Abstract

Central potentials V(r) are considered which admit the polar representation $V(r)=g(h(r)),$ where $h(r)={\rm sgn}(q)r\sp{q},$ 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 polar potential h(r). In many cases the optimal values $\{q\sb1,q\sb2\}$ of the power q can be chosen so that the corresponding polar functions $\{g\sb1,g\sb2\}$ have definite and opposite convexity. For such cases the spectral approximations provide both upper and lower bounds for the entire discrete spectrum. The example of the central potential $V(r)=ar\sp2+br\sp2/(1+cr\sp2)$ in $R\sp3$ is studied in detail: optimal bounds are determined for a wide range of the potential parameters. The method is applicable, essentially unchanged, for problems in any number of spatial dimensions.