Wave invariants at elliptic closed geodesics

TL;DR

This paper computes explicit wave invariants for non-degenerate elliptic closed geodesics on compact Riemannian manifolds by using quantum Birkhoff normal form techniques, linking spectral data to geometric properties.

Contribution

It introduces a new algorithm for calculating quantum Birkhoff normal form and expressing wave invariants in terms of this normal form, advancing spectral geometry methods.

Findings

Explicit formulas for wave invariants in terms of curvature and Jacobi fields.

A novel algorithm for quantum Birkhoff normal form calculation.

Connection between wave invariants and geometric data at elliptic geodesics.

Abstract

This paper concerns spectral invariants of the Laplacian on a compact Riemannian manifold (M,g) known as wave invariants. If U(t) denotes the wave group of (M,g), then the trace Tr U(t) is singular when t = 0 or when ti is the length of a closed geodesic. It has a special type of singularity expansion at each length and the coefficients are known as the wave invariants. Our main purpose is to calculate the wave invariants explicitly in terms of curvature, Jacobi fields etc. when the closed geodesic is non-degenerate elliptic. We do this by putting the Laplacian into quantum Birkhoff normal form at the closed geodesic. Such a normal form was previously introduced by V. Guillemin. We give a new algorithm for calculating it, and for expressing wave invariants in terms of normal form invariants.