Spectrum of the three dimensional fuzzy well

TL;DR

This paper extends quantum mechanics to three-dimensional fuzzy space, analyzing free particles and potential wells, revealing a high-energy cutoff, modified dispersion relations, and an ultraviolet/infrared duality.

Contribution

It develops a formalism for quantum mechanics on 3D fuzzy space and solves the Schrödinger equation for various wells, highlighting new spectral features and dualities.

Findings

High energy cutoff for free particle spectrum

Modification of high energy dispersion relation

Existence of an upper bound on energy eigenvalues

Abstract

We develop the formalism of quantum mechanics on three dimensional fuzzy space and solve the Schr\"odinger equation for a free particle, finite and infinite fuzzy wells. We show that all results reduce to the appropriate commutative limits. A high energy cut-off is found for the free particle spectrum, which also results in the modification of the high energy dispersion relation. An ultra-violet/infra-red duality is manifest in the free particle spectrum. The finite well also has an upper bound on the possible energy eigenvalues. The phase shifts due to scattering around the finite fuzzy potential well have been calculated.