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# Spectral gaps of one-dimensional Schrödinger operators with singular periodic potentials

### Abstract

The behavior of the lengths of spectral gaps $\{\gamma_{n}(q)\}_{n=1}^{\infty}$ of the Hill-Schrödinger operators $$S(q)u=-u''+q(x)u,\quad u\in \mathrm{Dom}\left(S(q) \right),$$ with real-valued 1-periodic distributional potentials $q(x)\in H_{1\mbox{-}{\operatorname{per}}}^{-1}(\mathbb{R})$ is studied. We show that they exhibit the same behavior as the Fourier coefficients $\{\widehat{q}(n)\}_{n=-\infty}^{\infty}$ of the potentials $q(x)$ with respect to the weighted sequence spaces $h^{s,\varphi}$, $s>-1$, $\varphi\in \mathrm{SV}$. The case $q(x)\in L_{1\mbox{-}{\operatorname{per}}}^{2}(\mathbb{R})$, $s\in \mathbb{Z}_{+}$, $\varphi\equiv 1$, corresponds to the Marchenko-Ostrovskii Theorem.

### Article Information

 Title Spectral gaps of one-dimensional Schrödinger operators with singular periodic potentials Source Methods Funct. Anal. Topology, Vol. 15 (2009), no. 1, 31-40 MathSciNet MR2502636 Copyright The Author(s) 2009 (CC BY-SA)

### Authors Information

Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs'ka, Kyiv, 01601, Ukraine

Volodymyr Molyboga
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs'ka, Kyiv, 01601, Ukraine

### Citation Example

Vladimir Mikhailets and Volodymyr Molyboga, Spectral gaps of one-dimensional Schrödinger operators with singular periodic potentials, Methods Funct. Anal. Topology 15 (2009), no. 1, 31-40.

### BibTex

@article {MFAT496,
AUTHOR = {Mikhailets, Vladimir and Molyboga, Volodymyr},
TITLE = {Spectral gaps of one-dimensional Schrödinger  operators with singular periodic potentials},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {15},
YEAR = {2009},
NUMBER = {1},
PAGES = {31-40},
ISSN = {1029-3531},
URL = {http://mfat.imath.kiev.ua/article/?id=496},
}