Three-dimensional superintegrable systems in a static electromagnetic field

Abstract

We consider a charged particle moving in a static electromagnetic field described by the vector potential $\vec{A}(\vec{x})$ and the electrostatic potential $V(\vec{x})$. We study the conditions on the structure of the integrals of motion of the first and second order in momenta, in particular how they are influenced by the gauge invariance of the problem. Next, we concentrate on the three possibilities for integrability arising from the first order integrals corresponding to three nonequivalent subalgebras of the Euclidean algebra, namely $(P_1,P_2)$, $(L_3,P_3)$ and $(L_1,L_2,L_3)$. For these cases we look for additional independent integrals of first or second order in the momenta. These would make the system superintegrable (minimally or maximally). We study their quantum spectra and classical equations of motion. In some cases nonpolynomial integrals of motion occur and ensure maximal superintegrability.

21 pages, 3 figures

References in corpus (5)

• Hamiltonians separable in cartesian coordinates and third-order integrals of motion
• Superintegrability with third order integrals of motion, cubic algebras and supersymmetric quantum mechanics I:Rational function potentials
• Integrable and superintegrable quantum systems in a magnetic field
• A New Superintegrable Hamiltonian
• SUSY approach to Pauli Hamiltonians with an axial symmetry

Cited by in corpus (19)

• Advances in QED with intense background fields
• Screw-symmetric gravitational waves: a double copy of the vortex
• Back-reaction on background fields: a coherent state approach
• Fourth order Superintegrable systems separating in Cartesian coordinates I. Exotic quantum potentials
• Superintegrable relativistic systems in spacetime-dependent background fields
• Maxwell's equations in homogeneous spaces for admissible electromagnetic fields
• Cylindrical type integrable classical systems in a magnetic field
• On superintegrability of 3D axially-symmetric non-subgroup-type systems with magnetic fields
• On rotationally invariant integrable and superintegrable classical systems in magnetic fields with non-subgroup type integrals
• Symmetries of Schroedinger equation with scalar and vector potentials
• New classes of quadratically integrable systems in magnetic fields: the generalized cylindrical and spherical cases
• Higher Order Quantum Superintegrability: a new "Painlevé conjecture"
• Superintegrable relativistic systems in scalar background fields
• Family of nonstandard integrable and superintegrable classical Hamiltonian systems in non-vanishing magnetic fields
• New classes of quadratically integrable systems with velocity dependent potentials: non-subgroup type cases
• Hydrogen Atom in Electric and Magnetic Fields: Dynamical Symmetries, Superintegrable and Integrable Systems, Exact Solutions
• Superintegrable families of magnetic monopoles with non-radial potential in curved background
• Hamiltonian integrable systems in a magnetic field and Symplectic-Haantjes geometry
• Cylindrical first order superintegrability with complex magnetic fields