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Precise asymptotics known for the Green function of the Laplacian have found their analogs for bounded below periodic elliptic operators of the second-order below and at the bottom of the spectrum. Due to the band-gap structure of the spectra of such operators, the question arises whether similar results can be obtained near or at the edges of spectral gaps. In a previous work, two of the authors considered the case of a spectral edge. The main result of this article is nding such asymptotics near a gap edge, for “generic” periodic elliptic operators of second-order with real coecients in dimension , when the gap edge occurs at a symmetry point of the Brillouin zone.
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Minh Kha, Peter Kuchment, Andrew Raich, Green's function asymptotics near the internal edges of spectra of periodic elliptic operators. Spectral gap interior. J. Spectr. Theory 7 (2017), no. 4, pp. 1171–1233DOI 10.4171/JST/188