Moment analysis for localization in random Schrödinger operators

Abstract

Abstract. We study localization effects of disorder on the spectral and dynamical properties of Schrödinger operators with random potentials. The new results include exponentially decaying bounds on the transition amplitude and related projection kernels, including in the mean. These are derived through the analysis of fractional moments of the resolvent, which are finite due to the resonance-diffusing effects of the disorder. The main difficulty which has up to now prevented an extension of this method to the continuum can be traced to the lack of a uniform bound on the Lifshitz-Krein spectral shift associated with the local potential terms. The difficulty is avoided here through the use of a weak-L 1 estimate concerning the boundary-value distribution of resolvents of maximally dissipative operators, combined with standard tools of relative compactness theory.

From Uncertainty Principles to Wegner Estimates

“…(In [1] operators with strictly positive imaginary part are called dissipative; we are facing operators with strictly negative imaginary part so that accretive seems to be an appropriate term; note, however that these notions are not standardized at all.) To do so, choose λ = 1 if B ≤ β < 1 and 0 < λ < 1 B else.…”

“…As discussed in [1], Appendix B and in [5], Section 3, the resolvent of a maximally accretive operator can always be written as the resolvent of a selfadjoint dilation.…”

An introduction to the mathematics of Anderson localization

“…It can also be deduced from (43) by considering g(h) = δ E (h), E ∈ I, using the result from [39] that almost surely all eigenvalues of h ω in I are non-degenerate. For details on this see Section 2.5 of [2], where a corresponding argument for the continuum Anderson model is provided which also applies to the discrete Anderson model considered here.…”

“…To make the central ideas behind the fractional moment method work in this setting required a much deeper understanding of some of the operator-theoretic aspects involved. Here we will follow the works [2] and [12], where these questions were settled. Earlier work in [26] extended certain aspects of the fractional moment method to continuum models, but still relied on finite-rank perturbation arguments by, for example, considering continuum models with random point interactions.…”

“…Entirely written in form of an outline is Section 8, in which we discuss the extension of the FMM to continuum Anderson models, as accomplished in [2] and [12]. This requires considerable technical effort and we merely point out the difficulties which had to be overcome and mention some of the tools which allowed to accomplish this.…”