Resonance Wave Expansions: Two Hyperbolic Examples

For scattering on the modular surface and on the hyperbolic cylinder, we show that the solutions of the wave equations can be expanded in terms of resonances, despite the presence of trapping. Expansions of this type are expected to hold in greater generality but have been understood only in non-trapping situations. 1. Introduction In this note we give two examples for which we can obtain, on compact sets, an asymptotic expansion of solutions to the wave equation with smooth, compactly supported initial data, although there is trapping. The expansions are given in terms of resonances and they generalize the standard \separation of variables" expansions in terms of eigenvalues. The examples are the modular surface where we can use detailed information about the zeta function (see Fig.1 and Theorem 1) and the hyperbolic cylinder where the resonances are particularly simple (see Fig.2(b) and Theorem 2). Resonances or scattering poles are deened as poles of the meromorphic continuation of the resolvent or the scattering matrix and they constitute a natural replacement of discrete spectral data for problems on exterior domains. That point of view was emphasized early by Lax-Phillips ((12]) { see 23] for a light-hearted overview of recent results. Although resonances are most frequently deened in the stationary framework of scattering theory they are a dynamical concept: the real part of a resonance describes the rest energy of a state and the imaginary part its rate of decay. Consequently they should be understood in terms of long time behaviour of solutions to evolution equations, and, in particular, to the wave equation. For the Schrr odinger evolution equation we refer to the recent paper by Sooer-Weinstein 17] and references given there; in that case one considers resonances which come from perturbing embedded eigenvalues.