Non-isometric domains with the same Marvizi-Melrose invariants

TL;DR

This paper constructs explicit examples of non-isometric convex planar domains that share identical Marvizi-Melrose spectral invariants, demonstrating these invariants do not uniquely determine the domain shape.

Contribution

It provides the first known counterexamples showing that Marvizi-Melrose invariants are not sufficient to uniquely identify convex domains up to isometry.

Findings

Two non-isometric domains share the same spectral invariants.

Each domain has infinitely many periodic orbits with matching periods and perimeters.

Spectral invariants alone do not determine domain shape uniquely.

Abstract

For any strictly convex planar domain $\Omega \subset \mathbb{R}^2$ with a $C^\infty$ boundary one can associate an infinite sequence of spectral invariants introduced by Marvizi-Merlose. These invariants can generically be determined using the spectrum of the Dirichlet problem of the Laplace operator. A natural question asks if this collection is sufficient to determine $\Omega$ up to isometry. In this paper we give a counterexample, namely, we present two non-isometric domains $\Omega$ and $\bar \Omega$ with the same collection of Marvizi-Melrose invariants. Moreover, each domain has countably many periodic orbits $\{S^n\}_{n \geqslant 1}$ (resp. $\{ \bar S^n\}_{n \geqslant 1}$) of period going to infinity such that $ S^n $ and $ \bar S^n $ have the same period and perimeter for each $ n $.