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.