The Landau Hamiltonian describes a quantum particle confined to a plane, interacting with a uniform magnetic field orthogonal to the plane. The spectrum of this operator was given independently by Fock (1928) and Landau (1930). It consists of an arithmetic progression of eigenvalues (so-called Landau levels), each with infinite multiplicity.

In the last decades many people have studied the spectrum of different perturbations of the Landau Hamiltonian. In this talk we will discuss perturbations by an electric field, by a magnetic field, and by a change of domain, focusing on the latter case.