  Random Schroedinger operators with imaginary vector potentials are studied in
dimension one. These operators are non-Hermitian and their spectra lie in the
complex plane. We consider the eigenvalue problem on finite intervals of length
n with periodic boundary conditions and describe the limit eigenvalue
distribution when n goes to infinity. We prove that this limit distribution is
supported by curves in the complex plane. We also obtain equations for these
curves and for the corresponding eigenvalue density in terms of the Lyapunov
exponent and the integrated density of states of a "reference" symmetric
eigenvalue problem. In contrast to these results, the spectrum of the limit
operator in l^2(Z) is a two dimensional set which is not approximated by the
spectra of the finite-interval operators.
