Absolutely continuous spectrum and spectral transition for some continuous random operators

Abstract

In this paper we consider two classes of random Hamiltonians on L 2 (R d ) one that imitates the lattice case and the other a Schrödinger operator with non-decaying, non-sparse potential both of which exhibit a.c. spectrum. In the former case we also know the existence of dense pure point spectrum for some disorder thus exhibiting spectral transition valid for the Bethe lattice and expected for the Anderson model in higher dimension.

AC spectrum for a class of random operators at small disorder

“…The absence of point spectrum in I follows from the Virial theorem (see a version given in Proposition 2.1, [15], the condition (1) there is easy since [A, H ω λ ] extends to a bounded operator from D(A) and ( 2) there holds for all vectors in ℓ 2 (Z ν ) of finite support, which are mapped into domain of A by (H ω λ ± z) −1 for |z| > H ω λ as can be seen from a use of Neumann series expansion).…”

“…We give examples of random potentials that satisfy the hypothesis 1.1, our examples are adapted from the continuum case given in Krishna [15] which are extensions of the Rodnianski-Schlag models [21]. We note that we use below the ℓ 2 -norm on R ν while we use the ℓ ∞ norm on Z ν , so that…”

Schrödinger operators with decaying randomness - Pure point spectrum

“…For 0 < α < 2 they showed that the operator H ω has infinitely many negative eigenvalues almost surely and provided a bound for N ω (E) := #{x ∈ σ(H ω ) : x < E} in terms of stationary model (i.e choosing a n = 1). Kirsch-Krishna-Obermeit [9] (see also [14,15] and [12]) considered H ω = −∆ + V ω on ℓ 2 (Z d ), where V ω (n) = a n ω n and ∆ is the adjacency operator for the graph Z d . Under certain restriction on {a n } n and µ they showed that σ(H ω ) = R and σ c (H ω ) ⊆ [−2d, 2d] almost surely.…”

Scattering Theory of Schrödinger Operators with Random Sparse Potentials

“…Moreover, since the work of Krishna [15], wave operators have been established for decaying random potentials (see e.g. [1,2,15,16,23]) and random sparse potentials (see e.g. [6,8,11,13,18,21]).…”