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Zelditch, Steve

Inverse spectral problem for analytic domains. II: \(\mathbb Z_2\)-symmetric domains. (English) Zbl 1196.58016

Ann. Math. (2) 170, No. 1, 205-269 (2009).
Summary: 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 \(\mathbb Z_2\) 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.
[For part I of this paper see the author, Commun. Math. Phys. 248, No. 2, 357–407 (2004; Zbl 1086.58016).]

Cited in 35 Documents

MSC:

58J53 Isospectrality
35R30 Inverse problems for PDEs
35P20 Asymptotic distributions of eigenvalues in context of PDEs

Keywords:

inverse spectral problems; stationary phase methods

Citations:

Zbl 1086.58016

Software:

OpenCourseWare
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References:

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