Absolutely continuous spectrum of a typical Schr\"odinger operator with a slowly decaying potential

TL;DR

This paper proves that for a class of multi-dimensional Schrödinger operators with slowly decaying potentials satisfying a specific integral condition, the absolutely continuous spectrum covers the entire positive real axis for almost all coupling constants.

Contribution

It establishes the presence of absolutely continuous spectrum for a broad class of Schrödinger operators with slowly decaying potentials, extending previous results to more general decay conditions.

Findings

Absolutely continuous spectrum covers [0,∞) for almost every t.

Potential decay satisfies a specific integral condition.

Results apply to multi-dimensional Schrödinger operators.

Abstract

We consider a family of multi-dimensional Schr\"odinger operators $-\Delta+t V$ with a real $t$. The potential $V$ in our model decays at infinity in a special way, so that it satisfies a certain integral condition. We prove that the absolutely continuous spectrum of this operator covers the interval $[0,\infty)$ for almost every $t$.