Band Edge Localization Beyond Regular Floquet Eigenvalues
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nnales Henri Poincar´ e
Band Edge Localization Beyond Regular Floquet Eigenvalues Albrecht Seelmann and Matthias T¨aufer Abstract. We prove that localization near band edges of multi-dimensional ergodic random Schr¨ odinger operators with periodic background potential in L2 (Rd ) is universal. By this, we mean that localization in its strongest dynamical form holds without extra assumptions on the random variables and independently of regularity or degeneracy of the Floquet eigenvalues of the background operator. The main novelty is an initial scale estimate the proof of which avoids Floquet theory altogether and uses instead an interplay between quantitative unique continuation and large deviation estimates. Furthermore, our reasoning is sufficiently flexible to prove this initial scale estimate in a non-ergodic setting, which promises to be an ingredient for understanding band edge localization also in these situations. Mathematics Subject Classification. Primary 47B80; Secondary 35J10, 35R60, 81Q15, 82B44.
1. Introduction and Results The Anderson model dates back to the work of Anderson in 1958 [2] in condensed matter physics who argued that the presence of disorder will drastically change the dynamics of electrons in a solid. This has triggered a huge research activity in mathematics and physics during the past 60 years. We refer to the monographs [3,38,42,48] for an overview on the mathematics literature. While Anderson’s original work was on a lattice model, analogous phenomena have since been studied for continuum Schr¨ odinger operators. The prototypical model investigated in this context is the ergodic Alloy-type or continuum Anderson model ωj u(· − j), Hωerg = −Δ + Vper + Vωerg = Hper + Vωerg , Vωerg = j∈Zd
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A. Seelmann and M. T¨ aufer
Ann. Henri Poincar´e
in L2 (Rd ), where Vper is a Zd -periodic potential, ω = (ωj )j∈Zd is a sequence of independent and identically distributed random variables with bounded density, and u is a bump function modelling the effective potential around a single atom. Under mild assumptions, this random family of self-adjoint operators has almost sure spectrum, which means that there exists a set Σ ⊂ R such that for almost every realization of ω the random operator Hωerg has spectrum Σ. The general philosophy is that randomness leads to Anderson localization, at least in a neighbourhood of the edges of Σ, or—in dimension one—on the whole of Σ. Anderson (or exponential ) localization in an interval I ⊂ Σ means that I almost surely consists of pure point spectrum of Hωerg , that is, I ⊂ σ pp (Hωerg ),
I ∩ σ c (Hωerg ) = ∅,
(1.1)
and all eigenfunctions corresponding to eigenvalues in I are exponentially decaying. This is a dramatic difference to the background operator Hper which has only absolutely continuous spectrum and no eigenvalues. There also exist stronger notions of localization such as dynamical localization, describing the non-spreading of wave packets, see, e.g. [22] for an overview. Its formally strongest form in [22, Definition 3.1 (vii)] (cf. [15, Eq. (1.6)]
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