A semiclassical Birkhoff normal form for symplectic magnetic wells

In this paper we construct a Birkhoff normal form for a semiclassical magnetic Schr{\"o}dinger operator with non-degenerate magnetic field, and discrete magnetic well, defined on an even dimensional riemannian manifold M. We use this normal form to get an expansion of the first eigenvalues in powers of h^{1/2}, and semiclassical Weyl asymptotics for this operator.


Introduction
The analysis of the magnetic Schrödinger operator, or magnetic Laplacian, on a Riemannian manifold L = (i d + A) * (i d + A) in the semiclassical limit → 0 has given rise to many investigations in the last twenty years.Asymptotic expansions of the lowest eigenvalues have been studied in many cases involving the geometry of the possible boundary of M and the variations of the magnetic field.For discussions about the subject, the reader is referred to the books and review [7], [8], [18].The classical picture associated with the Hamiltonian |p − A(q)| 2 has started being investigated to describe the semiclassical bound states (the eigenfunctions of low energy) of L , in [19] (on R 2 ) and [10] (on R 3 ).In these two papers, semiclassical Birkhoff normal forms were used to describe the first eigenvalues.In [20], Sjöstrand introduced the semiclassical Birkhoff normal form to study the spectrum of an electric Schrödinger operator, and some resonance phenomenons appeared.In [4], the resonant case for the same electric Schrödinger operator was tackled (see also [21] and [22]).In this paper, we adapt this method to L , following the ideas of [19].Some normal forms for magnetic Schrödinger operators also appear in [12].On a Riemannian manifold M, the magnetic Schrödinger operator is related to the Bochner Laplacian (see the recent papers [14] and [15], where bounds and asymptotic expansions of the first eigenvalues of Bochner Laplacians are given).
In this paper we get an expansion of the first eigenvalues of L in powers of 1/2 , and semiclassical Weyl asymptotics.It would be interesting to have a precise description of the eigenfunctions too, as was done in the 2D case by Bonthonneau-Raymond [3] (euclidian case) and Nguyen Duc Tho [17] (general riemannian metric).Moreover, we only have investigated the spectral theory of the stationary Schrödinger equation with a pure magnetic field ; it would be interesting to describe the longtime dynamics of the full Schrödinger evolution, as was done in the euclidian 2D case by Boil-Vu Ngoc [2].
1.1.Definition of the magnetic Schrödinger operator.Let (M, g) be a smooth d dimensional oriented Riemannian manifold, either without boundary or with smooth boundary.In particular we can take M = R d with the Euclidean metric, or M compact with boundary.For q ∈ M, g q is a scalar product on T q M. Since M is oriented, there is a canonical volume form, denoted either dx g or dq g .If f ∈ L 2 (M), we denote its norm by .
If p ∈ T q M * , we denote by |p| g ⋆ q or |p| the norm of p, defined by We denote by g * q the associated scalar product.The norm of a 1-form α on M is .
It is associated with a scalar product, denoted by brackets ., . .We denote by d the exterior derivative, associating to any p-form α a (p + 1)-form dα.Using the scalar products induced by the metric, we can define its adjoint d * , associating to any p-form α a (p − 1)-form d * α.
We take a 1-form A on M called the magnetic potential, and we denote by B = dA its exterior derivative.B is called the magnetic 2-form.The associated classical Hamiltonian is defined on T * M by: H(q, p) = |p − A(q)| 2 g * q , p ∈ T q M * .Using the isomorphism T q M ≃ T q M * given by the metric, we define the magnetic operator B(q) : T q M → T q M by: The norm of B(q) is |B(q)| = [Tr(B * (q)B(q))] 1/2 .
On the quantum side, for > 0, we define the magnetic quadratic form q on where L 2 Ω 1 (M) denotes the space of square-integrable 1-forms on M. By the Lax-Milgram theorem, this quadratic form defines a self-adjoint operator L on by the formula L u, v = q [u, v], ∀u, v ∈ C ∞ 0 (M), where q [., .] is the inner product associated with the quadratic form q (.).L is the magnetic Schrödinger operator with Dirichlet boundary conditions.1.2.Local coordinates.If we choose local coordinates q = (q 1 , ..., q d ) on M, we get the corresponding vector fields basis (∂ q 1 , ..., ∂ q d ) on T q M, and the dual basis (dq 1 , ..., dq d ) on T q M * .In these basis, g q can be identified with a symmetric matrix (g ij (q)) with determinant |g|, and g * q is associated with the inverse matrix (g ij (q)).We can write the 1-form A in the coordinates: the linear operator whose matrix is the Jacobian of A: In the coordinates, the 2-form B is Let us denote (B ij (q)) 1≤i,j≤d the matrix of the operator B(q) : T q M → T q M in the basis (∂q 1 , ..., ∂q d ).With this notation, equation (1.2) relating B to B can be rewritten: Also note that: Finally, in the coordinates H is given by: and L acts as the differential operator: (1.8) 1.3.Pseudodifferential operators.We refer to [16] and [24] for the general theory of -pseudodifferential operators.If m ∈ Z, we denote by If a depends on the semiclassical parameter , we require that the coefficients C αβ are uniform with respect to ∈ (0, 0 ].For a ∈ S m (R 2n ), we define its associated Weyl quantization Op w (a ) by the oscillatory integral and we denote: a = σ (A ).A pseudodifferential operator A on L 2 (M) is an operator acting as a pseudodifferential operator in coordinates.Then the principal symbol of A does not depend on the coordinates, and we denote it by σ 0 (A ).The subprincipal symbol σ 1 (A ) is also well-defined, up to imposing the charts to be volume-preserving (in other words, if we see A as acting on half-densities, its subprincipal symbol is well defined).
In any local coordinates, the coefficients A j of A (as a function of q ∈ R d ) are in S 1 (R 2d (q,p) ).Hence we see from (1.8) that L is a pseudodifferential operator on L 2 (M).Its principal and subprincipal Weyl symbols are: This is well-known, but we detail the computation of the subprincipal symbol in Appendix (Lemma A.1).
1.4.Assumptions.Since B(q), defined in (1.2), is a skew-symmetric operator for the scalar product g q , its eigenvalues are in iR.We define the magnetic intensity, which is equivalent to the trace-norm, by It is a continuous function of q, but not smooth in general.We also denote We first assume that the magnetic field satisfies the following inequality.
Assumption 1.We assume that there exist 0 > 0 and C 0 > 0 such that, for ∈ (0, 0 ], In the Appendix (Lemma A.4), we describe cases when Assumption 1 holds.In particular, it holds if for some C > 0. These results are adapted from [9].
We consider the case of a unique discrete magnetic well: Assumption 2. We assume that the magnetic intensity b admits a unique and nondegenerate minimum b 0 at q 0 ∈ M \ ∂M, such that 0 < b 0 < b ∞ .
Finally, we make a non-degeneracy assumption.Assumption 3. We assume that d is even and B(q 0 ) is invertible.
In particular, B(q) is invertible for q in a neighborhood of q 0 , which means that the 2-form B is symplectic near q 0 .Under this Assumption, the eigenvalues of B(q 0 ) can be written ±iβ 1 (q 0 ), . . ., ±iβ d/2 (q 0 ), with β j (q 0 ) > 0. We define the resonance order r 0 ∈ N * ∪ {∞} of the eigenvalues by We make a non-resonance assumption.Assumption 4. We assume that the eigenvalues of B(q 0 ) are simple (which is equivalent to assuming that r 0 ≥ 3).
In particular, there is a neighborhood Ω ⊂⊂ M \∂M of q 0 on which the eigenvalues of B(q) are simple, and defined by smooth positive functions We can choose Ω such that every β j is bounded from bellow by a positive constant on Ω.We can also find smooth orthonormal vectors on Ω: u 1 (q), v 1 (q), . . ., u d/2 (q), v d/2 (q) ∈ T q M, such that: B(q)u j (q) = −β j (q)v j (q), B(q)v j (q) = β j (q)u j (q).(1.10)We take Up to reducing Ω (depending on r), we also have (since r is finite), for 0 < |α| < r: α, β(q) = 0, ∀q ∈ Ω.
(1.12)Under Assumption 2, we can find b 0 < b1 < b ∞ such that Using the inequality in Assumption 1, it is proved in [9] that there exist 0 and c > 0 such that, for ∈ (0, 0 ], and so, for small enough, the spectrum of L below b 1 (for a given b 1 < b1 ) is discrete.
Our next aim is to construct a semiclassical Birkhoff normal form for L , that is to say a pseudodifferential operator N on L 2 (R d ), commuting with suitable harmonic oscillators such that: ) a microlocally unitary Fourier integral operator and R a remainder.We will contruct the remainder so that the first eigenvalues of L coincide with the first eigenvalues of N , up to a small error of order O( r/2−ε ), where r is defined in (1.11).More precisely, we prove the following theorem.the canonical variables.For ζ > 0 and ∈ (0, 0 ] small enough, there exist a Fourier integral operator , a smooth function f ⋆ (w, I 1 , ..., I d/2 , ), and a pseudodifferential operator R on R d such that: , ..., ) on a neighborhood of w = 0, (iv) U * U = I microlocally near (z, w) = 0, (v) U U * = I microlocally near (q, p) = (q 0 , A q 0 ), with , ..., I (d/2) , ) the normal form, and R the remainder.
Using microlocalization properties of the eigenfunctions of L and N , we prove that they have the same spectra in the following sense.We recall that b1 , defined in (1.13), is chosen such that {b(q) ≤ b1 } ⊂ Ω.
Finally, we deduce an expansion of the N > 0 first eigenvalues of L in powers of 1/2 .Theorem 1.5 (Expansion of the first eigenvalues).Let ε > 0 and N ≥ 1.There exist 0 > 0 and c 0 > 0 such that, for ∈ (0, 0 ], the N first eigenvalues of L : (λ j ( )) 1≤j≤N admit an expansion in powers of 1/2 of the form: where E j is the j-th eigenvalue of the d/2-dimensional harmonic oscillator Op w (Hess 0 (b • ϕ −1 )).
Note that, from Theorems 1.3 and 1.4, we deduce Weyl estimates for L .Some similar formulas appear in [12].Here N(L , b 1 ) denotes the number of eigenvalues λ of L such that λ ≤ b 1 , counted with multiplicities.

Corollary 1.1 (Weyl estimates). For any b
where The sum is finite because the β j are bounded from below by a positive constant on 1.6.Organization and strategy.In section 2, we construct a symplectomorphism which simplify H near its zero set Σ = H −1 (0) (Theorem 1.1).In the new coordinates, H becomes: In section 3, we construct a formal Birkhoff normal form: in the space of formal series in variables (x, ξ, ), we change Ĥ into ), and ρ a remainder of order r (Theorem 3.1).In section 4, we quantify the changes of coordinates constructed in section 2 and 3, and we get the semiclassical Birkhoff normal form (Theorem 1.2).In section 5, we reduce N (Theorem 1.4) and we deduce an expansion of its first eigenvalues.It remains prove that the spectra of L and N below b 1 coincide.Before doing it, we need microlocalization results proved in section 6.We prove that the eigenfunctions of L and N are microlocalized near the zero set of H, where our formal construction is valid.In section 7, we use the results of section 6, to prove that L and N have the same spectrum below b 1 (Theorem 1.3).This Theorem, together with the results of section 5, finishes the proof of Theorem 1.5.We also prove the Weyl estimates (Corollary 1.1) here.Finally, in section 8 we discuss what we can get in the case r 0 = ∞.

2.1.
A symplectic reduction of T * M. The zero set of H: is a d-dimensional smooth submanifold of the cotangent bundle T * M. We denote j : Ω → T * M the embedding j(q) = (q, A(q)).The symplectic structure on T * M is defined by the form In other words, for p ∈ T q M * and V ∈ T (q,p) (T * M), Where the map π * : T (q,p) (T * M) → T q M is the differential of the canonical projection Using local coordinates with the notations of section 1.2, at any point (q, p) ∈ T * M with p = p 1 dq 1 + ... + p d dq d , the tangent vectors V ∈ T (q,p) (T * M) are identified with (Q, P ) ∈ T q M × T q M * , with where ., .denotes the duality bracket between T q M and T q M * .
Proof.To say that Σ is a symplectic submanifold of T * M means that the restriction of ω to Σ is non-degenerate.Written with the embedding j, this restriction is j * ω.Actually, using the definition (2.1) of α with p = A q and V = d q j(Q), we get Since any j(q) is a critical point of H, the Hessian of H at j(q) is well defined and independant of any choice of coordinates.We now compute this Hessian according to the decomposition (2.2): Lemma 2.2.The Hessian T 2 j(q) H, as a bilinear form on T j(q) (T * M), can be written: Proof.Using local coordinates on M, we will denote every V ∈ T (q,p) (T * M), as (Q, P ) ∈ T q M × T q M * .In these coordinates, with the notations introduced in section 1.2, Σ ≡ {(q, A(q)), q ∈ R d } so that We can also describe T j(q) Σ ⊥ using these coordinates.Indeed, From the expression (1.7) of H in coordinates, we deduce that: so that the Hessian of H in coordinates is: H((Q, P ), (Q, P )) = 0, and from (2.4) and (1.5) that Let us rewrite this using B. Note that: and keeping in mind that (g ij ) is the inverse matrix of (g ij ) together with the relation (1.4) between B and B, we have and so We endow Ω × R d z with the symplectic form: with the notation z = (x, ξ).(Σ, B) is a d-dimensional symplectic submanifold of (T * M, ω).The following Darboux-Weinstein lemma claims that this situation is modelled on the submanifold Σ 0 = Ω × {0} of (Ω × R d z , ω 0 ).Lemma 2.3.There exists a local diffeomorphism In order to keep track on the construction of Φ 0 , we will give the proof of this result.
In other words, if we see e j and f j as vector fields on Σ using j(q), we extend them to a neighborhood of Σ.Then we consider the associated flows, defined on a neighborhood of Σ by: ∂φ x j j ∂x j (q, p) = êj (φ d/2 (j(q)) satisfies (2.6) and (2.7).Hence, if q ∈ Ω, the linear tangent map acts as: T q j 0 0 L q .
In particular, Φ * 0 ω = ω 0 on {z = 0} by (2.5) and lemma 2.1.By Weinstein lemma A.2 (Appendix), for ε > 0 small enough there exists a diffeomorphism S : It remains to compute a Taylor expansion of H in these coordinates.Using the Taylor Formula for Ĥ = H • Φ, we get: By the chain rule, we have (with q = ϕ −1 (w)): because T j(q) H = 0, and sends the canonical basis onto (e 1 (q), f 1 (q), ... , e d/2 (q), f d/2 (q)), so we get from Lemma 2.2: Hence (2.8) gives: 3. The Formal Birkhoff Normal Form 3.1.The Hamiltonian Ĥ.In the new coordinates given by Theorem 1.1, we have a Hamiltonian Ĥ(w, z) of the form: H 0 is defined for w ∈ V , but we extend the functions βj to R d w such that: This is just technical, since we will prove microlocalization results on V in section 6.Then we can construct a Birkhoff normal form, in the spirit of [20] and [19], with w as a parameter.
3.2.The space of formal series.We will work in the space of formal series We endow E with the Moyal product ⋆, compatible with the Weyl quantization (with respect to all the variables z and w).Given a pseudodifferential operator A = Op w (a) we will denote σ w,T (A) or [a] the formal Taylor series of a at zero, in the variables x, ξ, .With this notation, the compatibility of ⋆ with the Weyl quantization means σ w,T (AB) = σ w,T (A) ⋆ σ w,T (B).
The reader can find the main results on -pseudodifferential operators in [16] or [24].
We define the degree of x α ξ γ ℓ to be |α| + |γ| + 2ℓ.Hence, we can define the degree and valuation of a series κ, which depends on the point w ∈ R d .We denote O N the space of formal series with valuation at least N on V , and D N the space spanned by monomials of degree N on V (V ⊂ R d w is given by Theorem 1.1).We denote z j the formal series x j + iξ j .Thus every κ ∈ E can by written From formula (3.3), a simple computation yields to i ad 3.3.The formal normal form.In order to prove Theorem 1.2, we look for a pseudodifferential operator Q such that commutes with the harmonic oscillators 14).At the formal level, expression (3.5) becomes where H 0 + γ is the Taylor expansion of Ĥ, and τ = σ w,T (Q ).Moreover, so we want (3.6) to be equal to H 0 + κ, where [κ, |z j | 2 ] = 0, which is equivalent to say that κ is a series in ).This is possible modulo O r , as stated in the following theorem.We recall that r is the non-resonance order, defined in (1.11), and that we assumed r ≥ 3.
Theorem 3.1.If γ ∈ O 3 , there exist τ, κ, ρ ∈ O 3 such that: Assume that we have, for a τ N ∈ O 3 : ) and where R N ∈ D N .Using (3.2), we have for any τ ′ ∈ D N : Thus, we look for τ ′ and K N ∈ D N such that: To solve this equation, we need to study ad H 0 .Since Since βj only depends on w, Thus equation (3.7) can be rewritten with the notation From formula (3.4) we see that T acts on monomials as Thus, if we write ).The rest R N − K N is a sum of monomials of the form r αγℓ z α zγ ℓ with α = γ.As soon as 0 < |α − γ| < r, we have α − γ, β(w) = 0 (by (1.12) because r is lower than the resonance order (1.9)), so we can define the smooth coefficient Thus (3.9) yields to Hence we solved equation (3.8), and thus we can iterate until N = r − 1.The series ρ is the O r that remains:

The Semiclassical Birkhoff Normal Form
The next step is to quantize Theorems 1.1 and 3.1.

4.1.
Quantization of Theorem 1.1.Theorem 1.1 gives a symplectomorphism Φ reducing H to Ĥ = H • Φ.We can quantize this result in the following way.The Egorov Theorem (Thm 5.5.9 in [16]) implies the existence of a Fourier integral operator V : L 2 (R d (x,y) ) → L 2 (M), associated to the symplectomorphism Φ, and a pseudo-differential operator L with principal symbol Ĥ on V × B z (ε) and subprincipal symbol 0, such that: In particular, σ w,T L = H 0 + γ for some γ ∈ O 3 , with the notation of section 3.2.We want to construct a normal form using a bounded pseudodifferential operator Q : In Theorem 3.1, applied to γ, we have constructed formal series τ , κ, and ρ such that e i adτ (H 0 + γ) = H 0 + κ + ρ.
The idea is to choose pseudodifferential operators Q and N such that σ w,T (Q ) = τ and σ w,T (N ) = κ, and to check that they satisfy (4.5).Following this idea, we prove the following Theorem.
Theorem 4.1.For ∈ (0, 0 ] small enough, there exist a unitary operator a smooth function f ⋆ (w, I 1 , ..., I d/2 , ), and a pseudodifferential operator R such that: 1) , ..., I (d/2) , ) + R , (ii) f ⋆ has an arbitrarily small compact (I 1 , ..., I d/2 , )-support (containing 0), with I (j) = Op w (|z j | 2 ) and L 0 = Op w (H 0 ).We call We define functions: and arbitrarily small compact support in (I 1 , ..., I d/2 , ) (containing 0).Let c(w, z, ) be a smooth function with compact support with Taylor series τ , given by Theorem 3.1.Then by the Taylor formula, we have: By the Egorov Theorem and the fact that ad r i −1 Op w (c) : E → O r (see (3.2)), the integral remainder has a symbol with Taylor series in O r .Moreover, Thus, by the definition of f , there exists s(w, z, ) such that [s] ∈ O r and: Using the compatibility of the quantization with the Moyal product, we have σ w,T (f ⋆ (w, I 1) , ..., so we get: , ..., I (d/2) , )) + Op w (s), for a new symbol s(w, z, ) with [s] ∈ O r .Hence we get , ..., I (d/2) , )) + Op w (s), with U = e − i Op w (c) .To prove (iii) with R = Op w (s), note that Theorem 1.2 follows with the new operator Ũ = V U given by (4.1) and Theorem 4.1.Point (ii) of Theorem 1.2 is remaining.We prove it here, using that the function f ⋆ can be chosen with arbitrarily small compact support.Proposition 4.1.For any ζ ∈ (0, 1), up to reducing the support of f ⋆ , the normal form N of Theorem 4.1 satisfies for ∈ (0, 0 ] small enough: Proof.For a given K > 0, we can take a cutoff function χ supported in {λ ∈ R d/2 : λ ≤ K}, and change f ⋆ into χf ⋆ .Thus, for λ j ∈ sp(I (j) ), Hence, using functional calculus and the G • arding inequality, we deduce that 1) , ..., for K and small enough.

Spectral reduction of N
In this section, we prove an expansion of the first eigenvalues of N in powers of 1/2 .In order to prove Theorem 1.5, it will only remain to compare the spectra of N and L .This will be done in the next sections.
Let 1 ≤ j ≤ d/2.For n j ≥ 0, we denote h n j : R → R the n j -th Hermite function of the variable x j .In particular, for every 1 ≤ j ≤ d/2 we have: x , we define the functions h n for any n = (n 1 , ..., We have the following space decomposition: h n .
This follows directly from (5.1) and (4.6).Moreover, we can prove the following more precise inclusions of the spectra.Lemma 5.2.Let b 1 ∈ (b 0 , b1 ).There exist 0 , n max , c > 0 such that, for any ∈ (0, 0 ): and for any n ∈ N d/2 with 1 ≤ |n| ≤ n max : Proof.Remember that the functions βj are bounded from below by a positive constant.Thus, the G • arding inequality implies that there are 0 , c > 0 such that, for every ∈ (0, 0 ), For any n ∈ N d/2 , we have: because L 0 = j Op w ( βj )I (j) .Thus using (5.5) and the G • arding inequality, This proves (5.4) for a new c > 0.Moreover, if you take any eigenpair (λ, ψ) of N with λ ≤ b 1 , it is an eigenpair of some N (n) , with ψ = u ⊗ h n , and: Thus, there is a n max > 0 independent of , λ, ψ such that |n| ≤ n max .
Using the previous Lemma and the well-known expansion of the first eigenvalues of Op w ( b), we deduce an expansion of the first eigenvalues of N .
Theorem 5.1.Let ε > 0 and N ≥ 1.There exist 0 > 0 and c 0 > 0 such that, for ∈ (0, 0 ], the N first eigenvalues of N : (λ j ( )) 1≤j≤N admit an expansion in powers of 1/2 of the form: where E j is the j-th eigenvalue of the d/2-dimensional harmonic oscillator associated to the Hessian of b at 0, counted with multiplicity.
The first eigenvalues of a semiclassical pseudodifferential operator with principal symbol b (which admits a unique and non-degenerate minimum) have an expansion of the form: where E j is the j-th eigenvalue of the d/2-dimensional harmonic oscillator associated to the Hessian of b at the minimum.Let us recall the idea of the proof of this result.Since the minimum of b is non degenerate, we can write A linear symplectic change of coordinates changes Hess 0 b into for some positive numbers (ν j ) 1≤j≤d/2 .In these coordinates the symbol becomes and Helffer-Sjöstrand proved in [11] that the first eigenvalues of a pseudo-differential operator with such a symbol admits an expansion in powers of 1/2 .Sjöstrand [20] recovered this result using a Birkhoff normal form in the case where the coefficients (ν j ) j are non-resonant.Charles and Vu Ngoc also tackled the resonant case in [4].

Microlocalization results
In section 4, we have proved Theorem 1.2: We have constructed a normal form, which is only valid on a neighborhood U of Σ = H −1 (0) since the rest R can be large outside this neighborhood.Hence, we now prove that the eigenfunctions of L and N are microlocalized on a neighborhood of Σ.
6.1.Microlocalization of the eigenfunctions of L .We recall that For ε > 0, we denote The following Theorem states the well-known Agmon estimates (see Agmon's paper [1]), which gives exponential decay of the eigenfunctions of the magnetic Laplacian L outside the minimum q 0 of the magnetic intensity b.In particular, these eigenfunctions are localized in Ω. Theorem 6.1 (Agmon estimates).Let α ∈ (0, 1/2) and b 0 < b 1 < b1 .There exist C, 0 > 0 such that for all ∈ (0, 0 ] and for all eigenpair (λ, ψ) of L with λ ≤ b 1 , we have: In particular, if Proof.If Φ : M → R is a Lipschitz function such that e Φ ψ belongs to the domain of q , the Agmon formula (Theorem A.3 in Appendix), together with the Assumption 1, We split this integral into two parts: where χ m (t) = t for t < m, χ m (t) = 0 for t > 2m, and χ ′ m uniformly bounded with respect to m.Since Φ m (q) = 0 on K and b(q) − 1/4 C 0 ≥ 0, we have: Thus, up to changing the constant C 0 : and since Φ m = 0 on K: By Fatou's lemma in the limit m → +∞, To prove the second result, notice that Now we prove the microlocalization of the eigenfunctions of L near Σ.Theorem 6.2.Let ε > 0, δ ∈ (0, 1  2 ), and 0 < b 1 < b1 .Let χ 0 : M → [0, 1] be a smooth function being 1 on K ε .Let χ 1 : R → [0, 1] be a smooth compactly supported cutoff function being 1 near 0. Then for any normalized eigenpair (λ, ψ) of L such that λ ≤ b 1 we have: Proof.Using Theorem 6.1, we have is a bounded operator, we get: In fact, ψ = χ 1 ( −2δ L )ψ.Indeed, there exists a C > 0 such that and for ∈ (0, 0 ) small enough, 6.2.Microlocalization of the eigenfunctions of N .The next two theorems states the microlocalization of the eigenfunctions of the normal form.We recall that if ϕ is defined by Theorem 1.1, we have: We also recall the definition (6.1) of K ε .This first lemma gives a microlocalization result on the w variable.
Lemma 6.1.Let ∈ (0, 0 ] and b 1 ∈ (0, b1 ).Let χ 0 be a smooth cutoff function on R d w supported on V such that χ 0 = 1 on ϕ(K ε ).Then for any normalized eigenpair (λ, ψ) of N such that λ ≤ b 1 , we have: x,y ).Proof.Let χ = 1 − χ 0 , which is supported in ϕ(K ε ) c .The eigenvalue equation yields to because the principal symbol of N (n) is d/2 j=1 (2n j + 1) βj .Since the symbol of the commutator is of order and supported in suppχ, we have where χ is a small extension of χ, with value 1 on suppχ and 0 on a neighborhood of ϕ(K ε ).Moreover using Proposition 4.1, where we used the G • arding inequality because, the symbol of L 0 is greater than b1 on suppχ.Together with (6.2) and (6.3), we get supported on V such that χ 0 = 1 on ϕ(K ε ) and χ 1 a real cutoff function being 1 near 0. Then for any normalized eigenpair (λ, ψ) of N such that λ ≤ b 1 , we have: Proof.According to Lemma 6.1, ) is a bounded operator, we have It remains to prove that ψ = χ But χ 1 = 1 on a neighborhood of 0, so there is 0 > 0 such that, for any ∈ (0, 0 ] and any 0 ≤ |n| ≤ n max , 1 ( −2δ I )ψ.
6.3.Rank of the spectral projections.We want the microlocalization Theorems 6.2 and 6.3 to be uniform with respect to λ ∈ (−∞, b 1 ].That is why we need the rank of the spectral projections to be bounded by some finite power of −1 .If A is a bounded from below self-adjoint operator, and α ∈ R, we denote N(A, α) the number of eigenvalues of A smaller than α, counted with multiplicities.It is the rank of the spectral projection 1 ]−∞,α] (A).
The proof of the following estimate is inspired by the proof of Lemma A.4 in Appendix, adapted from [9].The idea is to locally approximate the magnetic field to a constant.Lemma 6.2.Let b 0 < b 1 < b1 .There exists C > 0 and 0 > 0 such that for all ∈ (0, 0 ], we have: Proof.Take (χ m ) m≥0 a smooth partition of unity, such that: is compact, there is a m 0 > 0 such that, for m > m 0 : for small enough.For 0 ≤ m ≤ m 0 , we can work like in R d using the charts, and we can find a new partition of unity on V m such that Thus we have for 0 ≤ m ≤ m 0 : On each B m,j , we will approximate the magnetic field by a constant.Up to a gauge transformation, we can assume that the vector potential vanishes at z m,j .In other words, we can find a smooth function ϕ m,j on B m,j such that Ã(z m,j ) = 0, where Ã = A + ∇ϕ m,j .The potential Ã defines the same magnetic field B as A.

Let us define
Then if q denotes the quadratic form for the new potential Ã, for v ∈ C ∞ 0 (B m,j ), q (v) = q lin (v) + ( Ã − A lin )v 2 + 2ℜ ( Ã − A lin )v, (i ∇ + A lin )v , and using (6.8) and the Cauchy-Schwarz inequality, We use 2|ab| ≤ ε 2 a 2 + ε −2 b 2 to get: and so Changing A into Ã amounts to conjugate the magnetic Laplacian by e i −1 ϕ j,m , so: q lin (v) is the quadratic form associated to a constant magnetic field operator.Now, we approximate the metric with a flat one: = (1 − C α )q f lat (v).(6.13)Hence, from (6.7) and (6.5) we get: and using (6.10) and (6.13): is the quadratic form associated to where L m,j is a Schrödinger operator with constant magnetic field acting on L 2 (B m,j ), and L m is the multiplication by is one-to-one, the space )-dimensional, and the min-max principle yields to: Since L m,j is a magnetic Laplacian with constant magnetic field, we know that, for small enough: With α = 3/8 and β = 1/8, K( ) = o( ), so we deduce: The same result holds for N : the smallest eigenvalues of N .The goal of this section is to prove the following theorem, using the results of section 6. Together with Theorem 5.1, this theorem concludes the proofs of Theorems 1.3 and 1.5.
Proof.We will prove that ν n ( ) ≤ λ n ( )+O( δr ), the other inequality being similar.Let 1 ≤ n ≤ N(L , b 1 ), and let us denote ψ 1, , ..., ψ n, the normalized eigenfunctions associated to the first eigenvalues of L .We also denote where χ 0 and χ 1 are defined in Theorem 6.2.We have the normal form: where Ũ = V U , is given by (4.1) and Theorem 4.1.
The proof follows the same lines as Theorem 6.1, with α = 1/4, K replaced by K , K ε replaced by K 0, , and Theorem 6.2 with no change.The uniformity with respect to (λ, ψ) follows from Lemma 6.2.
Proof.We follow the proof of Lemma 6.Since 2δ < η, we get a new C > 0 such that for small enough: Iterating with χ instead of χ, for δ < 1/3 we get Op w (χ)ψ = O( ∞ ).
The end of the proof is the same as the proof of Theorem 6.3.The uniformity with respect to (λ, ψ) comes from Lemma 6.