Singular inverse square potential in arbitrary dimensions with a minimal length: Application to the motion of a dipole in a cosmic string background

TL;DR

This paper analytically solves the N-dimensional inverse square potential Schrödinger equation with a minimal length, applying it to a dipole in a cosmic string background, revealing conditions for bound states and clarifying conflicting literature.

Contribution

It provides an exact solution using Heun's functions for the inverse square potential with a minimal length in arbitrary dimensions, and applies it to cosmic string physics.

Findings

Bound state exists only if the dipole angle exceeds π/4

Minimal length acts as a cosmic string radius

Clarifies previous conflicting results in the literature

Abstract

We solve analytically the Schr\"odinger equation for the N-dimensional inverse square potential in quantum mechanics with a minimal length in terms of Heun's functions. We apply our results to the problem of a dipole in a cosmic string background. We find that a bound state exists only if the angle between the dipole moment and the string is larger than {\pi}/4. We compare our results with recent conflicting conclusions in the literature. The minimal length may be interpreted as a radius of the cosmic string.