A fresh look at the Peierls-Onsager substitution

TL;DR

This paper extends the Peierls-Onsager substitution to more general magnetic fields and operators, providing a rigorous framework for analyzing spectral properties and time evolution in complex periodic systems.

Contribution

It introduces a generalized Peierls-Onsager substitution applicable to long-range magnetic fields and broad classes of operators, removing previous limitations.

Findings

Extended substitution to long-range magnetic fields

Proved existence of approximate time evolution with error bounds

Covered a large class of periodic pseudo-differential operators

Abstract

We formulate a general version of the Peierls-Onsager substitution for a finite family of Bloch eigenvalues under a local spectral gap hypothesis, via strongly localized tight-frames and magnetic matrices. This extends the existing results to long-range magnetic fields without any slow-variation hypothesis and without any triviality assumption for the associated Bloch sub-bundle. Moreover, our results cover a large class of periodic, elliptic pseudo-differential operators. We also prove the existence of an approximate time evolution for initial states supported inside the range of the isolated Bloch family, with a precise error control.