We consider the Schrödinger operator −∆ + V for negative potentials V , on open sets with positive first eigenvalue of the Dirichlet-Laplacian. We show that the spectrum of −∆ + V is positive, provided that V is greater than a negative multiple of the logarithmic gradient of the solution to the Lane-Emden equation −∆u = u q−1 (for some 1 ≤ q < 2). In this case, the ground state energy of −∆ + 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.