Schrödinger operators with negative potentials and Lane–Emden densities

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–Emde...

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Bibliographic Details
Published in:Journal of functional analysis 2018-03, Vol.274 (6), p.1825-1863
Main Authors: Brasco, Lorenzo, Franzina, Giovanni, Ruffini, Berardo
Format: Article
Language:English
Subjects:
Analysis of PDEs
Ground state energy
Hardy inequalities
Lane–Emden equation
Mathematics
Schrödinger operators
Spectral Theory
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Summary: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
ISSN:0022-1236
1096-0783
DOI:10.1016/j.jfa.2017.10.005