Integrable and superintegrable systems with spin in three-dimensional Euclidean space

TL;DR

This paper systematically identifies superintegrable quantum systems involving spin in three-dimensional space, revealing new exactly solvable models with explicit energy spectra and wave functions.

Contribution

It introduces new superintegrable Hamiltonians with spin, demonstrating their exact solvability and providing explicit solutions for bound states.

Findings

Several superintegrable systems with spin were identified.

Exact bound state energy formulas were derived.

Wave functions expressed in terms of Laguerre and Jacobi polynomials.

Abstract

A systematic search for superintegrable quantum Hamiltonians describing the interaction between two particles with spin 0 and 1/2, is performed. We restrict to integrals of motion that are first-order (matrix) polynomials in the components of linear momentum. Several such systems are found and for one non-trivial example we show how superintegrability leads to exact solvability: we obtain exact (nonperturbative) bound state energy formulas and exact expressions for the wave functions in terms of products of Laguerre and Jacobi polynomials.