The article surveys the known results and conjectures about the analytic properties of dispersion relations and Fermi surfaces for periodic equations of mathematical physics and their spectral incarnations.
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The specific choice of a co-compact abelian sub-group of shifts is irrelevant at this, and many others, moment.
99.
Infinite differentiability of the coefficients is a significant overkill, but we will not address the issues of minimal conditions on the coefficients.
100.
One can handle to some extent the hypo-elliptic, e.g., parabolic equations, while things get much harder there.50
101.
This labeling disappears when k is complex.
102.
It is saying that a solution vanishing in an open set is identically zero.
103.
It is interesting to compare the spider decoration with the zig-zag product used to produce expander graphs.77
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