  This paper considers the propagation of TE-modes in photonic crystal
waveguides. The waveguide is created by introducing a linear defect into a
periodic background medium. Both the periodic background problem and the
perturbed problem are modelled by a divergence type equation. A feature of our
analysis is that we allow discontinuities in the coefficients of the operator,
which is required to model many photonic crystals. Using the Floquet-Bloch
theory in negative order Sobolev spaces, we characterize the precise number of
eigenvalues created by the line defect in terms of the band functions of the
original periodic background medium for arbitrarily weak defects.
