Hypergeometric Expression for the Resolvent of the Discrete Laplacian in Low Dimensions
Hypergeometric Expression for the Resolvent of the Discrete Laplacian in Low Dimensions
复制标题
低维离散拉普拉斯求解的超几何表达式
DOI：
10.1007/s00020-021-02648-2
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发表时间：
2021
影响因子：
0.8
通讯作者：
Arne Jensen
中科院分区：
文献类型：
--
作者：
Kenichi Ito;Arne Jensen
我们给出了正方形格点上离散拉普拉斯算子的预解式的一个显式公式，并计算了其在低维阈值附近的渐近展开式。作为副产品，我们得到一个封闭的公式的基本解决方案的离散拉普拉斯。对于证明，我们表示在一般的维度上的Appell-Lauricella超几何函数的C型外的磁盘包围的频谱的预解式。在低维情况下，它简化为一个广义超几何函数，对于它，某些变换公式可用于所需的展开。
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 typeCoutside 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.