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