Symmetry and Localization for Magnetic Schrödinger Operators: Landau Levels, Gabor Frames and All That

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.