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arXiv · 1709.03816

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

Lorenzo Brasco·Giovanni Franzina·Berardo Ruffini
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Abstract

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.

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Lorenzo Brasco, Giovanni Franzina, Berardo Ruffini. 2017-09-12. Schr\"odinger operators with negative potentials and Lane-Emden densities. https://arxiv.org/abs/1709.03816

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