Ergodic Schrödinger Operators

Abstract

We now suppose that (a_n,b_n), n ε ZZ, is a stationary random process This means that the real random variables (a_n (ω), b_n(ω)) are defined on some complete probability space (Ω,a,ℙ) and that there exists an invertible measurable transformation θ of Ω, leaving ℙ invariant and such that a_n+1=a_nºθ, b_n+1, ºθ. In general we don’t write the variable ω and when a property depends only upon the common law of the sequence (a_n, b_n) we omit the index n and speak of the variables (a) and (b). We say that the family H(ω) of associated operators on H is ergodic if a θ invariant measurable subset of Ω is of zero or one ℙ measure. It’s easily seen that H° θ = U^-1 H U where U is the shift measure. It’s easily seen that H o θ = U^-1 H U where U is the shift operator on ZZ, (Uψ)_n = ψ_n-1.