JournalsprimsVol. 41 , No. 3DOI 10.2977/prims/1145475229

On the Spectrum of Curved Planar Waveguides

  • David Krejčiřík

    Czech Technical University in Prague, Czech Republic
  • Jan Kříž

    University Hradec Králové, Hradec Kralove, Czech Republic
On the Spectrum of Curved Planar Waveguides cover

Abstract

The spectrum of the Laplace operator in a curved strip of constant width built along an infinite plane curve, subject to three different types of boundary conditions (Dirichlet, Neumann and a combination of these ones, respectively), is investigated. We prove that the essential spectrum as a set is stable under any curvature of the reference curve which vanishes at infinity and find various sufficient conditions which guarantee the existence of geometrically induced discrete spectrum. Furthermore, we derive a lower bound to the distance between the essential spectrum and the spectral threshold for locally curved strips. The paper is also intended as an overview of some new and old results on spectral properties of curved quantum waveguides.