Spectral approximation by the polar transformation

Abstract

Central potentials V(r) are considered that admit the polar representation V(r) = g(h(r)), where h(r) = sgn (q)rq, q is fixed, and g is the polar transformation function. This representation allows the Schrödinger eigenvalues generated by V to be approximated in terms of those generated by the pure polar potential h(r). In many cases a pair of powers {q1, q2} can be chosen so that the corresponding polar functions {g1, g2} have definite and opposite convexity. For such cases, the spectral approximations provide both upper and lower bounds for the entire discrete spectrum. The example V(r) = ar2 + br2/(1 + cr2) is considered in detail. PACS No. 03.65Ge