A semiclassical Birkhoff normal form for symplectic magnetic wells

HAL Id: hal-02173445

https://hal.archives-ouvertes.fr/hal-02173445v2

Preprint submitted on 3 Sep 2020

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

magnetic wells

Léo Morin

To cite this version:

Léo Morin. A semiclassical Birkhoff normal form for symplectic magnetic wells. 2020. �hal- 02173445v2�

LÉO MORIN

Abstract. In this paper we construct a Birkhoff normal form for a semiclassical mag- netic Schrödinger operator with non-degenerate magnetic field, and discrete magnetic well, defined on an even dimensional riemannian manifoldM. We use this normal form to get an expansion of the first eigenvalues in powers of ~^1/2, and semiclassical Weyl asymptotics for this operator.

1. Introduction

The analysis of the magnetic Schrödinger operator, or magnetic Laplacian, on a Rie- mannian 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 ofM and the variations of the magnetic field. For discussions about the subject, the reader is referred to the books and review [10], [12], [23]. 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 [24] (on R²) and [14] (on R³). In these two papers, semiclassi- cal Birkhoff normal forms were used to describe the first eigenvalues. In [25], Sjöstrand introduced the semiclassical Birkhoff normal form to study the spectrum of an electric Schrödinger operator, and some resonance phenomenons appeared. In [5], the resonant case for the same electric Schrödinger operator was tackled (see also [26] and [27]). In this paper, we adapt this method to L_~, generalizing the results of [24] to higher dimensions and manifolds. Our normal forms give a great geometric interpretation of the semiclassical spectral asymptotics of L_~. Indeed, it enlightens the contributions of the cyclotron mo- tion (the first oscillator) and the variations of the magnetic field near its well (the second oscillator) in the Weyl asymptotics and the eigenvalue asymptotics. Some normal forms for magnetic Schrödinger operators also appear in [16]. On a Riemannian manifold M, the magnetic Laplacian is related to the Bochner Laplacian (see the recent papers [18], [19] and [20], 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 [22] (general riemannian metric). Moreover, we only have in- vestigated the spectral theory of the stationary Schrödinger equation with a pure magnetic field ; it would be interesting to describe the long-time dynamics of the full Schrödinger evolution, as was done in the euclidian 2D case by Boil-Vu Ngoc [2].

Key words and phrases. magnetic Laplacian, normal form, spectral theory, semiclassical limit, pseudo differential operators, microlocal analysis.

1

1.1. Definition of the magnetic Schrödinger operator. Let (M, g) be a smooth d dimensional oriented Riemannian manifold. We assume thatM is compact with boundary, or that M = R^d with the Euclidean metric. For q ∈ M,g_q is a scalar product on T_qM. Since M is oriented, there is a canonical volume form, denoted either dxg or dqg. If f ∈L²(M), we denote its norm by

kfk= Z

M

|f(q)|²dqg

1/2

. If p∈T_qM^∗, we denote by|p|_g^?

q or |p|the norm ofp, defined by

∀Q∈TqM, |Q|²_g

q =|g_q(Q, .)|²_g∗ q. (1.1)

We denote byg_q^∗ the associated scalar product. The norm of a1-formα on M is kαk=

Z

M

|α(q)|²dq_g 1/2

. It is associated with a scalar product, denoted by brackets h., .i.

We denote by d the exterior derivative, associating to anyp-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 a1-formA on M called the magnetic potential, and we denote by B= dA its exterior derivative. B is called the magnetic2-form. The associated classical Hamiltonian is defined onT^∗M by:

H(q, p) =|p−A(q)|²_g∗

q, p∈T_qM^∗.

Using the isomorphism T_qM ' T_qM^∗ given by the metric, we define the magnetic operator B(q) :T_qM →T_qM by:

B_q(Q₁, Q₂) =g_q(B(q)Q₁, Q₂), ∀Q₁, Q₂∈T_qM.

(1.2)

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 D(q_~) ={u∈L²(M),(i~d+A)u∈L²Ω¹(M), u∂M = 0},

by

q_~(u) = Z

M

|(i~d+A)u|²dq_g,

whereL²Ω¹(M)denotes the space of square-integrable1-forms onM. By the Lax-Milgram theorem, this quadratic form defines a self-adjoint operatorL_~ on

D(L_~) ={u∈L²(M),(i~d +A)^∗(i~d +A)u∈L²(M), u_∂M = 0}, by the formula

hL_~u, vi=q_~[u, v], ∀u, v∈ C₀^∞(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 = (q₁, ..., q_d) on M, we get the corresponding vector fields basis (∂q1, ..., ∂qd) on TqM, and the dual basis (dq1, ...,dqd) on T_qM^∗. 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:

A≡A₁dq₁+...+A_ddq_d, withA= (Aj)1≤j≤d∈ C^∞(R^d,R^d). We denote

T_qA:T_qM →T_qM^∗ the linear operator whose matrix is the Jacobian ofA:

(∇A(q))_ij =∂_jA_i(q).

In the coordinates, the2-form B is

B =X

i<j

Bijdqi∧dqj, with

Bij =∂iAj−∂jAi= (^t∇A− ∇A)_ij. (1.3)

Let us denote(B_ij(q))1≤i,j≤dthe matrix of the operator B(q) :T_qM → T_qM in the basis (∂q1, ..., ∂qd). With this notation, equation (1.2) relatingB to B can be rewritten:

∀Q,Q˜∈R^d, X

ijk

g_kjB_kiQ_iQ˜_j =X

ij

B_ijQ_iQ˜_j, which means that

∀i, j, Bij =X

k

g_kjB_ki. (1.4)

Also note that:

ι_QB=X

i<j

B_ij(Q_idq_j−Q_jdq_i) =X

j

X

i

B_ijQ_i

! dq_j (1.5)

=X

j

(^t∇A− ∇A)Q

jdq_j= (^tT_qA−T_qA)Q (1.6)

Finally, in the coordinatesH is given by:

H(q, p) =X

i,j

g^ij(q)(pi−Ai(q))(pj−Aj(q)), (1.7)

andL_~ acts as the differential operator:

L^coord

~ =

d

X

k,l=1

|g|^−1/2(i~∂_k+A_k)g^kl|g|^1/2(i~∂_l+A_l).

(1.8)

1.3. Pseudodifferential operators. We refer to [21] and [29] for the general theory of

~-pseudodifferential operators. Ifm∈Z, we denote by

S^m(R²ⁿ) ={a∈ C^∞(R²ⁿ),|∂_x^α∂_ξ^βa| ≤C_αβhξi^m−|β|, ∀α, β∈N^d}

the class of Kohn-Nirenberg symbols. If a depends on the semiclassical parameter ~, we require that the coefficientsCαβ are uniform with respect to~∈(0,~0]. Fora_~ ∈S^m(R²ⁿ), we define its associated Weyl quantizationOp^w_~(a_~) by the oscillatory integral

A_~u(x) =Op^w_~(a_~)u(x) = 1 (2π~)ⁿ

Z

R²ⁿ

e^~ihx−y,ξia_~

x+y 2 , ξ

u(y)dydξ,

and we denote:

a_~ =σ_~(A_~).

A pseudodifferential operator A_~ on L²(M) is an operator acting as a pseudodifferential operator in coordinates. Then the principal symbol of A_~ does not depend on the coordi- nates, and we denote it by σ0(A_~). The subprincipal symbol σ1(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).

If M is compact, in any local coordinates, the coefficients A_j of A (as a function of q ∈R^d) are inS⁰(R^2d_(q,p)). Hence we see from (1.8) that L_~ is a pseudodifferential operator on L²(M). Its principal and subprincipal Weyl symbols are:

σ₀(L_~) =H, σ₁(L_~) = 0.

This is well-known, but we detail the computation of the subprincipal symbol in Appendix (Lemma A.1).

If M =R^d, we assume thatA_j ∈S⁰(R^2d)for 1≤j≤d. We could also assume thatA_j belongs to some standard class of symbol defined by a general order function on R^2d. 1.4. Assumptions. Since B(q), defined in (1.2), is a skew-symmetric operator for the scalar product gq, its eigenvalues are in iR. We define the magnetic intensity, which is equivalent to the trace-norm, by

b(q) =Tr⁺B(q) = 1

2Tr([B^∗(q)B(q)]^1/2) = X

iβj∈sp(B(q)),β_j>0

βj. It is a continuous function of q, but not smooth in general. We also denote

b0 = inf

q∈Mb(q), and in the non-compact case M =R^d,

b∞= lim inf

|q|→+∞b(q).

We first assume that the magnetic field satisfies the following inequality.

Assumption 1. We assume that there exist~0 >0 and C0 >0 such that, for ~∈(0,~0],

∀u∈D(q_~), (1 +~^1/4C₀)q_~(u)≥ Z

M

~(b(q)−~^1/4C₀)|u(q)|²dq_g.

In [13], Helffer and Morame proved such an inequality in the case M compact. If M =R^d, they prove that it is sufficient to assume that

k∇B_ij(q)k ≤C(1 +|B(q)|), 1≤i, j≤d for some C >0to deduce the inequality.

We consider the case of a unique discrete magnetic well:

Assumption 2. We assume that the magnetic intensity b admits a unique and non- degenerate minimum b₀ at q₀ ∈M\∂M, such that 0< b₀ < b∞.

Finally, we make a non-degeneracy assumption.

Assumption 3. We assume that d is even and B(q₀) is invertible.

In particular, B(q) is invertible for q in a neighborhood of q0, which means that the 2-form B is symplectic near q₀. Under this Assumption, the eigenvalues of B(q₀) can be written

±iβ₁(q0), . . . ,±iβ_d/2(q0),

withβ_j(q₀)>0. We define the resonance orderr₀∈N^∗∪ {∞} of the eigenvalues by r₀ := min{|α|:α∈Z^d/2, α6= 0,hα, β(q₀)i= 0},

(1.9)

with the notations

|α|=

d/2

X

j=1

|α_j|, hα, β(q₀)i:=

d/2

X

j=1

αjβj(q0).

We make a non-resonance assumption.

Assumption 4. We assume that the eigenvalues of B(q0) are simple (which is equivalent to assuming thatr₀≥3).

In particular, there is a neighborhoodΩ⊂⊂M\∂M of q₀ on which the eigenvalues of B(q) are simple, and defined by smooth positive functions

βj : Ω→R^∗₊.

We can choose Ωsuch that every β_j is bounded from bellow by a positive constant on Ω.

We can also find smooth orthonormal vectors onΩ:

u1(q), v1(q), . . . , u_d/2(q), v_d/2(q)∈TqM, 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

r∈N∩[3, r₀].

(1.11)

Up to reducingΩ (depending onr), we also have (sincer is finite), for 0<|α|< r:

hα, β(q)i 6= 0, ∀q∈Ω.

(1.12)

Under Assumption 2, we can find b₀<˜b₁< b∞such that K :={b(q)≤˜b1} ⊂Ω.

(1.13)

In the caseM =R^d, using the inequality in Assumption 1, it is proved in [13] that there exist~0 andc >0 such that, for~∈(0,~0],

sp_ess(L_~)⊂[~(˜b1−c~^1/4),+∞),

and so, for~small enough, the spectrum of L_~ below~b₁ (for a given b₁<˜b₁) is discrete.

WhenM is compact,L_~ has compact resolvent, and its full spectrum is discrete.

1.5. Main results. On the classical part, we first prove the following reduction of the Hamiltonian. For z = (x, ξ) ∈ R^d and w = (y, η) ∈ R^d, we denote zj = (xj, ξj), wj = (y_j, η_j), and B_z(ε) = {|z| ≤ ε}. We use the notation R^d_z (or R^d_w) to emphasize that an element ofR^dis denoted z (orw).

Theorem 1.1. Under Assumptions 1,2,3 and 4, for Ωandε >0 small enough, there exist symplectomorphisms

ϕ: (Ω, B)→(V ⊂R^d_w,dη∧dy), and

Φ : (V ×Bz(ε),dη∧dy+ dξ∧dx)→(U ⊂T^∗M, ω), with Φ(ϕ(q),0) = (q, A(q)), under which the HamiltonianH becomes:

H(w, z) =ˆ H◦Φ(w, z) =

d/2

X

j=1

βˆj(w)|z_j|²+O(|z|³),

locally uniformly in w, with the notation βˆ_j(w) =β_j ◦ϕ⁻¹(w).

Our next aim is to construct a semiclassical Birkhoff normal form for L_~, that is to say a pseudodifferential operatorN_~ onL²(R^d), commuting with suitable harmonic oscillators such that:

U_~L_~U_~^∗=N_~+R_~,

with U_~ : L²(M) → L²(R^d) a microlocally unitary Fourier integral operator and R_~ a remainder. We will contruct the remainder so that the first eigenvalues ofL_~ 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.

Theorem 1.2 (Semiclassical Birkhoff normal form). We denote by z = (x, ξ) ∈ T^∗R^d/2x

and w= (y, η) ∈T^∗R^d/2_y the canonical variables. For ζ >0 and ~∈(0,~0]small enough, there exist a Fourier integral operator

U_~ :L²(R^d_(x,y))→L²(M),

a smooth function f^?(w, I₁, ..., I_d/2,~), and a pseudodifferential operator R_~ on R^d such that:

(i) U_~^∗L_~U_~ =N_~+R_~, (ii) N_~ =Op^w_~

H0+f^?(w,I⁽¹⁾

~ , ...,I^(d/2)

~ ,~)

,

(iii) σ_~^w(R_~)∈ O((|z|+~^1/2)^r) on a neighborhood of w= 0, (iv) U_~^∗U_~ =I microlocally near (z, w) = 0,

(v) U_~U_~^∗=I microlocally near (q, p) = (q₀, A_q0), (vi) (1−ζ)hOp^w

~H₀ψ, ψi ≤ hN_~ψ, ψi ≤(1 +ζ)hOp^w

~H₀ψ, ψi, ∀ψ∈ S(R^d), with

I^(j)

~ =Op^w_~(|z_j|²) =−~² ∂²

∂x²_j +x²_j, H0 =

d/2

X

j=1

βˆj(w)|z_j|². (1.14)

We call N_~ the normal form, andR_~ 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.

Theorem 1.3. Let ε >0 andb1 ∈(0,˜b1). We denote

λ₁(~)≤λ₂(~)≤... and ν₁(~)≤ν₂(~)≤...

the first eigenvalues of L_~ andN_~ respectively. Then λn(~) =νn(~) +O(~^r/2−ε), uniformly in n such that λ_n(~)≤~b₁ and ν_n(~)≤~b₁.

Thus, the eigenvalues of L_~ are given by the eigenvalues of N_~, which we can reduce since it commutes with harmonic oscillators.

Theorem 1.4. For k≥0, let us denote h_k the Hermite function, satisfying I^(j)

~ h_k(x_j) =~(2k+ 1)h_k(x_j).

For n = (n1, ..., n_d/2) ∈ N^d/2, there exists a pseudodifferential operator N⁽ⁿ⁾

~ acting on L²(R^d/2_y ) such that:

N_~(u⊗hn1 ⊗...⊗hn_d/2) =N⁽ⁿ⁾

~ (u)⊗hn1⊗...⊗hn_d/2, u∈ S(R^d/2_y ).

Its symbol is:

F⁽ⁿ⁾(w) =~

d/2

X

j=1

βˆ_j(w)(2n_j + 1) +f^?(w,~(2n+ 1),~), and we have:

sp(N_~) =[

n

sp(N⁽ⁿ⁾

~ ).

Moreover, the multiplicity of λas eigenvalue of N_~ is the sum overn of the multiplicities of λas eigenvalue of N⁽ⁿ⁾

~ .

Finally, we deduce an expansion of theN >0 first eigenvalues ofL_~ in powers of~^1/2. Theorem 1.5 (Expansion of the first eigenvalues). Let ε > 0 and N ≥ 1. There exist

~0 >0 andc0 >0 such that, for ~∈(0,~0], the N first eigenvalues of L_~ : (λj(~))1≤j≤N

admit an expansion in powers of ~^1/2 of the form:

λ_j(~) =~b₀+~²(E_j +c₀) +~^5/2c_j,5+...+~^(r−1)/2cj,r−1+O(~^r/2−ε), where ~E_j is thej-th eigenvalue of the d/2-dimensional harmonic oscillator

Op^w_~(Hess₀(b◦ϕ⁻¹)).

Remark. WhenM is compact but the2-formB is not exact, one can not define the mag- netic LaplacianL_~, but one can consider the Bochner Laplacian on a complex line bundle Lendowed with a connexion of curvatureiB. Then, the semiclassical limit corresponds to the high tensor product limit of L. Using quasimodes, Kordyukov [?] proved asymptotic expansions of the first eigenvalues of the Bochner Laplacian, in the case of non-degenerate magnetic wells. Thus, his result is closely related to Theorem 1.5. When M is compact, our case corresponds to the Bochner Laplacian of a trivial line bundle. However, our nor- mal form gives a geometrical interpretation of the coefficients, and also describes higher eigenvalues (semi-excited states).

Note that, from Theorems 1.3 and 1.4, we deduce Weyl estimates forL_~. Some similar formulas appear in [16]. HereN(L_~, b1~) denotes the number of eigenvaluesλof L_~ such thatλ≤b₁~, counted with multiplicities.

Corollary 1.1 (Weyl estimates). For anyb₁ ∈(b₀,˜b₁), N(L_~, b1~)∼ 1

(2π~)^d/2 X

n∈N^d/2

Z

b^[n](q)≤b1

B^d/2 (d/2)!. where

b^[n](q) =

d/2

X

j=1

(2n_j+ 1)β_j(q).

The sum is finite because the βj are bounded from below by a positive constant on Ω. In particular, if M =R^d, we get

N(L_~, b1~)∼ 1 (2π~)^d/2

X

n∈N^d/2

Z

b^[n](q)≤b1

β1(q)...β_d/2(q)dq.

Remark. In their works Demailly [7, 8] and Bouche [4] proved similar Weyl asymptotics for Bochner Laplacians on a compact complex manifold. They used an expansion of the associated heat kernel and an local approximation of the magnetic field by a constant.

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:

H(q, z) =ˆ

d/2

X

j=1

βj(q)|z_j|²+O(|z|³).

In section 3, we construct a formal Birkhoff normal form: in the space of formal series in variables (x, ξ,~), we change Hˆ into H⁰ +κ+ρ, with H⁰ = Pd/2

j=1βj|z_j|², κ a series in

|z_j|² (1≤j≤d/2), andρ a remainder of orderr (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_~ belowb1~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 b₁~ (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.

2. Reduction of the classical Hamiltonian 2.1. A symplectic reduction of T^∗M. The zero set of H:

Σ =H⁻¹(0) ={(q, A(q))∈T^∗M :q ∈Ω},

is a d-dimensional smooth submanifold of the cotangent bundleT^∗M. We denotej: Ω→ T^∗M the embedding

j(q) = (q, A(q)).

The symplectic structure on T^∗M is defined by the form ω= dp∧dq= dα, α=pdq.

In other words, for p∈TqM^∗ and V ∈T_(q,p)(T^∗M), α_(q,p)(V) =p(π∗V), (2.1)

Where the mapπ∗:T_(q,p)(T^∗M)→T_qM is the differential of the canonical projection π:T^∗M →M, π(q, p) =q.

Using local coordinates with the notations of section 1.2, at any point (q, p)∈T^∗M with p=p1dq1+...+p_ddq_d,

the tangent vectors V ∈T_(q,p)(T^∗M) are identified with(Q, P)∈T_qM ×T_qM^∗, with Q=Q1∂q1+...+Qd∂qd, P =P1dq1+...+Pddqd.

With this notation,

π∗(Q, P) =Q, α_(q,p)(Q, P) =p(Q),

ω_(q,p)((Q, P),(Q⁰, P⁰)) =hP⁰, Qi − hP, Q⁰i, whereh., .i denotes the duality bracket betweenT_qM andT_qM^∗.

Lemma 2.1. Σis a symplectic submanifold of (T^∗M, ω), and j^∗ω =B.

In particular, at each pointj(q)∈Σ,

T_j(q)(T^∗M) =T_j(q)Σ⊕T_j(q)Σ^⊥, (2.2)

where ⊥denotes the symplectic orthogonal for ω.

Proof. To say that Σis 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) ofα withp=Aq andV =dqj(Q), we get

∀Q∈T_qM, (j^∗α)_q(Q) =A_q(π∗d_qj(Q)) =A_q(Q).

Hence

j^∗α=A, so j^∗(dα) = dA=B.

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_j(q)² H, as a bilinear form onT_j(q)(T^∗M), can be written:

T_j(q)² H(V,V) = 0 if V ∈T_j(q)Σ, T_j(q)² H(V,V) = 2|B(q)π_∗V|²_g

q if V ∈T_j(q)Σ^⊥.

Proof. Using local coordinates on M, we will denote every V ∈ T_(q,p)(T^∗M), as (Q, P) ∈ T_qM ×T_qM^∗. In these coordinates, with the notations introduced in section 1.2,

Σ≡ {(q,A(q)), q∈R^d} so that

T_j(q)Σ ={(Q, P)∈T_qM×T_qM^∗, P =T_qA·Q}.

(2.3)

We can also describeT_j(q)Σ^⊥ using these coordinates. Indeed,

(Q, P)∈T_j(q)Σ^⊥⇔ ∀Q₀∈T_qM, ω((Q, P),(Q₀, T_qA·Q₀)) = 0

⇔ ∀Q₀∈T_qM, hP, Q₀i=hT_qA·Q₀, Qi

⇔P = ^tT_qA·Q.

Hence

T_j(q)Σ^⊥={(Q, P), P = ^tT_qA·Q}.

(2.4)

From the expression (1.7) ofH in coordinates, we deduce that:

T_(q,p)H(Q, P) = 2X

ij

g^ij(q)(pi−Ai(q))(Pj − ∇_qAj·Q)

+X

ijk

∂_kg^ij(q)Q_k(p_i−A_i(q))(p_j−A_j(q)), so that the Hessian ofH in coordinates is:

T_j(q)² H((Q, P),(Q, P)) = 2X

ij

g^ij(q)(P_i− ∇_qA_i·Q)(P_j − ∇_qA_j·Q)

= 2|P −T_qA·Q|²_g∗ q.

It follows from (2.3) that

∀(Q, P)∈T_j(q)Σ, T_j(q)² H((Q, P),(Q, P)) = 0, and from (2.4) and (1.5) that

∀(Q, P)∈T_j(q)Σ^⊥, T_j(q)² H((Q, P),(Q, P)) = 2|(^tT_qA−T_qA)Q|²_g∗ q

=|ι_QB|²_g∗ q. Let us rewrite this using B. Note that:

|ι_QB|²_g∗

q =X

ij

g^ij(q) X

ki

B_kiQ_k

!

 X

`j

B_{`j</sub>Q<sub>`}



=X

k`



 X

ij

g^ijB_kiB_`j



Q_kQ_`, and keeping in mind that(g^ij)is the inverse matrix of(g_ij)together with the relation (1.4) between B andB, we have

X

ij

g^ijB_kiB_`j = X

ijk⁰`⁰

g^ijg_k⁰_ig_{`</sub><sup>0</sup><sub>j</sub>B<sub>k</sub><sup>0</sup><sub>k</sub>B<sub>`}⁰_`=X

k⁰`⁰

g_k⁰_{`</sub><sup>0</sup>B<sub>k</sub><sup>0</sup><sub>k</sub>B<sub>`}⁰_`,

and so

|ι_QB|²_g∗

q =X

k⁰`⁰

gk⁰`⁰

X

k

Bk⁰kQk

! X

`

B`<sup>0</sup>`Q`

!

=|B(q)Q|²_gq.

We endowΩ×R^d_z with the symplectic form:

ω₀(q, z) =B⊕

d/2

X

j=1

dξ_j∧dx_j,

with the notationz= (x, ξ). (Σ, B)is ad-dimensional symplectic submanifold of(T^∗M, ω).

The following Darboux-Weinstein lemma claims that this situation is modelled on the submanifold Σ₀ = Ω× {0} of(Ω×R^d_z, ω₀).

Lemma 2.3. There exists a local diffeomorphism Φ0: Ω×R^d_z →T^∗M such that

Φ^∗₀ω=ω₀, andΦ₀(Σ₀) = Σ.

In order to keep track on the construction of Φ0, we will give the proof of this result.

Proof. Again, we use local coordinates onM to denote everyV ∈T_(q,p)(T^∗M)as(Q, P)∈ TqM×T_q^∗M. Forq ∈Ω, using the vectorsuj(q), vj(q)∈TqM defined in (1.10), we define the vectors

e_j(q) = 1

pβ_j(q) u_j(q), ^tT_qA u_j(q)

, f_j(q) = 1

pβ_j(q) v_j(q), ^tT_qA v_j(q) , which are in T_j(q)Σ^⊥ by (2.4). These vectors satisfy

ω_j(q)(ei(q), fj(q)) =δij, ω_j(q)(ei(q), ej(q)) = 0, ω_j(q)(fi(q), fj(q)) = 0.

(2.5)

Indeed, the first equality follows from ω_j(q)(ei, fj) =− 1

pβ_iβ_jh( ^tTqA−TqA)uj, vji

=− 1 pβiβj

B(u_i, v_j)

=− 1

pβ_iβ_jg_q(B(q)u_i, v_j)

= βi

pβ_iβ_jgq(vi, vj)

=δij, and the two others from similar calculations.

Let us construct aΦ˜0: Ω×R^d_z →T^∗M such that:

Φ˜₀(q,0) =j(q), (2.6)

∂_zΦ˜₀(q,0) =L_q, (2.7)

whereLq:R^d→T_j(q)Σ^⊥ is the linear map sending the canonical basis onto (e1(q), f1(q), ..., e_d/2(q), f_d/2(q)).

For this, we take local vector fieldseˆ_j(q, p),fˆ_j(q, p)∈T_(q,p)(T^∗M)defined in a neighborhood ofΣ, such that

ˆ

ej(j(q)) =ej(q), fˆj(j(q)) =fj(q).

In other words, if we seeej and fj as vector fields on Σ using j(q), we extend them to a neighborhood ofΣ. Then we consider the associated flows, defined on a neighborhood of Σby:

∂φ^x_j^j

∂xj

(q, p) = ˆe_j(φ^x_j^j(q, p)), x_j ∈R,

∂ψ^ξ_j^j

∂ξ_j (q, p) = ˆfj(ψ^ξ_j^j(q, p)), ξj ∈R, φ⁰_j(q, p) =ψ_j⁰(q, p) = (q, p).

Then

Φ˜₀(q, z) :=φ^x₁¹ ◦ψ^ξ₁¹◦...◦φ^x_d/2^d/2◦ψ^ξ_d/2^d/2(j(q)) satisfies (2.6) and (2.7). Hence, ifq ∈Ω,the linear tangent map

T_(q,0)Φ˜₀ :T_qM ⊕R^d→T_j(q)Σ⊕T_j(q)Σ^⊥ acts as:

T_qj 0 0 Lq

.

In particular, Φ˜^∗₀ω = ω0 on {z = 0} by (2.5) and lemma 2.1. By Weinstein lemma A.2 (Appendix), forε >0small enough there exists a diffeomorphismS : Ω×B_z(ε)→Ω×B_z(ε) such that S(q, z) = (q, z) +O(|z|²) and S^∗Φ˜^∗₀ω = ω0. Then Φ0 = ˜Φ0 ◦S is the desired

symplectomorphism.

2.2. Proof of Theorem 1.1. Now we can prove the normal form for the classical Hamil- tonian. Up to reducing Ω, we can take symplectic coordinatesw= (y, η)∈R^dto describe Ω, thanks to the Darboux lemma:

ϕ: Ω→V ⊂R^d_w. We get a new symplectomorphism

Φ :V ×B_z(ε)→U ⊂T^∗M, defined by

Φ(w, z) = Φ0(ϕ⁻¹(w), z).

It remains to compute a Taylor expansion of H in these coordinates. Using the Taylor Formula for Hˆ =H◦Φ, we get:

H(w, z) = ˆˆ H(w,0) +∂_zHˆ_|z=0(z) +1

2∂_z²Hˆ_|z=0(z, z) +O(|z|³).

(2.8)

By the chain rule, we have (with q=ϕ⁻¹(w)):

∂zHˆ|z=0(z) =T_j(q)H(∂zΦ|z=0(z)) = 0, because T_j(q)H= 0, and

∂_z²Hˆ_|z=0(z, z) =T_j(q)² H(∂_zΦ_|z=0(z), ∂_zΦ_|z=0(z)).

But ∂_zΦ_|z=0 sends the canonical basis onto (e₁(q), f₁(q), ... , e_d/2(q), f_d/2(q)), so we get from Lemma 2.2:

1

2∂²_zHˆ_|z=0(z, z) =

d/2

X

j=1

β_j(q)|z_j|². Hence (2.8) gives:

H(w, z) =ˆ H◦Φ(w, z) =

d/2

X

j=1

βˆ_j(w)|z_j|²+O(|z|³).

3. The Formal Birkhoff Normal Form

3.1. The Hamiltonian H.ˆ In the new coordinates given by Theorem 1.1, we have a Hamiltonian H(w, z)ˆ of the form:

H(w, z) =ˆ H⁰(w, z) +O(|z|³), whereH⁰(w, z) =

d/2

X

j=1

βˆ_j(w)|z_j|².

H⁰ is defined forw∈V, but we extend the functions βˆ_j to R^d_w such that:

d/2

X

j=1

βˆj(w)≥˜b1 for w∈V^c. (3.1)

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 [25] and [24], with w as a parameter.

3.2. The space of formal series. We will work in the space of formal series E=C^∞(R^d_w)[[x, ξ,~]].

We endow E with the Moyal product ?, compatible with the Weyl quantization (with respect to all the variablesz andw). Given a pseudodifferential operatorA=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 [21] or [29].

We define the degree ofx^αξ^γ~<sup>`</sup> to be|α|+|γ|+ 2`. 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 leastN on V, andD_N the space spanned by monomials of degreeN onV (V ⊂R^d_w is given by Theorem 1.1). We denotez_j the formal seriesx_j+iξ_j. Thus everyκ∈ E can by written

κ=X

αγ`

c<sub>αγ`</sub>(w)z<sup>α</sup>z¯<sup>γ</sup>~<sup>`</sup>, with the notation

z^α=z₁^α1...z_d/2^αd/2.

For κ₁, κ₂ ∈ E, we denote ad_κ1κ₂ = [κ₁, κ₂] = κ₁ ? κ₂−κ₂ ? κ₁. It is well known that [κ1, κ2]is of order~, so for N1+N2 ≥2, we have

1

~

[O_N1,O_N2]⊂ O_N1+N2−2. (3.2)

Explicitly, we have

[κ₁, κ₂](z, w,~) = 2 sinh ~

2i

(f(z⁰, w⁰,~)g(z⁰⁰, w⁰⁰,~))|_z⁰_=z⁰⁰_=z,w⁰_=w⁰⁰_=w, (3.3)

where[f] =κ1,[g] =κ2, and

=

d/2

X

j=1

∂_ξ⁰

j∂_x⁰⁰

j −∂_x⁰

j∂_ξ⁰⁰

j +∂_η⁰

j∂_y⁰⁰

j −∂_y⁰

j∂_η⁰⁰

j

. From formula (3.3), a simple computation yields to

i

~ad_|zj|²(z^αz¯^β~^{`</sup>) ={|z<sub>j</sub>|<sup>2</sup>, z<sup>α</sup>z¯<sup>γ</sup>~<sup>`}}= (αj−γj)z^αz¯^γ~^`. (3.4)

3.3. The formal normal form. In order to prove Theorem 1.2, we look for a pseudodif- ferential operator Q_~ such that

e^~iQ~Op^w_~Heˆ ^−~iQ~ (3.5)

commutes with the harmonic oscillators I^(j)

~ ,(1 ≤j ≤d/2)introduced in (1.14). At the formal level, expression (3.5) becomes

e^i~adτ(H⁰+γ), (3.6)

whereH0+γ is the Taylor expansion of H, andˆ τ =σ^w,T

~ (Q_~).Moreover, σ^w,T

~ (I^(j)

~ ) =|z_j|²,

so we want (3.6) to be equal to H⁰+κ, where [κ,|z_j|²] = 0, which is equivalent to say that κ is a series in (|z₁|², ...,|z_d/2|²,~). 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 ∈ O₃, there exist τ, κ, ρ∈ O₃ such that:

• e^~iadτ(H⁰+γ) =H⁰+κ+ρ,

• [κ,|z_j|²] = 0 for 1≤j≤d/2,

• ρ∈ O_r.

Proof. Let3≤N ≤r−1. Assume that we have, for a τN ∈ O₃:

e^~iadτN(H⁰+γ) =H⁰+K₃+...+KN−1+R_N+O_N+1,

where K_i ∈ D_i commutes with|z_j|² (1 ≤j ≤d/2) and where R_N ∈ D_N. Using (3.2), we have for anyτ⁰ ∈ D_N:

e^i~adτN+τ0(H⁰+γ) =e^~iadτ0 H⁰+K3+...+KN−1+RN +O_N+1

=H⁰+K₃+...+KN−1+R_N+ i

~ad_τ⁰H⁰+O_N+1. Thus, we look for τ⁰ andK_N ∈ D_N such that:

RN =KN + i

~ad_H⁰τ⁰ moduloO_N+1. (3.7)

To solve this equation, we need to studyad_H⁰. SinceH⁰ =P

jβˆj(w)|z_j|², i

~ad_H⁰τ⁰ =

d/2

X

j=1

βˆ_j(w)i

~ad_|zj|²(τ⁰) + i

~ad_βˆ

j(τ⁰)|z_j|²

. Since βˆ_j only depends onw,

i

~ad_βˆ

j(τ⁰)∈ O_N−1, (see formula (3.3)). Hence

i

~ad_H0τ⁰ =

d/2

X

j=1

βˆ_j(w)i

~ad_|zj|2(τ⁰) +O_N+1. Thus equation (3.7) can be rewritten

R_N =K_N +T(τ⁰) +O_N+1, (3.8)

with the notation

T =

d/2

X

j=1

βˆ_j(w)i

~ad_|zj|2. From formula (3.4) we see that T acts on monomials as

T(c(w)z^αz¯^γ) =hα−γ,β(w)ic(w)zˆ ^αz¯^γ. (3.9)

Thus, if we write

R_N = X

|α|+|γ|+2`=N

r<sub>αγ`</sub>(w)z<sup>α</sup>z¯<sup>γ</sup>~<sup>`</sup>, we choose

K_N =X

α=γ

r<sub>αγ`</sub>|z|<sup>2α</sup>~<sup>`</sup>,

which commutes with|z_j|² ( 1≤j ≤d/2 ). The restR_N −K_N is a sum of monomials of the form r<sub>αγ`</sub>z<sup>α</sup>z¯<sup>γ</sup>~<sup>`</sup> with α6=γ. As soon as 0<|α−γ|< r, we have hα−γ,β(w)i 6= 0ˆ (by (1.12) becauser is lower than the resonance order (1.9)), so we can define the smooth coefficient

c_{αγ`</sub>(w) = r<sub>αγ`}(w) hα−γ,β(w)iˆ . Thus (3.9) yields to

T(cαγ`z<sup>α</sup>z¯<sup>γ</sup>~<sup>`</sup>) =rαγ`(w)z<sup>α</sup>z¯<sup>γ</sup>~<sup>`</sup>,

so R_N −K_N is in the range of T modulo O_N+1 because N ≤ r −1. Hence we solved equation (3.8), and thus we can iterate until N = r −1. The series ρ is the O_r that remains:

e^i~−1adτN(H⁰+γ) =H⁰+K3+...+Kr−1+ρ.

4. 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 symplectomorphismΦreduc- ing H to Hˆ = H ◦Φ. We can quantize this result in the following way. The Egorov Theorem (Thm 5.5.9 in [21]) implies the existence of a Fourier integral operator

V_~ :L²(R^d_(x,y))→L²(M),

associated to the symplectomorphismΦ, and a pseudo-differential operator Lb_~ with prin- cipal symbolHˆ onV ×Bz(ε) and subprincipal symbol0, such that:

V_~^∗L_~V_~ =Lb_~, (4.1)

V_~^∗V_~ =I microlocally on V ×Bz(ε), (4.2)

and

V_~V_~^∗=I microlocally on U.

(4.3)

4.2. Proof of Theorem 1.2. By (4.1), we are reduced to the pseudodifferential operator Lb_~, which has a total symbol of the form

σ_~ = ˆH+~²r˜_~ on V ×B_z(ε).

(4.4)

In particular,σ^w,T

~

Lb_~

=H0+γ for some γ ∈ O₃, with the notation of section 3.2. We want to construct a normal form using a bounded pseudodifferential operatorQ_~:

e^~iQ~Lb_~e^−~iQ~ =N_~+R_~. (4.5)

In Theorem 3.1, applied toγ, we have constructed formal series τ,κ, and ρ such that e^~iadτ(H⁰+γ) =H⁰+κ+ρ.

The idea is to choose pseudodifferential operators Q_~ and N_~ such that σ^w,T

~ (Q_~) = τ andσ^w,T

~ (N_~) =κ, and to check that they satisfy (4.5). Following this idea, we prove the following Theorem.