Green's function asymptotics near the internal edges of spectra of periodic elliptic operators. Spectral gap interior

TL;DR

This paper derives precise asymptotics for the Green's function of generic second-order periodic elliptic operators near spectral gap edges, extending known results from the spectrum's bottom to internal edges.

Contribution

It provides the first known asymptotic analysis of Green's functions at internal spectral gap edges for generic periodic elliptic operators.

Findings

Asymptotics established near spectral gap edges at symmetry points.

Results apply to operators with real coefficients in dimensions d ≥ 2.

Extends classical Green's function asymptotics from spectrum bottom to internal edges.

Abstract

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 finding such asymptotics near a gap edge, for "generic" periodic elliptic operators of second-order with real coefficients in dimension $d \geq 2$, when the gap edge occurs at a symmetry point of the Brillouin zone.