  We consider the Schr\"odinger operator $-\Delta+V$ for negative potentials
$V$, on open sets with positive first eigenvalue of the Dirichlet-Laplacian. We
show that the spectrum of $-\Delta+V$ is positive, provided that $V$ is greater
than a negative multiple of the logarithmic gradient of the solution to the
Lane-Emden equation $-\Delta u=u^{q-1}$ (for some $1\le q< 2$). In this case,
the ground state energy of $-\Delta+V$ is greater than the first eigenvalue of
the Dirichlet-Laplacian, up to an explicit multiplicative factor. This is
achieved by means of suitable Hardy-type inequalities, that we prove in this
paper.
