Group classification of (1+3)-dimensional Schrödinger equations with position dependent mass

Abstract

Kinematical invariance groups of the 3d Schrödinger equations with position dependent masses (PDM) and arbitrary potentials are classified. It is shown that there exist 94 classes of such equations defined up to the generic equivalence group, and 70 classes defined up to the equivalence groupoid. The maximally extended kinematical invariance algebras of such equations appears to be eight dimensional. The specific symmetries connected with the presence of the ambiguity parameters are discussed and an extended class of systems which keep their forms for arbitrary or particular changes of these parameters is specified. The exact solution of the selected PDM Schrödinger equation is presented. This equation describes a deformed 3d isotropic harmonic oscillator and possesses extended continuous symmetries and hidden supersymmetries with two different superpotentials as well.

Additional equivalence transformations are presented

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• Equivalence groupoids and group classification of multidimensional nonlinear Schrödinger equations
• 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
• Symmetries of the Schroedinger-Pauli equation for neutral particles
• 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