Balian-Bloch Wave Invariants for Nearly Degenerate Orbits

Abstract

This paper is part I of a series in which we aim to show that the singular support of the wave trace and the length spectrum of a smooth, strictly convex, and bounded planar billiard table are generally distinct objects. We derive an asymptotic trace formula for the regularized resolvent which is dual to the wave trace and contains the same information. To do this, we consider a class of periodic orbits which have nearly degenerate Poincar\'e maps and study their leading order behavior as the deformation parameter goes to zero, generating large coefficients in the wave trace. We also keep careful track of the Maslov indices, which will allow us to match contributions of opposite signs in our subsequent paper. Each cancellation of coefficients in the resolvent trace corresponds to making the wave trace one degree smoother. The resolvent based approach is due to Balian and Bloch and was significantly expanded upon by Zelditch in a foundational series of papers [Zel09], [Zel04a], [Zel04c] and [Zel00].