Noncommutative quantum mechanics in the presence of delta-function potentials

TL;DR

This paper demonstrates that extending quantum mechanics with delta-function potentials to noncommutative space yields well-defined, finite eigenfunctions and eigenvalues, resolving UV divergencies present in the standard formulation.

Contribution

It introduces a noncommutative space framework for quantum systems with delta potentials, providing a new method to obtain finite, well-defined solutions.

Findings

Eigenfunctions and eigenvalues are finite in noncommutative space.

Standard UV divergencies are resolved through noncommutative extension.

Complete spectral sets are constructed using star product and Fock space formalism.

Abstract

Quantum mechanics in the presence of $\delta$-function potentials is known to be plagued by UV divergencies which result from the singular nature of the potentials in question. The standard method for dealing with these divergencies is by constructing self-adjoint extensions of the corresponding Hamiltonians. Two particularly interesting examples of this kind are nonrelativistic spin zero particles in $\delta$-function potential and Dirac particles in Aharonov-Bohm magnetic background. In this paper we show that by extending the corresponding Schr\"odinger and Dirac equations onto the flat noncommutative space a well-defined quantum theory can be obtained. Using a star product and Fock space formalisms we construct the complete sets of eigenfunctions and eigenvalues in both cases which turn out to be finite.