Superintegrable quantum mechanical systems with position dependent masses invariant with respect to two parametric Lie groups

Abstract

Quantum mechanical
systems with position dependent masses (PDM) admitting two parametric Lie symmetry groups are classified. Namely,
all PDM systems are specified which, in addition to their invariance w.r.t. a two
parametric Lie group, admit at least one second order integral of motion. The presented classification is partially extended to the more generic systems which do not accept any Lie group.

Superintegrable and Scale-Invariant Quantum Systems with Position-Dependent Mass

“…We will not discuss the determining equations for symmetries (139) and (140) but mention that both of them can be solved exactly also. However, for symmetry (139) the related solutions are functions of x 3 only, but for symmetry (140) we obtain the mass function which depends only on r. Such systems are out of the scop of the present paper since they were discussed in papers [35] and [39].…”

“…The systems admitting two-parametric Lie groups and second order integrals of motion have been classified in [39]. We again find a number of new systems in addition to the known maximally superintegrable ones.…”

“…Equations ( 9) , (10) and (11) give the necessary and sufficient conditions for commutativity of operators H (2) and Q (7) [39].…”

“…5 Identities in the extended enveloping algebra of c (3) It was noted in [39] that integrals of motion (7) where µ ab Killing tensors, i.e., solutions of equations can be represented as bilinear combinations of the basic elements of algebra c(3) (21) added by the special term with µ ab = δ ab g(x) and potential term η. In other words they admit the following representation:…”

“…However, our strategy is to solve it step by step for the systems admitting three, two and one parametric Lie groups, and, finally, for the systems which do not admit any Lie symmetry. The first two steps have been already done in papers [35] and [39]. The third step is the subject of the current paper.…”