Abstract

We study a concrete class of eigenvalue problems in mathematical physics, which arise from non-relativistic quantum mechanics and the Sturm-Liouville theory. We develop practical techniques to obtain reliable bounds for the eigenvalues of the Schrödinger operator [Special characters omitted.] . We introduce a three-parameter variational function, to determine an upper bound to the ground-state energy, of the supersingular spiked harmonic oscillator potentials [Special characters omitted.] . The entire parameter range n > 0 and [Special characters omitted.] is treated by a single formulation. We employ the method of potential envelopes to derive a simple energy lower bound formula, valid for all parameter ranges n > 0 and [Special characters omitted.] , and for all the discrete eigenvalues. The standard method of envelope potentials is extended and applied to analyse the discrete spectrum of the generalized singular potentials [Special characters omitted.] , where [Special characters omitted.] , Ì and Ý > 0 are arbitrary positive parameters. We analyse also the discrete spectrum of the generalized Kratzer's potentials [Special characters omitted.] . We obtain lower and upper bound expressions to the eigenvalues which are valid for all dimensions [Special characters omitted.] . We introduce the h-method to study smooth transformations [Special characters omitted.] , of the potentials [Special characters omitted.] , for which exact bound-state solutions of the Schrödinger equation are known for certain values of the positive parameter Ý. Eigenvalue approximation formulae thereby obtained provide lower or upper energy bounds, depending on whether the transformation function g is convex or concave. This enables us to give lower and upper bound expressions to the perturbed Coulomb potential [Special characters omitted.] , with arbitrary coefficients [Special characters omitted.] . Several new comparison theorems for the eigenvalues of a pair of Schrödinger equations [Special characters omitted.] , i = 1, 2, are introduced. These theorems allow the comparison function [Special characters omitted.] to intersect at a finite number of points within [- l, l ], while maintaining the eigenvalue comparisons. The extension to more general Sturm-Louvile problems is also discussed.