Potentials on the conformally compactified Minkowski spacetime and their application to quark deconfinement

Abstract

In this paper, we study a class of conformal metric deformations in the quasi-radial coordinate parametrizing the three-sphere in the conformally compactified Minkowski spacetime $S^1\times S^3$. Prior to reduction of the associated Laplace–Beltrami operators to a Schrödinger form, a corresponding class of exactly solvable potentials (each one containing a scalar and a gradient term) is found. In particular, the scalar piece of these potentials can be exactly or quasi-exactly solvable, and among them we find the finite range confining trigonometric potentials of Pöschl–Teller, Scarf and Rosen–Morse. As an application of the results developed in the paper, the large compactification radius limit of the interaction described by some of these potentials is studied, and this regime is shown to be relevant to a quantum mechanical quark deconfinement mechanism.