Russian Journal of Mathematical Physics

, Volume 15, Issue 2, pp 238–242 | Cite as

On the spectrum of narrow periodic waveguides

Article

Abstract

This is a continuation of [1] and [2]. We consider the spectrum of the Dirichlet Laplacian on the domain {(x, y) : 0 < y < εh(x)}, where h(x) is a positive periodic function. The main assumption is that h(x) has one point of global maximum on the period interval. We study the location of bands and prove that the band lengths decay exponentially as ε → 0.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    L. Friedlander and M. Solomyak, “On the Spectrum of the Dirichlet Laplacian in a Narrow Strip,” Israel J. Math., to appear.Google Scholar
  2. 2.
    L. Friedlander and M. Solomyak, “On the Spectrum of the Dirichlet Laplacian in a Narrow Strip, II,” preprint arXiv:0710.1886.Google Scholar
  3. 3.
    R. Shterenberg, “Absolute Continuity of the Spectrum of the Two-Dimensional Periodic Schrödinger Operator with Strongly Subordinate Magnetic Potential,” Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 303, Issled. po Linein. Oper. i Teor. Funkts. 31, 279–320, 325–326 (2003) [J. Math. Sci. (N. Y.) 129 (4), 4087–4109 (2005)].Google Scholar
  4. 4.
    A. Sobolev and J. Walthoe, “Absolute Continuity in Periodic Waveguides,” Proc. London Math. Soc. 85(3), 717–741 (2002).MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    T. Suslina and R. Shterenberg, “Absolute Continuity of the Spectrum of the Magnetic Schrödinger Operator with a Metric in a Two-Dimensional Periodic Waveguide,” Algebra i Analiz 14(2), 159–206 (2002) [St. Petersburg Math. J. 14 (2), 305–343 (2003)].MATHMathSciNetGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2008

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ArizonaTucsonUSA
  2. 2.Department of MathematicsThe Weizmann Institute of ScienceRehovotIsrael

Personalised recommendations