Inverse spectral problem for analytic domains I: Balian-Bloch trace formula

Steve Zelditch*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

12 Scopus citations

Abstract

This is the first in a series of papers [Z3, Z4] on inverse spectral/resonance problems for analytic plane domains ω. In this paper, we present a rigorous version of the Balian-Bloch trace formula [BEI, BB2], It is an asymptotic formula for the trace Tr l ω Rρ (k+i τ log k) of the regularized resolvent of the Dirichlet or Neumann Laplacian of ω as k → ∞ with τ > 0. When the support of p̂ contains the length LY of precisely one periodic reflecting ray y, then the asymptotic expansion of T r 1 ω Rρ (k + i τ log k) is essentially the same as the wave trace expansion at y. The raison d'ètre for this approach is that it leads to relatively simple explicit formulae for wave invariants. For example, we give the first formulae for wave invariants of bouncing ball orbits of plane domains (the details will appear in [Z3]). Although we only present details in dimension 2, the methods and results extend with few modifications to all dimensions.

Original languageEnglish (US)
Pages (from-to)357-407
Number of pages51
JournalCommunications in Mathematical Physics
Volume248
Issue number2
DOIs
StatePublished - Jul 2004

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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