The topological Bloch-Floquet transform and some applications

TL;DR

This paper introduces a generalized Bloch-Floquet transform that decomposes Schrödinger operators with symmetries into a Hilbert bundle, providing a foundation for analyzing topological invariants in quantum systems.

Contribution

It develops a new topological Bloch-Floquet transform that yields a Hilbert bundle structure, enabling rigorous analysis of topological quantum numbers in symmetric Schrödinger operators.

Findings

Constructive proof of the transform's properties

Explicit description of the fiber structure

Framework for analyzing topological invariants

Abstract

We investigate the relation between the symmetries of a Schr\"odinger operator and the related topological quantum numbers. We show that, under suitable assumptions on the symmetry algebra, a generalization of the Bloch-Floquet transform induces a direct integral decomposition of the algebra of observables. More relevantly, we prove that the generalized transform selects uniquely the set of "continuous sections" in the direct integral decomposition, thus yielding a Hilbert bundle. The proof is constructive and provides an explicit description of the fibers. The emerging geometric structure is a rigorous framework for a subsequent analysis of some topological invariants of the operator, to be developed elsewhere. Two running examples provide an Ariadne's thread through the paper. For the sake of completeness, we begin by reviewing two related classical theorems by von Neumann and Maurin.