Uniqueness of a spherically symmetric refractive index from modified transmission eigenvalues

Abstract

Abstract In the present work we study the modified transmission eigenvalue problem for the case of the spherically symmetric refractive index. This problem occurs as a modification of the classic transmission eigenvalue problem for a fixed wavenumber, by introducing a new spectral parameter. We prove the existence of infinite and discrete eigenvalues using asymptotic expressions and monotonicity properties of the characteristic functions. Next, we define the inverse spectral problem of determining the refractive index from the knowledge of modified transmission eigenvalues. We show that one eigenvalue corresponding to each characteristic function is sufficient to uniquely determine the fraction of the boundary Cauchy data. The knowledge of these data, in combination with a Dirichlet-to-Neumann mapping property allows us to establish uniqueness for the inverse problem.