Nonisometric Domains with the Same Marvizi – Melrose Invariants

Lev Buhovsky, Vadim Kaloshin

Research output: Contribution to journalArticlepeer-review


For any strictly convex planar domain Ω ⊂ R2 with a C boundary one can associate an infinite sequence of spectral invariants introduced by Marvizi–Merlose [5]. 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 Ω up to isometry. In this paper we give a counterexample, namely, we present two nonisometric domains Ω and Ω¯ with the same collection of Marvizi–Melrose invariants. Moreover, each domain has countably many periodic orbits {Sn}n≥1 (resp. {S¯n}n≥1) of period going to infinity such that Sn and S¯ n have the same period and perimeter for each n.

Original languageEnglish
Pages (from-to)54-59
Number of pages6
JournalRegular and Chaotic Dynamics
Issue number1
StatePublished - 1 Jan 2018


  • Laplace spectrum
  • Marvizi – Melrose spectral invariants
  • convex planar billiards
  • length spectrum

All Science Journal Classification (ASJC) codes

  • Mechanical Engineering
  • Applied Mathematics
  • Statistical and Nonlinear Physics
  • Mathematics (miscellaneous)
  • Mathematical Physics
  • Modelling and Simulation


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