Open AccessSome spectrally isolated convex planar regions
Author(s) -
Shahla Marvizi,
Richard Melrose
Publication year - 1982
Publication title -
proceedings of the national academy of sciences of the united states of america
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 5.011
H-Index - 771
eISSN - 1091-6490
pISSN - 0027-8424
DOI - 10.1073/pnas.79.22.7066
Subject(s) - countable set , mathematics , geodesic , dirichlet distribution , converse , pure mathematics , regular polygon , planar , euclidean space , poisson distribution , space (punctuation) , type (biology) , euclidean geometry , combinatorics , mathematical analysis , geometry , computer science , ecology , statistics , computer graphics (images) , biology , boundary value problem , operating system
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.