THE ELECTROSTATIC POTENTIAL OF PERIODIC CRYSTALS

The electrostatic potentials phi associated with neutral periodic crystals are defined by lattice sums that are never absolutely convergent. The sum depends on the order of summation. The mean zero periodic solution of Poisson's equation, denoted phi, is a natural potential. So is the potential obtained by the Mellin transform algorithm of [Borwein, Borwein, and Taylor, J. Math. Phys., 26 (1985), pp. 2999-3009]. We prove that these two are equal and are both equal to the potential obtained by Abel summation. The sum defining partial derivative(alpha)phi converges absolutely for vertical bar alpha vertical bar >= 3 to partial derivative(alpha)phi. The indeterminacy in the potential is at most a harmonic polynomial of degree 2.