Schr\"odinger Operators with Periodic Singular Potentials

TL;DR

This paper extends the theory of Schr"odinger operators to include singular periodic potentials, establishing their selfadjointness, spectral properties, and continuity with respect to potential variations.

Contribution

It introduces a framework for defining and analyzing Schr"odinger operators with singular periodic potentials in the space W^{-1}_{2,unif}(R), generalizing classical spectral results.

Findings

Operators are selfadjoint and bounded below.

Spectrum exhibits band and gap structure.

Spectral measures depend continuously on the potential.

Abstract

We show that formal Schr\"odinger operators with singular potentials from the space W^{-1}_{2,unif}(R) can be naturally defined to give selfadjoint and bounded below operators, which depend continuously in the uniform resolvent sense on the potential in the W^{-1}_{2,unif}(R)-norm. In the case of periodic singular potentials we also establish pure absolute continuity and a band and gap structure of the spectrum thus generalising some classical results for singular potentials of one-dimensional quasicrystal theory.