Green's function asymptotics of periodic elliptic operators on abelian coverings of compact manifolds

Abstract

Abstract. The main results of this article provide asymptotics at infinity of the Green's functions near and at the spectral gap edges for "generic" periodic secondorder, self-adjoint, elliptic operators on noncompact Riemannian co-compact coverings with abelian deck groups. Previously, analogous results have been known for the case of R n 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.

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“…One can ask the same question near and at the edges of the internal gaps of the spectrum, as long as the dispersion relation has a nondegenerate (parabolic) extremum there. Results of this type were obtained in the recent works [219,220,256].…”

On the structure of band edges of $2$-dimensional periodic elliptic operators

“…An immediate consequence of our result is that Liouville theorems (in the sense of [17,18]) hold for the operator (2.1) at all gap edges, see Corollary 2.2. Our result can also be used in studying Green's function asymptotics near spectral gap edges, see [10,19,9], and to obtain a "variable period" version of the non-degeneracy conjecture in 2D [20].…”

Generic properties of dispersion relations for discrete periodic operators

“…This in turn would trigger appearance of various properties analogous to those of the Laplace operator. One can mention, for instance, electron's effective masses in solid state theory [3,30], Green's function asymptotics [6,26,28,37], homogenization [8,14], Liouville type theorems [4,27,35,36,38], Anderson localization [1], perturbation of discrete spectra in gaps in general [11][12][13], and others.…”

“…The eigenvalues of the matrix A α (z) form the spectrum of Λ α (z). Dropping the subscript α, λ belongs to the spectrum of Λ(z) if and only if it satisfies the characteristic equation, (26) λ 2 − λTrA(z) + det A(z) = 0.…”