On the Contribution of Degenerate Periodic Trajectories to the Wave-Trace

Abstract

This paper is concerned with the wave trace Z(t)of a selfadjoint elliptic pseudodifferential operator on a compact manifold. It is devoted to the contribution of degenerate periodic trajectories of the associated Hamiltonian flow to the singularities of Z(t). Given a periodic trajectory γ and a point ν on it, we obtain a trace formula which relates microlocally the trace of the unitary group of A in a neighborhood of ν to an oscillatory integral, the phase and the amplitude of which are written in terms of the Poincare map of γ. For suitable isolated but degenerate γ, this enables us to obtain complete asymptotic expansions applying already known results about asymptotics of oscillatory integrals. As an application we obtain lower bounds on the resonances close to the real axis for the Laplace-Beltrami operator in Rn, n≥ 2, for suitable compactly supported perturbations of the Euclidean metric.