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