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

TL;DR

This paper analyzes the asymptotic behavior of the resolvent of the discrete Laplacian on multi-dimensional integer lattices near spectral thresholds, revealing dimension-dependent branching behaviors and providing explicit formulas.

Contribution

It introduces explicit asymptotic expansions of the resolvent at thresholds, demonstrating square-root or logarithmic branching depending on dimension, with less reliance on special functions.

Findings

Resolvent exhibits square-root branching in odd dimensions.

Resolvent exhibits logarithmic branching in even dimensions.

Explicit formulas involve Lauricella hypergeometric functions.

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.