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arXiv · 1608.03779

Branching form of the resolvent at threshold for multi-dimensional discrete Laplacians

Kenichi Ito·Arne Jensen
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Abstract

We consider the discrete Laplacian on $\mathbb Z^d$, and compute asymptotic expansions of its resolvent around thresholds embedded in continuous spectrum as well as those at end points. We prove that the resolvent has a square-root branching if $d$ is odd, and a logarithm branching if $d$ is even, and, moreover, obtain explicit expressions for these branching parts involving the Lauricella hypergeometric function. In order to analyze a non-degenerate threshold of general form we use an elementary step-by-step expansion procedure, less dependent on special functions.

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Kenichi Ito, Arne Jensen. 2016-08-12. Branching form of the resolvent at threshold for multi-dimensional discrete Laplacians. https://doi.org/10.1016/j.jfa.2019.05.018

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