Foliation of the space of periodic boundary-value problems by hypersurfaces corresponding to fixed lengths of the \(n\)th spectral lacuna (Q2821892)

The paper provides an analytic description of the space of second-order self-adjoint periodic boundary value problems (actually, considering the Sturm-Liouville equation on an interval, with periodic boundary conditions, and assuming the potential to be a continuous periodic function) in terms of hypersurfaces consisting of boundary-value problems, in which the \(n\)th spectral gap has some positive length supplemented by the subset characterized by the degeneration of the spectral gap. For this, a new parametrization is used, which is based on an ordered pair of eigenfunctions generated by the same differential operators and having the same oscillation. A complete description of the topology of this foliation is given. Particular attention is given to finding the linking numbers of the loops formed by shifts (preserving particular eigenvalues) and reflections of the argument in the boundary-value problem and the subset characterized by zero spectral gap.