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.