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We consider Schr\"odinger operators on the line with potentials that areperiodic with respect to the coordinate variable and real analytic with respectto the energy variable. We prove that if the imaginary part of the potential isbounded in the right half-plane, then the high energy spectrum is real, and thecorresponding asymptotics are determined. Moreover, the Dirichlet and Neumannproblems are considered. These results are used to analyze the good Boussinesqequation.

Hill's operators with the potentials analytically dependent on energy