Geodesic
On lacunas in the spectrum of the Laplacian with the Dirichlet boundary condition in a strip with oscillating boundary
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Complex Analysis. Mathematical Physics, Tome 162 (2019), pp. 3-14.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper, we consider the Laplace operator in a flat strip whose lower boundary periodically oscillates under the Dirichlet boundary condition. The period and the amplitude of oscillations are two independent small parameters. The main result obtained in the paper is the absence of internal lacunas in the lower part of the spectrum of the operator for sufficiently small period and amplitude. We obtain explicit upper estimates of the period and amplitude in the form of constraints with specific numerical constants. The length of the lower part of the spectrum, in which the absence of lacunas is guaranteed, is also expressed explicitly in terms of the period function and the amplitude.
Keywords: Bethe–Sommerfeld hypothesis, Laplacian, strip, oscillating boundary. 
@article{INTO_2019_162_a0,
     author = {D. I. Borisov},
     title = {On lacunas in the spectrum of the {Laplacian} with the {Dirichlet} boundary condition in a strip with oscillating boundary},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {3--14},
     publisher = {mathdoc},
     volume = {162},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2019_162_a0/}
}
TY  - JOUR
AU  - D. I. Borisov
TI  - On lacunas in the spectrum of the Laplacian with the Dirichlet boundary condition in a strip with oscillating boundary
JO  - Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
PY  - 2019
SP  - 3
EP  - 14
VL  - 162
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/INTO_2019_162_a0/
LA  - ru
ID  - INTO_2019_162_a0
ER  - 
%0 Journal Article
%A D. I. Borisov
%T On lacunas in the spectrum of the Laplacian with the Dirichlet boundary condition in a strip with oscillating boundary
%J Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
%D 2019
%P 3-14
%V 162
%I mathdoc
%U http://geodesic.mathdoc.fr/item/INTO_2019_162_a0/
%G ru
%F INTO_2019_162_a0
D. I. Borisov. On lacunas in the spectrum of the Laplacian with the Dirichlet boundary condition in a strip with oscillating boundary. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Complex Analysis. Mathematical Physics, Tome 162 (2019), pp. 3-14. http://geodesic.mathdoc.fr/item/INTO_2019_162_a0/

[1] Borisov D. I., “O lakunakh v spektre laplasiana v polose, vozmuschennogo ogranichennym periodicheskim operatorom”, Ufim. mat. zh., 10:2 (2018), 13–29

[2] Borisov D. I., “Ob otsutstvii lakun v nizhnei chasti spektra laplasiana s chastym cheredovaniem kraevykh uslovii v polose”, Teor. mat. fiz., 195:2 (2018), 225–239 | DOI | MR | Zbl

[3] Borisov D. I., “O lakunakh v nizhnei chasti spektra periodicheskogo magnitnogo operatora v polose”, Sovr. mat. Fundam. napr., 63, no. 3, 2017, 373–391 | MR

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

[5] Skriganov M. M., “Geometricheskie i arifmeticheskie metody v spektralnoi teorii mnogomernykh periodicheskikh operatorov”, Tr. Mat. in-ta im. V. A. Steklova AN SSSR., 171, 1985, 3–122

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

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

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

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

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

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

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

[13] Borisov D., Bunoiu R., Cardone G., “On a waveguide with an infinite number of small windows”, Comp. Rend. Math., 349:1-2 (2011), 53–56 | DOI | MR | Zbl

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

[15] Helffer B., Mohamed A., “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

[16] Karpeshina Y., “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

[17] Mohamed A., “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

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

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

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

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