On lacunas in the lower part of the spectrum of the periodic magnetic operator in a~strip

D. I. Borisov

Contemporary Mathematics. Fundamental Directions, Differential and functional differential equations, Tome 63 (2017) no. 3, pp. 373-391.

Résumé

We consider the Schrödinger periodic magnetic operator in an infinite flat straight strip. We prove that if the magnetic potential satisfies certain conditions and the period is small enough, then the lower part of the band spectrum has no inner lacunas. The length of the lower part of the band spectrum with no inner lacunas is calculated explicitly. The upper estimate for the small parameter allowing these results is calculated as a number as well.

Bibliographie

[1] D. I. Borisov, “On absence of lacunas in the lower part of Laplacian spectrum with fast alternation of boundary-value conditions in a strip”, Theor. Math. Phys. (in Russian)

[2] E. Lieb, M. Loss, Analysis, Nauchnaya kniga, Novosibirsk, 1998, (Russian translation)

[3] N. N. Senik, “Averaging of a periodic elliptic operator in a strip under various boundary conditions”, Algebra Anal., 25:4 (2013), 182–259 (in Russian) | MR | Zbl

[4] M. M. Skriganov, “Geometric and arithmetic methods in the spectral theory of multidimensional periodic operators”, Proc. Math. Inst. Russ. Acad. Sci., 171, 1985, 3–122 (in Russian) | MR | Zbl

[5] M. M. Skriganov, A. V. Sobolev, “Asymptotic estimates for spectral zones of periodic Schrödinger operators”, Algebra Anal., 17:1 (2005), 276–288, in Russian | MR | Zbl

[6] T. A. Suslina, “On averaging of periodic elliptic operator in a strip”, Algebra Anal., 16:1 (2004), 269–292 (in Russian) | MR | Zbl

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

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

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

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

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

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

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

[14] 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

[15] 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

[16] 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

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

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

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

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