  For the magnetic Laplacian on a bounded planar domain, imposing Neumann
boundary conditions produces eigenvalues below the lowest Landau level. If the
domain has two boundary components and one imposes a Neumann condition on one
component and a Dirichlet condition on the other, one gets fewer such
eigenvalues than when imposing Neumann boundary conditions on the two
components. We quantify this observation for two models: the strip and the
annulus. In both models one can separate variables and deal with a family of
fiber operators, thereby reducing the problem to counting band functions, the
eigenvalues of the fiber operators.
