Band Gap of the Spectrum in Periodically Curved Quantum Waveguides

Abstract

In this paper we study the spectral gap of the Dirichlet Laplacian on a periodically curved strip. We begin with the definition of the planar strip and the Dirichlet Laplacian. For γ ∈ C(R; R), let

$$ {k_\gamma }:R \ni s \mapsto \left( {{a_\gamma }\left( s \right),{b_\gamma }\left( s \right)} \right) \ni {R^2}$$

be a curve parameterized by its arc length whose curvature at s is γ(s). For d > 0, we define

$${\Omega _{d,\gamma }} \equiv \left\{ {\left( {{a_\gamma }\left( s \right) - u\frac{d}{{ds}}{b_\gamma }\left( s \right),{b_\gamma }\left( s \right) + u\frac{d}{{ds}}{a_\gamma }\left( s \right)} \right) \in {R^2};s \in R, - d < u < d} \right\}.$$