Stable and unstable almost-periodic problems

点击翻译按钮获取中文摘要

The main objective of our proposal is to study spectra of elliptic differential operators with almost-periodic coefficients in all dimensions. Such operators are interesting from physical points of view, since these operators describe various physical phenomena in crystalline media. Perhaps they are even more interesting from the mathematical prospective, since they represent the next step (after periodic operators) in the `natural scale of complexity' of ergodic operators. The random operators are located at the opposite end of this scale (they are the most complex ergodic operators). Whereas a lot is known about the spectral structure of either periodic or random operators, there exist very little information about the spectral structure of almost-periodic operators acting in dimensions higher than one. We intend to develop a new approach for working with such operators, which we have provisionally called the KAM-Floquet-Bloch decomposition. If we are successful, this approach will represent a very significant developement in the spectral theory of almost-periodic operators. We also plan to study a number of other problems, which we consider to be of lower risk than the main goal.