Elastic and piezoelectric waveguides may have infinite number of gaps in their spectra

Sergei A. Nazarov; Jari Taskinen

doi:10.1016/j.crme.2015.12.004

Elastic and piezoelectric waveguides may have infinite number of gaps in their spectra

Sergei A. Nazarov ^1,^2,³; Jari Taskinen ⁴

¹ Saint-Petersburg State University, Universitetskaya nab., 7–9, St. Petersburg, 199034, Russia
² Peter the Great Saint-Petersburg State Polytechnical University, Polytechnicheskaya ul., 29, St. Petersburg, 195251, Russia
³ Institute of Problems of Mechanical Engineering RAS, V.O., Bolshoj pr., 61, St. Petersburg, 199178, Russia
⁴ University of Helsinki, Department of Mathematics and Statistics, P.O. Box 68, 00014 Helsinki, Finland

Comptes Rendus. Mécanique, Volume 344 (2016) no. 3, pp. 190-194.

Abstract
Résumé

We consider elastic and piezoelectric waveguides composed from identical beads threaded periodically along a spoke converging at infinity. We show that the essential spectrum constitutes a non-negative monotone unbounded sequence and thus has infinitely many spectral gaps.

Nous considérons des guides d'ondes élastiques et piézoélectriques réalisés à partir de perles identiques agencées de façon périodique le long d'un rayon convergeant à l'infini. Nous montrons que le spectre essentiel est une suite croissante positive non bornée. Cela prouve l'existence d'un nombre infini de trous spectraux.

Received: 2015-12-04
Accepted: 2015-12-18
Published online: 2016-01-28

DOI: 10.1016/j.crme.2015.12.004

Keywords: Waveguide, Elasticity, Piezoelectricity, Essential spectrum, Spectral gap, Singular Weyl sequence, Parametrix
Mots-clés : Guide d'ondes, Élasticité, Piézoélectricité, Spectre essentiel, Troux spectraux, Suite de Weyl, Paramétrix

Author's affiliations:

Sergei A. Nazarov ^{1, 2, 3}; Jari Taskinen ⁴

¹ Saint-Petersburg State University, Universitetskaya nab., 7–9, St. Petersburg, 199034, Russia
² Peter the Great Saint-Petersburg State Polytechnical University, Polytechnicheskaya ul., 29, St. Petersburg, 195251, Russia
³ Institute of Problems of Mechanical Engineering RAS, V.O., Bolshoj pr., 61, St. Petersburg, 199178, Russia
⁴ University of Helsinki, Department of Mathematics and Statistics, P.O. Box 68, 00014 Helsinki, Finland

@article{CRMECA_2016__344_3_190_0,
     author = {Sergei A. Nazarov and Jari Taskinen},
     title = {Elastic and piezoelectric waveguides may have infinite number of gaps in their spectra},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {190--194},
     publisher = {Elsevier},
     volume = {344},
     number = {3},
     year = {2016},
     doi = {10.1016/j.crme.2015.12.004},
     language = {en},
}

TY  - JOUR
AU  - Sergei A. Nazarov
AU  - Jari Taskinen
TI  - Elastic and piezoelectric waveguides may have infinite number of gaps in their spectra
JO  - Comptes Rendus. Mécanique
PY  - 2016
SP  - 190
EP  - 194
VL  - 344
IS  - 3
PB  - Elsevier
DO  - 10.1016/j.crme.2015.12.004
LA  - en
ID  - CRMECA_2016__344_3_190_0
ER  -

%0 Journal Article
%A Sergei A. Nazarov
%A Jari Taskinen
%T Elastic and piezoelectric waveguides may have infinite number of gaps in their spectra
%J Comptes Rendus. Mécanique
%D 2016
%P 190-194
%V 344
%N 3
%I Elsevier
%R 10.1016/j.crme.2015.12.004
%G en
%F CRMECA_2016__344_3_190_0

Sergei A. Nazarov; Jari Taskinen. Elastic and piezoelectric waveguides may have infinite number of gaps in their spectra. Comptes Rendus. Mécanique, Volume 344 (2016) no. 3, pp. 190-194. doi : 10.1016/j.crme.2015.12.004. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2015.12.004/

References
Cited by

[1] S. Fliss; P. Joly Solutions of the time-harmonic wave equation in periodic waveguides: asymptotic behaviour and radiation condition, Arch. Ration. Mech. Anal. (2016) (in press) | DOI

[2] S.A. Nazarov Umov–Mandel'stam radiation conditions in elastic periodic waveguide, Mat. Sb., Volume 205 (2014) no. 7, pp. 43-72 (English transl.: Sb. Math. 205 (7) (2014) 953–982)

[3] S.A. Nazarov A gap in the essential spectrum of the Neumann problem for an elliptic system in a periodic domain, Funkc. Anal. Prilozh., Volume 43 (2009) no. 3, pp. 92-95 (English transl.: Funct. Anal. Appl. 43 (3) (2009) 239–241)

[4] S.A. Nazarov; K. Ruotsalainen; J. Taskinen Essential spectrum of a periodic elastic waveguide may contain arbitrarily many gaps, Appl. Anal., Volume 89 (2010) no. 1, pp. 109-124

[5] S.A. Nazarov On the plurality of gaps in the spectrum of a periodic waveguide, Mat. Sb., Volume 201 (2010) no. 4, pp. 99-124 (English transl.: Sb. Math. 201 (4) (2010) 569–594)

[6] S.A. Nazarov; J. Taskinen Spectral gaps for periodic piezoelectric waveguides, Z. Angew. Math. Phys., Volume 66 (2015), pp. 3017-3047

[7] V.N. Popov; M. Skriganov A remark on the spectral structure of the two dimensional Schrödinger operator with a periodic potential, Zap. Nauč. Semin. LOMI AN SSSR, Volume 109 (1981), pp. 131-133 (in Russian)

[8] M. Skriganov Geometrical and arithmetical methods in the spectral theory of the multi-dimensional periodic operators, Proc. Steklov Inst. Math., Volume 171 (1984)

[9] M. Skriganov The spectrum band structure of the three-dimensional Schrödinger operator with periodic potential, Invent. Math., Volume 80 (1985), pp. 107-121

[10] B. Helffer; A. Mohamed Asymptotics of the density of states for the Schrödinger operator with periodic electric potential, Duke Math. J., Volume 92 (1998), pp. 1-60

[11] L. Parnovski Bethe–Sommerfeld conjecture, Ann. Henri Poincaré, Volume 9 (2008) no. 3, pp. 457-508

[12] V.A. Kondratiev; O.A. Oleinik Boundary-value problems for the system of elasticity theory in unbounded domains. Korn's inequalities, Usp. Mat. Nauk, Volume 43 (1988) no. 5, pp. 55-98 (English transl. in Russ. Math. Surv. 43 (5) (1988) 65–119)

[13] M.Sh. Birman; M.Z. Solomyak Spectral Theory of Selfadjoint Operators in Hilbert Space, Leningrad Univ., Leningrad, 1980 (English transl.: Math. Appl. (Soviet Ser.), D. Reidel Publishing Co., Dordrecht, The Netherlands, 1987)

[14] V.G. Maz'ya; S.A. Nazarov; B.A. Plamenevsky Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains, vols. 1, 2, Birkhäuser, Basel, Switzerland, 2000

[15] S.A. Nazarov; B.A. Plamenevsky Elliptic Problems in Domains with Piecewise Smooth Boundaries, Walter de Gruyter, Berlin, New York, 1994

[16] F.L. Bakharev, J. Taskinen, Bands in the spectrum of a periodic elastic waveguide, submitted.

[17] V.Z. Parton; B.A. Kudryavtsev Electromagnetoelasticity, Piezoelectrics and Electrically Conductive Solids, Gordon and Breach Science Publishers, New York, 1988

[18] S.A. Nazarov Uniform estimates of remainders in asymptotic expansions of solutions to the problem on eigen-oscillations of a piezoelectric plate, Probl. Mat. Anal., Volume 25 (2003), pp. 99-188 (English transl.: J. Math. Sci. 114 (5) (2003) 1657–1725)

Cited by Sources:

Comments - Policy