Geometric relative Hardy inequalities and the discrete spectrum of Schrödinger operators on manifolds (Q368542)

Let \((M,g)\) be a complete non-compact Riemannian manifold of dimension \(n\). Let \(-\Delta_g\) be the associated Laplace-Beltrami operator. The authors study the finiteness and the infiniteness of the discrete spectrum of the Schrödingier operator \(-\Delta_g+V\). The authors use the Bochner technique and some geometric identities to obtain a sharp Hardy inequality if the manifold in question has a pole, i.e. if one of the ends of \(M\) has a compact connected smooth boundary such that the outward normal exponential map is a diffeomorphism under certain geometric conditions.