Discrete spectrum of cranked, branching, and periodic waveguides

Nazarov, S.

doi:10.1090/S1061-0022-2012-01200-8

Discrete spectrum of cranked, branching, and periodic waveguides
HTML articles powered by AMS MathViewer

by S. A. Nazarov
Translated by: A. Plotkin

St. Petersburg Math. J. 23 (2012), 351-379

DOI: https://doi.org/10.1090/S1061-0022-2012-01200-8

Published electronically: January 24, 2012

PDF | Request permission

Abstract:

The variational method is applied to the study of the spectrum of the Laplace operator with mixed boundary conditions and with Dirichlet conditions in planar or multidimensional domains (waveguides) with cylindrical or periodic exits to infinity. The planar waveguides of constant width are discussed completely, such as cranked, broken, smoothly bent, or branching waveguides. For them, the existence of eigenvalues below the continuous spectrum threshold is established. A similar result is obtained for the multidimensional cranked and branching waveguides, and also for some periodic ones. Several open questions are stated; in particular, they concern problems with Neumann boundary conditions, full multiplicity of the discrete spectrum, and planar waveguides with piecewise constant boundary.

References

P. Duclos and P. Exner, Curvature-induced bound states in quantum waveguides in two and three dimensions, Rev. Math. Phys. 7 (1995), no. 1, 73–102. MR 1310767, DOI 10.1142/S0129055X95000062
I. V. Kamotskiĭ and S. A. Nazarov, Elastic waves localized near periodic families of defects, Dokl. Akad. Nauk 368 (1999), no. 6, 771–773 (Russian); English transl., Dokl. Phys. 44 (1999), no. 10, 715–717. MR 1749046
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
D. S. Jones, The eigenvalues of $\nabla ^2u+\lambda u=0$ when the boundary conditions are given on semi-infinite domains, Proc. Cambridge Philos. Soc. 49 (1953), 668–684. MR 58086
F. Ursell, Mathematical aspects of trapping modes in the theory of surface waves, J. Fluid Mech. 183 (1987), 421–437. MR 919481, DOI 10.1017/S0022112087002702
Anne-Sophie Bonnet-Bendhia and Felipe Starling, Guided waves by electromagnetic gratings and nonuniqueness examples for the diffraction problem, Math. Methods Appl. Sci. 17 (1994), no. 5, 305–338. MR 1273315, DOI 10.1002/mma.1670170502
D. V. Evans, M. Levitin, and D. Vassiliev, Existence theorems for trapped modes, J. Fluid Mech. 261 (1994), 21–31. MR 1265871, DOI 10.1017/S0022112094000236
S. V. Sukhinin, Waveguide, anomalous, and whispering properties of a periodic chain of obstacles, Sib. Zh. Ind. Mat. 1 (1998), no. 2, 175–198 (Russian, with Russian summary). MR 1789383
I. Roitberg, D. Vassiliev, and T. Weidl, Edge resonance in an elastic semi-strip, Quart. J. Mech. Appl. Math. 51 (1998), no. 1, 1–13. MR 1610688, DOI 10.1093/qjmam/51.1.1
A. S. Bonnet-Ben Dhia, J. Duterte, and P. Joly, Mathematical analysis of elastic surface waves in topographic waveguides, Math. Models Methods Appl. Sci. 9 (1999), no. 5, 755–798. MR 1697393, DOI 10.1142/S0218202599000373
I. V. Kamotskiĭ and S. A. Nazarov, An augmented scattering matrix and exponentially decreasing solutions of an elliptic problem in a cylindrical domain, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 264 (2000), no. Mat. Vopr. Teor. Rasprostr. Voln. 29, 66–82, 322–323 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (New York) 111 (2002), no. 4, 3657–3666. MR 1796996, DOI 10.1023/A:1016377707919
S. A. Nazarov, A criterion for the existence of decaying solutions in the problem of a resonator with a cylindrical waveguide, Funktsional. Anal. i Prilozhen. 40 (2006), no. 2, 20–32, 95 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 40 (2006), no. 2, 97–107. MR 2256860, DOI 10.1007/s10688-006-0016-1
S. A. Nazarov, Trapped modes for a cylindrical elastic waveguide with a damping gasket, Zh. Vychisl. Mat. Mat. Fiz. 48 (2008), no. 5, 863–881 (Russian, with Russian summary); English transl., Comput. Math. Math. Phys. 48 (2008), no. 5, 816–833. MR 2433645, DOI 10.1134/S0965542508050102
I. V. Kamotskiĭ, On a surface wave traveling along the edge of an elastic wedge, Algebra i Analiz 20 (2008), no. 1, 86–92 (Russian); English transl., St. Petersburg Math. J. 20 (2009), no. 1, 59–63. MR 2411969, DOI 10.1090/S1061-0022-08-01037-6
S. A. Nazarov, A simple method for finding trapped modes in problems of the linear theory of surface waves, Dokl. Akad. Nauk 429 (2009), no. 6, 746–749 (Russian); English transl., Dokl. Math. 80 (2009), no. 3, 914–917. MR 2641034, DOI 10.1134/S1064562409060325
I. V. Kamotskiĭ and A. P. Kiselev, An energy approach to the proof of the existence of Rayleigh waves in an anisotropic elastic half-space, Prikl. Mat. Mekh. 73 (2009), no. 4, 645–654 (Russian, with Russian summary); English transl., J. Appl. Math. Mech. 73 (2009), no. 4, 464–470. MR 2584534, DOI 10.1016/j.jappmathmech.2009.08.003
C. M. Linton and P. McIver, Embedded trapped modes in water waves and acoustics, Wave Motion 45 (2007), no. 1-2, 16–29. MR 2441664, DOI 10.1016/j.wavemoti.2007.04.009
O. A. Ladyzhenskaya, Kraevye zadachi matematicheskoĭ fiziki, Izdat. “Nauka”, Moscow, 1973 (Russian). MR 0599579
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. Kamotskii and S. A. Nazarov, On eigenfunctions localized in a neighborhood of the lateral surface of a thin domain, J. Math. Sci. (New York) 101 (2000), no. 2, 2941–2974. Nonlinear equations and mathematical analysis. MR 1784687, DOI 10.1007/BF02672180
Sergey Nazarov, Properties of spectra of boundary value problems in cylindrical and quasicylindrical domains, Sobolev spaces in mathematics. II, Int. Math. Ser. (N. Y.), vol. 9, Springer, New York, 2009, pp. 261–309. MR 2484629, DOI 10.1007/978-0-387-85650-6_{1}2
R. L. Shult, D. G. Ravenhall, and H. D. Wyld, Quantum bound states in a classically unbounded system of crossed wires, Phys. Rev. B 39 (1989), no. 8, 5476–5479.
Y. Avishai, D. Bessis, B. G. Giraud, and G. Mantica, Quantum bound states in open geometries, Phys. Rev. B 44 (1991), no. 15, 8028–8034.
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
I. M. Gel′fand, Expansion in characteristic functions of an equation with periodic coefficients, Doklady Akad. Nauk SSSR (N.S.) 73 (1950), 1117–1120 (Russian). MR 0039154
S. A. Nazarov, Elliptic boundary value problems with periodic coefficients in a cylinder, Izv. Akad. Nauk SSSR Ser. Mat. 45 (1981), no. 1, 101–112, 239 (Russian). MR 607578
P. A. Kuchment, Floquet theory for partial differential equations, Uspekhi Mat. Nauk 37 (1982), no. 4(226), 3–52, 240 (Russian). MR 667973
Sergey A. Nazarov and Boris A. Plamenevsky, Elliptic problems in domains with piecewise smooth boundaries, De Gruyter Expositions in Mathematics, vol. 13, Walter de Gruyter & Co., Berlin, 1994. MR 1283387, DOI 10.1515/9783110848915.525
Peter Kuchment, Floquet theory for partial differential equations, Operator Theory: Advances and Applications, vol. 60, Birkhäuser Verlag, Basel, 1993. MR 1232660, DOI 10.1007/978-3-0348-8573-7
Tosio Kato, Perturbation theory for linear operators, 2nd ed., Grundlehren der Mathematischen Wissenschaften, Band 132, Springer-Verlag, Berlin-New York, 1976. MR 0407617
Peter Kuchment, The mathematics of photonic crystals, Mathematical modeling in optical science, Frontiers Appl. Math., vol. 22, SIAM, Philadelphia, PA, 2001, pp. 207–272. MR 1831334