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Schrödinger operators with potential waveguides on periodic graphs. (English) Zbl 07346627

The authors examine an appropriate Schrödinger operator on periodic graphs and assess the positions of the guided bands lying in gaps of the unperturbed operator. Also, they determine sufficient conditions for a class of potentials satisfying a few conditions under which the guided bands do not appear in gaps of the unperturbed problem.

MSC:

47B39 Linear difference operators
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
35J10 Schrödinger operator, Schrödinger equation
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
39A12 Discrete version of topics in analysis
81Q35 Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices
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[1] S. ALAMA, P.A. DEIFT, R. HEMPEL,Eigenvalue branches of the Schr¨odinger operator H−λW in a spectral gap of H, Comm. Math. Phys.121(1989), 291-321. · Zbl 0676.47032
[2] K. ANDO,Inverse scattering theory for discrete Schr¨odinger operators on the hexagonal lattice, Ann. Henri Poincar´e14(2013), 347-383. · Zbl 1262.81182
[3] K. ANDO, H. ISOZAKI, H. MORIOKA,Spectral properties of Schr¨odinger operators on perturbed lattices, Ann. Henri Poincar´e17(2016), 2103-2171. · Zbl 1347.81044
[4] J. ARAZY, L. ZELENKO,Finite-dimensional perturbations of self-adjoint operators, Integral Equations Operator Theory34, 2 (1999), 127-164. · Zbl 0959.47007
[5] M. SH. BIRMAN,Discrete spectrum in the gaps of a continuous one for perturbations with large coupling constant, Adv. Soviet Math.7(1991), 57-73. · Zbl 0754.35025
[6] J. BLANK, P. EXNER, M. HAVLICEK,Hilbert Space Operators in Quantum Physics, Springer Science and Business Media B.V., 2008. · Zbl 1163.47060
[7] A. H. CASTRONETO, F. GUINEA, N. M. R. PERES, K.S. NOVOSELOV, A. GEIM,The electronic properties of graphene, Rev. Mod. Phys.81(2009), 109-162.
[8] P. A. DEIFT, R.HEMPEL,On the existence of eigenvalues of the Schr¨odinger operator H−λW in a gap ofσ(H), Comm. Math. Phys.103(1986), 461-490. · Zbl 0594.34022
[9] J. S. FABILA-CARRASCO, F. LLEDO´, O. POST,Spectral gaps and discrete magnetic Laplacians, Linear Algebra Appl.547, 15 (2018), 183-216. · Zbl 1385.05047
[10] R. HALIR, P.J. BOCK, P. CHEBEN, A. ORTEGA-MONUX˜, C. ALONSO-RAMOS, J.H. SCHMID, J. LAPOINTE, D.-X. XU, J. G. WANGUEMERT-PEREZ, I. MOLINA-FERNANDEZ, S. JANZ,Waveguide sub-wavelength structures: a review of principles and applications, Laser Photon. Rev.9(2015), 25- 49.
[11] P. HARRIS,Carbon nano-tubes and related structure, Cambridge, Cambridge University Press, 2002.
[12] W.A. HARRISON,Electronic Structure and the Properties of Solids: The Physics of the Chemical Bond, Dover Publications, Inc., New York, 1989.
[13] R. HEMPEL,Eigenvalues in gaps and decoupling by Neumann boundary conditions, J. Math. Anal. Appl.169(1992), 229-259. · Zbl 0783.47063
[14] R. HEMPEL,Eigenvalues of Schr¨odinger operators in gaps of the essential spectrum – an overview, Contemp. Math.458(2008), 393-407. · Zbl 1152.35082
[15] Y. HIGUCHI, T. MATSUMOTO, O. OGURISU,On the spectrum of a discrete Laplacian onZwith finitely supported potential, Linear and multilinear algebra59, 8 (2011), 917-927. · Zbl 1230.39002
[16] Y. HIGUCHI, Y. NOMURA,Spectral structure of the Laplacian on a covering graph, European J. Combin.30, 2 (2009), 570-585. · Zbl 1247.05137
[17] Y. HIGUCHI, T. SHIRAI,The spectrum of magnetic Schr¨odinger operators on a graph with periodic structure, J. Funct. Anal.169(1999), 456-480. · Zbl 1073.47517
[18] Y. HIGUCHI, T. SHIRAI,A remark on the spectrum of magnetic Laplacian on a graph, the proceedings of TGT10, Yokohama Math. J.47(1999), Special issue, 129-142. · Zbl 0953.58028
[19] Y. HIGUCHI, T. SHIRAI,Weak Bloch property for discrete magnetic Schr¨odinger operators, Nagoya Math J.161(2001), 127-154. · Zbl 0985.58011
[20] Y. HIGUCHI, T. SHIRAI,Some spectral and geometric properties for infinite graphs, AMS Contemp. Math.347(2004), 29-56. · Zbl 1079.47035
[21] F. HIROSHIMA, I. SASAKI, T. SHIRAI, A. SUZUKI,Note on the spectrum of discrete Schr¨odinger operators, J. Math-for-Industry4(2012), 105-108. · Zbl 1302.47077
[22] R. HORN, C. JOHNSON,Matrix analysis, Cambridge University Press, 1985. · Zbl 0576.15001
[23] H. ISOZAKI, E. KOROTYAEV,Inverse problems, trace formulae for discrete Schr¨odinger operators, Ann. Henri Poincar´e13(2012), 751-788. · Zbl 1250.81124
[24] S. G. JOHNSON, P. R. VILLENEUVE, S. FAN, J. D.JOANNOPOULOS,Linear waveguides in photonic crystal slabs, Phys. Rev. B62(2000), 8212-8222.
[25] S. G. JOHNSON, J. D. JOANNOPOULOS,Photonic crystals. The road from theory to practice, Springer US, 2002.
[26] E. KOROTYAEV, N. SABUROVA,Schr¨odinger operators on periodic discrete graphs, J. Math. Anal. Appl.420, 1 (2014), 576-611. · Zbl 1297.47050
[27] E. KOROTYAEV, N. SABUROVA,Spectral band localization for Schr¨odinger operators on periodic graphs, Proc. Amer. Math. Soc.143(2015), 3951-3967. · Zbl 1333.47030
[28] E. KOROTYAEV, N. SABUROVA,Schr¨odinger operators with guided potentials on periodic graphs, Proc. Amer. Math. Soc.145, 11 (2017), 4869-4883. · Zbl 1417.47011
[29] E. KOROTYAEV, N. SABUROVA,Magnetic Schr¨odinger operators on periodic discrete graphs, J. Funct. Anal.,272(2017), 1625-1660. · Zbl 1355.81081
[30] E. KOROTYAEV, N. SABUROVA,Invariants for Laplacians on periodic graphs, Math. Ann.,377 (2020), 723-758. · Zbl 1454.47038
[31] A. KUTSENKO,Wave propagation through periodic lattice with defects, Comput. Mech.54(2014), 1559-1568. · Zbl 1309.74037
[32] A. KUTSENKO,Algebra of multidimensional periodic operators with defects, J. Math. Anal. Appl. 428, 1 (2015), 217-226. · Zbl 1325.35018
[33] A. KUTSENKO,Algebra of 2D periodic operators with local and perpendicular defects, J. Math. Anal. Appl.442, 2 (2016), 796-803. · Zbl 1347.47052
[34] K.S. NOVOSELOV, A.K. GEIM ET AL,Electricfield effect in atomically thin carbonfilms, Science 306, 5696 (2004), 666-669.
[35] G.G. OSHAROVICH, M.V. AYZENBERG-STEPANENKO,Wave localization in stratified square-cell lattices: The antiplane problem, J. Sound Vib.331(2012), 1378-1397. · Zbl 1329.74122
[36] V.S. RABINOVICH, S. ROCH,Essential spectra of difference operators onZn-periodic graphs, J. Phys. A40, 33 (2007), 10109-10128. · Zbl 1128.81011
[37] M. REED, B. SIMON,Methods of modern mathematical physics, vol.I. Functional analysis, Academic Press, New York, 1980. · Zbl 0459.46001
[38] M. REED, B. SIMON,Methods of modern mathematical physics, vol.IV. Analysis of operators, Academic Press, New York, 1978. · Zbl 0401.47001
[39] G.
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