the value of the functional f at the function ejk (in this case the series (7.22) converges in the weak topology of then the condition f c H'(Y") is equivalent to Show that if f keZ'
1112(1+Ik12)' 0. Then Ao is a selfadjoins operator in L2(M) and there exists in this space a complete orthonormal system {tp; }, j = 1, 2, ... of eigenfunctions of Ao. Here (pi E CO0(M), Acp; = Atp; and the eigenvalues A; are real, with IA; I + +oo as j  +oo. The spectrum or (A) coincides with the set of all eigenvalues. Proof. Note first of all that a (A) c IR in view of Proposition 8.5, since A is symmetric on C' (M) and can thus have no nonreal eigenvalues. Next, we want to show that a (A) * IR. Assuming a (A) = IR then we could for any A e IR find a function cpx e C°° (M), such that Aqpj = Aqp,, and II tpx II = 1
But then ((p,, cp,) = 0 for A * a by the symmetry of A, contradicting the separability of LZ(M). Now take A. a IRS a (A). By Theorem 8.2, R4 = (A  AO 1) ' is a compact selfadjoint operator in LZ (M). By a known theorem from functional analysis there is an orthonormal basis { v;) , of eigenfunctions, where the eigenvalues r,
tend to 0 as j + oo. Now note that r; 4 0 (since Ker (A  ).o 1) ' = 0). The condition R,,otp; = r; q7;
can therefore be rewritten in the form (AAoI)(p; = r1 'cp;
or
A(Q;=(r,i +Ao)rp;.
(8.11)
It is obvious from (8.11) that tp; E C °° (M) and the V, are eigenfunctions of A with
eigenvalues A, = r; ' + A0. It is also clear that I A; I . + oo as j' + oo. The remaining assertions of Theorem 8.3 are obvious. The fact that the spectrum a (A) coincides with the set of all eigenvalues {A;} follows from Proposition 8.5 and the selfadjointness from the representation A = Rxo + Aol.
The following theorem extends one of the statements of Theorem 8.3 to the nonselfadjoint case.
Theorem 8.4. Let A e HL'" h (M), 1  e 0. Then for the spectrum or (A), there are two possibilities:
a) a (A) = C (which, in particular, is the case if index A 4 0); b) a (A) is a discrete (maybe empty) subset of C (subset without limit points). If b) holds and AO e a (A) then there is a decomposition L2 (M) = E,, (D EL such that the following conditions are satisfied: 1) E,1, c C I (M), dim E,. < + oo, and E4 is an invariant subspace of A such that there exists a positive integer N > 0 with (A  A0 I)" E,. = 0 (in other words, the operator A I s,, has only the eigenvalue AO and is equal to the direct sum of Jordan cells of degree S N);
2) E,, is a closed subspace of L2(M), invariant with respect to AO (i.e. A (D,,.nE,) e and if we denote by A, the restriction AO I Ey, (understood as
Chapter 1. Foundations of'I'DO Theory
72
an unbounded operator in E;0 with domain DA° fl E;.O), then A'  AoI has a bounded inverse ( o r , in other w o r d s, A o ¢ a (A1Q)
Proof. 1. Let or (A) $ C. Let us prove that a (A) is a discrete subset in C. There is a point AOe C\ a (A) and we may, without loss of generality, assume that AO = 0, so that by Theorem 8.2 A0 has a compact inverse Ao '. Then since
A0  Al = (I).A0 ')A0 the inclusion A e a (A) is equivalent to A + 0 and A ' e a (Ao ' ). Discreteness of a (A) follows from the fact that a (A. ') may have
only 0 as an accumulation point. 2. Let a (A) * C, AO e a (A). Once again, without loss of generality, we may assume that A0 = 0. Let /'0 be a contour in the complex plane, encircling 0 and not containing any other points of a (A) (e. g. a circle, sufficiently small and with centre at the origin). Consider the operator 1
PO =
2I
I
(8.12)
Standard arguments (cf. Riesz, Sz.Nagy [1), ChapterXI) show that PO is a projection, of finite rank in view of the compactness of R, commuting with all the operators RA (and with AO in the sense that POAO c AOPO) and such that
if EEo = PO (L2 (M)), E2= (I  PO) (LZ (M)), then conclusions 1) and 2) of Theorem 8.4 hold. We leave it as exercise for the reader to take care about the details. We note only that the inclusion Eao C C°°(M) follows from Ao Ea° = 0 if we take into account the ellipticity of A and utilize the regularity Theorem 5.2. 8.4 Problems
Problem 8.1. Let E be a separable Hilbert space, nO (Fred (E, E)) the set of connected components of Fred (E, E) provided with the semigroup structure induced by the multiplication. Show that taking the index gives an isomorphism index: nO (Fred (E, E)) = Z.
Hint. An operator A of index 0 can be written in the form A = A0 + T, where A0 is invertible and T has finite rank. Show (by use of the polar decomposition) the connectedness of the group of all invertible operators in E.
In all the following problems M is a closed manifold and 1  g S S < g, m > 0.
Problem 8.2. Let A e HL'",1'(M). Prove that A is a Fredholm operator in
C(M), i.e. that dim Ker A < + oo, ACW (M) is closed in C(M) and dim CokerA < +oo, where Coker A= C°°(M)/AC°°(M). Show that AC`°(M) consists of all f c C`° (M) for which (f, g) = 0 for any g e Ker A' (here ( , ) is a scalar product determined by some smooth positive density and A* is the adjoint 'PDO with respect to this scalar product).
§8. The Fredholm Property, Index and Spectrum
73
Problem 8.3. Let A e HL!,":(M) and mo > 0. Let there be given on M a smooth positive density defining the scalar product(, )and the formal adjoint
'DO A. Assume A = A'. Let AO be the closure of the operator AIc.(M). Then Ao is selfadjoint in the Hilbert sense in the space L2(M). Problem 8.4. Let A e HLQ: s o(M), let A* be the formal adjoint operator and A0, Ao the closures of A I c. (M) and A I c.(M) in L2 (M) respectively. Show that AO
and Ao are adjoint to each other in the sense of the Hilbert space P(M). Hint. Consider the matrix of 'PDO'2I =
0 A
AO
'
Problem 8.5. Find an example of an operator A e HL ' (M) for which a (A) = C.
Problem 8.6. A sequence of Hilbert spaces E, and linear continuous operators d,: 0
°`1E0 d o E1
EN 1
°'
) EN
°" 60 (8.13)
is called a complex if d 1+
1
dj = 0 for all j = 0, 1, ..., N2. Put
Z'=Kerdt, B'=lmd,1, H'=Z'/B', j=0, 1, ..., N. (if (8.13) is a complex, BicZ'). The spaces H' are called the cohomology of the complex (8.13). The complex is called Fredholm if dim H' < oo for all
j=0, 1, ..., N. a) show that if the complex (8.13) is Fredholm, then the B' are closed subspaces of Z'. b) Let d; = b;df + dj1 b;1, where bj = d,*. The operators d; are called the Laplacians of the complex (8.13) (or the LaplaceHodge operators). Put I'' = Ker A, . Show that for the complex (8.13) to be Fredholm it is necessary and sufficient that all A, are Fredholm operators in Ej, j = 0, 1, 2, ... , N. In this case
dim H'= dim T'.
More precisely, I'' c Z' and the map T'  H' induced by the canonical projection
is an isomorphism (in the case of a Fredholm complex).
c) Put now N
X(E) _ E (1)'dimH' J=0
(the Euler characteristic of the Fredholm complex E). Prove that if N = 1, then the Euler characteristic of the complex
74
Chapter I. Foundations of `YDO Theory
0 . Eo °6 E1  0 is simply the index of do.
Prove that if dim Ej < +oo, j = 1, 2, ... , N, then N
X (E) _ E (1)j dim Ej. J=o
d) Show that X (E) does not change under a uniform deformation of all the operators d, if under this deformation the sequence (8.13) remains a Fredholm complex. Problem 8.7. Let V ; (j = 0, 1, ... , N) be vector bundles on a closed manifold M and H'(M, Vj) the Sobolev spaces of sections. Let d,: C°°(M, Vj)  C'
(M, V.1) be classical PDO of the same order m. Let To (M) be the cotangent
bundle over M without the zero section and no: To (M) * M the natural projection. Assume that the operators ...
0 ° _. C°°(M, Vo) a ' C_ (M' V1)
16
C°°(M, VN) , 0 (8.14)
form a complex. Let ad' no V + no Vj+, be the principal symbols of the operators d, (homogenous functions in of order m). The complex (8.14) is called elliptic if the sequence of vector bundles ,
o
0 + no Vo
o
no VN * 0
nO VI
is exact (i. e. an exact sequence of vector spaces at every point (x, ) a To (M)). a) Show that ellipticity of the complex (8.14) is equivalent to ellipticity
of all the Laplacians Aj = bjdj + dj_,8,_1, where 8j is the'PDO adjoint to d, with respect to some density on M and a Hermitean scalar product on the vector bundles Vj. b) Show that if (8.14) is an elliptic complex, then for any s e IR, the complex
H'(M, V1)....
0 d  HS(M, V0) d'
dam  HaN",(M, VN) °" . 0
is Fredholm and the dimension of its cohomology (and thus the Euler characteristic) does not depend on s. The cohomology itself can be defined also as the cohomology of the complex (8.14), i.e. putting Hi= Ker(dJIci,M.v,))/dj1(C00(M,Vj1))
Problem 8.8. Show that the de Rham complex on a real nmanifold M
0  A0(M)
? A' (M)  A2 (M) d ° i
_, ....
0 A"(M) ; 0
§8. The Fredholm Property, Index and Spectrum
75
(AJ (M) is the space of smooth exterior jforms on M, d is the exterior differential)
and the Dolbeault complex on a complex manifold M, dim, M = n,
0 , Aa. o(M)
. AP'(M)
L ...
, AP"(M)  0
(AD.9(M) is the space of smooth forms of type (p, q) on M and is the CauchyRiemannDolbeault operator) are elliptic complexes. Derive from this the finitedimensionality of the de Rham and Dolbeault cohomology in case of a closed M.
Chapter II Complex Powers of Elliptic Operators §9. Pseudodifferential Operators with Parameter. The Resolvent 9.1 Preliminaries. Let A be a subset of the complex plane (in the applications this will, as a rule, be an angle with the vertex at the origin). In spectral theory it is useful to consider operators depending on a parameter A E A
(an example of such an operator is the resolvent (A ).I) To begin with, we introduce some symbol classes. Let X be an open set in IR" and let a (x, 0, A) be a function on X x lR" x A,
xeX, OEIR, AEA. Definition 9.1. Let m, Q, b, d be real numbers with 0 5 S < g ;S 1, 0 < d < + oo. The class So o. d(X x R, A) consists of the functions a (x, 0, A) such that 1) a (x, 0, AO) E C°° (X x JR') for every fixed A0 E A;
2) For arbitrary multiindices a and f and for any compact set K C X there exist constants
such that
laeafa(x,9,A)j S
(9.1)
for x e K, 0 E lR1, A E A. As usual we put
S°°(XxIR",A) = n S'",Q a:a(XxIR",A) MCR
(the righthand side does not depend on g, b and d). If a(x, y, 4, A) E SQ J;d(X x X x IRA, A), we may construct a `PDO A,,, depending on the parameter). E A: (Ax u) (x) = l l e1 ,  y t a (x, y, , A) u (y) dy dd ,
(9.2)
for u E Co (X). In this case we will write A,ELQ a:d(X,A)
Note that A, a L Ic (X, A) if and only if the operator A, has a smooth kernel K, (x, y) for any fixed A E A and there exist constants C; s!K (K a compact set in X,
Chapter H. Complex Powers of Elliptic Operators
78
a and fi multiindices and N a positive integer) such that the following estimate holds
Iaxaf K2(x,y)I 5
Cvp'x(I+IA1)",
x, yEK.
(9.3)
Many of the statements about `PDO without a parameter (cf. §§37) can also be proved for the case with a parameter A. We indicate now some of these statements, which are necessary in what follows. First, note that the whole theory of asymptotic summation (Definition 3.4 and Propositions 3.5 and 3.6) carries over to symbols depending on a parameter. The corresponding formulations are obtained by changing SQ a (X x IR") to SQ b;d (X
X IR", A) and the proofs are almost verbatim repetitions of the arguments in 3.3 and are left for the reader as an exercise. We only state that the role of in these proofs (as in the following) is now played by (1+I©I2+ 1A121d),12
Further we will call an operator A,1 E LQ d;d(X, A) properly supported if it is uniformly properly supported in A, i. e. there exists a closed set L C X x X, having proper projections on each factor in X x X, such that supp KA, c L for all A E A.
Note that any operator A ELQ 6;d(X, A) can be decomposed into a sum A = A, + R1, where A 1 (depending on a parameter) is properly supported in the sense described and R, a Lo s d(X, A). For properly supported `PDO AA depending on a parameter, the symbol aA.(x, S) = aA (x, , A) is defined and a theorem of type 3.1 is valid. Naturally, we have to interpret formula (3.21) taking the parameter into account, i.e.
aA(x, S, A)  L « G A S Dy a(x, y, S,A)Iy_xe 121S"1
ESQ d
dd!"(XxIR", A).
In an analogous way Theorems 3.23.4 on the transpose and adjoint operato and composition can be generalized.
Exercise 9.1. Prove all the statements in sec. 3 in the case of operators an symbols depending on a parameter. Further, repeating the arguments of §4, for 1  Q < 6 < Q, we may introduce the classes LQ a:d (M, A) on a manifold M. Let us now pass to considering hypoellipticity and ellipticity. We introduce the class HS,',.' , (X x W, A) of symbols a (x, , A) (we will call them hypoelliptic with parameter), belonging to S',,,,, (X x 1R", A) and satisfying the estimates C1 (ICI+IAI"d)m, < l a(x,
C2(Ifl+IAi"d)
(9.4)
for x e K (K compact in X), I I + I A I ? R, C1 > 0 and R, C, , C2 may depend on K;
§9. Operators with Parameters
s
I [asaXa(x,
79
(9.5)
+ IA I z R (here, as above, R may depend on K). We will denote by HL"" a a (X, A) the class of properly supported `'DO (depending on the parameter I E A), whose symbols belong to HS0' a (X X for x e K, I
IRA, A). We have an analogue of Theorem 5.1: If A e HL Q (X, A), then there exists an operator B, a HLQ called the parametrix of the operator A,, such that
B,A,=!+Rx,
A,BA =I+Ri
(X, A)
(9.6)
where R;, Rx E Lw(X, A). The same statement is also true when X is a manifold.
Exercise 9.2. Prove this analogue of Theorem 5.1. It is natural also to consider classical `I'DO depending on a parameter. In this case A is assumed to be an angle with the vertex at 0. The corresponding symbols a(x, ! , A) admit asymptotic expansions (for I I + III' ? 1) of the form Tco
Y am i(x, ,2),
(9.7)
f=o
where am _, (x, , A) is positive homogeneous in
A"') of degree m j, i. e.
am1(x, t > 0, A e A and t') e A. Here m can be any complex number. This class of symbols will be denoted by CS, (X x IV, A) and the corresponding class of operators by CLd (X, A). This class is stable under composition, taking the transpose and taking the adjoint.
We will say that the operator A, e CLa (X, A) is elliptic with parameter if it is
properly supported and
a,, (x, , 2) 4 0
if
x e X and I f I + 12I"' * 0 .
(9.9)
It clearly follows that A, a HL; ; '" d (X, A). There exists a parametrix B, a CL; m(X, A) of a classical elliptic operator with parameter, which is also an elliptic operator with parameter. Example 9.1. Let A be a differential operator in X of degree m and I the unit
operator. Then A  Al a CL (X, C) and the principal symbol is given by the formula am (x, ) is the principal symbol of A. If A is a closed angle in the complex plane with vertex at the origin such that am (x, ) for I I = I does not take values
Chapter If. Complex Powers of Elliptic Operators
80
in A, then the operator A  2I is elliptic with parameter (and, in particular, belongs to HL,
m
(X, A)).
9.2 Norms of operators with parameter. In this subsection we will consider operators with parameter of two kinds: 1) operators Ax in IR", such that supp KAA lies in a fixed compact set k c IR2rt
(where 056 0,
/0,
A?bI,
4#0,
6>0,
(10.12) (10.13)
Let {(p}' 1 be a complete orthonorRemember that mal system of eigenfunctions for A with eigenvalues Aj. + oo as j  oo (cf. Theorem 8.3). It follows from (10.13) that A,? 6 > 0 for all j = 1, 2, .... Now, any distribution f e 9' (M) may be represented as a Fourier series r
i.e. (Au,u) z 6(u,u) for any
f > fapt(x),
where
xeM,
(10.14)
1=J
f=(f,w;)
(10.15)
Here, when f e L2 (M) we of course have the usual scalar product in LZ (M).
If, however f e 9' (M), then (f, 9j) denotes U; ipjdp>, where du is the fixed density on M (recall that the distributions are linear continuous functionals on the space of smooth densities). We now describe the properties of the Fourier series of smooth functions and distributions.
92
Chapter 11. Complex Powers of Elliptic Operators
Proposition 10.2. For a series OD
(10.16)
Y ci(pj(x)
j=1
with complex coefficients cj the following properties are equivalent: a) the series (10.16) converges in the C00(M)topology; b) the series (10.16) is the Fourier series of some function feC°°(M);
c) for any integer N Icj12AJN0;
U I'=0 = (P W;
(10.19)
has a unique solution in C°°(M) and .9'(M). Hint. The solution u (1, x) will necessarily be of the form u = e`Aco ,
(10.20)
where the operator e" is determined via the contour integral e'" = 21
r
Set'(A2I)1 dA.
(10.21)
Assuming that the spectrum a (A) is situated in the halfplane Re A < 0 (which can be achieved by changing A into A  CI or substituting u = ve' in (10.19)), it suffices to take r =1', u T2 , where r, and r2 are the following two rays:
A=re'lz` (+oo>r>0) on I',, A = re`
Z
(0 < r < + oo) on F2 .
The uniqueness of the generalized solution of the problem (10.19) is demonstrated using the Holmgren principle (cf. e. g. Gel'fand 1. M., gilov G.E. [1], vol. 3).
§ 11. The Structure of the Complex Powers of an Elliptic Operator 11.1 The symbol of the resolvent. Let A be an elliptic differential operator on a closed manifold M. We shall next construct in local coordinates the symbol of a special parametrix of the operator with parameter A  Al (which we view here as an operator in CL.(M, A), A a closed angle in C with vertex at 0). We assume that A satisfies the conditions for ellipticity with parameter relative to A, where
the angle A is as described in §10 (i.e. it satisfies (10.1') and (10.2'), and A contains the semiaxis ( oo, 01). The parametrix will be constructed in a chart XcM and we will identify X with an open set in 1R" using a coordinate system on X. Let the operator A on X be of the form
§ 11. The Structure of the Complex Powers
95
A = E aa(x)D*.
(11.1)
IaIS m
Its total symbol Y_ a. (x)
(11.2)
I*IS on
may be decomposed into homogeneous components
a; (x, ) _ Y as (x)
j = 0, 1.... , m .
(11.3)
I=I=;
The total symbol a (x, , A) = a (x, )  A of A  AI may be decomposed into components homogeneous in (, A"'") given by the formulas
a. (x, S, A) = a,, (x,0), a, (x,
(11.4)
j=0, 1, ..., m1.
(11.5)
The condition of ellipticity with parameter means that
for xEX, J.EA,
it 1+1A1'tm
0
(11.6)
It is natural to look for the symbol of the parametrix of A  ).1 in the form of
Denote these an asymptotic sum of functions homogeneous in functions by b°m_j (x, , A), j = 0, 1, 2, ... , where the lower index indicates the degree of homogeneity:
t>0, IfI+IAI'1'"*0.
(11.7)
These functions are recursively defined by the relations
am(x,
0,
(11.9)
k+I+IaI=J
I0,
r;$ 0.
(11.12)
To prove this, it is necessary to perform a change of variables in the integral (11.10) and use the homogeneity of bomi (x, 4, A): 2n
r = 2 f (tmp)zbMi(x, tm1) . tmdp r'
=
tmzi
t
mzI
i
_
2n 1µ t
2n r
mzj (z).0 fµzbmi(x,,p)dp=t 0 bmzi(x,
Here F` is the contour t"F (it has the same shape as F but the radius of the curved part is tmQ instead of Q). (x, ) to all z E C. This is Now it is necessary to extend the definition of done in the same way as the construction of AZ for Z E C in §10. The following analogue of Proposition 10.1 holds.
Proposition 11.1. a) For Re z < 0 and Re w < 0 we have the semigroup property
97
§ 11. The Structure of the Complex Powers as 1a1 +p+q = j
o b(2). '° (x ) xD41 b"1 o, (x )/I = b(2 { n,zP mwq ,n(2+w)j (x,
j=0,1,2,... j (x, l; ), j = 0, 1, 2, ... , is the set of b) If k e Z and k > 0, then the set b! homogeneous components of the parametrix of Ak. c) For any multiindices a, fi the derivative 64 '00 b'21_°j (x, c) is a holomorphic
function of z for Re z < 0 and t + 0.
Proof. This is achieved by repeating on the symbol level the proof of Proposition 10.1; recommended to the reader as a useful exercise.
L
In what follows it is convenient to denote by a;k) (x, ), k > 0 and integer, the homogeneous components (of degree j) of the symbol a(') (x, S) of the operator Ak, so that a(k)(x,
S) _ Mk > aak)(x, j=° FF
xx
If k is an integer and k < 0, then by a,") (x, ) we denote the homogeneous components of the symbol of the parametrix to the operator Ak, or, what is the
same thing, the homogeneous components of the symbol of A;. They are defined recursively by the relations a! ,'A(x, a(nik (x, S) ' a4akA j (x,
S) +
a a(mA(x,)
+
(11.14)
akA(x,
Dsa'"kA4
(x, )la! = 0,
j = ], 2. ...
(11.15)
P+q+ 1411 =j q1
(11.44)
But then the same holds for the symbols j(x, ) of the operators B(z),, the sum of which gives the operator B((N) (cf. the formulas (11.28)(11.30)). Taking the obvious estimates for the derivatives with respect to z into account, (11.40) is
obtained at once. Now, in view of (11.40), it is clear that it suffices to verify a memberthe symbol ship of the type (11.38) for 'R((N) = Az 'B(N). Denote by of 'R(()1 in some local coordinate system and by r(N)(x, , )) the symbol of R1N)(A)=(AAI)1_B(N#). Then
Chapter II. Complex Powers of Elliptic Operators
102
(11.45)
JA
and we have
2n r
J A (in A)' 01,V rN)(x,
(11.46)
But in view of Proposition 11.2, we have the estimate Ia
s
xeK,
(11.47)
where K is some compact set in the coordinate neighbourhood under consideration. From this it follows that
xeK,
I64aBr(N)(x, ,A)I
1. Standard arguments, already used before, show that the integral over the ball I 1 we may contract the curved part of I' to 0. Then we obtain for (12.31)
sinnz
n(mzj+n)
Srzbo_(x, I4I=1
o
The value of this expression for z = z j =1 e Z + equals exactly x, (x), where x, (x)
is given by the formula (12.6) or (12.9). Therefore 'B'';, (x, x) = x, (x).
9. Let us now note that the difference 'R( 'N') (x, y) = A. (x, y)
= Ids 2
'BiN'i (x, y)
fAze'(Xr)
(12.32)
1r
can be extended to a holomorphic function of z for Re z
0. b) K(t, x, x) has the following asymptotic expansion as t. +0: K(t, x, x)
E aj (x) t
j=o
(13.34)
where a j (x) e C °° (M). Express aj (x) in terms of yj (x) and x, (x) (cf. Theorem 12.1) and write down an expression for aj (x) in terms of the symbol of A.
120
Chapter 11. Complex Powers of Elliptic Operators
Verify that if A* = A, then
K(t,x,y)_
e 1,1(p;(x),pt(y)
(13.35)
0(t)= Y, e''= JK(t,x,x)dx
(13.36)
and the 6function
j=o
u
J=O
has the asymptotic expansion 0 (t)  2 aJ t M
.
(13.37)
i=o
Express index A in terms of the 0functions of the operators A*A and AA*.
Problem 13.5. Let E be a hermitean vector bundle on a closed manifold M with smooth positive density and let A be an elliptic selfadjoint differential operator mapping C a (M, E) into C °° (M, E) (not necessarily semibounded). Consider the function U
_
(sign A)
t,.Iz,
(13.38)
where the sum runs over all the eigenvalues of A. Show that the series (13.38) converges absolutely for Re z <  n/m and the function defined by it, >,4 (z), may be continued to the whole complex zplane as a meromorphic function with simple poles at z, = J  n , j = 0, 1, 2, .... Express the residues at these poles via m the symbol of A. Hint. Express 17A(z) in terms of (; (z) and C" (z) where C. (z) and C;; (z) are two Cfunctions of A, obtained by different choices of the branch for Az with cuts along the upper and lower semi axes of the imaginary axis.
§ 14. The Tauberian Theorem of Ikehara 14.1 Formulation. The Tauberian theorem of Ikehara allows us to deduce from the fact that the Cfunction is meromorphic asymptotic formulae for N (1) as t. + co or for A,% as k + + oo (cf. §13). Let us give its exact formulation. Theorem 14.1. Let N (t) be a nondecreasing junction equal to 0 jor t:5 1 and such that the integral
(z) = J tzdN(t)
(14.1)
§ 14. The Tauberian Theorem of Ikehara
121
converges for Re z < ko, where ko > 0 and the function
(z)+z+A
k0
can be extended by continuity to the closed halfplane Re z < ko. We will assume that A 0 0. Then, as t + +oo we have
N(t)  k t*
(14.2)
0
(recall that f, (t)  f2 (t) as t  + oo means that lim f, (t)/f2 (t) =1). ,.+,0 (The convergence of the integral in (14.1) for Re z <  k0 easily follows from a weaker condition. Namely, it suffices to suppose that the integral converges for
Re z <  k, for some k, and the function C (z) thus expressed can be holomorphically continued to the halfplane Re z < ko). Corollary 14.1. Suppose that the function C (z), defined for Re z <  ko by (14.1), can be meromorphically continued into the larger halfplane Re z <  ko + e, where e > 0, so that on the line Re z =  ko there is a single and moreover simple pole at  ko with residue  A. Then the asymptotic formula (14.2) holds. 14.2 Beginning of the proof of Theorem 14.1: The reductions.
1st reduction. It is convenient to consider instead of C (z) the function f (z) = C ( z). We then obtain
f(z) = J t=dN(t),
(14.3)
where the integral converges for Re z > ko and the function f (z) continuous for Re z ? ko.
 z Ako
is
2nd reduction: Reduction to the case ko = 1. By introducing the function f, (z) = f (ko z), we obtain
f,(z)= Jtk":dN(t)= TdN,(T), where N , (T) = N (T' /k"). Since
.f(koz)k
A
0z
=f(z)kA z1 0 1
k0
1
and since N(t) , A tr is equivalent to N, (T)  A T, then Theorem 14.1 reduces TO
to the following statement:
ko
122
Chapter 11, Complex Powers of Elliptic Operators
Let N(t) be a nondecreasing function and let the integral 00
f(z) = $ tzdN(t) be convergent for Re z > 1, where f(z)
N(t)  At
A
(14.4)
is continuous for Re z 2! 1. Then
z1
t'+oo.
as
Note that from the continuity of f(z) 
z
A
1
(14.5)
for Re z z 1 and the fact that
f(z) z 0 for real z;> 1, it follows that A > 0. Changing N(i) for A  1 N(t), which results in changing f (z) for A 1 f (z), we see that it suffices to show the statement for A = 1.
3rd reduction. Let us pass from the Melin transformation to the Laplace transformation, i. e. make a change of variables t = ex. Put N (ex) = 9 (x). We then see that rp (x) is a nondecreasing function, equal to zero for x < 0 and that the integral
f(z) = ! ezxd(p(x)
(14.6)
0
converges for Re z > 1 and f (z) z1 show that
is continuous for Re z z 1. We must
lim exgp(x)=1.
(14.7)
x.+w
4th reduction. Denote H(x) = ex(p(x). The So(x) is nondecreasing if and only if H(y)zH(x)exy
y> x.
for
(14.8)
Integrating by parts in (14.6) gives, for Re z > I
f(z) = zfe'x(p(x)dx =
ze(z1)x H(x)dx 0
0
Now put z =1 + s + it, where z > 0 and t is real. Note that °D
0
therefore, (14.9) implies
e(z ,)xdx = 1 z1
(14.9)
§ 14. The Tauberian Theorem of Ikehara
f(z) _ z
= $e
1
z
c=
123
.
0
Since
fZZ) Z
1= 1(f(z) Z
1
I
1
then, putting
h,(t)=Z(f(z) z
1
1
1)
(14.10) =+e+lf
we obtain 00 h.(t) = J e`x_"x(H(x) 1)dx.
(14.11)
0
We may now give the following reformulation of Theorem 14.1.
Theorem 14.1'. Let H(x) be a function, equal to 0 for x < 0 and satisfying (14.8) for all real x and y. Assume that the integral (14.11) converges absolutely for
any e > 0 and the function h, (t) defined by it, is such that the limit lim h, (t) = h (t)
(14.12)
exists and is uniform on any finite segment I t 15 22. Then
lim H(x) = 1.
(14.13)
Remark. If H(x) tends to I sufficiently quickly (if e.g. H(x) 1 EL' ([0, + oo))), then we obtain (14.12) from (14.13) by passing to the limit under the integral sign, which one may do in view of the dominated convergence theorem (the function h (t) then equals the Fourier transform of O (x) (H(x)  1), 6(x) the Heaviside function). In some sense, the Tauberian condition (14.8) allows one to invert this statement.
143 The basic lemma. It is clear that in order to prove Theorem 14.1' we have to somehow express H(x)  1 in terms of h (t) which, formally, is possible by the inverse Fourier transformation. However, we know nothing about the behaviour of h (t) as t . + oo or about the nature of the convergence of h. (t) to h (t) on the whole line and it is therefore necessary, to begin with, to multiply the limit equality (14.12) with a finite cutoff function j (t). These considerations, linked to the convenience of having transformations with positive kernels (of the Fejer type), demonstrate that it is convenient to consider p (t) to be the Fourier transform of a nonnegative function g (v) c L' (R):
Q(t)= Je ""go (v)dv.
(14.14)
Chapter 11. Complex Powers of Elliptic Operators
124
We shall assume that Q (t) is a continuous function with compact support such that a (0) = 1, p (v) >_ 0 and p (v) a Ll (IR). From this it follows that
f p(v)dv=1.
(14.15)
OD
The existence of a function B (t) of the type described may be shown in the same way as in 6.3 (at the beginning of the proof of Theorem 6.3 a function fi(t) E Co (IRl) is constructed which satisfies all these requirements). We can also explicitly define Q(t) , putting
1 121;
(t) =
A
0
f
111>2.
;
Then indeed, for a fixed v * 0 we have
p()v =
I2tll
J e
dt=
2 ertV
;,
/
d l
I 2t11
eu°
2
+2rziv
2
e"'
2
e"° sign =2J2iv  t dt
4nv
2
2
4av
10
2
I2) tII
""v
0
2
1
 1 cos2v 2nv 2
I since n
v
2
,
from which all the necessary properties of p (v) are obvious. Lemma 14.1. For any fixed A >/0 lim Y.
\
! H ` y  Al p (v) dv = 1
.
(14.16)
OD x
Proof. 1. Put e, (t) = B (t/A) and Q, (v) = Ap (A v) so that p, (t) is the Fourier
transform of p, (v). It is clear that + 00
I Hfp(v)dv= $ H(yv) p,(v)dv,
x
(14.17)
ac
a nd since p, (v) possesses the same properties as p (v), it suffices to prove (14.16)
forA=1. 2. Putting F(t) = e(t)ht(t), we compute the inverse Fourier transform of the function F (t) with compact support, taking into account that a (t) and ht (I)
are the Fourier transforms of the absolutely integrable functions p (v) and
0(v) (H(v)1)et 00
+
m
+ cc
ettYF.(t)dt = J eitvp(t) m
[1wx_ 1)eidx]dt 0
§ 14. The Tauberian Theorem of Ikehara
125
J Q(t) a""yx'dt dx
(H(x)1)e`x
J
(14.18)
00
o
J (H(x)_ 1) e`xQ(yx)dx. 0
As a result, as one might have anticipated, we obtain a convolution and we have made sure that (14.18) holds everywhere and in the usual sense (the change
in the order of integration is permitted by the Fubini's theorem). Let us now rewrite (14.18) in the form
J e"''F(t)dt+ Je`xQ(yx)dx = JH(x) e" Lg(yx)dx 00
(14.19)
0
0
and take the limit as a + 0. Since supp F c supp Q and F (t)  F(t) uniformly in t e supp j (here F(t) = Q (t) h (t)), then the first integral on the lefthand
side has a limit as ss +0 for any y. The same also holds for the second integral (e.g. by the dominated convergence theorem). Therefore, the integral on the righthand side of (14.19) has for any y a limit as a +0. Since
H(x)e"e(yx) converges monotonely as a++0 to H(x) e(yx), we get + ac
00
OD
J e"yF(t)dt+ f Q(yx)dx= JH(x)Q(yx)dx. 0
(14.20)
0
Now let y tend to + oo. By the Riemann lemma +m
lim
J e"y F(t) dt = 0.
In addition, it is clear that lim J Q (y  x) dx = 1. Therefore, it follows from .D o (14.20) that lim JH(x) y..+x; 0
Q(yx)dx= 1.
(14.21)
But ac
+W
+co
JH(x)Q(yx)dx= J H(x) Q(yx)dx= J H(yv) Q(v)dv,  CO
so that (14.21) implies the statement of the lemma.
126
Chapter 11. Complex Powers of Elliptic Operators
14.4 Proof of Theorem 14.1'. 1. First, we show that lim H (y) 0, it follows from Lemma 14.1 that
J H(yV) Q(v)dv 5 1
slim
.
(14.23)
Now, in view of the Tauberian condition (14.8) we have
H(yv)>H(yale 2i
for
ve[a,a].
Now, it follows from (14.23) that
/
2 _a a
lim H l y )/ e y.+m
A J Q (v) dv 5 1,
a
or 2a
lim H (y) < e z
a
(J
1
Q (v) dv)
(14.24)
a
Inequality (14.24) holds for any a > 0 and A > 0. Let a  + oo and A  + co in this inequality in such a way that a/A'0. Then we obtain the required estimate (14.22) from (14.24). 2. We will now verify that
lim H(y) z 1.
y.+m
(14.25)
To begin with, note that (14.22) implies the boundedness of H(y):
IH(y)I 0 lim j H (Y  v) Q (v) dv > 1  a (b). y.+m IvISb
\
(14.28)
Let us again use condition (14.8). We have
b\
2b
U
ve[b,b],
from which, in view of (14.28), it follows that
mH Cy+ b/ e'
y li
J Q(v)duz I a(b),
2b b b
or zz
lien H(y) z (1a(b)) e 7++m
b
(J Q (v) du)
(14.29)
b
Now let b + + oo and A + + oo, so that b/2  0. Then from (14.29) we obtain the
desired inequality (14.25). D
Problem 14.1. Let N(t) be a nondecreasing function, equal to 0 for t5 I and let the integral (14.1) be convergent for Re z <  ko, some ko > 0. Assume furthermore, that the function a (z), defined by (14.1), can be meromorphically continued to larger halfspace Re z <  ko + e, where e > 0 so that on the line Re z =  ko there is a single pole at  ko with principal part A (z + ko) ' in the Laurent expansion (here I is a positive integer, equal to the order of the pole at
ko). Show that
N(t) ' (1)'''A t,o(lnt)rt
(I1)!
as
Problem 14.2. Prove the Karamata Tauberian theorem: Let N(t) be a nondecreasing function oft E IR', equal to 0 for t < 1 and such that the integral
128
Chapter II. Complex Powers of Elliptic Operators
0(z) = f es`dN(t)
(14.30)
0
converges for all z > 0 and
0(z)  Az
as
z.+0
(14.31)
(here A > 0 and a > 0 are constants). Then
N(t)
r( +1)
to
as
t++oo.
(14.32)
§15. Asymptotic Behaviour of the Spectral Function and the Eigenvalues (Rough Theorem) 15.1 The spectral function and its asymptotic behaviour on the diagonal. Let M be a closed ndimensional manifold on which there is given a smooth positive
density and let A be a selfadjoint, elliptic operator on M such that (15.1)
Then A is semibounded. Denote by A its eigenvalues, enumerated in increasing order (counting multiplicities):
A,5A2;9A35..., By 47j(x) we denote the corresponding eigenfunctions, which constitute an orthonormal system. Let E, be the spectral projection of A (the orthogonal projection onto the linear hull of all eigenvectors with eigenvalues not exceeding t). It is clear that E,u = Y_ (u, apt) p,
(15.2)
x,St
Definition 15.1. The spectral junction of A is the kernel (in the sense of L. Schwartz) of the operator E, . Taking into account that on M there is a correspondence between functions and densities, we may assume that the spectral function is a function, not a density. From (15.2) it is obvious that this function, e(x, y, t), is given by the formula e (x, y, t) _ > (Pt (x) *P; (y) A's,
(15.3)
§ 15. Asymptotic Behaviour of the Spectral Function and Eigenvalues
129
and, in particular, belongs to C°°(M x M) for every fixed t. Let us note immediately the following properties of e (x, y, t): 1) e (x, x, t) is a nondecreasing function of t for any fixed x c M; 2) the function N(t) introduced in section 13.3, can be expressed in terms of e (x, x, t) by the formula
N(t)= $e(x,x,t)dx,
(15.4)
where dx is a fixed density on M. Now assume that local coordinates in a neighbourhood of x are so chosen, that the density coincides with the Lebesgue measure in these coordinates and put
I
V. (1) _
d
(15.5)
.
Theorem 15.1. For any x e M the following holds:
e(x,x,1)  V. (t)
as
(15.6)
Proof. 1.To begin with, note that without loss of generality we may assume Al ? 1. Indeed this is satisfied by the operator A I = A + MI for sufficiently large M. Now, if eI (x, y, 1) is the spectral function of AI , we have e (x, y, 1) = e 1(x, y, t + M). Therefore, the asymptotic formulae 1(x, x, t)  Vx (1)
implies e(x, x, t)  Vx(t+M). But Vx(t+M)=Vx(1) (t+M)"lm=Vx(1) t°Im(1+O(tI))_. Vx(1) t"""'=V.0), which implies (15.6).
2. Thus let A,;> 1. We may then define complex powers Az of A in accordance with the scheme of §10. Using (13.5), we may for x = y express the kernel Az (x, y) of A= in terms of the spectral function as follows 00
A, (x, x) = J t z de (x, x, t),
(15.7)
0
where d signifies the differentiation with respect to t (for a fixed x this is simply a Stieltjes integral). In view of Theorems 12.1 and 14.1 we obtain now fore (x, x, t)
the asymptotic formula
e(x,,x,t) I 11,
I 141 =1
afilm (x,
,)A,l J
.
t°/"g.
(15.8)
130
Chapter II. Complex Powers of Elliptic Operators
An elementary transformation of the righthand side of this formula, carried out in the proof of Lemma 13.1, shows that it equals V,,(t), implying (15.6). 15.2 Asymptotic behaviour of the Eigenvalues Theorem 15.2. Let A satisfy the conditions described at the beginning of this section. Then one has the following asymptotic relations
N(t)  V(t),
(15.9)
Ak  V(1)""k"",
(15.10)
where V(t) is defined by the formula (13.16).
Proof. In §13 we showed the equivalence of (15.9) and (15.10). (Proposition 13.1). Let us prove (15.9). This is done on the basis of the Tauberian theorem of Ikehara, by analogy with the proof of Theorem 15.1. Indeed, again we may assume that A, > 1. Then for Re a <  n/m, we clearly have the formula
S,(z)= f tzdN(t). It remains to use Theorems 13.1 and 14.1 and Lemma 13.1. Remark. One can derive (15.9) from (15.6) by integration over x. To justify this integration, it is necessary, however, to prove the uniformity in x of (15.6), which requires in several places (in particular, in the proof of the Ikehara theorem) the verification of uniformity in the parameter. To avoid this cumbersome verification, we have preferred to give an independent proof.
15.3 Problems
Problem 15.1. In the situation of this section prove the estimate
Ie(x,y,t)I SCt"1T, where x, y e M and the constant C > 0 does not depend on x, y and t (t z 1). Problem 15.2. Let A be an elliptic differential operator, on closed manifold M with smooth positive density, which is normal, i.e. A*A = AA*.
(15.12)
a) Show that A has an orthonormal basis of smooth eigenfunctions (p; (x),
j = 1, 2, . . ., with eigenvalues A; E C, such that IA;I
+oo
as
(15.13)
b) Show that if N (t) denotes the number of A,, such that I.ij I S t, and if V (t) is defined by the formula
§ 15. Asymptotic Behaviour of the Spectral Function and Eigenvalues
V(t) = (2n)""
f
131
(15.14)
la.(x, 0 1 0 for c + 0. Let A j be its eigenvalues and N, (1) the number of eigenvalues with Re.lj < t (here we take for the multiplicity of an eigenvalue Ao the dimension of the root subspace EA., cf. Theorem 8.4), N2 (t) the
number of eigenvalues with IA; < t. Show that
N,(t)N2(t)V(t)
as t++oo,
(15.16)
k ++oo
(15.17)
where V(t) is defined as before. Show that
2k _ V(1)nf"krl"
(this means, in particular, that Im Ak has a lower degree of growth than Re Ak).
Chapter III Asymptotic Behaviour of the Spectral Function §16. Formulation of the Hormander Theorem and Comments 16.1 Formulation and an example. Let M be a closed ndimensional manifold on which there is given a smooth positive density dx and let A be an elliptic, selfadjoint operator of degree m on M such that aM (x, ) > 0 for + 0. We will use the notations e (x, y, A), N (A), V. (A) and V (A) introduced in § 15. The
following theorem refines Theorems 15.1 and 15.2.
Theorem 16.1 (L. Hormander). The following estimate holds I e (x, x, A)  Vx (A) 15 CA(" ')/m,
A>_ 1,
xeM,
(16.1)
where the constant C > 0 is independent of x and A.
Corollary 16.1. The following asymptotic formula holds
N(A)=V(A)(1+O(A'/°'))
A +oo
as
(16.2)
Remark 16.1. In general the estimate of the remainder in (16.1), (16.2) cannot be improved. This can be seen by looking, for instance, at the operator
A=
on the circle S' =1R/2,t Z. The corresponding eigenfunctions are
of the form yrk (x) =
1
e'kx,
k = 0,
±1,
and the eigenvalues are Ak = k2, k = 0, ± 1, ± 2,
± 2, ... ,
... .
Further, since tWk(x)12= (2n)', then clearly e(x,x,A) =(21r)N(A). Since V(A) _ (2x) ' V (A) then (16.1) and (16.2) are equivalent. So it suffices to show that the estimate of remainder in (16.2) can not be improved.
But in this example (16.2) has the form N(A) = V(A) (1 +O(A'l2)) or N (A) = 2 i/ + O (1). The estimate O (1) can not be improved because N (A) has only integer values.
Chapter 111. Asymptotic Behaviour of the Spectral Function
134
Later, in §22, we will study a more interesting example, which is a generalization of the present one (the Laplace operator on the sphere) and shows
that (16.1) cannot be improved in the case of arbitrary n and m.
16.2 Sketch of the proof. First of all, the theory of complex powers of operators, allows a reduction to the case when A is a'f'DO of order 1. In this situation we will show, that for small t, e«4 is itself an FIO, with a phase function
which is a solution of a certain first order nonlinear equation. Let us now remark, that the kernel of eu4 is the Fourier transform (in A) of the spectral function of A. From this the asymptotic (16.1) is obtained, by invoking Tauberian type arguments for the Fourier transformation. The remainder of this chapter is as follows: § 17 contains some indispensible information on first order nonlinear equations; in § 18 an important theorem on
the action of `PDO on exponents is proved, from which, in particular, the composition formula for a `IMO with an FIO follows; in § 19 the class of phase functions corresponding to'I'DO is studied; in §20 we construct the operator e"4 in the form of an FI O for a first order operator A; in §21 Theorem 16.1 is proved in the general case (there is also information about e(x, y, A) for x * y); finally, §22 contains the definition of the Laplace operator on a Riemannian manifold and the computation of its spectral function in the case of a sphere.
Problem 16.1. Compute N (2) and e (x, x, A) for the operator
A=d= C a2
a2
02 axY+ax2+...+ax
on the torus T" = lR"/2nZ" and verify that the asymptotic formulae (16.1) and (16.2) hold.
§17. Nonlinear First Order Equations 17.1 Bicharacteristics. Let M be an ndimensional manifold and a (x, ) a smooth realvalued function, defined on an open subset of T M. Consider the Hamiltonian system on T*M, generated by as Hamiltonian: rx =
where a4 = S
as
as
a_
' ... , a ")' I
a{
a,,
(17.1)
as as and (z, )are the coordinates axl ' ... , ax"
on T' M, induced by a local coordinate system on M. It is wellkown, that the vector field on T' M, defined by the righthand side of (17.1), is independent of the choice of local coordinates on M (cf. e.g. V.I. Arnol'd [1]).
§ 17. Nonlinear First Order Equations
135
Definition 17.1. A solution curve (x(t), fi(t)) of (17.1) bicharacteristic of the function a (x, ).
is
called a
A bicharacteristic is not necessarily defined for all t E R. In this case we assume that it is defined on the maximal possible interval (concerning this consult also Problems 17.1 and 17.2).
Proposition 17.1. The function a (x, ) is a first integral of the system (17.1), i.e. if (x(t), l; (t)) is a bicharacteristic of the function a(x, ), then a(x(t), fi(t)) = const Proof. We have
d a(x(t),
(t)) = a,,.z + a, = a., a,  a. a, = 0. 1]
Proposition 17.1 makes sense of the following definition:
Definition 17.2. A bicharacteristic (x(t),1; (t)) of the function a(x, ) is called a nullbicharacteristic if a(x(t),: (t)) = 0. 17.2 The HamiltonJacobi equation. Consider the first order partial differential equation
a(x,(px(x)) = 0,
(17.2)
where (p is a smooth, realvalued function, defined on an open subset of M and gyp,, its gradient. Such an equation is called a HamiltonJacobi equation. For its treatment, it is convenient to introduce the graph of qpx, i.e. the set
F, = {(x, (px (x)), x e M} c T"M.
(17.3)
Proposition 17.2. If cp is a solution of (17.2), then the manifold F. is invariant under the phase flow of the system (17.1), i. e. if (x (t), c (t)) is a bicharacteristic of a, x (t) for t e [0, b] belongs to the domain of (p and (x (0), (0)) a f, then (x(t), fi(t)) ef, for all t c [0, b].
Proof. In view of the uniqueness theorem, it sufficies to verify that the Hamiltonian vector field (a,, a.,) is tangent to r. at all its points. This is equivalent to the following: if (x(t), (t)) is a bicharalcteristic and (x(0), (0)) e r, (i. e.
dt t)  4 (xId11=0 (t ))1 } = 0. But this follows (0) = cpx (x (0))), [fi (then
from the computation :
111
(dt R(t)(Px(x(t))11Ii3o=(4Px:
t =  ax (x (0),
(0))  coxx (x (0)) at (x (0),
a
=  ax
oI,o
[a (X, (P.(x))],, =X(0) = 0. C
(0))
Chapter III. Asymptotic Behaviour of the Spectral Function
136
In what follows the only important case for us is when a(x, ) is positively homogeneous with respect to of degree m, i.e. a (x,
tm a (x, ),
t>0, +0,
(17.4)
where m is any real number. Such functions are characterized by the Euler theorem:
a,=ma.
(17.5)
Proposition 17.3. Let a (x, ) be homogeneous of degree m and (P (x) a solution of (17.2). Then (p (x) is constant along the projections of the nullbicharacteristics of the function a (x, l;) belonging to I',, i.e. if (x (t), l; (t)) is a nullbicharacteristic and 4 (0) = V. (x (0)), then tp(x (t)) = const.
Proof. We have
dt
q,(x(t)) = cxx = (pxas = (px(x(t)) ag(x(t), fi(t))
= (px(x(t)) a,(x(t),cpx(x(t))) = ma(x(t),cpx(x(t))) = 0.
173 The Cauchy problem. The Cauchy problem for the HamiltonJacobi equation (17.2) consists in finding a solution rp (x) of this equation, subject to the condition
was=W,
(17.6)
where S is a hypersurface (submanifold of codimension 1) in M and W e C m (S). Locally, we may consider the hypersurface as a hyperplane, i.e. by choosing the local coordinate system in a neighbourhood of a point xo e S, we may achieve
that
S= {x:
(17.7)
so that yi = W (x'), where x' = (x, , . . . , x. _ ). In this coordinate system, it is convenient to formulate the condition of being noncharacteristic, guaranteeing local solvability of the Cauchy problem in a neighbourhood of the point x' eS: the equation a (x', 0, Wx (x'), A) = 0
(17.8)
has a simple root A, i. e. a root A a IR, which in addition to (17.8) satisfies as (x', 0, yrx (x'), A) 4 0.
(17.9)
§ 17. Nonlinear First Order Equations
137
Let a point 0 eS be fixed. Then by the implicit function theorem, the equation (17.10)
0
for I x I < s and I '  Wx (0) I < e has a solution A = a' (x, '), which is a smooth function of x and '. It is easy to verify that a' (x, ') is homogeneous of the first
order in i;', so we may assume that it is defined for Ix I < e and for all i' 0 0 in a conical neighbourhood of *,,(0). Equation (17.10) for Ix I < e and for a vector (4', A) close to the direction of (*j,(0), a'(0, *,(0))), may be represented in the form
A  a'(x, ') = 0.
(17.11)
Therefore the local Cauchy problem takes the following form: find a solution (p = cp(x) of (17.2), which satisfies (17.6) and, additionally, satisfies O(P
ax
(0, 0) =a ' ( 0 ,( 0 ) ) .
(17.12)
Since in this situation it is possible to pass from (17.10) to (17.11), our problem may be written in the following form
ax 
a
(a (P
= 0,
(17.13)
(17.14)
(P lx.o=W(x'), i.e. the matter reduces to the case
(17.]5)
Let us consider the bicharacteristics of a (x, ) of the form (17.15). Their equations are (17.16)
=a.' (x, 7 Consider a nullbicharacteristics (x(t), (t)) belonging to 1', and starting in S, i.e. such that 0. Then it is obvious from (17.16) that x. (t) = t. Fix another point x' = x' (0) a S. It is clear that the condition (x (0), (0)) e f, means the following
' (0) = W x (x'), , (0) _ W (x', 0) , X.
(17.17)
138
Chapter 111. Asymptotic Behaviour of the Spectral Function
and the condition a (x (0), (0)) = 0 gives
(0) = a'(x',0,Ws.(x')).
(17.18)
Therefore, the nullbicharacteristic belonging to r,, and such that x (0) = 0
and x'(0) = x', is uniquely defined. From (17.17) and (17.18) the smooth dependence on x' is clear. In addition, if we consider the transformation
g:
(17.19)
defined for I x I < c, then from the initial condition x'(0) = x', it follows that its Jacobian is I for x = 0, so that g is a local diffeomorphism. Now, from Proposition 17.3, it necessarily follows that
w(x)=W([g`(x)]'),
(17.20)
where [g' (x)]' is the vector, obtained from g `(x) by neglecting the last component (corresponding to the notation x' for x = (x', x.)). Therefore, we have shown the uniqueness of the solution of the local Cauchy problem and obtained a formula, (17.20), for this solution. The existence of this solution. is a simple verification. We recommend the reader to do the following exercise.
Exercise 17.1. Show that formula (17.20) actually gives a solution of the local Cauchy problem as described above. 17.4 Global formulation. We would like to formulate sufficient conditions for the existence of a solution of the Cauchy problem in a neighbourhood of S
without restricting to a small neighbourhood of a point on S (although the neighbourhood of the hypersurface S may be very small, in the sense of, for example, some distance from S). First, these conditions must of course, guarantee the existence of solutions of the local problem at any point x e S and secondly, roughly speaking, provide continuous dependence of the root d of
equation (17.8) on x. This means, that on S we may define a covector field _ 4 (x') E Ts M, continuously depending on x' c S and such that 1) i a (x') = Wx. (x'), where is S* M is the natural inclusion map and W.., (x') is the gradient of W (x') at x' c S, viewed as a covector on S (an element of T. S);
2) Introduce local coordinates as described in 17.3 in a neighbourhood of any point x' a S. Then
(x') = (x') is a root of (17.8), satisfying (17.9), i. e. satisfying all the conditions for the local solvability of the Cauchy problem. Let us note that (17.12) may be written, here without local coordinates as
cpx(x') = (x'), x'ES
(17.21)
§ 17. Nonlinear First Order Equations
139
Therefore, the final statement of the Cauchy problem goes as follows: find a solution of (17.2), defined on a connected neighbourhood of the hypersurface S, satisfying the initial condition (17.6) and the additional condition (17.21). In this form, the problem has a unique solution, depending smoothly on the parameters (if any), provided that the given quantities a, S, i, and also depend smoothly on these parameters.
Remark 17.1. Condition 1) is obviously necessary (assuming the rest is also fulfilled) for the solvability of the Cauchy problem and signifies simply the absence of topological obstructions to the global existence of a field (x'), the local existence and smoothness of which is ensured by solvability conditions of the local problems at the points x' a S. 17.5 Linear homogeneous equations. Equation (17.2) is called linear homo
genous if a (x, ) is linear in , i.e.
a (x, ) = V (x) ,
(17.22)
where V (x) is a vector field on M. The projections on M of the bicharacteristics, are in this case the solutions of the system x = V (x),
(17.23)
and the solutions of (17.2) are simply the first integrals of the system (17.23). The
same system (17.1) contains also, along with (17.23), the equations
4 =  V.(x) which are linear in . A standard growth estimate for
(17.24) I
(t) I shows that if
x (t) E K, where K is a compact set in M, then I (t) I is bounded on any finite interval on the taxis. Therefore a bicharacteristic is either defined for all t or its projection x(t) will leave any compact set K C M. The condition that S is noncharacteristic means, that V(x) is everywhere transversal to S. Let us consider the mapg mapping (x', t) into x (t) with x (t) a solution of the system (17.23) with the initial value x (0) = x'. If there exists e > 0 such that x(t) is defined for any x' for all I I I < e, then g determines a map
g: S x (  e, e)  M.
(17.25)
If g is a diffeomorphism, then the solution of the Cauchy problem with initial data on S is defined on the image of g. It is therefore important to be able to
estimate from below the number e > 0, for which the map (17.25) is a diffeomorphism. One important case, where such an estimate is possible will be shown below.
Chapter III. Asymptotic Behaviour of the Spectral Function
140
17.6 Nonhomogeneous linear equations. These are equations of the form V (x) (px (x) + b (x) (p (x) = f (x) ,
(17.26)
where b (x), f (x) E C °° (M), V (x) is a vector field on M, (p (x) is an unknown
function and (px its gradient. If x (t) is a solution of the system (17.23) then obviously dt (p(x(r)) +
b(x(t)) (p(x(t)) = f(x(t)),
from which (p (x(t)) can be found as a solution of an ordinary first order linear
differential equation, provided that (p (x (0)) is known. The basic feature following from this is that the domain, on which a solution of the Cauchy problem exists, depends only on V (x) and Sand is independent of the righthand side f(x) and the initial value V/ e C' (S). In particular, in what follows, we will need an equation of the special form
OX.

amp
a; (x) ;_1
ax,
+ b (x) 9 = J(x),
(17.27)
where x = (x', xp), x' c M' for some (n 1)dimensional closed manifold M' and x E ( a, a) with a > 0. The system (17.23) (for the corresponding homogeneous equation) is of the form
x' = V'(x),
(17.28)
The solutions x(1) of this system which start at x = 0, are defined for t e ( a, a) and if we put S = M'= {x: x = 0}, then the map g of the preceding section becomes a diffeomorphism g: M. M, where M = M' x ( a, a) and where it is clear from (17.28) that the "fiber" M' x xo is mapped onto itself diffeomorphically. Because of this, the Cauchy problem for (17.27) with initial condition W lx.=o = W W),
XE
,
(17.29)
has a solution (p e C°` (M).
In a number of cases one can carry out similar arguments also for noncompact M'. Problem 17.1. Let a (x , 4) be defined for x E M, $ 0 with degree of homogeneity I in . Show that if (x(t), fi(t)) is a bicharacteristic, then it is either defined for all t or x (t) will leave any compact set K C M. In particular, if M is compact, then all bicharacteristics are defined for all t E IR.
§ 18. The Action of an Operator on an Exponent
141
Problem 17.2. Show that the same holds for an arbitrary degree of homogeneity of a (x, 4), if condition of ellipticity holds:
for X40, xEM.
§18. The Action of a Pseudodifferential Operator on an Exponent 18.1 Formulation of the result. Here we describe the asymptotic behaviour as A  +oo of the expression A (e"*(,)), with A a 'PDO and t a smooth function without critical points. Theorem 18.1. Let X be an o p e n set i n 1R", A E LQ a(X), 1  Q S 8 < Q, A properly supported and with symbol a (x, ). Let Vi (x) e C' (X) and V,, (x) $ O for x e X (here Wx denotes the gradient of W). Then for any function f e C" (X) and
arbitrary integer N ? 0, for A > 1 we have
A(fe«.) = e14v
I
Y
a(a,(X,2W:(x)) D: (f(z) a!
(a! e, > 0
on
supp(1 X(Yx) X(CCx)],
we see that I (A)  7(A) =
A" J e'A9cy.0 LN [a (x, AC) (1 X (y 
x) X (C  C3))f(y)] dy dd
,
Chapter 111. Asymptotic Behaviour of the Spectral Function
144
and transforming this oscillatory integral into an absolutely convergent one
(cf. §1) we easily obtain the estimate (18.9) due to the factor A', in the expression for L. Analogously, one also obtains estimates for the xderivatives
from the difference I().) 1(2). However, note that they follow from the estimates of I(2)  1(X) with arguments similar to the proof of Proposition 3.6. In the sequel we shall omit estimates of the derivatives, leaving them to the reader. Thus, instead of I(2), we may consider 1(2). Making yet another change of coordinates { _ x + 2 ', . we obtain 1(2) _ !e'(xY) a
a (x, .lax + ry) in a Taylor series at j7 = 0: a (x, AS. + q) = Y a(a) (x, A4x) ! + rN (x 1, A) Ial N/2, then by the Dirichlet principle, in (18.15) there are no less than k  N/2 indices y, such that I y1 I = 1. But then, by the Hadamard lemma
(Dy'Q:(Y)) ... (Dy'e:(y)) _
gr(Y,x) (xY)r, bl2kN/2
wheregr (y, x) is a smooth function (in x and y), defined for y sufficiently close to
x. Inserting this expression into (18.13) and integrating by parts (utilizing the exponent allowing us to change (xy)r into (D")r), we see that r, (x, A, t) is a linear combination of terms of the form I2 ()) = A" j aar (x, n, A, t) e;(=P)  "
x) dy d11,
(18.16)
where ar = 87 as and AY, x) is smooth (in x and y) and supported in I y  x I S E. The indices k and y are related by I y I z k  N12. Taking into account that the volume of the domain of integration inn in (18.16) does not exceed C).", and using (18.14) we obtain for I1(A) the estimate II1(A)I < CA*+M(N+IYI) +"5 CA.+"N12,
146
Chapter III. Asymptotic Behaviour of the Spectral Function
which allows us to conclude the proof by applying the type of arguments used in the proof of Proposition 3.6.
18A The product of a pseudodifereatial operator and a Fourier integral operator. Let X, Y be open sets in 1R"x and IR"r and let P be an FIO of the form
Pu(x) = jp(x,y,0) e`'(.rO) u(y) dyd0,
(18.17)
where p (x, y, 9) a S" (X x Y x IR") and cp (x, y, 9) is an operator phase function
(cf. §2, Definition 2.3). Let there also be on X a properly supported PDO A e L',§ (X) with symbol a (x, ). Since P maps Co (Y) into CI (X) and d' (Y) into 9'(X) and A maps the spaces C°° (X) and 9'(X) into themselves, then the composition A P is defined as an operator, mapping COI(Y) into C°°(X) and
9'(Y) into 9'(X). Theorem 18.2. Let 1  Q 5 6 < Q 5 1. Then the composition Q = A P is also of the form (18.17) with the same phase function (p (x, y, 0) as P and with an amplitude of the form 9(x,y,9) = eio(x.r.9)a(x,Dx) (p(x,y,0) e+(Xr.8)],
(18.18)
with the asymptotic formula q (x, y, 0) ^.
a'°) (x, Px (x, y, 0))
D= (p(z Y 0)
a
as
e'ecx.x.r.ei)
a.
I91++oo
x=x
(18.19)
where g (z, x, y, 0) = q, (z, y, 0)  N (x, y, 0)  (z  x)  cx (x, y, 0).
Remark 18.3. Since ap (x, y, 0) is not smooth for 9 = 0, it is not immediately clear from (18.18) that Q is an F1 0. This is the case however, since adding an operator with smooth kernel to P we may assume that p (x, y, 0) = 0 for 101 < 1. Then (18.18) defines a smooth function in all the variables and the same holds for
all terms in the expansion (18.19), which has the usual meaning (cf. Definition 3.4). However, an operator with smooth kernel may always be written in the form (18.17) with an amplitude p (x, y, 9) which has compact support in 0 and equal 0 for 10 1 < I (cf. the hint to Exercise 2.4). Therefore Q is an FIO with phase function cp.
Proof of Theorem 18.2. Let us introduce the set
C. _ ((x,y,0): (Pa(x,y,0)=0) . used in §1 and §2. Note that cpx (x, y, 0) * 0 for (x, y, 0) a C, by the definition of an operator phase function. Changing P by adding an operator with a smooth
§19. Phase Functions
147
kernel, we may assume that suppp (x, y, 0) lies in an arbitrarily small conical neighbourhood of the set C, (cf. Proposition 2.1) and, in particular that qi * 0
on suppp. In addition and in accordance with Remark 18.3, assume that p(x,y,0)=0 for 101 0 i s s u f f i c i e n t l y s m a ll, then f o r I t I _ Q".
§23. The Algebra of Operators in IRO
177
If a (x, y, ) e DQ (lR3i), then a (x, x, ) E f'Q (1R2").
The most important example of an amplitude of the class 11 (]R3") is provided by the following
Proposition 23.3. Let a linear map p: 1R2". 1R" be such that the linear mapping (x, y) into (p(x,y),xy), is an isomorphism. Let b (x, ) E !'Q (R2 n). Define the amplitude a (x, y, 4) E C' (]R3") by the formula (23.10)
a (x, y, ) = b (p (x, y), ) . Then a E 17 (]R3n).
P r o o f. The functions I x I + I I and I p (x, y) I+ I x y I give equivalent norms on 1R2n. Therefore, for the proof of the proposition it remains to use the easily verified inequality
(1 + Ip(x,y) I +
I
SEIR,
I)' 
from which the estimates (23.9) follow for a (x, y, S) with m' = I m I. O
Corollary 23.1. If b e rQ (1R2n), then a (x, y, ) = b (x, t) and a (x, y, ) = b (y, ) belong to 17 (JR3e). 23.2 Function spaces and the action of the operator. Now we introduce the space Cp (1R") consisting of functions u e C°° (1R) such that (23.11)
10"u(X) I S C.
for any multiindex I al. The best constants C" in (23.11) constitute a family of seminorms for a given function, defining a Fre chet space structure on Cb (IR"). The operator A of (23.8) is conveniently studied in the space Cs (1R"). In order to give the correct definition of the oscillatory integral appearing in (23.8),
we shall have to proceed as in §1. For this purpose, let initially a (x, y, 4) E Co (1R3"). Then the integration in (23.8) in reality is performed over a compact
set and we may carry out an integration by parts, using the identities eit:y",
(23.12) (23.13)
where M, N are even nonnegative integers. From (23.8) one obtains
Au(x) = j
x [0,
we see from (23.14) that
(I+IxI)'IAu (x)IsCk for any k and a similar estimate holds if we replace Au (x) by 8j (Au (x)). From
this we also have Au e S (lR") for u e S (P') with an estimate of seminorms guaranteeing the continuity of the map (23.17) (which also could have been obtained from the closed graph theorem). Finally note, that since the transposed operator 'Au(y) =
JJeilxY"*4
v(x)dxdd
(23.18)
by similar reasoning, maps S (IR") into S (IR"), then A can be extended by duality
to a continuous map
A: S'(1R)+S'(1R"). Definition 23.4. The class of operators A of the form (23.8) with amplitudes a E 11Q (1R3") will be denoted by G' (IR") or simply by G,' (if the dimension n is
clear or unimportant).
§23. The Algebra of Operators in IR"
179
It is useful to have a description of the operators belonging to the intersection
G1= n GQ . We shall show that this intersection is independent of q and consists of operators with kernels K,, (x, y) e S (IR 2n). Clearly it suffices to consider the case Q < 1. Note that the operators with amplitudes a (x, y, ) and " (DD>' a (x, y, ) coincide, from which we see that if A E G  °°, then A can be determined by an amplitude a(x, y, ) satisfying (23.9) with arbitrarily small (arbitrarily close to  oo) numbers m and m'. But then A has the kernel (23.19)
KA(x,Y) =
belonging to S(1R2"). From this, it follows in particular that A defines a continuous map A : S' (IR") * S (IR") ,
(23.20)
given by the formula (23.21)
In the general case the kernel KA(x, y) is defined by the formula (KA,gi> =
a (x,
cpeS(IR2 ),
9, (x, y)
(23.22)
and is a distribution K4 ES'(IR2"). Exercise 23.2. Denote by C'O (IR") the space of functions u e C ' (IR"), with the property that for any multiindex a one can find constants C. and u., such that
Iasu(x)16 CQ"
(23.23)
Show that an operator A e G' defines a map A: CC ° (IR") + Cr°D (IR") .
(23.24)
Exercise 23.3. Let A E G, "(R") and KA the kernel of A. Show that KA a C°° (IR2"\ A), where A is the diagonal in 1R" x IR".
23.3 Left, right and Weyl symbols Theorem 23.1. An operator A e GQ of the form (23.8) can be written in any of the following three forms Au(x) = if ei(xY1
QA.r(x, )
(23.25)
Au (x) = j j ei (x  Y) . 4 UA., (Y, ) u (Y) dy dd ,
Au (x) = j j ei (x Y)
UA.
"
(!, \ u (Y) dy d . 1
(23.26) (23.27)
Chapter IV. Pseudodifferential Operator in IR"
180
Here 0,11, a4., and a4,,,, belong to I'Q (1R2"), are uniquely defined and have the following asymptotic expansions: aA., (x, a
Y
a.+., (Y, )
a!
ac!
as Dr a (x, y,
(23.28)
as(Dz)" a(x,y, )Is=r,
(23.29)
1
ft! Y!
r
2
,(23.30)
This theorem allows the introduction of Definition 23.5. The functions a4,,, a,,,, and a,,,.. from the formulae (23.25)(23.27) are called, respectively, the left, right and Weyl symbols of the operator A.
Although we shall not use any other symbols, let us show the following generalization of Theorem 23.1, containing a parameter t e lR and also allowing us to avoid repetitions in the proof of Theorem 23.1.
Theorem 23.2. Let A e GQ of the form (23.8) be given. Then for any t e IR A may be uniquely written as
Au(x) = JJ
er,:Y
4
b, has the following asymptotic expansion
b,
b,(x,t)^
1
/ICY.
T1B!(1T)!Y!aB+Y(Dx)ODYa(x,Y,(23.32) e r
Definition 23.6. The function b, (x, ) will be called the rsymbol of A. Proof of Theorem 23.2. Putting
V= (l T)x+Ty,
tl
(23.33)
w=xy,
we obtain
S x=v+Tw,
y=v(1T)w,
23.34)
a(x,y,4)=a(v+tw, v(1T)w,,;).
(23.35)
from which Let us now expand the righthand side of (23.35) at w = 0 in a Taylor series: TIM
IB+rI5N
Y!
Y
(23.36)
§23. The Algebra of Operators in IR"
181
where
rN(x,Y,)=
j(1t)NI
c6r(xy)°+Y
Y,
O
19+YI=N
(23.37)
x
and cd,, are constants.
In (23.36) the expression (axay a) (v, v, ) signifies that in the function ax a; a (x, y, ) it is necessary to take v = (1 r) x + r y instead of x and y. The expression (axaya) (v+trw, v 1(1 r) w, ) in formula (23.37) has a similar meaning.
Now note, that the operator with amplitude (xy)$+' (azaYa) (v, v, ) coincides with the one given via the amplitude (v, v,) = ( 1)191 *lYl(a{
+YDzDsa)
(v, v, ) .
Therefore it follows from (23.36) that A can be represented in the form of a sum A = AN + RN, where AN is an operator with rsymbol 1
I9+7ISNt Pt'yt'
x191(1r)IYIaft+Y(D.)PD a(x,Y, )ly=x,
and RN is an operator with amplitude rN (x, y, ). Note that RN is a linear combination of a finite number of terms having amplitudes of the form
O
III+yI=N. Let us show that this amplitude belongs to the class 17  2N2 (IR'"). For this it sufficies to show that this is true for the integrand, with all estimates uniform in t (note that this is obvious for each fixed t + 0 and true for t = 0 by Proposition 23.3). In view of the relations
v = (1 r) (v+trw)+r(v1(1r)w), tw = (v+trw)  (vt(1 r) w) it is obvious that Ct
0, is compact for any s e 1R. Exercise 25.6. Prove Propositions 25.3 and 25.4. 25.4 The Fredholm property. By analogy with Theorem 8.1 is proved Proposition 25.5. If A E HG,, M, then A e Fred (Q', Q'  M) for any s e lR. The space Im (A I Q.) in Q3 M is the orthogonal complement to Ker A * with respect to the scalar product ( ,  ) in L2 (lR").
Note that Ker (A I Q.) = Ker (A I s. (a.) = Ker (A I scat) for any A E HGQ MN.
(25.7)
196
Chapter IV. Pseudodifferential Operator in IR'
To extend Proposition 25.5 to operators A eHGQ''(with mo < m), it is but as an operator in the topological vector spaces S(IR"), S'(IR") and similar spaces or, as an unbounded operator necessary to regard A not as an operator from Q3 into Q'
A..:,: Q' Q',,
(25.8)
where s'2: s  mo, with the domain D,," consisting of those u e Q' such that
AueQs. Definition 25.4. Let E, and E2 be two topological vector spaces, A an unbounded operator from El into E2 with the domain DA. The operator A is called Fredholm operator if the following conditions are fulfilled: a) dim KerA < + oc ; b) ImA in a closed subspace in E2; c) dim Coker A < + co. Theorem 25.3. 1) The operator A eHGQ defines a Fredholm operator from S (IV) into S (W) and from S' (W) into S' (IR"). 2) The operators As,.. of the form (25.8) defined by A are, jor s' ? s  mo, also Fredholm operators.
Remark 25.1. We consider the weak topology in S'(IR"). Since in Definition 25.4 the topology appears only in b), it is clear that the Fredholm property also holds in all stronger topologies.
Proof of Theorem 25.3. Let the duality between S(IR") and S'(IR") be given by the extension of the scalar product ( , ) from L2 (1R"). Note that the finitedimensionality of Ker A and Ker A* follows from Theorem 25.1 since due to the inclusion
KerAc Ker BA a Ker(I+R1) the question reduces to the case A = I+ R1 for which everything is obvious. We shall now consider the inclusion A (S' (W)) z AB (S' (R")) = Y+ R2) (S' (W))
For the operator I+ R. the Fredholm property on S' (1R") follows from Proposition 25.5. Therefore the subspace A (S'(P.")) is closed in S'(IIt") and codim A (S' (IR")) < + co which proves the Fredholm property of A in S' (1R"). Let us prove the Fredholm property of A in S(IR"). It suffices to verify only conditions b) and c) in Definition 25.4. We shall show that A (S (IR")) _ {u: u e S (IR"), ul KerA* } ,
(25.9)
where orthogonality is in the sense of L2(IR"). First, note that
A(S'(IR"))= {u:ueS'(R"), ulKerA*},
(25.10)
§26. SelfAdjointness. Discreteness of the Spectrum
197
since A (S'(1R")) is closed in S'(lR"), and S(lR") is the dual of S'(IR"). But now (25.9) follows from (25.10) since A (S (IR")) = A (S' (1R")) n S (IR")
in view of Corollary 25.1. (25.9) shows the Fredholm property of A on S(IR"). Finally let us verify the Fredholm property of A,,,. for s'? s  mo . Once again, it only remains to verify that
Im A,.,. = {u: u eQ", ul KerA*) .
(25.11)
Let ueQ3, ue(KerA*)l. Then u = Av, where vES'(IR") in view of the already proven relation (25.10). But from Corollary 25.1 we then obtain that v E Qi.e. v e Q8, since s< s'+ mo. This shows (25.11). [Ti By analogy with Theorem 8.2 one proves Theorem 25.4. Let A E HGQ and Ker A = Ker A (0). which is the inverse to A. Then there is an operator A ' E HGQ
Exercise 25.7. Prove Theorem 25.4. Problem 25.1. Show that the operator A c HG'
is Fredholm in the space
C,°° (Dt").
Problem 25.2. Show that if a differential operator A with polynomial coefficients has a rsymbol a (z) elliptic in z = (x, ), then the symbol of its parametrix B has an asymptotic expansion in terms of homogeneous functions in z for IzI> 1.
§26. Essential SelfAdjointness. Discreteness of the Spectrum 26.1 Symmetric and selfadjoint operators. Let H1 and H. be Hilbert spaces and suppose we are given an, in general unbounded, operator
A: H1H2.
(26.1)
As usual, DA denotes the domain of A (it is understood that this domain is given with A, which is then a linear map from the linear subspace D,, into H2; note that writing (26.1) does not imply that A is defined on all of H1). The adjoint operator
A*: H2  H1
(26.2)
is defined if D,, is dense in H1 and, in this case, D,,. consists of all v E H2, for which
there exists a vector g e H1 with
(Au, v) = (u, g),
u E DA,
(26.3)
Chapter IV. Pseudodifferential Operator in IR"
198
(on the lefthand side of (26.3) is the scalar product in H2 and on the righthand side that in H,). It is clear that g is uniquely defined and by definition A*v = g. In particular, we have the identity (Au, v) = (u, A* v),
(26.4)
Definition 26.1. Let H be a Hilbert space. An operator A: H+H is called .symmetric if (Au, v) = (u, Av),
Definition 26.2. An operator A:
u, v EDA
(26.5)
is called selfadjoins if A = A*.
It is obvious that a selfadjoint operator is symmetric. The converse is in general not true. Definition 26.3. An operator A: H, + H2 is called closed, if the graph GA, consisting of all pairs {u, Au}, where u EDA, is a closed subspace in H, 9 H2.
Exercise 26.1. Show that if A* is defined, then it is closed.
Exercise 26.2. Let an operator A be bounded, i.e. there exists a constant C > 0, such that IlAu HH S C ]lull, u E D,, . Show that A is closed if and only if D,,
is a closed subspace of H, . The wellknown closedgraph theorem (cf. Rudin [11) states that if DA = H, and A is closed, then A is bounded. Obviously the same holds if DA is a closed subspace in H,. Let an operator A: H, H2 be given. We say that A has a closure A, if the closure G,, of the graph GA is again the graph of (closed) operator, which we denote by A. In particular, any symmetric operator A: H+ H has a closure if DA is dense. Indeed, it is enough to verify, that if u" is a sequence of vectors in DA, such that lim u" = 0 and lim Au = f, then f = 0. But for v ED,, we obtain R1
ac
R. 00
(f, v) = lim (Au., v) = lim (u", Av) = 0, a cc
R 00
from which we have f = 0. Note that if A is a symmetric operator, then so is A. Definition 26.4. An operator A: H' H is called essentially selfadjoins if DA is dense in H and A = A*.
In particular, A* is then an extension of A and, hence, A is symmetric. A criterion for essential selfadjointness is given by
Theorem 26.1. A symmetric operator A: H+H with dense domain is essentially selfadjoins if and only if the following inclusions hold Ker (A*  il) c D; F,
(26.6)
Ker (A* + il) c D; F.
(26.6')
§26. SelfAdjointness. Discreteness of the Spectrum
199
Proof. 1. The necessity of (26.6) and (26.6') is obvious. To verify their sufficiency, let us first note that since A* is an extension of A, it follows from
(26.6) that Ker(A*il) = Ker(Ail). But Ker(Ail)=0 since A is symmetric. Therefore, from (26.6) it follows that
Ker(A*il)=0.
(26.7)
Similarly, from (26.6') we find that
Ker(A*+iI) = 0
(26.7')
2. Let us now verify that (X il) ' (defined on (.4 i1) (H)) is bounded. We have II
(X_ il)f 112 = ((X_ il) j, (A il)f) = II Af II 2 +
11f112,
(26.8)
since (Af, f) is a real number in view of the fact that X is symmetric. It follows from (26.8) that 111112 s II(Ail)f1I2, i.e.
(((AiI)'gI) 5 Ilgh
,
ge(Ail)(H).
3. It is clear that A  it is closed. Therefore (X il)  'is also closed and since
is bounded, its domain (A  if) (H) is closed in H. However the orthogonal complement of (A  il) (H) is obviously equal to Ker (A  il)* = Ker (A* + il) = 0. Therefore (X il) ' is everywhere defined. By similar reasoning, (A+il)' is also everywhere defined.
4. Let us verify that (A il)and (A+iI)
are adjoint to each other. We
obviously have
u, veD2.
((Ail)u, v) = (u, (A+ if) v),
Denoting (Ail) u = f and (A+il) v = g, we obtain the required relation
(f,(A+il)'g) =
((Ail)'f
g),
f geH.
5. Let us finally verify that X= A*. We will use the following easily verified
fact: if B is an operator in H, such that (B')* and (B*) ' are defined, then (B')* = (B*)'. We have
A* =A* _ (A+il)*+i1= {[(A+i1)']`}*+il +i1=Ait+i1=A, = {[(A+il)'']*} as required.
200
Chapter IV. Pseudodifferential Operator in IR"
26.2 Essential selfadjointness of hypoelliptic symmetric operators. In this section we shall denote by A * the operator which is formally adjoint to an operator A e GQ , i.e. the operator A * E GQ , such that (Au, v) = (u, A+v),
u, v E Co (1R") .
In the preceding sections we have written A* instead of A+, but here the notation A* will be reserved for the adjoint operator in the sense of section 26.1.
Theorem 26.2. Let A E HGQ '"o, where mo > 0 and A=A. In L' (NV) consider the unbounded operator Ao, defined as the operator A on the domain Co (IR"). Then Ao is essentially selfadjoint and its closure coincides with the restriction of the operator A (defined on S'(lR")) to the set D,,,, = {u: u e L2 (IR"), Au E L2 (IR")}.
(26.9)
Proof. 1. Denote by D the righthand side of (26.9). Since (Au, v) = (u, Av),
uES(IR"),
veS'(IR),
(26.10)
it is clear that D (_ D4. and in addition
AID=AoID. Let us verify that indeed D = DA;. Let v ED, i.e. v e L2 (IR") and for some f e L2(lR") the identity
(Au, v) = (u, f),
u E Co (IR"),
(26.11)
holds. But it follows from (26.10) that the same identity holds if we replace f by A. Therefore Av = f, i. e. v e D as required. Thus we have demonstrated that the righthand side of (26.9) equals D.,.. 2. In order to now use Theorem 26.1, we will verify the inclusion
Ker (Ao  iI) cDuo .
(26.12)
From what we have already shown, it is clear that
Ker(A**iI) _ {u:uEL2(]R"), (Aif) u=0}. Taking into account that A  iI e HG' m0, it follows from Corollary 25.1 that Ker (A o  if) c S (RR"), from which (26.12) follows, since A maps S (IR") into S(IR") continuously and Q '(R") is dense in S(IR"). Similarly one proves the inclusion Ker (Ao +if) c D,o , which concludes the proof of Theorem 26.2. C]
§26. SelfAdjointness. Discreteness of the Spectrum
201
26.3 Discreteness of the spectrum
Theorem 26.3. Let A E HG; '
where mo > 0 and A + = A. Then A has
discrete spectrum in L2 (IR"). More precisely, there exists an orthonormal basis of eigenfunctions (pj (x) E S (IR"), j = 1, 2, ... , with eigenvalues Aj E IR, such that Al I + + oo as j  + ao. The spectrum a (A) of X = A* in L2 (IR") coincides with the set of all eigenvalues }At}.
Proof. The proof is similar to that of Theorem 8.3. In view of the separability of L2 (IR"), there exists a number Ao E 1R\ a (A). But then Theorem 25.4 implies
and, in particular we see that (A  AO !) ' is that (A  A0!) E HG; compact and selfadjoint in L2(IR"). The remainder of the proof is a verbatim repetition of the proof of Theorem 8.3. D Problem 26.1. Let A E GQ be such that there are numbers A* E C, such that Im A+ >0, Im A_ 0. Denote by Ao the operator A restricted to Co (1R"). Show that Ao = Ao and Ao = (Ao)*. Hint: Use the result of Problem 26.2, after extending the operators AO and A+ to S (IRO).
Remark 26.1. The result in Problem 26.3 means that the "strong and weak extensions coincide" for an operator A EHGQ ' for mo > 0: if u e L2 (1R") and Au E V(1111), then there exists a sequence u; E Co (IR") such that u; * u and Aul + Au as j + + oo in the L2 (1111)norm.
Problem 26.4. Prove analogue of Theorem 8.4 on the structure of the spectrum, eigenfunctions and associated functions for operators A
mo>0.
Chapter IV. Pseudodifferential Operator in IR"
202
Problem 26.5. Let an operator A have the antiWick symbol a (z) a 1'Q (lR2n
and let Aa be selfadjoint in L2 (1R"). Let a (z)  + oo as I z I  oo. Show that Ao has discrete spectrum in the sense of Theorem 26.3 and that A; + +oo as
j++o0. Problem 26.6. Let A eGQ be such that Ao is selfadjoint in L2 (IV) and has discrete spectrum such that A,j, + oo as j + + oo. Let c (z) be the Wick symbol of
the operator A. Show that c (z) + + oo as I z I . + oo.
§27. Trace and Trace Class Norm 27.1 The trace and the HilbertSchmidt norm expressed in terms of the symbol. Here we make use of notations and facts concerning HilbertSchmidt and trace class operators which are presented in Appendix 3. Let us begin with the formal expression for the trace in terms of the rsymbol. Let A e G', let b, (x, ) be the isymbol of A and K4 its kernel. We have formally
K4(x,Y)= ei(:A4
(27.1)
from which
K4(x,x)= and Sp A = $b1 (x, ) dx dd .
27.2)
Note that (27.2) means in particular, that its righthand side is independent
of r. Proposition A.3.2 yields
IIAIIi= jIK4(x,Y)l2dxdY= jIK4(x,x+z)I2dxdz.
(27.3)
But by (27.1)
KA(x,x+z) = je'"
(27.4)
Therefore we have formally
JIK4(x,x+z)I2dxdz= 1K4(x,x+z) K4(x,x+z)dxdz =
b,(x+rz,ry)ddrydxdz
= jer:"t) bt(x,) b=(x,n)ddpdxdz = jl je '=s bT(x,d 12dxdz=
(27.5)
§27. Trace and the Trace Class Norm
203
(we have here used the shift invariance of the integral and the Parceval identity for the Fourier transform). As a result we obtain (27.6)
11A 112 =
where again the righthand side is independent of r
Proposition 27.1. The correspondence between operators A E G  m and Tsymbols b, (x, ) E S (JR2") extends by continuity to an isometry between S2 (L2 (IR")) and L2 (IR") such that (27.6) holds. If A E G', then the condition A E S2 (L2 (]R")) is equivalent to b, E L2 (1R2 ")for some T and this then holds for all T
and the formula (27.6) also holds in this situation.
Proof. The computations (27.3)(27.6) are justified for A e G '0 or, what is the same, for K4 E S (II12w). Since G  °0 is dense in S2 (L2 (IR")) and S (IR2 ") is dense
in L2(IR2"), the existence and uniqueness of the required isometry is obvious. Finally, the last statement is obvious from the uniqueness of the rsymbol. Li Corollary 27.1. If A E GQ and m <  n, then A E S2 (L2 (IV)).
Proposition 27.2. 1) If A E GQ and m <  2 n, then A E Sl (L2 (Ill")) and for
any fixed m < 2n and T EIR there exist constants C and N, such that the following estimate holds
IIAII1 S C Y sup {I0 b,(z)I IyISN
",+Q1Y1}.
(27.7)
z
2) For A E GQ, m <  2n, formula (27.2) for the trace Sp A holds for any T E IR.
Proof. 1) Choose an operator P E for Ker P = Ker P` = 0, exists (the existence of an operator P of this type
so that P1 E HG;
follows, for instance, from Theorem 26.3). In view of Corollary 27.1, we have p2 E S1 (L2(IR")). But from the obvious representation A = P2(P2A) and
the fact that P2A E G°Q C
(L 2(I")), it follows that A E S, (L2(lU")).
Therefore the inclusion GQ c Sl (L2 (IR")) ,
m < 2n.
(27.8)
is proved. Let us now prove (27.7). It can be obtained in two ways: either by a direct sharpening of the arguments carried out so far (from similar estimates in the composition formula and the boundedness theorem) or from the closed graph theorem. The latter route is shorter and is commonplace for many argument of
this type, although it is also rougher. We will carry out carefully the corresponding arguments. Introduce in GQ a Fr6chet topology, defined by seminorms of the form IIAII(N)= I,ISN
sup {Ia"bT(z)I(27.9) i
204
Chapter IV. Pseudodifferential Operator in IR"
We have to show that the embedding (27.8) is continuous in the natural Banach space topology on S, (L2 (IR")). In view of the closed graph theorem (cf.
e. g. Rudin [I]) it is only necessary to show that this embedding has a closed graph. This is most conveniently proved constructing a Hausdorff space M such that GQ c S, (L2 (1R")) c M
(27.10)
where both embeddings GQ c M and S, (L2 (IR")) c M are continuous. Now as M we may, for instance, take S2( L2 (IR")), since the continuity of the embedding of GQ and S, (L2 (IR")) in S2 (L2 (R")) follows immediately from Propositions 27.1 (formula (27.6)) and A.3.7 (estimate (A.3.29)). 2) Now we will prove (27.2) for A E G,', m <  2n. Note that both its parts are continuous on G'. But for any m' > m, G  °° is dense in GQ in the topology of Go'. Therefore, it suffices to prove (27.2) for A e G
We would like to carry out carefully the argument from A.M. This is trivial, if we present A in the form A = L t o L 2, where the operators L I and L 2 have kernels with enough continuous and rapidly decreasing derivatives. But the latter representation can be constructed by an argument similar to the one used in 1) of this proof. 0 27.2 A more precise estimate of the trace class norm in terms of the rsymbol. The estimate (27.7) is not very convenient, since it contains a weightfunction increasing in z. At the same time, we see that II A II1 does not change if we shift the rsymbol by some vector zo = (xo , o) E 1R2n. Indeed, if b, (x, t) is the rsymbol of
A and if we denote by A, the operator with the rsymbol b, (x  xo,  o), then we obtain
A.u(x)= Jeu=r' =
r)x+ryxo, t0)u(y)dyd' t bi((1r) (xxo) u ((y  xo) + xo) dy d
= Je;c,
u(y'+xo)dy'dc', where x' = x  xo , Y' = Y  Yo , ' =  to. Denote by U the unitary operator, mapping u (x) into (Uu)(x) = e'to' x u (x+ xo), then we see that A,, = U from which IIA=,II, = IIAIII
AU,
(27.11)
An estimate of the trace class norm which is invariant relatively to the shifts of the rsymbol is given by the following
Proposition 27.3. There exist constants C and N, such that for A e Ge , m <  2n, the following estimate holds IIAII, s C E I 161b,(z) I dz. IrISN
(27.12)
§27. Trace and the Trace Class Norm
205
Proof. It suffices to show the estimate (27.12) in the case where b, (z) E Co (R2"). First let,
suppb,c {z: I z I S Ro},
(27.13)
where Ro is some fixed constant. Then it follows from Proposition 27.2 that there are constants Cl and M (depending on Ro) such that II A II 15 Cl E sup l a= b, (z) I . IYISM
(27.14)
=
But since for b (z) E Co (lR2n) we have b(z)
_
a2"b az
az2"
.
1
(t1, ..., 12")dtl
J I..... 2n
and consequently a2 n b (z)
sup lb(z)I
.1
aZl ... az2"
s
dz,
it follows from (27.14) that we have (27.12) (with N=M+2n), provided that (27.13) is satisfied. Now, using the invariance of the trace class norm (formula (27.11)) and the invariance of the righthand side of (27.12) with respect to shift in the argument
of b(z), we see that (27.12) always holds, with the same constants C and N, provided diam supp b, S Ro
(27.15)
Let us finally get rid of the condition (27.15). Take a partition of unity cc
1m
(Pi (z)
i=1
such that diam supp Bpi 5 Ro, there is a number I such that any ball of unit radius
does not intersect more than I sets suppvi, and, in addition, 101ip;(z)I 5 C,, j= 1, 2, ..., with constants C. not depending on j. Introducing the operators Ai with the rsymbols coj (z) b, (z), we obtain
IIAIIIS 1] IIA,IIISCY Y fla=(rpib,)(z)Idz i=1
j=1 IYISN
Cl
as required.
I IaYbjz)Idz, lit 5N
L]
206
Chapter IV. Pseudodifferential Operator in Qt"
Problem 27.1. Let A be determined by the antiWick symbol a (z) a rQ. Show that IIAII1 A (u, u),
it follows from (28.3) that
Ln(1E2)H= 0. But then E2 is injective as a mapping of L into E2H, from which it follows that
dim L 5 N (,k),
as required. U 28.2 Properties of the spectral projections. The spectral projections operators enjoy the following properties: 1) Ex = E2; 2) E,? = E2;
3) E2(A,11)E2 0 (meaning that the corresponding quadratic form is strictly greater than 0 on the nonzero vectors in DA); 5) Sp E2 = N (A).
Basic in what follows is
Proposition 28.1. Let E; be a family of operators, for which E,H c DA and which satisfies conditions 1)4). Then 5) is also fulfils, i.e. Sp E; = N (A) = Sp E2 .
(28.5)
Proof. It follows from 1) and 2) that Ex is an orthogonal projection. Putting L2 = EEH, MA = (I Ex) H, we have, in view of 3), that (Au, u) 5 A (u, u), u e L' a, from which, by Lemma 28.1, it follows that Sp EA' = dim L, ,;g N (A). Further,
from 4), we have (E2H) n M; = 0, which implies dim (E2H) = N (A) 5 dim L2 = SpEE, proving (28.5). 1
208
Chapter IV. Pseudodifferential Operator in IR"
Remark 28.1. Note that under the conditions of Proposition 28.1 we do not necessarily have E = Ez (cf. Problem 28.1). 28.3 Approximate spectral projection operator Theorem 28.1. Let A be an operator as in Lemma 28.1 and {4') z eR a family of operators such that S,,Hc D,, and that for some c > 0, 6 > 0, we have: 10.
2°. 'Z is a trace class operator and
.ice+oo,
as
A6)
(28.6)
where V (2) is some positive, nondecreasing function, defined for 2 z AO; 3°..9z(AA1) eSz 5 CA1
4°. (I8z)(AAI) (ItA) CA' 5°. Sp ofz = V(A) (1 +0 (Aa)) as 2 + oo. Let us also assume that the function V (A) appearing in 2° and 5°, is such that
[V(1+CA1`) V(A)]/V(,1) = O(Aa)
as
2> +co
(28.7)
for some C > 0. Then we have
N(A) = V(A) (1+O(Aa))
as
(28.8)
Proof. The idea is to apply Lemma 28.1 to the linear subspace L, spanned by the eigenvectors of 9z, having eigenvalues close to I (they are all close to either 1 or 0, as we shall see later). Let aj be eigenvalues of t. They are real by 1° and by 2° and 5° satisfy the conditions
j
aj  a;
0 (Z
V (A))
(28.9)
>a; = V(2) (1+O(Aa)). Lemma 28.2.
Y
aJ = V (A) ( 1 +0( A5 ) ) .
(28.10)
laj 1151/2
Proof. For I a;  I I > i we have I aj  a; I = I a; I I a;  1 I >_ i a; I. Therefore
Ia; ajI 112
lay 1 I> 112
which together with (28.9) implies (28.10).
J
1a;
a;I= O(XaV(X))
§28. The Approximate Spectral Projection
209
Lemma 28.3. Let N (A) be the number of eigenvalues of 8x in the interval 3/2]. Then
R(A)= V(A) (1+O(A8)).
(28.11)
Proof. Put Ej =1  aj. Then 2° can be rewritten as (28.12)
E IEj  Ej I = O (A' V (A)), J.
and the statement of Lemma 28.2 gives that
Y (1Ej)= V(A) (1+O(Ab)), iz,is1/2
or
('1)= V(.) (1+0(Aa))+ Y Ej. Ic,i.5t/2
But, as in Lemma 28.2, it follows from (28.12) that
Y IE;I 5 2 Y Ie E;I = O(A6 V(1)), {s,Is1/2
i.,I
I/2
giving also (28.11).
Let us continue the proof of Theorem 28.1. a) Let LA be the linear manifold spanned by the eigenvectors of !,, with eigenvalues a, such that I aj 11 < so that
R(1) = dimLA= V(A) (1+O(A"))
(28.13)
by Lemma 28.3. Condition 3° implies that
(4fx(AAI)4f,,u,u) 0 satisfies (28.20) for
Eza.
§28. The Approximate Spectral Projection
211
Proof of Proposition 28.2. Set V (A) = V, Wl V (A)
Then V(A) = V (A0) exp
(,(t)ir).
This gives V (A + CAI `)  V (A) = V (20)
= V(.I)
{ex(
{e(
A+ CA,
exP (f(t)dr)}
f cp (t) dt
A+CA''
f cp(t)dt  1
(28.21)
.
A
, then for e $ b we obtain
Since Icp(.t)I 5 CA'
CAI
A+CAI
I
cp(t)dt 5 C, A+ f
A
A
=
t`b dt = C2
C22`el(1+CA`)`d_1]
For E = S, we obtain the same estimate A+CA''
_
1 and J Xo (v) dv = 1.
Let now A have a real Weyl symbol (28.25)
such that for some C > 0 and Ro > 0
b(z) z ClzI'',
Izl
Ro
(28.26)
(it follows from (28.25), that (28.26) holds either for b (z) or for  b (z); we fix the sign in such a wav that A becomes semibounded from below, this fact should be
obvious from what follows. Put e (z, 1, x) = X (b (z),1, x)
and now, choosing x
(28.27)
where v > 0, define d';, as the operator with the
Weyl symbol e (z, 1) = X (b (z),),1'  Y),
(28.28)
where v > 0 will be chosen later. Let us note immediately that e (`'1)
_ (1 for b (z) 5 1,
10 forb
z).+21'"
(28.29)
§28. The Approximate Spectral Projection
213
Now we try to estimate the derivatives in z of e(;,..). Note that estimates f o r the class Hf Q with mo > 0 can be written in the form
Ialb(z)I 5
Cb(z)td"Y!,
Q'>0, IzI ? Ro,
(28.30)
where as Q' one can take, for instance, a'= p/m or possibly larger values. Now, differentiate (28.28):
8:e(z,A)=
...+hY=61z) cr,...Y,(dY'b(z))... (a''b(z))
A,
ark (t,
IY;I>0
(28.31)
where the sum runs over all possible decompositions of y into a sum y, + with an arbitrary number of terms k,5 I y I.
... +y,
Denote by Tk(z, k) the sum of all terms corresponding to a fixed k in (28.31). It follows from (28.31), (28.30) and (28.24) that
ITk(z,.l)I
7dC.
If K. (x, y) is the kernel of B, we obtain that then
(x+ x, c) a2 Ke(x, Y) =J aI I\ 2
2
\x1+Y dx, drtds .
x ei1(
(29.4)
Now using formula (23.39) (with r=), that yields an expression for the symbol in terms of the kernel b (x, ) = J e 'x, ' 1 KB (x +
2
,
X_ 2) dx2
(29.5)
Putting x2/2 = x3, we can also write
b(x,c) = 2 " Je2;x" KB(x+x3, xx3)dx3.
(29.6)
§29. Operators with Parameters
217
From this and (29.4) we find that
xlx37) 2
'7Zx,f[ dX, dX3dgds.
x
Instead of x, and x3 we introduce new integration variables x+X, +X3
x4 =
2
,
X5 =
X+X, X3 2
so that x, = x4 + x5  x, x3 = x4  x5. Observing that obtain
(XI, x3) = 2", we a (x4, x5)
b (x, c) = 22" [ a, (x4, S) a2 (xs, q) x ez'[(xx,1 ,+lx.x['7+1x,x.1'cl dx4 dx5 ds dq or
b(x,0 = 22n 1 a, (y, q) a2 (z, S) x
e2i[(xz)
7+u'x) c+(;y)
dydzdg dd
(29.7)
Note that the exponent in (29.7) may also be written as 1
=2i E
2i
j=1
1
1
xj yj z j
j qj 'j
From the form of this exponent, the possibility of integrating by parts follows, resulting in the appearance of decreasing factors of the type  ",  ",  ", 0 and ao does not depend on y. If A(1) is the operator with the Weyl symbol a(z, A), then for sufficiently large A we have A(il) > 0 (i.e. (A(.X)u, u) > 0 for u E S(lR")). For the proof, we need the following lemma which allows us to use the antiWick symbol. Lemma 29.1. Consider an operator B(A) with anti Wick symbol a (z, A) a I'Q and let b (z, A) be its Weyl symbol. Then
ab=
Y.
cY (c3Z a) + rN ,
00,
a>0.
(30.7)
We have to verify that all the conditions of Theorem 28.1 are fulfilled. The condition S,* = 6,, is obvious since e (z, 2) is realvalued. The fact that e2 belongs
to the trace class follows from Proposition 27.2. Denote the Weyl symbol of an arbitrary operator A by a (A). Obviously
(9x1)=
cas[araxe(z,2)]
[04axe(z,2)]+(e2e)+rN,
(30.8)
where rNCFMQ,2Na. Note that all terms in the sum, except for rN, are supported where A5 a(z) < 2(1 +2)."), and if we apply Proposition 27.3 to each term, we obtain the estimate 11 g'42
 gxl1I = O(V(2+22'") V(2)).
But it follows from Proposition 28.3 that
V'(2)/V(2) = O(2"'), which, by Proposition 28.2, gives the estimate
V(1+221") V(2) = 0 (2"" V(2)) . Therefore
II',i'x111=O(2" V(2)).
(30.9)
In addition, it follows from Proposition 27.2 that Sp 9,1 = (2n) 'S e (z, 2) dz
= V(2)+O(V(2+22'") V(2)) = V(2) (1+O(2' v)).
(30.10)
§30. Asymptotic Behaviour of the Eigenvalue
225
Note that we must take v < Q'. Choosing v = Q' ' e, where e > 0, we may rewrite (30.9) and (30.10) in the form Ii'f gx111
(30.11)
V(A))
Spd,, = V().)
(30.12)
2. Let us now verify requirement 3° of Theorem 28.1: fx (A AI) S3 5 CA'
We write this inequality in the form
49x(.1A)d,,+Cd'">0.
(30.13)
Next we compute the Weyl symbol of 'A (A.1 A) d' . We have 47 (d'A(AIA)) =
ca0 (axa4e)
4942 (Aa(z)))+rN.
la+oI O and o > 0, we have the estimates I(aY'e)(aYe)(aY(.1a)) I a(b,I+IAI+IY,I).
(30.15)
To begin with let y3 = 0. Then (30.15) holds in view of the fact that 12 a (z) I
2.1'
on the support of (aY' e) (aY, e). Let Y3 * 0, then
I (a1'e)(aye)(a,a) 15 Ci)(IY,I+IY,q
eoY,I+IY,D),e,IY,I,
226
Chapter IV. Pseudodifferential Operator in IR"
where a, = Q'  v. Taking into account, that the inequalities A:5 a _ 1. Remark A.1.1. The condition am(xo, 0) * 0 is sometimes called ellipticity of A at (x0i 0). It is easy to formulate and prove the hypoelliptic analogue of Proposition A.1.2. We leave this for the reader as an excercise. Corollary A.1.2. For AECLm(X) denote
char (A) = Then if Au = f e C°° (X) we have WF(u) a char (A). In particular, if char (A) = Q we have u e C aO (X).
Corollary A.1.3. If u e 9'(X), then
WF(u) _
(l
char (A).
(A.1.8)
AeCL°(X) Aye a C`(X)
This holds for u c 9'(X) if we take the intersection only over properly supported A.
The importance of Corollary A. 1.3 is that (A. 1.8) shows how to define WF(u) invariantly as a closed conic subset of T'X when X is a manifold. We now generalize Proposition A.1.2 even more, by weakening the requirement Au E C°° (X).
Proposition A.13. Again let A E CLm(X), u c91'(X) and either A be properly supported or u e o°'(X). Then, assuming am (xo, o) $ 0 and (x0, b°) t WF(Au), we have (x0 , o) + WF(u). In other words,
WF(u) c char (A) v WF(Au).
(A.1.9)
Proof. By proposition A. 1.1 there exists a properly supported P e CL° (X), with ap  1 (mod S °°) in a conic neighbourhood of (x0, c,0) and (PA) (u) e C°3 (X). But then, from Proposition A. 1.2 it obviously follows that (xo, 0) * WF(u).
Proposition A.1.4 (Pseudolocality of `PTO). Let u e 9'(X), A e LQ 6 (X) 0 < S < Q 0 0 Er,
eIR"\r,,
Ixx0I
= R(t,0).
(A.1.22)
from which, by the condition 1 n WF(f) = 0 and by Proposition A.1.5, it follows that for the values of t of interest to us, the estimate I R (t, 0) 1 5 C" 0 is the Planck constant; the operator A(h) is well defined on S(lR") for example, if the function b(z) belongs to F (IR2n). Classical mechanics is the limiting case of quantum mechanics, when the Planck constant can be considered to be negligible. This motivates an interest in
the asymptotic properties of operators of the form (A.2.1) as h0; the corresponding asymptotic analysis is called quasiclassical or semiclassical. A.2.1 Basic results
The change of variables  h ' transforms (A.2.1) into (x) = hn f
e+cxy
cIh b
rx 2 y ,
I u (y) dy d
(A.2.2)
where the symbol no longer contains the parameter h, which now is included in the exponent instead. We will say that b(z) is the Weyl hsymbol of A (h) or, briefly, the hsymbol (in this appendix, we will not use the vsymbols of chap. IV, which avoids any confusion). Clearly, the 1symbol is then the ordinary Weyl symbol.
Between the h and 1symbols the following relation exists. Making the change of variables x Vh_X, y. j y, in (A.2.2), this expression
lj
becomes (A(h)u)(x h+) = f
eux_
b(v x2 y,
v
)
(A.2.3)
In the space of functions on IR" introduce the dilatation operator
Th: f(x)  h"" f(
hx)
V
.
(A.2.4)
Quasiclassical Asymptotic Behaviour of Eigenvalues
241
It is easily seen that TI, is unitary on LZ(IR"). Using this operator, (A.2.3) can be written as ThA(h)u = A(*)Thu or
A(,,) = T, ' AiiiT ,
(A.2.5)
where A,i)) is the operator with the Isymbol b(')(z) = b( jz). Therefore the operator with hsymbol b(z) is unitarily equivalent to the operator with the 1symbol b('t(z).
We will be interested in the quasicalssical asymptotic behaviour of the eigenvalues.
Definition A.2.1. Let AM be a selfadjoint operator semibounded from below. N. (.1.) denotes the number of eigenvalues of the operator not exceeding ,l (counting multiplicities). If there are points from the continuous spectrum of A(,,)
in the interval ( oo, ).], then by definition N,,(2) = + oo. Remark A.2.1. The Glazman variational principle (28.1) remains valid also for N,,(1); the proof (cf. §28) can be taken over with minor changes to the case N (A).
To formulate the basic result, we need the following Proposition A.2.1. Let A(,,) be an operator with the real hsymbol 0. Then for any fixed h > 0 the operator A(,,) is essentially selfadjoint.
b(z) E Hr' "'O, m°
Proof. For m° > 0 the proposition follows from Theorem 26.2. An analysis of the proof of Theorem 26.2 shows that the strict inequality m° > 0 is only needed in order to ensure A ± iI E HG'r°. Under the assumptions of the proposition, for m° = 0 and h = 1 the inclusions A ± i I E HGQ 0 follow from the estimates
Ib(z)±iI>b(z), I01(b(z)±i) 16 C, Ib(z)I IzI1111 = C7 I b(z)±iI IzI1I11 Ib(z)I/Ib(z)±iI 5 C, Ib(z)±ii Iz11111.
For h * I one has to use (A.2.5) and the fact that b(') a Hf'o m. (as for b (z) but with other constants in the estimates of the derivatives). Let A(,,) have a real hsymbol b (z) a HFQ °. As in §30 changing the sign if necessary we may assume that b (z) z C > 0 for I z I z R°. Put
V(2) = (2n)" I dz. b(g) O for Q I z I z Rc . Let 1a be such that V (AO) < + co. Then, for almost all A < AO and arbitrary e > 0 we have the asymptotic formula
Nh(A) = h""(V(.)+O(h`"2`))
(A.2.7)
Remark A.2.2. Between the asymptotic formulae in h ash ' 0 and the ones in d as A  + oo there is an intimate relation which can be explicitly exhibited when b (z) is homogeneous: b (tz) = tsb(z), t > 0, s > 0. In accordance with (A.2.5) the operator with the hsymbol b(z) is conjugate to the operator with symbol Y" (z) = hs'2b(z), so that Nh(A) = N(hsnA).
Remark A.2.3. Theorem A.2.1 is analogous to Theorem 30.1. In the latter we assumed the essential inequality b (z) ? C I z I M0, c > 0, me > 0. Theorem A.2.1
states the weaker dependence of the asymptotic behaviour in h on the behaviour of the symbol at infinity.
A.2.2 The idea of proof of Theorem A.2.1. The proof of the theorem is based on the same considerations as the proof of Theorem 30.1. We will construct an approximate spectral projection .A,, in the following way. Let X,2 be the indicator function of the interval ( co, A]; we construct a family of functions Xh. a , converging to XA as h ' 0. The operator has the hsymbol X,,., (b(z)), where b(z) is the hsymbol of A,h,. We will show that the family .0h has the following properties:
1°. A*=.1h; 2°. Ah is a trace class operator and
112Ah111 =O(h 3°.
as
A,,(A(h)Al) . h 5 Ch";
40. (1Ah) (AM AI) (I.lh)>  Ch'; 5°. Sp Ah = h" V(2) (1+0(h"));
here 0 < x < 1/2 and the function V(2) in 5°, is a positive, nonvanishing function, defined on the interval [A, A+s] and differentiable from the right at A. Note also that Im Ah c DAM , where D., is the domain of A,h,. In the presence of a family of operators, with properties P5', theorem 28.1, reformulated in the new terminology is fundamental in obtaining the asymptotic formula (A.2.7). For convenience we formulate the following result.
Proposition A.2.2. Let A,h, be a family of essentially selfadjoint bounded from below operators; Ah a family of operators such that Im F. c D,, and having the properties 1 °5°. Then
N,(A)=h`(V(A)+0(h")) as
h0
Quasiclassical Asymptotic Behaviour of Eigenvalues
243
Proof. Similar to the proof of Theorem 28.1. Exercise A.2.1. Prove Proposition A.2.2. A.2.3 Symbols and operators with parameters
To study the approximate spectral projection, it is convenient to introduce class of symbols, depending on a parameter (cf. 29.1). Definition A.2.2. Denote by L the class of functions a(z, h), defined for z E IR2n, 0 < h ,Q,NQ,li+s't
=ClA+2h".
Let us estimate the zderivatives of e (z, h). Differentiating (A.2.31) with respect to z gives:
Quasiclassical Asymptotic Behaviour of Eigenvalues
249
k
a a (z, h) =
t" (t, , h) I, = NO
c,...... ,, (a'' b (z)) ... (ay b (z))
r.+...+r,r
(A.2.32)
The summation in (A.2.32) runs over all possible decompositions of y into a sum yt +... +yk, where k 5 l y I. Taking (A.2.30) and b E H I'Q 0 into account, we obtain for an individual term in (A.2.32) the estimate
(arb) ... (5 .b)
a:k
r =b(z)
/
Chkxlblk
I
0
Owing to (A.2.42) for the first term we have (3Y(p(z,h))(b(z)A)I S Ch'111
(A.2.45)
elYlhx;
and for the other summands of (A.2.44) we get
Ibi' I(a°b)/bI ChxI
Chxx'Yle'Y'.
(A.2.46)
The estimates (A.2.45) and (A.2.46) show that 4p (b  A) a E0; x , from which, taking (A.2.43) into account, (A.2.41) follows. Thus, the finite sum in (A.2.40) belongs to EQ x° '  x and the operator ,, (A,,,, A!) can be written in the form
.F,,(A,ti,AI) = Q, + R,
(A.2.47)
where IIR11=O(h'x)=o(hx) and a(Q,)=e  (bA). Using (A.2.47) we have
.tee(Aa)A!) 'F,,=Q,.F +R,,
(A.2.48)
IIR,II =o(hx).
We will compute the symbol of Q, .F,:
a(Q, tee) = e2 (bA) +
Y
c°ah12+61(0x8{ (e (bA)I)
00
< Chx" lYI)
_  Ch". Thus, we have for the principal part and consequently for the whole operator .F,,(At,,,1.1).Fh the estimate (A.2.52)
_f7h (A(h)  AI) .Fh 5 Ch'.
4. Now we will verify that (I  Fh) (Ach>  AI) (I ,Fh) Z  Ch".
(A.2.53)
The symbol of the left hand side of (A.2.53) (after getting rid of parentheses) is
(.$rh(AM .II).Fh)  a(. h(A(h)21))  a ((A(h)  Al) .Fh) + (b (z)  A).
(A.2.54)
In step 3 of this proof, it was shown that in the first two summands of (A.2.54) the principal terms are distinguished e2 (b  A) and e (2  b), and the operators corresponding to the remainders are estimated in norm by 0 (h'  x). The third summand in (A.2.54) is analogous to the second. Thus we obtain a ((I  _11,h) (AM  21) (I  `$h)) = (1 e) 2 (b 1.) + r ,
(A.2.55)
and the operator R with the hsymbol r admits the estimate
IIRII = O(h`") Now consider the operator P with hsymbol
q(z,h) = (I e(z,h))2 (b(z)A). Arguments similar to the ones used in section 29.3 in proving the positivity of an
operator with positive symbol show that
P=Qk+Ak.
(A.2.56)
Here the operator Qk has the hantiWick symbol q k (z, h) = q (z, h) +
Y_
251I h'"N.
(A.2.63)
By the Glazman lemma, we obtain then from (A.2.62) and (A.2.63) the inequality N,,(A + ch") > h"N which implies the result of the theorem. Introduce the set Q'= {z: b(z) S A). It is obvious from the definition of V(A), that V (A) = (2tr) "mes Q , so that under the conditions of the theorem we have
mes0''=+0o. Now let Qt be a family of open sets with smooth boundaries, satisfying the following conditions (1) 0= are bounded, 0,c QA; (2) W, = as a
i.e. the distance between 0, and 00'
is not less than e.
Now construct a smoothed characteristic function of Q. (this is possible along the lines of the construction in 28.6).
Quasiclassical Asymptotic Behaviour of Eigenvalues
255
Let 2hx < e (as always 0 < x < 1/2) and *E(z, h) the characteristic function of the (hx)thickening of the set . QE. Put Xa(z,h)=h2"xJWs(Y,h)
Xo((yz)h x)dy,
where Xo (v) a Co (I2"), x, Z 0, Xo(v)=O for I v I > i and J Xo (v) dv =1. It is obvious that
suppX,c W.
(A.2.64)
In addition, it is easily verified that 10= X, (z, h) I < Ch  xIYl, and from this estimate
and the compactness of the support of X, we have X, (z, h) E EQ, x . 0.
(A.2.65)
Let Fb be an approximate spectral projection as constructed in A.2.6. Denote by F,, the operator with the hantiWick symbol e(z,h), by d' the operator with the hsymbol X. (z, h) and by Et, ,, the operator with the hantiWick symbol L(z,h). The following relations hold between the operators . h, F,,, 8,.,,
and E,.,,: F,,  F,, =
RESQ.XQ.1 2x,
IIRII S Ch' 2x;
(A.2.67)
F,, > E,.,, ; R'eSQ.;°.1 2x,
8,. tiE,.,, =
(A.2.66)
Ch'2x.
IIR'II