The solution of the Schrödinger equation for Makarov potential and homogeneous manifold $SL (2, {\mathbb {C}})/GL (1, {\mathbb {C}})$

Abstract

In this study, we are going to obtain the energy spectrum and the corresponding solution of the non-central Makarov potential. In this case, we consider the arbitrary angular momentum with quantum number l. In order to calculate the energy spectrum and eigenfunction we use the factorisation method. The factorisation method leads us to discuss the shape-invariance condition with respect to any index as n and m. Here, we also achieve the shape invariance with respect to the main quantum number n. It leads to the quantum-solvable models on real forms of the homogeneous manifold $SL (2, {\mathbb {C}} )/ GL (1, {\mathbb {C}} )$ with infinite-fold degeneracy for $\gamma _v =0$ and $\gamma _v \ne 0$. These processes also help us to obtain raising and lowering operators of states on the above-mentioned homogeneous manifold.