*Submitted Paper*

**Inserted:** 13 apr 2017

**Last Updated:** 10 may 2017

**Pages:** 29

**Year:** 2017

**Abstract:**

We consider the Schrödinger 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.

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