Arbeitsgemeinschaft mit aktuellem Thema: Quasiperiodic Schr ¨ odinger Operators Mathematisches Forschungsinstitut Oberwolfach April 1–7, 2012 Organizers: Artur Avila David Damanik Svetlana Jitomirskaya Inst. de Math. de Jussieu Dept. of Mathematics Dept. of Mathematics CNRS UMR 7586 Rice University University of California 75252 Paris Houston, TX 77251 Irvine, CA 92664 France USA USA artur@math.sunysb.edu damanik@rice.edu szhitomi@math.uci.edu Introduction: The Arbeitsgemeinschaft will discuss quasiperiodic Schr¨odinger operators of the form [Hψ](n)= ψ(n + 1) + ψ(n − 1) + V (n)ψ(n) with potential V : Z → R given by V (n)= f (ω + nα), where ω,α ∈ T k = R k /Z k ,λ ∈ R and α =(α 1 ,...,α k ) is such that 1,α 1 ,...,α k are independent over the rational numbers, and f : T k → R is assumed to be at least continuous. H acts on the Hilbert space ℓ 2 (Z) as a bounded self-adjoint operator. While most of the talks will focus on this set- ting, occasionally we will also consider the multi-dimensional setting, where H acts on ℓ 2 (Z d ) and is again given by the sum of the discrete Laplacian and the multiplication operator generated by a quasiperiodic V : Z d → R. For the sake of simplicity, this introduction will consider the one-dimensional case. The associated unitary group {e −itH } t∈R describes the evolution of a quan- tum particle subjected to the quasiperiodic environment given by V . For any t ∈ R and n ∈ Z, |〈δ n ,e −itH ψ〉| 2 is the probability of finding the particle, 1