Non-fuchsian Singularities in Quantum Mechanics

The Schrödinger equation for stationary states with non-Fuchsian singularities both at the origin and at infinity can be studied with the help of a suitable change of independent variable, here taken to be of the form ρ = r , where γ is a real parameter greater than 1 and r is the original independent variable. Whenever the potential contains a finite number of negative and positive powers of r, the transformed stationary Schrödinger equation, expressed in terms of ρ, is found to ‘tend’ to an equation with Fuchsian singularities, if γ is sufficiently large. The three-dimensional isotropic harmonic oscillator is then considered, when the potential is modified by the addition of terms leading to non-Fuchsian singularities in the stationary Schrödinger equation either at the origin or at infinity. A general algorithm for the solution of such problems, relying on a suitable factorization of the wave function, is proposed, and a perturbative evaluation of the correction to the ground-state energy is performed.