Spectral gaps of the Hill-Schrödinger operators with distributional potentials. (English) Zbl 1324.47080

The authors study the Schrödinger operators whose potentials are periodic distributions from \(H^{-1}(\mathbb T,\mathbb R)\). More specifically, the potentials belong to the space \(H^\omega \subset H^{-1}(\mathbb T,\mathbb R)\) which coincides with the Hörmander space \(H^\omega (\mathbb T,\mathbb R)\) where the weight has the form \(\omega (\sqrt{1+\xi^2})\), and \(\omega\) belongs to certain subclasses of Avakumovich’s class \(OR\) of regularly varying functions. Here, \(\omega\) need not be monotonic; if \(\omega\) is the power function, the above space coincides with the Sobolev space.
The main result is a complete description of sequences arising as lengths of spectral gaps of operators of the above type.


47E05 General theory of ordinary differential operators
47A75 Eigenvalue problems for linear operators
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
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