Whispering gallery modes for a transmission problem

TL;DR

This paper constructs special eigenfunctions called whispering gallery modes for a Laplace operator with discontinuous coefficients, demonstrating exponential concentration near an interface and confirming the optimality of a previous unique continuation result.

Contribution

It introduces a new family of eigenfunctions with exponential concentration near interfaces, advancing understanding of wave behavior in singular media.

Findings

Eigenfunctions exhibit exponential concentration near the interface.

The constructed modes confirm the optimality of previous unique continuation results.

Provides a new method based on Agmon estimates for analyzing wave propagation in singular media.

Abstract

We construct a specific family of eigenfunctions for a Laplace operator with coefficients having a jump across an interface. These eigenfunctions have an exponential concentration arbitrarily close to the interface, and therefore could be considered as whispering gallery modes. The proof is based on an appropriate Agmon estimate. We deduce as a corollary that the quantitative unique continuation result for waves propagating in singular media proved by the author in [Fil22] is optimal.