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

Abstract

Quantum mechanical systems with position dependent masses (PDM) admitting four and more dimensional symmetry algebras are classified. Namely, all PDM systems are specified which, in addition to their invariance w.r.t. a three parametric Lie group, admit at least one second order integral of motion. The presented classification is partially extended to the more generic systems which admit one or two parametric Lie groups.

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

“…which is obviously correct in view of ( 5) and (5). However this identity makes it possible to reduce (19) to the following homogeneous system of linear algebraic equations for derivatives f a :…”

“…The number of linearly independent integrals of motion is more extended, but all of them can be found using the equivalence relations fixed in Section 4. In contrary, in paper [5] all linearly independent integrals of motion are represented.…”

“…Some elements of such classification can be found in paper [5] where the systems admitting the dilatation symmetry were studied. However, the classification presented in [5] was restricted to the integrals of motion which, up to potential terms, belong to the standard (non extended) enveloping algebra of c(3).…”

“…The fundamentals of quantum mechanics by definition include symmetries of its basic motion equations which have been discovered in famous papers [1], [2], [3] and [4]. We will not discuss these symmetries which are presented in our previous work [5] but restrict ourselves to note that there are new contemporary results in this very old field. And it is the case even for classical Lie symmetries [6,7,8].…”

“…The next step was to classify such systems which, in addition to the second order integrals of motion, are invariant with respect to three parametric Lie groups [5]. After that we plane to generalize this result to the case when the a priori requested invariance groups are two or at least one parametric.…”