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

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.