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


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.


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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


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

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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