Wave Invariants for Non-degenerate Closed Geodesics

TL;DR

This paper extends the concept of wave invariants and quantum Birkhoff normal forms from elliptic to all non-degenerate closed geodesics, providing new proofs and broader applicability in spectral geometry.

Contribution

It introduces a quantum Birkhoff normal form for the Laplacian at non-degenerate closed geodesics, generalizing previous results from elliptic cases.

Findings

Extended wave invariants to non-elliptic geodesics

Provided a new proof of Guillemin's results

Broadened the applicability of normal form techniques

Abstract

This paper generalizes the methods and results of our article xxx.lanl.gov math.SP/0002036 from elliptic to general non-degenerate closed geodesics. The main purpose is to introduce a quantum Birkhoff normal form of the Laplacian at a general non-degenerate closed geodesic in the sense of V.Guillemin. Guillemin proved that the coefficients of the normal form at an elliptic closed geodesic could be determined from the wave invariants of this geodesic. We give a new proof, and extend its range to any non-degenerate closed geodesic.