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

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 $K_{\mathrm{rad}}(r)=-1+\frac{o(1)}{r}$.

IMRN to appear

References in corpus (3)

• Noncompact complete Riemannian manifolds with dense eigenvalues embedded in the essential spectrum of the Laplacian
• Tosio Kato's Work on Non--Relativistic Quantum Mechanics
• Noncompact complete Riemannian manifolds with singular continuous spectrum embedded into the essential spectrum of the Laplacian, I. The hyperbolic case

Cited by in corpus (2)

• Noncompact complete Riemannian manifolds with dense eigenvalues embedded in the essential spectrum of the Laplacian
• Noncompact complete Riemannian manifolds with singular continuous spectrum embedded into the essential spectrum of the Laplacian, I. The hyperbolic case