Abstract:
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 ˉΩ with the same collection
of Marvizi – Melrose invariants. Moreover, each domain
has countably many periodic orbits {Sn}n⩾1 (resp.
{ˉSn}n⩾1) of period going to infinity such that
Sn and ˉSn have the same period and perimeter for each n.
VK acknowledges partial support of the NSF grant DMS-1402164 and the hospitality of the ETH Institute for Theoretical Studies and the support of Dr. Max Rössler, the Walter Haefner Foundation and the ETH Zurich Foundation.