Eigenvalues of the magnetic Dirichlet Laplacian with constant magnetic field on disks in the strong field limit

We consider the magnetic Dirichlet Laplacian with constant magnetic field on domains of finite measure. First, in the case of a disk, we prove that the eigenvalue branches with respect to the field strength behave asymptotically linear with an exponentially small remainder term as the field strength goes to infinity. We compute the asymptotic expression for this remainder term. Second, we show that for sufficiently large magnetic field strengths, the spectral bound corresponding to the P & oacute;lya conjecture for the non-magnetic Dirichlet Laplacian is violated up to a sharp excess factor which is independent of the domain.