Inverse spectral problem for analytic domains, II: ℤ2-symmetric domains

Steve Zelditch*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

50 Scopus citations


This paper develops and implements a new algorithm for calculating wave trace invariants of a bounded plane domain around a periodic billiard orbit. The algorithm is based on a new expression for the localized wave trace as a special multiple oscillatory integral over the boundary, and on a Feynman diagrammatic analysis of the stationary phase expansion of the oscillatory integral. The algorithm is particularly effective for Euclidean plane domains possessing a Z{double-struck}2 symmetry which reverses the orientation of a bouncing ball orbit. It is also very effective for domains with dihedral symmetries. For simply connected analytic Euclidean plane domains in either symmetry class, we prove that the domain is determined within the class by either its Dirichlet or Neumann spectrum. This improves and generalizes the best prior inverse result that simply connected analytic plane domains with two symmetries are spectrally determined within that class.

Original languageEnglish (US)
Pages (from-to)205-269
Number of pages65
JournalAnnals of Mathematics
Issue number1
StatePublished - 2009

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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