The radial curvature of an end that makes eigenvalues vanish in the essential spectrum II

TL;DR

This paper investigates how the radial curvature decay of an end influences the spectral properties of the Laplacian, specifically showing that under certain conditions, eigenvalues vanish from the essential spectrum.

Contribution

It extends previous work by deriving growth estimates for solutions to the eigenvalue equation under quadratic-decay conditions, demonstrating the absence of eigenvalues.

Findings

Eigenvalues vanish in the essential spectrum under quadratic-decay conditions

Growth estimates for solutions to the eigenvalue equation are established

The results generalize previous spectral analysis of manifolds with ends

Abstract

Under the quadratic-decay-conditions of the radial curvatures of an end, we shall derive growth estimates of solutions to the eigenvalue equation and show the absence of eigenvalues.