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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

A. G. Nikitin

Abstract

3d quantum mechanical systems with position dependent masses (PDM) admitting at least one second order integral of motion and symmetries with respect to dilatation or shift transformations are classified. Twenty-seven such systems are specified and the completeness of the classification results is proved.

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

Abstract

3d quantum mechanical systems with position dependent masses (PDM) admitting at least one second order integral of motion and symmetries with respect to dilatation or shift transformations are classified. Twenty-seven such systems are specified and the completeness of the classification results is proved.
Paper Structure (30 sections, 162 equations)

This paper contains 30 sections, 162 equations.

Table of Contents

  1. Introduction
  2. Formulation of the problem and algorithm for its solution
  3. Basic definitions
  4. Determining equations
  5. Linearization of the determining equations
  6. The algorithm for searching of general solution of the determining equations
  7. PDM systems invariant with respect to dilatation transformations
  8. Equivalence relations
  9. Decoupling of the determining equations
  10. Symmetries with differential terms $Q_{(n)}$ (\ref{['Q1']}) independent on spatial variables
  11. Symmetries with differential terms $Q_{(n)}$ (\ref{['Q0']}) linear in spatial variables
  12. Admissible combinations of arbitrary parameters
  13. Exact solutions for the arbitrary elements
  14. Symmetries with differential terms $Q_{(n)}$ (\ref{['Q1']}) quadratic in independent variables
  15. Vector integrals of motion
  16. ...and 15 more sections