Product integrals and the Schrödinger equation

Abstract

A brief introduction to product integration is given. The theory developed is used to give a simple and rigorous analysis of the asymptotic behavior (r→∞) of positive-energy solutions of the radial Schrödinger equation. Absence of positive-energy bound states is proved for various classes of potentials. It is shown that E=1 is the only positive energy for which the Wigner–von Neumann potential can have a positive-energy bound state. The results proved imply (as will be shown in a later publication) existence of the Mo/ller wave matrices for the potential V (R) = (sinr)/r and various related potentials. A brief discussion is given to justify the WKB approximation which gives the wavefunction asymptotically for large positive values of the energy E.