Group classification of Schrödinger equations with position dependent mass

Abstract

Maximal kinematical invariance groups of $2d$ Schrödinger equation with a position dependent mass and arbitrary potential are classified. It is demonstrated that there exist seven classes of such equations possessing non-equivalent continuous symmetry group. Three of these classes include arbitrary functions while the remaining ones are defined up to arbitrary parameters. In particular, for the case of a constant mass the class missing in the Boyer classification (Boyer C P 1974 Helv. Phys. Acta{\bf 47}, 450) is indicated. A constructive test of (non)equivalence of a PDM system to a constant mass system is proposed.

arXiv commentsignes for parameters αand γare corrected

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Cited by in corpus (7)

• Group classification of (1+3)-dimensional Schrödinger equations with position dependent mass
• Symmetries of Schroedinger equation with scalar and vector potentials
• The maximal 'kinematical' invariance group for an arbitrary potential revised
• 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