Growth of the eigensolutions of Laplacians on Riemannian manifolds I: construction of energy function

In this paper, we consider the eigen-solutions of $-Delta u+ Vu=lambda u$, where $Delta$ is the Laplacian on a non-compact complete Riemannian manifold. We develop Kato's methods on manifold and establish the growth of the eigen-solutions 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_{
m rad}(r)= -1+frac{o(1)}{r}$.

Growth of the eigensolutions of Laplacians on Riemannian manifolds I: construction of energy function Institutional Repository Document