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1 October 2001 The spectral bound and principal eigenvalues of Schrödinger operators on Riemannian manifolds
El Maati Ouhabaz
Duke Math. J. 110(1): 1-35 (1 October 2001). DOI: 10.1215/S0012-7094-01-11011-9

Abstract

Given a complete Riemannian manifold $M$ and a Schrödinger operator $-\Delta+m$ acting on $L^p(M)$, we study two related problems on the spectrum of $-\Delta+m$. The first one concerns the positivity of the $L^2$-spectral lower bound $s(-\Delta+m)$. We prove that if $M$ satisfies $L^2$-Poincaré inequalities and a local doubling property, then $s(-\Delta+m)>0$, provided that $m$ satisfies the mean condition

$\inf\substack {p\in M}\frac {1}{|B(p, r)|}\int \sb{B(p,r )}m(x)dx>0$

for some $r>0$. We also show that this condition is necessary under some additional geometrical assumptions on $M$.

The second problem concerns the existence of an $L^p$-principal eigenvalue, that is, a constant $\lambda\geq 0$ such that the eigenvalue problem $\Delta u=\lambda mu$ and equation above] has a positive solution $u\in L^p(M)$. We give conditions in terms of the growth of the potential $m$ and the geometry of the manifold $M$ which imply the existence of $L^p$-principal eigenvalues.

Finally, we show other results in the cases of recurrent and compact manifolds.

Citation

Download Citation

El Maati Ouhabaz. "The spectral bound and principal eigenvalues of Schrödinger operators on Riemannian manifolds." Duke Math. J. 110 (1) 1 - 35, 1 October 2001. https://doi.org/10.1215/S0012-7094-01-11011-9

Information

Published: 1 October 2001
First available in Project Euclid: 18 June 2004

zbMATH: 1015.58008
MathSciNet: MR1861087
Digital Object Identifier: 10.1215/S0012-7094-01-11011-9

Subjects:
Primary: 58J50
Secondary: 35P15 , 47F05 , 58J05

Rights: Copyright © 2001 Duke University Press

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Vol.110 • No. 1 • 1 October 2001
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