  Let P be a linear, second order, elliptic operator satisfying a Hardy
inequality with potential W (i.e. $P-W\geq0$) and best constant $\alpha$. We
give conditions so that the spectrum of $W^{-1}P$ is $[\alpha,\infty)$. We
apply this to several well-known Hardy inequalities: (improved) Hardy
inequalities on a bounded convex domain with potential involving the distance
to the boundary, and Hardy inequalities for minimal submanifolds of the
Euclidean space.
