Graded colour Lie superalgebras for solving L\'evy-Leblond equations

TL;DR

This paper explores advanced graded colour Lie superalgebras associated with the Lévy-Leblond equation, extending previous symmetry algebra results and demonstrating their utility in solving and analyzing the equation with various potentials.

Contribution

It constructs higher-graded colour Lie superalgebras for the Lévy-Leblond equation and shows their application in solution construction and spectrum computation.

Findings

Higher-graded colour Lie superalgebras exist for the Lévy-Leblond equation.

These algebras help in constructing solutions for the free equation.

Ladder operators generate graded superalgebras used to compute spectra.

Abstract

The L\'evy-Leblond equation with free potential admits a symmetry algebra that is a $ \mathbb{Z}_2\times\mathbb{Z}_2 $-graded colour Lie superalgebra (see arXiv:1609.08224). We extend this result in two directions by considering a time-independent version of the L\'evy-Leblond equation. First, we construct a $ \mathbb{Z}_2^3 $-graded colour Lie superalgebra containing operators that leave the eigenspaces invariant and demonstrate the utility of this algebra in constructing general solutions for the free equation. Second, we find that the ladder operators for the harmonic oscillator generate a $ \mathbb{Z}_2\times\mathbb{Z}_2 $-graded colour Lie superalgebra and we use the operators from this algebra to compute the spectrum. These results illustrate two points: the L\'evy-Leblond equation admits colour Lie superalgebras with gradings higher than $ \mathbb{Z}_2\times\mathbb{Z}_2 $ and colour Lie superalgebras appear for potentials besides the free potential.