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

Abstract

We give a rigorous version of the classical Balian-Bloch trace formula, a semiclassical expansion around a periodic reflecting ray of the (regularized) resolvent of the Dirichlet Laplacian on a bounded smooth plane domain. It is equivalent to the Poisson relation (or wave trace formula) between spectrum and closed geodesics. We view it primarily as a computational device for explicitly calculating wave trace invariants. Its effectiveness will be illustrated in subsquent articles in the series in which concrete inverse spectral results are proved.

First in a series on the inverse spectral problem for analytic plane domains. 53 pages, 1 figure. Added some references

References in corpus (3)

• The inverse resonance problem for ${\mathbb{Z}}_2$-symmetric analytic obstacles in the plane
• Inverse spectral problem for analytic plane domains II: ${\mathbb{Z}}_2$-symmetric domains
• Birkhoff Normal Forms in Semi-Classical Inverse Problems

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