Non-isometric domains with the same Marvizi-Melrose invariants

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 $\{S^n{\}}_{n⩾1}$ (resp. $\{{\bar{S}}^n{\}}_{n⩾1}$) of period going to infinity such that $S^n$ and ${\bar{S}}^n$ have the same period and perimeter for each $n$.

References in corpus (1)

• Survey of the inverse spectral problem

Cited by in corpus (1)

• Chaotic properties for billiards in circular polygons