Globally Coupled Inverse Spectral Reconstruction for Discrete Sturm–Liouville Equations with Multiple Interior Discontinuities

This paper investigates inverse spectral problems for a class of discrete Sturm–Liouville operators with multiple transmission interfaces. The presence of several interfaces generates a coupled spectral structure in which the reconstruction of the operator coefficients is determined by a common generalized spectral function. A generalized spectral framework adapted to the multi-interface setting is introduced, and explicit reconstruction formulas for the associated tridiagonal coefficient matrix are derived. Using the corresponding Hankel moment determinants, it is shown that the interface coefficients are spectrally linked through the same moment sequence, producing a globally coupled reconstruction mechanism. In contrast to the single-interface case, the reconstruction cannot be decomposed into independent local procedures. A constructive recovery algorithm for the operator coefficients is obtained, and the role of the transmission parameters in the spectral representation is analyzed. An example illustrating the reconstruction process for multiple interfaces is also presented.