Geodesic
On spectral gaps of a Laplacian in a strip with a bounded periodic perturbation
Ufa mathematical journal, Tome 10 (2018) no. 2, pp. 14-30 Cet article a éte moissonné depuis la source Math-Net.Ru

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In the work we consider the Laplacian subject to the Dirichlet condition in an infinite planar strip perturbed by a periodic operator. The perturbation is introduced as an arbitrary bounded periodic operator in $L_2$ on the periodicity cell; then this operator is extended periodically on the entire strip. We study the band spectrum of such operator. The main obtained result is the absence of the spectral gaps in the lower part of the spectrum for a sufficiently small potential. The upper bound for the period ensuring such result is written explicitly as a certain number. It also involves a certain characteristics of the perturbing operator, which can be nonrigorously described as “the maximal oscillation of the perturbation”. We also explicitly write out the length of the part of the spectrum, in which the absence of the gaps is guaranteed. Such result can be regarded as a partial proof of the strong Bethe-Sommerfeld conjecture on absence of internal gaps in the band spectra of periodic operators for sufficiently small periods.
Keywords: periodic operator, Schrödinger operator, strip, Bethe-Sommerfeld conjecture. 
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     title = {On spectral gaps of a {Laplacian} in a strip with a bounded periodic perturbation},
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D. I. Borisov. On spectral gaps of a Laplacian in a strip with a bounded periodic perturbation. Ufa mathematical journal, Tome 10 (2018) no. 2, pp. 14-30. http://geodesic.mathdoc.fr/item/UFA_2018_10_2_a1/

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

[2] L. Parnovski, “Bethe-Sommerfeld conjecture”, Ann. H. 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. Helvetici., 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 Arithm., 120:3 (2005), 245–267 | DOI | MR | Zbl

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

[7] Y. Karpeshina, “Spectral properties of the periodic magnetic Schrödinger operator in the high-energy region. Two-dimensional case”, Comm. 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”, Comm. Part. Diff. Equat., 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] Suslina T.A., “Ob usrednenii periodicheskogo ellipticheskogo operatora v polose”, Alg. an., 16:1 (2004), 269–292 | Zbl

[13] Senik N.N., “Usrednenie periodicheskogo ellipticheskogo operatora v polose pri razlichnykh granichnykh usloviyakh”, Alg. an., 25:4 (2013), 182–259 | MR

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

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

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

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

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

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

[20] E. Krätzel, Lattice Points, Kluwer Academic Publishers, Dordrecht, 1988 | MR | Zbl