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We study a class of conformal metric deformations in the quasi-radialcoordinate parameterizing the 3-sphere in the conformally compactifiedMinkowski spacetime $S^1\times S^3$. After reduction of the associatedLaplace-Beltrami operators to a Schr\"odinger form, a corresponding class ofexactly solvable potentials (each one containing a scalar and a gradient term)is found. In particular, the scalar piece of these potentials can be exactly orquasi-exactly solvable, and among them we find the finite range confiningtrigonometric potentials of P\"oschl-Teller, Scarf and Rosen-Morse. As anapplication of the results developed in the paper, the large compactificationradius limit of the interaction described by some of these potentials isstudied, and this regime is shown to be relevant to a quantum mechanical quarkdeconfinement mechanism.

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