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 Ufimsk. Mat. Zh., 2013, Volume 5, Issue 1, Pages 36–55 (Mi ufa185)

On analytic properties of Weyl function of Sturm–Liouville operator with a decaying complex potential

Kh. K. Ishkin

Bashkir State University, Faculty of Mathematics and Information Technologies

Abstract: We study the spectral properties of the operator $L_\beta$ associated with the quadratic form $\mathcal{L}_\beta=\int\limits_{0}^{\infty}(|y'|^2-\beta x^{-\gamma}|y|^2)dx$ with the domain ${Q_0=\{y\in W_2^1(0,+\infty): y(0)=0\}}$, $0<\gamma<2$, $\beta\in \mathbf{C}$, as well as of the perturbed operator $M_\beta=L_\beta+W$. Under the assumption $(1+x^{\gamma/2})W\in L^1(0,+\infty)$ we prove the existence of finite quantum defect of the discrete spectrum that was established earlier by L. A. Sakhnovich as $\beta>0$, $\gamma=1$ and for real $W$ satisfying a more strict decaying condition at infinity. The main result of the paper is the proof of necessity (with some reservations) of the sufficient conditions for $W(x)$ obtained earlier by Kh. Kh. Murtazin under which the Weyl function of the operator $M_\beta$ possesses an analytic continuation on some angle from non-physical sheet.

Keywords: spectral instability, localization of spectrum, quantum defect, Weyl function, Darboux transformation.

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English version:
Ufa Mathematical Journal, 2013, 5:1, 36–55 (PDF, 483 kB); https://doi.org/10.13108/2013-5-1-36

Bibliographic databases:

Document Type: Article
UDC: 517.9

Citation: Kh. K. Ishkin, “On analytic properties of Weyl function of Sturm–Liouville operator with a decaying complex potential”, Ufimsk. Mat. Zh., 5:1 (2013), 36–55; Ufa Math. J., 5:1 (2013), 36–55

Citation in format AMSBIB
\Bibitem{Ish13} \by Kh.~K.~Ishkin \paper On analytic properties of Weyl function of Sturm--Liouville operator with a decaying complex potential \jour Ufimsk. Mat. Zh. \yr 2013 \vol 5 \issue 1 \pages 36--55 \mathnet{http://mi.mathnet.ru/ufa185} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3429949} \elib{http://elibrary.ru/item.asp?id=18929625} \transl \jour Ufa Math. J. \yr 2013 \vol 5 \issue 1 \pages 36--55 \crossref{https://doi.org/10.13108/2013-5-1-36}