Green's Function Estimates for Lattice Schrödinger Operators and Applications
This book deals mainly with the localization problem (i.e. with a description of the properties of the spectrum, the eigenstates and the resolvent) for quasi-periodic lattice Schrödinger operators. The origins of the field can be traced to the seminal works of Sinai and collaborators in the eighties and to the work of Fröhlich, Spencer and Wittwer. The main goal of the book is to give an overview of some of the important lines of current research in the field and to present it in a unified way. Some parts of the text thus follow original research literature, in particular recent articles written by the author and his collaborators.
This determines the organization of the book, which consists of twenty short chapters. Both nonperturbative and perturbative results are treated. Ideas on the latter (from chapter 14 onwards) come from KAM theory and they have a wide range of applications. The book also contains new results, e.g. results concerning regularity properties of the Lyapunov exponent and integrated density of states in chapter 8. Chapters 18 - 20 deal with the problem of quasi-periodic solutions for infinite Hamiltonian systems given by nonlinear Schrödinger or wave equations. A new general method presented by the author can be used to consider the problem for systems that are perturbations of the linear ones in any dimension. The Figotin-Pastur approach for Schrödinger operators associated to strongly mixing dynamical systems is explained in the appendix. The whole subject is still an ongoing area of research and some methods (the renormalization group by Helfer and Sjöstrand) are omitted in the book. Still, it gives an up to date review of this important field of current research.