We prove that the two dimensional free magnetic Schrödinger operator, with a fixed constant magnetic field and Dirichlet boundary conditions on a planar domain with a given area, attains its smallest possible eigenvalue if the domain is a disk. This generalizes the classical Faber-Krahn inequality for magnetic fields. The result is used to determine the low energy asymptotic behaviour of the integrated density of states of the magnetic Schrödinger operator with Poissonian random potential.
Heinz Siedentop