Localization at threshold in noncommutative space

TL;DR

This paper investigates how noncommutative quantum mechanics introduces a scale that enables the existence of a threshold bound state in a scale symmetric system, revealing new features of conformal symmetry and algebraic structure.

Contribution

It demonstrates the use of noncommutative parameters as a scale to achieve threshold bound states and analyzes the algebraic structure of the system in noncommutative space.

Findings

Existence of a threshold bound state at E=0 due to noncommutativity

Noncommutative parameter  acts as a scale for the system

The so(2,1) algebra is deformed but restores in the limit  to the commutative case

Abstract

The ground state energy of a scale symmetric system usually does not possess any lower bound, thus making the system quantum mechanically unstable. Self-adjointness and renormalization techniques usually provide the system a scale and thus making the ground state bounded from below. We on the other hand use noncommutative quantum mechanics and exploit the noncommutative parameter \Theta as a scale for a scale symmetric system. The resulting Hamiltonian for the system then allows an unusual bound state at the threshold of the energy, E=0. Apart from the Hamiltonian \hat{H} we also compute the other two generators of the so(2,1) algebra, the dilation \hat{D} and the conformal generator \hat{K} in the noncommutative space. The so(2,1) algebra is not closed in the noncommutative space, but the limit \Theta\to 0 smoothly goes to the so(2,1) algebra restoring the conformal symmetry. We also discuss the system for large noncommutative parameter.