  We consider Schr\"odinger operators on the line with potentials that are
periodic with respect to the coordinate variable and real analytic with respect
to the energy variable. We prove that if the imaginary part of the potential is
bounded in the right half-plane, then the high energy spectrum is real, and the
corresponding asymptotics are determined. Moreover, the Dirichlet and Neumann
problems are considered. These results are used to analyze the good Boussinesq
equation.
