On the eigenvalue problem for arbitrary odd elements of the Lie superalgebra gl(1|n) and applications

TL;DR

This paper solves the eigenvalue problem for arbitrary odd elements in the Lie superalgebra gl(1|n) within unitary representations, using a basis decomposition approach, with applications in Wigner quantum mechanics.

Contribution

It provides a method to determine eigenvalues and eigenvectors for odd elements of gl(1|n) in any unitary irreducible representation, including explicit solutions for Fock and ladder representations.

Findings

Eigenvalues are obtained via decomposition with respect to gl(1|1) + gl(n-1).

Eigenvectors are explicitly constructed using properties of the Gel'fand-Zetlin basis.

Solutions are illustrated for specific classes of representations.

Abstract

In a Wigner quantum mechanical model, with a solution in terms of the Lie superalgebra gl(1|n), one is faced with determining the eigenvalues and eigenvectors for an arbitrary self-adjoint odd element of gl(1|n) in any unitary irreducible representation W. We show that the eigenvalue problem can be solved by the decomposition of W with respect to the branching gl(1|n) --> gl(1|1) + gl(n-1). The eigenvector problem is much harder, since the Gel'fand-Zetlin basis of W is involved, and the explicit actions of gl(1|n) generators on this basis are fairly complicated. Using properties of the Gel'fand-Zetlin basis, we manage to present a solution for this problem as well. Our solution is illustrated for two special classes of unitary gl(1|n) representations: the so-called Fock representations and the ladder representations.