Liouville theorems and spectral edge behavior on abelian coverings of compact manifolds
Peter Kuchment, Yehuda Pinchover
TL;DR
The paper analyzes Liouville-type theorems for periodic elliptic operators on abelian covers of compact manifolds, linking the finiteness of polynomial-growth solution spaces to the dispersion relation at spectral edges. Using Floquet–Gelfand theory, it shows that the space of solutions growing no faster than a polynomial of degree N, V_N(P), is finite-dimensional exactly when the real Fermi surface is finite, with such solutions expressible as linear combinations of Floquet solutions of order N and a homogenized operator Λ(D) providing a one-to-one correspondence with polynomial solutions. The dimensions and representations are computable from dispersion data under genericity, yielding explicit formulas tied to dispersion curves and extending to overdetermined elliptic systems, holomorphic functions on abelian covers, and to abelian coverings of combinatorial or quantum graphs. Concrete applications include Schrödinger operators, periodic magnetic Schrödinger operators, and nonselfadjoint cases, where dimensions like h_{n,N} and q_{n-1,N} illustrate how spectral-edge behavior governs Liouville-type phenomena.
Abstract
The paper describes relations between Liouville type theorems for solutions of a periodic elliptic equation (or a system) on an abelian cover of a compact Riemannian manifold and the structure of the dispersion relation for this equation at the edges of the spectrum. Here one says that the Liouville theorem holds if the space of solutions of any given polynomial growth is finite dimensional. The necessary and sufficient condition for a Liouville type theorem to hold is that the real Fermi surface of the elliptic operator consists of finitely many points (modulo the reciprocal lattice). Thus, such a theorem generically is expected to hold at the edges of the spectrum. The precise description of the spaces of polynomially growing solutions depends upon a `homogenized' constant coefficient operator determined by the analytic structure of the dispersion relation. In most cases, simple explicit formulas are found for the dimensions of the spaces of polynomially growing solutions in terms of the dispersion curves. The role of the base of the covering (in particular its dimension) is rather limited, while the deck group is of the most importance. The results are also established for overdetermined elliptic systems, which in particular leads to Liouville theorems for polynomially growing holomorphic functions on abelian coverings of compact analytic manifolds. Analogous theorems hold for abelian coverings of compact combinatorial or quantum graphs.