Two- and Many-Dimensional Quasi-Exactly Solvable Models With An Inhomogeneous Magnetic Field

Abstract

Let group generators having finite-dimensional representation be realized as Hermitian linear differential operators without nhomogeneous terms as takes place, for example, for the SO(n) group. Then orresponding group Hamiltonians containing terms linear in generators (along with quadratic ones) give rise to quasi-exactly solvable models with a magnetic field in a curved space. In particular, in the two-dimensional case such models are generated by quantum tops. In the three-dimensional one for the SO(4) Hamiltonian with an isotropic quadratic part the manifold within which a quantum particle moves has the geometry of the Einstein universe.