Nonisometric Domains with the Same Marvizi – Melrose Invariants

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.