  We formulate a general sufficiency criterion for discreteness of the spectrum
of both supersymmmetric and non-su-persymmetric theories with a fermionic
contribution. This criterion allows an analysis of Hamiltonians in complete
form rather than just their semiclassical limits. In such a framework we
examine spectral properties of various (1+0) matrix models. We consider the BMN
model of M-theory compactified on a maximally supersymmetric pp-wave
background, different regularizations of the supermembrane with central charges
and a non-supersymmetric model comprising a bound state of N D2 with m D0.
While the first two examples have a purely discrete spectrum, the latter has a
continuous spectrum with a lower end given in terms of the monopole charge.
