Inverse spectral problem for analytic $(Z/2Z{)}^n$-symmetric domains in $R^n$

Abstract

In this paper we show that bounded analytic domains in ${\mathbb{R}}^n$ with mirror symmetries across all coordinate axes are spectrally determined among other such domains. Our approach builds on finding concrete formulas for the wave invariants at a bouncing ball orbit. The wave invariants are calculated from a stationary phase expansion applied to a well-constructed microlocal parametrix for the trace of the resolvent.

29 pages; some typos corrected and references added

References in corpus (3)

• Inverse spectral problem for analytic plane domains I: Balian-Bloch trace formula
• Inverse Spectral Problem for Schrödinger Operators
• Inverse spectral problem for analytic plane domains II: ${\mathbb{Z}}_2$-symmetric domains

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• Spectral invariants of convex billiard maps:a viewpoint of Mather's beta function
• Trace singularities in obstacle scattering and the Poisson relation for the relative trace