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 $1-n(r)$.