Horia D Cornean

AND Bernard Helffer

Radu Purice

A BEALS CRITERION FOR MAGNETIC PSEUDODIFFERENTIAL OPERATORS PROVED WITH MAGNETIC GABOR FRAMES

First, we give a new proof for the Beals commutator criterion for non-magnetic Weyl pseudo-differential operators based on classical Gabor tight frames. Second, by introducing a modified 'magnetic' Gabor tight frame, we naturally derive the magnetic analogue of the Beals criterion originally considered by Iftimie-Mȃntoiu-Purice.

Introduction and main results

1.1. Introduction. The Beals criterion [B] naturally characterizes pseudo-differential operators by their commutation properties with fundamental objects like multiplication and differentiation operators; the basics of Weyl pseudo-differential calculus can be found in e.g. [START_REF] Hörmander | The Analysis of Linear Partial Differential Operators[END_REF]. To the best of our knowledge, all existent proofs of Beals' criterion use in an essential way some special properties of the Fourier transform and the translation invariance of the seminorms in S (R 2d ), see for example Lemma 2.2 in [B97] or Proposition 8.2 in [Di-Sj]. In recent years it appeared useful to introduce a magnetic pseudo-differential calculus (see [START_REF] Iftimie | Magnetic pseudodifferential operators[END_REF][START_REF] Iftimie | Commutator criteria for magnetic pseudo-differential operators[END_REF][START_REF] Iftimie | Magnetic Fourier integral operators[END_REF] and references therein) which is adapted to the presence of long-range magnetic fields. The main motivation behind this particular class of operators was the need of highlighting the magnetic flux effects and building up a gauge covariant calculus. Therefore, it was natural to search for a magnetic Beals-like criterion where the commutation with the plain momentum operators should be replaced by the commutation with their magnetic counterparts. Such a criterion was indeed proved by Iftimie-Mȃntoiu-Purice [START_REF] Iftimie | Commutator criteria for magnetic pseudo-differential operators[END_REF] and one of the technically heavy points in that work was the extension of Bony's lemma to the magnetic case. The main motivation of our paper is to propose an alternative proof of Beals' classical criterion based on the use of a normalized tight Gabor frame and to show how this approach can be quite naturally extended to the magnetic case and recover the criterion established in [START_REF] Iftimie | Commutator criteria for magnetic pseudo-differential operators[END_REF]. Note that no a-priori knowledge of the magnetic calculus is needed in order to understand the current manuscript.

1.2. The non-magnetic case. Let X j be the multiplication operator by x j , 1 ≤ j ≤ d, while

D j := -i∂ xj . We introduce W k := X k when 1 ≤ k ≤ d and W k := D k-d when d + 1 ≤ k ≤ 2d. The operators W k leave the Schwartz space S (R d ) invariant. Let us consider a bounded map T : S (R d ) → S ′ (R d ). Seen as maps from S (R d ) to S ′ (R d ), the following multiple commutators [W j1 , [W j2 , ..., [W jm , T ]]...] , m ≥ 1, j ℓ ∈ {1, 2, ..., 2d}, (1.1)
are also bounded. Then the classical Beals criterion [B] reads as follows:

Theorem 1.1. Let us assume that both T and all possible commutators as in (1.1) can be extended to bounded operators on L 2 (R d ). Then there exists a symbol a 0 (x, ξ) ∈ S 0 0,0 (R 2d ) such that for every Ψ, Φ ∈ S (R d ) we have:

Ψ, T Φ L 2 (R d ) = (2π) -d R d R 2d e iξ•(x-x ′ ) Ψ(x)a 0 ((x + x ′ )/2, ξ)Φ(x ′ )dxdx ′ dξ, i.e. T = Op w (a 0 ) is the Weyl quantization of a 0 .
In this paper, the scalar product of L 2 (R d ) is linear in the second variable and we use the standard notation for Hörmander type symbols (see Section 7.8 in [START_REF] Hörmander | The Analysis of Linear Partial Differential Operators[END_REF]).

1.3. The magnetic case. Let d ≥ 2. Consider a 2-form B(x) = 1≤j,k≤d B jk (x)dx j ∧dx k where B jk = -B kj are in BC ∞ (R d ) (i.e. the space of C ∞ bounded functions together with all their derivatives). We assume that the form is "magnetic" , i.e. that ∂ j B kℓ + ∂ k B ℓj + ∂ ℓ B jk ≡ 0 holds. This simply expresses that the 2-form is closed. Given any fixed y ∈ R d one can construct a 1-form A(•, y) such that B = dA(•, y) and

A j (x, y) = - d k=1 1 0 s (x k -y k ) B jk (y + s(x -y))ds.
(1.2)

We observe that A j (x, y) grows at most linearly in |x -y|, and this fact remains true for all its derivatives in x. Let Γ y,x denote the straight oriented segment joining y with x. Since A(•, 0) -A(•, y) is closed and exact, we have the identity

A j (x, 0) -A j (x, y) = ∂ xj ϕ(x, y), 1 ≤ j ≤ d , (1.3) ϕ(x, y) = Γy,x (A(•, 0) -A(•, y)) = Γy,x A(•, 0) . (1.4)
Here A(•, y) does not contribute to the integral because it is "orthogonal" to the integration path. The same orthogonality property allows us to identify ϕ(x, y) with the circulation of A(•, 0) on the oriented triangle generated by the origin, y and x. Stokes' theorem implies that ϕ(x, y) is equal with the magnetic flux through this triangle.

We denote the magnetic flux through the oriented triangle ∆(u, v, w) having vertices at u, v, w ∈ R d as:

f(u, v, w) := ∆(u,v,w) B , f(x, y, 0) = ϕ(x, y).
We note the identities

ϕ(x, y) = -ϕ(y, x) and ϕ(u, v) + ϕ(v, w) -ϕ(u, w) = f(u, v, w).
(1.5)

Now we can formulate the magnetic version of Beals' criterion as stated in Theorem 1.1 of [START_REF] Iftimie | Commutator criteria for magnetic pseudo-differential operators[END_REF]. Let Π j := D j -A j (•, 0) be the "magnetic" momenta which also leave S (R d ) invariant. We denote by

W k either X k if 1 ≤ k ≤ d, or Π k-d if d + 1 ≤ k ≤ 2d. Let us consider a bounded map T : S (R d ) → S ′ (R d
) and all possible commutators as in (1.1) but with the new W k 's. Here is the magnetic Beals criterion: Theorem 1.2. Let us assume that both T and all the "magnetic" commutators as in (1.1) can be extended to bounded operators on L 2 (R d ). Then there exists a symbol a 0 (x, ξ) ∈ S 0 0,0 (R 2d ) such that for every Ψ, Φ ∈ S (R d ) we have:

Ψ, T Φ L 2 (R d ) = (2π) -d R d R 2d e iϕ(x,x ′ ) e iξ•(x-x ′ ) Ψ(x)a 0 ((x + x ′ )/2, ξ)Φ(x ′ )dxdx ′ dξ .
The rest of this manuscript is as follows: in Section 2 we construct a family of tight frames which generalizes the classical Gabor case, in Section 3 we give the proof of Theorem 1.1, and in Section 4 we prove Theorem 1.2.

A magnetic normalized Gabor tight frame

Let g ∈ C ∞ 0 (R d ; R) such that supp g ⊂ (-1, 1) d and g γ (x) := g(x -γ), γ∈Z d g 2 γ (x) = 1 , ∀x ∈ R d . (2.1) Let ψ m (x) := e im•x , for any m ∈ Z d . Denote by τ z f (x) := f (x -z) the translation with z ∈ R d . Lemma 2.1. The functions G ϕ γ,m (x) := (2π) -d/2 e iϕ(x,γ) (τ γ gψ m )(x) : γ, m ∈ Z d , (2.2)
with ϕ defined in (1.4), satisfy the identity

f (x) = γ,m G ϕ γ,m (x) G ϕ γ,m , f L 2 (R d ) , ∀f ∈ S (R d ) , (2.3)
where the series is absolutely convergent. In particular, these functions generate a normalized tight frame in L 2 (R d ) (see [Gr, Ch]).

Proof. We have

G ϕ γ,m , f L 2 (R d ) = (2π) -d/2 {max d j=1 |xj-γj |≤1} g(x -γ)e -im•(x-γ) e -iϕ(x,γ) f (x) dx = (2π) -d/2 {max d j=1 |xj|≤1} e -im•x g(x) τ -γ e -iϕ(•,γ) f (x) dx =: F gτ -γ e -iϕ(•,γ) f (m) (2.4)
where gτ -γ e -iϕ(•,γ) f ∈ C ∞ 0 (-1, 1) d may be naturally considered, via its (2πZ) d -periodic extension to R d , as a function in C ∞ (T d ), and where the right hand side of (2.4) is nothing but the m'th Fourier coefficient of gτ -γ e -iϕ(•,γ) f .

Integrating by parts in (2.4), using (1.3) and the fact that f is a Schwartz function, then given any N ≥ 1 we may find a constant C f,N such that for every m and γ we have:

| G ϕ γ,m , f L 2 (R d ) | ≤ C f,N < γ > -N < m > -N . (2.5)
By the Fourier inversion formula and (2.4) we obtain:

gτ -γ e -iϕ(•,γ) f = (2π) -d/2 m∈Z d ψ m G ϕ γ,m , f L 2 (R d )
where the series converges absolutely due to (2.5) where we fix for example N ≥ 2d. Translating by γ ∈ Z d we obtain:

g γ (x)f (x) = (2π) -d/2 m∈Z d e iϕ(x,γ) (τ γ ψ m )(x) G ϕ γ,m , f L 2 (R d ) ,
which coupled with (2.1) leads to:

f = γ∈Z d g γ (g γ f ) = γ∈Z d m∈Z d G ϕ γ,m G ϕ γ,m , f L 2 (R d ) .
This proves (2.3).

Proof of Theorem 1.1

In order to simplify notation, in the non-magnetic case (ϕ ≡ 0) we denote the Gabor frame by G γ,m . By assumption, T can be extended to a bounded operator on L 2 (R d ) with norm T , thus:

T γ,γ ′ ;m,m ′ := G γ,m , T G γ ′ ,m ′ , T γ,γ ′ ;m,m ′ ≤ (2π) -d g 2 L 2 (R d ) T . (3.1)
For every N ∈ N, an application of the form

S (R d ) × S (R d ) ∋ (Φ, Ψ) → max(|γ|,|m|,|γ ′ |,|m ′ |)≤N T γ,γ ′ ;m,m ′ Ψ, G γ,m L 2 (R d ) G γ ′ ,m ′ , Φ L 2 (R d ) defines a tempered distribution on R d × R d .
Then the distribution kernel of the bounded operator

T : L 2 (R d ) → L 2 (R d ) is given by the series T = lim N ր∞ max(|γ|,|m|)≤N max(|γ ′ |,|m ′ |)≤N T γ,γ ′ ;m,m ′ G γ,m ⊗ G γ ′ ,m ′ , (3.2)
where each finite sum belongs to BC ∞ (R d ×R d ) and converges weakly in the space of the tempered distributions on R d × R d . If we restrict the distribution T to a compact in R d × R d there exists a finite number of non-zero contributions from the series in γ and γ ′ , but generally, the series in m and m ′ are not absolutely convergent. In order to remedy that difficulty, we make a regularization and define for ε > 0:

Tε (x, x ′ ) := (2π) -d γ,γ ′ m,m ′ T γ,γ ′ ;m,m ′ g γ (x)g γ ′ (x ′ )e -ε(|m| 2 +|m ′ | 2 ) e im•(x-γ) e -im ′ •(x ′ -γ ′ ) . (3.3)
Due to (3.1), it is not difficult to see that for a fixed ε > 0 the function Tε is jointly continuous. We will later see that it is much more regular. We start by proving an estimate which is stronger than (3.1).

Lemma 3.1. Given any pair N, M ≥ 1, there exists a constant C N,M such that

|T γ,γ ′ ;m,m ′ | ≤ C N,M < γ -γ ′ > -N < m -m ′ > -M .
(3.4)

Proof. The decay in γ -γ ′ is a consequence of the fact that all commutators of T with the position operators are bounded, see (1.1). For example:

(γ 1 -γ ′ 1 )T γ,γ ′ ;m,m ′ = (2π) -d (γ 1 -X 1 )g γ ψ m (• -γ), T g γ ′ ψ m ′ (• -γ ′ ) +(2π) -d g γ ψ m (• -γ), [X 1 , T ]g γ ′ ψ m ′ (• -γ ′ ) +(2π) -d g γ ψ m (• -γ), T (X 1 -γ ′ 1 )g γ ′ ψ m ′ (• -γ ′ ) .
The decay in m -m ′ is due to the boundedness of the commutators of T with the momentum operators (one has to integrate by parts). For example:

(m 1 -m ′ 1 )T γ,γ ′ ;m,m ′ = (2π) -d m 1 g γ ψ m (• -γ), T g γ ′ ψ m ′ (• -γ ′ ) -(2π) -d g γ ψ m (• -γ), T m ′ 1 g γ ′ ψ m ′ (• -γ ′ ) = (2π) -d g γ (D x1 ψ m )(• -γ), T g γ ′ ψ m ′ (• -γ ′ ) -(2π) -d g γ ψ m (• -γ), T g γ ′ (D x1 ψ m ′ )(• -γ ′ ) = (2π) -d (D x1 g) γ ψ m (• -γ), T g γ ′ ψ m ′ (• -γ ′ ) -(2π) -d g γ ψ m (• -γ), T (D x1 g) γ ′ ψ m ′ (• -γ ′ ) +(2π) -d g γ ψ m (• -γ), [D x1 , T ]g γ ′ ψ m ′ (• -γ ′ ) .
The next Lemma will show that the approximating kernel Tε has a fast off-diagonal decay.

Lemma 3.2. Let ε > 0. Then for every fixed t ∈ R d , the function

R d ∋ s → Tε (t + s/2, t -s/2) ∈ C belongs to S (R d ).
Proof. Given x and x ′ , the only γ's and γ ′ 's contributing to (3.3) must obey the conditions |x -γ| ≤ √ d and |x ′ -γ ′ | ≤ √ d. Given t = (x + x ′ )/2, the only γ's and γ ′ 's contributing to (3.3) must also obey |(γ + γ ′ )/2 -t| ≤ √ d. Thus:

Tε (t + s/2, t -s/2) = (2π) -d m,m ′ T γ,γ ′ ;m,m ′ g(t + s/2 -γ)g(t -s/2 -γ ′ ) × e -ε(|m| 2 +|m ′ | 2 ) e im•(t+s/2-γ) e -im ′ •(t-s/2-γ ′ ) . (3.5)
The series in m and m ′ are absolutely convergent due to the regularizing Gaussians, while the sum in the direction of γ -γ ′ is convergent due to (3.4). We can also differentiate as many times as we want with respect to s in (3.5), and the series remain absolutely convergent. Given s = x -x ′ , the only γ's and γ ′ 's contributing to (3.5) must obey |(γ -γ ′ ) -s| ≤ 2 √ d, thus when we estimate s α D β s Tε (t + s/2, t -s/2) we may replace |s| with |γ -γ ′ | + 2 √ d and obtain something bounded (actually independent of t). More precisely, given multi-indices α and β, there exists a constant C(α, β, ε) such that, for any t ∈ R,

sup s∈R d |s α D β s Tε (t + s/2, t -s/2)| ≤ C(α, β, ε).
Let us consider the symbol associated by the Weyl quantization with the distribution kernel Tǫ :

a ε (t, ξ) := R d e -iξ•s Tε (t + s/2, t -s/2) ds. (3.6)
Due to Lemma 3.2, for fixed t ∈ R d and ε > 0, the function ξ → a ε (t, ξ) is a Schwartz function.

Lemma 3.3. The function a ε (t, ξ) converges uniformly on compact sets of R 2d to a smooth function a 0 (t, ξ). More precisely:

sup t∈R d sup ξ∈R d |D α t D β ξ a ε (t, ξ)| ≤ C(α, β), ∀α, β ∈ N d , ε ≥ 0 ,
and given any compact K ⊂ R 2d we have

lim εց0 sup (t,ξ)∈K |D α t D β ξ {a ε (t, ξ) -a 0 (t, ξ)}| = 0 , ∀α, β ∈ N d .
In particular, a 0 ∈ S 0 0,0 (R 2d ). Remark. Before proving the lemma, let us show how we can conclude the proof of Theorem 1.1.

If Ψ, Φ ∈ C ∞ 0 (R d ) we have: Ψ, T Φ = lim εց0 R 2d Ψ(x) Tε (x, x ′ )Φ(x ′ )dxdx ′ = (2π) -d lim εց0 R d R 2d e iξ•(x-x ′ ) Ψ(x)a ε ((x + x ′ )/2, ξ)Φ(x ′ )dxdx ′ dξ = (2π) -d R d R 2d e iξ•(x-x ′ ) Ψ(x)a 0 ((x + x ′ )/2, ξ)Φ(x ′ )dxdx ′ dξ ,
where the last equality follows from the Lebesgue dominated convergence theorem applied to the ξ integral, for which we use Lemma 3.3. Then the identity can be extended to S (R d ) because a 0 ∈ S 0 0,0 (R 2d ). Proof of Lemma 3.3. Let us introduce the notation

κ := (γ + γ ′ )/2 ∈ 2 -1 Z d , κ ′ := γ -γ ′ ∈ Z d , n := (m + m ′ )/2 ∈ 2 -1 Z d , n ′ := m -m ′ ∈ Z d .
Using (3.3) and (3.5) we obtain

a ε (t, ξ) =(2π) -d |κ-t|≤ √ d κ ′ n,n ′ R d e -i(ξ-n)•s g(t -κ + (s -κ ′ )/2)g(t -κ -(s -κ ′ )/2)ds × e in ′ •t e -i(n•κ ′ +n ′ •κ) e -ε(2|n| 2 +|n ′ | 2 /2) T κ,κ ′ ;n,n ′ ,
where in order to simplify notation we write T κ,κ ′ ;n,n ′ instead of T γ,γ ′ ;m,m ′ . The estimate (3.4) insures a strong localization in both the κ ′ and n ′ series. The only series which apparently still needs ε > 0 in order to converge, is the series in n. Define

F (t -κ, ξ -n, κ ′ ) := (2π) -d R d e -i(ξ-n)•s g(t -κ + (s -κ ′ )/2)g(t -κ -(s -κ ′ )/2)ds (3.7) so that a ε (t, ξ) = |κ-t|≤ √ d κ ′ n,n ′ e in ′ •t F (t -κ, ξ -n, κ ′ )e -i(n•κ ′ +n ′ κ) e -ε(2|n| 2 +|n ′ | 2 /2) T κ,κ ′ ;n,n ′ .
It is important to remember that in the integral of (3.7), the integrand is different from zero only if s is of the order of κ ′ , i.e. |s -

κ ′ | ≤ 2 √ d.
By differentiating F with respect to ξ we produce a polynomial growth in s which can be traded off with a growth in |κ ′ |. Also, by standard partial integration with respect to s we can generate a strong localization in |ξ -n|. In conclusion, one can prove the following statement: given any two multi-indices α, β ∈ N d , there exists a constant C(α, β) < ∞ such that

|D α t D β ξ F (t -κ, ξ -n, κ ′ )| ≤ C(α, β) < ξ -n > -2d < κ ′ > |β| . (3.8)
The growth in κ ′ is controlled by the decay of the matrix element T κ,κ ′ ;n,n ′ , while due to (3.8) the series in n converges absolutely without any help from the ε-dependent Gaussian. Now we can take ε to zero and define

a 0 (t, ξ) := |κ-t|≤ √ d κ ′ n,n ′ e in ′ •t F (t -κ, ξ -n, κ ′ )e -i(n•κ ′ +n ′ κ) T κ,κ ′ ;n,n ′ .
The limit can be taken uniformly on compacts in R 2d , and remains valid for all possible derivatives with respect to both ξ and t. This concludes the proof of the lemma.

Proof of Theorem 1.2

This time we let ϕ = 0 in (2.2) and have

G ϕ γ,m (x) = g(x -γ)e iϕ(x,γ) (2π) -d/2 ψ m (x -γ), γ, m ∈ Z d . Then (3.2) reads as: T (x, x ′ ) = γ,γ ′ m,m ′ G ϕ γ,m (x)G ϕ γ ′ ,m ′ (x ′ )T ϕ γ,γ ′ ;m,m ′ , T ϕ γ,γ ′ ;m,m ′ := G ϕ γ,m , T G ϕ γ ′ ,m ′ . (4.1)
Let us prove that T ϕ γ,γ ′ ;m,m ′ obeys exactly the same type of localization as in (3.4). The localization in γ -γ ′ follows just like before from the boundedness of commutators with the position operators, while the localization in m -m ′ is obtained by integration by parts and the use of the gauge covariance (1.3). For example, we have

(m 1 -m ′ 1 )T ϕ γ,γ ′ ;m,m ′ = (2π) -d m 1 e iϕ(•,γ) g γ ψ m (• -γ), T g γ ′ e iϕ(•,γ ′ ) ψ m ′ (• -γ ′ ) -(2π) -d g γ e iϕ(•,γ) ψ m (• -γ), T m ′ 1 g γ ′ e iϕ(•,γ ′ ) ψ m ′ (• -γ ′ ) = (2π) -d g γ e iϕ(•,γ) (D x1 ψ m )(• -γ), T g γ ′ e iϕ(•,γ ′ ) ψ m ′ (• -γ ′ ) -(2π) -d e iϕ(•,γ) g γ ψ m (• -γ), T e iϕ(•,γ ′ ) g γ ′ (D x1 ψ m ′ )(• -γ ′ ) = (2π) -d e iϕ(•,γ) (D x1 g) γ ψ m (• -γ), T e iϕ(•,γ ′ ) g γ ′ ψ m ′ (• -γ ′ ) -(2π) -d e iϕ(•,γ) g γ ψ m (• -γ), T e iϕ(•,γ ′ ) (D x1 g) γ ′ ψ m ′ (• -γ ′ ) +(2π) -d e iϕ(•,γ) g γ ψ m (• -γ), (-i∂ x1 -∂ x1 ϕ(•, γ)) T e iϕ(•,γ ′ ) g γ ′ ψ m ′ (• -γ ′ ) -(2π) -d e iϕ(•,γ) g γ ψ m (• -γ), T (-i∂ x1 -∂ x1 ϕ(•, γ ′ ))e iϕ(•,γ ′ ) g γ ′ ψ m ′ (• -γ ′ ) = (2π) -d e iϕ(•,γ) (D x1 g) γ ψ m (• -γ), T e iϕ(•,γ ′ ) g γ ′ ψ m ′ (• -γ ′ ) -(2π) -d e iϕ(•,γ) g γ ψ m (• -γ), T e iϕ(•,γ ′ ) (D x1 g) γ ′ ψ m ′ (• -γ ′ ) +(2π) -d e iϕ(•,γ) g γ ψ m (• -γ), [D x1 -A 1 (•, 0), T ] e iϕ(•,γ ′ ) g γ ′ ψ m ′ (• -γ ′ ) -(2π) -d e iϕ(•,γ) A 1 (•, γ))g γ ψ m (• -γ), T e iϕ(•,γ ′ ) g γ ′ ψ m ′ (• -γ ′ ) +(2π) -d e iϕ(•,γ) g γ ψ m (• -γ), T e iϕ(•,γ ′ ) A 1 (x, γ ′ )g γ ′ ψ m ′ (• -γ ′ ) .
Here the last formula was obtained by integration by parts. We also used (1.3) for writing ∂ x1 ϕ(x, γ) = A 1 (x, 0) -A 1 (x, γ) and the fact that on the support of g γ the function A(•, γ) is bounded uniformly in γ.

We now regularize the distributional kernel in (4.1) and introduce:

T ϕ ε t + s 2 , t - s 2 := (2π) -d γ,γ ′ m,m ′ g γ (t + s/2)e iϕ(t+s/2,γ) e -iϕ(t-s/2,γ ′ ) g γ ′ (t -s/2) × e i(m-m ′ )•t e i(m+m ′ )•s/2 e -im•γ e im ′ •γ ′ e -ε(|m| 2 +|m ′ | 2 ) T ϕ γ ,γ ′ ;m,m ′ . (4.2)
This function has a rapid decay in s and is smooth in both t and s when ε > 0. Using twice the second identity of (1.5) we obtain:

ϕ(t + s/2, t -s/2) = ϕ(t + s/2, γ) + ϕ(γ, t -s/2) -f(t + s/2, γ, t -s/2) = ϕ(t + s/2, γ) + ϕ(γ, γ ′ ) + ϕ(γ ′ , t -s/2) -f(γ, γ ′ , t -s/2) -f(t + s/2, γ, t -s/2). (4.3)
Let us introduce the quantity (we use (4.3) in the second equality):

a ε (t, ξ) := R d e -iϕ(t+s/2,t-s/2) e -iξ•s T ϕ ε (t + s/2, t -s/2)ds (4.4) =(2π) -d |(γ+γ ′ )/2-t|≤ √ d e -iϕ(γ,γ ′ ) × m,m ′ R d
e if(γ,γ ′ ,t-s/2) e if(t+s/2,γ,t-s/2) e -i[ξ-(m+m ′ )/2]•s g γ (t + s/2)g γ ′ (t -s/2)ds × e i(m-m ′ )•t e -im•γ e im ′ •γ ′ e -ε(|m| 2 +|m ′ | 2 ) T ϕ γ,γ ′ ;m,m ′ .

As in the non-magnetic case, the only series which apparently poses convergence problems is the one with respect to the "direction" (m + m ′ )/2. It turns out (as in the non-magnetic case) that the integral: R d e if(γ,γ ′ ,t-s/2) e if(t+s/2,γ,t-s/2) e -i[ξ-(m+m ′ )/2]•s g γ (t + s/2)g γ ′ (t -s/2)ds is the one which insures decay in that direction. In order to prove it, let us notice that the fluxes f(t + s/2, γ, t -s/2) and f(γ, γ ′ , t -s/2) grow like the area of the corresponding triangle, hence only like |γ -γ ′ | because both t + s/2 -γ and t -s/2 -γ ′ have a length of order one on the joint support of g γ and g γ ′ ; the same is true for their derivatives with respect to both t and s.

Integrating by parts with respect to s we can generate a decay of the type < ξ -(m + m ′ )/2 > -2d at the price of a polynomial growth in |γ -γ ′ |, a growth which is taken care of by the decay of the matrix element T ϕ γ,γ ′ ;m,m ′ . Thus the same strategy which was used in the previous section concerning the limit ε ց 0 can be repeated. We conclude that a ε (t, ξ) ∈ S 0 0,0 (R d ) uniformly in ε ≥ 0 and thus the symbol we are looking for is: e if(γ,γ ′ ,t-s/2) e if(t+s/2,γ,t-s/2) e -i[ξ-(m+m ′ )/2]•s g γ (t + s/2)g γ ′ (t -s/2)ds × e i(m-m ′ )•t e -im•γ e im ′ •γ ′ T ϕ γ,γ ′ ;m,m ′ .

|(γ+γ ′ )/2-t|≤ √ d

E-mail address: radu.purice@imar.ro


Horia D Cornean

Bernard Helffer

Radu Purice

Peierls' substitution for low lying spectral energy windows

We consider a 2d magnetic Schrödinger operator perturbed by a weak magnetic field which slowly varies around a positive mean. In a previous paper we proved the appearance of a 'Landau type' structure of spectral islands at the bottom of the spectrum, under the hypothesis that the lowest Bloch eigenvalue of the unperturbed operator remained simple on the whole Brillouin zone, even though its range may overlap with the range of the second eigenvalue. We also assumed that the first Bloch spectral projection was smooth and had a zero Chern number.

In this paper we extend our previous results to the only two remaining possibilities: either the first Bloch eigenvalue remains isolated while its corresponding spectral projection has a non-zero Chern number, or the first two Bloch eigenvalues cross each other.

Introduction

1.1 Some history of the problem and the structure of the paper Since the pioneering work of Peierls [START_REF] Peierls | Quantum Theory of Solids[END_REF] and Luttinger [START_REF] Luttinger | The Effect of a Magnetic Field on Electrons in a Periodic Potential[END_REF], physicists consider Peierls' substitution to be the right way of taking into account external magnetic fields in periodic solid state systems. Very roughly, the receipt is as follows: consider a non-magnetic periodic Schrödinger Hamiltonian having an energy band λ0(θ) which is periodic with respect to the Brillouin zone; then in the presence of an external magnetic vector potential A, the right physical behaviour of the magnetically perturbed system should be given by λ0(θ -A(x)), whatever that means.

From a rigorous mathematical point of view the situation is more complicated, and here we will only discuss the case in which the external magnetic perturbation is weak and slowly variable. The smallness of the magnetic perturbation is generically represented by a parameter ǫ ≥ 0, where ǫ = 0 corresponds to the unperturbed objects. The first natural question one asks in this context is whether one can construct an effective Hamiltonian which is close in some sense to the original one, either dynamically or spectrally. The second question is to analyse the effective model and draw conclusions on its spectral and dynamical properties.

To the best of our knowledge, all the effective models constructed until now suppose that the unperturbed Bloch spectrum under investigation remains isolated from the rest; this does not require a gap in the full spectrum, only a non-empty gap at each fixed quasi-momentum θ of the Brillouin zone (the non-crossing, range-overlapping case). On top of that, with the notable exception of a recent work by Freund and Teufel [START_REF] Freund | Peierls substitution for magnetic Bloch bands[END_REF], all existing papers require that the Bloch projection associated to the Bloch spectrum under investigation is topologically trivial. The triviality condition leads to the existence of a basis consisting of spatially localized Wannier functions [START_REF] Cornean | On the construction of composite Wannier functions[END_REF][START_REF] Cornean | On the construction of Wannier functions in topological insulators: the 3D case[END_REF][START_REF] Fiorenza | Construction of real-valued localized composite Wannier functions for insulators[END_REF][START_REF] Helffer | On diamagnetism and the de Haas-van Alphen effect[END_REF][START_REF] Nenciu | Bloch electrons in a magnetic field, rigorous justification of the Peierls-Onsager effective Hamiltonian[END_REF][START_REF] Nenciu | Dynamics of Bloch electrons in electric and magnetic fields, rigorous justification of the effective Hamiltonians[END_REF] which plays an essential role in the reduction procedure. We note that although [START_REF] Freund | Peierls substitution for magnetic Bloch bands[END_REF] can treat non-trivial projections, only bounded perturbations are allowed.

When the external magnetic field is constant and the range of λ0 does not overlap with the other ones (the gapped case), one can construct a smooth and periodic symbol λǫ(ξ) such that the Weyl quantization of λǫ(ξ -Aǫ(x)) is isospectral with the "true" magnetic band Hamiltonian, provided ǫ is small enough. This is essentially due to Nenciu and Helffer-Sjöstrand [START_REF] Helffer | On diamagnetism and the de Haas-van Alphen effect[END_REF][START_REF] Nenciu | Dynamics of Bloch electrons in electric and magnetic fields, rigorous justification of the effective Hamiltonians[END_REF][START_REF] Sjöstrand | Microlocal analysis for the periodic magnetic Schrödinger equation and related questions[END_REF]. Concerning the spectral analysis of these effective magnetic pseudodifferential operators there exists a large amount of literature which classifies the spectral gaps induced by the magnetic field; see the works by Bellissard [START_REF] Bellissard | C * algebras in solid state physics. 2D electrons in a uniform magnetic field[END_REF][START_REF] Bellissard | Lipschitz continuity of gap boundaries for Hofstadter-like spectra[END_REF], Bellissard-Rammal [START_REF] Rammal | An algebraic semi-classical approach to Bloch electrons in a magnetic field[END_REF], Helffer-Sjöstrand [START_REF] Helffer | Equation de Schrödinger avec champ magnétique et équation de Harper[END_REF][START_REF] Helffer | On diamagnetism and the de Haas-van Alphen effect[END_REF] and references therein.

When the magnetic perturbation comes from a non-constant but slowly variable magnetic field, still under the assumption that a localized Wannier basis exists, one can again construct an effective Hamiltonian up to any order in ǫ (see [START_REF] Cornean | Peierls substitution and magnetic pseudodifferential calculus[END_REF][START_REF] Nittis | Applications of Magnetic ΨDO Techniques to Space-adiabatic Perturbation Theory[END_REF][START_REF] Panati | Effective dynamics for Bloch electrons: Peierls substitution and beyond[END_REF]) but which typically lives in an ǫ-dependent subspace. Thus the spectral analysis of the effective operator seems to be as complicated as the original one.

In our previous paper [START_REF] Cornean | Low lying spectral gaps induced by slowly varying magnetic fields[END_REF] we proved (for the first time in the non-constant case) the appearance of a 'Landau type' structure at the bottom of the spectrum consisting of spectral islands of width of order ǫ, separated by gaps of roughly the same order of magnitude; in [START_REF] Cornean | Low lying spectral gaps induced by slowly varying magnetic fields[END_REF] we still worked under the hypothesis that the lowest Bloch eigenvalue of the unperturbed operator remained simple on the whole Brillouin zone and a localized Wannier basis can be constructed.

The current manuscript extends our previous results to the only two remaining possibilities: either the first Bloch eigenvalue remains isolated while its corresponding spectral projection has a non-zero Chern number, or the first two Bloch eigenvalues cross each other.

Our main result is Theorem 2.2. It roughly states that in a certain narrow energy window near the bottom of the spectrum of the "true" perturbed Hamiltonian, the perturbed spectrum is well approximated by the spectrum of a pseudo-differential operator corresponding to a magnetically quantized periodic symbol which only depends on ξ. This result is new also for the constant magnetic case since we no longer need global non-crossing and/or triviality conditions on the lowest Bloch eigenvalue.

Another important consequence is that we can prove that gaps of order ǫ open near the bottom of the perturbed spectrum, due to the fact that the spectrum of our effective Hamiltonian has the same property. For more details see Corollaries 2.3 and 2. [START_REF] Cornean | On the construction of composite Wannier functions[END_REF].

The paper is organized as follows:

• In the rest of this section we introduce some of the objects we want to study and we briefly review the Bloch-Floquet theory.

• In Section 2 we state our main results (Theorem 2.2 and Corollaries 2.3 and 2.4). In subsection 2.3 we outline the main ideas behind the proofs.

• In Section 3 we construct a Wannier-like localized basis which spans a certain energy window of the unperturbed Hamiltonian; in this window, the lowest Bloch eigenvalue has to be non-degenerate.

• In Section 4 we describe a general reduction argument based on the Feshbach-Schur map which permits the construction of an effective Hamiltonian in a narrow window near the bottom of the spectrum.

• In Section 5 we review the so-called magnetic pseudo-differential calculus where the symbol composition rule is a twisted Moyal product, while in Section 6 we construct the magnetic counterpart of the Wannier-like projection from Section 3.

• In Section 7 we show that the magnetic Hamiltonian satisfies the conditions of the abstract reduction procedure which we developed in Section 4.

• In Section 8 we analyse the structure of the effective magnetic matrix constructed in Section 7. In particular, we show in Proposition 8.12 that the spectrum of this magnetic matrix is close to the spectrum of a magnetic pseudo-differential operator whose symbol λ -ǫ (ξ) is smooth and periodic in ξ while independent of x. In particular, this completes the proof of Theorem 2.2(i). In Proposition 8.13 we show that the function λ -ǫ (•) is close to the unperturbed Bloch eigenvalue λ0 near its minimum; this ends the proof of Theorem 2.2. Finally, we prove Corollaries 2.3 and 2.4.

Notation and conventions

For any real finite dimensional Euclidean space V we shall denote by:

1. BC ∞ (V) (respectively C ∞ pol (V)) the family of smooth complex functions on V that are bounded (respectively polynomially bounded) together with all their derivatives, 2. τv the translation by -v ∈ V on any class of distributions on V, 3. the "Japanese bracket" is denoted by < v >:= 1 + |v| 2 for any v ∈ V with |v| ∈ R+ its Euclidean norm, 4. S (V) the space of Schwartz functions on V and with S ′ (V) the space of tempered distributions on V.

For any Banach space B we denote by L(B) the Banach space of continuous linear operators in B. For any Hilbert space H and any pair of unit vectors (u, v) in it we use the physics notation |u v| for the the projector

H ∋ w → |u v|w := v, w Hu . (1.1)
Our scalar products are anti-linear in the first factor. As we are working in a two-dimensional framework, we use the following notation for the vector product of two vectors u and v in R 2 :

u ∧ v := u1v2 -u2v1 . (1.2)
and the -π/2 rotation of v: v ⊥ := v2, -v1 .

(1.3)

The Hamiltonian

Let Γ ⊂ R 2 ≡ X be a regular periodic lattice. We consider a smooth Γ-periodic potential VΓ : X → R and a smooth Γ-periodic magnetic field B Γ : X → R with vanishing flux through the unit cell of the lattice. Let us denote by A Γ : X → R 2 a smooth Γ-periodic vector potential generating the magnetic field B Γ . We consider the Hilbert space H := L 2 (X ) and define the periodic Schrödinger operator

H 0 := (-i∂x 1 -A Γ 1 ) 2 + (-i∂x 2 -A Γ 2 ) 2 + VΓ , (1.4)
as the unique self-adjoint extension in H of the above differential operator initially defined on S (V) (see [START_REF] Reed | Methods of Modern Mathematical Physics[END_REF]). As a perturbation of the periodic case, we consider a family of magnetic fields indexed by (ǫ, κ)

∈ [0, 1] × [0, 1]: B ǫ,κ (x) := ǫB0 + κǫB(ǫx) . (1.5)
Here B0 > 0 is constant, while B : X → R is of class BC ∞ (X ). We choose the vector potential A 0 of the constant magnetic field B0 in the form (transverse gauge):

A 0 (x) = (B0/2) -x2, x1 (1.6)
and some vector potential A : X → R 2 , that we can choose to be of class C ∞ pol (X ) such that:

B0 = ∂1A 0 2 -∂2A 0 1 , B = ∂1A2 -∂2A1 . (1.7)
Similarly, for A ǫ,κ (x) := ǫA 0 (x) + κA(ǫx) ,

we have B ǫ,κ = ∂1A ǫ,κ 2 -∂2A ǫ,κ 1 . (1.8)
We shall also use the notation:

B ǫ := ǫB0 , A ǫ := ǫA0, Bǫ(x) := B(ǫx) , Aǫ(x) := A(ǫx) . (1.9)
Definition 1.1. We consider the magnetic Schrödinger operators defined as the self-adjoint extension in H of the following differential operators:

H ǫ,κ := (-i∂x 1 -A Γ 1 -A ǫ,κ 1 ) 2 + (-i∂x 2 -A Γ 2 -A ǫ,κ 2 ) 2 + VΓ . (1.10)
H ǫ := H ǫ,0 = (-i∂x 1 -A Γ 1 -A ǫ 1 ) 2 + (-i∂x 2 -A Γ 2 -A ǫ 2 ) 2 + VΓ . (1.11)
When κ = ǫ = 0, we recover H 0 .

The Bloch-Floquet Theory

We consider the quotient space IR 2 /Γ which is canonically isomorphic to the two dimensional torus T 2 ≡ T and denote by IR 2 ∋ x → x ∈ T the canonical projection onto the quotient. We choose two generators {e1, e2} for the lattice Γ and the associate elementary cell :

EΓ = y = 2 j=1 tjej ∈ IR 2 | -(1/2) ≤ tj < (1/2) , ∀j ∈ {1, 2} , (1.12)
so that we have a bijection

X ∋ x → γ(x), y(x) ∈ Γ × EΓ . The dual lattice of Γ is then defined as Γ * := {γ * ∈ X * | < γ * , γ >∈ 2π Z , ∀γ ∈ Γ} .
Considering the dual basis {e * 1 , e * 2 } ⊂ X * of {e1, e2}, which is defined by < e * j , e k >= 2π δ jk , we have Γ * := ⊕j=1,2 Ze * j . We similarly define T * := X * /Γ * and EΓ * ≡ E * . We recall that the Floquet unitary map UΓ is defined by:

S (X ) ∋ φ → UΓφ ∈ C ∞ X × T * with UΓφ (x, θ) := γ∈Γ e i<θ,γ> φ(x -γ) , (1.13)
so that we have the following property

UΓφ (x + γ, θ) = e i<θ,γ> UΓφ (x, θ) , ∀γ ∈ Γ , ∀(x, θ) ∈ X × T * . (1.14)
For any θ ∈ T * we introduce

F θ := v ∈ L 2 loc (X ) | v(x + γ) = e i<θ,γ> v(x) , ∀γ ∈ Γ (1.15)
and notice that it is a Hilbert space for the scalar product

v , w F θ := E v(x) w(x) dx . (1.16)
For each θ ∈ T * , we define V θ : F θ → L 2 (T) as the multiplication with the function σ θ (x) := e -i<θ,x> . We can then form the associated direct integral of Hilbert spaces

F := ⊕ T * F θ dθ , (1.17)
and the following unitary map from

⊕ T * F θ dθ onto L 2 (E * ) ⊗ L 2 (
T) defined by:

V T * := ⊕ T * V θ dθ . (1.18)
Then UΓ defines a unitary operator L 2 X → F . Note that a bounded operator X ∈ L(H) commutes with all the translations with elements from Γ if and only if there exists a measurable family T * ∋ θ → X(θ) ∈ L(F θ ) such that

UΓXU -1 Γ = ⊕ T * X(θ) dθ .
The following statements are well-known (see for example [START_REF] Kuchment | An overview of periodic elliptic operators[END_REF][START_REF] Reed | Methods of Modern Mathematical Physics[END_REF]):

1. We have the direct integral decomposition:

Ĥ0 := UΓH 0 U -1 Γ = ⊕ T * Ĥ0 (θ) dθ , (1.19) with Ĥ0 (θ)
:= (-i∂x 1 -A Γ 1 ) 2 + (-i∂x 2 -A Γ 2 )
2 + VΓ , whose domain in F θ is the following local Sobolev space:

H 2 θ := v ∈ H 2 loc (X ) | v(x + γ) = e i<θ,γ> v(x) , ∀γ ∈ Γ ⊂ F θ . (1.20)
2. The family Ĥ0 (θ) (θ ∈ T * ) is unitarily equivalent with an analytic family of type A in the sense of Kato [START_REF] Kato | Perturbation Theory for Linear Operators[END_REF]. The unitary operator is given by the multiplication with V θ which maps F θ to L 2 (T). The rotated operator is essentially self-adjoint on the set of smooth and periodic functions in L 2 (E) and acts on them as:

H 0 (θ) := (-i∇ -A Γ + θ) 2 + VΓ, θ ∈ E * .
3. There exists a family of continuous functions T * ∋ θ → λj(θ) ∈ R indexed by j ∈ N, called the Bloch eigenvalues, such that λj(θ) ≤ λj+1(θ) for every j ∈ N and θ ∈ T * , and

σ Ĥ0 (θ) = j∈N {λj (θ)} = σ H 0 (θ) . (1.21) 4.
For each fixed θ we can define the Riesz spectral projections using complex Cauchy integrals of the resolvent (z -Ĥ0 (θ)) -1 around each eigenvalue λj (θ). If the lowest eigenvalue λ0(θ) is always non-degenerate, it is also smooth in θ and its corresponding Riesz projection π0(θ) is globally smooth on the dual torus and has rank one. If λ0 can cross with the other higher eigenvalues, then by convention, at these crossing points π0(θ) is defined as the higher-rank Riesz projection which encircles the still lowest, but now degenerate eigenvalue. In this way, the projection π0(θ) is well defined on the whole dual torus but it is no longer continuous.

2 The main results

(Non)existence of localized Wannier functions

In our previous work [START_REF] Cornean | Low lying spectral gaps induced by slowly varying magnetic fields[END_REF] we imposed the condition that λ0 had to remain non-degenerate on the whole dual torus, while the unperturbed periodic operator H 0 had to be time-reversal invariant (i.e. commuting with the complex conjugation operator). Under these conditions, the orthogonal projection family π0(θ) is smooth and time-reversal symmetric in the sense that C π0(θ) = π0(-θ)

C
where C is the anti-unitary operator given by the complex conjugation. In particular, this implies that the Chern number of this family equals zero and that one can construct [START_REF] Helffer | On diamagnetism and the de Haas-van Alphen effect[END_REF][START_REF] Nenciu | Dynamics of Bloch electrons in electric and magnetic fields, rigorous justification of the effective Hamiltonians[END_REF][START_REF] Fiorenza | Construction of real-valued localized composite Wannier functions for insulators[END_REF][START_REF] Cornean | On the construction of Wannier functions in topological insulators: the 3D case[END_REF] a global smooth section φ0 :

T * → F such that π0(θ) = | φ0(θ) φ0(θ)| . (2.1)
Thus we have the orthogonal projection in H

π0 := U -1 Γ π0 UΓ , (2.2)
which commutes with H 0 but is not necessarily a spectral projection of it (unless the range of λ0 is isolated from the rest of the spectrum). When a smooth global section φ0 exists, the principal Wannier function φ0 is defined by:

φ0(x) := U -1 Γ φ0 (x) = |E * | -1 2 T * φ0(x, θ) dθ (2.3)
and has rapid decay. The family φγ := τγ φ0 generated from φ0 by translations over the lattice Γ is an orthonormal basis for the subspace π0H, consisting of (exponentially) localized functions.

As we have already mentioned, the main spectral result in [START_REF] Cornean | Low lying spectral gaps induced by slowly varying magnetic fields[END_REF] was obtained under the hypothesis that a global smooth section as in (2.1) exists, or equivalently, that there exists a localized Wannier basis for π0. Our main concern in the present paper is to show that roughly the same spectral results can be proved demanding neither the global simplicity of λ0, nor the symmetry with respect to ξ → -ξ of the symbol of H 0 . In other words, we no longer demand the existence of a localized Wannier basis for the range of π0 .

Main statements

Without any loss of generality we may assume that inf(σ(H 0 )) = 0 . Let us state our only additional hypothesis concerning the bottom of the spectrum of the unperturbed operator H 0 . Hypothesis 2.1. The map λ0 : T * → R has a unique non-degenerate global minimum value realized for θ0 ∈ T * and λ0(θ0) = 0 .

When A Γ = 0 the above hypothesis is always satisfied [START_REF] Kirsch | Comparison theorems for the gap of Schrödinger operators[END_REF][START_REF] Cornean | Low lying spectral gaps induced by slowly varying magnetic fields[END_REF] with θ0 = 0 and moreover, λ0(θ) = λ0(-θ). Also, under this hypothesis and if b is small enough, the set

Σ b := λ -1 0 ([0, b)) ⊂ T *
becomes a simply connected neighbourhood of the minimum.

Before we can state the main result of our paper we need some more notation:

• Given a vector potential A with components of class C ∞ pol (X ) and a symbol F (x, ξ) we can define the following Weyl-magnetic pseudodifferential operator:

Op A (F )u (x) := (2π) -2 X X * e i ξ,x-y e -i [x,y] A F x + y 2 , ξ u(y) dξ dy . (2.4)
The magnetic pseudodifferential calculus was developed in [START_REF] Mȃntoiu | The magnetic Weyl calculus[END_REF]; for the convenience of the reader we summarized its main features in Section 5.

• dH (M1, M2) denotes the Hausdorff distance between the subsets M1 and M2 in R.

Theorem 2.2. Let H ǫ,κ , with (ǫ, κ) ∈ [0, 1] × [0, 1], be the family introduced in (1.10) for a Hamiltonian H 0 satisfying Hypothesis 2.1 and a magnetic field of the form (1.5). Then there exists b > 0 and a smooth real function λ -ǫ : T * → R such that:

(i) For any N ∈ N * there exist some constant C0 > 0 and some (ǫ0, κ0) ∈ (0, b/N ) × (0, 1), such that, for any (ǫ, κ) ∈ (0, ǫ0] × (0, κ0],

dH σ H ǫ,κ [0, N ǫ] , σ Op ǫ,κ ( λ -ǫ ) [0, N ǫ] ≤ C0 κǫ + ǫ 2 , (2.5)
where λ -ǫ : X * → R is the periodic extension of λ -ǫ : T * → R considered as a function on Ξ constant along the directions in X × {0}, and Op ǫ,κ ( λ -ǫ ) ≡ Op A ǫ,κ ( λ -ǫ ) is the magnetic quantization of λ -ǫ as in (2.4).

(ii) For every m ∈ N there exists Cm > 0 such that for all θ ∈ Σ b and |α| ≤ m we have

|∂ α λ -ǫ (θ) -∂ α λ0(θ)| ≤ Cm ǫ .
Combining the above theorem with the results from [START_REF] Cornean | Low lying spectral gaps induced by slowly varying magnetic fields[END_REF] regarding the existence of spectral gaps for magnetic pseudo-differential operators like Op ǫ,κ ( λ -ǫ ), we will prove in Subsection 8.5 the following two corollaries: Corollary 2.3. Assume B Γ = 0 . Then for any integer N ≥ 1, there exist some constants C0, C1, C2 > 0 , and (ǫ0, κ0) ∈ (0, b/N ) × (0, 1) , such that, for any (ǫ, κ) ∈ (0, ǫ0] × (0, κ0], there exist a0 < b0 < a1 < • • • < aN < bN , with a0 = inf{σ(H ǫ,κ )} so that:

σ(H ǫ,κ ) ∩ [a0, bN ] ⊂ N k=0 [a k , b k ] , dim RanE [a k ,b k ] (H ǫ,κ ) = +∞ , b k -a k ≤ C0 ǫ κ + C1 ǫ 1/3 , 0 ≤ k ≤ N , and a k+1 -b k ≥ 1 C2 ǫ , 0 ≤ k ≤ N -1 . (2.6)
Corollary 2.4. Assume B Γ = 0 . Under Hypothesis 2.1, for any integer N ≥ 1, there exist positive constants C0, C1, C2 , and (ǫ0, κ0) ∈ (0, b/N ) × (0, 1), such that for any (ǫ, κ) ∈ (0, ǫ0] × (0, κ0] there exist a0 < b0 < a1 < • • • < aN < bN , with a0 = inf{σ(H ǫ,κ )} so that:

σ(H ǫ,κ ) ∩ [a0, bN ] ⊂ N k=0 [a k , b k ] , dim RanE [a k ,b k ] (H ǫ,κ ) = +∞ , (2.7) b k -a k ≤ C0ǫ κ + C1 ǫ 1/5 for 0 ≤ k ≤ N , and a k+1 -b k ≥ 1 C2 ǫ , for 0 ≤ k ≤ N -1 .
As a final remark, let us notice that in the case when the global minimum of the band λ0 is attained at several non degenerate points, one can repeat the argument given in Section 4 of [START_REF] Cornean | Low lying spectral gaps induced by slowly varying magnetic fields[END_REF] making cut-offs around each minimum and the results remain true.

A roadmap of the proof

1. We start by constructing a global smooth section ψ0 : T * → F of norm 1 vectors in F θ that coincides with the Bloch eigenvector φ0(θ) on the neighborhood Σ b of the global minimum θ0 of λ0. Then the range of its associated orthogonal projection π := U -1 Γ π UΓ in H will have an orthonormal basis of localized functions given by all the Γ-translations of ψ0 := U -1 Γ ψ0. This will be presented in Section 3. An important observation is that although the projection π does not commute with H 0 , it still contains information about the lowest part of the spectrum of H 0 . This is because the bounded operator πH 0 π has exactly one Bloch band λ0(θ) which coincides with λ0(θ) on Σ b (a fact proved in Proposition 8.13).

2. We notice that the operator π ⊥ H 0 π ⊥ seen as acting in π ⊥ H is bounded from below by b. Up to a use of the Feshbach-Schur argument (see for example [START_REF] Gustafson | Mathematical Concepts of Quantum Mechanics (2-nd edition)[END_REF]) and keeping the spectral parameter E in the interval [0, E0] with E0 < b/2 , the operator H 0 -E is invertible if and only if the reduced operator

S(E) := π(H 0 -E)π -πH 0 π ⊥ [π ⊥ (H 0 -E)π ⊥ ] -1 π ⊥ H 0 π (2.8)
is invertible in πH . Since we are only interested in values of E which are close to zero, we may expand the reduced resolvent around E = 0 and obtain:

S(E) = πH 0 π -πH 0 π ⊥ [π ⊥ H 0 π ⊥ ] -1 π ⊥ H 0 π -E(π + πH 0 π ⊥ [π ⊥ H 0 π ⊥ ] -2 π ⊥ H 0 π) + O(E 2 0 ) .
We observe that the operator coefficient of E, i.e.

Y := π + πH 0 π ⊥ [π ⊥ H 0 π ⊥ ] -2 π ⊥ H 0 π (2.9) is bounded and satisfies Y ≥ π. Thus S(E) is invertible in πH iff Y -1/2 S(E)Y -1/2
is invertible in πH, and some elementary arguments (see the proof of Proposition 4.2) allow to obtain an estimation of the Hausdorff distance between the spectrum of the 'dressed' operator

H := Y -1/2 {πH 0 π -πH 0 π ⊥ [π ⊥ H 0 π ⊥ ] -1 π ⊥ H 0 π}Y -1/2 (2.10)
restricted to the interval [0, E0] and the spectrum of H 0 in the same interval.

We notice that due to our construction of a special Wannier-like basis for the range of π, the 'dressed' operator H can be identified with a matrix acting in ℓ 2 (Γ). The details will be presented in Section 4 as a consequence of a more abstract theorem.

3. The crucial step (explained below) is to extend these objects to the magnetic case and construct a magnetic matrix acting in ℓ 2 (Γ), whose spectrum has not only gaps, but it also lies close enough to the spectrum of the full operator H ǫ,κ near zero such that some gaps must also appear in the spectrum of H ǫ,κ . In Section 6 we use the procedure developed in [START_REF] Nenciu | Dynamics of Bloch electrons in electric and magnetic fields, rigorous justification of the effective Hamiltonians[END_REF][START_REF] Helffer | Equation de Schrödinger avec champ magnétique et équation de Harper[END_REF][START_REF] Cornean | On the construction of composite Wannier functions[END_REF][START_REF] Cornean | Peierls substitution and magnetic pseudodifferential calculus[END_REF] in order to define a magnetic quasi-band projection π ǫ,κ which is the magnetic version of π and its range is again spanned by some localized functions which are indexed by Γ. An important observation is that (π ǫ,κ ) ⊥ H ǫ,κ (π ǫ,κ ) ⊥ is still bounded from below by b/2 if ǫ is small enough.

4. We can repeat the above Feshbach-Schur argument for the pair H ǫ,κ , π ǫ,κ with E0 = N ǫ where N is a fixed large number and construct a dressed magnetic matrix H ǫ,κ acting in ℓ 2 (Γ), such that its spectrum in the interval [0, N ǫ] is at a Hausdorff distance of order ǫ 2 from the spectrum of the full operator H ǫ,κ . Hence if we can prove that the matrix H ǫ,κ has gaps or order ǫ in the interval [0, N ǫ], the same must be true for H ǫ,κ .

5.

In Section 8 we construct a 'quasi-band' periodic and smooth function λ -ǫ (θ) which is close to λ0 on Σ b such that the Hausdorf distance between the spectrum of H ǫ,κ (acting on ℓ 2 (Γ)) and the spectrum of the magnetic quantization Op ǫ,κ ( λ -ǫ ) (acting on L 2 (X )) is of order ǫκ. This leads to (2.5).

The quasi Wannier system

Let us spell out a straightforward but important consequence of Hypothesis 2.1: Lemma 3.1. There exists b > 0 such that, for every 0 < b ≤ b, we have:

• The set Σ b is diffeomorphic to the open unit disc in R 2 ,
has a smooth boundary and contains θ0 .

• The function λ0 is smooth on Σ b and its Hessian matrix is positive.

• For θ outside of Σ b we have Ĥ0 (θ) ≥ b .
Let us fix some b ∈ (0, b) and consider the local smooth section φ0

Σ b : Σ b → F | Σ b .
Proposition 3.2. There exists a global smooth section of norm 1 vectors

ψ0 : T * → F , (3.1)
such that ψ0(θ) = φ0(θ) for any θ ∈ Σ b .

Proof. Let fj ∈ C ∞ 0 (X ), j ∈ {1, 2}, with unit norms in L 2 (X ) such that both of them have support in |x| ≤ 1/10 and moreover, f1(x)f2(x) = 0 for all x. Let us define fj (x, θ) = γ∈Γ e i<θ,γ> fj (xγ).

These functions are smooth in x and belong to F θ . We can check that f1(x, θ) f2(x, θ) = 0 for all x ∈ E, thus their scalar product in F θ equals zero. Moreover, both have norm one at fixed θ.

When θ is close to θ0, the eigenvector φ(x, θ) can be chosen to be smooth as a function of x due to elliptic regularity. It is also smooth as a function of θ on a small neighborhood of θ0. Now the vector φ(•, θ0) is certainly not parallel with both fj (•, θ0) and there must exist a j (assume without loss of generality that j = 1) such that

| φ(θ0), f1(•, θ0) | ≤ 1/ √ 2 .
Due to continuity in θ, there exists a small ball Br(θ0) ⊂ Σ b where we have

| φ(θ), f1(•, θ) | ≤ 3/4 , ∀θ ∈ Br(θ0) .
Let 0 ≤ g(θ) ≤ 1 with support in Br(θ0), equal to one on B r/2 (θ0). Define ĥ(x, θ)

:= g(θ) φ(x, θ) + (1 -g(θ)) f (x, θ) .
Then ĥ(•, θ) 2 ≥ 1/8 hence ψ0(x, θ) := ĥ(x, θ)/ ĥ(•, θ) ∈ F θ is a smooth extension of φ in both x and θ.

We are now ready to define the principal quasi Wannier function ψ0 by the formula

ψ0 := U -1 Γ ψ0 ∈ L 2 (X ) , (3.2)
where UΓ has been introduced in (1.13). The above function has norm 1 in H, it is smooth on X and has fast decay together with all its derivatives. Also, the corresponding family

ψγ := τγ ψ0 ∈ L 2 (X ) , γ ∈ Γ , (3.3)
forms an orthonormal system in L 2 (X ) that we shall call the quasi Wannier system associated with the energy window [0, b]. We shall also denote by π ∈ L(H) the orthogonal projection on the closed linear subspace generated by the quasi Wannier system:

H0 := Span{ψγ : γ ∈ Γ} ⊂ L 2 (X ). (3.4)
We shall use the notation:

π(θ) := | ψ0(θ) ψ0(θ)|. ( 3.5)
The following statement can be proven by the same arguments as Proposition 3.12 of [START_REF] Cornean | Low lying spectral gaps induced by slowly varying magnetic fields[END_REF].

Proposition 3.3. The vector ψ0 defined in (3.2) belongs to the Schwartz space S(X ). Moreover, if we consider the Weyl symbol p0 in S (Ξ) of the orthogonal projection |ψ0 ψ0|

p0(x, ξ) = (2π) -1 X e i<ξ,v> ψ0(x + (v/2))ψ0(x -(v/2)) dv ,
then the series γ∈Γ p0(xγ, ξ) converges pointwise with all its derivatives to a Γ-periodic symbol p ∈ S -∞ (Ξ) and we have

π = Op(p) , Kp(x, y) := γ∈Γ ψ0(x -γ)ψ0(y -γ) , UΓπU -1 Γ = ⊕ T * | ψ0(θ) ψ0(θ)|dθ , (3.6)
where Kp denotes the distribution kernel of π. Proof. Due to the properties of ψ0 listed after its definition in (3.2) and due to the fact that H 0 is a differential operator with polynomially bounded coefficients, the quasi Wannier function ψ0 belongs to D(H 0 ). Moreover, due to the rapid decay of the quasi Wannier function and its derivatives, it follows that H0 (defined in (3.4)) belongs in fact to D(H 0 ). We conclude that H0 = πH ⊂ D(H 0 ) and thus H 0 π is a well defined bounded linear operator on H. Moreover, the operator πH 0 : D(H 0 ) → H is a restriction of the adjoint H 0 π * and thus has a bounded closure in H .

An abstract reduction argument

We consider a more abstract setting and give the details of the arguments sketched in the third item of Subsection 2.3 that gives a procedure to apply the Feshbah-Schur method in a more complicate situation when the commutator H, π is no longer controlled by a small parameter (as was the situation in our previous paper [START_REF] Cornean | Low lying spectral gaps induced by slowly varying magnetic fields[END_REF]).

Hypothesis 4.1. We consider a triple (H, Π, β) where H is a positive self-adjoint operator, Π is an orthogonal projection such that HΠ is bounded (as well as ΠH), β > 0 and

Π ⊥ HΠ ⊥ ≥ 2βΠ ⊥ . (4.1) This implies that Π ⊥ (H -E)Π ⊥ is invertible in Π ⊥ H for E ∈ [0, 2β
) and we denote by R ⊥ (E) ∈ L(Π ⊥ H) its inverse. The spectral theorem gives:

sup E∈[0,β] R ⊥ (E) ≤ β -1 . (4.2)
Starting from

Y := Π + ΠHΠ ⊥ R ⊥ (0) 2 Π ⊥ HΠ , (4.3)
and noting that Y ≥ Π , we introduce

H := Y -1/2 ΠHΠ -ΠHΠ ⊥ R ⊥ (0)Π ⊥ HΠ Y -1/2 ∈ L(ΠH). (4.4)
Proposition 4.2. For any β ′ < β we have:

dH {σ(H) ∩ [0, β ′ ], σ( H) ∩ [0, β ′ ]} ≤ HΠ 2 (β ′ ) 2 β -3 , (4.5)
where H is defined in (4.4).

Proof. Using the Feshbach-Schur reduction we get that if E ∈ [0, β] then the operator H -E is invertible if and only if the operator

S(E) := Π H -E Π -ΠHR ⊥ (E)HΠ (4.6)
is invertible in ΠH. In this case we have the identity:

Π(H -E) -1 Π = S(E) -1 . (4.7)
Using the resolvent equation, we have, if

E ∈ [0, β ′ ), R ⊥ (E) = R ⊥ (0) + R ⊥ (0) 2 + X(E) , X(E) := E 2 R ⊥ (0) 2 R ⊥ (E) , X(E) ≤ E 2 β -3 ≤ β ′2 β -3 .
Thus we can write S(E) as:

S(E) = ΠHΠ -ΠHR ⊥ (0)HΠ -E(Y -Π) -EΠ + ΠHX(E)HΠ = Π [H -HR ⊥ (0)H] Π -EY + ΠHX(E)HΠ.
Since Y in (4.3) is bounded from below by 1, we can define the positive operator Y -1/2 in L(ΠH). Then (use the notation (4.4) in the above identity) S(E) is invertible if and only if

Y -1/2 S(E)Y -1/2 = H -E + Y -1/2 ΠHX(E)HΠY -1/2 (4.8)
is invertible in ΠH with bounded inverse. We are now prepared to prove Proposition 4.2 and we do it in two steps.

Step 1. Assume that E ∈ [0, β ′ ] is in the resolvent set of H. Then (4.8) implies:

S(E) = Y 1/2 {1 l + Y -1/2 ΠHX(E)HΠY -1/2 ( H -E) -1 )}( H -E)Y 1/2 .
We have:

Y -1/2 ΠHX(E)HΠY -1/2 ( H -E) -1 ≤ β ′2 HΠ 2 β -3 dist(E, σ( H)) , if E ∈ [0, β ′ ]. Thus, if dist(E, σ( H)) > β ′2 HΠ 2 β -3 , S(E) is invertible, hence E is in the resolvent set of H.
In other words, no element of σ(H) ∩ [0, β ′ ] can be situated at a distance which is larger than

β ′2 HΠ 2 β -3 from σ( H) ∩ [0, β ′ ).
Step 2. Assume that E ∈ [0, β ′ ) is in the resolvent set of H hence (4.7) holds. From (4.8) we see that H -E is invertible when the operator:

S(E) -ΠHX(E)HΠ = {1 l -ΠHX(E)HΠ(H -E) -1 Π}S(E) is invertible.
As before we have:

ΠHX(E)HΠ(H -E) -1 Π ≤ β ′2 HΠ 2 β -3 dist(E, σ(H)) , E ∈ [0, β ′ ).
Hence

H -E is invertible when dist(E, σ(H)) > β ′2 HΠ 2 β -3 . Equivalently, no element of σ( H) ∩ [0, β ′ ) can be situated at a distance larger than β ′2 HΠ 2 β -3 from σ(H) ∩ [0, β ′ ]. Corollary 4.3. Let 0 < D1 < D2 < β ′ < β and assume (D1, D2) ∩ σ( H) = ∅. Then, if HΠ 2 (β ′ ) 2 β -3 < 1 2 (D2 -D1)
we have

(D1 + HΠ 2 (β ′ ) 2 β -3 , D2 -HΠ 2 (β ′ ) 2 β -3 ) ∩ σ(H) = ∅ .
It will be interesting to apply this corollary for a family indexed by η ∈ [0, ǫ1] of triples (H(η), Π(η), β) satisfying (4.1). This leads to:

D := sup η∈[0,ǫ 1 ] H(η)Π(η) 2 (β ′ ) 2 β -3 < 1 2 (D2 -D1) ,
we have

(D1 + D , D2 -D) ∩ σ(H(η)) = ∅ . Proof. It is a direct consequence of Corollary 4.3 .
Finally, the next proposition gives sufficient conditions for the appearance of gaps of order η in the spectrum of H(η). Proposition 4.5. Suppose that that there exist two numerical constants 0 < C1 < C2 such that the interval (C1, C2) belongs to the resolvent set of η -1 H(η) for all 0 < η ≤ min(ǫ1, β C 2 +1 ).

Let C0 := (C2 +1) 2 (sup η∈[0,ǫ 1 ] H(η)Π(η) ) 2 β -3 and ǫ0 := min(ǫ1, β C 2 +1 , 1 2 (C2 -C1)/C0) . Then, (C1η + C0η 2 , C2η -C0η 2 ) ∩ σ(H(η)) = ∅ , ∀η ∈ (0, ǫ0) .
Proof. Let D1 = C1η , D2 = C2η , β ′ = (C2 + 1)η . We then apply Corollary 4.4.

The magnetic pseudodifferential calculus

Let us recall from [START_REF] Cornean | Low lying spectral gaps induced by slowly varying magnetic fields[END_REF] the following two notations:

Λ A (x, y) := e -i [x,y] A , Ω B (x, y, z) := e -i <x,y,z> B , (5.1)
where [x, y] denotes the oriented interval from x to y and < x, y, z > denotes the oriented triangle with vertices x, y, z, with the integrals being the usual invariant integrals of k-forms for k = 1, 2.

We shall also use the following shorthand notation:

Λ ǫ ≡ Λ ǫA 0 , Λ ǫ,κ ≡ Λ A ǫ,κ , Ω ǫ ≡ Ω ǫB 0 , Ω ǫ,κ ≡ Ω B ǫ,κ , Λ ǫ,κ ≡ Λ κA ǫ , Ω ǫ,κ ≡ Ω κǫBǫ . (5.2)
We notice for future use that

Λ A (x, z)Λ A (z, y)Λ A (y, x) = Ω B (x, z, y) , (5.3)
and that there exists C > 0 such that, for any magnetic field B of class BC ∞ (X ) ,

Ω B (x, y, z) -1 ≤ C B ∞ |(y -x) ∧ (z -x)| . (5.4)
We notice also that for our choice of gauge for A0 we have the relations:

Λ ǫ (x, y) = exp {i(B0/2) x ∧ y} (5.5) Ω ǫ (x, y, z) = exp {iB0 (x -y) ∧ (z -y)} . (5.6)
Let us recall the gauge covariance property which states that whatever magnetic potential A we choose for a given magnetic field B(x) := B12(x) = -B21(x) we have:

{-i∇ -A(x)} 2 Λ A (x, z) = Λ A (x, z) {-i∇ -a(x, z)} 2 , (5.7)
where

aj (x, z) = k (x -z) k 1 0
B jk z + s(xz) s ds for j = 1, 2 .

(5.8)

Let X * ∼ = R 2 be the dual of our configuration space X ≡ R 2 and denote by < • , • > the duality map X * × X → R between these two spaces. Let Ξ := X × X * be the associated phase space with the canonical symplectic structure σ (x, ξ), (y, η) :=< ξ, y > -< η, x >, ∀ (x, ξ), (y, η) ∈ Ξ × Ξ .

(5.9)

We will use the following class of Hörmander type symbols.

Definition 5.1. For any s ∈ R we denote by Definition 5.3. We consider a vector potential A with components of class C ∞ pol (X ) . For any F ∈ S (Ξ) we can define ( [START_REF] Mȃntoiu | The magnetic Weyl calculus[END_REF]) the following linear operator:

S s (Ξ) := {F ∈ C ∞ (Ξ) | ν s n,m (F ) < +∞ , ∀(n, m) ∈ N × N} , where ν s n,m (f ) := sup (x,ξ)∈Ξ |α|≤n |β|≤m ξ -s+m ∂ α x ∂ β ξ f (x, ξ) ,
u → Op A (F )u (x) := (2π) -2 X X * e i ξ,x-y e -i [x,y] A F x + y 2 , ξ u(y) dξ dy , (5.11) ∀u ∈ S (X ).
This operator is continuous on S (X ) and has a natural extension by duality to S ′ (X ). For A = 0 we obtain the usual Weyl calculus. In [START_REF] Mȃntoiu | The magnetic Weyl calculus[END_REF] it is proven that Proposition 5.4. For any vector potential A with components of class C ∞ pol (X ) the quantization Op A defines a linear and topological isomorphism between S ′ (Ξ) and the linear topological space of continuous operators from S (X ) to S ′ (X ).

We shall denote by S A its inverse, i.e. the application that associates to a given operator the tempered distribution which is its symbol for the magnetic pseudodifferential calculus defined by the vector potential A:

S A • Op A = I .
(5.12)

In the same spirit as the Calderon-Vaillancourt theorem for classical pseudo-differential operators, Theorem 3.1 in [START_REF] Iftimie | Magnetic pseudodifferential operators[END_REF] states that any symbol F ∈ S 0 0 (Ξ) defines a bounded operator Op A (F ) in L 2 (X ) with an upper bound of the operator norm given by some seminorm of F as Hörmander type symbol. We denote by F B the operator norm of Op A (F ) in L(L 2 (X )) :

F B := ||Op A (F )|| L(L 2 (X )) .
(5.13)

This norm only depends on the magnetic field B and not on the choice of the vector potential (different choices being unitary equivalent).

We notice that

H 0 = Op A Γ (h), h(x, ξ) := ξ 2 + VΓ(x) .
(5.14)

and for the perturbed Hamiltonians we will use the following shorthand notations

H ǫ,κ = Op A ǫ,κ (h) =: Op ǫ,κ (h), H ǫ = Op A ǫ (h) =: Op ǫ (h). (5.15)
We also recall from [START_REF] Mȃntoiu | The magnetic Weyl calculus[END_REF] that for any two test functions f and g in S (Ξ) the product of the linear operators Op A (f ) and Op A (g) induces a twisted Moyal product, also called magnetic Moyal product, such that

Op A (f ) Op A (g) = Op A (f ♯ B g) .
This product depends only on the magnetic field B and is given by an explicit oscillating integral. The following relation proven in [START_REF] Mȃntoiu | The magnetic Weyl calculus[END_REF] (Lemme 4.14):

Ξ dX f ♯ B g (X) = Ξ dXf (X)g(X) (5.16)
allows to extend the magnetic Moyal product to a class of tempered distributions that leave S (Ξ) invariant by magnetic Moyal composition. In [START_REF] Mȃntoiu | The magnetic Weyl calculus[END_REF] it is proven that this class forms a *-algebra that we call the magnetic Moyal algebra and denote by M B . Moreover this class contains all the Hörmander type symbols (see [START_REF] Mȃntoiu | The magnetic Weyl calculus[END_REF][START_REF] Iftimie | Magnetic pseudodifferential operators[END_REF]) and the usual composition of symbols theorem is still valid (Theorem 2.2 in [START_REF] Iftimie | Magnetic pseudodifferential operators[END_REF]).

For any invertible symbol F in M B , we denote by F - B its inverse. It is shown in Subsection 2.1 of [START_REF] Mȃntoiu | Spectral and propagation results for magnetic Schrödinger operators; a C * -Algebraic Approach[END_REF] that, for any m > 0 and for a > 0 large enough (depending on m) the symbol sm(x, ξ) :=< ξ > m +a, has an inverse in M B . We shall use the shorthand notation s B -m instead of sm -B and extend it to any m ∈ R (thus for m > 0 we have s B m ≡ sm). The following results have been established in [START_REF] Iftimie | Commutator criteria for magnetic pseudodifferential operators[END_REF] (Propositions 6.2 and 6.3): Proposition 5.5.

1. If F ∈ S 0 (Ξ) is invertible in M B , then the inverse F - B also belongs to S 0 (Ξ) . 2. For m < 0, if f ∈ S m (Ξ) is such that 1+f is invertible in M B , then (1+f ) - B -1 ∈ S m (Ξ) . 3. For m > 0, if G ∈ S m (Ξ) is invertible in M B , with Op A sm ♯ B G - B ∈ L L 2 (X ) , then G - B ∈ S -m (Ξ) .
From the explicit form of the magnetic Moyal product (see in [START_REF] Mȃntoiu | The magnetic Weyl calculus[END_REF]) we easily notice the following fact. Proposition 5.6. Let us consider the lattice Γ * ⊂ X * . If f and g are Γ * -periodic symbols, then their magnetic Moyal product is also Γ * -periodic.

About the resolvent

By Theorem 4.1 in [START_REF] Iftimie | Magnetic pseudodifferential operators[END_REF], for any real elliptic symbol h ∈ S m (Ξ) (with m > 0) and for any A in C ∞ pol (X , R 2 ), the operator Op A (h) has a closure H A in L 2 (X ) that is self-adjoint on a domain H m A (a magnetic Sobolev space) and lower semibounded. Thus we can define its resolvent (H Az) -1 for any z / ∈ σ(H A ) and Theorem 6.5 in [START_REF] Iftimie | Commutator criteria for magnetic pseudodifferential operators[END_REF] states that it exists a symbol

r B z (h) ∈ S -m (Ξ) such that (H A -z) -1 = Op A (r B z (h)) .

Integral kernels and symbols

For symbols of class S 0 (Ξ), we have seen that the associated magnetic pseudodifferential operator is bounded in H and is self-adjoint if and only if its symbol is real. In that case we can also define its resolvent and the results in [START_REF] Iftimie | Commutator criteria for magnetic pseudodifferential operators[END_REF], cited above, show that it is also defined by a symbol of class S 0 (Ξ) .

Given any tempered distribution T ∈ S ′ (X × X ) we shall denote by Int (T ) : S (X ) → S ′ (X ) the integral operator having the distribution kernel T , given by the formula

u, Int (T ) v H := T, u ⊗ v , ∀(u, v) ∈ S (X ) × S (X ) .
(5.17)

For any tempered distribution F ∈ S ′ (Ξ) we denote by KF ∈ S ′ (X × X ) the integral kernel of its Weyl quantization, i.e. Op(F ) = Int (KF ). Let us recall that there exists a linear bijection W : S ′ (Ξ) → S ′ (X × X ) defined by

WF (x, y) := (2π) -2 X * e i<ξ,x-y> F x + y 2 , ξ dξ , (5.18)
such that Op(F ) = Int(WF ) .

In the magnetic calculus, we have the equality

Op A (F ) = Int(Λ A WF ) . (5.19)
6 The magnetic quasi-band projections

Following step by step the arguments in Section 3 of [START_REF] Cornean | Low lying spectral gaps induced by slowly varying magnetic fields[END_REF] we can associate with the quasi-band orthogonal projection π ∈ L(H) some magnetic versions of it: π ǫ and π ǫ,κ . Starting from the principal quasi Wannier function ψ0 introduced in Definition 3.2 and the magnetic field introduced in (1.5) we define the following objects: Definition 6.1.

1. φǫ γ (x) := Λ ǫ (x, γ)ψ0(x -γ) , G ǫ αβ := φǫ α , φǫ β H , F ǫ := G ǫ -1/2 ∈ L ℓ 2 (Γ) , 2. ψ ǫ 0 (x) := α∈Γ F ǫ α 0 Ω ǫ (α, 0, x)ψ0(x -α) , φ ǫ γ (x) := Λ ǫ (x, γ) ψ ǫ 0 (x -γ). 3. π ǫ := γ∈Γ |φ ǫ γ φ ǫ γ | = (π ǫ ) 2 = (π ǫ ) * , 4. φǫ,κ γ (x) := Λ ǫ,κ (x, γ)ψ ǫ 0 (x -γ) , G ǫ,κ αβ := φǫ,κ α , φǫ,κ β H , F ǫ,κ := G ǫ,κ -1/2 ∈ L ℓ 2 (Γ) , 5. φ ǫ,κ γ := α∈Γ F ǫ,κ αγ φǫ,κ α , π ǫ,κ := γ∈Γ |φ ǫ,κ γ φ ǫ,κ γ | = (π ǫ,κ ) 2 = (π ǫ,κ ) * .
Using the notation introduced in (1.8)-(5.2) we notice that Λ ǫ,κ (x, y) = Λ ǫ,κ (x, y)Λ ǫ (x, y) (6.1) so that

• φ ǫ,κ γ (x) = Λ ǫ,κ (x, γ)φ ǫ γ (x) (6.2)
If we simply replace the Wannier function φ0 in Section 3 of [START_REF] Cornean | Low lying spectral gaps induced by slowly varying magnetic fields[END_REF] with the quasi Wannier function ψ0, all the results of Section 3 of [START_REF] Cornean | Low lying spectral gaps induced by slowly varying magnetic fields[END_REF] remain true due to the fact that the only properties of φ0 that are used are its smoothness and rapid decay together with all its derivatives, and these facts are true for the quasi Wannier function too. Thus we obtain the following statement, extending the results in Section 3 of [3]: Proposition 6.2. There exists ǫ0 > 0 such that:

1. F ǫ belongs to L ℓ 2 (Γ) L ℓ ∞ (Γ) for any ǫ ∈ [0, ǫ0] , and satisfies:

∀m ∈ N, ∃Cm s.t. < α -β > m F ǫ αβ -δ αβ ≤ Cm ǫ , ∀(α, β) ∈ Γ 2 , ∀ǫ ∈ [0, ǫ0] ,
and there exists a rapidly decaying function F ǫ : Γ → C such that

F ǫ αβ = Λ ǫ (α, β)F ǫ (α -β) .
2. For any m ∈ N and any a ∈ N 2 , there exists Cm,a > 0 such that

sup x∈X < x > m ∂ a ψ ǫ 0 (x) -∂ a ψ0 (x) ≤ Cm,aǫ , ∀ǫ ∈ [0, ǫ0] .
3. For any m ∈ N , there exists Cm > 0 such that

sup (α,β)∈Γ 2 < α -β > m F ǫ,κ α,β -δ αβ ≤ Cm κǫ , ∀(ǫ, κ) ∈ [0, ǫ0] × [0, 1] . (6.3)
Replacing ψ0 with ψ ǫ 0 in the proof of Proposition 3.4 we obtain the following result: Proposition 6.3. There exists ǫ0 > 0 such that for any (ǫ, κ)

∈ [0, ǫ0] × [0, 1] we have π ǫ,κ H ⊂ D(H ǫ,κ ) , while H ǫ,κ π ǫ,κ ∈ L(H) and π ǫ,κ H ǫ,κ has a bounded closure.
We can also construct the Γ-periodic symbol pǫ ∈ S -∞ (Ξ) such that π ǫ = Op ǫ (pǫ) and the symbol pǫ,κ ∈ S -∞ (Ξ) such that π ǫ,κ = Op ǫ,κ (pǫ,κ) and the same proof as in [START_REF] Cornean | Low lying spectral gaps induced by slowly varying magnetic fields[END_REF] gives the following result: Proposition 6.4. There exists ǫ0 > 0 such that for any seminorm ν on S -∞ (Ξ), there exists Cν > 0 such that

ν(p ǫ -p) ≤ Cν ǫ and ν(p ǫ,κ -p ǫ ) ≤ Cν κǫ , ∀(ǫ, κ) ∈ [0, ǫ0] × [0, 1] .
Note that in this case the commutator H ǫ,κ , π ǫ,κ is no longer small (due to the arbitrary deformation which was made in constructing the quasi Wannier function).

7 Checking the conditions of the abstract reduction procedure for the magnetic operators

We want to apply Propositions 4.2 and 4.5 for magnetic pseudodifferential operators and to control the behavior of their symbols. The following abstract result concerns the symbol of the "reduced resolvent" R ⊥ (E) which is introduced after (4.1). In our applications below we shall replace H with H -E1 l for some E < inf σ(H).

Suppose given some real elliptic symbol h ∈ S m 1 (Ξ) ell for some m > 0 and consider the magnetic pseudodifferential calculus, denoted by Op A , associated with a smooth polynomially bounded vector potential A such the magnetic field B = curl A is of class BC ∞ (X ). Let

H = Op A (h)
and suppose also given an orthogonal projection Π ≡ Op A (p) with p ∈ S -∞ (Ξ) and such that ΠH ⊂ D(H). Let us also introduce

H ⊥ := Π ⊥ HΠ ⊥ with Π ⊥ := 1 l -Π . Proposition 7.1. With the above notation, if H ⊥ is invertible on Π ⊥ H with bounded inverse R ∈ L(Π ⊥ H), then there exists r ∈ S -m (Ξ) such that Π ⊥ RΠ ⊥ = Op A (r). Proof. We notice that Π ⊥ RΠ ⊥ = Π ⊥ Π + Π ⊥ HΠ ⊥ -1 Π ⊥ (7.1)
and use point (3) of Proposition 5.5.

Let us verify that the pair of operators (H ǫ,κ , π ǫ,κ ) satisfies Hypothesis 4.1.

Proposition 7.2. There exist ǫ0 > 0 and b0 ∈ (0, b) (with b > 0 introduced in Lemma 3.1) such that for any (ǫ, κ) ∈ [0, ǫ0] × [0, 1] the pair of operators (H ǫ,κ , π ǫ,κ ) verify the following properties:

1. H ǫ,κ π ǫ,κ is bounded on H, uniformly in ǫ and κ .

2.

(1 l -π ǫ,κ )H ǫ,κ (1 l -π ǫ,κ ) ≥ b0 (1 l -π ǫ,κ ) .
Proof. The first statement is a direct consequence of Proposition 6.3 and we focus on the second statement.

Let b ′ < b < b with b > 0 introduced in Lemma 3.1 so that the closure of Σ b ′ is included in Σ b . We choose a function g ∈ C ∞ 0 (Σ b ) such that 0 ≤ g(θ) ≤ 1 and g(θ) = 1 on Σ b ′ .
Then we define successively (using (3.5)):

K0 (θ) := Ĥ0 (θ) + g(θ)π(θ) , K 0 := U -1 Γ ⊕ E * K0 (θ) dθ UΓ , (7.2)
W 0 := K 0 -H 0 , w := S(W 0 ), W ǫ,κ := Op ǫ,κ (w), K ǫ,κ := H ǫ,κ + W ǫ,κ . (7.3)
By construction, we have K 0 ≥ b ′ 1 l, which implies:

π ⊥ H 0 π ⊥ = π ⊥ K 0 π ⊥ ≥ b ′ π ⊥ . (7.4)
Using the regularity of the spectrum (see Corollary 1.6 in [START_REF] Cornean | Spectral edge regularity of magnetic Hamiltonians[END_REF]), we conclude that for ǫ0 > 0 small enough, K ǫ,κ ≥ (3/4)b ′ for every ǫ ∈ [0, ǫ0] and 0 ≤ κ ≤ 1, and this implies:

1 l -π ǫ,κ K ǫ,κ 1 l -π ǫ,κ ≥ (3/4)b ′ 1 l -π ǫ,κ . (7.5)
By construction, we have that W 0 = πW 0 π. We show that their magnetic counterparts are also close in norm. We write: π ǫ,κ W ǫ,κ π ǫ,κ -W ǫ,κ = Op ǫ,κ (p ǫ,κ ♯ ǫ,κ w ♯ ǫ,κ p ǫ,κw) , while p ǫ,κ ♯ ǫ,κ w ♯ ǫ,κ p ǫ,κp ♯ w ♯ p = p ǫ,κ ♯ ǫ,κ wp ǫ,κ ♯ w ♯ ǫ,κ p ǫ,κ + p ǫ,κ ♯ w ♯ ǫ,κ p ǫ,κw ♯ p + p ǫ,κp ♯ w ♯ p is a symbol of class S 0 1 (Ξ) ⊂ C ǫ,κ (Ξ) having the norm in C ǫ,κ (Ξ) (i.e. the operator norm of its magnetic quantization Op ǫ,κ ) of order ǫ and thus there exist C and ǫ0 such that

π ǫ,κ W ǫ,κ π ǫ,κ -W ǫ,κ ≤ C ǫ , (7.6)
for any (ǫ, κ)

∈ [0, ǫ0] × [0, 1]. Finally, writing (1 l -π ǫ,κ )H ǫ,κ (1 l -π ǫ,κ ) = (1 l -π ǫ,κ )K ǫ,κ (1 l -π ǫ,κ ) -(1 l -π ǫ,κ )(W ǫ,κ -π ǫ,κ W ǫ,κ π ǫ,κ )(1 l -π ǫ,κ )
and using (7.5) and (7.6) we see that if ǫ is small enough then we can choose b0 = b ′ /2 . This achieves the proof.

Thus Hypothesis 4.1 is satisfied when H and Π are replaced by H ǫ,κ and π ǫ,κ respectively, and β = b ′ /2 . Hence Proposition 4.5 can be applied and fixing N > 1 and working with ǫ ≥ 0 such that ǫN < b ′ /2 , we deduce that H ǫ,κ has N gaps of order ǫ in its lower spectrum if the operator H ǫ,κ associated with the pair (H ǫ,κ , π ǫ,κ ) as in (4.3) and (4.4) also has N gaps of order ǫ in its lower spectrum. We shall use the notations Y ǫ,κ ∈ L(H) and H ǫ,κ ∈ L(π ǫ,κ H) for the operators defined as in (4.3) and (4.4) for H replaced by H ǫ,κ and Π replaced by π ǫ,κ .

The one band effective Hamiltonian

The conclusion of Section 7 implies that in order to finish the proof of our main result in Theorem 2.2 we have to analyze the bottom of the spectrum of the operator π ǫ,κ H ǫ,κ π ǫ,κ . We follow essentially the arguments and ideas from Subsection 3.3 in [START_REF] Cornean | Low lying spectral gaps induced by slowly varying magnetic fields[END_REF], but due to the fact that in the present situation the calculus is much more involved, we prefer to present rather complete proofs.

Preliminaries

As we shall start with the situation with constant magnetic field, let us recall the following results concerning the Zak magnetic translations [START_REF] Zak | Magnetic translation group[END_REF] (see also [START_REF] Nenciu | Dynamics of Bloch electrons in electric and magnetic fields, rigorous justification of the effective Hamiltonians[END_REF][START_REF] Helffer | Equation de Schrödinger avec champ magnétique et équation de Harper[END_REF][START_REF] Cornean | On the construction of composite Wannier functions[END_REF]).

Proposition 8.1. In the case of a constant magnetic field of the form Bǫ = ǫB0 the family of unitary operators

T ǫ γ := Λ ǫ (•, γ)τγ , γ ∈ Γ (8.1)
satisfies the following properties:

1. T ǫ α T ǫ β = Λ ǫ (β, α)T ǫ α+β . 2.
The operator Op ǫ (F ) commutes with all the {T ǫ γ }γ∈Γ if and only if F ∈ S ′ (Ξ) is Γperiodic with respect to the variable in X .

On the other hand, looking at the definition of the magnetic Moyal product and using the second point of the above proposition one can prove the following statement. Proposition 8.2. For a constant magnetic field Bǫ := ǫB0 the following statements hold true:

1. The subspace M Bǫ Γ of the Γ-periodic distributions of M Bǫ is a subalgebra. 2. If F ∈ M Bǫ Γ is invertible in M Bǫ , its inverse is in M Bǫ Γ .
Having in mind Proposition 5.4 and with the shorthand notation (see (5.12))

S ǫ,κ := S A ǫ,κ , S ǫ := S A ǫ , (8.2)
we introduce the following symbols:

Definition 8.3. • h ǫ,κ • := S ǫ,κ π ǫ,κ H ǫ,κ π ǫ,κ ; h ǫ • := S ǫ π ǫ H ǫ π ǫ ; • h ǫ,κ ⊥ := S ǫ,κ 1 l -π ǫ,κ H ǫ,κ 1 l -π ǫ,κ ; h ǫ ⊥ := S ǫ 1 l -π ǫ H ǫ 1 l -π ǫ ; • h ǫ,κ • := S ǫ,κ π ǫ,κ H ǫ,κ 1 l -π ǫ,κ ; h ǫ • := S ǫ π ǫ H ǫ 1 l -π ǫ . • rǫ,κ := S ǫ,κ (1 l -π ǫ,κ )R ǫ,κ ⊥ (1 l -π ǫ,κ ) ; rǫ := S ǫ (1 l -π ǫ )R ǫ ⊥ (1 l -π ǫ ) ; • y ǫ,κ := S ǫ,κ π ǫ,κ Y ǫ,κ ) -1/2 π ǫ,κ ; y ǫ := S ǫ π ǫ Y ǫ -1/2 π ǫ ;
• z ǫ,κ := y ǫ,κp ǫ,κ ; z ǫ := y ǫp ǫ Proposition 8.4. We have the following properties:

1. {rǫ,κ, rǫ} ⊂ S -2 (Ξ) , 2. y ǫ,κ , z ǫ,κ , y ǫ , z ǫ ⊂ S -∞ (Ξ) .
Proof. For the first two conclusions we just use Proposition 7.1 with m = 2. For the third conclusion we notice that π ǫ,κ and π ǫ have symbols of class S -∞ (Ξ).

Main approximation

8.2.1

Reduction to the constant field ǫB 0

The next proposition states that H ǫ,κ is close in operator norm to a magnetic matrix in which the non-constant magnetic field appears only through the phase Λ ǫ,κ .

Proposition 8.5. With the definitions and notations of this section, the following approximation holds, for any (α, β) ∈ Γ × Γ,

φ ǫ,κ α , H ǫ,κ φ ǫ,κ β H = Λ ǫ,κ (α, β) φ ǫ α , Op ǫ (h ǫ )φ ǫ β H + κǫ O(|α -β| -∞ ) , (8.3)
with hǫ := h ǫ • + h ǫ • ♯ ǫ z ǫ + z ǫ ♯ ǫ h ǫ • ♯ ǫ y ǫ + y ǫ ♯ ǫ h ǫ • ♯ ǫ rǫ ♯ ǫ h ǫ • ♯ ǫ y ǫ . Moreover φ ǫ α , Op ǫ (h ǫ )φ ǫ β H = Λ ǫ (α, β) φ ǫ α-β , Op ǫ (h ǫ )φ ǫ 0 H . (8.4)
Proof. Having in mind the notation of Section 6 (see Definition 6.1, item 5), we get:

φ ǫ,κ α , π ǫ,κ H ǫ,κ π ǫ,κ φ ǫ,κ β H = Σ1(α, β) + Σ2(α, β) , (8.5)
where, using (4.4) we can write

Σ1(α, β) := α ′ ∈Γ β ′ ∈Γ F ǫ,κ α ′ α F ǫ,κ β ′ β Y ǫ,κ -1 2 Λ ǫ,κ (•, α ′ )τ α ′ ψ ǫ 0 , H ǫ,κ Y ǫ,κ -1 2 Λ ǫ,κ (•, β ′ )τ β ′ ψ ǫ 0 H (8.6) and Σ2(α, β) := α ′ ∈Γ β ′ ∈Γ F ǫ,κ α ′ α F ǫ,κ β ′ β Y ǫ,κ -1 2 Λ ǫ,κ (•, α ′ )τ α ′ ψ ǫ 0 , K ǫ,κ Y ǫ,κ -1 2 Λ ǫ,κ (•, β ′ )τ β ′ ψ ǫ 0 H (8.7) with K ǫ,κ := π ǫ,κ H ǫ,κ R ǫ,κ ⊥ (0)H ǫ,κ π ǫ,κ ∈ L(π ǫ,κ H) .
Using Formula (6.2) and Proposition 6.2 (point 3), we will express the series appearing in Σ1(α, β) and Σ2(α, β) in terms of elements associated with π ǫ H in the constant magnetic field ǫB0.

In Subsection 8.2.2 we will prove the following estimation contained in Lemma 8.6:

Σ1(α, β) = Λ ǫ,κ (α, β) φ ǫ α , H ǫ + Op ǫ h ǫ,κ • ♯ ǫ,κ z ǫ,κ + Op ǫ z ǫ,κ ♯ ǫ,κ h ǫ,κ • ♯ ǫ,κ y ǫ,κ φ ǫ β H +κǫ O(|α -β| -∞ ) .
For the term Σ2(α, β) we will prove in Subsection 8.2.3 (Lemma 8.11) that

Σ2(α, β) = Λ ǫ,κ (α, β) φ ǫ α , Op ǫ y ǫ,κ ♯ ǫ,κ h ǫ,κ • ♯ ǫ,κ rǫ,κ ♯ ǫ,κ h ǫ,κ • ♯ ǫ,κ y ǫ,κ φ ǫ β H +κǫ O(|α-β| -∞ ) .
We continue by noticing that Proposition 6.4 and the properties of the magnetic composition of symbols (see Proposition B.12 in [START_REF] Cornean | Low lying spectral gaps induced by slowly varying magnetic fields[END_REF]) imply that for any seminorm ν on S -∞ (Ξ) there exists Cν > 0 such that

ν(h ǫ,κ • -h ǫ • ) ≤ Cν κǫ , ν(h ǫ,κ • -h ǫ • ) ≤ Cν κǫ , ν(h ǫ,κ ⊥ -h ǫ ⊥ ) ≤ Cν κǫ . (8.8)
Then let us consider the difference rǫ,κrǫ ∈ S -2 (Ξ) and compute

h ǫ,κ ⊥ ♯ ǫ,κ rǫ,κ -rǫ ♯ ǫ h ǫ ⊥ = (1 -p ǫ,κ ) ♯ ǫ h ǫ ⊥ -h ǫ,κ ⊥ ♯ ǫ,κ rǫ ♯ ǫ h ǫ ⊥ . (8.9)
Using Propositions B.12 and B.14 in [START_REF] Cornean | Low lying spectral gaps induced by slowly varying magnetic fields[END_REF] and the estimates in Proposition 6.4 we conclude that

h ǫ,κ ⊥ ♯ ǫ,κ rǫ ♯ ǫ h ǫ ⊥ = h ǫ,κ ⊥ ♯ ǫ rǫ ♯ ǫ h ǫ ⊥ + κǫr1,ǫ,κ = h ǫ,κ ⊥ ♯ ǫ (1 -p ǫ ) + κǫ r1,ǫ,κ, (8.10)
and

h ǫ,κ ⊥ ♯ ǫ,κ rǫ,κ -rǫ ♯ ǫ h ǫ ⊥ = (1 -p ǫ,κ ) ♯ ǫ h ǫ ⊥ -h ǫ,κ ⊥ ♯ ǫ (1 -p ǫ ) -κǫ r1,ǫ,κ = h ǫ ⊥ -h ǫ,κ ⊥ + κǫ r2,ǫ,κ = (1 -p ǫ ) ♯ ǫ h ♯ ǫ (1 -p ǫ ) -(1 -p ǫ,κ ) ♯ ǫ,κ h ♯ ǫ,κ (1 -p ǫ,κ ) -κǫ r3,ǫ,κ , (8.11
) with r1,ǫ,κ , r2,ǫ,κ and r3,ǫ,κ in S 0 (Ξ) uniformly for (ǫ, κ) ∈ [0, ǫ0] × [0, 1]. Using Proposition 6.4 several times we obtain that the difference rǫ,κrǫ belongs to S -4 (Ξ) and for any seminorm ν on S -4 (Ξ) there exists Cν > 0 such that ν rǫ,κrǫ ≤ Cν κǫ .

(8.12)

From Proposition 6.4 and the magnetic version of Calderon-Vaillancourt Theorem (see Theorem 3.1 in [START_REF] Iftimie | Magnetic pseudodifferential operators[END_REF]) we deduce that

Op ǫ p ǫ,κ -p ǫ ≤ C κǫ . (8.13)
Meanwhile, from Proposition B.14 in [START_REF] Cornean | Low lying spectral gaps induced by slowly varying magnetic fields[END_REF], we also deduce that for any symbols f and g in S 0 (Ξ) there exists C > 0 such that

Op ǫ f ♯ ǫ,κ g -f ♯ ǫ g ≤ C κǫ . (8.14)
Putting all these results together, using (8.8) and (8.12) in order to control the difference z ǫ,κz ǫ and the usual result concerning the estimate of the difference of the two magnetic products ♯ ǫ,κ and ♯ ǫ (see Proposition B.14 in [START_REF] Cornean | Low lying spectral gaps induced by slowly varying magnetic fields[END_REF]) we finally obtain:

Σ1(α, β) = Λ ǫ,κ (α, β) φ ǫ α , Op ǫ h ǫ • + h ǫ • ♯ ǫ z ǫ + z ǫ ♯ ǫ h ǫ • ♯ ǫ y ǫ φ ǫ β H + κǫ O(|α -β|| -∞ ) , (8.15) and Σ2
(α, β) = Λ ǫ,κ (α, β) φ ǫ α , Op ǫ y ǫ ♯ ǫ h ǫ • ♯ ǫ rǫ ♯ ǫ h ǫ • ♯ ǫ y ǫ φ ǫ β H + κǫ O(|α -β|) -∞ . (8.16)
This gives us (8.3) in the proposition. From Proposition 8.2 it follows that the symbol

h ǫ := h ǫ • + h ǫ • ♯ ǫ z ǫ + z ǫ ♯ ǫ h ǫ • ♯ ǫ y ǫ + y ǫ ♯ ǫ h ǫ • ♯ ǫ rǫ ♯ ǫ h ǫ • ♯ ǫ y ǫ (8.17)
is Γ-periodic in the X -variable. Thus, using the results in Proposition 8.1, Op ǫ (h ǫ ) commutes with all the operators {Λ ǫ (γ, •)τγ }γ∈Γ and we get (8.4), achieving the proof of the proposition.

Control of Σ 1 (α, β).

The main result here is the following lemma:

Lemma 8.6. We have

Σ1(α, β) = Λ ǫ,κ (α, β) φ ǫ α , H ǫ + Op ǫ h ǫ,κ • ♯ ǫ,κ z ǫ,κ + Op ǫ z ǫ,κ ♯ ǫ,κ h ǫ,κ • ♯ ǫ,κ y ǫ,κ φ ǫ β H + κǫ O(|α -β| -∞ ) .
The rest of this paragraph is dedicated to the proof of the above statement. We start by considering the scalar products appearing in the double series in the expression of Σ1(α, β) in (8.6).

H ǫ,κ α ′ β ′ := Y ǫ,κ -1/2 Λ ǫ,κ (•, α ′ )τ α ′ ψ ǫ 0 , H ǫ,κ Y ǫ,κ -1/2 Λ ǫ,κ (•, β ′ )τ -β ′ ψ ǫ 0 H = φ ǫ α ′ , Λ ǫ,κ (α ′ , •)H ǫ,κ Λ ǫ,κ (•, β ′ )φ ǫ β ′ H + S1(α ′ , β ′ ) + S2(α ′ , β ′ ) , (8.18)
with

S1(α ′ , β ′ ) := φ ǫ α ′ , Λ ǫ,κ (α ′ , •)H ǫ,κ Y ǫ,κ -1/2 -1 l Λ ǫ,κ (•, β ′ )φ ǫ β ′ H (8.19)
and

S2(α ′ , β ′ ) := φ ǫ α ′ , Λ ǫ,κ (α ′ , •) Y ǫ,κ -1/2 -1 l H ǫ,κ Y ǫ,κ -1/2 Λ ǫ,κ (•, β ′ )φ ǫ β ′ H . (8.20)
Lemma 8.7. Let m ∈ N. There exists Cm > 0 such that for all α ′ , β ′ ∈ Γ and 0 < ǫ, κ ≤ 1 we have

φ ǫ α ′ , Λ ǫ,κ (α ′ , •)H ǫ,κ Λ ǫ,κ (•, β ′ )φ ǫ β ′ H -Λ ǫ,κ (α ′ , β ′ ) φ ǫ α ′ , H ǫ φ ǫ β ′ H ≤ Cm κǫ < α ′ -β ′ > -m .
Proof. We use (5.7) for the pair of vector potentials (A ǫ,κ , A ǫ ), in the form

(-i∇ -A ǫ,κ (x)) 2 Λ ǫ,κ (x, β) = Λ ǫ,κ (x, β) -i∇ -A ǫ (x) + κaǫ(x, β)
Λ ǫ,κ (x, α) -1 Λ ǫ,κ (x, β) = Λ ǫ,κ ( α, β) Ω ǫ,κ ( α, x, β) , (8.23)
and we know from (5.2) and (5.4) that

| Ω ǫ,κ ( α, x, β) -1 l| ≤ C κǫ |x -α| |x -β| .
The fast decay of the quasi Wannier function (Proposition 3.3) allows us to finish the proof.

Lemma 8.8. Let us consider an operator of the form π ǫ,κ Op ǫ,κ (F )π ǫ,κ with F ∈ S - 1 (Ξ) (see (5.10)). Then for any m ∈ N, there exists a seminorm νm such that,

∀ (α ′ , β ′ ) ∈ Γ × Γ and ∀(ǫ, κ) ∈ [0, 1] × [0, 1], φ ǫ α ′ , Λ ǫ,κ (α ′ , •)π ǫ,κ Op ǫ,κ (F )π ǫ,κ Λ ǫ,κ (•, β ′ )φ ǫ β ′ H -Λ ǫ,κ (α ′ , β ′ ) φ ǫ α ′ , π ǫ Op ǫ (F )π ǫ φ ǫ β ′ H ≤ νm(F ) κǫ < α ′ -β ′ > -m .
Proof. Using Proposition B.8 in [START_REF] Cornean | Low lying spectral gaps induced by slowly varying magnetic fields[END_REF] we may conclude that for F ∈ S - 1 (Ξ) and for any k ∈ N there exists a seminorm ν k :

S - 1 (Ξ) → R+ such that sup x∈X X |x -y| k |KF (x, y) dy| ≤ ν k (F ) . (8.24)
Let us compute for u ∈ H:

Λ ǫ,κ (α ′ , •)Op ǫ,κ (F ) Λ ǫ,κ (•, β ′ )u (x) = Λ ǫ,κ (α ′ , •) Int ( Λ ǫ,κ Λ ǫ KF )( Λ ǫ,κ (•, β ′ )u) (x) = Λ ǫ,κ (α ′ , β ′ ) Ω ǫ,κ (α ′ , x, β ′ ) X dy Ω ǫ,κ (x, y, β ′ )Λ ǫ (x, y)KF (x, y)u(y) , (8.25)
and take into account the estimates (from (5.4))

Ω ǫ,κ (α ′ , x, β ′ ) -1 ≤ C κǫ |x -α ′ | |x -β ′ | , (8.26)
and

Ω ǫ,κ (x, x + z, β ′ ) -1 ≤ C κǫ |z| |x -β ′ | . (8.27)
Thus we obtain

φ ǫ α , Λ ǫ,κ (α, •)π ǫ,κ Op ǫ,κ (F )π ǫ,κ Λ ǫ,κ (•, β)φ ǫ β H -Λ ǫ,κ (α, β) φ ǫ α , π ǫ Op ǫ (F )π ǫ φ ǫ β H ≤ κǫ νm(F ) < α ′ -β ′ > m , (8.28
) for any m ∈ N, with some νm : S 0 0 (Ξ) → R+ a seminorm on S 0 0 (Ξ).

Lemma 8.9. For any m ∈ N there exists Cm > 0 such that

H ǫ,κ α ′ β ′ ≤ Cm < α ′ -β ′ > -m .
Proof. We have

H ǫ,κ α ′ β ′ = Y ǫ,κ -1/2 Λ ǫ,κ (•, α ′ )τ α ′ ψ ǫ 0 , H ǫ,κ Y ǫ,κ -1/2 Λ ǫ,κ (•, β ′ )τ -β ′ ψ ǫ 0 H = φ ǫ,κ α ′ , Op ǫ,κ y ǫ,κ ♯ ǫ,κ h• ♯ ǫ,κ y ǫ,κ φ ǫ,κ β ′ H . (8.29)
We notice that h• ♯ ǫ,κ y ǫ,κ ∈ S -∞ (Ξ) so that using once again Proposition B.8 from [START_REF] Cornean | Low lying spectral gaps induced by slowly varying magnetic fields[END_REF] as in the begining of the proof of Lemma 8.8 and the rapid decay of the magnetic distorted Wannier functions we conclude that for any m ∈ N there exist Cm > 0, C ′ m > 0 and a seminorm νm on S -∞ (Ξ), such that

H ǫ,κ α ′ β ′ ≤ C ′ m νm y ǫ,κ ♯ ǫ,κ h• ♯ ǫ,κ y ǫ,κ X X dx dy < x -α ′ > -m-3 < x -y > -m < y -β ′ > -m-3 ≤ Cm < α ′ -β ′ > -m .
Remark 8.10. Let us notice that

α ′ ∈Γ β ′ ∈Γ F ǫ,κ α ′ α F ǫ,κ β ′ β H ǫ,κ α ′ ,β ′ = H ǫ,κ α,β + α ′ ∈Γ F ǫ,κ -1 l α ′ α H ǫ,κ α ′ ,β + α ′ ∈Γ β ′ ∈Γ F ǫ,κ α ′ α F ǫ,κ -1 l β ′ β H ǫ,κ α ′ ,β ′
(8.30) and due to Proposition 6.2 (point 3) we deduce that for any (ǫ, κ) ∈ [0, ǫ0] × [0, 1] for some small enough ǫ0 > 0 and for any (α, β) ∈ Γ × Γ we have that for any m ∈ N there exists some Cm > 0 such that

α ′ ∈Γ β ′ ∈Γ F ǫ,κ α ′ α F ǫ,κ β ′ β H ǫ,κ α ′ ,β ′ -H ǫ,κ α,β ≤ Cm κǫ < α -β > -m . (8.31)
Coming to the last two terms (8.19) and (8.20) we notice that they may be written as:

S1 = φ ǫ α ′ , Λ ǫ,κ (α ′ , •)π ǫ,κ Op ǫ,κ h ǫ,κ • ♯ ǫ,κ z ǫ,κ π ǫ,κ Λ ǫ,κ (•, β ′ )φ ǫ β ′ H , (8.32)
and

S2 = φ ǫ α ′ , Λ ǫ,κ (α ′ , •)π ǫ,κ Op ǫ,κ z ǫ,κ ♯ ǫ,κ h ǫ,κ • ♯ ǫ,κ y ǫ,κ π ǫ,κ Λ ǫ,κ (•, β ′ )φ ǫ β ′ H . (8.33)
Thus we notice that h ǫ,κ • ♯ ǫ,κ z ǫ,κ and z ǫ,κ ♯ ǫ,κ h ǫ,κ • ♯ ǫ,κ y ǫ,κ are symbols of class S -∞ (Ξ) and we may use Lemma 8.8 in order to obtain that:

φ ǫ α ′ , Λ ǫ,κ (α ′ , •)H ǫ,κ Y ǫ,κ -1/2 -1 l Λ ǫ,κ (•, β ′ )φ ǫ β ′ H -Λ ǫ,κ (α ′ , β ′ ) φ ǫ α ′ , Op ǫ h ǫ,κ • ♯ ǫ,κ z ǫ,κ φ ǫ β ′ H ≤ Cm κǫ < α ′ -β ′ > -m , and
φ ǫ α ′ , Λ ǫ,κ (α ′ , •) Y ǫ,κ -1/2 -1 l H ǫ,κ Y ǫ,κ -1/2 Λ ǫ,κ (•, β ′ )φ ǫ β ′ H -φ ǫ α ′ , Op ǫ z ǫ,κ ♯ ǫ,κ h ǫ,κ • ♯ ǫ,κ y ǫ,κ φ ǫ β ′ H ≤ Cm κǫ < α ′ -β ′ > -m .

Control of Σ 2 (α, β).

Going back to the double series appearing in (8.7) we write

R ǫ,κ α ′ β ′ := = Y ǫ,κ -1/2 Λ ǫ,κ (•, α ′ )τ α ′ ψ ǫ 0 , π ǫ,κ H ǫ,κ R ǫ,κ ⊥ (0)H ǫ,κ π ǫ,κ Y ǫ,κ -1/2 Λ ǫ,κ (•, β ′ )τ -β ′ ψ ǫ 0 H = φ ǫ α ′ , Λ ǫ,κ (α ′ , •) Y ǫ,κ -1/2 π ǫ,κ H ǫ,κ R ǫ,κ ⊥ (0)H ǫ,κ π ǫ,κ Y ǫ,κ -1/2 Λ ǫ,κ (•, β ′ )φ ǫ β ′ H = φ ǫ α ′ , Λ ǫ,κ (α ′ , •)Op ǫ,κ y ǫ,κ ♯ ǫ,κ h ǫ,κ • ♯ ǫ,κ rǫ,κ ♯ ǫ,κ h ǫ,κ • ♯ ǫ,κ y ǫ,κ Λ ǫ,κ (•, β ′ )φ ǫ β ′ H , (8.34)
and notice that

α ′ ∈Γ β ′ ∈Γ F α ′ α F β ′ β R ǫ,κ α ′ β ′ = R ǫ,κ α,β + α ′ ∈Γ F ǫ,κ -1 l α ′ α R ǫ,κ α ′ ,β + α ′ ∈Γ β ′ ∈Γ F ǫ,κ α ′ α F ǫ,κ -1 l β ′ β R ǫ,κ α ′ ,β ′ .
Noticing further that h ǫ,κ

• := p ǫ,κ ♯ ǫ,κ h ♯ ǫ,κ (1 -p ǫ,κ ) ∈ S -∞ (
Ξ) and using again Lemma 8.8 we obtain that for any m ∈ N there exists Cm > 0 such that

R ǫ,κ α ′ β ′ -Λ ǫ,κ (α ′ , β ′ ) φ ǫ α ′ , Op ǫ y ǫ,κ ♯ ǫ,κ h ǫ,κ • ♯ ǫ,κ rǫ,κ ♯ ǫ,κ h ǫ,κ • ♯ ǫ,κ y ǫ,κ φ ǫ β ′ H ≤ Cmκǫ < α ′ -β ′ > -m . (8.35)
Moreover the proof of Lemma 8.9 applies in this situation also and we obtain that for any m ∈ N there exists Cm > 0 such that

R ǫ,κ α ′ β ′ ≤ Cm < α ′ -β ′ > -m . (8.36)
Taking into account all these results ((8.34)-(8.36)) and point 3 of Proposition 6.2 we obtain the following result.

Lemma 8.11. We have

Σ2(α, β) = Λ ǫ,κ (α, β) φ ǫ α , Op ǫ y ǫ,κ ♯ ǫ,κ h ǫ,κ • ♯ ǫ,κ rǫ,κ ♯ ǫ,κ h ǫ,κ • ♯ ǫ,κ y ǫ,κ φ ǫ β H +κǫ O(|α-β| -∞ ) .
8.3 The modified energy band and proof of Theorem 2.2 (i)

Having in mind the result of Proposition 8.5 let us introduce

k ǫ : Γ → C, k ǫ (γ) := φ ǫ α , Op ǫ (h ǫ )φ ǫ β H ; λ -ǫ (θ) := γ∈Γ e -i<θ,γ> k ǫ (γ). (8.37)
Proceeding as in [START_REF] Cornean | Spectral edge regularity of magnetic Hamiltonians[END_REF] and [START_REF] Cornean | Low lying spectral gaps induced by slowly varying magnetic fields[END_REF] we define the magnetic matrix acting in ℓ 2 (Γ):

M ǫ,κ (α, β) := φ ǫ,κ α , Op ǫ (h ǫ )φ ǫ,κ β H = Λ ǫ,κ (α, β)k ǫ (α -β) . ( 8

.38)

Let us consider the smooth function λ -ǫ : T * → R as a periodic smooth function on Ξ constant along the directions in X × {0} and write its magnetic quantization as an integral operator Op ǫ,κ (λ

-ǫ ) := Int Λ ǫ,κ K λ -ǫ , (8.39) with K λ -ǫ (x, y) = (2π) -2 X * e i<ξ,x-y> λ -ǫ (ξ)dξ = γ∈Γ k ǫ (γ)δ0(x -y -γ) . ( 8

.40)

Let us define the following unitary map

WΓ : L 2 (X ) → ℓ 2 (Γ) ⊗ L 2 (E), WΓF (α, x) := F (α + x) (8.41)
and compute the integral kernel

K of WΓ (Op ǫ,κ (λ -ǫ )) W -1 Γ in this representation K(α + x, β + y) = Λ ǫ,κ α + x, β + x k ǫ (α -β)δ0(x -y) . (8.42)
In order to compare it with (8.38) we shall consider two unitary gauge transformations:

Uǫ : ℓ 2 (Γ) ⊗ L 2 (E) → ℓ 2 (Γ) ⊗ L 2 (E), UǫΦ (α, x) := Λ ǫ (x, α)Φ(α, x) Uǫ,κ : ℓ 2 (Γ) ⊗ L 2 (E) → ℓ 2 (Γ) ⊗ L 2 (E), Uǫ,κΦ (α, x) := Λ ǫ,κ (α, α + x)Φ(α, x).
We notice that, using also (5.3), the kernel K of Uǫ,κUǫWΓ (Op ǫ,κ (λ -ǫ )) Uǫ,κUǫWΓ -1 is given by:

K (α, x); (β, y) = Λ ǫ,κ (α, α + x)Λ ǫ (x, α)Λ ǫ,κ (α + x, β + x)Λ ǫ (β, x) Λ ǫ,κ (β + x, β)k ǫ (α -β)δ0(x -y) = Λ ǫ,κ (α, β) Ω ǫ,κ (α, α + x, β + x) Ω ǫ,κ (α, β + x, β)k ǫ (α -β)δ0(x -y) .
Proposition 8.12. The Hausdorff distance between the spectra of the operator Op ǫ,κ (λ -ǫ ) in L H and the hermitian operator associated with the matrix M ǫ,κ (α, β) in the orthonormal basis {φ ǫ,κ γ }γ∈Γ of π ǫ,κ H is of order κǫ.

Proof. If we denote by x := α + x and x ′ := β + x and use (5.4) we obtain that

1 -Ω ǫ,κ (α, x, x ′ ) ≤ C κǫ |α -β| |x| ≤ C1 |α -β| κǫ, ( 8
(λ -ǫ )) Uǫ,κUǫWΓ -1 - (α,β)∈Γ×Γ M ǫ,κ (α, β) |eα e β | ⊗ 1 l L 2 (E) ≤ Cκǫ.
On the other hand it is evident that the two operators:

(α,β)∈Γ×Γ M ǫ,κ (α, β) |eα e β | ⊗ 1 l L 2 (E) ; H ǫ,κ = (α,β)∈Γ×Γ M ǫ,κ (α, β)|φ ǫ,κ α φ ǫ,κ β |
have the same spectrum.

Putting together the Propositions 8.5, 8.12 and Proposition 3.19 in [START_REF] Cornean | Low lying spectral gaps induced by slowly varying magnetic fields[END_REF] we obtain the third conclusion of our Theorem 2.2.

Behavior of the modified energy band function in the chosen energy window and proof of Theorem 2.2 (ii)

We will only prove Theorem 2.2 (ii) in the case when m = 0; the general case is similar due to the rapid decay of the quasi Wannier function ψ0. Proof. Let us also define the smooth function

λ -ǫ • (θ) := γ∈Γ e -i<θ,γ> φ ǫ γ , Op ǫ (h ǫ • )φ ǫ 0 H (8.45)
and estimate the above difference by a two step procedure:

λ -ǫ (θ) -λ0(θ) = λ -ǫ (θ) -λ -ǫ • (θ) + λ -ǫ • (θ) -λ0(θ) . ( 8

.46)

Step 1. We begin by computing

λ -ǫ • (θ) -λ0(θ) = γ∈Γ e -i<θ,γ> φ ǫ γ , Op ǫ (h ǫ • )φ ǫ 0 H -λ0(θ).
Using Proposition 8.1 we obtain

φ ǫ γ , Op ǫ (h ǫ • )φ ǫ 0 H = φ ǫ γ , Op ǫ (h)φ ǫ 0 H = Λ ǫ (α, β) φ ǫ α-β , Op ǫ (h)φ ǫ 0 H .
Taking also into account that we have fixed the gauge for A0 as in (1.6), we conclude that

φ ǫ γ , Op ǫ (h)φ ǫ 0 H = ψγ , H 0 ψ0 H + ǫ φ ǫ γ , D • A0(•) + A0(•) • D + ǫA0(•) 2 ψ ǫ 0 H + τγ(ψ ǫ 0 -ψ0) , H 0 ψ ǫ 0 H + τγ ψ ǫ 0 , H 0 ψ ǫ 0 -ψ0 H + τγ ψ ǫ 0 , Ω ǫ (γ, 0, •) -1 (D + ǫa0(•, 0)) 2 + V (•) ψ ǫ 0 H . (8.47)
We recall from Definition 6.1 that, for any γ ∈ Γ,

φ ǫ γ = Λ ǫ (•, γ)τγ β∈Γ Ω ǫ (β, 0, •)F ǫ (β)τ β ψ0 .
Taking into account the rapid decay of F (see Proposition 6.2) and of ψ0, we conclude that for any m ∈ N there exists Cm > 0 such that for any ǫ ∈ [0, ǫ0]:

φ ǫ γ , D • A0(•) + A0(•) • D + ǫA0(•) 2 ψ ǫ 0 H ≤ Cm < γ > -m , ∀γ ∈ Γ,
and

τγ ψ ǫ 0 , Ω ǫ (γ, 0, •) -1 (D + ǫA0(•)) 2 + V (•) ψ ǫ 0 H ≤ Cm ǫ < γ > -m , ∀γ ∈ Γ.
Using Definition 6.1 and point (2) in Proposition 6.2 we get also that for any m ∈ N there exists Cm > 0 such that for any ǫ ∈ [0, ǫ0]: τγ(ψ ǫ 0 -ψ0) , H 0 ψ ǫ 0 H + τγ ψ ǫ 0 , H 0 ψ ǫ 0 -ψ0 H ≤ Cm ǫ < γ > -m , ∀γ ∈ Γ.

We have thus obtained: Step 2. We have to study the difference

λ -ǫ (θ) -λ -ǫ • (θ) = γ∈Γ e -i<θ,γ> φ ǫ γ , Op ǫ (h ǫ )φ ǫ 0 H -φ ǫ γ , Op ǫ (h ǫ • )φ ǫ 0 H .
Thus let us analyse the symbol:

h ǫ -h ǫ • = h ǫ • ♯ ǫ z ǫ + z ǫ ♯ ǫ h ǫ • + z ǫ ♯ ǫ h ǫ • ♯ ǫ z ǫ + y ǫ ♯ ǫ h ǫ ⊥ ♯ ǫ rǫ ♯ ǫ h ǫ ⊥ ♯ ǫ y ǫ ,
where y ǫ and z ǫ are defined in the last item in Definition 8.3. Let h• ∈ S -∞ (Ξ) be the symbol of the bounded operator

πH 0 π = U -1 Γ ⊕ T *
π(θ) Ĥ0 (θ)π(θ) dθ UΓ , h• ∈ S -∞ (Ξ) be the symbol of the operator H• := πH 0 π ⊥ , and h * • ∈ S -∞ (Ξ) be the symbol of its adjoint. Moreover, using (7.4) and then Proposition 7.1, we denote by R ⊥ the inverse of π ⊥ H 0 π ⊥ as operator acting in π ⊥ H and let r ∈ S -m 1 (Ξ) be its symbol. Using Proposition 8.1 and Proposition 6.2 and the fast decay of the modified Wannier function we can prove: Here we also used that Λ ǫ (γ, 0) = 1 for any γ ∈ Γ.

φ ǫ γ , Op ǫ (h ǫ • ♯ ǫ z ǫ + z ǫ ♯ ǫ h ǫ • ) + z ǫ ♯ ǫ h ǫ • ♯ ǫ z ǫ + y ǫ ♯ ǫ h ǫ • ♯ ǫ rǫ ♯ ǫ (h ǫ • ) * ♯ ǫ y ǫ φ ǫ
The operator Z has the following form:

Z = H 0 Y -1/2 -1 l + Y -1/2 -1 l H 0 + Y -1/2 -1 l H 0 Y -1/2 -1 l +Y -1/2 πH 0 π ⊥ R ⊥ π ⊥ H 0 πY -1/2 , (8.50)
and recall from (4.3) that Y := πH 0 π ⊥ R 2 ⊥ π ⊥ H 0 π . All the operators appearing in (8.50) have evidently Γ-periodic symbols and thus also Z and Y . More precisely we have the following direct integral decomposition: First, an application of Proposition 8.12 and Proposition 4.5 with η = ǫ shows that the spectrum of H ǫ,κ must have gaps of order ǫ in the interval [0, N ǫ], provided the same is true for Op ǫ,κ ( λ -ǫ ). Second, one has to perform the spectral analysis of Op ǫ,κ ( λ -ǫ ) applying the results in [START_REF] Cornean | Low lying spectral gaps induced by slowly varying magnetic fields[END_REF] and prove the existence of gaps of order ǫ independent of κ in its spectrum restricted to the interval [0, N ǫ], provided κ and ǫ are small enough.

Y = πH 0 π ⊥ R 2 ⊥ π ⊥ H 0 π ≡ H•R 2 ⊥ H• = U -1 Γ ⊕ T * dθ Ŷ (θ) UΓ,
• Assume B Γ = 0. Then we notice that the function λ -ǫ : T * → R has exactly the same properties as the function λ ǫ : T * → R defined in Definition 3.18 in [START_REF] Cornean | Low lying spectral gaps induced by slowly varying magnetic fields[END_REF] with the only difference that the estimate |∂ α λ ǫ (θ) -∂ α λ0(θ)| ≤ Cmǫ which was valid for all θ ∈ T * (as stated in Proposition 4.1 in [START_REF] Cornean | Low lying spectral gaps induced by slowly varying magnetic fields[END_REF]) is now only valid on Σ b . Nevertheless, if we choose an upper bound δ0 for the cut-off parameter δ > 0 in Subsection4.3 (Paragraph 'Cut-off functions near the minimum') in [START_REF] Cornean | Low lying spectral gaps induced by slowly varying magnetic fields[END_REF] such that {θ ∈ E * | |θ| ≤ 2m -1 1 δ0} ⊂ Σ b , all the results of Section 4 of [START_REF] Cornean | Low lying spectral gaps induced by slowly varying magnetic fields[END_REF] remain true for our function λ -ǫ : T * → R and Corollary 2.3 follows.

• Assume B Γ = 0. Then the situation is rather similar, but the function λ0 : T * → R may no longer be symmetric around its local minimum θ0. In this case, a third order term may appear in the Taylor expansion (4.1) in [START_REF] Cornean | Low lying spectral gaps induced by slowly varying magnetic fields[END_REF] and thus formula (4.3) in [START_REF] Cornean | Low lying spectral gaps induced by slowly varying magnetic fields[END_REF] becomes

λ ǫ (θ) -λ ǫ (θ ǫ ) = 1≤j,k≤2
a ǫ jk (θjθ ǫ j )(θ kθ ǫ k ) + O(|θθ ǫ | 3 ). (8.51)

As a consequence, the exponent µ > 0 relating the cut-off parameter δ > 0 with the intensity of the magnetic field ǫ > 0 through condition (4.20) in [START_REF] Cornean | Low lying spectral gaps induced by slowly varying magnetic fields[END_REF] may vary only in the interval (2, 3) in R. A 'symmetric choice', similar to the one in [START_REF] Cornean | Low lying spectral gaps induced by slowly varying magnetic fields[END_REF] is thus µ = 2.5 and the only effect of this new choice is that in the second formula (4.28) in Proposition 4.5 in [START_REF] Cornean | Low lying spectral gaps induced by slowly varying magnetic fields[END_REF] we now have the estimate Op ǫ,κ (r δ,a ) ≤ C ǫ 1/5 .

This ends the proof of Corollary 2.4.

Proposition 3 . 4 .

 34 With H 0 introduced in (1.4) and π in (3.6), we have πH ⊂ D(H 0 ), H 0 π ∈ L(H) and πH 0 has a bounded closure.

Corollary 4 . 4 .

 44 Let 0 < D1 < D2 < β ′ < β and assume (D1, D2) ∩ σ( H(η)) = ∅ , for all η ∈ [0, ǫ1] . Then, if

Definition 5 . 2 .

 52 A symbol F in S s (Ξ) is called elliptic if there exist two positive constants R and C such that |F (x, ξ)| ≥ C ξ s ,for any (x, ξ) ∈ Ξ with |ξ| ≥ R .

2 , 1 0

 21 writing:aǫ(x, z)j = k (xγ) k ǫB jk ǫγ + sǫ(xγ) s ds for j = 1, 2 ,(8.21) and noticing that |aǫ(x, γ)| ≤ C ǫ < xγ > . (8.22) Using (5.3) we have

.43) and 1 -

 1 Ω ǫ,κ (α, x ′ , β) ≤ Cκǫ|α -β| |x| ≤ C2 |α -β| κǫ. (8.44) Taking into account the rapid decay of k ǫ (γ) for |γ| → ∞ and considering the canonical orthonormal basis {eγ }γ∈Γ of ℓ 2 (Γ) defined by eγ (α) := δγ,α we obtain Uǫ,κUǫWΓ (Op ǫ,κ

Proposition 8 . 13 .

 813 For b ∈ (0, b) with b as in Lemma 3.1 there exists ǫ0 > 0 and C > 0 such that, for any θ ∈ Σ b and any ǫ ∈ [0, ǫ0], |λ -ǫ (θ) -λ0(θ)| ≤ C ǫ.

2 ,

 2 ) -λ0(θ) = γ∈Γ e -i<θ,γ> ψγ , H 0 ψ0 H -λ0(θ) + O(ǫ) . Let us now use the Bloch-Floquet representation and the properties of ψ0 as constructed in Section 3 in order to write: ψγ , H 0 ψ0 H = T * dω e i<ω,γ> ψ0(ω) , n≥0 λn(ω)| φn(ω) φn(ω)| ψ0(ω) Fω = n∈N T * dω e i<ω,γ> λn(ω) ψ0(ω) , φn(ω) Fω and we notice that for ω ∈ Σ b we have ψ0(ω) = φ0(ω). Finally we can write: γ∈Γ e -i<θ,γ> ψγ , H 0 ψ0 H = n∈N λn(θ) ψ0(θ) , φn(θ) ,γ> ψγ , H 0 ψ0 H = λ0(θ) for θ ∈ Σ b .

0 H=

 0 τγψ0 , Zψ0 H + ǫ O(< γ > -m ) , (8.48) with Z := Op (h• ♯ z + z ♯ h•) + z ♯ h• ♯ z + y ♯ h• ♯ r ♯ h * • ♯ y . (8.49)

  with Ŷ (θ) := Ĥ•(θ) R⊥ (θ) 2 Ĥ•(θ) * , and Ĥ•(θ) maps π⊥ (θ)F θ into π(θ)F θ (see(3.5) for the definition of π(θ)) and is defined byĤ•(θ) = π(θ) n∈N λn(θ)| φn(θ) φn(θ)| π(θ) ⊥ . Thus Y = πH 0 π ⊥ R 2 ⊥ π ⊥ H 0 π = U -1 Γ ⊕ T * dθ Ŷ (θ)| ψ0(θ) ψ0(θ)| , UΓ with Ŷ ∈ C ∞ (T * ; R+).Taking into account the definitions and arguments in Section 3, we obtain that, for θ ∈ Σ b ,Ĥ•(θ) = | φ0(θ) φ0(θ)| n∈N λn(θ)| φn(θ) φn(θ)| 1 l -| φ0(θ) φ0(θ)| = 0 ,so that we conclude that Ŷ (θ) = 0 for θ ∈ Σ b . Now the operator Z in (8.49) is also Γ-decomposable andZ = U -1 Γ ⊕ T * dθ Ẑ(θ) UΓ,where Ẑ ∈ C ∞ (T * ) satisfies Ẑ(θ) = 0, ∀θ ∈ Σ b . Finally, we obtain thatλ -ǫ (θ)λ -ǫ • (θ) = γ∈Γ e -i<θ,γ> φ ǫ γ , Op ǫ (k ǫh ǫ • )φ ǫ 0 H = γ∈Γ e -i<θ,γ> τγ ψ0, Zψ0 H + O(ǫ) = Ẑ(θ) + O(ǫ) = O(ǫ), ∀θ ∈ Σ b ,which ends the proof of the proposition and of Theorem 2.2 (ii).

8. 5

 5 Proof of Corollaries 2.3 and 2.4

Acknowledgements. Two of the authors (BH and RP) had an extended stay at the Department of Mathematical Sciences, Aalborg University, period during which this work had been finalized. The financial support is gratefully acknowledged.