Quantum Bound States with Zero Binding Energy

TL;DR

This paper explores zero-energy quantum states in specific potentials, revealing conditions under which they are bound or normalizable, including unusual cases with repulsive potentials and higher-dimensional spaces.

Contribution

It provides explicit solutions for zero-energy states in power-law potentials and analyzes their normalizability and bound state conditions across different dimensions.

Findings

Solutions are normalizable for certain potentials with u > 2 and l > 0.

States are normalizable even for repulsive potentials when u < -2.

Higher dimensions (D > 4) enable normalizable zero-energy states for l=0.

Abstract

After reviewing the general properties of zero-energy quantum states, we give the explicit solutions of the \seq with $E=0$ for the class of potentials $V=-|\gamma|/r^{\nu}$, where $-\infty < \nu < \infty$. For $\nu > 2$, these solutions are normalizable and correspond to bound states, if the angular momentum quantum number $l>0$. [These states are normalizable, even for $l=0$, if we increase the space dimension, $D$, beyond 4; i.e. for $D>4$.] For $\nu <-2$ the above solutions, although unbound, are normalizable. This is true even though the corresponding potentials are repulsive for all $r$. We discuss the physics of these unusual effects.