Rayleigh-type isoperimetric inequality with a homogeneous magnetic field

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 a Poissonian random potential.