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

Abstract

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 ‫ޚ‬ 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.

Sojourn times, manifolds with infinite cylindrical ends, and an inverse problem for planar waveguides

“…Some results similar to this, though for singularities of the wave trace at periodic orbits, can be found in [18,Section 5].…”

“…Thus, when ξ 2 < (1+h 2 )η 2 , there are solutions to (18) and (19) with initial condition a j (0, y, ξ, η) = 0, and we can find the Taylor expansions for a j at x = 0. Note that using a cut-off function to excise a conic neighborhood of η = 0 produces a smooth error, sinceΦ is rapidly decreasing there.…”

“…In this way, they are related to results of Zelditch [16,17,18] which use a detailed study of the singularities of the wave trace associated to a single isolated periodic geodesic to recover certain planar analytic domains from the spectra, or resonances, of the Dirichlet or Neumann Laplacian. There is also a relationship to results of Majda [9] for recovering a convex obstacle from backscattering data in all directions.…”

C∞spectral rigidity of the ellipse

“…If it were, then Siburg's results would imply that the marked length spectrum is preserved [Siburg 1999;2004]. In [Zelditch 2009; it is shown that analytic domains with one symmetry are spectrally determined if the length of the minimal bouncing ball orbit and one iterate is a -isospectral invariant. The prior results on -isospectral deformations that we are aware of are contained in the articles [Guillemin and Melrose 1979a;Popov and Topalov 2003; and concern deformations of boundary conditions.…”

Inverse Problems for Obstacles in a Waveguide

“…We give some extensions of this result to other planar waveguide settings in §7, to which we refer for specific statements. Theorem 1.1 bears some resemblance to results of Zelditch [9,10,11] on recovering planar domains with one symmetry from the spectrum of the Laplacian. Zelditch's proof uses the study of singularities of the fundamental solution of the wave equation which propagate along a single isolated periodic broken geodesic.…”