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

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 selfadjoint 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.