The Neumann isospectral problem for trapezoids

TL;DR

This paper proves that trapezoids with the same Neumann spectral data are congruent, using heat and wave trace invariants, highlighting differences from the Dirichlet case.

Contribution

It introduces new wave trace invariants for Neumann spectra and demonstrates spectral rigidity for trapezoids, a novel result in spectral geometry.

Findings

Neumann spectra determine trapezoid congruence

New wave trace invariants for Neumann boundary conditions

Wave trace is more singular for Neumann case

Abstract

We show that trapezoids with identical Neumann spectra are congruent up to rigid motions of the plane. The proof is based on heat trace invariants and some new wave trace invariants associated to certain diffractive billiard trajectories. The reason we can only treat the Neumann case is that the wave trace is "more singular" for the Neumann case compared to the Dirichlet case. This is a new observation which is interesting on its own.