Explicit finite volume criteria for localization in continuous random media and applications

Abstract

We give finite volume criteria for localization of quantum or classical waves in continuous random media. We provide explicit conditions, depending on the parameters of the model, for starting the bootstrap multiscale analysis. A simple application to Anderson Hamiltonians on the continuum yields localization at the bottom of the spectrum in an interval of size Cλ for large λ, where λ stands for the disorder parameter. A more sophisticated application proves localization for two-dimensional random Schrödinger operators in a constant magnetic field (random Landau Hamiltonians) up to a distance C log B B from the Landau levels for large B, where B is the strength of the magnetic field.

Unique Continuation Principle for Spectral Projections of Schrödinger Operators and Optimal Wegner Estimates for Non-ergodic Random Schrödinger Operators

“…(The bootstrap multiscale analysis can be adapted for crooked Anderson Hamiltonians [R1, R2].) By complete localization on an interval I we mean that for all E ∈ I there exists δ(E) > 0 such that we can perform the bootstrap multiscale analysis on the interval (E − δ(E), E + δ(E)), obtaining Anderson and dynamical localization; see [GK1,GK2,GK3].…”

Ground state energy of trimmed discrete Schrödinger operators and localization for trimmed Anderson models

“…The Wegner type estimate (1.17) allows us to establish localization for Γ-trimmed Anderson models at the bottom of the spectrum. By complete localization on an interval I we mean that for all E ∈ I there exists δ(E) > 0 such that we can perform the bootstrap multiscale analysis on the interval (E − δ(E), E + δ(E)), obtaining Anderson and dynamical localization; see [GK1,GK2,GK3]. (Note that by this definition we always have complete localization in (−∞, E ∅ (H 0 )).…”

Untitled

“…It follows from standard results [Klein and Molchanov 2006;Carmona and Lacroix 1990;Pastur and Figotin 1992] that there exist fixed subsets , pp , ac and sc of ‫ޒ‬ so that the spectrum σ (H ω ) of H ω , as well as its pure point, absolutely continuous, and singular continuous components, are equal to these fixed sets with probability one. With our normalization, the nonrandom spectrum of an Anderson Hamitonian H ω satisfies [Kirsch and Martinelli 1982] An Anderson Hamiltonian H ω exhibits Anderson and dynamical localization at the bottom of the spectrum [Martinelli and Holden 1984;Combes and Hislop 1994;Klopp 1995;Kirsch et al 1998;Germinet and De Bièvre 1998;Damanik and Stollmann 2001;Germinet and Klein 2001;2003a;Aizenman et al 2006]. More precisely, there exists an energy E 1 > 0 such that [0, E 1 ] ⊂ CL , where CL is the region of complete localization for the random operator H ω [Germinet and Klein 2004;.…”

“…It is also natural to consider dynamical localization, where the moments of a wave packet, initially localized both in space and in energy, should remain uniformly bounded under time evolution. For the multidimensional continuum Anderson Hamiltonian, localization has been proved by a multiscale analysis [Martinelli and Holden 1984;Combes and Hislop 1994;Klopp 1995;Kirsch et al 1998;Germinet and De Bièvre 1998;Damanik and Stollmann 2001;Germinet and Klein 2001;2003a], and, in the case when we have the covering condition δ − ≥ 1, also by the fractional moment method [Aizenman et al 2006]. These methods give more than just Anderson or dynamical localization, although they imply both.…”