Kac is to determine a Euclidean domain R
n
up
to isometry from the spectrum Spec
B
() of its Laplacian
B
with Dirichlet,
Neumann or more general boundary conditions B. The physical motivation
is to identify physical objects from the light or sound they emit, which may
be all that is observable of remote objects such as stars or atoms.
The inverse spectral problem is just one among many kinds of inverse
problems whose goal is to determine a metric, domain or scatterer from phys-
ically relevant invariants. A comparison with other inverse problems shows
just how small a set of invariants the spectrum is. For instance, the boundary
inverse problem asks to determine the metric g on a xed bounded domain
M of a Riemannian manifold (M, g) from the spectrum of the Dirichlet
Laplacian on L
2
(), and from the Cauchy data | of its eigenfunctions,
or equivalently from its Dirichlet-to-Neumann operator
BK, KKL, LU
[BK, KKL, LU]. The
inverse scattering problem seeks to determine an obstacle from its scattering
amplitude
Ma, Ma2
[Ma, Ma2], or from its scattering length spectrum
St, St2
[St, St2], a set
of lengths parametrized by S
n1
× S
n1
. By comparison, the inverse spectral
problem has to make do with just a discrete unformatted set of eigenvalues
(or resonance poles in the open case). Research partially supported by NSF grant #DMS-0071358.
1 It is probably a consequence of this relative poverty of invariants that
most of the results in inverse spectral theory over the last two decades (es-
pecially since the appearance of Sunadas aricle
Su
[Su]) are negative results,
showing that one cannot determine metrics or domains by their spectra. The
collection of non-isometric isospectral pairs of Riemannian manifolds would
require a lengthy survey of its own and for that we refer to the lectures of C.
Gordon (cf.
Gor, Gor2
[Gor, Gor2]). By comparison, the number of positive results
showing that one can indeed recover a domain or metric is rather small.
This survey is devoted to the positive results. The emphasis is on relations
between the spectrum of the Laplacian and the dynamics of the geodesic ow
G
t
: S M S M. Most of the new material concerns wave trace invariants
and their applications to solving concrete inverse spectral problems. The
wave group is the quantization of the geodesic ow, and so wave trace meth-
ods often reduce inverse spectral problems to inverse dynamical problems.
Because of their relevance, we have attempted to describe inverse dynamical
problems and results.
1.1
Some basic inverse spectral problems
Let us introduce some basic terminology. The spectrum of a compact Rie-
mannian manifold denes a map
Spec : M R
N
+
, (g, B) Spec(
g,B
) = {
0
<
2
1

2
2
· · · }
from some class of metrics M on a manifold M to the spectrum of its Lapla-
cian,
j
=
2
j j
, i
,
j
=
ij
B
j
= 0 on M,
with boundary conditions B : C (M) C (M ) if M = . Here,
denotes the positive Laplacian
= 1
g
n
i,j=1 x
i
g
ij
g
x
j
of a Riemannian manifold (M, g), where g
ij
= g( x
i
, x
j
), [g
ij
] is the inverse
matrix to [g
ij
] and g = det[g
ij
]. We will only consider Dirichlet Bu = u|
M
2 and Neumann Bu = u|
M
. Eigenvalues are repeated according to their
multiplicities.
Two metrics or domains are called isospectral if they have the same spec-
trum. The main problem in inverse spectral theory is to describe the possible
spectra R
N
of Laplacians and, for each possible spectrum, to describe
the metrics or domains in the spectral class
Spec
1
().
(1.1)
Somewhat simpler is to describe the possible smooth curves in the isospectral
class, since it apriori eliminates irregular subsets. An isospectral deformation
of a Riemannian manifold (possibly with boundary) is one-parameter family
of metrics satisfying Spec(M, g
t
) = Spec(M, g
0
) for each t. Similarly, an
isospectral deformation of a domain with a xed background metric g
0
and
boundary conditions B is a family
t
with Spec
B
(
t
) = Spec
B
(). One
could also pose the inverse spectral problems for boundary conditions (while
holding the other data xed) as in
GM2, PT
[GM2, PT].
The inverse spectral and isospectral deformation problems are dicult
because the map Spec is highly nonlinear. The linearization of the problem
is to nd innitesimal isospectral deformations, i.e. deformations for which
the eigenvalue variations vanish to rst order. By rst order perturbation
theory, the variations of the eigenvalues under a variation of the metric are
given by j
= d
dt
j
(t)|
t=0
=
j
,
j
,
(1.2)
DOTLAMBDA
where
is the variation of the Laplacian and where
j
=
j
(0) is an or-
thonormal basis of eigenfunctions which varies smoothly in t (such a basis
exists by the Kato-Rellich theory). As will be recalled below, lengths L of
closed geodesics are spectral invariants (at least, when there is at most one
closed geodesic of each length), so innitesimal iso-length-spectral deforma-
tions are those where
L = 0, .
(1.3)
LE
The deformation of the metric is a symmetric 2-tensor g, so the linearized
problem is to determine the space of g S
2
T M (modulo tensors arising
from dieomorphisms t
(g)) for which gds = 0
j
,
j
= 0, (j).
(1.4)
INF
3 The operator
= Op( g) is the dierential operator with symbol g. The
linearized problem is still very dicult because it requires a study of the
asymptotic behavior of the expressions (
INF
1.4) as the lengths or eigenvalues
tend to innity. This is tantamount to the study of the equidistribution
theory of closed geodesics and eigenfunctions.
The basic distinctions in inverse spectral theory are the following. We
say that
a metric or domain is spectrally determined (within M) if it is the unique
element of M with its spectrum;
it is locally spectrally determined if there exists a neighborhood of the
metric or domain in M on which it is spectrally determined;
a metric or domain is spectrally rigid in M if it does not admit an
isospectral deformation within the class;
the inverse spectral problem is solvable in M if Spec|
M
is 1 1, i.e. if
any other metric or domain in M with the same spectrum is isometric
to it. If not, one has found a counterexample.
There are analogous problems for Laplacians on non-compact Riemannian
manifolds, which often have continuous spectra as well as discretely occurring
eigenvalues. In place of eigenvalues, one considers the resonances Res() of
, i.e. the poles of the analytic continuation of its resolvent
R(z) = ( + z
2
)
1
.
Depending on whether the dimension is odd or even, Res() is a discrete
subset of C or of the logarithmic plane. The inverse spectral problems above
have natural analogues for resonance poles. We refer to Zworskis expository
articles
Zw2, Zw3
[Zw2, Zw3] for background.
To illustrate the current state of knowledge, we note that even the sim-
plest special metrics are not known to be spectrally determined at the present
time (at least to the authors knowledge). It is not known:
if the standard metric g
0
on S
n
is determined by its spectrum (in di-
mensions 7), i.e. if (M, g) (or even (S
n
, g)) is isospectral to (S
n
, g
0
)
then it is isometric to it. This has been proved in dimensions 6
T
[T].
4 if ellipses in the plane are determined by their Dirichlet spectra, or even
if they are spectrally rigid, i.e. if there exist isospectral deformations
of ellipses (with Dirichlet boundary conditions).
if hyperbolic manifolds are determined by their spectra in dimensions
3. I.e. if (M
0
, g
0
) is hyperbolic and (M, g) is isospectral to it, then
is (M, g) hyperbolic? This is of course true in dimension 2. Is (M, g)
isometric to (M
0
, g
0
)? In dimension 2, this is known to be false for
some hyperbolic surfaces.
if at metrics are determined by their spectra in the sense that if (M, g
0
)
is at and (M, g) is isospectral to it, then (M, g) is at (it is known
that this is true in dimensions 6 or in all dimensions if additionally
g is assumed to lie in a suciently small neighborhood of g
0
Ku3
[Ku3]);
it is also classical that there are non-isometric at tori with the same
spectra.
These special cases are tests of the strength of the known methods. An-
other test is given by the two-dimensional inverse spectral problem. One-
dimensional problems are comparatively well understood because the eigen-
value problems are ordinary dierential equations and the underlying dy-
namics consists of just one orbit (an interval)! Two-dimensional problems
are already rich in spectral and dynamical complexities, as illustrated by the
classical dynamics of twist maps or geodesic ows on Riemannian surfaces.
In general, the inverse spectral problem grows rapidly in diculty with the
dimension, and is already quite open for analytic surfaces and domains in
two dimensions. This motivates our concentration on two-dimensional prob-
lems for much of the survey. The following simple-sounding problems are
still apparently beyond the reach of known methods:
Are convex analytic domains determined by their spectra among other
such domains? Are they spectrally rigid?
Are convex analytic surfaces of revolution determined by their spectra
among all metrics on S
2
? Are they spectrally rigid?
These problems are in some ways analogous to each other in that the
unknown is a function of one variable (the boundary or the prole curve),
and that is limit of what wave trace invariants at one orbit can hope to
5 recover. Surfaces of revolution are simpler than plane domains since the
geodesic ow is integrable, while billiards on plane domains could have any
dynamical type. On the other hand, in the domain problem the class M only
consists of