Inverse spectral problem for analytic (ℤ/2ℤ)n-symmetric domains in ℝn

Hamid Hezari, Steve Zelditch*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

15 Scopus citations

Abstract

We prove that bounded real analytic domains in ℝn, with the symmetries of an ellipsoid and with one axis length fixed, are determined by their Dirichlet or Neumann eigenvalues among other bounded real analytic domains with the same symmetries and axis length. Some non-degeneracy conditions are also imposed on the class of domains. It follows that bounded, convex analytic domains are determined by their spectra among other such domains. This seems to be the first positive result for the well-known Kac problem, "Can one hear the shape of a drum?", in higher dimensions.

Original languageEnglish (US)
Pages (from-to)160-191
Number of pages32
JournalGeometric and Functional Analysis
Volume20
Issue number1
DOIs
StatePublished - Jun 1 2010

Keywords

  • Kac's problem
  • Monodromy operator
  • Trace formula
  • Wave invariants

ASJC Scopus subject areas

  • Analysis
  • Geometry and Topology

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