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arXiv · 1402.3816

Particle in a field of two centers in prolate spheroidal coordinates: integrability and solvability

Willard Miller, Jr·Alexander V Turbiner
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Abstract

We analyze one particle, two-center quantum problems which admit separation of variables in prolate spheroidal coordinates, a natural restriction satisfied by the H$_2^+$ 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 Schr\"odinger operator in the space of prolate spheroidal coordinates, including one for the H$_2^+$ 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 4-parameter expression for a model potential that is always integrable and is conformally superintegrable for some parameter choices.

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Willard Miller, Jr, Alexander V Turbiner. 2014-02-16. Particle in a field of two centers in prolate spheroidal coordinates: integrability and solvability. https://doi.org/10.1088/1751-8113/47/19/192002

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