Weyl type bound on positive Interior Transmission Eigenvalues

TL;DR

This paper establishes a Weyl-type lower bound on the number of positive interior transmission eigenvalues, confirming their infinite existence and extending results to media with obstacles, using novel estimates for the D-t-N operator.

Contribution

It provides the first Weyl-type lower bound for positive interior transmission eigenvalues, including cases with obstacles, advancing spectral theory in inverse problems.

Findings

Proves an infinite set of positive interior transmission eigenvalues exists.

Derives a Weyl-type lower bound for the counting function of these eigenvalues.

Develops estimates for the D-t-N operator for positive λ.

Abstract

This paper contains a lower bound of the Weyl type on the counting function of the positive eigenvalues of the interior transmission eigenvalue problem which justifies the existence of an infinite set of positive interior transmission eigenvalues. We consider the classical transmission problem as well as the case where the inhomogeneous medium contains an obstacle. One of the essential components of the proof is an estimate for the D-t-N operator for the Helmholtz equation for positive $\lambda$ that replaces the standard parameter-elliptic estimate valid outside of the positive semi-axis.