Spectral approximation by the polar transformation

Central potentials V(r) are considered that admit the polar representation V(r) = g(h(r)), where h(r) = sgn(q)r(q), q is fixed, and g is the polar transformation function. This representation allows the Schrodinger 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 (q(1), q(2)) can be chosen so that the corresponding polar functions (g(1), g(2)) 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) = ar(2) + br(2)/(1 + cr(2)) is considered in detail.