Unique reconstruction of the potential for the interior transmission eigenvalue problem for spherically stratified media

Abstract

This work deals with the interior transmission eigenvalue problem for a spherically stratified medium supported in {x: |x| = r = b}, which can be formulated as −y″ + q(x)y = λ2y with boundary conditions y(0) = 0 = y′(1) cos(aλ) − y(1) sin(aλ)/λ, where a = b/B with B being the size of the wave speed, measured by an integral. We provide a necessary and sufficient condition for the existence issue by giving a new method that allows unique reconstruction of a potential of this Sturm–Liouville problem from the spectrum of the problem and the set of the norming constants corresponding to the real eigenvalues when 0 < a < 1; and from the spectrum together with one additional piece of information when a = 1. The method is based on Mittag-Leffler expansions which can help us to decompose an entire function of exponential type into two functions of more smaller exponential types. This decomposition provides us a well-suited situation for utilizing Levin–Lyubarski interpolation formula to reconstruct the potential for our problem.