Denis Borisov

Francisco Hoecker-Escuti

Ivan Veseli

EXPANSION OF THE ALMOST SURE SPECTRUM IN THE WEAK DISORDER REGIME

The spectrum of random ergodic Schrödinger-type operators is almost surely a deterministic subset of the real line. The random operator can be considered as a perturbation of a periodic one. As soon as the disorder is switched on via a global coupling constant, the spectrum expands. We estimate how much the spectrum expands at its bottom for operators on ℓ 2 (Z d ).

Due to the self-averaging property of ergodic Schrödinger operators the resulting spectrum is almost surely a fixed subset of the real line. If a random operator is a perturbation of a periodic operator, it is of interest to know how the spectrum expands once we switch on the disorder via a global coupling constant. Apart from the genuine interest to identify the location of the spectrum, this is also of central importance when identifying energy regions corresponding to localized wavepackets.

Otherwise it may happen that one proves a Wegner estimate, a Lifschitz tail bound or a similar statement related to localization, and then later discovers that the considered energy regime belongs to the resolvent set.

In this paper we consider an ǫ-small random perturbation of a discrete translation-invariant operator and we study how the bottom of its spectrum behaves. By symmetry, similar estimates apply to the location of the maximum of the spectrum, in a weak disorder regime. To fix the ideas, let us introduce a prototypical example. Let H = ℓ 2 (Z d ) and ∆ Z d : H → H the (negative definite) discrete Laplacian on Z d , i.e.

(∆ Z d u) (n) := |n-m|∞=1 (u(m) -u(n)) .
We define the operator H 0 : H → H by

H 0 := -∆ Z d + W,
where W is the multiplication operator by a real-valued function, which we also denote by W and which we assume periodic with respect to the subgroup γ := NZ d .

Let := [0, N -1] d ⊂ Z d and V ∈ ℓ ∞ (Z d ) be a non-trivial, compactly supported single-site potential satisfying supp(V ) ⊂ .

Let (ω k ) k∈γ be a sequence of non-trivial, bounded, independent, identically distributed random variables. For the sake of the introduction, assume that {-1, 1} ∈ supp ω 0 ⊂ [-1 , 1]. From now on we denote by V ω : ℓ 2 (Z d ) → ℓ 2 (Z d ) the diagonal operator defined, for f ∈ ℓ 2 (Z d ), as [START_REF] Aizenman | Localization at weak disorder: some elementary bounds[END_REF] (

V ω f )(x) = k∈N Z d ω k V (x -k)f (x).
To motivate our results, let us consider the following discrete alloy-type random Schrödinger operator defined by

H ω,ǫ := H 0 + ǫV ω . (2)
Under the stated assumptions, this operator is ergodic, and thus there exists a set Σ ǫ ⊂ R such that σ(H ω,ǫ ) = Σ ǫ with probability 1 (see e.g. [START_REF] Pastur | Spectra of random and almost-periodic operators[END_REF]). From now on we refer to Σ ǫ as the almost-sure spectrum of H ω,ǫ . The best known example of this kind of operators is the celebrated Anderson model, where H 0 is the discrete Laplacian on Z d (i.e. W ≡ 0), V = δ 0 and N = 1. In this case, it is not hard to see ( [START_REF] Pastur | Spectra of random and almost-periodic operators[END_REF]) that the bottom of the spectrum of the perturbed operator E ǫ := inf (Σ ǫ ) moves away from the bottom of the spectrum of the free operator E 0 := inf (Σ 0 ) as

E ǫ = E 0 -ǫ.
If one considers instead, for example, the dipole model, i.e. V = δ 0 -δ e 1 with e 1 = (1, 0, . . . , 0), it is proven in [START_REF] Cao | The weak localization for the alloy-type Anderson model on a cubic lattice[END_REF] that

E ǫ E 0 -Cǫ 2 .
In this note we study this question for a very general, wide class of operators (see assumptions in Section 2). More precisely, we prove some upper bounds of the quantity E ǫ -E 0 , which in turns gives us information on the location of the spectrum of the perturbed operator. We also discuss some partial results on the lower bound.

In order to state the result in this setting, we need to consider the operator H 0 with NZ d -periodic boundary conditions. Because of the translation invariance, the subspace of NZ d -periodic functions in ℓ ∞ (Z d ) is invariant under the action of H 0 . This subspace is N ddimensional, so that the action of the operator corresponds to a matrix we denote by

(3) H 0 : ℓ 2 ( ) → ℓ 2 ( ).
We now state the result.

Theorem 1.1. Let H ω,ǫ be the alloy-type random Schrödinger operator defined by (2) and E ǫ the bottom of its corresponding almost-sure spectrum. To the NZ d -periodic operator H 0 we associate a Hermitian matrix H 0 ∈ C N d ×N d , defined as in (3), and we let ψ 1 ∈ ℓ 2 ( ) be the (unique normalized) positive ground state of H 0 . Define

A 1 := -ψ 1 , V ψ 1 ℓ 2 ( ) .
There exists A 2 0 such that for ǫ > 0 small enough

E ǫ E 0 + ǫA 1 + ǫ 2 A 2 .
Furthermore, if A 1 = 0 then |A 2 | is non-zero and larger than the spectral gap of H 0 , i.e. the difference between its two smallest eigenvalues.

We provide an explicit formula for the constant A 2 only in the next section as it requires the introduction of additional notation. We have an analogous estimate for (fibers of) periodic operators, see Theorems 5.1 and 5.8. In fact, the estimate for periodic operators is one step in the proof of Theorem 1.1. In the context of periodic operators we have a related, complementary lower bound, see Lemmas 5.7 and 5.9.

We would like to make some remarks on the relevance of this result. First, the location of the bottom of the spectrum with respect to the coupling constant has been the subject of several papers: with periodic potentials in dimension one [START_REF] Titchmarsh | Eigenfunction expansions associated with second-order differential equations[END_REF] and in arbitrary dimension [START_REF] Kirsch | Comparison theorems for the gap of Schrödinger operators[END_REF], [START_REF] De Verdière | Sur les singularités de Van Hove génériques[END_REF], as well as with random positive potentials [START_REF] Kirsch | Localization for random perturbations of periodic Schrödinger operators[END_REF] and under some generic assumptions on W [START_REF] Klopp | Weak disorder localization and Lifshitz tails: continuous Hamiltonians[END_REF]. Understanding the spectrum provides valuable information on the solutions of partial differential equations. In particular, if one considers the Schrödinger equation for the Hamiltonian H ω,ǫ , the spectral type of the Anderson model characterizes the transport properties of the underlying disordered medium. For this model, the spectrum is expected to exhibit a transition from localized states at the bottom of the spectrum (pure point spectrum with exponentially localized eigenfunctions) to extended states (absolutely continuous spectrum) in the bulk of the spectrum. This Anderson transition is still a conjecture in the setting of this article. The existence of localized states at the bottom of the spectrum has been studied in many papers. We invite the reader to consult the monographs [START_REF] Carmona | Spectral theory of random Schrödinger operators[END_REF], [START_REF] Pastur | Spectra of random and almost-periodic operators[END_REF], [START_REF] Stollmann | Caught by disorder. Bound states in random media[END_REF], [START_REF] Hislop | Lectures on Random Schrödinger Operators[END_REF] and their extensive bibliography. The perturbative regime ǫ ≪ 1 has attracted much attention [START_REF] Aizenman | Localization at weak disorder: some elementary bounds[END_REF], [START_REF] Wang | Localization and universality of Poisson statistics for the multidimensional Anderson model at weak disorder[END_REF], [START_REF] Klopp | Weak disorder localization and Lifshitz tails[END_REF], [START_REF] Klopp | Weak disorder localization and Lifshitz tails: continuous Hamiltonians[END_REF], [START_REF] Elgart | Lifshitz tails and localization in the three-dimensional Anderson model[END_REF], [START_REF] Cao | The weak localization for the alloy-type Anderson model on a cubic lattice[END_REF], [START_REF] Hoecker-Escuti | Le modèle d'Anderson en régime de faible désordre[END_REF], [START_REF] Elgart | Anderson localization for a class of models with a sign-indefinite single-site potential via fractional moment method[END_REF], [START_REF] Borisov | Low lying spectrum of weak-disorder quantum waveguides[END_REF], [START_REF] Borisov | Low lying eigenvalues of randomly curved quantum waveguides[END_REF]. In this regime one can prove very precise estimates of the interval of localization, namely that states with energies in

I η (ǫ) := (-∞, -C 0 ǫ η ] ∩ Σ ǫ = (-∞, -C 0 ǫ η ] ∩ [E ǫ , +∞)
are localized. In [START_REF] Elgart | Lifshitz tails and localization in the three-dimensional Anderson model[END_REF] it was proved that in dimension d = 3 one may take η to be as large as 2 and in [START_REF] Hoecker-Escuti | Le modèle d'Anderson en régime de faible désordre[END_REF] that for d 2 this holds with η < 2. These results are meaningful, as for the Anderson model E ǫ = -C 1 ǫ. If we now consider different potentials, we may have a quadratic expansion of the bottom of the spectrum E ǫ , and understanding where the bottom of the spectrum lies appears to be crucial, so that the interval of localization is non-trivial. Some of the issues addressed in this note were already explored in [START_REF] Klopp | Weak disorder localization and Lifshitz tails: continuous Hamiltonians[END_REF] where it is assumed that the single-site potential has a non-zero mean and the Floquet eigenvalues of the underlying periodic discrete Schrödinger operator H 0 = -∆+W are assumed to be non-degenerated, as well as in [START_REF] Cao | The weak localization for the alloy-type Anderson model on a cubic lattice[END_REF] for the dipole potential. These are special cases of our models. The general operator we study corresponds roughly to tridiagonal block matrices of the form (4)

         . . . . . . . . . B * A B 0 . . . 0 B * A B 0 . . . 0 B * A B . . . . . . . . .          +          . . . . . . . . . 0 ω n-1 V 0 . . . . . . 0 ω n V 0 . . . . . . 0 ω n+1 V 0 . . . . . . . . .         
, where A and V are Hermitian matrices and {ω n } i.i.d. random variables. We introduce in Section 2 the general framework in which our results are obtained.

To complete the description of the obtained results, let us briefly address the question of the optimality of the lower bound (or at least its exponent). As far as the authors know, there is no result in the literature in this direction in a discrete setting (apart from the Anderson model, where the bottom of the spectrum is known explicitly). One may naively expect, from perturbation theory, that the behavior should be linear or quadratic. The question turns out to be more subtle as the behavior may depend on the speed at which the Floquet eigenvalues associated to the bottom of the spectrum approach their minimum, as the following example shows. Theorem 1.2. For H 0 := (-∆ Z ) 2 defined on ℓ 2 (Z) and V the multiplication operator given by the following single-site potential:

V := - 1 2 δ -1 + δ 0 - 1 2 δ 1 .
Set as before H ω,ǫ := H 0 + ǫV ω , cf. (1) and (2). Pick some ξ > 1/4. Then for ǫ > 0 small enough we have

E ǫ := inf σ(H 0 + ǫV ω ) - 1 6 ǫ 1+2ξ . ( 5
)
For this example, which is of the form (4), the coefficient A 1 corresponding to the linear term vanishes. The bound in Theorem 1.2 is nevertheless better than quadratic thanks to the quartic behavior of the Floquet eigenvalues in a neighbourhood of their minimum. Unfortunately, apart from the trivial linear bound, we have no corresponding lower bound, although some results in this article provide a first step in this direction.

This work can be extended in several directions. It would be very interesting to find the corresponding lower bounds, or at least conditions under which the infimum of the spectrum does not expand linearly. A related question concerns the expansion of the spectrum near a band edge, where one can also prove Anderson localisation. Note that if one studies the expansion of the spectrum from a band edge instead of the bottom of the spectrum, the Floquet eigenvalues may vanish faster than quadratically when approaching the edge, even for the operator defined by [START_REF] Borisov | Low lying spectrum of weak-disorder quantum waveguides[END_REF]. Rather than a pathological example, Theorem 1.2 provides a model for this situation. Another question of interest is the study of overlapping single-site potentials. Under some non-degeneracy condition (see Remark 4.5) the results stated here can be extended to this situation, but a full understanding needs to consider periodic approximations of every order, something we also believe necessary to prove the lower bounds complementing Theorem 1.1.

In a forthcoming project we consider the same questions for operators of Schrödinger type in the continuum setting, i.e. for operators acting on (dense subspaces) of L 2 (R d ). Most of our findings are similar. In the continuum, it is more natural to define the operators via quadratic forms, and then formulate appropriate regularity conditions. Also, certain additional compactness arguments are necessary, due to the infinite dimensionality of the Hilbert space over the periodicity cell. On the other hand, in specific situations, better results are possible in the continuum setting, due to unique continuation principles for solutions of partial differential equations.

General model

Let d 1 be the space dimension, D = Z d be the physical space and γ = NZ d a sub-lattice of D. We denote by its periodicity cell, i.e.

:= [0, N -1] d ∩ Z d . Note that D = k∈γ {x ∈ D : x -k ∈ }.
We also denote the reciprocal periodicity cell as * := [0, 2π N ) d . From now on we assume the following hypotheses to hold. (HA) Let H 0 : ℓ 2 (D) → ℓ 2 (D) be a bounded, non-negative Hermitian operator defined by the matrix

H 0 := (H 0 (k, k ′ )) k,k ′ ∈D ,
satisfying the following properties:

• for all k, k ′ ∈ D, we have H 0 (k, k ′ ) = H 0 (k ′ , k); • there exists k 0 = 0 such that H 0 (0, k 0 ) = 0; • the associated operator is γ-invariant, i.e. for every k ∈ γ τ k u, H 0 τ k v = u, H 0 v ,
where u, v ∈ ℓ 2 (D) and τ k is the translation by k ∈ γ operator; and • the associated operator is of finite hopping range with hopping range

N, i.e. if |k -k ′ | N then H 0 (k, k ′ ) = 0.
• Through a global energy shift we may assume, with no loss of generality, that E 0 := inf σ(H 0 ) = 0. Note that if an operator is of finite hopping range with hopping range R, for some R > 0, then it also is of finite hopping range with hopping range R ′ for any R ′ > R. On the other hand, any γ-invariant operator is also nγ-invariant, n ∈ N. This means that we can always assume that R = N, without loss of generality. (HB) Let V :

→ R be a non-trivial Hermitian matrix (we call it the single-cell potential, even when V is not diagonal). For any bounded sequence (ω k ) k∈γ of real numbers, we define the block diagonal operator

V ω : ℓ 2 (D) → ℓ 2 (D) V ω := k∈γ ω k τ -k V τ k .
For any real number q ∈ R, we denote also by q the constant sequence indexed by γ, equal to q on every site in Z d . We thus have, for example, that (6)

V q := q k∈γ τ -k V τ k and V q is γ-invariant.
From now on, the values of ω will be drawn from a sequence of bounded, non-trivial, independent and identically distributed random variables with distribution measure µ. We will write S µ := supp µ and we assume that

{s -, s + } ∈ S µ ⊂ [s -, s + ],
where s -and s + satisfy one of the following alternatives:

(HC) The random variables change sign, i.e. s -< 0 < s + .

(HC ′ ) The random variables are positive, i.e. 0 s -< s + .

The methods in this paper may also be adapted to negative random variables.

Remark 2.1. It looks tempting, in order to achieve s -= 0, to renormalize the random variables by adding and substracting some periodic potential, but in this case the underlying non-random operator depends on ǫ. On the other hand, it is indeed allowed to rescale the random variables by absorbing the scaling factor in the single site potential V .

Let us now define our object of study. For each ǫ > 0, we let

H ω,ǫ := H 0 + ǫV ω
which is a self-adjoint, ergodic operator. We denote its almost-sure spectrum by Σ ǫ and by [START_REF] Elgart | Lifshitz tails and localization in the three-dimensional Anderson model[END_REF] E ǫ := inf Σ ǫ the bottom of the spectrum. We also write H q,ǫ := H 0 + ǫV q the corresponding operator with V ω replaced by the periodic potential V q (defined as in [START_REF] De Verdière | Sur les singularités de Van Hove génériques[END_REF]) and E q,ǫ := inf σ(H q,ǫ ). In the following we will study the bottom of the spectrum E ǫ of the random operator for small ǫ.

We define a finite dimensional matrix associated to the above objects. Define the

(| | × | |)-matrix H 0 (θ) by its coefficients H 0 (θ) (k, k ′ ) := m∈γ e iθ•m H 0 (k, k ′ -m) (8) = |m| N m∈N Z d e iθ•m H 0 (k, k ′ -m),
where k, k ′ ∈ . Note that the second line is a consequence of the finite hopping range and the sum in ( 8) is thus finite. Now define the matrix H q,ǫ (θ) by

H q,ǫ (θ) := H 0 (θ) + ǫqV .
Remark 2.2. The matrix H q,ǫ represents the action of H q,ǫ on the fiber of θ-quasiperiodic functions in the Floquet-Bloch direct integral decomposition. More precisely, let (abusing notation) ϕ ∈ ℓ 2 ( ) ⊂ ℓ 2 (Z d ). Then, regarding H q,ǫ as an operator ℓ

∞ (Z d ) → ℓ ∞ (Z d ), (9) H ǫ,q (θ)ϕ = χ H ǫ,q m∈γ e iθ•m τ m ϕ ∈ ℓ 2 ( ), (abusing notation)
where χ is the indicator function of ⊂ Z d .

Main results

Recall that, by the continuity of the Floquet-Bloch eigenvalues ( [START_REF] Krueger | Periodic and limit-periodic discrete Schrödinger operators[END_REF], [START_REF] Reed | Analysis of Operators[END_REF]), there exists some θ such that

inf σ(H 0 (θ)) = E 0 := inf σ(H 0 ) = 0.
We denote by Θ ⊂ * the compact set of θ for which the last equality holds. From now on we fix some θ ∈ Θ, so the quantities below will depend on θ. Let V 0 be the eigenspace of H 0 (θ) associated to the eigenvalue E 0 = 0 and p its multiplicity. Choose an orthonormal basis ψ j , j = 1, . . . , p spanning V 0 and diagonalizing the Hermitian matrix A ∈ C p×p , given by the coefficients

A ij := ψ i , V ψ j .
We take the eigenvalues of the matrix A in ascending order counting multiplicities so that P 1 := A 11 = ψ 1 , V ψ 1 is the minimal eigenvalue and P p := A pp = ψ p , V ψ p is the maximal eigenvalue of A.

Our result for sign-changing random variables reads as follows.

Theorem 3.1. Assume (HA), (HB) and (HC). Fix θ ∈ Θ and define

(10) A 1 := inf q∈Sµ inf ψ∈V 0 ψ ℓ 2 ( ) =1
q ψ, V ψ = min(s + P 1 , s -P p ) 0, and (11)

A 2 := -max(s 2 -, s 2 + ) sup ψ∈V 0 ψ ℓ 2 ( ) =1 sup ϕ∈V ⊥ 0 ϕ ℓ 2 ( ) =1 ψ, V ϕ 2 H 0 (θ)ϕ, ϕ 0.
For any ǫ > 0 small enough the following holds: if

A 1 = 0, E ǫ ǫA 1 , whereas if A 1 = 0, then E ǫ ǫ 2 A 2 + O(ǫ 3 ). Finally, if A 1 = A 2 = 0, then E ǫ 0.
Our result for positive random variables reads as follows.

Theorem 3.2. Assume (HA), (HB) and (HC ′ ). Fix θ ∈ Θ. Let us define the subspace V 01 ⊂ V 0 as

V 01 := span {ψ i : i ∈ N, P i = P 1 },
i.e. the eigenspace of A associated to its minimal eigenvalue P 1 . Define

(12) A ′ 1 := inf q∈Sµ inf ψ∈V 01 ψ ℓ 2 ( ) =1
q ψ, V ψ = min(s + P 1 , s -P 1 ) ∈ R, and

A ′ 2 := -s 2 + sup ψ∈V 01 ψ ℓ 2 ( ) =1 sup ϕ∈V ⊥ 0 ϕ ℓ 2 ( ) =1 ψ, V ϕ 2 H 0 (θ)ϕ, ϕ 0. (13)
For any ǫ > 0 small enough the following holds: if

P 1 = 0, E ǫ ǫA ′ 1 , whereas if P 1 = 0, then E ǫ ǫ 2 A ′ 2 + O(ǫ 3 ). Finally, if P 1 = A ′ 2 = 0, then E ǫ 0.
Note that A 1 is always non-positive but A ′ 1 may be positive. Both Theorems 3.1 and 3.2 are a consequence of similar upper bounds for perturbations of periodic operators restricted to a fiber, see Theorems 5.1 and 5.8. For these upper bounds, we present complementary lower bounds in Lemmas 5.7 and 5.9.

Periodic comparison operators

In the present section we reduce the problem of studying E ǫ to that of understanding certain adapted operators which are periodic with respect to a sublattice. Define

n := m∈γ |m| nN + m and χ n := χ n , i.e. χ n (x) = 1 for x ∈ n 0 otherwise.
Note that 0 = and that n is just the collection of (2n+1) d disjoint translates of . Let us start by stating the following lemma.

Lemma 4.1. Let u be a θ-quasi-γ-periodic function, i.e. such that for all n ∈ Z d and k ∈ γ we have

u(n + k) = e -iθ•k u(n).
Define,

u n := χ n u, n = 0, 1, 2, . . . . Then lim n→∞ u n , H q,ǫ u n ℓ 2 (D) u n 2 ℓ 2 (D) = u 0 , H q,ǫ (θ)u 0 ℓ 2 ( ) u 0 2 ℓ 2 ( )
.

The proof of this lemma is found in the appendix.

For the definition of periodic comparison operators we introduce ( 14) Ω n per := {ω ∈ Ω : ω is periodic w.r.t. nγ}. We now state the first comparison theorem. Theorem 4.2. Assume (HA), (HB) and either (HC) or (HC ′ ). Let ǫ 0, n ∈ N and let ω ∈ Ω n per be a nγ-periodic sequence of real numbers satisfying ω ∈ (S µ ) γ , i.e. ω k ∈ S µ for all k ∈ γ. Then, we have

σ(H ω,ǫ ) ⊂ Σ ǫ .
We immediately deduce the following upper bound on the minimum of the spectrum.

Corollary 4.3. Assume (HA), (HB) and either (HC) or (HC ′ ). Let ǫ 0, then E ǫ inf q∈Sµ E q,ǫ .

Proof of Theorem 4.2. For the calculation below, we need a Weyl sequence of compactly supported functions. This can indeed be done, since we only deal with bounded operators. Fix ω ∈ Ω n per and E ∈ σ(H ω,ǫ ). By Floquet-Bloch theory, there exists some θ and some normalized state f ∈ ℓ 2 ( ) for which

E = H q,ǫ (θ)f, f .
We extend f as a θ-quasi-γ-periodic function, i.e. for any x ∈ Z d let k ∈ γ such that x -k ∈ and let

f (x) := e iθ•k f (x -k). Using Lemma 4.1, extract a sub-sequence {f n } from the sequence of functions χ n f χ n f 2 , such that | (H ω,ǫ -E)f n , f n | 1/n
and satisfying, for a sequence l n ∈ N,

supp f n ⊂ Λ ln ,
where Λ ln is a cube centered at zero and sidelength l n . For x ∈ γ we define

Ω(x, n) := {ω ′ ∈ Ω : ∀k ∈ (x + Λ ln ) ∩ γ : |ǫ(ω ′ k -ω k )| 1/n} . Now, since ω ∈ (S µ ) γ , P [Ω(x, n)] > 0,
and for x, y ∈ γ satisfying |x -y| > l n , the events Ω(x, n) and Ω(y, n) are independent (and identically distributed). Using Borel-Cantelli lemma, we see that the event

Ω ′ := n∈N x∈γ Ω(x, n)
has probability one.

From the definition of Ω(x, n), we have that given ω ′ ∈ Ω ′ and n ∈ N, there exists a x(n, ω ′ ) such that ω ′ ∈ Ω(x, n). We write from now on τ x(n,ω ′ ) f n for the translated function f n (• -x(n, ω ′ )). Let ω ′ ∈ Ω ′ and n ∈ N, and calculate

(H ω ′ ,ǫ -E)τ x(n,ω ′ ) f n , τ x(n,ω ′ ) f n = (H 0 -E)τ x(n,ω ′ ) f n , τ x(n,ω ′ ) f n + ǫ V ω ′ τ x(n,ω ′ ) f n , τ x(n,ω ′ ) f n = (H 0 -E)f n , f n + ǫ V ω τ x(n,ω ′ ) f n , τ x(n,ω ′ ) f n + ǫ V ω ′ -ω τ x(n,ω ′ ) f n , τ x(n,ω ′ ) f n = (H ω,ǫ -E)f n , f n + ǫ V ω ′ -ω τ x(n,ω ′ ) f n , τ x(n,ω ′ ) f n Note that |ǫV ω ′ -ω (x)| V ∞ /n if x ∈ supp f n (• -x(n, ω ′ )), so that (H ω ′ ,ǫ -E)τ x(n,ω ′ ) f n , τ x(n,ω ′ ) f n -(H ω,ǫ -E)f n , f n 1 n V ∞ .
In particular, we see that τ x(n,ω ′ ) f n is a Weyl sequence.

Remark 4.4. This is an adaptation of a well known argument of Kirsch and Martinelli [START_REF] Kirsch | On the spectrum of Schrödinger operators with a random potential[END_REF] in the continuous setting, with S µ connected and V a multiplication operator.

Remark 4.5. When the random potential is diagonal (as in the introduction), the proof above can be adapted to overlapping, but compactly supported single-site potentials V ∈ ℓ ∞ (D) as long as

n∈γ V (• -n) ≡ 0.
Note that if this condition does not hold then H q = H 0 for all q. One way around this problem would be to consider periodic (non-constant) sequences of coupling constants ω n such that the resulting periodic potential is not zero.

To prove the following converse to Theorem 4.2 we define

Ω per := {ω ∈ Ω : ∃n ∈ N such that ω is periodic w.r.t. nγ} = n∈N Ω n per .
Lemma 4.6. Denote by Σ ǫ the almost sure spectrum of H ω,ǫ . Then:

Σ ǫ ⊂ ω∈Ωper σ(H ω,ǫ ).
Proof. Let n ∈ N and set

ω (n) k = ω k for f ∈ N ω (n) k = ω j if j -k ∈ Nγ. Let C 0 (Z d ) be the set of compactly supported functions in ℓ 2 (Z d ). Choose any ϕ ∈ C 0 (Z d ). Then lim n→∞ H ω,ǫ ϕ -H ω (n) ,ǫ ϕ = 0,
i.e. we have strong convergence H ω (n) → H ω . Since the operators H ω are bounded, the set C 0 is an operator core for H ω . This implies that we have strong convergence on the whole ℓ 2 (Z d ).

By the resolvent equation, for any

E ∈ R Σ, (H ω,ǫ -E) -1 -(H ω (n) ,ǫ -E) -1 =(H ω,ǫ -E) -1 (V ω -V ω (n) )(H ω (n) ,ǫ -E) -1 =(H ω (n) ,ǫ -E) -1 (V ω -V ω (n) )(H ω,ǫ -E) -1 ,
which converges strongly to 0. We know that if E ∈ R Σ, then (H ω,ǫ -E) -1 ϕ ∈ ℓ 2 (D) for any ϕ ∈ ℓ 2 (D) and that, using Theorem 4.2, the inclusion σ(H ω,ǫ (n)) ⊂ Σ holds for any ω in the support of the product measure D µ. To conclude, we apply Theorem VIII.24 in [START_REF] Reed | Analysis of Operators[END_REF] which tells us that

σ(H ω,ǫ ) ⊂ n∈N σ(H ω (n) ).
This finishes the proof.

In particular we obtain the following corollary.

Corollary 4.7. As before we set

Ω per := {ω ∈ Ω : ∃n ∈ N such that ω is periodic w.r.t. nγ}
and denote by Σ ǫ the almost sure spectrum of H ω,ǫ . Then:

inf Σ ǫ = inf ω∈Ωper σ(H ω,ǫ ).

Perturbation calculation

For the readers convenience we recall the definition of the constants A 1 and A 2 , the notation and the statement of the theorems before the proofs. By the continuity of the Floquet-Bloch eigenvalues there exists some θ such that

E 0 := inf σ(H 0 ) = inf σ(H 0 (θ)) = 0.
We denote by Θ ⊂ * the compact set of θ for which the last equality holds. From now on we fix some θ ∈ Θ, so the quantities below will depend on θ. Let V 0 be the eigenspace of H 0 (θ) associated to the eigenvalue E 0 = 0, p its multiplicity and choose an orthonormal basis ψ j , j = 1, . . . , p spanning V 0 and diagonalizing the Hermitian matrix A ∈ C p×p , given by the coefficients

A ij := ψ i , V ψ j .
We take the eigenvalues of the matrix A in the ascending order counting multiplicities so that P 1 := A 11 = ψ 1 , V ψ 1 is the minimal eigenvalue and P p := A pp = ψ p , V ψ p is the maximal eigenvalue of A.

Sign-changing random variables.

In this subsection we assume (HC) to hold. We will only treat this case in detail as the calculation for positive random variables is very similar. Recall from (HC) that s -< 0 < s + . We define the following quantities :

(15) A 1 := inf q∈Sµ inf ψ∈V 0 ψ ℓ 2 ( ) =1
q ψ, V ψ = min(s + P 1 , s -P p ) 0, and ( 16)

A 2 := -max(s 2 -, s 2 + ) sup ψ∈V 0 ψ ℓ 2 ( ) =1 sup ϕ∈V ⊥ 0 ϕ ℓ 2 ( ) =1 ψ, V ϕ 2 H 0 (θ)ϕ, ϕ 0.
Note that the sign of A 1 and A 2 is fixed. We will prove the following theorem, which is only a restatement of Theorem 3.1.

Theorem 5.1. Assume (HA), (HB) and (HC). Fix θ ∈ Θ. Then, if

A 1 = 0, E ǫ ǫA 1 , whereas if A 1 = 0, then E ǫ ǫ 2 A 2 + O(ǫ 3 ). Finally, if A 1 = A 2 = 0, then E ǫ 0. Remark 5.2.
• We remind that we have fixed θ to simplify notations, but A 1 and A 2 depend on θ. The best bound for the behavior of the bottom of the spectrum is obtained by looking at each θ ∈ Θ and taking the minimum. • We see that our bound on the bottom of the spectrum behaves linearly, quadratically or it doesn't move with ǫ. In the analogous setting in continuum space, if the unique continuation principle is not violated, then the analogous result does not allow the third case A 1 = A 2 = 0. This leaves only the cases of a linear or a quadratic bound.

• The definition of the quantities A 1 , A 2 may seem complicated at first sight, but these choices are optimal, in the sense of Lemma 5.7 below, which is a converse of Lemma 5.6 in the regime ǫ ≪ 1.

Before proving the theorem, let us provide a much simpler, nonoptimal upper bound for A 2 as well as a condition ensuring that |A 1 | + |A 2 | = 0.

5.2.

A simple non-degeneracy condition. Theorem 5.1 tells us that if A 2 = 0, then the expansion of the bottom of the spectrum is at least quadratic, but if A 1 = A 2 = 0, we can only say that the spectrum starts at zero. When V is diagonal this only happens if the support of the single-cell potential and the eigenfunctions ψ 1 , . . . , ψ p are disjoint (the ψ i were defined at the beginning of this section).

Note, that in the continuous configuration space this can only happen if the potential violates the unique continuation principle. For a discussion on the validity of the unique continuation principle see for instance [START_REF] Wolff | Recent work on sharp estimates in second-order elliptic unique continuation problems[END_REF].

Let us discuss the condition in our general setting. First let us remark that if A 1 = 0, then the matrix A ∈ C p×p vanishes identically, i.e. ( 17)

A 1 = 0 =⇒ sup ψ∈V 0 ψ ℓ 2 ( ) =1 ψ, V ψ = 0. and thus (∀ψ ∈ V 0 ) V ψ ∈ V ⊥ 0 .
The operator H 0 is invertible on V ⊥ 0 and thus there exists some ϕ ∈ V ⊥ 0 such that

(18) H 0 (θ)ϕ = V ψ * .
Hence, we have that

V ϕ, ψ * = H 0 (θ)ϕ, ϕ .
Now, assume there exists some ψ * ∈ V 0 such that

(19) V ψ * = 0.
Then ϕ in [START_REF] Meyer | Matrix analysis and applied linear algebra[END_REF] does not vanish and

A 2 -max(s 2 -, s 2 + ) V ϕ, ψ * 2 H 0 (θ)ϕ, ϕ = -max(s 2 -, s 2 + ) H 0 (θ)ϕ, ϕ < 0, because ϕ ∈ ker H 0 (θ).
Remark 5.3. Formally, we have

A 2 -max(s 2 -, s 2 + ) ψ * , V H 0 (θ) -1 V ψ * when A 1 = 0.
In the converse direction, A 1 = 0 together with A 2 = 0 implies that (∀ψ ∈ V 0 and ∀ϕ ∈ ℓ 2 ( )) V ψ, ϕ = 0, i.e. that (∀ψ ∈ V 0 ) V ψ = 0.

We summarize the above discussion as follows.

Lemma 5.4. Under the assumptions of Theorem 5.1 we have that

A 1 = 0 and A 2 = 0 if and only if (∀ψ * ∈ V 0 ) V ψ * = 0.
5.3. Proof of Theorem 5.1. We subdivide the proof of Theorem 5.1 into two lemmas. The first covers both types of sign assumptions on the random variables.

Lemma 5.5. Assume (HA), (HB), and either (HC) or (HC ′ ). Let u ∈ ℓ 2 ( ) and E ǫ as in [START_REF] Elgart | Lifshitz tails and localization in the three-dimensional Anderson model[END_REF]. Then,

E ǫ inf q∈Sµ inf u∈ℓ 2 ( ) H ǫ,q (θ)u, u u ℓ 2 ( )
for any θ ∈ * Proof. By Corollary 4.3 it is enough to consider the periodic realizations of the potential. By the Courant-Weyl-Fischer min-max principle, (20)

E ǫ E q,ǫ = min σ(H ǫ,q ) = inf a∈ℓ 2 (Z d ) a 2 =1
H ǫ,q a, a .

Finally, by Lemma 4.1, ( 21) inf

a∈ℓ 2 (Z d ) a =1
H ǫ,q a, a inf

u∈ℓ 2 ( ) H ǫ,q (θ)u, u u ℓ 2 ( ) .
This proves the lemma.

We state now the second lemma. It applies to the case of signchanging random variables. Lemma 5.6. Let A 1 , A 2 as in [START_REF] Hislop | Lectures on Random Schrödinger Operators[END_REF], [START_REF] Hoecker-Escuti | Le modèle d'Anderson en régime de faible désordre[END_REF], assume (HA), (HB), and (HC) and fix θ ∈ Θ. Then, for ǫ > 0 small enough, if

A 1 = 0, inf q∈Sµ inf u ℓ 2 ( ) =1 H ǫ,q (θ)u, u ǫA 1 , whereas if A 1 = 0, then inf q∈Sµ inf u ℓ 2 ( ) =1 H ǫ,q (θ)u, u ǫ 2 A 2 + O(ǫ 3 ) Finally, if A 1 = A 2 = 0, then inf q∈Sµ inf u ℓ 2 ( ) =1
H ǫ,q (θ)u, u 0.

Proof. It is enough to show that for some q ∈ S µ , there is some normalized state u ∈ ℓ 2 ( ) satisfying H ǫ,q (θ)u, u ǫA 1 or ǫ 2 A 2 + O(ǫ 3 ) or 0 resp.. Let ψ ∈ V 0 and ϕ ∈ V ⊥ 0 , to be chosen later, and u = ψ + ǫqϕ. We assume furthermore ψ = 1. We expand

u 2 = ψ 2 + ǫ 2 q 2 ϕ 2
and thus [START_REF] Titchmarsh | Eigenfunction expansions associated with second-order differential equations[END_REF] 1/ u 2 = 1 -ǫ 2 q 2 ϕ 2 + O(ǫ 4 ϕ 4 ).

We calculate the kinetic energy of this state, i.e. ( 23)

H 0 (θ)u, u = H 0 (θ)ψ, ψ + 2ǫq Re H 0 (θ)ψ, ϕ + ǫ 2 q 2 H 0 (θ)ϕ, ϕ .
Because ψ ∈ V 0 and E 0 = 0, we see that [START_REF] Wang | Localization and universality of Poisson statistics for the multidimensional Anderson model at weak disorder[END_REF] becomes

H 0 (θ)u, u = ǫ 2 q 2 H 0 (θ)ϕ, ϕ .
We expand the potential energy as

ǫq V u, u = ǫq V ψ, ψ + 2ǫ 2 q 2 Re V ϕ, ψ + ǫ 3 q 3 V ϕ, ϕ .
Thus,

H ǫ,q (θ)u, u =ǫq V ψ, ψ + ǫ 2 q 2 H 0 (θ)ϕ, ϕ + 2 Re ψ, V ϕ + ǫ 3 q 3 V ϕ, ϕ . ( 24
)
Case A 1 = 0. Note that in this case P 1 P p = 0. From now on we assume that s + P 1 s -P p . If this is not the case, we can always replace V → -V and ω n → -ω n to get an equivalent model. In this case, we take ψ = ψ 1 , ϕ = 0 and q = s + . Then, [START_REF] Wolff | Recent work on sharp estimates in second-order elliptic unique continuation problems[END_REF] becomes

H ǫ,q (θ)u, u = ǫq V ψ 1 , ψ 1 = ǫs + P 1 ,
which proves the result in this case, as u is normalized.

Case A 1 = 0 and A 2 = 0. First let us remark that if A 1 = 0 then the matrix A ∈ C p×p vanishes identically, i.e.

(25)

A 1 = 0 =⇒ sup ψ∈V 0 ψ ℓ 2 ( ) =1 ψ, V ψ = 0.
In this case we have that, for any ψ ∈ V 0 and ϕ ∈ V ⊥ 0 , the expansion (24) becomes (26)

H ǫ,q (θ)u, u = ǫ 2 q 2 H 0 (θ)ϕ, ϕ + 2 Re ψ, V ϕ +ǫ 3 q 3 V ϕ, ϕ .
Note that, for ψ ∈ V 0 and ϕ ∈ V ⊥ 0 such that

ψ 2 = ϕ 2 = 1 the map (ϕ, ψ) → ψ, V ϕ 2 H 0 (θ)ϕ, ϕ
is continuous. Given that the spaces involved are finite-dimensional and their respective unit balls thus compact, we know that there exists a couple (ψ * , ϕ * ) maximizing this quantity, i.e.

ψ * , V ϕ * 2 H 0 (θ)ϕ * , ϕ * = sup ψ∈V 0 ψ ℓ 2 ( ) =1 sup ϕ∈V ⊥ 0 ϕ ℓ 2 ( ) =1 ψ, V ϕ 2 H 0 (θ)ϕ, ϕ .
Let ψ = ψ * and ϕ = λϕ * in the definition of u, where

λ = - ψ * , V ϕ * H 0 (θ)ϕ * , ϕ * ∈ C.
Replacing, we see that

H 0 (θ)ϕ, ϕ + 2Re ψ, V ϕ =|λ| 2 H 0 (θ)ϕ * , ϕ * + 2Re λ ψ * , V ϕ * = ψ * , V ϕ * 2 H 0 (θ)ϕ * , ϕ * -2 ψ * , V ϕ * 2 H 0 (θ)ϕ * , ϕ * = - ψ * , V ϕ * 2 H 0 (θ)ϕ * , ϕ * .
Using this in (26) and letting q 2 = max(s 2 -, s 2 + ), we obtain

H ǫ,q (θ)u, u = -ǫ 2 max(s 2 -, s 2 + ) ψ * , V ϕ * 2 H 0 (θ)ϕ * , ϕ * + O(ǫ 3 q 3 ϕ * 2 ) = ǫ 2 A 2 + O(ǫ 3 q 3 ϕ 2 ).
Normalizing u by multiplying by [START_REF] Titchmarsh | Eigenfunction expansions associated with second-order differential equations[END_REF] gives the result.

Case A 1 = 0 and A 2 = 0. Choose ϕ = 0 and any normalized ψ ∈ V 0 . The development using u in this case gives H ǫ,q (θ)u, u = ǫ 3 q 3 V ϕ, ϕ = 0 and this yields the desired result.

We prove the following converse lemma.

Lemma 5.7. Let A 1 , A 2 as in (10), [START_REF] Hoecker-Escuti | Le modèle d'Anderson en régime de faible désordre[END_REF], assume (HA), (HB) and (HC), and fix θ ∈ Θ. Then, for ǫ > 0 small enough, if

A 1 = 0, inf q∈Sµ inf u ℓ 2 ( ) =1 H ǫ,q (θ)u, u ǫA 1 + O(ǫ 3/2 ), whereas if A 1 = 0, then inf q∈Sµ inf u ℓ 2 ( ) =1 H ǫ,q (θ)u, u ǫ 2 A 2 + O(ǫ 3 ) Finally, if A 1 = A 2 = 0, then inf q∈Sµ inf u ℓ 2 ( ) =1
H ǫ,q (θ)u, u 0.

Proof. Fix ǫ > 0 and let q ∈ S µ be the value which minimizes the map

q → inf u ℓ 2 ( ) =1
H ǫ,q (θ)u, u , i.e. q ∈ {s -, s + }. We lower bound this quantity by minimizing over a larger set by writing inf

u ℓ 2 ( ) =1
H ǫ,q (θ)u, u inf

ψ∈V 0 ψ ℓ 2 ( ) 1 inf ϕ∈V ⊥ 0 ϕ ℓ 2 ( ) 1 H ǫ,q (θ)(ψ + ϕ), (ψ + ϕ) .
By continuity and compactness, there exists some pair (ψ * , ϕ * ) in V 0 × V ⊥ 0 realizing the infimum on the right hand side. We see that (27)

H 0 (θ)(ψ * + ϕ * ), (ψ * + ϕ * ) = H 0 (θ)ϕ * , ϕ * g ϕ * 2 ℓ 2 ( ) ,
where the constant g is the spectral gap of H 0 . Due to our normalization g coincides with the (positive) second eigenvalue of H 0 . We study the different cases.

Case A 1 = 0. From Lemma 5.6, we know already that (28)

|A 1 | + |A 2 | = 0 =⇒ H ǫ,q (θ)(ψ * + ϕ * ), (ψ * + ϕ * ) < 0.
Using ( 27) and (28) we get that

ϕ * 2 ℓ 2 ( ) -g -1 ǫq V (ψ * + ϕ * ), (ψ * + ϕ * ) 4g -1 V ∞ ǫq.
We deduce then that inf

u ℓ 2 ( ) =1 H ǫ,q (θ)u, u ǫq V ψ * , ψ * + 2ǫq Re V ϕ * , ψ * + ǫq V ϕ * , ϕ * ǫA 1 -4g -1/2 ǫ 3/2 q 3/2 V 3/2 ∞ -4g -1 ǫ 2 q 2 V 2 ∞ . Case A 1 = 0 and A 2 = 0. In this case H ǫ,q (θ)(ψ * + ϕ * ), (ψ * + ϕ * ) = H 0 (θ)ϕ * , ϕ * + 2ǫq Re V ϕ * , ψ * + ǫq V ϕ * , ϕ * .
Using (28) we see that ϕ * = 0. Furthermore, ( 27) and (28) together imply that

ϕ * 2 ℓ 2 ( ) ǫqg -1 V ∞ (2 ϕ * ℓ 2 ( ) + ϕ * 2 ℓ 2 ( ) ) 3ǫqg -1 V ∞ ϕ * ℓ 2 ( ) . Note that ϕ * ℓ 2 (
) is on both sides of the inequality. Simplifying, (29)

ϕ * ℓ 2 ( ) 3ǫqg -1 V ∞ .
Expanding as ǫ → 0, employing (29) and then simply multiplying by 1 = |λ| 2 /|λ| 2 = λ/λ, we write inf

u ℓ 2 ( ) =1 H ǫ,q (θ)u, u H 0 (θ)ϕ * , ϕ * + 2 Re ǫq V ψ * , ϕ * + ǫq V ϕ * , ϕ * = |λ| 2 H 0 (θ)ϕ * , ϕ * |λ| 2 + 2 Re λǫq V ψ * , ϕ * λ + O(ǫ 3 )
We choose λ as

λ = - H 0 (θ)ϕ * , ϕ * V ψ * , ϕ * .
It is well defined for small ǫ. Indeed, using ( 27) and ( 28) we see that

-2ǫq Re V ϕ * , ψ * H 0 (θ)ϕ * , ϕ * + ǫq V ϕ * , ϕ * c ϕ * 2 ℓ 2 ( ) -ǫq V ∞ ϕ * 2 ℓ 2 ( )
and we know that ϕ * = 0. We see that λ = 0 for ǫ small enough.

Using our choice of λ gives

c ϕ * 2 ℓ 2 ( ) -ǫq V ∞ ϕ * 2 ℓ 2 ( ) = (|λ| 2 -2 Re λǫq) V ψ * , ϕ * 2 H 0 (θ)ϕ * , ϕ * + O(ǫ 3 ) -ǫ 2 A 2 + O(ǫ 3 ),
where we have used that |λ| 2 -2 Re λǫq |λ| 2 -2|λ|ǫq -ǫ 2 q 2 .

Case A 1 = 0 and A 2 = 0. In this case 0 inf

u ℓ 2 ( ) H ǫ,q (θ)u, u = H 0 (θ)ϕ * , ϕ * + ǫq V ϕ * , ϕ * (30) c ϕ * 2 -O(ǫ) ϕ * 2 , ( 31
)
where the first inequality comes from Lemma 5.6. It is now clear that ψ * = 0 and this finishes the proof.

5.4.

Positive random variables. We study in this subsection the case involving positive random variables. We remind the reader the definition of the constants involved, for which we use the functions ψ i , the matrix A, its eigenvalues P i and the linear space V 0 , which can be found at the beginning of this section. We define the subspace V 01 ⊂ V 0 as V 01 := span

{i:P i =P 1 } ψ i ,
i.e. the eigenspace of A associated to its minimal eigenvalue P 1 .

We recall the following quantities :

(32)

A ′ 1 := inf q∈Sµ inf ψ∈V 01 ψ ℓ 2 ( ) =1
q ψ, V ψ = min(s + P 1 , s -P 1 ) ∈ R, and (33)

A ′ 2 := -s 2 + sup ψ∈V 01 ψ ℓ 2 ( ) =1 sup ϕ∈V ⊥ 0 ϕ ℓ 2 ( ) =1 ψ, V ϕ 2 H 0 (θ)ϕ, ϕ 0.
Note that, unlike the other coefficients, A ′ 1 may take on both signs. We also restate Theorem 3.2 for the reader's convenience.

Theorem 5.8. Assume (HA), (HC) and (HC ′ ). Fix θ ∈ Θ. Then, for ǫ > 0 small enough, if P 1 = 0,

E ǫ ǫA ′ 1 ,
whereas if P 1 = 0, then E ǫ ǫ 2 A ′ 2 + O(ǫ 3 ). Finally, if P 1 = A ′ 2 = 0, then E ǫ 0.
Sketch of proof. The proof of this theorem is very similar to the proof of Theorem 5.1. Indeed, Lemma 5.5 is also valid in this setting. We proceed then as in Lemma 5.6 up to equation [START_REF] Wolff | Recent work on sharp estimates in second-order elliptic unique continuation problems[END_REF]. If A ′ 1 = 0 we let

u = ψ 1 , ϕ = 0, q = s + if P 1 < 0 s -if P 1 0
in [START_REF] Wolff | Recent work on sharp estimates in second-order elliptic unique continuation problems[END_REF]. If A ′ 1 = 0 but A ′ 2 = 0, then we find ψ * ∈ V 01 and ϕ ∈ V 0 realizing the supremum in the definition of A ′ 2 and then we proceed as in Lemma 5.6. Finally, if A ′ 1 = A ′ 2 = 0 we take u = ψ 1 in [START_REF] Wolff | Recent work on sharp estimates in second-order elliptic unique continuation problems[END_REF]. We prove the following converse lemma. Lemma 5.9. Let A ′ 1 , A ′ 2 as in (12), [START_REF] Kirsch | Comparison theorems for the gap of Schrödinger operators[END_REF], assume (HA), (HB) and (HC ′ ), and fix θ ∈ Θ. Then, for ǫ > 0 small enough, if

A ′ 1 = 0, inf q∈Sµ inf u ℓ 2 ( ) =1 H ǫ,q (θ)u, u ǫA ′ 1 + O(ǫ 3/2 ),
whereas if P 1 = 0, then inf q∈Sµ inf u ℓ 2 ( ) =1 H ǫ,q (θ)u, u ǫ 2 A ′ 2 + O(ǫ 3 ) Finally, if P 1 = A ′ 2 = 0, then inf q∈Sµ inf u ℓ 2 ( ) =1
H ǫ,q (θ)u, u 0.

Proof. We adapt here the proof of Lemma 5.7. Fix ǫ > 0 and let q ∈ S µ be the value which minimizes the map

q → inf u ℓ 2 ( ) =1
(H ǫ,q (θ) -ǫA ′ 1 )u, u , i.e. q ∈ {s -, s + }. We lower bound this quantity by minimizing over a larger set by writing inf

u ℓ 2 ( ) =1 (H ǫ,q (θ) -ǫA ′ 1 )u, u inf ψ∈V 0 ψ ℓ 2 ( ) 1 inf ϕ∈V ⊥ 0 ϕ ℓ 2 ( ) 1 (H ǫ,q (θ) -ǫA ′ 1 )(ψ + ϕ), (ψ + ϕ) .
By continuity and compactness, there exists some pair (ψ * , ϕ * ) in V 0 × V ⊥ 0 realizing the infimum on the right hand side. We see that (34)

H 0 (θ)(ψ * + ϕ * ), (ψ * + ϕ * ) = H 0 (θ)ϕ * , ϕ * g ϕ * 2 ℓ 2 ( ) ,
where the constant g is the spectral gap of H 0 , which is also its (positive) second eigenvalue.

We study the different cases.

Case A ′ 1 = 0. We know already that (35) (H ǫ,q (θ) -ǫA ′ 1 )(ψ * + ϕ * ), (ψ * + ϕ * ) 0.

Using (34), (35) and

|A ′ 1 | q V we get ϕ * 2 ℓ 2 ( ) -g -1 ǫ (qV -A ′ 1 )(ψ * + ϕ * ), (ψ * + ϕ * ) 4g -1 V ∞ ǫq. Note that q V ψ * , ψ * q inf ψ∈V 0 ψ ℓ 2 ( ) 1 V ψ, ψ = q inf ψ∈V 00 ψ ℓ 2 ( ) 1 V ψ, ψ E ′ 1 . It implies inf u ℓ 2 ( ) =1 H ǫ,q (θ)u, u ǫq V ψ * , ψ * + 2ǫq Re V ϕ * , ψ * + ǫq V ϕ * , ϕ * ǫA ′ 1 -4g -1/2 ǫ 3/2 q 3/2 V 3/2 ∞ -4g -1 ǫ 2 q 2 V 2 ∞ .
Case A ′ 1 = 0 and A ′ 2 = 0. We know that (36) H ǫ,q (θ)(ψ * + ϕ * ), (ψ * + ϕ * ) < 0.

We will decompose further

ψ * = ψ * 01 + ψ * 0⊥ ∈ V 0 , with ψ * 01 ∈ V 01 and ψ * 0⊥ ∈ V ⊥ 01 . We have (37) V u, u 2 Re V ψ * 01 , ϕ * + O ϕ * 2
Using (39) and then simply multiplying by 1 = |λ| 2 /|λ| 2 = λ/λ, we get inf

u ℓ 2 ( ) =1 H ǫ,q (θ)u, u |λ| 2 H 0 (θ)ϕ * , ϕ * |λ| 2 + 2 Re λǫq V ψ * 01 , ϕ * λ + O(ǫ 3 )
In view of this inequality and (36), we see that V ψ * 01 , ϕ * = 0 for small ǫ. We choose λ as

λ = - H 0 (θ)ϕ * , ϕ * V ψ * 01 , ϕ * . It is well defined for small ǫ. Using our choice of λ, it gives inf u ℓ 2 ( ) =1 H ǫ,q (θ)u, u (|λ| 2 -2 Re λǫq) V ψ * 01 , ϕ * 2 H 0 (θ)ϕ * , ϕ * + O(ǫ 3 ) -ǫ 2 A ′ 2 + O(ǫ 3
), where we have used that |λ| 2 -2 Re λǫq |λ| 2 -2|λ|ǫq -ǫ 2 q 2 .

Case A 1 = 0 and A 2 = 0. In this case, re-using (37) and employing

A ′ 2 = 0, 0 inf u ℓ 2 ( ) =1 H ǫ,q (θ)u, u = H 0 (θ)ϕ * , ϕ * + 2ǫq Re V ψ * 01 , ϕ * + ǫqO ϕ * 2 = H 0 (θ)ϕ * , ϕ * + ǫqO ϕ * 2 c ϕ * 2 -|O(ǫ)| ϕ * 2 ,
where the first inequality comes from Lemma 5.6. It is now clear that ψ * = 0 and this finishes the proof. Let us check property [START_REF] Pastur | Spectra of random and almost-periodic operators[END_REF] for the operator

H 0 (0) := -∆ + W + C W ,
where ∆ is the Laplacian on with periodic boundary conditions and W is the restriction of W to . To check the property we use Perron-Frobenius theorem [START_REF] Meyer | Matrix analysis and applied linear algebra[END_REF]. For m > | |, we verify that

δ x , (∆ -W + W ∞ + 2d + 1) m δ y δ x , (∆ + 2d + 1) m δ y 1.
This implies that the largest eigenvalue of the matrix ∆ -W + W ∞ + 2d + 1 is simple and its corresponding eigenfunction ψ 1 is positive (i.e. (∀n ∈ ) ψ 1 (n) > 0). Because of this strict positivity, condition ( 19) is satisfied as soon as V ≡ 0. The subspace V 0 is thus one-dimensional and contains only ψ 1 . The theorem is now proven, by simply stating the consequences of Theorem 5.1. We know recall Theorem 2.4 in [START_REF] Kirsch | Comparison theorems for the gap of Schrödinger operators[END_REF], with our notations. It implies that 0 is the unique θ ∈ * realizing the minimum of the spectrum.

Theorem 5.10. Let H 0 = -△ Z d + W with W a periodic potential with respect to γ = NZ d , and E 0 (θ) be the smallest eigenvalue of H 0 (θ). Then

(a -/a + ) 2 2d - d i=1 cos(θ i ) E 0 (θ) -E 0 (0) 2d - d i=1 cos(θ i ) .
Here, a ± = ± max ±ψ 1 and ψ 1 is the positive ground state of H 0 (0).

The band structure of the spectrum of discrete alloy type models has been analyzed in [START_REF] Elgart | Discrete Schrödinger operators with random alloy-type potential[END_REF] 6. Appendix 6.1. An interesting example: Proof of Theorem 1.2. Let H 0 := ∆ 2 Z defined on ℓ 2 (Z). This operator has hopping range N = 3 (see (HA)) and thus = {-1, 0, 1, }. We define V as the multiplication operator given by the following single-site potential:

V : ℓ 2 ( ) → R V (n) := - 1 2 δ -1 (n) + δ 0 (n) - 1 2 δ 1 (n).
With these definitions, we see that, for θ ∈ [-π/3, π/3) d , H 0 (θ) =   6 -4 + e -3iθ 1 -4e -3iθ -4 + e 3iθ 6 -4 + e -3iθ 1 -4e -3iθ -4 + e 3iθ 6   , after [START_REF] Elgart | Discrete Schrödinger operators with random alloy-type potential[END_REF]. This matrix has a simple ground state ψ 0 (θ) := (e -iθ , 1, e iθ )/ √ 3

with eigenvalues E 0 (θ) = (2 -2 cos(θ)) 2 . Let now fn (θ) = χ n ψ0 (θ) ∈ ℓ 2 (Z d ) where ψ0 (θ) is the θ-quasi-γ-periodic extension of ψ 0 (θ). Finally, for ξ > 1/4, let u n := f n (0) + ǫ ξ f n (ǫ ξ ).

Let us calculate the kinetic energy. We see that (40) H 0 u n , u n = H 0 f n (0), f n (0) + 2ǫ ξ Re H 0 f n (0), f n (ǫ ξ ) + ǫ 2ξ H 0 f n (ǫ ξ ), f n (ǫ ξ ) .

Let δ > 0 and pick n so large so that f n (0), H 0 f n (0) ℓ 2 (D)

f n (0) 2

f n (0) 2 ℓ 2 (D) = f n (0), H 2 0 f n (0) ℓ 2 (D) f n 2 ℓ 2 (D) δ.
Then, from (40) we see that

H 0 u n , u n δ f n (0) 2 + 2ǫ ξ δ f n (ǫ ξ ) + ǫ 2ξ ψ 0 (ǫ ξ ), H 0 (θ)ψ 0 (ǫ ξ ) ℓ 2 ( ) f n (θ) 2 3δ + ǫ 2ξ E 0 (ǫ ξ ) f n (θ) 2 .
Letting n → ∞ and δ → 0 we see that

H 0 u n , u n ǫ 2ξ E 0 (ǫ ξ ) Cǫ 6ξ .
Now let us calculate the potential energy.

ǫ V q u n , u n = ǫ V q f n (0), f n (0) + 2ǫ 1+ξ Re V q f n (0), f n (ǫ ξ ) + ǫ 1+2ξ V q f n (ǫ ξ ), f n (ǫ ξ ) = 2ǫ 1+ξ Re V q f n (0), f n (ǫ ξ ) .

Now we can calculate explicitly

V q f n (ǫ ξ ), f n (ǫ ξ ) = 1 6 -e -iǫ ξ + 2 -e iǫ ξ = -

1 3 ǫ ξ + O(ǫ 2ξ ).
This shows that, for small ǫ, H ǫ,q u n , u n Cξ 6ξ -

1 3 ǫ 1+2ξ + O(ǫ 1+2ξ ) - 1 6 ǫ 1+2ξ ,
where we have used that 6ξ > 1 + 2ξ.

6.2. Proof of Lemma 4.1. As the V q is block-diagonal, it is enough to do the calculation for the free operator H 0 . Let us first calculate some norms. Because of the quasi-periodicity, we easily see that We see from this calculation and (42) thus that H 0 u n , u n -(2n -3) d H 0 (θ)u 0 , u 0 Cn d-1 u 0 2 ℓ 2 ( ) . As (2n -3)/(2n -1) → 1, dividing by u n 2 ℓ 2 (D) , using (41) and taking the limit proves the lemma.

5. 5 .

 5 Application to the discrete alloy type model. Proof of Theorem 1.1. It is enough to verify that the assumptions of Theorem 5.1 are satisfied. Let C W := inf σ(-∆ Z d + W ). It is clear that the operator H 0 := -∆ Z d + W -C W satisfies hypothesis (HA). In this case the set Θ consists of the single point θ = 0 (see Theorem 5.10 below).

ℓ 2 2 ℓ 2 2 ℓ 2

 22222 (D) -ψ 0 (0), H 0 (θ)ψ 0 (0) ℓ 2 ( ) ǫ ξ ), H 0 f n (ǫ ξ ) ℓ 2 (D) f n (ǫ ξ ) (D) -ψ 0 (ǫ ξ ), H 0 (θ)ψ 0 (ǫ ξ ) ℓ 2 ( ) ψ 0 (ǫ ξ )

u n 2 ℓ 2 ( 2 ℓ 2 Cn d-1 u 0 2 ℓ 2 2 ℓ 2 1 =

 222222221 D) = (2n + 1) d u 0 ( ) . So we have thatH 0 u n , u n = H 0 u n , u n-1 + O(n d-1 ) u 0 ( ) .(42)For any k ∈ n-1 and k ′ ∈ Z d n , we have that |k -k ′ | N and thus, because of the finite hopping range (assumption (HB)),H 0 u n , u n-1 = H 0 u, u n-1 .Now, we developH 0 u, u n-k∈Z d k ′ ∈Z d H 0 (k, k ′ )u(k ′ )u n-1 (k) (43) = m∈γ |m| (n-1)N m ′ ∈γ k∈ +m k ′ ∈ +m ′ H 0 (k, k ′ )u(k ′ )u n-1 (k) = m∈γ |m| (n-1)N m ′ ∈γ k∈ k ′ ∈ H 0 (k + m, k ′ + m ′ )u(k ′ + m ′ )u n-1 (k + m)Using the translation invariance (assumption (HB)), the last quantity is equal to (44)m∈γ |m| (n-1)N m ′ ∈γ k∈ k ′ ∈ H 0 (k, k ′ + m ′ -m)u(k ′ + m ′ -m)u n-1 (k) = m∈γ |m| (n-1)N m ′ ∈γ k∈ k ′ ∈ e iθ•(m-m ′ ) H 0 (k, k ′ -m ′ + m)u 0 (k ′ )u 0 (k) = m∈γ |m| (n-1)N m ′′ ∈γ k∈ k ′ ∈ e iθ•m ′′ H 0 (k, k ′ -m ′′ )u 0 (k ′ )u 0 (k) = (2n -3) d H 0 (θ)u 0 , u 0 .

Acknowledgment

The research of D.B. was supported by Russian Scientific Foundation (project no. 14-11-00078). F.H-E and I.V. were supported by DFG-Projects Zufällige und periodische Quantengraphen and Eindeutige-Fortsetzungsprinzipien und Gleichverteilungseigenschaften von Eigenfunktionen.

We thank M. Täufer for reading a previous version of the manuscript.

Indeed, using the definition of u, we see that,

Let us also note that

Hence if V ψ * 0⊥ , ϕ * = 0, we immediately get (37) from (38). Assume that V ψ * 0⊥ , ϕ * = 0. For each µ ∈ C,

Using this result, we obtain

H 0 (θ)ϕ * , ϕ * + 2ǫq Re V ψ * 01 , ϕ * + O ǫq ϕ * 2 Equality (38) and inequalities (34) and (36) together imply that

) . Note that ϕ * ℓ 2 ( ) is involved in both sides of the inequality. Simplifying, (39) ϕ * ℓ 2 ( ) 3ǫqg -1 V ∞ .


Amaël Caillard

Stéphane Cuynet

Thomas Lecas

Maxime Mikikian

Pascal Andreazza

Pascal Brault

Gas aggregation source based on pulsed plasma sputtering for the synthesis of PtX catalytic nanoclusters

Gas aggregation source based on pulsed plasma sputtering for the synthesis of PtX catalytic nanoclusters

A. Caillard (*)1 , S. Cuynet 1 , T. Lecas 1 , M. Mikikian 1 , P. Andreazza 2 , P. Brault 1 1 1Groupe de Recherches sur l'Energétique des Milieux Ionisés (GREMI) UMR7344, CNRS -Université d'Orléans, BP6744, 14 rue d'Issoudun, 45067 Orléans, France 2 Centre de Recherche sur la Matière Divisée, FRE 3520, CNRS -Université d'Orléans, 1b rue de la férollerie 45071 Orléans, France (*) amael.caillard@univ-orleans.fr

The gas-aggregation source technique with a planar magnetron is used for the production of PtX (X = Pd or Au) catalyst nanoclusters on polymeric substrate used in low temperature fuel cell. The target material is sputtered by employing either a DC power supply or a high-power impulse magnetron sputtering power device known to enhance the degree of ionization of the target species. The nanocluster properties are studied as a function of the power characteristics by diagnostics like time-of-flight mass spectrometer and post deposition methods (Grazing Incidence X-Ray Diffraction, Transmission Electron Microscopy) compared to those obtained by standart DC magnetron sputtering.

The gas-aggregation source (GAS) with a planar magnetron may be a convenient technique for the production of pure metallic nanoclusters (NC) [1] compared to wet chemical techniques needing a final calcination step that induces cluster growth and substrate deterioration. Such NC may have important applications such as in heterogeneous catalysis, gas sensor technology and microelectronics. The nanocluster source is already employed for the production of a wide variety of metal NCs; e.g. Co, C, Ti, Pd, Pt… and of a few bimetallic materials. Proton exchange membrane (PEM) fuel cells require the production of such electrocatalysts directly on a polymeric membrane. The properties (shape, structure and size) of these catalytic NCs must be well controlled in order to promote their catalytic activity towards the oxygen reduction reaction.

The present study aims to synthesize PtX (X = Pd or Au) catalysts on the proton exchange membrane by the GAS technique with a planar magnetron. The experimental setup is made of three aligned components : the GAS (0.3 mbar in pure argon), a high vaccum deposition chamber (about 0.003 mbar) and a time-of-flight mass spectrometer. Pt and X atoms are sputtered by argon ions from the PtX target mounted on the two inch magnetron in the GAS. Due to the relative high pressure, PtX NCs are formed. They are ejected from the GAS through a 4 mm in diameter aperture and land on the substrate placed in the high vaccum chamber. When the substrate holder is removed, the mass spectrometer may be used to characterize the NC size. In the past 2 decades, a DC power supply was applied to the magnetron for the production of a temporally stable NC beam. Adjustable parameters such as aggregation length and gas flow rates were used for controlling the beam intensity and the NC size. In this study, these two parameters are kept constant and the target is sputtered by employing a pulsed power supply using low frequency. Such power supply, called high-power impulse magnetron sputtering power (HIPIMS) supply is known to enhance the degree of ionization of the target species [2]. Two recent publications provide insight into the potential benefit of using pulsed power supply for the synthesis of metallic nanoclusters [3,4].

The PtX NCs grown by HiPIMS based GAS are compared to those obtained by conventional GAS employing DC power supply. The size of the PtX NCs is measured by Transmission Electron Microscopy and Mass spectrometer. The nanoclusters have a mean diameter of 5 nm, depending of the operating parameters. The influence of the HiPIMS power characteristics (pulse power, duty cycle and repetition rate) on the NC growth mechanism is studied. For very low frequencies, a unique HiPIMS pulse induces an individual group of nanoclusters. This group is not correlated to the previous and to the following one making possible the control of the NC size. Grazing Incidence X-Ray Diffraction (Soleil synchrotron, SixS) is used to study the cristalline structure of PtPd nanoclusters.