Hypergeometric expression for the resolvent of the discrete Laplacian in low dimensions
Mathematical Physics
2022-03-16 v1 Functional Analysis
math.MP
Abstract
We present an explicit formula for the resolvent of the discrete Laplacian on the square lattice, and compute its asymptotic expansions around thresholds in low dimensions. As a by-product we obtain a closed formula for the fundamental solution to the discrete Laplacian. For the proofs we express the resolvent in a general dimension in terms of the Appell--Lauricella hypergeometric function of type outside a disk encircling the spectrum. In low dimensions it reduces to a generalized hypergeometric function, for which certain transformation formulas are available for the desired expansions.
Keywords
Cite
@article{arxiv.2004.05866,
title = {Hypergeometric expression for the resolvent of the discrete Laplacian in low dimensions},
author = {Kenichi Ito and Arne Jensen},
journal= {arXiv preprint arXiv:2004.05866},
year = {2022}
}