The discrete spectrum of cross-shaped waveguides
HTML articles powered by AMS MathViewer
- by
F. L. Bakharev, S. G. Matveenko and S. A. Nazarov
Translated by: A. Plotkin - St. Petersburg Math. J. 28 (2017), 171-180
- DOI: https://doi.org/10.1090/spmj/1444
- Published electronically: February 15, 2017
- PDF | Request permission
Abstract:
The discrete spectrum of the Dirichlet problem for the Laplace operator on the union of two circular unit cylinders whose axes intersect at the right angle consists of a single eigenvalue. For the threshold value of the spectral parameter, this problem has no bounded solutions. When the angle between the axes reduces, the multiplicity of the discrete spectrum grows unboundedly.References
- Peter Kuchment, Graph models for waves in thin structures, Waves Random Media 12 (2002), no. 4, R1–R24. MR 1937279, DOI 10.1088/0959-7174/12/4/201
- —(Ed.), Quantum graphs and their applications, a special issue of Waves Random Media 14 (2004), no. 1. MR (2005h:81148)
- L. Pauling, The diamagnetic anisotropy of aromatic molecules, J. Chem. Phys. 4 (1936), 672–678.
- Peter Kuchment and Hongbiao Zeng, Asymptotics of spectra of Neumann Laplacians in thin domains, Advances in differential equations and mathematical physics (Birmingham, AL, 2002) Contemp. Math., vol. 327, Amer. Math. Soc., Providence, RI, 2003, pp. 199–213. MR 1991542, DOI 10.1090/conm/327/05815
- Pavel Exner and Olaf Post, Convergence of spectra of graph-like thin manifolds, J. Geom. Phys. 54 (2005), no. 1, 77–115. MR 2135966, DOI 10.1016/j.geomphys.2004.08.003
- Daniel Grieser, Spectra of graph neighborhoods and scattering, Proc. Lond. Math. Soc. (3) 97 (2008), no. 3, 718–752. MR 2448245, DOI 10.1112/plms/pdn020
- S. A. Nazarov, Trapped modes in a $T$-shaped waveguide, Akust. Zh. 56 (2010), no. 6, 747–758; English transl., Acoust. Phys. 56 (2010), 1004–1015.
- S. A. Nazarov, Bounded solutions in a T-shaped waveguide and the spectral properties of the Dirichlet ladder, Comput. Math. Math. Phys. 54 (2014), no. 8, 1261–1279. MR 3250876, DOI 10.1134/S0965542514080090
- S. A. Nazarov, Asymptotics of eigenvalues of the Dirichlet problem in a skewed ${\scr I}$-shaped waveguide, Comput. Math. Math. Phys. 54 (2014), no. 5, 811–830. MR 3211884, DOI 10.1134/S0965542514050121
- S. A. Nazarov, The structure of the spectrum of a lattice of quantum waveguides and bounded solutions of a model problem on the threshold, Dokl. Akad. Nauk 458 (2014), no. 6, 636–640 (Russian); English transl., Dokl. Math. 90 (2014), no. 2, 637–641. MR 3408325, DOI 10.1134/s1064562414050305
- S. A. Nazarov, Discrete spectrum of cross-shaped quantum waveguides, J. Math. Sci. (N.Y.) 196 (2014), no. 3, Problems in mathematical analysis. No. 73 (Russian), 346–376. MR 3391299, DOI 10.1007/s10958-014-1662-0
- S. A. Nazarov, K. Ruotsalainen, and P. Uusitalo, The Y-junction of quantum waveguides, ZAMM Z. Angew. Math. Mech. 94 (2014), no. 6, 477–486. MR 3223792, DOI 10.1002/zamm.201200255
- —, Asymptotics of the spectrum of the Dirichlet Laplacian on a thin carbon nano-structure, C. R. Mecanique 343 (2015), 360–364.
- M. Š. Birman and M. Z. Solomjak, Spektral′naya teoriya samosopryazhennykh operatorov v gil′bertovom prostranstve, Leningrad. Univ., Leningrad, 1980 (Russian). MR 609148
- I. V. Kamotskiĭ and S. A. Nazarov, Exponentially decreasing solutions of the problem of diffraction by a rigid periodic boundary, Mat. Zametki 73 (2003), no. 1, 138–140 (Russian); English transl., Math. Notes 73 (2003), no. 1-2, 129–131. MR 1993547, DOI 10.1023/A:1022186320373
- Rolf Leis, Initial-boundary value problems in mathematical physics, B. G. Teubner, Stuttgart; John Wiley & Sons, Ltd., Chichester, 1986. MR 841971, DOI 10.1007/978-3-663-10649-4
- V. G. Osmolovskii, Nonlinear Sturm–Liouville problem, St.-Petersburg Univ., St.-Petersburg, 2003. (Russian)
- G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematics Studies, No. 27, Princeton University Press, Princeton, N. J., 1951. MR 0043486
Bibliographic Information
- F. L. Bakharev
- Affiliation: St. Petersburg State University, Universitetskii pr. 28, Petrodvorets, 198504 St. Petersburg; P. L. Chebyshev laboratory, 14 liniya V.O. 29b, 199178 St. Petersburg, Russia
- Email: fbakharev@yandex.ru, f.bakharev@spbu.ru
- S. G. Matveenko
- Affiliation: National Research University Higher School of Economics, ul. Kantemirovskaya 3A, office 417, 194100 St. Petersburg; P. L. Chebyshev Laboratory, St. Petersburg State University, 14 liniya V.O. 29b, 199178 St. Petersburg, Russia
- Email: matveis239@gmail.com
- S. A. Nazarov
- Affiliation: St. Petersburg State University, Universitetskii pr. 28, Petrodvorets, 198154 St. Petersburg; Peter the Great St. Petersburg Polytechnical University, Polytekhnicheskaya yl., 29, 195251 St. Petersburg; Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, Bol′shoi pr. V.O. 61, 199178 St. Petersburg, Russia
- MR Author ID: 196508
- Email: srgnazarov@yahoo.co.uk, s.nazarov@spbu.ru
- Received by editor(s): October 29, 2015
- Published electronically: February 15, 2017
- Additional Notes: The research was done in the framework of the SPbGU project no. 0.38.237.2014
The first two authors were supported by the P. L. Chebyshev Laboratory of SPbGU (RF Governement grant no. 11.G34.31.0026) - © Copyright 2017 American Mathematical Society
- Journal: St. Petersburg Math. J. 28 (2017), 171-180
- MSC (2010): Primary 81Q37
- DOI: https://doi.org/10.1090/spmj/1444
- MathSciNet review: 3593003