I. S. Lobanov, V. Yu. Lotoreichik, I. Yu. Popov, “Lower bound on the spectrum of the two-dimensional Schrödinger operator with a $\delta$-perturbation on a curve”, TMF, 162:3 (2010), 397–407; Theoret. and Math. Phys., 162:3 (2010), 332

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Teoreticheskaya i Matematicheskaya Fizika, 2010, Volume 162, Number 3, Pages 397–407
DOI: https://doi.org/10.4213/tmf6477 (Mi tmf6477)

This article is cited in 13 scientific papers (total in 13 papers)

Lower bound on the spectrum of the two-dimensional Schrödinger operator with a

$\delta$ -perturbation on a curve

I. S. Lobanov, V. Yu. Lotoreichik, I. Yu. Popov

St.~Petersburg State University of Information Technologies, Mechanics, and Optics, St.~Petersburg, Russia

Full-text PDF (416 kB) Citations (13)

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DOI: https://doi.org/10.4213/tmf6477

Abstract: We consider the two-dimensional Schrödinger operator with a

$\delta$ -potential supported by curve. For the cases of infinite and closed finite smooth curves, we obtain lower bounds on the spectrum of the considered operator that are expressed explicitly in terms of the interaction strength and a parameter characterizing the curve geometry. We estimate the bottom of the spectrum for a piecewise smooth curve using parameters characterizing the geometry of the separate pieces. As applications of the obtained results, we consider curves with a finite number of cusps and general “leaky” quantum graph as the support of the

$\delta$ -potential.

Keywords: Schrödinger operator, singular potential, spectral estimate, Birman–Schwinger transformation.

Received: 01.08.2009

English version:
Theoretical and Mathematical Physics, 2010, Volume 162, Issue 3, Pages 332–340
DOI: https://doi.org/10.1007/s11232-010-0025-3

Bibliographic databases:

Language: Russian

Citation: I. S. Lobanov, V. Yu. Lotoreichik, I. Yu. Popov, “Lower bound on the spectrum of the two-dimensional Schrödinger operator with a

$\delta$ -perturbation on a curve”, TMF, 162:3 (2010), 397–407; Theoret. and Math. Phys., 162:3 (2010), 332–340

Citation in format AMSBIB

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https://www.mathnet.ru/eng/tmf6477

https://doi.org/10.4213/tmf6477

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This publication is cited in the following 13 articles:

S Kondej, “Bound states asymptotics in the system with quantum wires in ℝ3”, J. Phys. A: Math. Theor., 56:3 (2023), 035202
Mantile A., Posilicano A., Sini M., “Self-adjoint elliptic operators with boundary conditions on not closed hypersurfaces”, J. Differ. Equ., 261:1 (2016), 1–55
Behrndt J., Exner P., Lotoreichik V., “Schrodinger Operators With Delta- and Delta `-Interactions on Lipschitz Surfaces and Chromatic Numbers of Associated Partitions”, Rev. Math. Phys., 26:8 (2014), 1450015
Lobanov I.S., Popov I.Yu., Popov A.I., Gerya T.V., “Numerical Approach To the Stokes Problem With High Contrasts in Viscosity”, Appl. Math. Comput., 235 (2014), 17–25
Behrndt J., Langer M., Lotoreichik V., “Schrodinger Operators with Delta and Delta `-Potentials Supported on Hypersurfaces”, Ann. Henri Poincare, 14:2 (2013), 385–423
Eremin D.A. Ivanov D.A. Popov I.Yu., “Regular potential approximation for $\delta$-perturbation supported by curve of the Laplace–Beltrami operator on the sphere”, Z. Anal. Anwend., 31:2 (2012), 125–137
Kaynak B.T., Turgut O.T., “A Klein-Gordon particle captured by embedded curves”, Ann. Phys., 327:11 (2012), 2605–2616
Popov S.I., Gavrilov M.I., Popov I.Yu., “Two interacting particles in deformed nanolayer: discrete spectrum and particle storage”, Phys. Scr., 86:3 (2012), 035003
Kaynak B.T., Turgut O.T., “Singular interactions supported by embedded curves”, J. Phys. A, 45:26 (2012), 265202, 14 pp.
S.I. Popov, M.I. Gavrilov, I.Yu. Popov, 2012 Proceedings of the International Conference Days on Diffraction, 2012, 203
Cisło J., Kondej S., “Upper bound for the number of bound states induced by the curvature of singular potential”, Rep. Math. Phys., 68:2 (2011), 225–240
Eremin D.A., Popov I.Yu., “Kvantovoe koltso s provodnikom:model dvukhchastichnoi zadachi”, Nanosistemy: fizika, khimiya, matematika, 2:2 (2011), 15–31
Gavrilov M.I., Popov I.Yu., Popov S.I., “Mnogochastichnye sostoyaniya v iskrivlennykh sloistykh nanostrukturakh”, Nauchno-tekhnicheskii vestnik informatsionnykh tekhnologii, mekhaniki i optiki, 2011, 45–47

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

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