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In this paper, we characterize singularity of the $n$-th eigenvalue ofself-adjoint discrete Sturm-Liouville problems in any dimension. For a fixedSturm-Liouville equation, we completely characterize singularity of the $n$-theigenvalue. For a fixed boundary condition, unlike in the continuous case, the$n$-th eigenvalue exhibits jump phenomena and we describe the singularity undera non-degenerate assumption. Compared with the continuous case in [8, 12], thesingular set here is involved heavily with coefficients of the Sturm-Liouvilleequations. This, along with arbitrariness of the dimension, causes difficultywhen dividing areas in layers of the considered space such that the $n$-theigenvalue has the same singularity in any given area. We study the singularityby partitioning and analyzing the local coordinate systems, and provide aHermitian matrix which can determine the areas' division. To prove theasymptotic behavior of the $n$-th eigenvalue, we generalize the methoddeveloped in [21] to any dimension. Finally, by transforming theSturm-Liouville problem of Atkinson type in any dimension to a discrete one, wecan not only determine the number of eigenvalues, but also apply our approachabove to obtain the complete characterization of singularity of the $n$-theigenvalue for the Atkinson type.

Discrete Sturm-Liouville problems: singularity of the $n$-th eigenvalue with application to Atkinson type