Nonisometric Domains with the Same Marvizi–Melrose Invariants

    2018, Volume 23, Number 1, pp.  54-59

    Author(s): Buhovsky L., Kaloshin V.

    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, Volume 23, Number 1, pp. 54-59

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