Neumann series of Bessel functions in direct and inverse spherically symmetric transmission eigenvalue problems

The transmission eigenvalue problem (TEP) plays a central role in inverse scattering theory. Despite substantial theoretical progress, the numerical solution of direct and inverse TEP in spherically symmetric domains with variable refractive index- covering real and complex eigenvalues-remains challenging. This study introduces a novel Neumann Series of Bessel Functions (NSBF) methodology to address this challenge. After reformulating the TEP as a Sturm-Liouville equation via a Liouville transformation, we expand its characteristic function in an NSBF whose coefficients are computed by simple recursive integration. In the direct problem, eigenvalues-real or complex-are found by root-finding on a truncated NSBF partial sum, yielding high accuracy with a few coefficients, as demonstrated with various examples. For the inverse problem, we develop a two-step approach: first, recovering the transformed interval length δ from spectral data via a new NSBF-based algorithm, and second, reconstructing the refractive index n ( r ) by solving a linear system for the first NSBF coefficients. A spectrum completion technique is also implemented to complete the spectrum and solve the corresponding inverse problem when eigenvalue data is limited. Numerical examples confirm the method’s robustness and accuracy across a wide range of refractive indices, with no a priori assumptions on δ or the sign of the contrast 1 − n ( r ) .