Uniqueness of solutions, radiation conditions, and complexity of the metric at infinity

TL;DR

This paper proves the uniqueness of solutions to eigenvalue equations with radiation conditions on Riemannian manifolds and explores the complexity of metrics affecting spectral properties of Laplacians.

Contribution

It establishes a uniqueness theorem for drift Laplacian eigenvalue solutions under radiation conditions and analyzes metric growth affecting spectral properties.

Findings

Uniqueness theorem for eigenvalue solutions with radiation conditions

Identification of manifolds with no embedded eigenvalues

Analysis of metric growth and spectral properties

Abstract

The purpose of this paper is to prove the uniqueness theorem of solutions of eigenvalue equations on one end of Riemannian manifolds for drift Laplacians, including the standard Laplacian as a special case; we shall impose "a sort of radiation condition" at infinity on solutions. We shall also provide several Riemannian manifolds whose Laplacians satisfy the absence of embedded eigenvalues and besides the absolutely continuity, although growth orders of their metrics on ends are very complicated.