Integral Representations and Liouville Theorems for Solutions of Periodic Elliptic Equations

Abstract

The paper discusses relations between the structure of the complex Fermi surface below the spectrum of a second order periodic elliptic equation and integral representations of certain classes of its solutions. These integral representations are analogs of those previously obtained by S. Agmon, S. Helgason, and other authors for solutions of the Helmholtz equation (i.e., for generalized eigenfunctions of Laplace operator). In a previous joint work with Y. Pinchover we described all solutions that can be represented as integrals of positive Bloch solutions over the imaginary Fermi surface, with a hyperfunction as a "measure". Here we characterize the class of solutions such that the corresponding hyperfunction is a distribution on the Fermi surface.

Liouville theorems and spectral edge behavior on abelian coverings of compact manifolds

“…In fact, we have the following stronger result which extends Lemma 15 in [59] to nonreal λ. In addition, the proof of the statement below is more elementary than in that lemma.…”

“…This transform is the main tool in the Floquet theory for PDEs (e.g., [52,78,82,85] [59,85], recast into the abelian covering form: Theorem 2.1.…”

“…It is a generalization of Lemma 25 in [59], where the Fredholm rather than the semi-Fredholm property is assumed. For the results of this section, Lemma 25 in [59] would be sufficient, while we need the full strength of the lemma below to treat overdetermined systems and holomorphic functions in Section 6. …”

“…It is shown in [59] that if Λ(0) ≥ 0, then the Fermi surface F L can touch the real space only at the origin (modulo the reciprocal lattice G * = (2πZ) n ) and in this case Λ(0) = 0.…”

“…The reader can find all the necessary preliminary information in the next two sections. In comparison with [59], the present paper provides explicit formulas for the dimensions of the corresponding spaces, where [59], in general, contains only an implicit algorithm to calculate these numbers. Moreover, multiplicities at spectral edges, as well as overdetermined systems (in particular, ∂-operators and Liouville theorems for holomorphic functions) are now allowed.…”