In this dissertation, we study some spectral problems for periodic elliptic operators arising in solid state physics, material sciences, and differential geometry. More precisely, we are interested in dealing with various effects near and at spectral edges of such operators. We use the name "threshold effects" for the features that depend only on the infinitesimal structure (e.g., a finite number of Taylor coefficients) of the dispersion relation at a spectral edge. We begin with an example of a threshold effect by describing explicitly the asymptotics of the Green's function near a spectral edge of an internal gap of the spectrum of a periodic elliptic operator of second-order on Euclidean spaces, as long as the dispersion relation of this operator has a non-degenerate parabolic extremum there. This result confirms the expectation that the asymptotics of such operators resemble the case of the Laplace operator. Then we generalize these results by establishing Green's function asymptotics near and at gap edges of periodic elliptic operators on abelian coverings of compact Riemannian manifolds. The interesting feature we discover here is that the torsion-free rank of the deck transformation group plays a more important role than the dimension of the covering manifold. Finally, we provide a combination of the Liouville and the Riemann-Roch theorems for periodic elliptic operators on abelian co-compact coverings. We obtain several results in this direction for a wide class of periodic elliptic operators. As a simple application of our Liouville-Riemann-Roch inequalities, we prove the existence of non-trivial solutions of polynomial growth of certain periodic elliptic operators on noncompact abelian coverings with prescribed zeros, provided that such solutions grow fast enough.
