On the spectrum of narrow periodic waveguides

This is a continuation of [1] and [2]. We consider the spectrum of the Dirichlet Laplacian on the domain {( x, y ) : 0 < y < εh ( x )}, where h ( x ) is a positive periodic function. The main assumption is that h ( x ) has one point of global maximum on the period interval. We study the location of bands and prove that the band lengths decay exponentially as ε → 0.