Schrodinger Equation weakly attractive 1/r^2 Potential Eigenfunctions

TL;DR

This paper demonstrates the existence of bound states in the weakly attractive 1/r^2 potential for all angular momentum states, correcting previous misconceptions and providing explicit solutions and conditions.

Contribution

It shows that bound states do exist in the weak coupling regime for the 1/r^2 potential, contradicting earlier claims, and derives explicit conditions and solutions for these states.

Findings

Existence of bound states for all angular momentum states in weak coupling.

Correction of previous theoretical misconceptions about the potential.

Explicit formulas for interior well strength and bound state conditions.

Abstract

Bound state solutions of the Schrodinger Equation for the $-a/r^2$ potential have been presented recently for both the weak and strong coupling cases. However, Shortley in 1931 and Landau and Lifshitz in 1958 claimed that no bound state solutions exist for the weak coupling case when $0 < 2ma/\hbar^2 <= (l + 1/2)^2$. We demonstrate that one bound state solution can exist for each angular momentum state l, and that a complete orthogonal set of continuum eigenfunctions orthogonal to the bound state eigenfunction can be constructed when $l(l+ 1) < 2ma/\hbar^2 <= (l + 1/2)^2$. We show that Shortley's argument is spurious due to his neglecting a boundary term arising from the momentum operator and that the Landau and Lifshitz claim is based on a restrictive fitting of the exterior solution to an interior spherical well. Instead, to each weak coupling, a, we find a unique interior well strength $V_{in} = -a'/ro^2$ which yields a finite bound state. In particular for the l = 0 case the well strength is given by: $2ma'/\hbar^2 = 1.3734 + 2.2265*(1/4 - 2ma'/\hbar^2)^{1/2}$.