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

TL;DR

This paper investigates the growth behavior of eigen-solutions of Laplacians on non-compact Riemannian manifolds, establishing criteria for the absence of eigenvalues based on geometric and potential conditions.

Contribution

It extends Kato's methods to Riemannian manifolds and provides new criteria for eigenvalue absence, including a novel proof related to the essential spectrum.

Findings

Eigen-solutions grow at rates determined by elta r and V(x)

Criteria established for absence of eigenvalues in certain geometric conditions

New proof for absence of embedded eigenvalues in the essential spectrum

Abstract

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_{\rm rad}(r)= -1+\frac{o(1)}{r}$.