Singular discrete Sturm–Liouville problems with transmission conditions
DOI:
https://doi.org/10.31489/2026m3/19-32Keywords:
difference equations, discrete Sturm–Liouville problem, boundary conditions, Weyl theory and its generalizations, limit circle case, limit point case, transmission conditions, singular caseAbstract
This paper examines singular discrete Sturm–Liouville equations subject to transmission conditions, focusing on their fundamental spectral and geometric properties. In mathematical physics and operator theory, understanding the asymptotic behavior of solutions at singular boundaries is highly essential. While the classical spectral theory for continuous systems has been comprehensively analyzed, the structural nuances of their discrete counterparts containing interior discontinuities require a more detailed investigation. For the discrete equations under consideration, Weyl’s famous alternative classification is successfully established, dividing the underlying equations into limit-circle and limit-point cases based on the behavior of their solutions. By incorporating transmission conditions at an interior point, this study bridges a critical gap between discrete calculus and boundary value problems with transmission interactions. We construct the corresponding Hilbert space framework and thoroughly investigate the core properties of the corresponding Weyl function. Furthermore, the geometric characteristics of Weyl circles are systematically derived. Ultimately, the generalized spectral characteristics established herein offer comprehensive insights into discrete systems with transmission conditions, contributing significantly to the ongoing advancement of the spectral analysis of discontinuous difference operators.
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