Growth of the Eigensolutions of Laplacians on Riemannian Manifolds I: Construction of Energy Function

Abstract In this paper, we consider the eigensolutions of $-Delta u+ Vu=lambda u$, where $Delta $ is the Laplacian on a non-compact complete Riemannian manifold. We develop Katos methods on manifold and establish the growth of the eigensolutions as r goes to infinity based on the asymptotical behaviors of $Delta r$ and V (x), where r = r(x) is the distance function on the manifold. As applications, we prove several criteria of absence of eigenvalues of Laplacian, including a new proof of the absence of eigenvalues embedded into the essential spectra of free Laplacian if the radial curvature of the manifold satisfies $ K_{ extrm{rad}}(r)= -1+frac{o(1)}{r}$.

Growth of the Eigensolutions of Laplacians on Riemannian Manifolds I: Construction of Energy Function Academic Article