New Classes of Potentials for which the Radial Schrodinger Equation can be solved at Zero Energy

TL;DR

This paper introduces a method to generate new solvable potentials for the radial Schrödinger equation at zero energy by combining known potentials, allowing for indefinite construction of such potentials with controlled bound state properties.

Contribution

The authors present a novel construction technique to generate an infinite family of potentials solvable at zero energy, extending previous solutions and applicable to regular, singular, and long-range potentials.

Findings

Constructed new potentials explicitly from known ones.

Extended the method to singular and long-range potentials.

Provided explicit examples demonstrating the construction.

Abstract

Given two spherically symmetric and short range potentials $V_0$ and V_1 for which the radial Schrodinger equation can be solved explicitely at zero energy, we show how to construct a new potential $V$ for which the radial equation can again be solved explicitely at zero energy. The new potential and its corresponding wave function are given explicitely in terms of V_0 and V_1, and their corresponding wave functions \phi_0 and \phi_1. V_0 must be such that it sustains no bound states (either repulsive, or attractive but weak). However, V_1 can sustain any (finite) number of bound states. The new potential V has the same number of bound states, by construction, but the corresponding (negative) energies are, of course, different. Once this is achieved, one can start then from V_0 and V, and construct a new potential \bar{V} for which the radial equation is again solvable explicitely. And the process can be repeated indefinitely. We exhibit first the construction, and the proof of its validity, for regular short range potentials, i.e. those for which rV_0(r) and rV_1(r) are L^1 at the origin. It is then seen that the construction extends automatically to potentials which are singular at r= 0. It can also be extended to V_0 long range (Coulomb, etc.). We give finally several explicit examples.