Some spectrally isolated convex planar regions.
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.
Marvizi
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School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540.
Melrose
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