  The ground state energy of a quantum field in the background of classical
field configurations is considered.
  The subject of the ground state energy in framework of the quantum field
theory is explained.
  The short review of calculation methods (generalized zeta function and heat
kernel expansion) and their mathematical foundations is given. We use the
zeta-functional regularization 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.
  The investigation for three models has been carried out: scalar quantum field
on the background of scalar string with rectangular shape, spinor vacuum
polarized by magnetic string of the similar shape and spinor vacuum interacting
with the Nielsen-Olesen vortex. Using the uniform asymptotic expansion of the
special functions entering the Jost function we are also able to calculate
higher order heat kernel coefficients.
  Several features of vacuum energy have been investigated numerically. We
discuss corresponding numerical results.
