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Inverse problem for the Rayleigh system with spectral data

Maarten V. de Hoop, Alexei Iantchenko

Abstract

We analyze an inverse problem associated with the time-harmonic Rayleigh system on a flat elastic half-space concerning the recovery of Lamé parameters in a slab beneath a traction-free surface. We employ the Markushevich substitution, while the data are captured in a Jost function, and point out parallels with a corresponding problem for the Schrödinger equation. The Jost function can be identified with spectral data. We derive a Gel'fand-Levitan type equation and obtain uniqueness with two distinct frequencies.

Inverse problem for the Rayleigh system with spectral data

Abstract

We analyze an inverse problem associated with the time-harmonic Rayleigh system on a flat elastic half-space concerning the recovery of Lamé parameters in a slab beneath a traction-free surface. We employ the Markushevich substitution, while the data are captured in a Jost function, and point out parallels with a corresponding problem for the Schrödinger equation. The Jost function can be identified with spectral data. We derive a Gel'fand-Levitan type equation and obtain uniqueness with two distinct frequencies.
Paper Structure (23 sections, 19 theorems, 287 equations)

This paper contains 23 sections, 19 theorems, 287 equations.

Table of Contents

  1. Introduction
  2. Rayleigh system
  3. Markushevich transform to two adjoint matrix Sturm-Liouville problems
  4. Jost and Weyl solutions, Jost function and Weyl matrix
  5. Spectral data
  6. Cauchy integral
  7. Even extension
  8. Data for the original Rayleigh system
  9. Weyl matrix
  10. Jost solution
  11. Jost function
  12. Unique recovery of a potential of Lamé type
  13. Asymptotic expansions
  14. Jost solutions
  15. Weyl matrix
  16. ...and 8 more sections

Key Result

Lemma 4.1

Let ${\bf F}$, ${\bf F}_0$, ${\bf F}^{\rm a}$ and ${\bf F}^{\rm a}_0$ be given by (eq:Jostsoldef) and (eq:Jostsoladef), respectively. Then

Theorems & Definitions (38)

  • Remark 3.1
  • Definition 3.1
  • Remark 4.1
  • Remark 4.2
  • Lemma 4.1
  • Remark 4.3
  • Lemma 5.1
  • proof
  • Remark 5.1
  • Lemma 5.2
  • ...and 28 more