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Journal of Mathematical Physics / Volume 53 / Issue 2 / ARTICLES / Methods of Mathematical Physics

J. Math. Phys. 53, 023503 (2012); http://dx.doi.org/10.1063/1.3681895 (22 pages)

Planar waveguide with “twisted” boundary conditions: Small width

Denis Borisov1 and Giuseppe Cardone2

1Institute of Mathematics of Ufa Scientific Center of RAS, Chernyshevskogo st. 112, 450008, Ufa, Russia Federation and Bashkir State Pedagogical University, October St. 3a, 450000 Ufa, Russian Federation
2University of Sannio, Department of Engineering, Corso Garibaldi, 107, 82100 Benevento, Italy

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(Received 7 December 2011; accepted 13 January 2012; published online 7 February 2012)

We consider a planar waveguide with “twisted” boundary conditions. By twisting we mean a special combination of Dirichlet and Neumann boundary conditions. Assuming that the width of the waveguide goes to zero, we identify the effective (limiting) operator as the width of the waveguide tends to zero, establishes the uniform resolvent convergence in various possible operator norms, and gives the estimates for the rates of convergence. We show that studying the resolvent convergence can be treated as a certain threshold effect and we present an elegant technique which justifies such point of view.

© 2012 American Institute of Physics

Article Outline

  1. INTRODUCTION
  2. FORMULATION OF THE PROBLEM AND THE MAIN RESULT
  3. PRELIMINARIES
  4. PROOF OF THEOREM 2.1
  5. PROOF OF THEOREM 2.2

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KEYWORDS and PACS

Keywords

boundary-value problems, waveguides

PACS

  • 02.60.Lj

    Ordinary and partial differential equations; boundary value problems

  • 84.40.Az

    Waveguides, transmission lines, striplines

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.3681895

PUBLICATION DATA

ISSN

0022-2488 (print)  
1089-7658 (online)

Publisher

$abstract.society.value is a Member of CrossRef American Institute of Physics

For access to fully linked references, you need to log in.
    Albeverio, S. and Finco, D., “Coupling in the singular limit of thin quantum waveguides,” J. Math. Phys. 48, 032103 (2007)JMAPAQ000048000003032103000001.

    Borisov, D., Exner, P., and Gadyl'shin, R., “Geometric coupling thresholds in a two-dimensional strip,” J. Math. Phys. 43, 6265–6278 (2002)JMAPAQ000043000012006265000001.

    Borisov, D. and Cardone, G., “Planar waveguide with “twisted” boundary conditions: Discrete spectrum,” J. Math. Phys. 52, 123513 (2011)JMAPAQ000052000012123513000001.

    Dittrich, J. and Kříž, J., “Bound states in straight quantum waveguides with combined boundary conditions,” J. Math. Phys. 43, 3892–3915 (2002)JMAPAQ000043000008003892000001.

    Exner, P. and Post, O., “Convergence of resonances on thin branched quantum wave guides,” J. Math. Phys. 48, 092104 (2007)JMAPAQ000048000009092104000001.


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