Inverse spectral problem for analytic plane domains II: $\Z_2$-symmetric domains

TL;DR

This paper advances the inverse spectral problem for symmetric analytic plane domains by explicitly calculating wave trace invariants and proving spectral determination for domains with a specific symmetry.

Contribution

It introduces explicit calculations of wave trace invariants for bouncing ball orbits and demonstrates spectral uniqueness for symmetric analytic domains with one symmetry.

Findings

Spectral data determines domains with one symmetry

Explicit wave trace invariants are computed for bouncing ball orbits

Domains with certain symmetries are uniquely identified by their spectra

Abstract

This is part of a series of papers on the inverse spectral problem for bounded analytic plane domains. Here, we use the trace formula established in the first paper (`Balian-Bloch trace formula') to explicitly calculate wave trace invariants associated to bouncing ball orbits and dihedral rays. We use these invariants to prove that simply connected bounded analytic plane domains with one symmetry (which reverses a bouncing ball orbit of fixed length L) are spectrally determined in this class.