论文标题
离散拉普拉斯在低维度的分解的超几何表达
Hypergeometric expression for the resolvent of the discrete Laplacian in low dimensions
论文作者
论文摘要
我们提出了一个明确的公式，用于在平方晶格上的离散拉普拉斯式的分解，并在低维度以阈值围绕阈值计算其渐近扩展。作为副产品，我们获得了用于离散拉普拉斯的基本解决方案的封闭公式。对于证据，我们以appell-lauricella的超几何函数在圆盘外的圆盘外的appell-lauricella超几何函数方面表达了分解。在低维度中，它减少到广义的超几何函数，为此，某些转换公式可用于所需的扩展。
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 $C$ 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.
