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 Izv. RAN. Ser. Mat., 2017, Volume 81, Issue 1, Pages 31–92 (Mi izv8380)

The spectra of rectangular lattices of quantum waveguides

S. A. Nazarovabc

a Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, St. Petersburg
b Saint-Petersburg State Polytechnical University
c St. Petersburg State University, Department of Mathematics and Mechanics

Abstract: We obtain asymptotic formulae for the spectral segments of a thin ($h\ll 1$) rectangular lattice of quantum waveguides which is described by a Dirichlet problem for the Laplacian. We establish that the structure of the spectrum of the lattice is incorrectly described by the commonly accepted quantum graph model with the traditional Kirchhoff conditions at the vertices. It turns out that the lengths of the spectral segments are infinitesimals of order $O(e^{-\delta/h})$, $\delta>0$, and $O(h)$ as $h\to+0$, and gaps of width $O(h^{-2})$ and $O(1)$ arise between them in the low-frequency and middle-frequency spectral ranges respectively. The first spectral segment is generated by the (unique) eigenvalue in the discrete spectrum of an infinite cross-shaped waveguide $\Theta$. The absence of bounded solutions of the problem in $\Theta$ at the threshold frequency means that the correct model of the lattice is a graph with Dirichlet conditions at the vertices which splits into two infinite subsets of identical edges-intervals. By using perturbations of finitely many joints, we construct any given number of discrete spectrum points of the lattice below the essential spectrum as well as inside the gaps.

Keywords: quantum waveguide, thin rectangular lattice, Dirichlet problem, gaps, Kirchhoff transmission conditions, discrete spectrum, asymptotic analysis.

 Funding Agency Grant Number Saint Petersburg State University 0.38.237.2014 This paper was written with the financial support of St.-Petersburg State University, project no. 0.38.237.2014.

DOI: https://doi.org/10.4213/im8380

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English version:
Izvestiya: Mathematics, 2017, 81:1, 29–90

Bibliographic databases:

UDC: 517.956.328+517.956.225+517.956.8
MSC: 35P20, 35J25, 47A55, 81Q15, 81Q37, 82D77

Citation: S. A. Nazarov, “The spectra of rectangular lattices of quantum waveguides”, Izv. RAN. Ser. Mat., 81:1 (2017), 31–92; Izv. Math., 81:1 (2017), 29–90

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/izv8380
• https://doi.org/10.4213/im8380
• http://mi.mathnet.ru/eng/izv/v81/i1/p31

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

This publication is cited in the following articles:
1. S. A. Nazarov, “Open waveguides in a thin Dirichlet ladder: I. Asymptotic structure of the spectrum”, Comput. Math. Math. Phys., 57:1 (2017), 156–174
2. F. L. Bakharev, S. G. Matveenko, S. A. Nazarov, “Examples of plentiful discrete spectra in infinite spatial cruciform quantum waveguides”, Z. Anal. Anwend., 36:3 (2017), 329–341
3. V. A. Kozlov, S. A. Nazarov, “Model meshkovidnoi anevrizmy bifurkatsionnogo uzla arterii”, Matematicheskie voprosy teorii rasprostraneniya voln. 47, Zap. nauchn. sem. POMI, 461, POMI, SPb., 2017, 174–194
4. S. A. Nazarov, “Spectrum of a problem of elasticity theory in the union of several infinite layers”, Russ. J. Math. Phys., 25:1 (2018), 73–87
5. F. L. Bakharev, S. A. Nazarov, “Asimptotika sobstvennykh chisel dlinnykh plastin Kirkhgofa s zaschemlennymi krayami”, Matem. sb., 210:4 (2019), 3–26
6. S. A. Nazarov, “Asimptotika sobstvennykh chisel i funktsii tonkoi kvadratnoi reshetki Dirikhle s iskrivlennoi peremychkoi”, Matem. zametki, 105:4 (2019), 564–588
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