Geodesic
Absence of gaps in a lower part of the spectrum of a Laplacian with frequent alternation of boundary conditions in a strip
Teoretičeskaâ i matematičeskaâ fizika, Tome 195 (2018) no. 2, pp. 225-239 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We consider the Laplacian in a planar infinite straight strip with frequent alternation of boundary conditions. We show that for a sufficiently small alternation period, there are no gaps in a lower part of the spectrum. In terms of certain numbers and functions, we write an explicit upper bound for the period and an expression for the length of the lower part of the spectrum in which the absence of gaps is guaranteed.
Keywords: Bethe–Sommerfeld conjecture, gap, periodic operator, alternation of boundary conditions, Laplacian, infinite strip. 
@article{TMF_2018_195_2_a4,
     author = {D.I. Borisov},
     title = {Absence of gaps in a~lower part of the~spectrum of {a~Laplacian} with frequent alternation of boundary conditions in a~strip},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {225--239},
     year = {2018},
     volume = {195},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2018_195_2_a4/}
}
TY  - JOUR
AU  - D.I. Borisov
TI  - Absence of gaps in a lower part of the spectrum of a Laplacian with frequent alternation of boundary conditions in a strip
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2018
SP  - 225
EP  - 239
VL  - 195
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TMF_2018_195_2_a4/
LA  - ru
ID  - TMF_2018_195_2_a4
ER  - 
%0 Journal Article
%A D.I. Borisov
%T Absence of gaps in a lower part of the spectrum of a Laplacian with frequent alternation of boundary conditions in a strip
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2018
%P 225-239
%V 195
%N 2
%U http://geodesic.mathdoc.fr/item/TMF_2018_195_2_a4/
%G ru
%F TMF_2018_195_2_a4
D.I. Borisov. Absence of gaps in a lower part of the spectrum of a Laplacian with frequent alternation of boundary conditions in a strip. Teoretičeskaâ i matematičeskaâ fizika, Tome 195 (2018) no. 2, pp. 225-239. http://geodesic.mathdoc.fr/item/TMF_2018_195_2_a4/

[1] M. M. Skriganov, A. V. Sobolev, “Asimptoticheskie otsenki dlya spektralnykh zon periodicheskikh operatorov Shredingera”, Algebra i analiz, 17:1 (2005), 276–288 | DOI | MR | Zbl

[2] L. Parnovski, “Bethe–Sommerfeld conjecture”, Ann. Henri Poincaré, 9:3 (2008), 457–508 | DOI | MR | Zbl

[3] B. E. J. Dahlberg, E. Trubowitz, “A remark on two dimensional periodic potentials”, Comment. Math. Helv., 57:1 (1982), 130–134 | DOI | MR | Zbl

[4] B. Helffer, A. Mohamed, “Asymptotics of the density of states for the Schrödinger operator with periodic electric potential”, Duke Math. J., 92:1 (1998), 1–60 | DOI | MR | Zbl

[5] M. M. Skriganov, A. V. Sobolev, “Variation of the number of lattice points in large balls”, Acta Arith., 120:3 (2005), 245–267 | DOI | MR | Zbl

[6] M. M. Skriganov, “Geometricheskie i arifmeticheskie metody v spektralnoi teorii mnogomernykh periodicheskikh operatorov”, Tr. MIAN, 171 (1985), 3–122 | MR | Zbl

[7] Y. Karpeshina, “Spectral properties of the periodic magnetic Schrödinger operator in the high-energy region. Two-dimensional case”, Commun. Math. Phys., 251:3 (2004), 473–514 | DOI | MR | Zbl

[8] A. Mohamed, “Asymptotic of the density of states for the Schrödinger operator with periodic electromagnetic potential”, J. Math. Phys., 38:8 (1997), 4023–4051 | DOI | MR | Zbl

[9] L. Parnovski, A. Sobolev, “On the Bethe–Sommerfeld conjecture for the polyharmonic operator”, Duke Math. J., 107:2 (2001), 209–238 | DOI | MR | Zbl

[10] G. Barbatis, L. Parnovski, “Bethe–Sommerfeld conjecture for pseudo-differential perturbation”, Commun. Partial Differ. Equ., 34:4 (2009), 383–418 | DOI | MR | Zbl

[11] L. Parnovski, A. V. Sobolev, “Bethe–Sommerfeld conjecture for periodic operators with strong perturbations”, Invent. Math., 181:3 (2010), 467–540 | DOI | MR | Zbl

[12] C. B. E. Beeken, Periodic Schrödinger operators in dimension two: constant magnetic fields and boundary value problems, PhD thesis, University of Sussex, Brighton, 2002

[13] D. Borisov, G. Cardone, T. Durante, “Homogenization and uniform resolvent convergence for elliptic operators in a strip perforated along a curve”, Proc. Roy. Soc. Edinburgh Sect. A, 146:6 (2016), 1115–1158 | DOI | MR | Zbl

[14] D. Borisov, G. Cardone, L. Faella, C. Perugia, “Uniform resolvent convergence for a strip with fast oscillating boundary”, J. Differ. Equ., 255:12 (2013), 4378–4402 | DOI | MR | Zbl

[15] D. Borisov, R. Bunoiu, G. Cardone, “Waveguide with non-periodically alternating Dirichlet and Robin conditions: homogenization and asymptotics”, Z. Angew. Math. Phys., 64:3 (2013), 439–472 | DOI | MR | Zbl

[16] D. Borisov, R. Bunoiu, G. Cardone, “On a waveguide with frequently alternating boundary conditions: homogenized Neumann condition”, Ann. Henri Poincaré, 11:8 (2010), 1591–1627 | DOI | MR | Zbl

[17] D. Borisov, G. Cardone, “Homogenization of the planar waveguide with frequently alternating boundary conditions”, J. Phys. A.: Math. Gen., 42:36 (2009), 365205, 21 pp. | DOI | MR | Zbl

[18] D. Borisov, G. Cardone, T. Durante, “Norm-resolvent convergence for elliptic operators in domain with perforation along curve”, C. R. Math. Acad. Sci. Paris, 352:9 (2014), 679–683 | DOI | MR | Zbl

[19] D. Borisov, R. Bunoiu, G. Cardone, “Homogenization and asymptotics for a waveguide with an infinite number of closely located small windows”, Probl. matem. analiza, 58:6 (2011), 59–68 | MR | Zbl