Influence of the bound states in the Neumann Laplacian in a thin waveguide

We study the Neumann Laplacian operator $-\Delta_\Omega^N$ restricted to a twisted waveguide $\Omega$. The goal is to find the effective operator when the diameter of $\Omega$ tends to zero. However, when $\Omega$ is "squeezed" there are divergent eigenvalues due to the transverse oscillations. We show that each one of these eigenvalues influences the action of the effective operator in a different way. In the case where $\Omega$ is periodic and sufficiently thin, we find information about the absolutely continuous spectrum of $-\Delta_\Omega^N$ and the existence and location of band gaps in its structure.