Duke Mathematical Journal

The spectral bound and principal eigenvalues of Schrödinger operators on Riemannian manifolds

El Maati Ouhabaz

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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.

Article information

Source
Duke Math. J. Volume 110, Number 1 (2001), 1-35.

Dates
First available: 18 June 2004

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1087574811

Mathematical Reviews number (MathSciNet)
MR1861087

Digital Object Identifier
doi:10.1215/S0012-7094-01-11011-9

Zentralblatt MATH identifier
1015.58008

Subjects
Primary: 58J50: Spectral problems; spectral geometry; scattering theory [See also 35Pxx]
Secondary: 35P15: Estimation of eigenvalues, upper and lower bounds 47F05: Partial differential operators [See also 35Pxx, 58Jxx] (should also be assigned at least one other classification number in section 47) 58J05: Elliptic equations on manifolds, general theory [See also 35-XX]

Citation

Ouhabaz, El Maati. The spectral bound and principal eigenvalues of Schrödinger operators on Riemannian manifolds. Duke Mathematical Journal 110 (2001), no. 1, 1--35. doi:10.1215/S0012-7094-01-11011-9. http://projecteuclid.org/euclid.dmj/1087574811.


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