Transmission eigenvalues for operators with constant coefficients

TL;DR

This paper investigates the properties of transmission eigenvalues for constant coefficient differential operators, establishing conditions for their discreteness and existence, with applications to biharmonic operators and Dirac systems.

Contribution

It provides new results on the discreteness and existence of transmission eigenvalues for operators with constant coefficients, extending previous theories and applying to specific operators.

Findings

Transmission eigenvalues are discrete under certain growth conditions.

Sufficient conditions for the existence of transmission eigenvalues are derived.

Connections to scattering theory are established in the hypoelliptic case.

Abstract

In this paper we study the interior transmission problem and transmission eigenvalues for multiplicative perturbations of linear partial differential operator of order $\ge 2$ with constant real coefficients. Under suitable growth conditions on the symbol of the operator and the perturbation, we show the discreteness of the set of transmission eigenvalues and derive sufficient conditions on the existence of transmission eigenvalues. We apply these techniques to the case of the biharmonic operator and the Dirac system. In the hypoelliptic case we present a connection to scattering theory.