  We identify explicit conditions on geometry and material contrast for
creating band gaps in 2-d photonic and 3-d acoustic crystals. This approach is
novel and makes use of the electro-static and quasi-periodic source free
resonances of the crystal. The source free modes deliver a spectral
representation for solution operators associated with propagation of
electromagnetic and acoustic waves inside periodic high contrast media. An
accurate characterization of the quasi-periodic and electrostatic resonance
spectrum in terms of the shape and geometry of the scatters is possible. This
information together with the Dirichlet and a Neumann like spectra associated
with the inclusions deliver conditions sufficient for opening band gaps at
finite contrast. The theory provides a systematic means for the identification
of photonic and phononic band gaps within a specified frequency range.
