Schr\"odinger operators with negative potentials and Lane-Emden densities

We consider the Schr"odinger operator $-\mathrm{\Delta }+V$ for negative potentials $V$, on open sets with positive first eigenvalue of the Dirichlet-Laplacian. We show that the spectrum of $-\mathrm{\Delta }+V$ is positive, provided that $V$ is greater than a negative multiple of the logarithmic gradient of the solution to the Lane-Emden equation $-\mathrm{\Delta }u=u^{q-1}$ (for some $1\le q<2$). In this case, the ground state energy of $-\mathrm{\Delta }+V$ 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.