We consider the quantum mechanical problem of the motion of a spinless charged relativistic particle with mass , described by the Klein-Fock-Gordon equation with equal scalar and vector Coulomb plus ring-shaped potentials. It is shown that the system under consideration has both a discrete at and a continuous at energy spectra. We find the analytical expressions for the corresponding complete wave functions. A dynamical symmetry group for the radial wave equation of motion is constructed. The algebra of generators of this group makes it possible to find energy spectra in a purely algebraic way. It is also shown that relativistic expressions for wave functions, energy spectra, and group generators in the limit go over into the corresponding expressions for the nonrelativistic problem.

中文翻译：

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库仑和环形势之和的Klein-Fock-Gordon方程的近似解

我们考虑一个非旋转的运动的量子力学问题装入质量相对论性粒子，通过克莱-福克-Gordon方程等于标量描述和矢量库仑加上环形的电位。结果表明，所考虑的系统在 和一个连续的 能谱。我们找到了相应完整波函数的解析表达式。动力对称群构造了径向波运动方程。该组发生器的代数使以纯代数的方式找到能谱成为可能。还显示了极限中的波函数，能谱和群发生器的相对论表达式 讨论非相对论问题的对应表达式。