We present a unified treatment of three cases of quasi-exactly-solvable problems, namely, a charged particle moving in Coulomb and magnetic fields for both the Schrödinger and the Klein-Gordon case, and the relative motion of two charged particles in an external oscillator potential. We show that all these cases are reducible to the same basic equation, which is quasiexactly solvable owing to the existence of a hidden sl 2 algebraic structure. A systematic and unified algebraic solution to the basic equation using the method of factorization is given. Analytical expressions of the energies and the allowed frequencies for the three cases are given in terms of roots of one and the same set of Bethe ansatz equations.

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The Dirac equation for an electron in two spatial dimensions in the Coulomb and homogeneous magnetic fields is an example of the so-called quasi-exactly solvable models.The solvable parts of its spectrum was previously solved from the recursion relations.In this work we present a purely algebraic solution based on the Bethe ansatz equations. It is realised that, unlike the corresponding problems in the Schrödinger and the Klein-Gordon case, here the unknown parameters to be solved for in the Bethe ansatz equations include not only the roots of wave function assumed, but also a parameter from the relevant operator. We also show that the quasi-exactly solvable differential equation does not belong to the classes based on the algebra sl 2 .

The Dirac equation for an electron in two spatial dimensions in the Coulomb and homogeneous magnetic fields is a physical example of quasi-exactly solvable systems.This model, however, does not belong to the classes based on the algebra sl(2) which underlies most one-dimensional and effectively one-dimensional quasi-exactly solvable systems. In this paper we demonstrate that the quasi-exactly solvable differential equation possesses a hidden osp(2, 2) superalgebra. PACS: 03.65. Pm, 31.30.Jv, 03.65.Fd

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