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

Given a complete Riemannian manifold M and a Schrodinger 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-Poincare inequalities and a local doubling property, then s (-Delta + m) > 0, provided that m. satis fies the mean condition inf(p is an element ofM) 1/\B(p,r)\ integral (B(p,r))m(x)dx > 0 for some r > 0. We also show that this condition is necessary under some ad ditional geometrical assumptions on M. The second problem concerns the existence of an L-p-principal eigenvalue, t hat is, a constant lambda greater than or equal to 0 such that the eigenval ue problem Deltau = lambda mu has a positive solution u is an element of 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 eige nvalues. Finally, we show other results in the cases of recurrent and compact manifo lds.