Product integrals and the Schrödinger equation

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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.

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