O ct 2 00 4 INVERSE SPECTRAL PROBLEM FOR ANALYTIC DOMAINS II : Z 2 - SYMMETRIC DOMAINS STEVE

This paper develops and implements a new algorithm for calculating wave trace invariants of a bounded plane domain around a periodic billiard orbit. The algorithm is based on a new expression for the localized wave trace as a special multiple oscillatory integral over the boundary, and on a Feynman diagrammatic analysis of the stationary phase expansion of the oscillatory integral. The algorithm is particularly effective for Euclidean plane domains possessing a Z 2 symmetry which reverses the orientation of a bouncing ball orbit. It is also very effective for domains with dihedral symmetries. For simply connected analytic Euclidean plane domains in either symmetry class, we prove that the domain is determined within the class by either its Dirichlet or Neumann spectrum. This improves and generalizes the previous best inverse result of [Z1, Z2, ISZ] that simply connected analytic plane domains with two symmetries are spectrally determined within that class. This paper is part of a series (cf. [Z5, Z4]) devoted to the inverse spectral problem for simply connected analytic Euclidean plane domains Ω. The motivating problem is whether generic analytic Euclidean drumheads are determined by their spectra. All known counterexamples to the question, 'can you hear the shape of a drum?', are plane domains with corners [GWW1], so it is possible, according to current knowledge, that analytic drumheads are spectrally determined. Our main results give the strongest evidence to date for this conjecture by proving it for two classes of analytic drumheads: (i) those with an up/down symmetry, and (ii) those with a dihedral symmetry. This improves and generalize the best prior results that simply connected analytic domains with the symmetries of an ellipse and a bouncing ball orbit of prescribed length L are spectrally determined within this class [Z1, Z2, ISZ]. The proofs of the inverse results involve three new ingredients. The first is a simple and precise expression (cf. Theorem 3.1) for the localized trace of the wave group (or dually the resolvent), up to a given order of singularity, as a finite sum of special oscillatory integrals over the boundary ∂Ω of the domain with transparent dependence on the boundary defining function. Theorem 3.1 is a general result combining the Balian-Bloch approach to the wave trace expansion of [Z5] with a reduction to boundary integral operators explained in [Z4]. Presumably it could be obtained by other methods, such as the monodromy operator method of Iantchenko, Sjöstrand and …