A note on quantum Bohlin transformation

Abstract

It is shown, that the reduction of the circular quantum oscillator by the $Z_2$-group action results to the two systems: a two-dimensional hydrogen atom, and a ``charge - charged magnetic vortex" one, with the spin $\frac{1}{2}$. Analogously, the $Z_N$-reduction of the two-dimensional system with the central potential $r^{2(N-1)}$ results into $N$ bound ``charge - magnetic vertex" systems with the interaction potential $r^{2(1/N-1)}$ and spins $\sigma =\frac{k}{N}$, $k=0,1,...,(N-1)$.

6 pages, LaTeX

Cited by in corpus (30)

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• Non-Relativistic Holography -- A Group-Theoretical Perspective
• Second Hopf map and Yang-Coulomb system on 5d (pseudo)sphere
• ${\mathbb{C}}^N$-Smorodinsky-Winternitz system in a constant magnetic field
• Symmetries in superintegrable deformations of oscillator/Coulomb systems: "holomorphic factorization"
• Elements of (super-)Hamiltonian Formalism
• The Stark effect in the charge-dyon system
• 3D Oscillator and Coulomb Systems reduced from Kahler spaces
• Spheroidal analysis of the generalized MIC-Kepler system
• The charge-dyon bound system in the spherical quantum well
• Multi-center MICZ-Kepler systems
• How to relate the oscillator and Coulomb systems on spheres and pseudospheres?
• Bohlin-Arnold-Vassiliev's duality and conserved quantities
• Quantum oscillator as 1D anyon
• Hopf maps and Wigner's little groups
• Self-Adjointness of Generalized MIC-Kepler System
• Geometry and integrability in $\mathcal{N}=8$ supersymmetric mechanics
• Conformally related vacuum gravitational waves, and their symmetries
• The laws of planetary motion, derived from those of a harmonic oscillator (following Arnold)
• Duality properties of Gorringe-Leach equations
• Application of regularization maps to quantum mechanical systems in 2 and 3 dimensions
• Four dimensional quantum oscillator and magnetic-monopole with U(1) dynamical group
• Reductions related with Hopf maps
• Superintegrable deformations of oscillator and Coulomb systems