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Journal Articles Journal of Functional Analysis Year : 2018

SCHRODINGER OPERATORS WITH NEGATIVE POTENTIALS AND LANE-EMDEN DENSITIES

Abstract

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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Dates and versions

hal-01819321 , version 1 (20-06-2018)

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Lorenzo Brasco, Giovanni Franzina, Berardo Ruffini. SCHRODINGER OPERATORS WITH NEGATIVE POTENTIALS AND LANE-EMDEN DENSITIES. Journal of Functional Analysis, 2018. ⟨hal-01819321⟩
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