Bound state energies and phase shifts of a non-commutative well

TL;DR

This paper investigates the quantum behavior of a non-commutative well, calculating bound states and phase shifts, revealing subtle effects of non-commutativity on scattering properties.

Contribution

It provides a clear definition of the non-commutative well and computes its bound states and phase shifts, highlighting non-uniform convergence and strong parameter dependence.

Findings

Results closely match commutative case for large wells or small non-commutative parameters

Phase shifts show strong dependence on non-commutative parameter at certain energies

Convergence to commutative results is not uniform

Abstract

Non-commutative quantum mechanics can be viewed as a quantum system represented in the space of Hilbert-Schmidt operators acting on non-commutative configuration space. Within this framework an unambiguous definition can be given for the non-commutative well. Using this approach we compute the bound state energies, phase shifts and scattering cross sections of the non- commutative well. As expected the results are very close to the commutative results when the well is large or the non-commutative parameter is small. However, the convergence is not uniform and phase shifts at certain energies exhibit a much stronger then expected dependence on the non-commutative parameter even at small values.