  In this work a spectral theory for 2-dimensional, simply periodic,
complex-valued solutions u of the sinh-Gordon equation is developed. Spectral
data for such solutions are defined (following Hitchin and Bobenko) and the
space of spectral data is described by an asymptotic characterization. Using
methods of asymptotic estimates, the inverse problem for the spectral data is
solved along a line, i.e. the solution u is reconstructed on a line from the
spectral data. Finally a Jacobi variety and Abel map for the spectral curve is
constructed; they are used to describe the change of the spectral data under
translation of the solution u.
