# Asymptotic expansions for the transmission eigenvalues of periodic scatterers of bounded support

## TL;DR

This paper derives first-order asymptotic corrections to transmission eigenvalues for bounded periodic scatterers, using two-scale homogenization and boundary correctors, advancing understanding of wave scattering in periodic media.

## Contribution

It introduces a novel asymptotic expansion method for transmission eigenvalues in periodic media, incorporating boundary correctors for improved accuracy.

## Key findings

- Derived first-order correction formulas for transmission eigenvalues.
- Established convergence estimates with boundary correctors.
- Applied two-scale asymptotics to biharmonic operators with periodic coefficients.

## Abstract

We consider the transmission eigenvalues for a bounded scatterer with a periodically varying index of refraction, and derive the first order corrections to the limiting transmission eigenvalues. We assume the scatterer contrast to be of one sign, in which case the transmission eigenvalue problem can be written in terms of operators corresponding to a fourth order PDE with periodic coefficients. We perform two-scale asymptotics for this biharmonic type homogenization problem and show convergence estimates which require a boundary corrector function, and this boundary corrector function appears in the formula for the transmission eigenvalues correction.