# Symmetry and localization for magnetic Schroedinger operators: Landau levels, Gabor frames and all that

## TL;DR

This paper explores how broken time-reversal symmetry affects localization in Landau models, linking symmetries, Gabor frames, and the Balian-Low Theorem to quantum localization phenomena.

## Contribution

It provides a representation-independent proof of the Balian-Low Theorem applied to Landau eigenstates, connecting symmetry, localization, and Gabor analysis.

## Key findings

- Established the relation between symmetry breaking and localization.
- Applied an abstract Balian-Low Theorem to Landau eigenstates.
- Provided a novel, representation-independent proof of the Balian-Low Theorem.

## Abstract

We investigate the relation between broken time-reversal symmetry and localization of the electronic states, in the explicitly tractable case of the Landau model. We first review, for the reader's convenience, the symmetries of the Landau Hamiltonian and the relation of the latter with the Segal-Bargmann representation of Quantum Mechanics. We then study the localization properties of the Landau eigenstates by applying an abstract version of the Balian-Low Theorem to the operators corresponding to the coordinates of the centre of the cyclotron orbit in the classical theory. Our proof of the Balian-Low Theorem, although based on Battle's main argument, has the advantage of being representation-independent.