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

Jani Lukkarinen
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Abstract

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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Jani Lukkarinen. 2006-04-20. Asymptotics of resolvent integrals: The suppression of crossings for analytic lattice dispersion relations. https://doi.org/10.1016/j.matpur.2006.11.003

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