Abstract
We study one-dimensional Schrödinger operators $\mathrm{S}(q)$ on the space $L^{2}(\mathbb{R})$ with potentials $q$ being complex-valued generalized functions from the negative space $H_{{\operatorname{unif}}}^{-1}(\mathbb{R})$. Particularly the class $H_{{\operatorname{unif}}}^{-1}(\mathbb{R})$ contains periodic and almost periodic $H_{{\operatorname{loc}}}^{-1}(\mathbb{R})$-functions. We establish an equivalence of the various definitions of the operators $\mathrm{S}(q)$, investigate their approximation by operators with smooth potentials from the space $L_{{\operatorname{unif}}}^{1}(\mathbb{R})$ and prove that the spectrum of each operator $\mathrm{S}(q)$ lies within a certain parabola.
Key words: 1-D Schr¨odinger operator, complex potential, distributional potential, resolvent approximation, localization of spectrum.
Full Text
Article Information
| Title | Schrödinger operators with complex singular potentials |
| Source | Methods Funct. Anal. Topology, Vol. 19 (2013), no. 1, 16-28 |
| MathSciNet |
MR3088075 |
| zbMATH |
1289.47093 |
| Milestones | Received 22/11/2012 |
| Copyright | The Author(s) 2013 (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, Schrödinger operators with complex singular potentials, Methods Funct. Anal. Topology 19
(2013), no. 1, 16-28.
BibTex
@article {MFAT677,
AUTHOR = {Mikhailets, Vladimir and Molyboga, Volodymyr},
TITLE = {Schrödinger operators with complex singular potentials},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {19},
YEAR = {2013},
NUMBER = {1},
PAGES = {16-28},
ISSN = {1029-3531},
MRNUMBER = {MR3088075},
ZBLNUMBER = {1289.47093},
URL = {http://mfat.imath.kiev.ua/article/?id=677},
}
