Eigenvalue bounds for a class of singular potentials

We study smooth transformations V(x) = g(x(2)) + f(1/x(2)) of the solvable potentials lambda x(2) + mu/x(2). Eigenvalue approximation formulae are obtained which provide lower or upper energy bounds for all the discrete energy eigenvalues E-n, n = 0,1,2,..., accordingly as the transformation functions g and f are both convex or both concave. Detailed results are presented For the special case of two-term singular potentials of the form V(x) = lambda x(beta) + mu/x(alpha), alpha, beta > 0, and also for the potentials V(x) = lambda x(1.9) + mu/x(1.9) and V(x) = lambda x(2.1) + mu/x(2.1), lambda > 0, mu > 0, for 0 less than or equal to n less than or equal to 10.