The main novelty is that we consider wave guides i.e. a medium that is periodic in cer-
tain directions of space only.

It is well known that for a medium periodic in all directions both the Schrödinger and
the Maxwell operator exhibit only purely absolutely continuous spectrum (see e.g. work
by  M. Birman, A. Morame, L. Thomas, A. Sobolev, Z. Shen, T. Suslina, etc). 

This is the case d = 3 
The first general idea underlying all these works is Floquet theory.

Using the translational invariance of the coefficients of the operator, one can write it as
a direct integral of operators acting on the fundamental cell:
consider the Floquet-Gelfand transformation


It is a unitary operator