Measures of Fermi surfaces and absence of singular continuous spectrum for magnetic Schroedinger operators

TL;DR

This paper investigates measures related to Fermi surfaces in periodic operators, demonstrating the absence of singular continuous spectrum in various magnetic Schrödinger operators and related models.

Contribution

It introduces new measure-theoretic tools for Fermi surfaces and proves the absence of singular continuous spectrum for a broad class of magnetic Schrödinger operators.

Findings

Absence of singular continuous spectrum in periodic magnetic Schrödinger operators

Measures connected to Fermi surfaces have specific measure-theoretic properties

Results apply to Schrödinger, Pauli, and Dirac-type operators with periodic magnetic fields

Abstract

Fermi surfaces are basic objects in solid state physics and in the spectral theory of periodic operators. We define several measures connected to Fermi surfaces and study their measure theoretic properties. From this we get absence of singular continuous spectrum and of singular continuous components in the density of states for symmetric periodic elliptic differential operators acting on vector bundles. This includes Schroedinger operators with periodic magnetic field and rational flux, as well as the corresponding Pauli and Dirac-type operators.