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

Mikhailets, Vladimir; Molyboga, Volodymyr

Methods Funct. Anal. Topol. 20, No. 4, 321-327 (2014).

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.

Reviewer: Anatoly N. Kochubei (Kyïv)

MSC:

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

Keywords:

Hill-Schrödinger operators; distributional potentials; spectral gaps; Avakumovich’s class