MULTIPOLE MATRIX ELEMENTS FOR DH-SYSTEMS AND THEIR ASYMPTOTICS

Abstract

A “DH-system” is defined as a multidimensional hydrogen atom (or its one-particle analogue), D≥1. Investigating many Coulomb problems in ℝD it is necessary to know exact analytical expressions of multipole matrix elements <q|rk|q'>D for DH-systems, where q=(N, µ) is a set of parameters, N —"principal” and µ — "orbital” quantum numbers. The paper deals with the new method for the evaluation of similar matrix elements using new properties of Appell’s function F2(x, y) to the vicinity of the singular point (1, 1). Such approach allows: 1) to get exact analytical expressions of these matrix elements (considering the selection rules) by means of Appell’s F2 (or Clausen’s 3F2) functions; 2) to reveal “latent” symmetry of diagonal matrix elements with respect to the point k0=−3/2, the above symmetry is connected with the property of Appell’s function F2 (1,1) mirror-like symmetry; 3) to find (exact) asymptotics of the off-diagonal matrix elements in terms of Horn’s function ψ1 (x, y); 4) to prove that the orthogonality of radial functions fNµ (D, r) over N and μ for DH-systems is connected with the properties of Appell’s F2 function to the vicinity of the singular point (1, 1), it generalizes the known result for 3H-atom by Pasternack and Sternheimer, J. Math. Phys.3, 1280 (1962).