Asymptotics of resolvent integrals: The suppression of crossings for analytic lattice dispersion relations

Details Asymptotics of resolvent integrals: The suppression of crossings for analytic lattice dispersion relations Asymptotics of resolvent integrals: The suppression of crossings for analytic lattice dispersion relations

2006-04-20

arxiv.org/abs/math-ph/0604049v1

J. Math. Pures Appl. 87 (2007) 193-225

Physics

Mathematical Physics

47 pages

Scientific paper

10.1016/j.matpur.2006.11.003

We study the so called crossing estimate for analytic dispersion relations of periodic lattice systems in dimensions three and higher. Under a certain regularity assumption on the behavior of the dispersion relation near its critical values, we prove that an analytic dispersion relation suppresses crossings if and only if it is not a constant on any affine hyperplane. In particular, this will then be true for any dispersion relation which is a Morse function. We also provide two examples of simple lattice systems whose dispersion relations do not suppress crossings in the present sense.

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