We consider the Schrödinger operator −Δ + V for negative potentials V , on open sets with positive first eigenvalue of the Dirichlet–Laplacian. We show that the spectrum of −Δ+V is positive, provided that V is greater than a negative multiple of the logarithmic gradient of the solution to the Lane–Emden equation −Δu = uq−1 (for some 1 ≤ q < 2). In this case, the ground state energy of −Δ + 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.
Schrodinger operators with negative potentials and Lane-Emden densities / Brasco L; Franzina G; Ruffini B. - In: JOURNAL OF FUNCTIONAL ANALYSIS. - ISSN 0022-1236. - STAMPA. - 274:6(2018), pp. 1825-1863. [10.1016/j.jfa.2017.10.005]
Schrodinger operators with negative potentials and Lane-Emden densities
Ruffini B
2018
Abstract
We consider the Schrödinger operator −Δ + V for negative potentials V , on open sets with positive first eigenvalue of the Dirichlet–Laplacian. We show that the spectrum of −Δ+V is positive, provided that V is greater than a negative multiple of the logarithmic gradient of the solution to the Lane–Emden equation −Δu = uq−1 (for some 1 ≤ q < 2). In this case, the ground state energy of −Δ + 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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
