ABSTRACTS OF PAPERS TO APPEAR IN FUTURE ISSUES

Abstract

The analytical properties of the lattice Green functionG(α, w)=1π3∫π0∫π0∫π0dθ1 dθ2 dθ3w−cosθ1−cosθ2−αcosθ3are investigated, where w=u+iv is a complex variable in the (u, v) plane and α is a real parameter in the interval (0, ∞). In particular, it is shown that the function yG(α, z)≡wG(α, w), where z=1/w2, is a solution of a fourth-order linear differential equation of the type∑j=04fj(α, z)D4−jy=0,where fj(α, z) is a polynomial in the variables α and z and D≡d/dz. It is then proved that the solutions of this differential equation can all be expressed in terms of a product of two functions H1(α, z) and H2(α, z) which satisfy second-order linear differential equations of the normal type[D2+U+(α, z)]y=0,[D2+U−(α, z)]y=0,respectively, where U±(α, z) are complicated algebraic functions of α and z. Next Schwarzian transformation theory is used to reduce both these second-order differential equations to the standard Gauss hypergeometric differential equation. From this result it is deduced thatwG(α, w)=21−(2−α)2z+1−(2+α)2z[2πK(k+)][2πK(k−)],wherek2±≡k2±(α, z)=12−12[1−(2−α)2z+1−(2+α)2z]−3[1+(2−αz1−(2+α)z+1−(2−α)z1+(2+α)z]{±16z+1−α2z[1+(2−α)z1+(2+α)z+1−(2−α)z1−(2+α)z]2}and K(k) denotes the complete elliptic integral of the first kind with a modulus k. This basic formula is valid for all values of w=u+iv which lie in the (u, v) plane, provided that a cut is made along the real axis from w=−2−α to w=2+α. In the remainder of the paper exact series expansions for G(α, w) are derived which are valid in a sufficiently small neighbourhood of the branch-point singularities at w=2+α, w=α, and w=|2−α|. In all cases it is shown that the real and imaginary parts of the coefficients in the analytic part of these expansions can be expressed in terms of complete elliptic integrals of the first and second kinds, while the coefficients in the singular part of the expansions can be expressed in terms of rational functions of α. The behaviour of G(α, w) in the immediate neighbourhood of w=0 is also investigated in a similar manner. Finally, several applications of the results are made in lattice statistics.