Particle in a field of two centers in prolate spheroidal coordinates: integrability and solvability
TLDR
In this paper, the authors give a general 4-parameter expression for a model potential that is always integrable and is conformally superintegrable for some parameter choices.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 Schrodinger 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.read more
Citations
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Symmetry and Separation of Variables (Encyclopaedia of Mathematics and its Applications Vol 4)
TL;DR: Miller as discussed by the authors used Lie algebraic techniques to discuss the method of separation of variables and its application to the Laplace, wave, Helmholtz, Schrodinger and diffusion equations.
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Approximate solutions to the quantum problem of two opposite charges in a constant magnetic field
TL;DR: In this article, the authors consider two particles of equal mass and opposite charge in a plane subject to a perpendicular constant magnetic field and show that the solution is given by two fourth degree Hill differential equations which involve the energy as well as a second constant of motion.
Journal ArticleDOI
Eisenhart lift for Euler’s problem of two fixed centers
Hai Zhang,Qing-Yi Hao +1 more
TL;DR: The approach of Eisenhart lift is applied to the classical Euler’s bicentric system, yielding a three-dimensional geodesic system, with explicit forms of the Killing tensors on the associated Riemannian manifolds.
Journal ArticleDOI
Photoelectron holography of the H2+ molecule
TL;DR: In this paper, the spatial interference pattern is created by the coherent superposition of electronic wave packets emitted at the same time, but following different paths, and the location of the interference minima in the spectra is dominantly determined by the target's ionization energy, however, clear differences were observed for the molecular potential relative to the central potentials.
References
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Quantum Integrable Systems Related to Lie Algebras
TL;DR: In this article, a review of quantum integrable finite-dimensional systems related to Lie algebras is presented, which contains results such as the forms of spectra, wave functions, S-matrices and quantum integrals of motion.
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TL;DR: The relationship between symmetries of a linear second-order partial differential equation of mathematical physics, the coordinate systems in which the equation admits solutions via separation of variables, and the properties of the special functions that arise in this manner are discussed in this paper.
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Quasi-exactly-solvable problems and sl(2) algebra
TL;DR: In this paper, it was shown that the bilinear formh=a αβ J α J β +b α J α (α stand for the generators) allows one to generate a set of quasi-exactly-solvable problems of different types, including those that are already known.
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Momentum Maps and Hamiltonian Reduction
Juan-Pablo Ortega,Tudor S. Ratiu +1 more
TL;DR: In this paper, the Symplectic Slice Theorem and Singular reduction and the stratification theorem are used to define a regular symplectic reduction theory for Lie Group actions.
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