The classical Liouville theorem says that any entire function in C^n that grows polynomially, is a polynomial. Thus, for a fixed rate of polynomial growth, the space of such functions is finite dimensional, and its dimension can be easily computed. There is also a classical version of this theorem that deals with harmonic functions in R^n. A program of extending this theorem to solutions of Laplace-Beltrami equations on more general Riemannian manifolds (or to more general elliptic operators in Euclidean space) was started by S. T Yau about 30 years ago. Major work on this has been done by Colding and Minicozzi and P. Li with his co-authors.
A new development since the beginning on 1990s was devoted to Liouville theorems for periodic elliptic operators in R^n, where amazingly explicit dimension formulas could be obtained. This was due to work by Avellaneda and Lin, Moser and Struwe, and later P. Li and Wang.
At the same time, similar questions about validity of Liouville type theorems have been asked in the settings of holomorphic functions on coverings of complex manifolds (e.g., work by P. Lin and A. Brudnyi) and discrete harmonic functions on graph coverings with compact bases (G. Margulis).
In this talk, we provide an overview of recent results by the authors that treat all these three situations simultaneously for periodic operators on abelian coverings of compact bases. This includes elliptic operators on Riemannian manifolds, Cauchy-Riemann operators on analytic manifolds, discrete equations on graphs, and differential operators on quantum graphs. One can give necessary and sufficient conditions for the validity of Liouville theorems, as well as explicit dimension formulas. One can also mention striking similarities with the formulas obtained in the work by Gromov and Shubin concerning Riemann-Roch theorems for elliptic operators.
Albeit the problem does not look like being related to spectral theory, the answers (as well as the tools) are given in terms of the structure of the dispersion relation near edges of the spectrum of the operator.
- http://arxiv.org/abs/math-ph/0503010 - preprint of the paper to appear