Superintegrable and shape invariant systems with position dependent mass

Abstract

Second order integrals of motion for 3d quantum mechanical systems with position dependent masses (PDM) are classified. Namely, all PDM systems are specified which, in addition to their rotation invariance, admit at least one second order integral of motion. All such systems appear to be also shape invariant and exactly solvable. Moreover, some of them possess the property of double shape invariance and can be solved using two different superpotentials. Among them there are systems with double shape invariance which present nice bridges between the Coulomb and isotropic oscillator systems. A simple algorithm for calculation the discrete spectrum and the corresponding state vectors for the considered PDM systems is presented and applied to solve five of the found systems.

misprints are corrected and discussion is extended

References in corpus (9)

• Quantum mechanics on spaces of nonconstant curvature: the oscillator problem and superintegrability
• Integrable and superintegrable systems with spin
• Superintegrable systems with position dependent mass
• Superintegrable Oscillator and Kepler Systems on Spaces of Nonconstant Curvature via the Stäckel Transform
• Superintegrable and shape invariant systems with position dependent mass
• An exactly solvable deformation of the Coulomb problem associated with the Taub-NUT metric
• A Family of Exactly Solvable Radial Quantum Systems on Space of Non-Constant Curvature with Accidental Degeneracy in the Spectrum
• Laplace-Runge-Lenz vector with spin in any dimension
• SU(2) Symmetry and Degeneracy From SUSY QM of a Neutron in the Magnetic Field of a Linear Current

Cited by in corpus (15)

• Position-dependent mass, finite-gap systems, and supersymmetry
• Superintegrable systems with position dependent mass
• Superintegrable and shape invariant systems with position dependent mass
• Group classification of (1+3)-dimensional Schrödinger equations with position dependent mass
• Group classification of Schrödinger equations with position dependent mass
• New infinite families of $N$th-order superintegrable systems separating in Cartesian coordinates
• Symmetries of Schroedinger equation with scalar and vector potentials
• Symmetries of the Schroedinger-Pauli equations for charged particles and quasirelativistic Schroedinger equations
• Superintegrable quantum mechanical systems with position dependent masses invariant with respect to two parametric Lie groups
• Superintegrable relativistic systems in scalar background fields
• Symmetries of the Schroedinger-Pauli equation for neutral particles
• Mapping of Two-Dimensional Schrödinger Equation under the Point Transformation
• Exact solvability of PDM systems with extended Lie symmetries
• Integrable and superintegrable quantum mechanical systems with position dependent masses invariant with respect to one parametric Lie groups. 2. Systems with dilatation and shift symmetries
• Symmetries and Supersymmetries of Generalized Schrödinger equations