Preface

This monograph is devoted to the analysis of some inverse problems concerning the spectrum of the Laplace operator in a bounded domain , and of the scattering length spectrum (SLS) (the set of sojourn times of reflecting rays) of the scattering kernel associated with scattering in the exterior of a bounded obstacle . In both cases our aim is to obtain some geometric information about (resp. K from spectral (resp. scattering) data. We treat both inverse problems by using similar techniques based on properties of the generalized geodesic flow in and on microlocal analysis of the corresponding mixed problems.

Let , be a closed bounded domain with C8 smooth boundary , and let A be the self-adjoint operator in related to the Laplacian

in with Dirichlet boundary condition on . The spectrum of A is given by a sequence

of eigenvalues for which the problem

has a non-trivial solution . The counting function

where every eigenvalue is counted with its multiplicity, admits a polynomial bound

Moreover, it is known (see [Se] [H4] [SaV]) that N(?) has a Weyl type asymptotic

as ? 8. Thus, from the spectrum (0.1) we can recover the volume of . In 1911, Weyl [W] conjectured that for every bounded domain in with smooth boundary we have

as ? 8. Ivrii [Ivl] proved that if the points for which there exists a periodic billiard trajectory in issued from x in direction v form a subset of Lebesgue measure zero in the space , then the asymptotic (0.4) holds. Therefore, for such domain becomes another spectral invariant. It is not known so far if the assumption in Ivrii's result is alwayssatisfied.

To obtain more information from the knowledge of the spectrum , it is convenient to examine some distributions determined by the sequence (0.1). The distribution

has the asymptotic

and the constants cj are spectral invariants. Moreover, one can recover and from c0 and c1.

In his classical work Kac [Kac] posed the problem of recovering the shape of a strictly convex domain from the spectrum (0.1). This article has had a big influence on the investigations of various inverse spectral problems for manifolds with and without boundary as well as on the analysis of the so-called isospectral manifolds, that is manifolds for which the spectra of the corresponding Laplace-Beltrami operators coincide.

To determine a strictly convex planar domain , modulo Euclidean transformations, it suffices to know the curvature of at each point . In general, the spectral data , given by (0.5), is not sufficient to determine the function . Let us mention that the distribution t(t) is singular only at t = 0. A distribution related to having a larger singular set is

This distribution is singular at 0 and

(see [Me3] [Iv2]). The constants dj provide other spectral invariants, and the first two determine again and .

It turns out that the set of singularities of s(t) is related to the so-called length spectrum of . By definition, is the set of periods (lengths) of all periodic generalized geodesics in . Let us mention that the generalized geodesics are the projections in of the generalized bicharacteristics of the wave operator in defined by Melrose and Sjöstrand ([MS1] [MS2]). We refer to Chapter 1 for the precise definitions. The so-called Poisson relation for manifolds with boundary has the form

For strictly convex (concave) domains this relation has been established by Anderson and Melrose [AM]. Its proof for general domains is based on the results in [MS2] on the propagation of C8 singularities. A relation similar to (0.6) was first established for Riemannian manifolds without boundary. This was achieved independently by Chazarain [Ch2] and Duistermaat and Guillemin [DG]. Moreover, under certain assumptions on the periodic geodesics with period T, the leading singularity at T was examined in [DG].

It is natural to investigate the inverse inclusion in (0.7), however in the general case, very little is known so far. For certain strictly convex planar domains Marvizi and Melrose [MM] found a sequence of closed billiard trajectories in whose lengths belong to sing supp s(t). It was expected ([Cl] [GM3]) that for generic strictly convex domains in the inclusion (0.7) could become an equality. Such a result was established in [PS2] (see also [PSl]) for all generic domains (not necessarily convex). Its analogue in the case n > 2 is proved only for strictly convex domains [S3]. The results, just mentioned, form one of the main topics in this book.

If the equality

holds for some domain , then the lengths of the periodic geodesics in can be considered as spectral invariants. From them one can determine various spectral invariants. The reader may consult [MM] [Cl] [Pol] [Po2] [Po3] [PoT] [HeZ] and [Z] for more information and further results in this direction.

Let be the set of all periodic geodesics in . For we denote by c01-math10}}}}} the period (length) of ?. There are three types of elements of : periodic reflecting rays (i.e. closed billiard trajectories in ), closed geodesics on and periodic geodesics of mixed type, containing both linear segments in and geodesic segments on . Amongst the periodic reflecting rays we will distinguish those without segments tangent to the boundary ; such rays will be called ordinary. Similarly to the case of closed geodesics on , for each ordinary periodic reflecting ray ? one can naturally define a Poincaré map