From two-dimensional (super-integrable) quantum dynamics to (super-integrable) three-body dynamics

Abstract

It is shown that planar quantum dynamics can be related to 3-body quantum dynamics in the space of relative motion with a special class of potentials. As an important special case the $O(d)$ symmetry reduction from $d$ degrees of freedom to one degree is presented. A link between two-dimensional (super-integrable) systems and 3-body (super-integrable) systems is revealed. As illustration we present number of examples. We demonstrate that the celebrated Calogero-Wolfes 3-body potential has a unique property: two-dimensional quantum dynamics coincides with 3-body quantum dynamics on the line at $d=1$; it is governed by the Tremblay-Turbiner-Winternitz potential for parameter $k=3$.

arXiv comment13 pages, 24 refs; title modified, several explicit examples added, Abstract and Introduction extended, some statements clarified

References in corpus (15)

• Classical and Quantum Superintegrability with Applications
• An infinite family of solvable and integrable quantum systems on a plane
• Exact Solvability of Superintegrable Systems
• Superintegrability of the Caged Anisotropic Oscillator
• Superintegrability and higher order constants for classical and quantum systems
• Periodic orbits for an infinite family of classical superintegrable systems
• An infinite family of superintegrable deformations of the Coulomb potential
• Superintegrability of the Tremblay-Turbiner-Winternitz quantum Hamiltonians on a plane for odd $k$
• Coupling constant metamorphosis and Nth order symmetries in classical and quantum mechanics
• Classification of quantum superintegrable systems with quadratic integrals on two dimensional manifolds
• Tools for Verifying Classical and Quantum Superintegrability
• A nonseparable quantum superintegrable system in 2D real Euclidean space
• Three-body problem in $d$-dimensional space: ground state, (quasi)-exact-solvability
• Few-electron atomic ions in non-relativistic QED: the Ground state energy
• Three-body closed chain of interactive (an)harmonic oscillators and the algebra $sl(4)$

Cited by in corpus (4)

• New infinite families of $N$th-order superintegrable systems separating in Cartesian coordinates
• 3-body harmonic molecule
• Dynamical symmetry algebra of two superintegrable two-dimensional systems
• $\mathfrak{gl}(3)$ Polynomial Integrable System: Different Faces of the 3-Body/${\mathcal{A}}_2$ Elliptic Calogero Model