Regularization of the Singular Inverse Square Potential in Quantum Mechanics with a Minimal length

TL;DR

This paper investigates how a minimal length modifies the inverse square potential in quantum mechanics, showing it becomes regular and solvable analytically using Heun's functions, with implications for the system's dimensionality.

Contribution

It introduces a regularization of the inverse square potential via a generalized uncertainty principle and provides analytical solutions for bound states in this framework.

Findings

The inverse square potential is regularized in the minimal length framework.

Analytical solutions for bound states are obtained using Heun's functions.

The minimal length relates to the effective dimension of the quantum system.

Abstract

We study the problem of the attractive inverse square potential in quantum mechanics with a generalized uncertainty relation. Using the momentum representation, we show that this potential is regular in this framework. We solve analytically the s-wave bound states equation in terms of Heun's functions. We discuss in detail the bound states spectrum for a specific form of the generalized uncertainty relation. The minimal length may be interpreted as characterizing the dimension of the system.