New quantum solvable models on the homogeneous manifold SL(2, c)/ GL(1, c) with infinite-fold degeneracy

Abstract

Using the idea of shape invariance with respect to the main quantum number n, we construct here new generators for the realization of commutative relations of Lie algebra gl(2, c). By an appropriate parametrization, new quantum solvable Hamiltonians with dynamical symmetry group GL(2, c) and infinite-fold degeneracy are derived. These models correspond to the motion of a free particle on SL(2, c)/ GL(1, c). It is shown that the related quantum states satisfy shape invariance equations with respect to n, and they also represent the Lie algebra gl(2, c).