Vacuum polarization by a magnetic flux of special rectangular form

We consider the ground state energy of a spinor field in the background of a square well shaped magnetic flux tube. We use the zeta-function regu-larization and express the ground state energy as an integral involving the Jost function of a two dimensional scattering problem. We perform the renormalization by subtracting the contributions from first several heat kernel coefficients. The ground state energy is presented as a convergent expression suited for numerical evaluation. We discuss corresponding numerical calculations. Using the uniform asymptotic expansion of the special functions entering the Jost function we are able to calculate higher order heat kernel coefficients. 1 Introduction Since the classical work [1] of H.B.G. Casimir, where the energy of vacuum polarized by two conducting planes was calculated, a number of similar problems was investigated for various external conditions (boundary conditions, background potentials etc.), see, for example, the recent review [2] or the books [3, 4]. No general rule for the dependence of the vacuum energy on the background properties has been found so far. In particular, it is unknown how to forecast the sign of the energy. An interesting problem is the calculation of the vacuum energy of a spinor field in the background of a magnetic field. This problem was first considered as early as in 1936 by W. Heisenberg and H. Euler [5] who were interested in the effective action in the background of a homogeneous magnetic field.