TY - JOUR

T1 - Particle in a field of two centers in prolate spheroidal coordinates

T2 - Integrability and solvability

AU - Miller, Willard

AU - Turbiner, Alexander V.

N1 - Publisher Copyright:
© 2014 IOP Publishing Ltd.

PY - 2014

Y1 - 2014

N2 - 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.

AB - 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.

KW - elliptic coordinates

KW - integrability

KW - one particle two centers

KW - solvability

UR - http://www.scopus.com/inward/record.url?scp=84937057204&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84937057204&partnerID=8YFLogxK

U2 - 10.1088/1751-8113/47/19/192002

DO - 10.1088/1751-8113/47/19/192002

M3 - Article

AN - SCOPUS:84937057204

SN - 1751-8113

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

IS - 19

M1 - 192002

ER -