On the eigenvalue problem for arbitrary odd elements of the Lie superalgebra and applications

Abstract

In a Wigner quantum mechanical model, with a solution in terms of the Lie superalgebra , one is faced with determining the eigenvalues and eigenvectors for an arbitrary self-adjoint odd element of 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 . The eigenvector problem is much harder, since the Gel'fand–Zetlin basis of W is involved, and the explicit actions of 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 representations: the so-called Fock representations and the ladder representations.