Remarks on interior transmission eigenvalues, Weyl formula and branching billiards

TL;DR

This paper derives a Weyl formula for interior transmission eigenvalues in parameter-elliptic cases, constructs branching billiard trajectories, and estimates the second term of Weyl asymptotics based on periodic billiard trajectories.

Contribution

It introduces a Weyl formula for the interior transmission problem and provides estimates for the second asymptotic term using branching billiard trajectories.

Findings

Weyl formula established for parameter-elliptic interior transmission problem

Construction of branching billiard trajectories

Upper estimates for the second Weyl asymptotic term

Abstract

The paper contains the Weyl formula for the counting function of the interior transmission problem when the latter is parameter-elliptic. Branching billiard trajectories are constructed, and the second term of the Weyl asymptotics is estimated from above under some conditions on the set of periodic billiard trajectories.