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It is natural to consider continuous dependence of the $n$-th eigenvalue on$d$-dimensional ($d\geq2$) Sturm-Liouville problems after the results on$1$-dimensional case by Kong, Wu and Zettl [14]. In this paper, we find all theboundary conditions such that the $n$-th eigenvalue is not continuous, and givecomplete characterization of asymptotic behavior of the $n$-th eigenvalue. Thisrenders a precise description of the jump phenomena of the $n$-th eigenvaluenear such a boundary condition. Furthermore, we divide the space of boundaryconditions into $2d+1$ layers and show that the $n$-th eigenvalue iscontinuously dependent on Sturm-Liouville equations and on boundary conditionswhen restricted into each layer. In addition, we prove that the analytic andgeometric multiplicities of an eigenvalue are equal. Finally, we obtainderivative formula and positive direction of eigenvalues with respect toboundary conditions.

Singularity of the $n$-th eigenvalue of high dimensional Sturm-Liouville problems