Precise Spectral Asymptotics for Elliptic Operators Acting in Fiberings over Manifolds with Boundary

243 Pages·1984·4.465 MB·English

by  Victor Ivrii (auth.)

Lecture Notes ni
Mathematics
Edited by A. Dold and B. Eckmann
:seiresbuS RSSU
Adviser: L.D. ,veeddaF dargnineL
1100
Victor Ivrii
Precise Spectral Asymptotics
for Elliptic Operators Acting ni
Fiberings over Manifolds with Boundary
galreV-regnirpS
Berlin Heidelberg New York Tokyo 1984
Author
Victor Ivrii
Dept. of Mathematics, Institute of Mining and Metallurgy
Magnitogorsk 455000, USSR
Consulting Editor
Olga A. Ladyzhenskaya
Leningrad Branch of the Steklov V.A. Mathematical Institute
Fontanka 27, Leningrad ,110191 USSR
AMS Subject Classification (1980): 35 20, P 58 G ,71 58 20, G 58 G 25
ISBN 3-540-13361-5 Springer-Verlag Berlin Heidelberg New Tokyo York
ISBN 0-387-13361-5 Springer-Verlag New York Heidelberg Berlin Tokyo
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TABLE OF CONTENTS
Introduction . . . . . . . . . . . . . . . . . . . . . . . . IV
PART I. THE ASYMPTOTICS FOR SECOND-ORDER OPERATORS ..... I
O. Main theorems . . . . . . . . . . . . . . . . . . . I
.I The L.Boutet de Monvel operator algebra
and selfadjoint projectors . . . . . . . . . . . . . 19
2. The "Wave" equation: finite speed of
propagation of singularities . . . . . . . . . . . . 44
3. The normality of the "great" singularity . . . . . . 61
4. Calculation of the "great" singularity . . . . . . . 96
5. Proofs of the main theorems . . . . . . . . . . . 150
Appendices A - E . . . . . . . . . . . . . . . . . 160
Part IL THE ASYMPTOTICS FOR MISCELLANEOUS PROBLEMS .... 177
6. The asymptotics for first-order operators .... 177
7. The asymptotics for second-order spectral
problems . . . . . . . . . . . . . . . . . . . . . 183
8. The asymptotics for higher order spectral problems
related to the Laplace-Beltrami operator ..... 190
Appendix F . . . . . . . . . . . . . . . . . . . . 225
List of notations . . . . . . . . . . . . . . . . . . . . 226
Subject index . . . . . . . . . . . . . . . . . . . . . . 226
References . . . . . . . . . . . . . . . . . . . . . . . . 229
INTRODUCTION
This book is devoted to the determination of the precise asympto-
tics for eigenvalues of certain elliptic selfadjoint operators acting
in fiberings over compact m~n~folds with boundary and for more
general elliptic selfadJoint spectral problems. The precise asympto-
tics for restriction to the diagonal of the Schwartz kernels of the
corresponding spectral projectors is derived too. These asymptotics
for closed manifolds were determined in author's paper ~57S using the
same method; therefore one can consider C57~ as a simple and short in-
troduction to the methods and ideas of this book. More general re-
sults were obtained, for example, in E32 - 35J, but with much weaker
remainder estlmAtes.
In Part I we derive these asymptotics for the second-order ellip-
tic selfadjoint differential operators acting in fiberings over m~n~-
folds with boundary on which an elliptic boundary condition is
given; this condition must be either the Dirichlet condition or the
generalized Neumann condition; the latter means that the boundary va-
lue of the derivative of the function with respect to the direction
transversal to the boundary is expressed through the boundary value
of the function by means of first-order pseudo-differential operator
on the boundary; examples show that for more general elliptic bounda-
ry conditions our results may not be valid.
We shall derive the following asymptotics for the eigenvalue dis-
tribution function:
)k(N = o~ + ~ k O(k~-4),
N(k) = ,,~ k ~-~ + O(k J'-2) or
N(k)= O(4)
where ~ is the dimension; the second asymptotics may occur only if
the boundary is not empty. It is impossible to derive stronger estima-
tes for remainder without some condition of global nature. Pot the
Laplace - Beltrami operator this condition is:
The set of all points periodic with respect to the geodesic flow
with reflection at the boundary has measure zero.
In a general case if certain strong conditions involving some glo-
bal condition are satisfied then the following asymptotics hold:
N(k)=~ok+~k +o(k N(k)=~k +%k +o(k ).
) or
Under some conditions the asymptotics for the restriction to the
diagonal of the Schwartz kernels of the spectral projectors are deri-
red too.~Inside these asymptotics have the same nature:
%0(,~)~ + 0 (~-4) or 0(~) but near the boundary the lea-
dlng part of th~ asymptotics contains the term of boundary-layer type
of the order k w. If and only if this term exhausts the leading part
of the asymptotics then we have the asymptotics ~ (~) ----
~I 1-~k + O ~ k~-2 ) for the eigenvalue distribution functions.
In Part I the methods and ideas of E56, 572 are generalized and
improved upon and in Part II these methods and ideas are applied to
certain new situations and the results of Part I are extended to these
situations. In 26 the asymptotics for the elliptic selfadjoint first-
order operators are derived. In ~7 we derive the asymptotics for the
spectral problem
= o
where A is the elliptic selfadJoint second-order positive definite
operator acting in the fiberingovera manifold withboundary and ~ is
non-degenerate Hermitianmatrix, acting in the fibers of this fibering.
The apposite case when ~ is positive definite A is not necessarily
I
positive or negative definite may be reduced to the case ~= in-
vestigated in Part .I In 8 ~ we study the problem (*), when A and
differ from (-A) ~, (-A)~ respectively only by lower order
terms where ~> 0 ~ ~ and A is the Laplace - Beltrami operator.
We must note that the latter case is the most complicated and in this
case we need some improvements and modifications of our methods.
We intend to write a few papers in future; in these papers the
spectral asymptotics for global and partially global operators 326~
and the quasiclassical spectral asymptotics for ~ -(pseudo)-differen-
tial operators ~63S will be derived and applications of these results
to the problem )*( where ~ may degenerate in a definite manner, will
be given.
ACKNOWLEDGEMENTS. I express my sincere thanks to B.M.Levitan and
M.A, Subin who drew my attention to the spectral asymptotics (to H.Weyl
conjecture), to M.S.Agranovi~, V.M.Babi~, L.Pridlender, G.Grubb,
V.B.Lidskii, G.V.Rosenblum, M.Z.Solomjak and to M.A.Subin V again for
useful, stimulating and clarifying discussions and to my wife Olga
for her understanding, encouragement and support. ! express my sincere
thanks to Yu.N.Kovalik for assistance in translation.
Part I. THE ASYMPTOTICS FOR SECOND-ORDER OPERATORS
O. Main theorems
O.1. Let X be a compact ~-dimensional C°°-manifold with
the boundary y~ C °° , ~2, ~0~ a C Co -density on X , E a Her-
mitian D-dimensional C -fibering over X • Let ~: C°°(X,
be a second-order elliptic differential operator, for-
mally selfadjoint with respect to inner product in Lg(X~E) ; if
y=~ ~then ~ may be a classical pseudo-differential operator. Let
be ~ a G vector field transversal to y at every point, % the
operator of restriction to Y , B=~ or B= ~V ~ IB ~ a boundary
operator, where B1:C°°(y~E) ~- C (Y~) E is a first-order classical
pseudo-differential operator on y ; certainly, only ~ but not its
components is an invariant object. We suppose that (E). Operator
[~,~:CO°(X~E) --~ C°°(X,E) ~ Co°~Y ~ E) satisfies the Sa-
piro - Lopatinskii condition; (S) ~5 - the restriction of ~ to the
~et ~ (i.e. operator with the domain D(~B) = COO(X , E)
QX,E) L
0 ~g% B) is a symmetric operator in
Let A5 : L2(X, E) = L~ (X,E) be a closure of %B in
L~(X, E) ; then AB is selfadjoint, its spectrum is discrete, with
finite multiplicity and tends either to + oC , or to + oc , or to
-co. Without the loss of generality one can suppose that 0 is not+
an eigenvalue of 5 A . Then, if X is a closed manifold, then ~--
selfadjoint projectors to positive(negative)invariante subspaces of
B A are zero-order classical pseudo-differential operators on X •
If X is a manifold with+ boundary then ~- belong to Boutet
de Monvel algebra, i.e. ~- = ~°+-+ ~/~- where ~ -+° are zero-
order classical pseudo-differential operators with the transmission
property on X and ~'± are zero-order classical singular Green
÷
operators (see [143 or I ~ of this book). N- - the principal symbols
of +-~ - are selfadjoint projectors to positive and negative inva-
riant subspaces of ~ G ; is the principal symbol of ~ . If N±=0
then ± n is a zero-order classical singular Green operator. It
should be pointed out that for a more general boundary operator
v
even satisfying the Sapiro - Lopatinskii condition, these statements
+
may not be valid: ~- may be operators of a more general nature;
are likely to belong to Rempel - Schulze algebra F83].
Let H be a closed subspace in I ~X,E) such that N -selfad-
joint projector to H - belongs to Boutet de Monvel algebra (if X
is a closed manifold then ~ is a classical zero-order pseudo-dif-
~BA
ferential operator). We suppose that ~AB -c ; this means
that H is an invariant subspace of A& . Then AB, H " H--~ H ,
~CAB,H) = ~ (A B) ~ H - the restriction of B A to H is a
selfadjoint operator; its spectrum is discrete, with finite multipli-
city and tends either to ±oo , or to +oo , or to -co.
Let us introduce the eigenvalue distribution functions for
N±ik)= N ~kiH
,sA H : is eht rebmun fo fo ,BA H lying
eigenvalues
between 0 and ±k.
Let E~$) ÷be spectral selfadjoint projectors of 8~+ ~ ~H = ($)
HE-)2k±(
=NEC5) He
and &H(5) = ±(+E H CO)) ; let (OC, ~,k)
be Schwartz kernels of -e H )kC ; then
H N )kC : tt ,OoC ,Co k)dog.
H e )kC =
X + +
eW era interested in eht asymptotlcs of NH(k ) dna )k,c0,~oC~
k
as tends to O0 assuming~ of course, that +-ABH is not semiboun-
ded from above.
5
0.2. Let ~ and ~O be the principal and subprincipal soymbols
of ~ , F]- and lF the principal symbols of ~ Oz and ~ ~(5) ~
+
spectral selfadjoint projectors of ~ , ~-= • (~C-+I)-6CO)).
0.3. We start from the results of F57] concerning asymptotics
+
on closed manifolds. One can assume that+ rIFl-~O on some connective
component of T*X\O ; otherwise ~- is of order )~-~ , there-
fore N~-+ is finite-dimensional projector and +-AB,H is semi-
bounded from above.
THEOR~ 0.1. If X is closed manifold then the following asymp-
totics hold:
+ _ + ~ )~-~
kCHN ) =~k CO+ k oo>--
(O.1) ~s
)2.o( 6Hi~,~,k)=~C~) + + +,~k kCO )4-a
9X
as ~ -,00 uniformly with respect to ~CE
where
S
~ =C~)-~ ~ , ~ is the measure no ~; X , generated by
the la~lLLtan measure ~)~ no T~X and by fixed measure ~ on X;
+
;~o );d[ > 0 +on ~ -connective component of X ,if
cone s~p ~-~ @ ; otherwise
±
(0.2) H& (:~,~,k) = 00) sa k---,.-oo
uniformly with respect to 0 ~ ;
X*T
X
R~ARK. If X is a manifold with the boundary Y then the asym-
ptotics (0.2) holds for ~EX\Y uniformly on every compact sub-
set of X\Y. -4 eo _+
R~JuRK. It is obvious that o~J E C ; all coefficients ~. ~)
which will be introduced in future will also be smooth.
To obtain the second term of the asymptotics for the eigenvalue
distribution function we need some condition of a global character.
Let 0 < ~I 2 <~ ~2 ~ °" °~<~D+ be all the positive eigenvalues of
_+~ ; D± may depend on connective component of T'X\ 0 . Let
~ be a conical with respect to ~ closed nowhere dense subsets
fo~-- T~X\0 such+that for everyJ ~; has the constant multiplici-
ty outside of ~ ; this multiplicity may depend on the connective
J i +
component of TX\0\~T . These subsets exist without question.
Consider the bicharac~eristics J of ~d - the curves along which
±
cp), p = pit) \ X 0 \ 7 i ,
'
erehw ~M = ~ ~ ~> -- < E ' -~ > is the Hamil-
tonian field generated by ~ ~ . bicharacteristic is called periodic
if there exists T e~ 0 such that ~(t ÷ T) = ~(~) T ; is called the
period of bicharacteristic.
THEOR~ 0.2. Let X be a closed manifold and the following con-
dition hold: + ±
(HI~ There exists a set ~ of measure zero, Uj ~jT~A ?
\ X 0 , such that through each point of \
for every I there ,sessap a nonperiodic bicharacteristic of
lying in ~T*X \ 0 \ ~; , which is infinite in both directions.
Then the following Q asymptotics holds:
_ + _ + ~ - + J,-~ ~-~
(0.5) NHCk)=%k +~ k +oCk ) k . oo
where
X T;
,
R V . I f A is a differential operator and fl(x,-s)= (~,5)
then = 0 where depends only on a, 17 ( see Appendices D , F$
0.4. Now let X be a manifold with the bo.u ndary. Let xI€C
X, = 0 and d ~# ,0 at Y , X,2 0 in X \ y Consider the charaote-
ristic sp~boi% + (p, 7) = det (z2 ia cp).
DEFINITION. hhe point p E T*X\ Y has the multiplicity
E=CF(f)if
(ii is negative if
(iii) is tangential if
...,
vi 0, 1-1;
=
(iv) is indefinite in the remaining cases. a
Here 5, is the dual to 3, variable, -= -Hq
851
j : ~ * x I y T*Y
let be a natural mapping.
nn-
THEOREM 0.3. Let be not a singular Green operator (i.e.
±
the asymptotics (0.1) holds with the same coefficient $80.
THEOR~ 0.4. Let ]-[ 7[ ± be a singular Green operator. Then the
following asymptotics holds:
*)
±
(o.7) NH(k)= l k +O(k ) sa k ~00
+ !
+
where Bg<O l- 1 depends only on ~]Y and on (~ ~-)0 - the prin-
cipal symbol of singular Green operator (~+-')/ ; ~ is the
principal symbol of ~.
0.5. To obtain the second term of the asymptotics for the eigen-
value distribution function we need some condition of a global na-
ture.
Suppose first that
(H.2) ,
=~(~,~)
where ~(~) is a positive definite quadratic form in
(i.e. a Riemannian metric on X ) and ~(#, T) does net vanish.
~X
Then on -tangent spheres bundle of the Riemannianmanifold
X (more precisely on its subset of the complete measure) one can
define a continious measure - preserving ~eodesic flow with reflec-
tion at the boundary. More particulary: consider S*X with iden-
tified points ~ and p1 such that J~:J~. i
By a ~eodesic we mean a curve lying in S X , along which
(i.e. the bicharacteristic of ~ ;) the geodesic, minus, per-
haps, its endpoints, must lie in S (X\Y) ; $ (the length) is
the natural parameter along the geodesic.
By a geodesic billiard we mean a curve lying in S*X consisting
of segments of geodesics; the endpoint of the precedent segment and
the starting point of the consequent must belong to S*X YI and
be equivalent (i.e. identified); ~ (the length) is the natural para-
meter along geodesic billiards.
It is easy to show that there exists a set ~ ~ S X , of
first Baire category and measure zero, such that through each point
of ~*X \~- a geodesic billiard of infinite length in both direc-
tions can be passed in such a way that each of its intervals of fi-
*) Here O(k °)~ O(~k)