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. Acta47, 450) is indicated. A constructive test of (non)equivalence of a PDM system to a constant mass system is proposed.

The Maximal "Kinematical" Invariance Group for an Arbitrary Potential Revised

“…Its terms are products of functions of different independent variables which makes it possible to make the effective separation of these variables and reduce the problem to solution of systems of ordinary differential equations for time dependent functions α, ν a and f . Then the corresponding potentials are easily calculated integrating equations (12) with found functions of t.…”

“…In any case, Charles Boyer was the first who made the group classification of 3d Schrödinger equations with arbitrary potentials. Moreover, he deduced the determining equations (12) for arbitrary number of spatial variables. Up to minor misprints, the list of non-equivalent symmetries presented by him is correct but incomplete, and I appreciate the chance to make a small addition to these well known results.…”

“…Thus to classify point Lie symmetries of equation (1) it is necessary to find all nonequivalent solutions of equation (12). The evident equivalence transformations for (1), i.e., transformations which keep the generic form of this equation but can change the potential, are shifts, rotations and scalings of independent variables.…”

“…Just this case is missing in Boyer classification. Solving such defined class of equations (12) we obtain potentials presented in Items 1-5 of Table 1. In Item 4 we set κ = 0 since this parameter can be reduced to zero using mapping inverse to (24).…”

On the Solution of the Schrödinger Equation with Position-Dependent Mass

“…The equation can be considered with time and position-dependent mass. In this case, Schrödinger equation is written as Lψ = i ∂ ∂t − H k − V ψ = 0 [49]. The fact that the equation depends on both position-dependent mass and time makes its solution even more difficult.…”