The band-edge behavior of the density of surfacic states

This paper is devoted to the asymptotics of the density of surfacic states near the spectral edges for a discrete surfacic Anderson model. Two types of spectral edges have to be considered : fluctuating edges and stable edges. Each type has its own type of asymptotics. In the case of fluctuating edges, one obtains Lifshitz tails the parameters of which are given by the initial operator suitably “reduced” to the surface. For stable edges, the surface density of states behaves like the surface density of states of a constant (equal to the expectation of the random potential) surface potential. Among the tools used to establish this are the asymptotics of the surface density of states for constant surface potentials. 0. Introduction On Zd (d = d1 + d2, d1 > 0, d2 > 0), we consider random Hamiltonians of the form Hω = − 1 2 ∆ + Vω where • −∆ is the free Laplace operator, i.e., −(∆u)(n) = |m−n|=1 u(m); • Vω is a random potential concentrated on the sub-lattice Zd1 × {0} ⊂ Zd of the form (0.1) Vω(γ1, γ2) = { ωγ1 if γ2 = 0, 0 if γ2 6= 0. , γ = (γ1, γ2) ∈ Z1 × Z2 = Z. and (ωγ1)γ1∈Zd1 is a family of i.i.d. bounded random variables. For the sake of simplicity, let us assume that the random variables are uniformly distributed in [a, b] (a < b). To keep the exposition as simple as possible in the introduction, we use these quite restrictive assumptions. We will deal with more general models in the next section. The operator Hω is bounded for almost every ω. It is ergodic with respect to shifts parallel to the surface. So we know there exists Σ the almost sure spectrum of Hω (see e.g. [14, 23]. For Hω, one defines the integrated density of surface states (the IDSS in the sequel), in the following way (see e.g. [8, 2, 3, 20]): for φ ∈ C∞ 0 (R), we set (0.2) (φ, ns) = E(tr(Π1[φ(Hω)− φ(− 1 2 ∆)]Π1)) where Π1 is the orthogonal projector on the subspace Cδ0 ⊗ `2(Zd2) ⊂ `2(Zd). Here, δ0 denotes the vector with components (δ0j)j∈Zd1 . Obviously, equation (0.2) defines the integrated density of surface states ns only up to a constant. We choose this constant so that ns vanishes below Σ ∪ Σ0 where Σ0 is the spectrum of − 12∆. We will see later on that, up to addition of a well controlled distribution, ns is a positive measure. One knows that Σ = σ(− 2∆)∪supp(dns) (see [8, 9, 2]. We will study the behavior of ns at the edges of Σ. To simplify this set as much as possible, we will assume that the support of the random variables (ωγ1)γ1∈Zd1 is connected. Under this assumption, we know that Lemma 0.1. Σ is a compact interval given by (0.3) Σ = σ(− 2 ∆d1) + ⋃ ω0∈[a,b] σ(− 2 ∆d2 + ωΠ 2 0) where Π0 is the projector on the unit vector δ 2 0 ∈ `2(Zd2). 1 This is a consequence of a standard characterization of Σ in terms of periodic potentials (see [14, 23]). The assumption that the random variables have connected support can be relaxed; more connected components for the support of the random variables will in general give rise to more spectral edges (as in the case of bulk randomness, see [16]). For the value of Σ, two different possibilities occur : (1) Σ = σ(− 2∆) + [−α, β] = [−d− α, d+ β] where α = α(a), β = β(b) and α+ β > 0; this occurs • if d2 ≤ 2 and either a < 0, in which case α(a) > 0, or b > 0, in which case β(b) > 0, • if d2 ≥ 3 and a > a0 or b > b0, where, by (0.3), the thresholds a0 and b0 are uniquely determined by the family of operators (− 12∆d2 + tΠ0)t∈R. If α > 0 (resp. β > 0), we say that the left (resp. right) edge is a “fluctuation edge” or “fluctuation boundary” (see [23]). If α = 0 (resp. β = 0), we will speak of a “stable edge” or “stable boundary”. (2) Σ = σ(− 2∆); this occurs only in d2 ≥ 3 and if a is not too large, that is, if a ∈ (0, a0]. In this case, both spectral edges are stable. On the other hand, it is well known (see [24]) that, • if d2 = 1, 2, then, for a > 0, σ(− 12∆d2 − aΠ0) = [−d2, d2] ∪ {λ(a)}, and the spectrum in [−d2, d2] is purely absolutely continuous and λ(a) is a simple eigenvalue; • if d2 ≥ 3, there exists a0 > 0 such that – if 0 < a < a0, then, σ(− 2∆d2 − aΠ0) = [−d2, d2], and the spectrum is purely absolutely continuous; – if a = a0, then ∗ if d2 = 3, 4, then σ(−2∆d2 − aΠ0) = [−d2, d2], the spectrum is purely absolutely continuous, and −d2 is a resonance for − 1 2∆d2 − aΠ0; ∗ if d2 ≥ 5, then σ(− 2∆d2 −aΠ0) = [−d2, d2], the spectrum is purely absolutely continuous in [−d2, d2), and −d2 is a simple eigenvalue for − 12∆d2 − aΠ0; – if a > a0, then, σ(−2∆d2 − aΠ0) = [−d2, d2] ∪ {λ(a)}, and the spectrum in [−d2, d2] is purely absolutely continuous and λ(a) is a simple eigenvalue; For the operator − 1 2∆d2 + bΠ0, we have a symmetric situation. Our aim is to study the density of surface states near the edges of Σ. In the present case, both edges are obviously symmetric. So we will only describe the lower edge. One has to distinguish between the case of fluctuation and stable edges. The behavior in the two cases are radically different. 0.1. The stable edge. As the discussion for lower and upper edge are symmetric, let us assume the lower edge is stable and work near that edge. In the case of a stable edge, it is convenient to modify the normalization of the IDSS. Therefore, we introduce the operator Ht = − 1 2 ∆ + t1⊗Π0. As above, let a be the infimum of the random variables (ωj)j . For φ ∈ C∞ 0 (R), define (φ, ns,norm) = E(tr(Π1[φ(Hω)− φ(Ha)]Π1)) The advantage of this renormalization is that the IDSS ns,norm is the distributional derivative of a positive measure. Indeed, for φ ∈ C∞ 0 (R), define (φ, dNs,norm) = −E(tr(Π1[P (φ)(Hω)− P (φ)(Ha)]Π1))