Functional model for generalised resolvents and its application to time-dispersive media

TL;DR

This paper develops a functional model for generalized resolvents of differential operators with contrasting coefficients, enabling spectral analysis and scattering operator construction in wave propagation through time-dispersive media.

Contribution

It introduces a novel functional model for generalized resolvents, facilitating spectral and scattering analysis in media with highly contrasting properties.

Findings

Spectral representation of generalized resolvents

Construction of scattering operators for wave propagation

Application to time-dispersive media

Abstract

Motivated by recent results concerning the asymptotic behaviour of differential operators with highly contrasting coefficients, which have involved effective descriptions involving generalised resolvents, we construct the functional model for a typical example of the latter. This provides a spectral representation for the generalised resolvent, which can be utilised for further analysis, in particular the construction of the scattering operator in related wave propagation setups.