Exact solution of inverse-square-root potential $V\left( r\right) =-\frac{\alpha}{\sqrt{r}}$

TL;DR

This paper provides an exact analytical solution for the quantum mechanical problem involving the inverse-square-root potential, covering both scattering states and bound states.

Contribution

It presents the first exact solutions for the inverse-square-root potential in three dimensions, including bound and scattering states.

Findings

Derived explicit wavefunctions for bound states.

Obtained scattering phase shifts analytically.

Enhanced understanding of inverse-square-root potential behavior.

Abstract

An exact solution of the three-dimensional spherical symmetry inverse-square-root potential $V\left( r\right) =-\frac{\alpha}{\sqrt{r}}$, including scattering and bound-state solutions, is presented.