  According to Anderson's orthogonality catastrophe, the overlap of the
$N$-particle ground states of a free Fermi gas with and without an (electric)
potential decays in the thermodynamic limit. For the finite one-dimensional
system various boundary conditions are employed. Unlike the usual setup the
perturbation is introduced by a magnetic (vector) potential. Although such a
magnetic field can be gauged away in one spatial dimension there is a
significant and interesting effect on the overlap caused by the phases. We
study the leading asymptotics of the overlap of the two ground states and the
two-term asymptotics of the difference of the ground-state energies. In the
case of periodic boundary conditions our main result on the overlap is based
upon a well-known asymptotic expansion by Fisher and Hartwig on Toeplitz
determinants with a discontinuous symbol. In the case of Dirichlet boundary
conditions no such result is known to us and we only provide an upper bound on
the overlap, presumably of the right asymptotic order.
