Particle in a field of two centers in prolate spheroidal coordinates: integrability and solvability
Particle in a field of two centers in prolate spheroidal coordinates: integrability and solvability
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DOI：
10.1088/1751-8113/47/19/192002
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发表时间：
2014-02
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影响因子：
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通讯作者：
Willard Miller Jr;A. Turbiner
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作者：
Willard Miller Jr;A. Turbiner
我们分析了一个粒子，两个中心的量子问题，承认分离的变量在长椭球坐标，一个自然的限制，满足H2+？>分子离子对称算子被明确地构造。我们给出了详细的哈密顿约化的3D系统的2D系统与修改后的潜力，是可分离的椭圆坐标系。在长球坐标系中，薛定谔算子具有双周期性的势函数，其中包括H2+的势函数。分子离子。我们研究可能的潜力，承认确切的可解性，以及我们已知的所有模型与分离方程的（准）确切的可解性。我们发现二阶超可积和共形超可积系统与这些易处理的问题之间有着深刻的联系。特别是，我们推导出一个一般的四参数表达式的模型势，总是精确可解和可积的，是共形超可积的一些参数的选择。
We analyze one particle, two-center quantum problems which admit separation of variables in prolate spheroidal coordinates, a natural restriction satisfied by the H2+?> molecular ion. The symmetry operator is constructed explicitly. We give the details of the Hamiltonian reduction of the 3D system to a 2D system with modified potential that is separable in elliptic coordinates. The potentials for which there is double-periodicity of the Schrödinger operator in the space of prolate spheroidal coordinates, including one for the H2+?> molecular ion, are indicated. We study possible potentials that admit exact-solvability is as well as all models known to us with the (quasi)-exact-solvability property for the separation equations. We find deep connections between second-order superintegrable and conformally superintegrable systems and these tractable problems. In particular we derive a general four-parameter expression for a model potential that is always exactly-solvable and integrable and is conformally superintegrable for some parameter choices.