Superintegrable and shape invariant systems with position dependent mass

Abstract

Second order integrals of motion for 3d quantum mechanical systems with position dependent masses (PDM) are classified. Namely, all PDM systems are specified which, in addition to their rotation invariance, admit at least one second order integral of motion. All such systems appear to be also shape invariant and exactly solvable. Moreover, some of them possess the property of double shape invariance and can be solved using two different superpotentials. Among them there are systems with double shape invariance which present nice bridges between the Coulomb and isotropic oscillator systems.A simple algorithm for calculation the discrete spectrum and the corresponding state vectors for the considered PDM systems is presented and applied to solve five of the found systems.

Group classification of Schrödinger equations with position dependent mass

“…Equation (11) defines the conformal Killing vector whose generic form is given by the following formula:…”

Mapping of two-dimensional Schrödinger equation under the point transformation

“…However, keeping in mind possible physical applications, it is desirable to extend the class of solvable models onto cases of higher dimensionality of space. Some results in this direction were already obtained in the literature [32], [18], [33], [34], [20], [35], [21], [22], [13], [36], but, as a rule, the systems with separation of variables were considered.…”

SUSY method for the three-dimensional Schrödinger equation with effective mass

“…These models (also called as models with the effective mass) appear in many branches of physics very distinct from each other: quantum liquids [59], nuclear physics [61], quantum wires and dots [60], physics of semiconductors [62]- [65], and some others. The SUSY QM approach was used successfully to investigate the wide class of such models -see for example [66], [67], [68], [69], [70]. Although most of the papers on the effective mass were focused on one-dimensional systems, there were few two-dimensional [66], [67], [68], three-dimensional [69] and (2 + 1)-nonstationary [71] exclusions.…”

“…The SUSY QM approach was used successfully to investigate the wide class of such models -see for example [66], [67], [68], [69], [70]. Although most of the papers on the effective mass were focused on one-dimensional systems, there were few two-dimensional [66], [67], [68], three-dimensional [69] and (2 + 1)-nonstationary [71] exclusions. In particular, in [68] the general solution of SUSY intertwining relations of first order in derivatives was built for two-dimensional Schrödinger operators with position-dependent (effective) mass in terms of four arbitrary functions.…”

“…The paper [66], [67] mainly deals with two-dimensional systems but with a mass function depending only on one coordinate. The paper [69] was devoted to the three-dimensional rotationally invariant PDM models starting from the condition of superintegrability. The complete set of such systems was built and classified.…”