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.
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David Krejčiřík, Jan Kříž, On the Spectrum of Curved Planar Waveguides. Publ. Res. Inst. Math. Sci. 41 (2005), no. 3, pp. 757–791DOI 10.2977/PRIMS/1145475229