On absolute continuity of the spectrum of periodic Schr\"odinger operators

TL;DR

This paper establishes a new condition on real periodic potentials ensuring the associated Schrödinger operator has a purely absolutely continuous spectrum, using resolvent estimates, spectral projections, and oscillatory integral techniques.

Contribution

It introduces a novel criterion for absolute continuity of the spectrum of periodic Schrödinger operators based on advanced analytical estimates.

Findings

Spectrum is purely absolutely continuous under the new potential condition

Resolved spectral projection estimates in weighted L^2 spaces

Developed oscillatory integral estimates for spectral analysis

Abstract

In this paper we find a new condition on a real periodic potential for which the self-adjoint Schr\"odinger operator may be defined by a quadratic form and the spectrum of the operator is purely absolutely continuous. This is based on resolvent estimates and spectral projection estimates in weighted $L^2$ spaces on the torus, and an oscillatory integral theorem.