On the simple cubic lattice Green function

Abstract

The analytical properties of the simple cubic lattice Green function G ( t ) = 1 π 3 ∫ ∫ ∫ 0 π [ t − ( cos x 1 + cos x 2 + cos x 3 ) ] − 1 d x 1 d x 2 d x 3 are investigated. In particular, it is shown that tG(t) can be written in the form t G ( t ) = [ F ( 9 , − 3 4 ; 1 4 , 3 4 , 1 , 1 2 ; 9 / t 2 ) ] 2 , where F ( a , b ;α, β, γ, β; z) denotes a Heun function. The standard analytic continuation formulae for Heun functions are then used to derive various expansions for the Green function G − ( s ) ≡ G R ( s ) + i G I ( s ) = lim ∈→ 0 + G ( s − i ∈ ) ( 0 ≤ s < ∞ ) about the points s = 0,1 and 3. From these expansions accurate numerical values of G R ( s ) and G I ( s ) are obtained in the range 0≤ s ≤3, and certain new summation formulae for Heun functions of unit argum ent are deduced. Quadratic transformation formulae for the Green function G(t) are discussed, and a connexion between G(t) and the Lamé-Wangerin differential equation is established. It is also proved that G(t) can be expressed as a product of two complete elliptic integrals of the first kind. Finally, several applications of the results are made in lattice statistics.

Heun functions and diagonals of rational functions

“…The simple cubic lattice Green function [20] 1 (2 π) 3 · 2 π 0 2 π 0 2 π 0 dθ 1 dθ 2 dθ 3 1 − x · (cos(θ 1 ) + cos(θ 2 ) + cos(θ 3 ))…”

A Survey on Spectra of infinite Graphs

“…..d0L A \ (7.4) n Jo Jo l-2z(cos a i +...+cosa r f ) l "' compare, for example, with [70,96,103]. The analytic behaviour of W u u (z) near the 'principal' singularity z = r(A) = 2dis as follows; see, for example, [70] for d = 3 and [19, Lemma 4] in general:…”

Lattice Green's Function for the Diamond Lattice

“…Green's function for the diamond lattice outside the band is given by (Guttmann and Prellberg, 1993;Ishioka and Koiwa, 1978;Joyce, 1973;Koiwa and Ishioka, 1979):…”