Exterior Steklov eigenvalues and modified exterior Steklov eigenvalues in inverse scattering

Abstract

The inverse scattering problem for anisotropic media with data measured inside a cavity is considered. We aim to introduce the exterior Steklov eigenvalues (ESEs) and the modified exterior Steklov eigenvalues (MESEs) to detect changes in the material properties of the medium from a knowledge of a modified near field operator each. First, a connection between a modified near field operator and the exterior Steklov eigenvalue problem is established. Then the ESEs are proved to be discrete and lie in the lower half-plane. Since the existence of the ESEs is unknown due to non-self-adjointness, we propose the modified exterior Steklov eigenvalue problem for which infinitely many eigenvalues exist. Based on the factorizations of both the modified near field operator and the auxiliary near field operator, a nonsymmetric version of generalized linear sampling method (GLSM) is employed to compute the ESEs. This method also allows for the computation of the MESEs by adjusting the definitions of some associated operators. Numerical examples are presented to demonstrate the viability of this method and show the potential use of these eigenvalues as new classes of target signatures in nondestructive testing.