No eigenvectors embedded in the singular continuous spectrum of Schr\"odinger operators

TL;DR

This paper investigates the spectral properties of Schr"odinger operators with sparse potentials, providing conditions to exclude eigenvalues at the spectrum's edge and constructing examples with purely singular continuous spectrum.

Contribution

It offers a sufficient condition to prevent eigenvalues at the spectrum edge and constructs examples with singular continuous spectrum lacking eigenvalues or having a single negative eigenvalue.

Findings

Edge of singular continuous spectrum can be free of eigenvalues under certain conditions.

Examples of Schr"odinger operators with purely singular continuous spectrum and no eigenvalues.

Construction of operators with a single negative eigenvalue.

Abstract

It is known that the spectrum of Schr\"odinger operators with sparse potentials consists of singular continuous spectrum. We give a sufficient condition so that the edge of the singular continuous spectrum is not an eigenvalue and construct examples with singular continuous spectrum which have no eigenvalues and which have a single negative eigenvalue.