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.

在本文中，我们研究了准径向坐标中的一类共形度量变形，参数化了共形紧致明可夫斯基时空中的三球体 $S^1\times S^3$ 。在将相关的拉普拉斯-贝尔特拉米算子简化为薛定谔形式之前，找到一类相应的完全可解势（每个势包含一个标量和一个梯度项）。特别是，这些势的标量部分可以是精确或准精确可解的，其中我们发现了 Pöschl-Teller、Scarf 和 Rosen-Morse 的有限范围限制三角势。作为本文研究结果的应用，研究了由其中一些势描述的相互作用的大致密半径极限，并且该状态被证明与量子力学夸克解禁闭机制相关。