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

Willard Miller, Alexander V. Turbiner

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We analyze one particle, two-center quantum problems which admit separation of variables in prolate spheroidal coordinates, a natural restriction satisfied by the H+2molecular 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 Schrodinger operator in the space of prolate spheroidal coordinates, including one for the H+2molecular ion, are indicated. We study possible potentials that admit exactsolvability is as well as allmodels 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.

Original languageEnglish (US)
Article number192002
JournalJournal of Physics A: Mathematical and Theoretical
Issue number19
StatePublished - 2014

Bibliographical note

Publisher Copyright:
© 2014 IOP Publishing Ltd.


  • elliptic coordinates
  • integrability
  • one particle two centers
  • solvability


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