Some spectrally isolated convex planar regions

摘要

The basic question raised by M. Kac as to whether a domain in Euclidean space is determined by its Dirichlet spectrum remains open. In this note, dealing only with convex planar regions, we introduce a new countable family of (generic) spectral invariants of wave type, discuss some asymptotic properties of the distribution of closed geodesics, describe a partial converse to the Poisson relation, and thereby construct a two-parameter family of spectrally isolated regions, including the circles.