We consider Schrodinger 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.

中文翻译：

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具有分析依赖于能量的势能的希尔算子

我们考虑在线上的薛定谔算子，其势对于坐标变量是周期性的，对于能量变量是实解析的。我们证明，如果势的虚部有界于右半平面，则高能谱是实数，相应的渐近线就确定了。此外，还考虑了 Dirichlet 和 Neumann 问题。这些结果用于分析良好的 Boussinesq 方程。