Eigenvalue bounds for a class of singular potentials in N dimensions

The eigenvalue bounds obtained earlier (Hall R L and Saad N 1998 J. Phys. A : Math. Gen. 31 963) for smooth transformations of the form V(x) = g(x(2)) + f(1/x(2)) are extended to N dimensions. In particular a simple formula is derived which bounds the eigenvalues for the spiked harmonic oscillator po tential V(x) = x(2) lambda/x(alpha), alpha > 0, lambda > 0, and is valid fo r all discrete eigenvalues, arbitrary angular momentum l and spatial dimens ion N.