Pré-Publication, Document De Travail Année : 2017

Schrodinger operators with negative potentials and Lane-Emden densities

Résumé

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.

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hal-01510816 , version 1 (19-04-2017)

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  • HAL Id : hal-01510816 , version 1

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Lorenzo Brasco, Giovanni Franzina, Berardo Ruffini. Schrodinger operators with negative potentials and Lane-Emden densities. 2017. ⟨hal-01510816⟩
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