The Peierls-Onsager Effective Hamiltonian in a complete gauge covariant setting: Description of the spectrum

TL;DR

This paper develops a gauge covariant effective Hamiltonian for periodic pseudodifferential operators in magnetic fields, accurately describing the spectrum without restrictions on the vector potential or adiabatic assumptions.

Contribution

It introduces a comprehensive gauge covariant framework for the effective Hamiltonian, extending previous methods to more general magnetic and spectral settings.

Findings

Constructed an effective Hamiltonian for spectral analysis.

Applicable to a broad class of periodic pseudodifferential Hamiltonians.

Operates without restrictions on vector potential or adiabaticity.

Abstract

Using the procedures in \cite{Bu} and \cite{GMS} and the magnetic pseudodifferential calculus we have developped in \cite{MP1,MPR1,IMP1,IMP2} we construct an effective Hamitonian that describes the spectrum in any compact subset of the real axis for a large class of periodic pseudodifferential Hamiltonians in a bounded smooth magnetic field, in a completely gauge covariant setting, without any restrictions on the vector potential and without any adiabaticity hypothesis.