  We study a class of conformal metric deformations in the quasi-radial
coordinate parameterizing the 3-sphere in the conformally compactified
Minkowski spacetime $S^1\times S^3$. Prior to reduction of the associated
Laplace-Beltrami operators to a Schr\"odinger 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 P\"oschl-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.
