Foliation of the space of periodic boundary-value problems by hypersurfaces corresponding to fixed lengths of the $n$th spectral lacuna

The space of one-dimensional stationary Schrödinger equations with a real-valued periodic potential and periodic boundary conditions is considered. An analytic and topological description of its foliation by hypersurfaces defined by the condition that the $n$th spectral lacuna has fixed length is given. The case when a lacuna degenerates into a point gives the Schwarzian derivative and the Arnold manifold. In the nondegenerate case, the linking number of the loop formed by potentials with shifted argument and an Arnold manifold is calculated.
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