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

Abstract

Abstract. We prove that two particular entries in the scattering matrix for the Dirichlet Laplacian on R × (−γ, γ) \ O determine an analytic strictly convex obstacle O. With an additional symmetry assumption, one entry suffices. Part of the proof is an integral identity involving an entry in the scattering matrix and a distribution related to the fundamental solution of the wave equation. This identity holds for general manifolds with infinite cylindrical ends. A consequence of this is a relationship between the singularities of the Fourier transform of an entry in the scattering matrix and the sojourn times of certain geodesics.

Inverse Problems for Obstacles in a Waveguide

“…We remark that an entry in the scattering matrix is a scalar function; see (1.4) and subsequent discussion for the definition. The results of this paper extend the inverse results of [2], both by allowing higher-dimensional waveguides and by considering either Dirichlet or Neumann boundary conditions on ∂O.…”

“…Identification of x 0 will follow from the scattering data. Theorem 1.1 extends Theorems 1.1 and 5.1 of [2], which deal with obstacles in planar cylindrical waveguides. In such a case, hypothesis (1.8) becomes the hypothesis of bilateral symmetry of O about its axis (taken in [2] to run down the middle of X = R × [γ 1 , γ 2 ]).…”

“…Theorem 1.1 extends Theorems 1.1 and 5.1 of [2], which deal with obstacles in planar cylindrical waveguides. In such a case, hypothesis (1.8) becomes the hypothesis of bilateral symmetry of O about its axis (taken in [2] to run down the middle of X = R × [γ 1 , γ 2 ]). Another way in which the current result improves on the results of [2] is that we treat the Neumann boundary condition on ∂O as well as the Dirichlet boundary condition.…”

“…Another way in which the current result improves on the results of [2] is that we treat the Neumann boundary condition on ∂O as well as the Dirichlet boundary condition. Theorem 1.2 of [2] is a result about unique determination of O ⊂ R × (γ 1 , γ 2 ) without bilateral symmetry, given knowledge of two elements of the scattering matrix, namely S 11 (λ) and S 13 (λ) in the terminology of [2]. We give some extensions of this result to other planar waveguide settings in §7, to which we refer for specific statements.…”

“…In such a case, hypothesis (1.8) becomes the hypothesis of bilateral symmetry of O about its axis (taken in [2] to run down the middle of X = R × [γ 1 , γ 2 ]). Another way in which the current result improves on the results of [2] is that we treat the Neumann boundary condition on ∂O as well as the Dirichlet boundary condition. Theorem 1.2 of [2] is a result about unique determination of O ⊂ R × (γ 1 , γ 2 ) without bilateral symmetry, given knowledge of two elements of the scattering matrix, namely S 11 (λ) and S 13 (λ) in the terminology of [2].…”