Eigenvalue bounds for transformations of solvable potentials

We study smooth transformations V(r) = h(0)(r) + g(h(beta r)) of potentials V-0(r) = h(0)(r)+ h(beta r) for which exact bound-state solutions of Schrodinger's equation are known. Eigenvalue approximation formulae are obtained which provide lower or upper energy bounds according to whether the transformation function g is convex or concave. Detailed results are presented for perturbed Coulomb potentials of the form V(r) = a/r + br + cr(2) and V(r) = -1/r + mu ln(r + r(2)).