Weakly Resonant Tunnelling Interactions for Adiabatic Quasi-Periodic Schrödinger Operators
In this book, the authors collect together the developments they have recently achieved in the Anderson problem, i.e. the question of characterization of nature of spectrum in the theory of Schrödinger operators (on a line). They consider a "slow" adiabatic quasiperiodic perturbation of the one dimensional Schrödinger operator with a periodic potential and they study the corresponding resonance effects affecting the nature of the spectrum (which, in the unperturbed case, consists of spectral "bands", i.e. intervals of absolutely continuous spectra, separated by spectral gaps). They show how the presence of a quasiperiodic perturbation changes the classical behaviour of the spectrum. Namely, it creates resonance effects similar to the ones appearing in the problem of interacting quantum wells. These effects create additional layers, where a singular spectrum appears. The authors investigate the delicate nature of these phenomena and the interplay between the absolutely continuous and singular parts of the spectrum. The book explains in a concise but essentially self-contained way all the needed prerequisites and basic concepts (the monodromy matrix, density of states, Lyapunov exponents, Bloch solutions, etc.) needed for understanding these important and technically complicated phenomena.