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arXiv · 1801.00952

Non-isometric domains with the same Marvizi-Melrose invariants

Lev Buhovsky·Vadim Kaloshin
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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 $.

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Lev Buhovsky, Vadim Kaloshin. 2018-01-03. Non-isometric domains with the same Marvizi-Melrose invariants. https://doi.org/10.1134/s1560354718010057

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