M L Glasser

Elliptic integral evaluations of Bessel moments and applications

We record what is known about the closed forms for various Bessel function moments arising in quantum field theory, condensed matter theory and other parts of mathematical physics. More generally, we develop formulae for integrals of products of six or fewer Bessel functions. In consequence, we are able to discover and prove closed forms for c(n,k) := Int_0 inf tk K_0 n(t) dt, with integers n = 1, 2, 3, 4 and k greater than or equal to 0, obtaining new results for the even moments c3,2k and c4,2k . We also derive new closed forms for the odd moments s(n,2k+1) := Int_0 inf t(2k+1) I_0(t) K_0n(t) dt,with n = 3, 4 and for t(n,2k+1) := Int_0 inf t(2k+1) I_02(t) K_0(n-2) dt, with n = 5, relating the latter to Green functions on hexagonal, diamond and cubic lattices. We conjecture the values of s(5,2k+1), make substantial progress on the evaluation of c(5,2k+1), s(6,2k+1) and t(6,2k+1) and report more limited progress regarding c(5,2k), c(6,2k+1) and c(6,2k). In the process, we obtain 8 conjectural evaluations, each of which has been checked to 1200 decimal places. One of these lies deep in 4-dimensional quantum field theory and two are probably provable by delicate combinatorics. There remains a hard core of five conjectures whose proofs would be most instructive, to mathematicians and physicists alike.

Exact values for the cubic lattice Green functions

It is shown that the value of the cubic lattice Green functions G(z;l,m,n) at the upper critical point z = 3 (for the simple cubic and face-centred cubic lattices) or z = 1 (for the body-centred cubic lattice) is expressible rationally in terms of the known value of G(3;0,0,0) or G(1;0,0,0), and ?.

Lattice sums then and now

The study of lattice sums began when early investigators wanted to go from mechanical properties of crystals to the properties of the atoms and ions from which they were built (the literature of Madelung’s constant). A parallel literature was built around the optical properties of regular lattices of atoms (initiated by Lord Rayleigh, Lorentz and Lorenz). For over a century many famous scientists and mathematicians have delved into the properties of lattices, sometimes unwittingly duplicating the work of their predecessors. Here, at last, is a comprehensive overview of the substantial body of knowledge that exists on lattice sums and their applications. The authors also provide commentaries on open questions, and explain modern techniques which simplify the task of finding new results in this fascinating and ongoing field. Lattice sums in one, two, three, four and higher dimensions are covered. How To Order

Exact solutions of anisotropic diffusion-limited reactions with coagulation and annihilation

We report exact results for one-dimensional reaction-diffusion modelsA+A→inert,A+A→A, andA+B→inert, where in the latter case like particles coagulate on encounters and move as clusters. Our study emphasizes anisotropy of hopping rates; no changes in universal properties are found, due to anisotropy, in all three reactions. The method of solution employs mapping onto a model of coagulating positive integer charges. The dynamical rules are synchronous, cellular-automaton type. All the asymptotic large-time results for particle densities are consistent, in the framework of universality, with other model results with different dynamical rules, when available in the literature.

New exactly solvable periodic potentials for the Dirac equation

A new exactly solvable relativistic periodic potential is obtained by the periodic extension of a well-known transparent scalar potential. It is found that the energy band edges are determined by a transcendental equation which is very similar to the corresponding equation for the Dirac Kronig–Penney model. The solutions of the Dirac equation are expressed in terms of elementary functions.

Lattice Green function (at 0) for the 4D hypercubic lattice

The generating function for recurrent Polya walks on the four-dimensional hypercubic lattice is expressed as a Kampe-de Feriet function. Various properties of the associated walks are enumerated.

Integral representations for the exceptional univariate Lommel functions

An elementary integral representation for the Lommel function S−1, 0(z) is given and extended to other exceptional cases.

Second-order Darboux displacements

The potentials for a one-dimensional Schrodinger equation that are displaced along the x-axis under second-order Darboux transformations, called 2-SUSY invariant, are characterized in terms of a differential–difference equation. The solutions of the Schrodinger equation with such potentials are given analytically for any value of the energy. The method is illustrated by a two-soliton potential. It is proved that a particular case of the periodic Lame–Ince potential is 2-SUSY invariant. Both Bloch solutions of the corresponding Schrodinger equation are found for any value of the energy. A simple analytic expression for a family of two-gap potentials is derived.

Exchange Energy of an Electron Gas of Arbitrary Dimensionality

Procedures are presented for obtaining the complete low temperature asymptotic behavior of fractional integrals (of Riemann–Liouville type) of squares of Fermi–Dirac integrals. These integrals occur in consideration of the properties of electron systems. The development is presented in the context of calculating the exchange thermodynamic potential for a d-dimensional neutralized homogeneous electron gas interacting via a

A general integral identity

1/r