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.
Domaines
Mathématiques [math]
Origine : Fichiers produits par l'(les) auteur(s)
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