A boundary integral equation method for the transmission eigenvalue problem

Abstract

We propose a new integral equation formulation to characterize and compute transmission eigenvalues for constant refractive index that play an important role in inverse scattering problems for penetrable media. As opposed to the recently developed approach by Cossonnière and Haddar [1,2] which relies on a two by two system of boundary integral equations our analysis is based on only one integral equation in terms of Dirichlet-to-Neumann or Robin-to-Dirichlet operators which results in a noticeable reduction of computational costs. We establish Fredholm properties of the integral operators and their analytic dependence on the wave number. Further we employ the numerical algorithm for analytic non-linear eigenvalue problems that was recently proposed by Beyn [3] for the numerical computation of transmission eigenvalues via this new integral equation.

Keywords:

• Boundary integral equations
• transmission eigenvalues
• inverse scattering
• inhomogeneous media

AMS Subject Classifications:

• 35R30
• 35J25
• 35P25
• 35P05

Acknowledgements

This research was initiated while R.K. was visiting F.C. at the University of Delaware. The hospitality and the support are gratefully acknowledged.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research of F.C. was supported in parts by the AFOSR [grant number FA9550-13-1-0199]; NSF [grant number DMS1602802].