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Hypergeometric expression for the resolvent of the discrete Laplacian in low dimensions. (English) Zbl 07367380

The authors have obtained some closed formulae for lattice Green functions of the form \[ G(z,n)=(2\pi)^{-d}\int_{\mathbb{T}^d}\dfrac{e^{in\theta}}{2d-2\cos(\theta_1)-\ldots-2\cos(\theta_d)-z}d\theta.\]
Such investigation was mainly restricted to dimensions \(d=1,2\). In introduction they started to depict that \(2dG(0,n)\) (\(z=0\)) shall be represented as the expectation value \(\displaystyle \mathbb{E}[n]=\sum_{k=0}^\infty P(X_k=n)\) that counts the number of times that a walker visits \(n\in \mathbb{Z}^d\). To rid the fact that \(\mathbb{E}[n]\) is divergent for dimensions \(d=1,2\) (see also Appendix B), they propose a renormalization technique to approximate \(\mathbb{E}[n]\) by \(\displaystyle \mathbb{E}[\epsilon,n]=\frac{2d}{1-\epsilon}G\left(\frac{-2d\epsilon}{1-\epsilon},n\right)\), for values of \(\epsilon\in (0,1]\).
In this way they succeed in representing \(G(z,n)\) as a convergent series (see e.g. Theorem 2.2. and Theorem 2.3.). Such analysis goes far beyond the asymptotic analysis, in the limit \(z\rightarrow 0\), considered by so many authors in the past.
As a whole, this paper is complementary to authors’s previous paper [J. Funct. Anal. 277, No. 4, 965–993 (2019; Zbl 07066831)] on which the authors have shown that may \(G(z,n)\) admits, for each threshold \(4q\), \(q=0,\ldots,n\) the splitting formula \[ G(z,n)=\mathcal{E}_q(z,n)+f_q(z)\mathcal{F}_q(z,n), \] whereby \(\mathcal{F}_q(z,n)\) – the singular part of \(G(z,n)\) – was represented in terms of the so-called Appell-Lauricella hypergeometric function of type \(B\) ,\(F_B^{(d)}\). Further comparisons between both approaches may be found in Appendix A.

MSC:

47A10 Spectrum, resolvent
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
39A12 Discrete version of topics in analysis
30G35 Functions of hypercomplex variables and generalized variables

Citations:

Zbl 07066831

Software:

DLMF
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References:

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