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On the spectrum of curved planar waveguides. (English) Zbl 1113.35143
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.
Reviewer: Zhiqin Lu (Irvine)

MSC:
35Q40 PDEs in connection with quantum mechanics
35P15 Estimates of eigenvalues in context of PDEs
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
78A50 Antennas, waveguides in optics and electromagnetic theory
47F05 General theory of partial differential operators
47N50 Applications of operator theory in the physical sciences
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