Spectrum of the Schrِdinger Operator on a Complete Manifold

The spectrum and essential spectrum or the Schrödinger operator Δ + V on a complete manifold are studied. As applications, we determine the index of the catenoid of any dimension and the essential spectrum for several minimal submanifolds in the Euclidean space of the Jacobi operator arising from the second variation for the volume of minimal submanifolds.