Vacuum polarization in a strong field

We study the vacuum charge density induced by a deep square well potential with special emphasis on the evaluation of the contour integration of the Green function over the continuum states. We investigate the relationship of the Levinson theorem for Dirac particles and the vacuum charges by defining a relative phase shift between the positive and negative frequency modes, and clarify a role of bound states in vacuum polarization. The vacuum charge densities for various depths of a square well potential are calculated nonperturbatively from the imaginary part of the Green function that can be defined as a physical density of states. We present the results of the contour integration of the Green function for the vacuum charges and the Levinson theorem obtained from a numerical study. We compare our results with the vacuum charge density calculated by adopting Wichmann-Kroll formalism and discuss the role of the free particle-like Green function inside the potential well in the contour integration