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 Mat. Zametki, 2017, Volume 102, Issue 5, Pages 761–774 (Mi mz11780)

Essential Spectrum of Schrödinger Operators with $\delta$-Interactions on Unbounded Hypersurfaces

V. S. Rabinovich

Instituto Politecnico Nacional, ESIME–Zacatenco

Abstract: Let $\Gamma$ be a simply connected unbounded $C^{2}$-hypersurface in $\mathbb{R}^{n}$ such that $\Gamma$ divides $\mathbb{R}^{n}$ into two unbounded domains $D^{\pm}$. We consider the essential spectrum of Schrödinger operators on $\mathbb{R}^{n}$ with surface $\delta_{\Gamma}$-interactions which can be written formally as
$$H_{\Gamma}=-\Delta+W-\alpha_{\Gamma}\delta_{\Gamma},$$
where $-\Delta$ is the nonnegative Laplacian in $\mathbb{R}^{n}$, $W\in L^{\infty}(\mathbb{R}^{n})$ is a real-valued electric potential, $\delta_{\Gamma}$ is the Dirac $\delta$-function with the support on the hypersurface $\Gamma$ and $\alpha_{\Gamma}\in L^{\infty}(\Gamma)$ is a real-valued coupling coefficient depending of the points of $\Gamma$. We realize $H_{\Gamma}$ as an unbounded operator $\mathcal{A}_{\Gamma}$ in $L^{2}(\mathbb{R}^{n})$ generated by the Schrödinger operator
$$H_{\Gamma}=-\Delta+W\qquad on\quad \mathbb{R}^{n}\setminus\Gamma$$
and Robin-type transmission conditions on the hypersurface $\Gamma$. We give a complete description of the essential spectrum of $\mathcal{A}_{\Gamma}$ in terms of the limit operators generated by $A_{\Gamma}$ and the Robin transmission conditions.

Keywords: surface $\delta$-interaction, self-adjoint realization, Robin transmission conditions, limit operators, essential spectra.

 Funding Agency Grant Number CONACYT - Consejo Nacional de Ciencia y Tecnología CB-179872 National System of Researchers in Mexico (SNI) The work was supported by the National System of Investigators of Mexico and the CONACYT project CB-179872.

DOI: https://doi.org/10.4213/mzm11780

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English version:
Mathematical Notes, 2017, 102:5, 698–709

Bibliographic databases:

UDC: 517

Citation: V. S. Rabinovich, “Essential Spectrum of Schrödinger Operators with $\delta$-Interactions on Unbounded Hypersurfaces”, Mat. Zametki, 102:5 (2017), 761–774; Math. Notes, 102:5 (2017), 698–709

Citation in format AMSBIB
\Bibitem{Rab17} \by V.~S.~Rabinovich \paper Essential Spectrum of Schr\"{o}dinger Operators with $\delta$-Interactions on Unbounded Hypersurfaces \jour Mat. Zametki \yr 2017 \vol 102 \issue 5 \pages 761--774 \mathnet{http://mi.mathnet.ru/mz11780} \crossref{https://doi.org/10.4213/mzm11780} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3716509} \elib{https://elibrary.ru/item.asp?id=30512316} \transl \jour Math. Notes \yr 2017 \vol 102 \issue 5 \pages 698--709 \crossref{https://doi.org/10.1134/S0001434617110098} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000418838500009} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85039418838} 

• http://mi.mathnet.ru/eng/mz11780
• https://doi.org/10.4213/mzm11780
• http://mi.mathnet.ru/eng/mz/v102/i5/p761

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This publication is cited in the following articles:
1. Rabinovich V., “Electromagnetic Schrodinger Operators on Periodic Graphs With General Conditions At Vertices”, Russ. J. Math. Phys., 26:2 (2019), 185–205
2. V. S. Rabinovich, “Suschestvennyi spektr odnomernykh operatorov Diraka s delta-vzaimodeistviyami”, Funkts. analiz i ego pril., 54:2 (2020), 90–94
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