Progress in Inverse Spectral Geometry

201 Pages·1997·15.51 MB·English

by  Stig I. Andersson

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Progress in
Inverse Spectral
Geometry
Stig 1. Andersson
Michel L. Lapidus
Editors
Springer Basel AG
Editors' addresses:
Stig 1. Andersson Michel L. Lapidus
CECIL Department of Mathematics
Blâ Hallen, Eriksberg University of California
S-41 7 64 G6teborg Riverside, CA 92521
Sweden USA
1991 Mathematical Subject Classification 58G30, 35P20
A CIP catalogue record for this book is available from
the Library of Congress, Washington D.C., USA
Progress in inverse spectral geometry /Stig 1. Andersson ; Michel L.
Lapidus, editors.
p. cm. --(Trends in mathematics)
Includes bibliographical references and index.
In English and French.
ISBN 978-3-0348-9835-5 ISBN 978-3-0348-8938-4 (eBook)
DOI 10.1007/978-3-0348-8938-4
1. Spectral geometry /2. Inverse problems (Differential
equations) 1. Andersson, S. 1. (Stig Ingvar), 1945- .
II. Lapidus, Michel L. (Michel Laurent), 1956- . III. Series
QA614.95.P78 1997
516.3'62--dc21
Deutsche Bibliothek Cataloging-in-Publication Data
Progress in inverse spectral geometry/Stig 1. Andersson ; Michel
L. Lapidus, ed. -Basel ; Boston; Berlin: Birkhăuser, 1997
ISBN 978-3-0348-9835-5
This work is subject to copyright. AII rights are reserved, whether the whole or part of the material is con
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© 1997 Springer Basel AG
Originally published by Birkhăuser Verlag in 1997
Softcover reprinl of Ihe hardcover 1s I edilion 1997
987654321
Contents
Stig I. Andersson and Michel L. Lapidus
Spectral Geometry: An Introduction and Background Material
for this Volume .................................................. 1
Jeffrey M. Lee
Geometry Detected by a Finite Part of the Spectrum 15
Carolyn S. Gordon and Ruth Garnet
Spectral Geometry on Nilmanifolds 23
S. Frankel and J. Tysk
Upper Bounds for the Poincare Metric Near a Fractal Boundary 51
Pierre Berard et Hubert Pesce
Construction de Varietes Isospectrales
Autour du Theoreme de T. Sunada 63
Seiki Nishikawa, Philippe Tondeur and Lieven Vanhecke
Inverse spectral theory for Riemannian
foliations and curvature theory .................................... 85
Cheryl A. Griffith, Michel L. Lapidus
Computer Graphics and the Eigenfunctions for the
Koch Snowflake Drum ....... ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
Robert Brooks
Inverse Spectral Geometry ............................................. 115
P. Buser
Inverse Spectral Geometry on Riemann Surfaces ........................ 133
Toshikazu Sunada
Quantum Ergodicity ................................................... 175
Index ............................................................... 197
Spectral Geometry:
An Introduction and Background
Material for this Volume
Stig I. Andersson and Michel L. Lapidus
o Introduction
Inverse spectral geometry (ISG) has for the last couple of decades exhibited a
very strong dynamics. A good deal of this dynamics stems from the fact that ISG
is a melting pot for ideas and results from as diverse areas as global and local
differential geometry, algebraic topology, analysis of pseudodifferential operators
on manifolds, group theory and, last but not least, mathematical physics.
Results over the last decades have been very rich in structure and frequently
of an impressive depth and variety. In particular, if one keeps in mind that ISG
has one of its roots in the apparently «harmless» looking question by Mark Kac in
1966: «Can one hear the shape of a drum?» [Ka]. In fact, the circle of problems
commonly treated in ISG has its origin even further back in time, the above being
a very natural question to ask in studying eigenvalue problems. One can at least
trace the origin back to Lorentz and Weyl [We] in papers from the beginning of
this century.
Still, Kac's paper in some sense revived the whole area and was the starting
point for much of the work being done up till now.
Generally speaking, a great deal of information as to what may be «heard»,
i.e., may be gained from spectral information in some form, is now available. This
information forces us on the one hand to give a negative answer to Kac's original
question and, on the other hand, opens up a very rich structure, with an interesting
and essentially intrinsic interplay between geometry and analysis as one of the
characteristic features.
1 The Smooth Case
Obviously, the general spectra of nontrivial differential operators on even simple
domains are rarely known; one just needs to consider the Laplacian on a general
triangle to appreciate this statement. ISG therefore essentially deals with various
forms of asymptotic spectral information, using various spectral functions as the
analytic ingredient. Common spectral functions are the O-function, the 1]-function
and the (-function, which we shall define below. However, let us first establish a
Trends in Mathematics, © 1997 Birkhauser Verlag Basel
2 Stig I. Andersson and Michel L. Lapidus
fairly standard framework for ISG in the smooth case, so as to be able to be more
precise and keep the discussion somewhat definite.
Let X be a closed, n-dimensional compact manifold with a strictly positive
density dx (so that, e.g., the densities COO(X, fln) ~ COO(X), where fln is the
line bundle of densities of order ex) and let P be a bounded from below elliptic
pseudodifferential operator of order m. Let E -> X be a suitable vector bundle
and lei us consider the relationships between spectral analysis for
P : COO(E) -> COO(E)
(acting on COO(E), the smooth sections of E) and the geometry, topology of
E -> X. With a suitable domain, P will be symmetric and hence have selfadjoint
extensions. Let us from now on work with one of these.
The spectrum of P is discrete (since X is compact), and we denote it by
O'(P) = {Adk=o' where the eigenvalues Ak are counted according to multiplicity
and arranged in nondecreasing order:
The aforementioned spectrai functions attached to O'(P) are then defined (for t > 0)
as follows:
I0:0> -tAk, L00
(}p(t) = (p(t) = Xkt ,
k=O k=O
and (if we assume a nonpositive spectral part)
L00
17P(t) := sign(Ak)IAkl-t.
k=O
Here, {}p and (p are most relevant and it is easily seen that they are connected by
a so-called (inverse) Mellin transformation; i.e.,
r
(}p(z) = -12. z-S(p(s)r(s)ds, for c large enough.
7fl iRe(s)=c
Also, provided e-tP and p-t are of trace class, one establishes easily that in fact
(}p(t) = trace(e-tp) and (p(t) = trace(p-t).
These functions have attractive analytical properties. For example, (p (t) has a
meromorphic extension to C (the complex plane) with simple poles at Zk =
n-;/ '
k = 0, 1,2, ... (where n = dim X and m = order P) having residues W(~,~;)
r m
(where Wk (P) can be expressed in terms of the symbol of P). Further, (p has at
Spectral Geometry: An Introduction and Background 3
most polynomial growth on every half-space Re (z) ::::: c. Moreover, Op(t) depends
holomorphically on t for Re t > O.
General references for much of the material on the derivation of spectral
functions, asymptotic expansions and analytic properties of spectral functions are
[A-P-S] and [Sh], especially Chapter 2.
To study the spectral functions and their relation to the geometry and topology
of X, one could, for example, take the natural associated parabolic problem as a
starting point. That is, consider the 'heat equation':
(%t p)
{ + u(x, t) = 0
u(x,O) = Uo(x),
which is solved by means of the (heat) semi group V(t) = e-tP; namely, u(·, t) =
V(t)uoU·
Assuming that V(t) is of trace class (which is guaranteed, for instance, if P
has a positive principal symbol), it has a Schwartz kernel
K E COO(X x X x Rt,E* ®E),
locally given by
00
K(x,y; t) = L>-IAk(~k ® 'Pk)(X,y),
k=O
for a complete set of orthonormal eigensections 'Pk E COO(E). Taking the trace,
we then obtain:
20:0: >-
Op(t) = trace(V(t)) = tA
k.
k=O
Now, using, e.g., the Dunford calculus formula
(where C is a suitable curve around a(P)) as a starting point and the standard for
malism of pseudodifferential operators, one easily derives asymptotic expansions
for the spectral functions, in this case for Op.
Assume for simplicity that P is a partial differential operator and that, as
before, X has no boundary. Then we have the following asymptotic expansion:
00
O,,(t) t-n/mLQjtj/m (where n=dimX and m= order of P).
rv
1--;0+
j=o
(1.1)
4 Stig I. Andersson and Michel L. Lapidus
Here,
r
aj = Jx aj(x), with aj(-): X -+ E* ® E
and, via the calculus of pseudodifferential operators, these quantities are nicely
expressible in terms of the symbol of P. For instance,
jdt{3
oo-(X) = ~ (_1)r-1 -(x t)e-Pm(x,{;)
j ~ (r - I)! <" r,j , <" ,
r=1
where (3r,j(x,~) is positively homogenous of degree m(r - 1) - j in ~ and can
he seen to be a certain locally invariant polynomial in the symbol of P and its
covariant derivatives. Further, Pm(x,() denotes the principal symbol of P.
Analogous types of asymptotic expansions and formulas can be written down
in much more general cases:
• for P a genuine pseudo differential operator (not necessarily elliptic), in which
case the asymptotic expansion contains logarithmic terms (see, e.g., [Gru]),
• for X noncompact,
• for X having a boundary (using standard reduction to the boundary tech-
niques),
• for X being a singular manifold.
From the geometric point of view, the interesting operators are the so-called geo
metric ones; i.e., d (the differential acting on the de Rahm complex, as well as the
+
associated Laplacian dd* d*d), the Dirac operator acting on the spin complex,
and [) (acting on the Dolbeault complex); cf. [A-B-P), Sections 5 and 6 for details
on these operators.
The technique briefly described above has been applied to many different
mathematical structures, viz:
1.A. Inverse Spectral Geometry
This is the topic of this volume. An enormous wealth of information has been
accumulated as to what geometric or topological properties of X are determined
(audible) or not determined (nonaudible) from spectral data related to P. The basic
question here is that of isospectrality: Let Xl, X2 be two Riemannian manifolds
and let P be a geometric selfadjoint operator acting on some suitable domain. If
O"(PfXI) = 0"(PfX2) (that is, if Xl and X2 are isospectral, relative to P), then
does it follow necessarily that Xl and X2 are isometric? The first counterexample
that is, isospectral manifolds which are not isometric - was constructed by Milnor
[Mi] in 1964 (16-dimensional tori) and by now counterexamples are known in all
dimensions;::: 2; see the well-known monograph [B-G-M] which still makes for
very interesting reading aside from an exposition of Milnor's and related examples.
(The n = 2 case for Euclidean space - which corresponds to Kac's original
question - was recently solved in the negative by Gordon, Webb and Wolpert in
Spectral Geometry: An Introduction and Background 5
[GWW], where two (piecewise smooth) bounded planar domains were constructed
that are isospectral (both for the Dirichlet and Neumann Laplacians) but are not
isometric. These results relied, in particular, on work of Sunada [Su] as extended
by Berard in [Be2].)
[Often, this question is asked for the Laplacian (acting on the functions or
more generally, the differential forms) of Xj' j = 1,2. It follows from Weyl's
asymptotic formula for the eigenvalue distribution [We] (or, equivalently, from
the leading term of Equation (l.1) above) that necessarily, XI and X2 must have
the same dimension and the same volume.]
It is worthwhile mentioning that:
• all known examples of pairs of mutually isospectral manifolds have a common
Riemannian covering and are hence locally isometric;
• it is an open question whether there exist isospectral, nonisometric and simply
connected manifolds.
Note that 7TI (X), the fundamental group of X, is not an audible quantity.
This was established in 1980 by Marie-France Vigneras [Vi], who constructed
isospectral manifolds having non isomorphic fundamental groups.
In 1991, DeTurck et al. [dTGGW] formulated the question «How can a drum
change shape while sounding the same?» and showed that, for instance, suitably
defined volumes of Hj(X), the homology classes of X, are nonaudible.
Another very interesting approach to the audibility - nonaudibility question
has been pursued by Steve Zelditch [Z]. He has asked the following very straight
forward question: «What generic conditions should be fulfilled in order for a
Riemannian manifold (M, g) to be spectrally determined?» A partial answer was
given by Zelditch in terms of Fourier integral operator conjugacy classes of Rie
mannian manifolds.
Recently, yet another attempt to understand the isospectrality question has
been made, using microlocal analysis (cf. [AnD. The basic idea here is that since
isospectral manifolds are locally isometric, interesting differences may perhaps be
visible on the next level of refinement, i.e., microlocally (locally in the cotangent
bundle or «phase space»). Analytically, the starting point in this approach is the
observation that for every nonsingular operator F,
Hence, localizing the trace function and letting F be a suitable Fourier integral
-Ix;;!
operator, F P F -I can be microlocally reduced to very simple operators, like
Thus the spectral geometry question can be microlocalized. Work in this direction
is in progress.