The band-edge behavior of the density of surfacic states

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The band-edge behavior of the density of surfacic states

Werner Kirsch, Frédéric Klopp

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Werner Kirsch, Frédéric Klopp. The band-edge behavior of the density of surfacic states. 2004.

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THE BAND-EDGE BEHAVIOR OF THE DENSITY OF SURFACIC STATES

WERNER KIRSCH AND FR ´ED´ERIC KLOPP

Abstract. 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

OnZ^d (d=d₁+d₂,d₁ >0,d₂>0), we consider random Hamiltonians of the form H_ω =−1

2∆ +V_ω where

• −∆ is the free Laplace operator, i.e., −(∆u)(n) =P

|m−n|=1u(m);

• V_ω is a random potential concentrated on the sub-latticeZ^d1× {0} ⊂Z^d of the form (0.1) V_ω(γ₁, γ₂) =

(ω_γ1 ifγ₂ = 0,

0 ifγ₂ 6= 0., γ= (γ₁, γ₂)∈Z^d1×Z^d2 =Z^d.

and (ω_γ1)_γ1∈Z^d1 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 H_ω 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 ofH_ω (see e.g. [14, 23].

ForH_ω, 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∈ C0^∞(R), we set

(0.2) (ϕ, n_s) =E(tr(Π₁[ϕ(H_ω)−ϕ(−1

2∆)]Π₁))

where Π₁ is the orthogonal projector on the subspace Cδ₀⊗ℓ²(Z^d2) ⊂ℓ²(Z^d). Here, δ₀ denotes the vector with components (δ0j)_j∈Z^d1.

Obviously, equation (0.2) defines the integrated density of surface states n_sonly up to a constant. We choose this constant so thatn_s 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,nsis a positive measure.

One knows that Σ =σ(−¹₂∆)∪supp(dn_s) (see [8,9,2]. We will study the behavior ofn_s at the edges of Σ. To simplify this set as much as possible, we will assume that the support of the random variables (ω_γ1)_γ1∈Z^d1 is connected. Under this assumption, we know that

Lemma 0.1. Σ is a compact interval given by

(0.3) Σ =σ(−1

2∆_d1) + [

ω0∈[a,b]

σ(−1

2∆_d2 +ωΠ²₀) where Π²₀ is the projector on the unit vectorδ²₀ ∈ℓ²(Z^d2).

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(−¹₂∆) + [−α, β] = [−d−α, d+β] where α=α(a), β =β(b) and α+β >0; this occurs

• ifd₂ ≤2 and either a <0, in which case α(a)>0, orb >0, in which case β(b)>0,

• ifd2 ≥3 anda > a0 orb > b0, where, by (0.3), the thresholdsa0andb0are uniquely determined by the family of operators (−¹₂∆_d2 +tΠ²₀)_t∈R.

Ifα >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(−¹₂∆); this occurs only ind₂ ≥3 and if ais not too large, that is, ifa∈(0, a₀].

In this case, both spectral edges are stable.

On the other hand, it is well known (see [24]) that,

• ifd₂ = 1,2, then, for a >0,σ(−¹₂∆_d2−aΠ²₀) = [−d₂, d₂]∪ {λ(a)}, and the spectrum in [−d₂, d₂] is purely absolutely continuous and λ(a) is a simple eigenvalue;

• ifd₂ ≥3, there exists a⁰ >0 such that

– if 0 < a < a⁰, then, σ(−¹₂∆_d2 −aΠ²₀) = [−d₂, d₂], and the spectrum is purely absolutely continuous;

– ifa=a⁰, then

∗ ifd2 = 3,4, thenσ(−¹₂∆_d2−aΠ²₀) = [−d2, d2], the spectrum is purely absolutely contin- uous, and −d₂ is a resonance for −¹₂∆_d2 −aΠ²₀;

∗ ifd₂ ≥5, then σ(−¹₂∆_d2−aΠ²₀) = [−d₂, d₂], the spectrum is purely absolutely continuous in [−d2, d2), and−d2 is a simple eigenvalue for−¹₂∆_d2 −aΠ²₀;

– ifa > a⁰, then, σ(−¹₂∆_d2 −aΠ²₀) = [−d₂, d₂]∪ {λ(a)}, and the spectrum in [−d₂, d₂] is purely absolutely continuous andλ(a) is a simple eigenvalue;

For the operator−¹₂∆_d2+bΠ²₀, 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

H_t=−1

2∆ +t1⊗Π²₀.

As above, letabe the infimum of the random variables (ωj)j. Forϕ∈ C0^∞(R), define (ϕ, n_s,norm) =E(tr(Π₁[ϕ(H_ω)−ϕ(H_a)]Π₁))

The advantage of this renormalization is that the IDSSn_s,normis the distributional derivative of a positive measure. Indeed, forϕ∈ C0^∞(R), define

(ϕ, dN_s,norm) =−E(tr(Π₁[P(ϕ)(H_ω)−P(ϕ)(H_a)]Π₁)) where

P(ϕ)(x) = Z +∞

x

ϕ(t)dt.

Clearly,dNs,norm is independent of the anti-derivative of ϕchosen to define it; it is a positive measure and n_s,norm=− d

dEdN_s,norm.

2

Letn^t_s be the IDSS forH_t. As above, one can define a anti-derivative ofn^t_s; denote it by −dN_s^t. Letn^t_s,norm be the normalized version ofn^t_s, i.e. n^t_s,norm=n^t_s−n^a_s. One has

(0.4) ns,norm+ns =n^a_s.

One problem one encounters when studying n_s is that very little is known about its regularity for random surfacic models (see nevertheless [21]). Thanks to (0.4), we know thatn_sis the difference of two distributions each of which is the derivative of a signed measure. So we can take the counting function of dN_s as dN_s=dN_s,norm−dN_s^a is the difference of two measures. Thus, we define its counting function

(0.5) N_s(E) =

Z _E

−d

dN_s(e).

An obvious consequence of (0.4) is the Proposition 0.1. One has

(0.6) N_s^a(E)≤N_s(E)≤N_s^b(E).

In section 5.1, we study the asymptotics forN_s^t. As a consequence of this study, we prove Theorem 0.1. Assume d2= 1 or 2. Then, one has

N_s(E) ∼

E→−d E>−d











Vol(S^d1−1)

d₁(d₁+ 2)(2π)^d1 ·(E+d)^1+d1/2 if d₂ = 1, 2Vol(S^d1−1)

d₁(d₁+ 2)(2π)^d1 ·(E+d)^1+d1/2

|log(E+d)| if d₂ = 2.

where S^d1−1 is the d1−1 dimensional unit sphere.

Ifa >0, this result is an immediate consequence of Proposition0.1and of Theorem1.1giving the asymptotics of the IDSS for constant surface potential (see also section 5.1). Ifa= 0, one needs to improve upon (0.6) as the left hand side of this inequality vanishes making it unusable. This is the purpose of Theorem1.2.

Whend₂ ≥3, the situation becomes more complicated and we are only able to use Proposition0.1to get the two-sided estimate

(0.7) Ca(1 +o(1))

(1 +aI) ≤ (2π)^d

s(E+d)(E+d)^1+d1/2 ·N_s(E)≤Cb(1 +o(1)) (1 +bI)

whereC is a positive constant depending only on the dimensionsd1 and d2 (see section5.1) and s(x) = 1

2|x|^d22−2, I = 1

2 sup

θ1∈T^d1

Z

θ2∈T^d2



d−

d1

X

j=1

cos(θ^j₁)−

d2

X

j=1

cos(θ₂^j)





−1

dθ2.

Here, and in the sequel, the measuredθ_α (α∈ {1,2}) is the Haar measure on the torusT^dα, i.e. the Lebesgue measure normalized to have total mass equal to one.

Let us note that, if a < 0 < b, the inequality (0.7) does not give much information of the actual behavior ofN_s(E) when d₂ ≥3.

0.2. The fluctuation edge. Here, we assume that E₀= infσ(H_ω) is strictly below−d= infσ(−¹₂∆). In this case, E0 is a fluctuation edge of the spectrum.

Below the spectrum of −¹₂∆, the density of surface states n_s is positive; hence, it is a Borel measure and the integrated density of surface statesN_s(E) can be defined as its distribution function, i.e. N_s(E) = n_s((−∞, E)) forE <−d. We will prove Lifshitz type behavior forN_s(E) forEց E₀which is characteristic for fluctuation edges. However, the Lifshitz exponent, in the homogeneous case typically equal to −^d₂, is given by−^d₂¹ in our case. More precisely, we will show

EցElim0

ln|ln(Ns(E))|

ln(E−E₀) =−d1

2 .

3

1. The main results

Let us now describe the general model we consider. LetH be a translational invariant Jacobi matrix with exponential off-diagonal decay that is H= ((h_γ−γ′))_γ,γ′∈Z^d such that,

(H0.a): h_−γ =h_γ forγ ∈Z^d and for some γ 6= 0,h_γ 6= 0.

(H0.b): There exists c >0 such that, forγ ∈Z^d,

|h_γ| ≤ 1 ce^−c|γ|.

The infinite matrixH defines a bounded self-adjoint operator on ℓ²(Z^d). Using the Fourier transform, it is easily seen thatH is unitarily equivalent to the multiplication by the functionθ7→h(θ) defined by

h(θ) =X

γ∈Z

hγe^iγθ whereθ= (θ1, . . . , θ_d),

acting as an operator onL²(T^d) whereT^d=R^d/(2πZ^d) (the Lebesgue measure onT^dis normalized so that the constant function 1 has norm 1). The functionh is real analytic on T^d. We normalize it so that it be non-negative and 0 be its minimum.

As both ends of the spectrum of our operator play symmetric parts, we only study what happens at a left edge, i.e. near the bottom of the spectrum. All our assumptions will reflect this fact.

1.1. The case of a constant surface potential. We will start with a study of the density of surface states when the surfacic potentialV is constant, i.e. V =tΠ²₀. We define the operator H_t=H+t1⊗Π²₀. We prove two results on Ht. The first one is a criterion for the positivity of Ht and a description of its infimum when it is negative; the other result describes the density of the density of surface states near 0 whenHt is non-negative.

In the present section, we assume

(H1): the functionh: T^d→Radmits a unique minimum; it is quadratic non-degenerate.

IfH is −¹₂∆, thenh=h₀ where

(1.1) h₀(θ) := cos(θ₁) +· · ·+ cos(θ_d).

In this case, assumption (H1) is satisfied. Below, we give an example why considering more general Hamil- tonians can be of interest.

For the sake of definiteness, we assume the minimum ofhto be 0. This amounts to adding a constant toH.

We start with a characterization of the infimum of the spectrum ofH_t. Therefore, writeh(θ) =h(θ₁, θ₂) whereθ= (θ₁, θ₂),θ₁ ∈T^d1,θ₂ ∈T^d2. Define

(1.2) I(θ₁, z) =

Z

T^d2

1

h(θ₁, θ₂)−zdθ₂.

We recall that the measuresdθ₂ is normalized so that the measure ofT^d2 be equal to 1.

We prove

Proposition 1.1. Assume (H0) and (H1) are satisfied.

H_t is non negative if and only if tsatisfies

(1.3) 1 +tI_∞≥0 where I_∞:= sup

θ1∈T^d1

Z

T^d2

1 h(θ₁, θ₂)dθ₂ Assume now that1 +tI∞<0. Then, there exists a unique E0 ∈(−∞,0]such that

∀θ₁ ∈T^d1, 1 +tI(θ₁, E₀)≥0 and ∃θ₁∈T^d1, 1 +tI(θ₁, E₀) = 0.

Moreover,E₀ is the infimum of the spectrum ofH_t. Proposition1.1is proved in section5.

Criterion (1.3) immediately gives the obvious fact that ift≥0 thenH_tis non-negative. As we assumed thath has only non degenerate minima, ifd₂= 1,2 and t <0, then H_t is not non-negative.

4

We now turn to our second result. It describes the asymptotics of N_s^t near 0 when (1.3) is satisfied.

Recall thatN_s^t is the density of surface states ofH_t. Theorem 1.1. Assume t satisfies condition (1.3). Define

I = Z

T^d2

1

h(0, θ₂)dθ2. One has

• if d2 = 1:

Z E 0

dN_s^t(e) ∼

E→0⁺

Vol(S^d1−1) d₁(d₁+ 2)(2π)^d1

q

Det(Q₁−RQ⁻¹₂ R^∗)

·E^1+d1/2

• if d2 = 2:

Z _E

0

dN_s^t(e) ∼

E→0⁺

2Vol(S^d1−1) d₁(d₁+ 2)(2π)^d1

q

Det(Q₁−RQ⁻¹₂ R^∗)

E^1+d1/2

|logE| If d₂ ≥3 and 1 +t·I >0, then, one has

Z _E

0

dN_s^t(e) ∼

E→0⁺

c(d₁, d₂)Vol(S^d2−1)Vol(S^d1−1) d(2π)^d√

DetQ · t

1 +tI ·s(E)E^1+d1/2 If d₂ ≥ 3 and 1 +t·I = 0, if we assume, moreover, that θ₁ 7→I(θ₁,0) :=

Z

T^d2

(h(θ₁, θ₂))⁻¹dθ₂ has a local maximum forθ₁ = 0, then one has

• if d₂ = 3:

Z _E

0

dN_s^t(e)de ∼

E→0⁺

Z

|θ1|≤1

Arg(−i|1−θ₁²|^1/2+ ˜g(θ₁))dθ₁ d₁(d₁+ 2)π(2π)^d1

q

Det(Q₁−RQ⁻¹₂ R^∗) ·E^1+d1/2

• if d₂ = 4:

Z _E

0

dN_s^t(e) ∼

E→0⁺− 2Vol(S^d1−1) d₁(d₁+ 2)(2π)^d1

q

Det(Q₁−RQ⁻¹₂ R^∗)

E^1+d1/2

|logE|

• if d₂ ≥5:

Z _E

0

dN_s^t(e) ∼

E→0⁺

c(d₁, d₂)Vol(S^d2−1)Vol(S^d1−1) d(2π)^d√

DetQ ·−1

J ·s(E)E^d1/2 Here, we used the following notations:

• Arg(·) denotes the principal determination of the argument of a complex number,

• for n∈ {d₁, d₂}, Sⁿ⁻¹ is the n−1 dimensional unit sphere,

• g˜is a linear form defined below,

• the functions and the constants c(d₁, d₂) and J are defined by s(x) = 1

2|x|^d22−2, c(d₁, d₂) = Z ₁

0

r^d1−1(1−r²)^(d2−2)/2dr, J = Z

T^d2

1 h²(0, θ₂)dθ₁

• Q is the Hessian matrix of h at 0 that can be decomposed asQ=

Q1 R^∗ R Q₂

.

5

About the function ˜g, it is defined as follows. We assumed₂≥3 and 1+tI = 0. Leth₂(θ₁) = inf

θ2∈T^d2

h(θ₁, θ₂).

In section5.1, we show that the functionθ₁ 7→

Z

T^d2

(h(θ₁, θ₂)−h₂(θ₁))⁻¹dθ₂is real analytic in a neighborhood of 0. Using the Taylor expansion of this function near 0, one obtains

1 +t Z

T^d2

1

h(θ₁, θ₂)−h₂(θ₁)dθ₂=tg(θ₁) +O(|θ₁|²).

This defines the linear formg uniquely. Then, ˜g is defined by

˜

g(θ) := (2π)^d2p

Det (Q₂)g((Q₁−RQ⁻¹₂ R^∗)^−1/2θ₁).

If the variables (θ₁, θ₂) separate in h, i.e., ifh(θ₁, θ₂) = ˜h₁(θ₁) + ˜h₂(θ₂), the function ˜g is identically 0.

1.2. The case of a random surface potential. Let V_ω be a random potential concentrated on the sub-lattice Z^d1 × {0} ⊂Z^d (d₁ is chosen as in section 0) of the form

(1.4) V_ω(γ₁, γ₂) =

(ω_γ1 ifγ₂ = 0,

0 ifγ₂ 6= 0., γ= (γ₁, γ₂)∈Z^d1×Z^d2 =Z^d. and (ω_γ1)_γ1∈Z^d1 is a family of i.i.d. bounded, non constant random variables.

Let ω_± be respectively the maximum and minimum of the random variables (ω_γ1)_γ1∈Zd1, and let ω be its expectation.

Finally, we define the random surfacic model by

(1.5) H_ω=H+V_ω,

and its IDSS by

(n_s, ϕ) =E(tr(Π₁[ϕ(H_ω)−ϕ(H)]Π₁)) Following section 0, one regularizes n_s intoN_s as in (0.5).

Remark 1.1. An interesting case which can be brought back to a Hamiltonian of the form (1.5) with H andV_ω as above is the following.

Consider Γ, a sub-lattice of Z^d obtained in the following way Γ = G({0} ×Z^d2) where G is a matrix in GSL_d(Z), the d-dimensional special linear group over Z, i.e. the multiplicative group of invertible matrices with coefficients inZand unit determinant. One easily shows that the random operator

H_ω(Γ) =−1

2∆ +X

γ∈Γ

ω_γΠ_γ

(where Π_γis the projector onto the vector δ_γ ∈ℓ²(Z^d)) is unitarily equivalent toH+V_ω whereV_ω is defined in (1.4) and h(θ) =h₀(G^′·θ); here, h₀ is defined in (1.1) and G^′ is the inverse of the transpose of G, i.e.

G^′ = ^tG⁻¹.

Definition 1.1. We say that E, an edge (or boundary) of the spectrum of H_ω, is stableif it is an edge of the spectrum ofH+tV_ω for allt∈[0,1]. If an edge is not stable, we call it a fluctuation edge.

Note that in the case of the introduction, this definition is equivalent to that given there.

As in the introduction, one has to distinguish between

(1) stable boundaries : at these boundaries, the IDSS is given by the IDSS of a model operator computed from the random model.

(2) fluctuation boundaries: at these boundaries, one has standard Lifshitz tails.

To complete this section, let us give a very simple description of the spectrum ofH_ω. One has Proposition 1.2. Let H_ω be defined as above. Then

σ(H_ω) = [

t∈supp(P0)

σ(H_t).

Here and in the followingP₀ denotes the common distribution of the random variables (ω_γ2)_γ2.

6

1.3. The stable boundaries. The stable boundary we are studying is the lower boundary that we assumed to be 0. Let us first give a criterion for the lower edge of the spectrum ofH (that we assume to be equal to 0) to be a stable edge. We prove

Proposition 1.3. Writeh(θ) =h(θ₁, θ₂) whereθ= (θ₁, θ₂), θ₁∈T^d1, θ₂ ∈T^d2. Then,0 is a stable spectral edge if and only if ω₋ satisfies condition (1.3).

Proposition1.3 is an immediate consequence of Proposition 1.1 and Proposition 1.2. It gives the obvious fact that, ifω₋≥0, then 0 is a stable edge. As we assumed that hhas only non degenerate minima, we see that ifd₂ = 1,2 andω₋<0, then 0 is never a stable edge. Actually, it need not be an edge of the spectrum ofH_ω.

Using the same notations as above, we prove

Theorem 1.2. Assume (H0) and (H1) are verified. Assume, moreover, that 0 is a stable spectral edge for H_ω. Then, one has

(1.6) if ω >0, then lim inf

E→0⁺

N_s(E)

N_s^ω(E) ≥1 and ifω <0, then lim sup

E→0⁺

N_s(E) N_s^ω(E) ≤1

where N_s^ω is the IDSS of the operator with constant surface potential ω, the common expectation value of the random variables(ω_γ1)_γ1.

This result admits an immediate corollary

Theorem 1.3. Assume (H0) and (H1) hold. Assume, moreover, that 0 is a stable spectral edge for H_ω. Then,

• if d2 = 1:

N_s(E) ∼

E→0⁺

Vol(S^d1−1) d₁(d₁+ 2)(2π)^d1

q

Det(Q₁−RQ⁻¹₂ R^∗)

·E^1+d1/2;

• if d₂ = 2:

N_s(E) ∼

E→0⁺

2Vol(S^d1−1) d1(d1+ 2)(2π)^d1

q

Det(Q1−RQ⁻¹₂ R^∗)

E^1+d1/2

|logE|. Theorem1.3is an immediate consequence of Theorem1.2 and the bound

N_s^ω−(E)≤Ns(E)≤N_s^ω+(E).

As noted in the introduction, Theorem1.2 is only necessary whenω−= 0 (in which caseω >0). Moreover, one obtains the analogue of (0.7) in the present case for d₂≥3.

The above results may lead to the belief that N_s(E) ∼

E→0N_s^ω(E)

for all dimensions d₂. Let us now explain why this result, if true, is not obtained for dimension d₂ ≥ 3.

Therefore, we explain the heuristics behind the proof of Theorem1.2; it is very similar to that of standard Lifshitz tails with one big difference whend₂ ≥3.

RestrictH_ω to some large cube. One wants to estimate the IDSS for H_ω; for this restriction, this comes up to estimating the differences between the integrated density of states (the usual one) of the operatorH_ωand the integrated density of the operator Hω₋ (see Lemma 2.2). So we want to count the eigenvalues of Hω

below energyE, say, subtract the number of eigenvalues of H_ω− below energy E, divide by the volume of the cube, and see how this behaves whenE gets small. Assume ϕis a normalized eigenfunction associated to an eigenvalue of H_ω below E. Then, one has h(H+V_ω)ϕ, ϕi ≥ E. Assume for a moment that V_ω is non negative. Then, we see that one must have both hHϕ, ϕi ≥ E and hV_ωϕ, ϕi ≥ E. The first of these conditions guarantees thatϕis localized in momentum. So it has to be extended in space. If one plugs this information into the second condition, one sees that hV_ωϕ, ϕi ∼ω with a large probability. So that, to ϕ, H_ω roughly looks likeH+ωΠ²₀. There is one problem with this reasoning:

asV_ω only lives on a hyper-surface, and as ϕis flat, it only sees a very small part ofϕ; a simple calculation

7

shows that kΠ²₀ϕk ∼ E^d2/2; on the other hand, when one says that ϕ is roughly constant, one makes an error of size E^α (for some 0< α <1); hence, for dimension d₂ ≥3, this error is much larger than the term we want to estimate, namely, hV_ωϕ, ϕi. In other words, because ϕ is very flat, we can modify it on the hyper-surface (e.g. localize the part of it living on the hyper-surface) with almost no change to the total energy ofϕ; hence, we cannot guarantee that ϕis also flat on the hyper-surface, which implies thathV_ωϕ, ϕi need not be closeω with a large probability.

1.4. The fluctuation boundaries. In this section we assume that the infimum of Σ which we call E₀ is (strictly) below inf(σ(H)), so that E₀ is a fluctuation edge. In this case, we consider a “reduced” operator H˜ which acts on ℓ²(Z^d1). In Fourier representation this operator is multiplication by the function ˜h given by:

(1.7) ˜h(θ1) =

Z

T^d2

1

h(θ₁, θ₂)−E₀ dθ2

−1

+E0

We will reduce the proof of Lifshitz tails forH_ω =H+V_ωto a proof of Lifshitz tails for the reduced operator H˜_ω = ˜H+ ˜V_ω (where ˜V_ω is a diagonal matrix with entries (ω_γ1)_γ1). To prove Lifshitz tail behavior for ˜H_ω we have to impose a condition on the behavior of ˜h near its minimum. We either suppose:

(H2): the function ˜h: T^d1 →Radmits a unique quadratic minimum.

or we assume the weaker hypothesis:

(H2’): the function ˜h: T^d→Ris not constant.

Moreover, we always assume that the random variablesω_γ1 defining the potential (0.1) are independent with a common distributionP₀. We set ω₋= inf(supp(P₀)) and assume:

(H3): P₀ is not concentrated in a single point andP₀([ω₋, ω₋+ε))≥C ε^k for somek.

We will prove below:

Theorem 1.4. If (H2) and (H3) are satisfied then

EցElim0

ln|ln(N_s(E))|

ln(E−E₀) =−d₁ 2 . We have an additional result for low dimension of the surface:

Theorem 1.5. Assume (H2’) and (H3) hold. If d₁ = 1 then

EցElim0

ln|ln(N_s(E))|

ln(E−E₀) =− lim

EցE0

ln(n(E)) (E−E₀) where n(E) is the integrated density of states for H.˜

If d₂ = 2, then

EցElim0

ln|ln(N_s(E))| ln(E−E₀) =−α where the computation ofα is explained below.

For the sake of simplicity, let us assumeE₀= 0. The Lifshitz exponentα will depend on the way ˜hvanishes atS ={θ₁|˜h= 0} and on the curvature of S.

To describe it precisely, we need to introduce some objects from analytic geometry (see [19] for more details). IfE is a set contained in the closed first quadrant inR² then itsexterior convex hullis the convex hull of the union of the rectanglesR_xy = [x,∞)×[y,∞), where the union is taken over all (x, y)∈ E.

Pickθ₀ ∈ S and consider the Newton diagram of ˜h at θ₀, i.e., (1) Express ˜h as a Taylor series at θ₀, ˜h(θ¹, θ²) =P

ija_ij(θ¹−θ₀¹)ⁱ(θ²−θ₀²)^j,θ= (θ¹, θ²).

(2) Form the exterior convex hull of the points (i, j) with a_ij 6= 0. This is a convex polygon, called the Newton polygon.

(3) The boundary of the polygon is theNewton diagram.

8

The Newton decay exponent is then defined as follows. The Newton diagram consists of certain line segments.

Extend each to a complete line and intersect it with the diagonal lineθ¹ =θ². This gives a collection of points (a_k, a_k), one for each boundary segment. Take the reciprocal of the largesta_k and call this number ˜α(˜h, θ₀);

it is theNewton decay exponent. Defineα(˜h, θ₀) = min{α(˜˜ h◦T₀, θ₀) : T₀(·) =θ₀+T(· −θ₀), T ∈SL(2,R)}. Similarly, defineα(˜h, θ) ifθ is any other point inS, the zero set of ˜h. Then, theLifshitz exponent α is defined by

(1.8) α= min

θ∈S α(˜h, θ).

The Lifshitz exponent α is positive as θ 7→ α(˜h, θ) is a positive, lower semi-continuous function and S is compact (see [19]).

Remark 1.2. Let us return to the example given in Remark 1.1. In the section 6, we check that (H.2’) holds in this case; so ford=d1+d2 = 3, Theorem1.5 applies.

2. Approximating the IDSS

To approximate the IDSS, we use a method that has proved useful to approximate the density of states of random Schr¨odinger operators, the periodic approximations. We shall show that the IDSS is well approximated by the suitably normalized density of states of a well chosen periodic operator.

2.1. Periodic approximations. Let (ω_γ1)_γ1∈Z^d1 be a realization of the random variables defined above.

FixN ∈N^∗. We defineH_ω^N, a periodic operator acting onℓ²(Z^d) by H_ω^N =H+V_ω^N =H+ X

γ1∈Z^d1

2N+1

ω_n X

β1∈(2N+1)Z^d1

β2∈(2N+1)Z^d2

|δ_γ1+β1 ⊗δ_β2ihδ_γ1+β1 ⊗δ_β2|.

Here, Z^d˜

2N+1 = Z^d˜/(2N + 1)Z^d˜, δ_l = (δ_jl)_j∈Zd˜ is a vector in the canonical basis ofℓ²(Z^d˜) where δ_jl is the Kronecker symbol and, ˜d= d₁ or ˜d=d₂, the choice being clear from the context. As usual, |uihu| is the orthogonal projection on a unit vectoru.

By definition,H_ω^N is periodic with respect to the (non degenerate) lattice (2N+ 1)Z^d. We define the density of states denoted byn^N_ω as usual for periodic operators: forϕ∈ C^∞0 (R),

(ϕ, dn^N_ω) = Z

R

ϕ(x)dn^N_ω(x) = lim

L→+∞

1 (2L+ 1)^d

X

γ∈Z^d

|γ|≤L

hδ_γ, ϕ(H_ω^N)δ_γi.

This limit exists (see e.g. [4,23]). In a similar way, one can define the density of states ofH; we denote it by dn₀. The operators (H_ω^N)_ω,N are uniformly bounded; hence, their spectra are contained in a fixed compact set, sayC. This set also contains the spectrum of H_ω andH. We prove

Lemma 2.1. Pick U ⊂Ra relatively compact open set such that C ⊂ U. There existsC >1 such that, for ϕ∈ C0^∞(R), for K∈N,K ≥1, and N ∈N^∗, we have

(2.1)

(ϕ, dn)−(2N + 1)^d2E{(ϕ,[dn^N_ω −dn0])} ≤

CK N

K

sup

0≤J≤K+d+2x∈U

d^Jϕ d^Jx(x)

.

Proof of Lemma 2.1 Fix ϕ∈ C0^∞(R). As the spectra of the operators H_ω^N are contained in U, we may restrict ourselves toϕ supported inU which we do from now on. By the definition (0.2), one has

(2.2) (ϕ, n_s) =E



 X

γ∈Z^d2

hδ₀⊗δ_γ2,[ϕ(H_ω)−ϕ(H)]δ₀ ⊗δ_γ2i



=M_N(ϕ) +R_N(ϕ)

9

where

MN(ϕ) =E









 X

γ2∈Z^d2

|γ2|≤N

hδ0⊗δγ2,[ϕ(Hω)−ϕ(H)]δ0⊗δγ2i









 ,

R_N(ϕ) =E









 X

γ2∈Z^d2

|γ2|>N

hδ₀⊗δ_γ2,[ϕ(H_ω)−ϕ(H)]δ₀⊗δ_γ2i









 .

Let us now show that

(2.3) |R_N(ϕ)| ≤

CK N

K

sup

0≤J≤K+d+2x∈U

d^Jϕ d^Jx(x)

.

Therefore, we use some ideas from the proof of Lemma 1.1 in [17]. Helffer-Sj¨ostrand’s formula ([10]) reads ϕ(H_ω) = i

2π Z

C

∂ϕ˜

∂z(z)·(z−H_ω)⁻¹dz∧dz.

where ˜ϕis an almost analytic extension ofϕ(see [22]), i.e. a function satisfying (1) forz∈R, ˜ϕ(z) =ϕ(z);

(2) supp( ˜ϕ)⊂ {z∈C; |Im(z)|<1}; (3) ˜ϕ∈ S({z∈C; |Im(z)|<1});

(4) the family of functions x7→ ∂ϕ˜

∂z(x+iy)· |y|⁻ⁿ (for 0<|y|<1) is bounded in S(R) for any n∈ N; more precisely, there exists C >1 such that, for allp, q, r∈N, there exists C_p,q >0 such that

(2.4) sup

0<|y|≤1

sup

x∈R

x^p ∂^q

∂x^q

|y|^−r·∂ϕ˜

∂z(x+iy)

≤C^rC_p,q sup

q^′≤r+q+2 p^′≤p

sup

x∈R

x^p′∂^q′ϕ

∂x^q′(x) .

As we are working with ϕ with compact support inU, its almost analytic extension can be taken to have support in (U + [−1,1]) +i[−1,1] (see e.g. [6]).

We estimateE(|hδ0⊗δγ2,[ϕ(Hω)−ϕ(H)]δ0⊗δγ2i|) for|γ2|> N. Using the fact that the random variables (ω_γ2)_γ2 are bounded, we get

E(|hδ₀⊗δ_γ2,[ϕ(H_ω)−ϕ(H)]δ₀⊗δ_γ2i|)

≤ 1 4πE

Z

C

∂ϕ˜

∂z(z)

|hδ0⊗δγ2, (z−H_ω^N)⁻¹−(z−H)⁻¹

δ0⊗δγ2i|dxdy

≤C X

γ1∈Z^d1

Z

C

∂ϕ˜

∂z(z)

·E |hδ₀⊗δ_γ2,(z−H_ω^N)⁻¹δ_γ1 ⊗δ₀i| · |hδ_γ1 ⊗δ₀,(z−H)⁻¹δ₀⊗δ_γ2i|

dxdy

wherez=x+iy.

By a Combes-Thomas argument (see e.g. [18]), we know that there exists C > 1 such that, uniformly in (ω_γ)_γ,γ₁∈Z^d1 and N ≥1, we have, for Im(z)6= 0,

(2.5) |hδ_γ1⊗δ_γ2,(z−H_ω^N)⁻¹δ_γ^′

1 ⊗δ_γ^′

2i|+|hδ_γ1 ⊗δ_γ2,(z−H)⁻¹δ_γ^′

1⊗δ_γ^′

2i| ≤

≤ C

|Im(z)|e−|Im(z)|(|γ1−γ₁^′|+|γ2−γ₂^′|)/C 10

Hence, for someC >1,

|R_N(ϕ)| ≤C X

γ1∈Z^d1

Z

C

∂ϕ˜

∂z(z) · 1

|Im(z)|²e^{−|Im(z)(|γ1|+|γ2|)|/C}dxdy

≤C Z

C

∂ϕ˜

∂z(z)

1

|Im(z)|^d+2e−|Im(z)N|/Cdxdy.

Taking into account the properties of almost analytic extensions (2.4), for some C > 1, for K ≥ 1 and N ≥1, we get

|R_N(ϕ)| ≤C^K+1 Z

(U+[−1,1])+i[−1,1]|y|^Ke^−|yN|/Cdxdy sup

0≤J≤K+d+2x∈U

d^Jϕ d^Jx(x)

≤ CK

N K

sup

0≤J≤K+d+2x∈U

d^Jϕ d^Jx(x)

.

This completes the proof of (2.3).

We now compare MN(ϕ) to (2N + 1)^d2E{(ϕ,[dn^N_ω −dn0])}. Therefore, we rewrite this last term as follows. Using the (2N + 1)Z^dperiodicity of H_ω^N and H, we get

X

γ∈Z^d

|γ|≤N+L(2N+1)

hδ_γ, ϕ(H_ω^N)δ_γi= (2L+ 1)^d X

γ∈Z^d

|γ|≤N

hδ_γ, ϕ(H_ω^N)δ_γi.

This gives

(2.6) (2N + 1)^d(ϕ, dn^N_ω) =E









 X

γ∈Z^d

|γ|≤N

hδ_γ, ϕ(H_ω^N)δ_γi









 .

On the other hand, as the random variables (ωγ2)γ2 are i.i.d. and as H is Z^d-periodic, as in [18], one computes

E









 X

γ∈Z^d

|γ|≤N

hδγ, ϕ(H_ω^N)δγi











=E











X

γ1∈Z^d1,|γ1|≤N γ2∈Z^d2,|γ2|≤N

hδγ1 ⊗δγ2, ϕ(H_ω^N)δγ1 ⊗δγ2i











= (2N+ 1)^d1E









 X

γ2∈Z^d2

|γ2|≤N

hδ₀⊗δ_γ2, ϕ(H_ω^N)δ₀⊗δ_γ2i











Combining this with (2.6), we get

(2N+ 1)^d2E[(ϕ, dn^N_ω)] =E









 X

γ2∈Z^d2

|γ2|≤N

hδ₀⊗δ_γ2, ϕ(H_ω^N)δ₀⊗δ_γ2i











11

Of course, such a formula also holds when H_ω^N is replaced with H. In view of (0.2), (2.3) and (2.2), to complete the proof of Lemma2.1, we need only to prove

(2.7) E

X

γ2∈Z^d2

|γ2|≤N

hδ₀⊗δ_γ2,[ϕ(H_ω^N)−ϕ(H_ω)]δ₀⊗δ_γ2i

≤ CK

N K

sup

0≤J≤K+d+2x∈U

d^Jϕ d^Jx(x)

.

forϕ,K J and N as in Lemma 2.1.

Proceeding as above, forγ₂ ∈Z^d2,|γ₂| ≤N, we estimate

|hδ₀⊗δ_γ2,[ϕ(H_ω^N)−ϕ(H_ω)]δ₀⊗δ_γ2i|

≤C











X

γ₁^′∈Z^d1

γ₂^′∈((2N+1)Z^d2)^∗

+ X

γ₁^′∈Z^d1,|γ₁^′|>N γ₂^′=0









 Z

C

∂ϕ˜

∂z(z)

dxdy·

E

|hδ₀⊗δ_γ2,(z−H_ω^N)⁻¹δ_γ′ 1 ⊗δ_γ′

2i|·

|hδ_γ′ 1 ⊗δ_γ′

2,(z−H_ω)⁻¹δ₀⊗δ_γ2i|

.

Here we used the fact that the operatorsHω and H_ω^N coincide in the cube{|γ| ≤N}.

As H_ω satisfies the same Combes-Thomas estimate (2.5) as H_ω^N, doing the same computations as in the estimate forR_N(ϕ), we obtain (2.7). This completes the proof of Lemma2.1.

Obviously, one has an analogue of (2.1) for n_s,norm, n^t_s or n^t_s,norm. One needs to replace H_ω^N and H with their obvious counterparts i.e., choose the random variables (ω_γ2)_γ2 to be the appropriate constant.

This enables us to prove

Lemma 2.2. FixI, a compact interval. Pickα >0. There existsν₀ >0and ρ >0such that, for γ ∈[0,1], E∈I, ν ∈(0, ν₀) and N ≥ν^−ρ, one has

(2.8) E(N_ω^N(E−ν))−e^−ν−α ≤N_s(E)≤E(N_ω^N(E+ν)) +e^−ν−α where N_norm,ω^N =N_ω^N−N_ω^N

−, and N_ω^N (resp. N_ω^N

−) is the integrated density of states ofH_ω^N (resp H_ω^N

−, i.e.

H_ω^N where ω_γ=ω₋ for all γ).

Let us note here that one can prove a similar result for the approximation of N_s,norm by N_norm,ω^N or that of N_s^t by N_s^t,N.

ProofLet us now prove Lemma 2.2. Pick ϕa Gevrey class function of Gevrey exponent α >1 (see [11]);

assume, moreover, thatϕhas support in (−1,1), that 0≤ϕ≤1 and that ϕ≡1 on (−1/2,1/2]. LetE ∈I andν ∈(0,1),and set

ϕ_E,ν(·) =1_[0,E]∗ϕ· ν

.

Then, by Lemma 2.1 and the Gevrey estimates on the derivatives of ϕ, there exist C > 1 such that, for N ≥1,k≥1 and 0< ν <1, we have

(2.9) |E((ϕ_E,ν, dN_ω^N))−(ϕ_E,ν, dN_s)| ≤C(N ν)³

Ck^1+α N ν

k

.

We optimize the right hand side of (2.9) in k and get that, there exist C > 1 such that, for N ≥ 1 and 0< ν <1, we have

|E((ϕ_E,ν, dN_ω^N))−(ϕ_E,ν, dN_s)| ≤C(N +ν⁻¹)³e^{−(N ν/C)1/(1+α)+C(N ν/C)−1/(1+α)} Now, there existν0 >0 such that, for 0< ν < ν0 and N ≥ν^−1−η, we have

(2.10) |E((ϕ_E,ν, dN_ω^N))−(ϕ_E,ν, dN_s)| ≤e^{−ν−η/(2α)}.

12