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.

Full Text

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

Vladimir Mikhailets
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},
  MRNUMBER = {MR2502636},
       URL = {http://mfat.imath.kiev.ua/article/?id=496},
}