Asymptotics of Green's functions of periodic elliptic operators on abelian Riemannian covers
Green’s function behavior near and at a spectral edge of a periodic operator is one of what was called by M. Birman and T. Suslina “threshold properties.”. These properties depend upon the infinitesimal structure of the dispersion relation at the spectral edge. For a "generic" symmetric periodic second-order elliptic operators on a co-compact abelian cover, we will describe the asymptotics at infinity of the Green's functions near and at the spectral gap edge as long as the dispersion relation of the operator has a non-degenerate extremum there. Previously, analogous results have been known for the Euclidean case only. One of the interesting features discovered is that the rank of the deck group plays more important role than the dimension of the manifold. We also discuss the case of non-symmetric periodic elliptic operators.