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Vadim Kaloshin

College Park, MD, 20740, USA
Department of Mathematics, University of Maryland


Buhovsky L., Kaloshin V.
Nonisometric Domains with the Same Marvizi–Melrose Invariants
2018, vol. 23, no. 1, pp.  54-59
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~\cite{MM}. 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 nonisometric 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 $.
Keywords: convex planar billiards, length spectrum, Laplace spectrum, Marvizi–Melrose spectral invariants
Citation: Buhovsky L., Kaloshin V.,  Nonisometric Domains with the Same Marvizi–Melrose Invariants, Regular and Chaotic Dynamics, 2018, vol. 23, no. 1, pp. 54-59

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