  Our main goal in the present paper is to expand the known class of open
manifolds over which the $L^2$-spectrum of a general Dirac operator and its
square is maximal. To achieve this, we first find sufficient conditions on the
manifold so that the $L^p$-spectrum of the Dirac operator and its square is
independent of $p$ for $p\geq 1$. Using the $L^1$-spectrum, which is simpler to
compute, we generalize the class of manifolds over which the $L^p$-spectrum of
the Dirac operator is the real line for all $p$. We also show that by applying
the generalized Weyl criterion, we can find large classes of manifolds with
asymptotically nonnegative Ricci curvature, or which are asymptotically flat,
such that the $L^2$-spectrum of a general Dirac operator and its square is
maximal.
