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

TL;DR

This paper analyzes leaky modes and wavenumber resonances in the Rayleigh system within a half space, using complex analysis tools to characterize their distribution and properties, with applications in seismology.

Contribution

It introduces a novel framework using Jost solutions, boundary matrices, and reflection matrices on a Riemann surface to analyze Rayleigh resonances.

Findings

Resonances are characterized as poles of the meromorphic continuation of the resolvent.

Resonances are shown to appear on nonphysical sheets of the Riemann surface.

The analysis provides a detailed distribution of wavenumber resonances in the system.

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 Schr\"{o}dinger 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.