Necessary Conditions on Potential Functions for Nonrelativistic Bound States

Abstract

It is shown that $\ensuremath{\int}{|V(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}})|}^{\frac{3}{2}}{d}^{3}x>(\frac{{\ensuremath{\pi}}^{2}}{4}){(\frac{3{\ensuremath{\hbar}}^{2}}{2m})}^{\frac{3}{2}}$ is a necessary condition for one or more normalizable bound-state solutions to the nonrelativistic Schr\"odinger energy-eigenvalue equation for any potential function $V(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}})$ such that the integral is finite. Moreover, for potentials such that $\ensuremath{\int}{|V(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}})|}^{\frac{3\ensuremath{\gamma}}{2}}{d}^{3}x$ is finite for a value or values of $\ensuremath{\gamma}>1$, the magnitude of a negative energy eigenvalue is related to the latter integral by a general inequality derived here.