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Zbl 1113.35143
Krejčiř\'ik, David; Kř\'iž, Jan
On the spectrum of curved planar waveguides. (English)
[J] Publ. Res. Inst. Math. Sci. 41, No. 3, 757-791 (2005). ISSN 0034-5318; ISSN 1663-4926/e

The authors study the spectrum properties of the Laplacian on 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, respectively). Under certain natural conditions, the authors prove that (Theorem 4.1), the essential spectrum of the strip is a connected set, and moreover, under the Neumann boundary conditions, the essential spectrum is $[0,+\infty)$ and no discrete spectrum exists (Theorem 4.2). Under the Dirichlet boundary condition, with certain natural conditions, the ground state exists (Theorem 4.3). In the more general situation of mixed boundary conditions, sufficient conditions for the upper and lower bounds of the infimum of the spectrum were given (Theorem 4.4). The authors also estimate the number of bound states and spectral thresholds in the cases of mixed boundary conditions. In addition, various examples of are given in the paper. The paper is also intended as an overview of some new and old results on spectral properties of curved quantum waveguides.
[Zhiqin Lu (Irvine)]

MSC 2000:: *35Q40 PDE from quantum mechanics
35P15 Estimation of eigenvalues for PD operators
81Q10 Selfadjoint operator theory in quantum theory
78A50 Antennas, wave-guides
47F05 Partial differential operators
47N50 Appl. of operator theory in quantum physics

Keywords: essential spectrum; ground state; bound state; planar waveguide

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