A fresh look at the Peierls-Onsager substitution
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.
Keywords
Cite
@article{arxiv.2601.16745,
title = {A fresh look at the Peierls-Onsager substitution},
author = {Horia D. Cornean and Bernard Helffer and Radu Purice},
journal= {arXiv preprint arXiv:2601.16745},
year = {2026}
}
Comments
This paper is a revised version of our preprint "A rigorous Peierls-Onsager effective dynamics for semimetals in long-range magnetic fields" (arXiv 2411.14171); the main results are the same but the proof strategy has been simplified putting more emphasise on Parseval frames. arXiv admin note: text overlap with arXiv:2411.14171