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

TL;DR

This paper investigates the asymptotic behavior of Green's functions near spectral edges of periodic elliptic operators, extending known results from the Laplace operator to more complex spectral gap scenarios in higher dimensions.

Contribution

It establishes the possibility of deriving Green's function asymptotics at spectral edges of periodic elliptic operators in dimensions greater than two, expanding spectral analysis understanding.

Findings

Asymptotics are obtainable at spectral edges in dimensions d>2.

Results extend classical Laplace operator asymptotics to periodic elliptic operators.

Spectral edge behavior is characterized for higher-dimensional cases.

Abstract

Precise asymptotics known for the Green's function of the Laplace operator have found their analogs for periodic elliptic operators of the second order at and below 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. As the result of this work shows, this is possible at a spectral edge in dimensions d>2.