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.

中文翻译：

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马卡罗夫势和齐次流形的薛定谔方程的解 $$SL(2, {\mathbb {C}})/GL (1, {\mathbb {C}})$$SL(2,C)/GL( 1,C)

在本研究中，我们将获得非中心马卡罗夫势能的能谱和相应的解。在这种情况下，我们考虑量子数为 l 的任意角动量。为了计算能谱和本征函数，我们使用因式分解方法。分解方法引导我们讨论关于任何指数为 n 和 m 的形状不变性条件。在这里，我们还实现了关于主量子数 n 的形状不变性。它导致在齐次流形 $$SL (2, {\mathbb {C}} )/ GL (1, {\mathbb {C}} )$$ 的实数形式上的量子可解模型具有无限倍简并性$$\gamma _v =0 $$ 和 $$\gamma _v \ne 0$$。这些过程也有助于我们在上述齐次流形上获得状态的升降算符。