A weak Gordon type condition for absence of eigenvalues of one-dimensional Schr\"odinger operators

TL;DR

This paper introduces an improved criterion based on weak local periodicity for determining the absence of eigenvalues in one-dimensional Schrödinger operators with complex measure potentials, providing sharp bounds and applications to quasiperiodic measures.

Contribution

It presents a novel weak Gordon type condition that enhances existing criteria for eigenvalue absence in complex measure potentials.

Findings

Established a new criterion for eigenvalue absence involving weak local periodicity.

Derived sharp quantitative bounds on eigenvalues.

Applied the criterion to quasiperiodic measure potentials.

Abstract

We study one-dimensional Schr\"odinger operators with complex measures as potentials and present an improved criterion for absence of eigenvalues which involves a weak local periodicity condition. The criterion leads to sharp quantitative bounds on the eigenvalues. We apply our result to quasiperiodic measures as potentials.