Integral representations of 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.