HEAT KERNEL-EXPANSION FOR FERMIONIC BILLIARDS IN AN EXTERNAL MAGNETIC-FIELD

Using Seeley's heat kernel expansion, the authors compute the asymptotic density of states of the Dirac operator coupled to a magnetic field on a two-dimensional manifold with boundary ('fermionic billiard'). Local boundary conditions compatible with vector current conservation depend on a free parameter alpha . It is shown that the perimeter correction identically vanishes for alpha =0. In that case, the next-order constant term is found to be proportional to the Euler characteristic of the manifold. These results are independent of the external magnetic field and of the shape of the billiard, provided the boundary is sufficiently smooth. For the flat circular billiard, the constant term is found to be -1/2, in agreement with a numerical result of Berry and Mondragon.