Asymptotics of spectral gaps in a regularly perturbed periodic waveguide
Since the spectrum of a periodic waveguide is the union of a countable family of closed bounded segments (spectral bands), it can contain opened spectral gaps, i.e., intervals in the real positive semi-axis which are free of the spectrum but have both endpoints in it. A cylindrical waveguide has the intact spectrum that is a closed ray. We consider a small periodic perturbation of the waveguide wall and, by means of an asymptotic analysis of the eigenvalues in the model problem on the periodicity cell, we show how a spectral gap opens when the cylindrical waveguide converts into a periodic one. Indeed, a cylindrical waveguide can be interpreted as a periodic one with arbitrary period, but all its spectral bands touch each other. A periodic perturbation of the waveguide wall provides the split of band edges. This effect is known in the physical literature for waveguides of different shapes, and in the paper we provide a rigorous mathematical proof of the effect. We investigate several variants of the edge split (alone and coupled, simple and multiple knots). We derive explicit formulas for a two-dimensional waveguide.