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 ${\mathrm{sl}}_2$ 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.

• Received 30 October 2000

DOI:https://doi.org/10.1103/PhysRevA.63.062105