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Analysis of leaky modes or wavenumber resonances for the Rayleigh system in a half space

Maarten V. de Hoop, Alexei Iantchenko

TL;DR

The paper develops a rigorous spectral-theoretic framework for Rayleigh-wave resonances in an isotropic elastic half-space with a finite slab, framing wavenumber resonances as poles of the resolvent on a four-sheet Riemann surface. It constructs Jost solutions, a boundary matrix, and the Rayleigh determinant, and introduces a reflection matrix to describe boundary scattering, all connected through symmetry and conjugation relations. By expressing the resolvent kernel and transporting the problem to a matrix Schrödinger form via the Markushevich transform, the authors establish that key spectral data assemble into entire functions of Cartwright class, with the zeros of a product function $F$ encoding the wavenumber resonances and enabling a precise counting law and asymptotics. The results yield both a detailed description of resonance distribution and a solid foundation for inverse problems aiming to recover Lamé parameters from resonance data, with direct relevance to seismology and layered elastic media.

Abstract

We present a comprehensive analysis of wavenumber resonances or leaky modes associated with the Rayleigh operator in a half space containing a heterogeneous slab, being motivated by seismology. To this end, we introduce Jost solutions on an appropriate Riemann surface, a boundary matrix and a reflection matrix in analogy to the studies of scattering resonances associated with the Schrödinger operator. We analyze their analytic properties and characterize the distribution of these wavenumber resonances. Furthermore, we show that the resonances appear as poles of the meromorphic continuation of the resolvent to the nonphysical sheets of the mentioned Riemann surface as expected.

Analysis of leaky modes or wavenumber resonances for the Rayleigh system in a half space

TL;DR

The paper develops a rigorous spectral-theoretic framework for Rayleigh-wave resonances in an isotropic elastic half-space with a finite slab, framing wavenumber resonances as poles of the resolvent on a four-sheet Riemann surface. It constructs Jost solutions, a boundary matrix, and the Rayleigh determinant, and introduces a reflection matrix to describe boundary scattering, all connected through symmetry and conjugation relations. By expressing the resolvent kernel and transporting the problem to a matrix Schrödinger form via the Markushevich transform, the authors establish that key spectral data assemble into entire functions of Cartwright class, with the zeros of a product function encoding the wavenumber resonances and enabling a precise counting law and asymptotics. The results yield both a detailed description of resonance distribution and a solid foundation for inverse problems aiming to recover Lamé parameters from resonance data, with direct relevance to seismology and layered elastic media.

Abstract

We present a comprehensive analysis of wavenumber resonances or leaky modes associated with the Rayleigh operator in a half space containing a heterogeneous slab, being motivated by seismology. To this end, we introduce Jost solutions on an appropriate Riemann surface, a boundary matrix and a reflection matrix in analogy to the studies of scattering resonances associated with the Schrödinger operator. We analyze their analytic properties and characterize the distribution of these wavenumber resonances. Furthermore, we show that the resonances appear as poles of the meromorphic continuation of the resolvent to the nonphysical sheets of the mentioned Riemann surface as expected.
Paper Structure (16 sections, 21 theorems, 207 equations)

This paper contains 16 sections, 21 theorems, 207 equations.

Table of Contents

  1. Introduction
  2. Rayleigh system
  3. Branch cuts and Riemann surface
  4. Conjugation properties of the Jost solutions and the boundary matrix
  5. Reflection matrix and its properties
  6. Reflected Jost solutions
  7. Representation of reflection matrix in terms of boundary matrix
  8. Flux normalization and the Rayleigh determinant revisited
  9. Resolvent, its analytic continuation and poles
  10. Analytic properties of the boundary matrix
  11. Analytic properties of the Jost solutions on the Riemann surface
  12. Analytic properties of the Jost function
  13. Cartwright class of entire functions
  14. Analytic properties of the relevant entire functions and proof of the main results
  15. Discussion
  16. ...and 1 more sections

Key Result

Lemma 4.1

On the Riemann surface, $\mathcal{R}$, the following holds true On the projection of the Riemann surface $\mathcal{R}$ to the cut plane $\mathcal{K}$, we have On the branch cut, $\xi\in\left[-\frac{\omega}{\sqrt{\sigma_0}},\frac{\omega}{\sqrt{\sigma_0}}\right]$, and on the branch cut, $\xi \in \mathrm{i}\mathbb{R}$,

Theorems & Definitions (37)

  • Remark 2.1
  • Definition 2.1
  • Lemma 4.1: Conjugation of Jost solutions
  • proof
  • Lemma 4.2: Conjugation of boundary matrix
  • proof
  • Remark 4.1
  • Remark 4.2
  • Remark 4.3
  • Remark 4.4
  • ...and 27 more