Schrodinger operators with slowly decaying Wigner--von Neumann type potentials

TL;DR

This paper studies Schrödinger operators with slowly decaying Wigner--von Neumann type potentials, proving spectral properties like absence of singular continuous spectrum and characterizing embedded eigenvalues.

Contribution

It establishes spectral results for a broad class of potentials including slowly decaying oscillatory types and constructs explicit examples with embedded eigenvalues.

Findings

Absence of singular continuous spectrum for these operators.

Embedded eigenvalues can only occur at finitely many explicit points.

Constructed examples with precise asymptotics for eigenfunctions.

Abstract

We consider Schr\"odinger operators with potentials satisfying a generalized bounded variation condition at infinity and an $L^p$ decay condition. This class of potentials includes slowly decaying Wigner--von Neumann type potentials $\sin(ax)/x^b$ with $b>0$. We prove absence of singular continuous spectrum and show that embedded eigenvalues in the continuous spectrum can only take values from an explicit finite set. Conversely, we construct examples where such embedded eigenvalues are present, with exact asymptotics for the corresponding eigensolutions.