A semiclassical Birkhoff normal form for symplectic magnetic wells
aa r X i v : . [ m a t h . SP ] J u l A SEMICLASSICAL BIRKHOFF NORMAL FORM FORSYMPLECTIC MAGNETIC WELLS
LÉO MORIN
Abstract.
In this paper we construct a Birkhoff normal form for a semiclassicalmagnetic Schrödinger operator with non-degenerate magnetic field, and discretemagnetic well, defined on an even dimensional riemannian manifold M . We usethis normal form to get an expansion of the first eigenvalues in powers of ~ / ,and semiclassical Weyl asymptotics for this operator. Introduction
The analysis of the magnetic Schrödinger operator, or magnetic Laplacian, on aRiemannian manifold L ~ = ( i ~ d + A ) ∗ ( i ~ d + A ) in the semiclassical limit ~ → has given rise to many investigations in the lasttwenty years. Asymptotic expansions of the lowest eigenvalues have been studied inmany cases involving the geometry of the possible boundary of M and the variationsof the magnetic field. For discussions about the subject, the reader is referred to thebooks and review [7], [8], [18]. The classical picture associated with the Hamiltonian | p − A ( q ) | has started being investigated to describe the semiclassical bound states (the eigen-functions of low energy) of L ~ , in [19] (on R ) and [10] (on R ). In these twopapers, semiclassical Birkhoff normal forms were used to describe the first eigenval-ues. In [20], Sjöstrand introduced the semiclassical Birkhoff normal form to studythe spectrum of an electric Schrödinger operator, and some resonance phenomenonsappeared. In [4], the resonant case for the same electric Schrödinger operator wastackled (see also [21] and [22]). In this paper, we adapt this method to L ~ , followingthe ideas of [19]. Some normal forms for magnetic Schrödinger operators also appearin [12]. On a Riemannian manifold M , the magnetic Schrödinger operator is relatedto the Bochner Laplacian (see the recent papers [14] and [15], where bounds andasymptotic 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 ~ / , and semiclassical Weyl asymptotics. It would be interesting to have a precisedescription 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 Schrödingerequation 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 2Dcase by Boil-Vu Ngoc [2]. Key words and phrases. magnetic Laplacian, normal form, spectral theory, semiclassical limit,pseudo differential operators, microlocal analysis.
Definition of the magnetic Schrödinger operator.
Let ( M, g ) be a smooth d dimensional oriented Riemannian manifold, either without boundary or with smoothboundary. In particular we can take M = R d with the Euclidean metric, or M com-pact 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 d x g or d q g . If f ∈ L ( M ) , we denoteits norm by k f k = (cid:18)Z M | f ( q ) | d q g (cid:19) / . If p ∈ T q M ∗ , we denote by | p | g ⋆q or | p | the norm of p , defined by ∀ Q ∈ T q M, | Q | g q = | g q ( Q, . ) | g ∗ q . (1.1)We denote by g ∗ q the associated scalar product. The norm of a -form α on M is k α k = (cid:18)Z M | α ( q ) | d q g (cid:19) / . It is associated with a scalar product, denoted by brackets h ., . i .We denote by d the exterior derivative, associating to any p -form α a ( p + 1) -form d α . Using the scalar products induced by the metric, we can define its adjoint d ∗ ,associating to any p -form α a ( p − -form d ∗ α .We take a -form A on M called the magnetic potential, and we denote by B = d A its exterior derivative. B is called the magnetic -form. The associated classicalHamiltonian is defined on T ∗ M by: H ( q, p ) = | p − A ( q ) | g ∗ q , p ∈ T q M ∗ . Using the isomorphism T q M ≃ T q M ∗ given by the metric, we define the magneticoperator B ( q ) : T q M → T q M by: B q ( Q , Q ) = g q ( B ( q ) Q , Q ) , ∀ Q , Q ∈ T q M. (1.2)The norm of B ( q ) is | B ( q ) | = [ Tr ( B ∗ ( q ) B ( q ))] / . On the quantum side, for ~ > , we define the magnetic quadratic form q ~ on D ( q ~ ) = { u ∈ L ( M ) , ( i ~ d + A ) u ∈ L Ω ( M ) , u ∂M = 0 } , by q ~ ( u ) = Z M | ( i ~ d + A ) u | d q g , where L Ω ( M ) denotes the space of square-integrable -forms on M . By the Lax-Milgram theorem, this quadratic form defines a self-adjoint operator L ~ on D ( L ~ ) = { u ∈ L ( M ) , ( i ~ d + A ) ∗ ( i ~ d + A ) u ∈ L ( M ) , u ∂M = 0 } , by the formula hL ~ u, v i = q ~ [ u, v ] , ∀ u, v ∈ C ∞ ( M ) , where q ~ [ ., . ] is the inner product associated with the quadratic form q ~ ( . ) . L ~ is themagnetic Schrödinger operator with Dirichlet boundary conditions. SEMICLASSICAL BIRKHOFF NORMAL FORM 3
Local coordinates.
If we choose local coordinates q = ( q , ..., q d ) on M , weget the corresponding vector fields basis ( ∂ q , ..., ∂ q d ) on T q M , and the dual basis (d q , ..., d q 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 -form A in the coordinates: A ≡ A d q + ... + A d d q d , with A = ( A j ) ≤ j ≤ d ∈ C ∞ ( R d , R d ) . We denote T q A : T q M → T q M ∗ the linear operator whose matrix is the Jacobian of A : ( ∇ A ( q )) ij = ∂ j A i ( q ) . In the coordinates, the -form B is B = X i Pseudodifferential operators. We refer to [16] and [24] for the general theoryof ~ -pseudodifferential operators. If m ∈ Z , we denote by S m ( R n ) = { a ∈ C ∞ ( R n ) , | ∂ αx ∂ βξ a | ≤ C αβ h ξ i m −| β | , ∀ α, β ∈ N d } the class of Kohn-Nirenberg symbols. If a depends on the semiclassical parameter ~ , we require that the coefficients C αβ are uniform with respect to ~ ∈ (0 , ~ ] . For a ~ ∈ S m ( R n ) , we define its associated Weyl quantization Op w ~ ( a ~ ) by the oscillatoryintegral A ~ u ( x ) = Op w ~ ( a ~ ) u ( x ) = 1(2 π ~ ) n Z R n e i ~ h x − y,ξ i a ~ (cid:18) x + y , ξ (cid:19) u ( y )d y d ξ, and we denote: a ~ = σ ~ ( A ~ ) . A pseudodifferential operator A ~ on L ( M ) is an operator acting as a pseudodiffer-ential operator in coordinates. Then the principal symbol of A ~ does not depend onthe coordinates, and we denote it by σ ( A ~ ) . The subprincipal symbol σ ( A ~ ) is alsowell-defined, up to imposing the charts to be volume-preserving (in other words, ifwe 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 ) arein S ( R d ( q,p ) ) . Hence we see from (1.8) that L ~ is a pseudodifferential operator on L ( M ) . Its principal and subprincipal Weyl symbols are: σ ( L ~ ) = H, σ ( L ~ ) = 0 . This is well-known, but we detail the computation of the subprincipal symbol inAppendix (Lemma A.1).1.4. Assumptions. Since B ( q ) , defined in (1.2), is a skew-symmetric operator forthe scalar product g q , its eigenvalues are in i R . We define the magnetic intensity,which is equivalent to the trace-norm, by b ( q ) = Tr + B ( q ) = 12 Tr ([ B ∗ ( q ) B ( q )] / ) = X iβ j ∈ sp ( B ( q )) ,β j > β j . It is a continuous function of q , but not smooth in general. We also denote b = inf q ∈ M b ( q ) ,b ∞ = lim inf | q |→ + ∞ b ( q ) . We first assume that the magnetic field satisfies the following inequality. Assumption 1. We assume that there exist ~ > and C > such that, for ~ ∈ (0 , ~ ] , ∀ u ∈ D ( q ~ ) , (1 + ~ / C ) q ~ ( u ) ≥ Z M ~ ( b ( q ) − ~ / C ) | u ( q ) | d q g . In the Appendix (Lemma A.4), we describe cases when Assumption 1 holds. Inparticular, it holds if M is compact. If M = R d , it is true if we assume that k∇ B ij ( q ) k ≤ C (1 + | B ( q ) | ) , for some C > . These results are adapted from [9]. SEMICLASSICAL BIRKHOFF NORMAL FORM 5 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 b at q ∈ M \ ∂M , such that < b < b ∞ . Finally, we make a non-degeneracy assumption. Assumption 3. We assume that d is even and B ( q ) is invertible. In particular, B ( q ) is invertible for q in a neighborhood of q , which means thatthe -form B is symplectic near q . Under this Assumption, the eigenvalues of B ( q ) can be written ± iβ ( q ) , . . . , ± iβ d/ ( q ) , with β j ( q ) > . We define the resonance order r ∈ N ∗ ∪ {∞} of the eigenvalues by r := min {| α | : α ∈ Z d/ , α = 0 , h α, β ( q ) i = 0 } , (1.9)with the notation h α, β ( q ) i := d/ X j =1 α j β j ( q ) . We make a non-resonance assumption. Assumption 4. We assume that the eigenvalues of B ( q ) are simple (which isequivalent to assuming that r ≥ ). In particular, there is a neighborhood Ω ⊂⊂ M \ ∂M of q on which the eigenvaluesof B ( q ) are simple, and defined by smooth positive functions β j : Ω → R ∗ + . We can choose Ω such that every β j is bounded from bellow by a positive constanton Ω . We can also find smooth orthonormal vectors on Ω : u ( q ) , v ( q ) , . . . , u d/ ( q ) , v d/ ( 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 r ∈ N ∩ [3 , r ] . (1.11)Up to reducing Ω (depending on r ), we also have (since r is finite), for < | α | < r : h α, β ( q ) i 6 = 0 , ∀ q ∈ Ω . (1.12)Under Assumption 2, we can find b < ˜ b < b ∞ such that K := { b ( q ) ≤ ˜ b } ⊂ Ω . (1.13)Using the inequality in Assumption 1, it is proved in [9] that there exist ~ and c > such that, for ~ ∈ (0 , ~ ] , sp ess ( L ~ ) ⊂ [ ~ (˜ b − c ~ / ) , + ∞ ) , and so, for ~ small enough, the spectrum of L ~ below ~ b (for a given b < ˜ b ) isdiscrete. LÉO MORIN Main results. On the classical part, we first prove the following reduction ofthe Hamiltonian. For z = ( x, ξ ) ∈ R d , we denote z j = ( x j , ξ j ) and B z ( ε ) = {| z | ≤ ε } . Theorem 1.1. Under Assumptions 1,2,3 and 4, for Ω and ε > small enough,there exist symplectomorphisms ϕ : (Ω , B ) → ( V ⊂ R dw , d η ∧ d y ) , and Φ : ( V × B z ( ε ) , d η ∧ d y + d ξ ∧ d x ) → ( U ⊂ T ∗ M, ω ) , with Φ( ϕ ( q ) , 0) = ( q, A ( q )) , under which the Hamiltonian H becomes: ˆ H ( w, z ) = H ◦ Φ( w, z ) = d/ X j =1 ˆ β j ( w ) | z j | + O ( | z | ) , locally uniformly in w , with the notation ˆ β j ( w ) = β j ◦ ϕ − ( w ) . Our next aim is to construct a semiclassical Birkhoff normal form for L ~ , that is tosay a pseudodifferential operator N ~ on L ( R d ) , commuting with suitable harmonicoscillators such that: U ~ L ~ U ∗ ~ = N ~ + R ~ , with U ~ : L ( M ) → L ( R d ) 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/ − 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/ x and w = ( y, η ) ∈ T ∗ R d/ y the canonical variables. For ζ > and ~ ∈ (0 , ~ ] small enough, there exist a Fourier integral operator U ~ : L ( R d ( x,y ) ) → L ( M ) , a smooth function f ⋆ ( w, I , ..., I d/ , ~ ) , and a pseudodifferential operator R ~ on R d such that: ( i ) U ∗ ~ L ~ U ~ = L ~ + Op w ~ f ⋆ ( w, I (1) ~ , ..., I ( d/ ~ , ~ ) + R ~ , ( ii ) (1 − ζ ) hL ~ ψ, ψ i ≤ hN ~ ψ, ψ i ≤ (1 + ζ ) hL ~ ψ, ψ i , ∀ ψ ∈ S ( R d ) , ( iii ) σ w ~ ( R ~ ) ∈ O (( | z | + ~ / ) r ) on a neighborhood of w = 0 , ( iv ) U ∗ ~ U ~ = I microlocally near ( z, w ) = 0 , ( v ) U ~ U ∗ ~ = I microlocally near ( q, p ) = ( q , A q ) , with I ( j ) ~ = Op w ~ ( | z j | ) = − ~ ∂ ∂x j + x j , L ~ = Op w ~ d/ X j =1 ˆ β j ( w ) | z j | . (1.14) We call N ~ = L ~ + Op w ~ f ⋆ ( w, I (1) ~ , ..., I ( d/ ~ , ~ ) the normal form, and R ~ the remainder. SEMICLASSICAL BIRKHOFF NORMAL FORM 7 Using microlocalization properties of the eigenfunctions of L ~ and N ~ , we provethat they have the same spectra in the following sense. We recall that ˜ b , defined in(1.13), is chosen such that { b ( q ) ≤ ˜ b } ⊂ Ω . Theorem 1.3. Let ε > and b ∈ (0 , ˜ b ) . We denote λ ( ~ ) ≤ λ ( ~ ) ≤ ... and ν ( ~ ) ≤ ν ( ~ ) ≤ ... the first eigenvalues of L ~ and N ~ respectively. Then λ n ( ~ ) = ν n ( ~ ) + O ( ~ r/ − ε ) , uniformly in n such that λ n ( ~ ) ≤ ~ b and ν n ( ~ ) ≤ ~ b . We also reduce N ~ according to harmonic oscillators. Theorem 1.4. For k ≥ , let us denote h k the Hermite function, satisfying I ( j ) ~ h k ( x j ) = ~ (2 k + 1) h k ( x j ) . For n = ( n , ..., n d/ ) ∈ N d/ , there exists a pseudodifferential operator N ( n ) ~ actingon L ( R d/ y ) such that: N ~ ( u ⊗ h n ⊗ ... ⊗ h n d/ ) = N ( n ) ~ ( u ) ⊗ h n ⊗ ... ⊗ h n d/ , u ∈ S ( R d/ y ) . Its symbol is: F ( n ) ( w ) = ~ d/ X j =1 ˆ β j ( w )(2 n j + 1) + f ⋆ ( w, ~ (2 n + 1) , ~ ) , and we have: sp ( N ~ ) = [ n sp ( N ( n ) ~ ) . Moreover, the multiplicity of λ as eigenvalue of N ~ is the sum over n of the multi-plicities of λ as eigenvalue of N ( n ) ~ . Finally, we deduce an expansion of the N > first eigenvalues of L ~ in powers of ~ / . Theorem 1.5 (Expansion of the first eigenvalues) . Let ε > and N ≥ . Thereexist ~ > and c > such that, for ~ ∈ (0 , ~ ] , the N first eigenvalues of L ~ : ( λ j ( ~ )) ≤ j ≤ N admit an expansion in powers of ~ / of the form: λ j ( ~ ) = ~ b + ~ ( E j + c ) + ~ / c j, + ... + ~ ( r − / c j,r − + O ( ~ r/ − ε ) , where ~ E j is the j -th eigenvalue of the d/ -dimensional harmonic oscillator Op w ~ ( Hess ( b ◦ ϕ − )) . Note that, from Theorems 1.3 and 1.4, we deduce Weyl estimates for L ~ . Somesimilar formulas appear in [12]. Here N ( L ~ , b ~ ) denotes the number of eigenvalues λ of L ~ such that λ ≤ b ~ , counted with multiplicities. LÉO MORIN Corollary 1.1 (Weyl estimates) . For any b ∈ ( b , ˜ b ) , N ( L ~ , b ~ ) ∼ π ~ ) d/ X n ∈ N d/ Z b [ n ] ( q ) ≤ b B d/ ( d/ . where b [ n ] ( q ) = d/ X j =1 (2 n j + 1) β j ( q ) . The sum is finite because the β j are bounded from below by a positive constant on Ω . In particular, if M = R d , we get N ( L ~ , b ~ ) ∼ π ~ ) d/ X n ∈ N d/ Z b [ n ] ( q ) ≤ b β ( q ) ...β d/ ( q )d q. Organization and strategy. In section 2, we construct a symplectomorphismwhich simplify H near its zero set Σ = H − (0) (Theorem 1.1). In the new coordi-nates, H becomes: ˆ H ( q, z ) = d/ X j =1 β j ( q ) | z j | + O ( | z | ) . In section 3, we construct a formal Birkhoff normal form: in the space of formalseries in variables ( x, ξ, ~ ) , we change ˆ H into H + κ + ρ , with H = P d/ j =1 β j | z j | , κ a series in | z j | ( ≤ j ≤ d/ ), and ρ a remainder of order r (Theorem 3.1). Insection 4, we quantify the changes of coordinates constructed in section 2 and 3, andwe 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 remainsprove that the spectra of L ~ and N ~ below b ~ coincide. Before doing it, we needmicrolocalization 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 isvalid. In section 7, we use the results of section 6, to prove that L ~ and N ~ have thesame spectrum below b ~ (Theorem 1.3). This Theorem, together with the resultsof 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 = ∞ . 2. Reduction of the classical Hamiltonian A symplectic reduction of T ∗ M . The zero set of H : Σ = { ( q, A ( q )) ∈ T ∗ M : q ∈ Ω } , 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 ω = d p ∧ d q = d α, α = p d q. In other words, for p ∈ T q M ∗ and V ∈ T ( q,p ) ( T ∗ M ) ,α ( q,p ) ( V ) = p ( π ∗ V ) , (2.1) SEMICLASSICAL BIRKHOFF NORMAL FORM 9 Where the map π ∗ : T ( q,p ) ( T ∗ M ) → T q M 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 = p d q + ... + p d d q d , the tangent vectors V ∈ T ( q,p ) ( T ∗ M ) are identified with ( Q, P ) ∈ T q M × T q M ∗ , with Q = Q ∂q + ... + Q d ∂q d , P = P d q + ... + P d d q d . With this notation, π ∗ ( Q, P ) = Q,α ( q,p ) ( Q, P ) = p ( Q ) ,ω ( q,p ) (( Q, P ) , ( Q ′ , P ′ )) = h P ′ , Q i − h P, Q ′ i , where h ., . i denotes the duality bracket between T q M and T q M ∗ . Lemma 2.1. Σ is a symplectic submanifold of ( T ∗ M, ω ) , and j ∗ ω = B. In particular, at each point j ( 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 Σ is a symplectic submanifold of T ∗ M means that the restrictionof 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 ∀ Q ∈ T q M, ( j ∗ α ) q ( Q ) = A q ( π ∗ d q j ( Q )) = A q ( Q ) . Hence j ∗ α = A, so j ∗ (d α ) = d A = B. (cid:3) Since any j ( q ) is a critical point of H , the Hessian of H at j ( q ) is well defined andindependant of any choice of coordinates. We now compute this Hessian accordingto the decomposition (2.2): Lemma 2.2. The Hessian T j ( q ) H , as a bilinear form on T j ( q ) ( T ∗ M ) , can be written: T j ( q ) H ( V , V ) = 0 if V ∈ T j ( q ) Σ ,T j ( q ) H ( V , V ) = 2 | B ( q ) π ∗ V| 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 q M × T q M ∗ . In these coordinates, with the notations introduced insection 1.2, Σ ≡ { ( q, A ( q )) , q ∈ R d } so that T j ( q ) Σ = { ( Q, P ) ∈ T q M × T q M ∗ , P = T q A · Q } . (2.3) We can also describe T j ( q ) Σ ⊥ using these coordinates. Indeed, ( Q, P ) ∈ T j ( q ) Σ ⊥ ⇔ ∀ Q ∈ T q M, ω (( Q, P ) , ( Q , T q A · Q )) = 0 ⇔ ∀ Q ∈ T q M, h P, Q i = h T q A · Q , Q i⇔ P = t T q A · Q. Hence T j ( q ) Σ ⊥ = { ( Q, P ) , P = t T q A · Q } . (2.4)From the expression (1.7) of H in coordinates, we deduce that: T ( q,p ) H ( Q, P ) = 2 X ij g ij ( q )( p i − A i ( q ))( P j − ∇ q A j · Q )+ X ijk ∂ k g ij ( q ) Q k ( p i − A i ( q ))( p j − A j ( q )) , so that the Hessian of H in coordinates is: T j ( q ) H (( Q, P ) , ( Q, P )) = 2 X ij g ij ( q )( P i − ∇ q A i · Q )( P j − ∇ q A j · Q )= 2 | P − T q A · Q | g ∗ q . It follows from (2.3) that ∀ ( Q, P ) ∈ T j ( q ) Σ , T j ( q ) H (( Q, P ) , ( Q, P )) = 0 , and from (2.4) and (1.5) that ∀ ( Q, P ) ∈ T j ( q ) Σ ⊥ , T j ( q ) H (( Q, P ) , ( Q, P )) = 2 | ( t T q A − T q A ) Q | g ∗ q = | ι Q B | g ∗ q . Let us rewrite this using B . Note that: | ι Q B | g ∗ q = X ij g ij ( q ) X ki B ki Q k ! X ℓj B ℓj Q ℓ ! = X kℓ X ij g ij B ki B ℓj ! Q k Q ℓ , 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 X ij g ij B ki B ℓj = X ijk ′ ℓ ′ g ij g k ′ i g ℓ ′ j B k ′ k B ℓ ′ ℓ = X k ′ ℓ ′ g k ′ ℓ ′ B k ′ k B ℓ ′ ℓ , and so | ι Q B | g ∗ q = X k ′ ℓ ′ g k ′ ℓ ′ X k B k ′ k Q k ! X ℓ B ℓ ′ ℓ Q ℓ ! = | B ( q ) Q | g q . (cid:3) We endow Ω × R dz with the symplectic form: ω ( q, z ) = B ⊕ d/ X j =1 d ξ j ∧ d x j , SEMICLASSICAL BIRKHOFF NORMAL FORM 11 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 ismodelled on the submanifold Σ = Ω × { } of (Ω × R dz , ω ) . Lemma 2.3. There exists a local diffeomorphism Φ : Ω × R dz → T ∗ M such that Φ ∗ ω = ω , and Φ (Σ ) = Σ . In order to keep track on the construction of Φ , we will give the proof of thisresult. Proof. Again, we use local coordinates on M to denote every V ∈ T ( q,p ) ( T ∗ M ) as ( Q, P ) ∈ T q M × T ∗ q M . For q ∈ Ω , using the vectors u j ( q ) , v j ( q ) ∈ T q M defined in(1.10), we define the vectors e j ( q ) = 1 p β j ( q ) (cid:0) u j ( q ) , t T q A u j ( q ) (cid:1) , f j ( q ) = 1 p β j ( q ) (cid:0) v j ( q ) , t T q A v j ( q ) (cid:1) , which are in T j ( q ) Σ ⊥ by (2.4). These vectors satisfy ω j ( q ) ( e i ( q ) , f j ( q )) = δ ij , ω j ( q ) ( e i ( q ) , e j ( q )) = 0 , ω j ( q ) ( f i ( q ) , f j ( q )) = 0 . (2.5)Indeed, the first equality follows from ω j ( q ) ( e i , f j ) = − p β i β j h ( t T q A − T q A ) u j , v j i = − p β i β j B ( u i , v j )= − p β i β j g q ( B ( q ) u i , v j )= β i p β i β j g q ( v i , v j )= δ ij , and the two others from similar calculations.Let us construct a ˜Φ : Ω × R dz → T ∗ M such that: ˜Φ ( q, 0) = j ( q ) , (2.6) ∂ z ˜Φ ( q, 0) = L q , (2.7)where L q : R d → T j ( q ) Σ ⊥ is the linear map sending the canonical basis onto ( e ( q ) , f ( q ) , ..., e d/ ( q ) , f d/ ( q )) . For this, we take local vector fields ˆ e j ( q, p ) , ˆ f j ( q, p ) ∈ T ( q,p ) ( T ∗ M ) defined in a neigh-borhood of Σ , such that ˆ e j ( j ( q )) = e j ( q ) , ˆ f j ( j ( q )) = f j ( q ) . In other words, if we see e j and f j as vector fields on Σ using j ( q ) , we extendthem to a neighborhood of Σ . Then we consider the associated flows, defined on aneighborhood of Σ by: ∂φ x j j ∂x j ( q, p ) = ˆ e j ( φ x j j ( q, p )) , x j ∈ R ,∂ψ ξ j j ∂ξ j ( q, p ) = ˆ f j ( ψ ξ j j ( q, p )) , ξ j ∈ R ,φ j ( q, p ) = ψ j ( q, p ) = ( q, p ) . Then ˜Φ ( q, z ) := φ x ◦ ψ ξ ◦ ... ◦ φ x d/ d/ ◦ ψ ξ d/ d/ ( j ( q )) satisfies (2.6) and (2.7). Hence, if q ∈ Ω , the linear tangent map T ( q, ˜Φ : T q M ⊕ R d → T j ( q ) Σ ⊕ T j ( q ) Σ ⊥ acts as: (cid:18) T q j L q (cid:19) . In particular, ˜Φ ∗ ω = ω on { z = 0 } by (2.5) and lemma 2.1. By Weinstein lemma A.2(Appendix), for ε > small enough there exists a diffeomorphism S : Ω × B z ( ε ) → Ω × B z ( ε ) such that S ( q, z ) = ( q, z ) + O ( | z | ) and S ∗ ˜Φ ∗ ω = ω . Then Φ = ˜Φ ◦ S is the desired symplectomorphism. (cid:3) Proof of Theorem 1.1. Now we can prove the normal form for the classicalHamiltonian. Up to reducing Ω , we can take symplectic coordinates w = ( y, η ) ∈ R d to describe Ω , thanks to the Darboux lemma: ϕ : Ω → V ⊂ R dw . We get a new symplectomorphism Φ : V × B z ( ε ) → U ⊂ T ∗ M, defined by Φ( w, z ) = Φ ( ϕ − ( w ) , z ) . It remains to compute a Taylor expansion of H in these coordinates. Using theTaylor Formula for ˆ H = H ◦ Φ , we get: ˆ H ( w, z ) = ˆ H ( w, 0) + ∂ z ˆ H | z =0 ( z ) + 12 ∂ z ˆ H | z =0 ( z, z ) + O ( | z | ) . (2.8)By the chain rule, we have (with q = ϕ − ( w ) ): ∂ z ˆ H | z =0 ( z ) = T j ( q ) H ( ∂ z Φ | z =0 ( z )) = 0 , because T j ( q ) H = 0 , and ∂ z ˆ H | z =0 ( z, z ) = T j ( q ) H ( ∂ z Φ | z =0 ( z ) , ∂ z Φ | z =0 ( z )) . But ∂ z Φ | z =0 sends the canonical basis onto ( e ( q ) , f ( q ) , ... , e d/ ( q ) , f d/ ( q )) , so weget from Lemma 2.2: ∂ z ˆ H | z =0 ( z, z ) = d/ X j =1 β j ( q ) | z j | . SEMICLASSICAL BIRKHOFF NORMAL FORM 13 Hence (2.8) gives: ˆ H ( w, z ) = H ◦ Φ( w, z ) = d/ X j =1 ˆ β j ( w ) | z j | + O ( | z | ) . The Formal Birkhoff Normal Form The Hamiltonian ˆ H . In the new coordinates given by Theorem 1.1, we havea Hamiltonian ˆ H ( w, z ) of the form: ˆ H ( w, z ) = H ( w, z ) + O ( | z | ) , where H ( w, z ) = d/ X j =1 ˆ β j ( w ) | z j | .H is defined for w ∈ V , but we extend the functions ˆ β j to R dw such that: d/ X j =1 ˆ β j ( w ) ≥ ˜ b for w ∈ V c . (3.1)This is just technical, since we will prove microlocalization results on V in section6. 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 E = C ∞ ( R dw )[[ x, ξ, ~ ]] . 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 Weylquantization 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 | + 2 ℓ . Hence, we can define thedegree 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 spacespanned by monomials of degree N on V ( V ⊂ R dw is given by Theorem 1.1). Wedenote z j the formal series x j + iξ j . Thus every κ ∈ E can by written κ = X αγℓ c αγℓ ( w ) z α ¯ z γ ~ ℓ , with the notation z α = z α ...z α d/ d/ . For κ , κ ∈ E , we denote ad κ κ = [ κ , κ ] = κ ⋆ κ − κ ⋆ κ . It is well known that [ κ , κ ] is of order ~ , so for N + N ≥ , we have ~ [ O N , O N ] ⊂ O N + N − . (3.2) Explicitly, we have [ κ , κ ]( z, w, ~ ) = 2 sinh (cid:18) ~ i (cid:3) (cid:19) ( f ( z ′ , w ′ , ~ ) g ( z ′′ , w ′′ , ~ )) | z ′ = z ′′ = z,w ′ = w ′′ = w , (3.3)where [ f ] = κ , [ g ] = κ , and (cid:3) = d/ X j =1 (cid:16) ∂ ξ ′ j ∂ x ′′ j − ∂ x ′ j ∂ ξ ′′ j + ∂ η ′ j ∂ y ′′ j − ∂ y ′ j ∂ η ′′ j (cid:17) . From formula (3.3), a simple computation yields to i ~ ad | z j | ( z α ¯ z β ~ ℓ ) = {| z j | , z α ¯ z γ ~ ℓ } = ( α j − γ j ) z α ¯ z γ ~ ℓ . (3.4)3.3. The formal normal form. In order to prove Theorem 1.2, we look for apseudodifferential operator Q ~ such that e i ~ Q ~ Op w ~ ˆ He − i ~ Q ~ (3.5)commutes with the harmonic oscillators I ( j ) ~ , (1 ≤ j ≤ d/ introduced in (1.14). Atthe formal level, expression (3.5) becomes e i ~ ad τ ( H + γ ) , (3.6)where H + γ is the Taylor expansion of ˆ H , and τ = σ w, T ~ ( Q ~ ) . Moreover, σ w, T ~ ( I ( j ) ~ ) = | z j | , so we want (3.6) to be equal to H + κ , where [ κ, | z j | ] = 0 , which is equivalent tosay that κ is a series in ( | z | , ..., | z d/ | , ~ ) . This is possible modulo O r , as statedin the following theorem. We recall that r is the non-resonance order, defined in(1.11), and that we assumed r ≥ . Theorem 3.1. If γ ∈ O , there exist τ, κ, ρ ∈ O such that: • e i ~ ad τ ( H + γ ) = H + κ + ρ, • [ κ, | z j | ] = 0 for ≤ j ≤ d/ , • ρ ∈ O r . Proof. Let ≤ N ≤ r − . Assume that we have, for a τ N ∈ O : e i ~ ad τN ( H + γ ) = H + K + ... + K N − + R N + O N +1 , where K i ∈ D i commutes with | z j | ( ≤ j ≤ d/ ) and where R N ∈ D N . Using(3.2), we have for any τ ′ ∈ D N : e i ~ ad τN + τ ′ ( H + γ ) = e i ~ ad τ ′ (cid:0) H + K + ... + K N − + R N + O N +1 (cid:1) = H + K + ... + K N − + R N + i ~ ad τ ′ H + O N +1 . Thus, we look for τ ′ and K N ∈ D N such that: R N = K N + i ~ ad H τ ′ modulo O N +1 . (3.7) SEMICLASSICAL BIRKHOFF NORMAL FORM 15 To solve this equation, we need to study ad H . Since H = P j ˆ β j ( w ) | z j | , i ~ ad H τ ′ = d/ X j =1 (cid:18) ˆ β j ( w ) i ~ ad | z j | ( τ ′ ) + i ~ ad ˆ β j ( τ ′ ) | z j | (cid:19) . Since ˆ β j only depends on w , i ~ ad ˆ β j ( τ ′ ) ∈ O N − , (see formula (3.3)). Hence i ~ ad H τ ′ = d/ X j =1 ˆ β j ( w ) i ~ ad | z j | ( τ ′ ) + O N +1 . Thus equation (3.7) can be rewritten R N = K N + T ( τ ′ ) + O N +1 , (3.8)with the notation T = d/ X j =1 ˆ β j ( w ) i ~ ad | z j | . From formula (3.4) we see that T acts on monomials as T ( c ( w ) z α ¯ z γ ) = h α − γ, ˆ β ( w ) i c ( w ) z α ¯ z γ . (3.9)Thus, if we write R N = X | α | + | γ | +2 ℓ = N r αγℓ ( w ) z α ¯ z γ ~ ℓ , we choose K N = X α = γ r αγℓ | z | α ~ ℓ , which commutes with | z j | ( ≤ j ≤ d/ ). The rest R N − K N is a sum ofmonomials of the form r αγℓ z α ¯ z γ ~ ℓ with α = γ . As soon as < | α − γ | < r , we have h α − γ, ˆ β ( w ) i 6 = 0 (by (1.12) because r is lower than the resonance order (1.9)), sowe can define the smooth coefficient c αγℓ ( w ) = r αγℓ ( w ) h α − γ, ˆ β ( w ) i . Thus (3.9) yields to T ( c αγℓ z α ¯ z γ ~ ℓ ) = r αγℓ ( w ) z α ¯ z γ ~ ℓ , so R N − K N is in the range of T modulo O N +1 because N ≤ r − . Hence we solvedequation (3.8), and thus we can iterate until N = r − . The series ρ is the O r thatremains: e i ~ − ad τN ( H + γ ) = H + K + ... + K r − + ρ. (cid:3) The Semiclassical Birkhoff Normal Form The next step is to quantize Theorems 1.1 and 3.1. Quantization of Theorem 1.1. Theorem 1.1 gives a symplectomorphism Φ reducing H to ˆ H = 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 integraloperator V ~ : L ( R d ( x,y ) ) → L ( M ) , associated to the symplectomorphism Φ , and a pseudo-differential operator b L ~ withprincipal symbol ˆ H on V × B z ( ε ) and subprincipal symbol , such that: V ∗ ~ L ~ V ~ = b L ~ , (4.1) V ∗ ~ V ~ = I microlocally on V × B z ( 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 pseudodifferentialoperator b L ~ , which has a total symbol of the form σ ~ = ˆ H + ~ ˜ r ~ on V × B z ( ε ) . (4.4)In particular, σ w, T ~ (cid:16) b L ~ (cid:17) = H + γ for some γ ∈ O , with the notation of section 3.2.We want to construct a normal form using a bounded pseudodifferential operator Q ~ : e i ~ Q ~ b L ~ e − i ~ Q ~ = N ~ + R ~ . (4.5)In Theorem 3.1, applied to γ , we have constructed formal series τ , κ , and ρ suchthat e i ~ ad τ ( H + γ ) = H + 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 provethe following Theorem. Theorem 4.1. For ~ ∈ (0 , ~ ] small enough, there exist a unitary operator U ~ : L ( R d ) → L ( R d ) , a smooth function f ⋆ ( w, I , ..., I d/ , ~ ) , and a pseudodifferential operator R ~ suchthat: ( i ) U ∗ ~ b L ~ U ~ = L ~ + Op w ~ f ⋆ ( w, I (1) ~ , ..., I ( d/ ~ , ~ ) + R ~ , ( ii ) f ⋆ has an arbitrarily small compact ( I , ..., I d/ , ~ ) -support (containing 0), ( iii ) σ w, T ~ ( R ~ ) ∈ O r and σ w, T ~ ( U ~ R ~ U ∗ ~ ) ∈ O r . with I ( j ) ~ = Op w ~ ( | z j | ) and L ~ = Op w ~ ( H ) . We call N ~ = L ~ + Op w ~ f ⋆ ( w, I (1) ~ , ..., I ( d/ ~ , ~ ) (4.6) the normal form, and R ~ the remainder. SEMICLASSICAL BIRKHOFF NORMAL FORM 17 Proof. The pseudodifferential operator b L ~ defined by (4.1) has a symbol of the form σ ~ = ˆ H + ~ ˜ r ~ on V × B z ( ε ) , so σ ~ = H + r ~ with γ := [ r ~ ] ∈ O . We apply Theorem 3.1 with this γ ∈ O . Theformal series κ ∈ O that we get commutes with | z j | ( ≤ j ≤ d/ ), so by formula(3.4) we can write it κ = X k ≥ X l + | m | = k c l,m ( w ) | z | m ... | z d/ | m d/ ~ l , and we can change the coefficients to get κ = X k ≥ X l + | m | = k c ⋆l,m ( w )( | ˆ z | ) ⋆m ... ( | z d/ | ) ⋆m d/ ~ l . We define functions: f ( w, I , ..., I d/ , ~ ) with Taylor series X k ≥ X l + | m | = k c l,m ( w ) I m ...I m d/ d/ ~ l ,f ⋆ ( w, I , ..., I d/ , ~ ) with Taylor series X k ≥ X l + | m | = k c ⋆l,m ( w ) I m ...I m d/ d/ ~ l , and arbitrarily small compact support in ( I , ..., I d/ , ~ ) (containing ).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: e i ~ Op w ~ ( c ) Op w ~ ( H + r ~ ) e − i ~ Op w ~ ( c ) = r − X n =0 n ! ad ni ~ − Op w ~ ( c ) Op w ~ ( H + r ~ )+ Z r − − t ) r − e it ~ − Op w ~ ( c ) ad ri ~ − Op w ~ ( c ) Op w ~ ( H + r ~ ) e − it ~ − Op w ~ ( c ) d t. By the Egorov Theorem and the fact that ad ri ~ − Op w ~ ( c ) : E → O r (see (3.2)), theintegral remainder has a symbol with Taylor series in O r . Moreover, σ w, T ~ r − X n =0 n ! ad ni ~ − Op w ~ ( c ) Op w ~ ( H + r ~ ) ! = r − X n =0 n ! ad ni ~ − τ ( H + γ )= e i ~ ad τ ( H + γ ) + O r = H + κ + O r . Thus, by the definition of f , there exists s ( w, z, ~ ) such that [ s ] ∈ O r and: e i ~ Op w ~ ( c ) Op w ~ ( H + r ~ ) e − i ~ Op w ~ ( c ) = Op w ~ ( H ) + Op w ~ ( f ( w, | z | , ..., | z d/ | , ~ )) + Op w ~ ( s ) . Using the compatibility of the quantization with the Moyal product, we have σ w, T ~ ( f ⋆ ( w, I (1) ~ , ..., I ( d/ ~ , ~ )) = [ f ( w, | z | , ..., | z d/ | , ~ )] , so we get: e i ~ Op w ~ ( c ) Op w ~ ( H + r ~ ) e − i ~ Op w ~ ( c ) = Op w ~ ( H ) + Op w ~ ( f ⋆ ( w, I (1) ~ , ..., I ( d/ ~ , ~ )) + Op w ~ (˜ s ) , for a new symbol ˜ s ( w, z, ~ ) with [˜ s ] ∈ O r . Hence we get U ∗ ~ b L ~ U ~ = Op w ~ ( H ) + Op w ~ ( f ⋆ ( w, I (1) ~ , ..., I ( d/ ~ , ~ )) + Op w ~ (˜ s ) , with U ~ = e − i ~ Op w ~ ( c ) . To prove ( iii ) with R ~ = Op w ~ (˜ s ) , note that σ w, T ~ ( R ~ ) = [˜ s ] ∈ O r and σ w, T ~ ( U ~ R ~ U ∗ ~ ) = e i ~ ad τ ([˜ s ]) ∈ O r . (cid:3) Theorem 1.2 follows with the new operator ˜ U ~ = V ~ U ~ given by (4.1) and Theorem4.1. 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 , , up to reducing the support of f ⋆ , the normalform N ~ of Theorem 4.1 satisfies for ~ ∈ (0 , ~ ] small enough: (1 − ζ ) hL ~ ψ, ψ i ≤ hN ~ ψ, ψ i ≤ (1 + ζ ) hL ~ ψ, ψ i , ∀ ψ ∈ S ( R d ) . Proof. For a given K > , we can take a cutoff function χ supported in { λ ∈ R d/ : k λ k ≤ K } , and change f ⋆ into χf ⋆ . Thus, for λ j ∈ sp ( I ( j ) ~ ) , | χf ⋆ ( w, λ , ..., λ d/ , ~ ) | ≤ CK k λ k≤ CK X j β j ˆ β j ( w ) λ j ≤ ˜ CK X j ˆ β j ( w ) λ j . Hence, using functional calculus and the G ◦ arding inequality, we deduce that |h Op w ~ f ∗ ( w, I (1) ~ , ..., I ( d/ ~ , ~ ) ψ, ψ i| ≤ ˜ CK hL ~ ψ, ψ i + c ~ k ψ k ≤ ζ hL ~ ψ, ψ i , for K and ~ small enough. (cid:3) Spectral reduction of N ~ In this section, we prove an expansion of the first eigenvalues of N ~ in powers of ~ / . 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 ≤ j ≤ d/ . For n j ≥ , we denote h n j : R → R the n j -th Hermite functionof the variable x j . In particular, for every ≤ j ≤ d/ we have: I ( j ) ~ h n j ( x j ) = ~ (2 n j + 1) h n j ( x j ) . (5.1)Moreover, ( h n j ) n j ≥ is a Hilbertian basis of L ( R x j ) : L ( R x j ) = M n j ≥ h h n j i . On R d/ x , we define the functions h n for any n = ( n , ..., n d/ ) ∈ N d/ by h n ( x ) = h n ⊗ ... ⊗ h n d/ ( x ) = h n ( x ) ...h n d/ ( x d/ ) . SEMICLASSICAL BIRKHOFF NORMAL FORM 19 We have the following space decomposition: L ( R d/ x ) = M n ∈ N d/ h h n i . In particular, we have: L ( R dx,y ) = M n ∈ N d/ (cid:0) L ( R d/ y ) ⊗ h h n i (cid:1) . (5.2)Since N ~ commutes with the harmonic oscillators I ( j ) ~ (1 ≤ j ≤ d/ , it is reducedin the decomposition (5.2). More precisely, Lemma 5.1. For n = ( n , ..., n d/ ) ∈ N d/ , there exists a classical pseudodifferentialoperator N ( n ) ~ acting on L ( R d/ y ) such that: N ~ ( u ⊗ h n ⊗ ... ⊗ h n d/ ) = N ( n ) ~ ( u ) ⊗ h n ⊗ ... ⊗ h n d/ , ∀ u ∈ S ( R d/ y ) . Its symbol is: F ( n ) ( w ) = ~ d/ X j =1 ˆ β j ( w )(2 n j + 1) + f ⋆ ( w, ~ (2 n + 1) , ~ ) , and we have: sp ( N ~ ) = [ n sp ( N ( n ) ~ ) . Moreover, the multiplicity of λ as eigenvalue of N ~ is the sum over n of the multi-plicities of λ as eigenvalue of N ( n ) ~ . This follows directly from (5.1) and (4.6). Moreover, we can prove the followingmore precise inclusions of the spectra. Lemma 5.2. Let b ∈ ( b , ˜ b ) . There exist ~ , n max , c > such that, for any ~ ∈ (0 , ~ ) : sp ( N ~ ) ∩ ( −∞ , b ~ ] ⊂ [ ≤| n |≤ n max sp ( N ( n ) ~ ) , (5.3) and for any n ∈ N d/ with ≤ | n | ≤ n max : sp ( N ( n ) ~ ) ⊂ [ ~ ( b + c | n | ) , + ∞ ) . (5.4) Proof. Remember that the functions ˆ β j are bounded from below by a positive con-stant. Thus, the G ◦ arding inequality implies that there are ~ , c > such that, forevery ~ ∈ (0 , ~ ) , h Op w ~ ( ˆ β j ) u, u i ≥ c k u k , ∀ u ∈ L ( R d/ y ) . (5.5)For any n ∈ N d/ , we have: hN ( n ) ~ u, u i = hN ~ ( u ⊗ h n ) , u ⊗ h n i≥ (1 − ζ ) hL ~ ( u ⊗ h n ) , u ⊗ h n i by Proposition . 1= (1 − ζ ) d/ X j =1 ~ (2 n j + 1) h Op w ~ ( ˆ β j ) u, u i because L ~ = P j Op w ~ ( ˆ β j ) I ( j ) ~ . Thus using (5.5) and the G ◦ arding inequality, hN ( n ) ~ u, u i ≥ ~ (1 − ζ )(2 c | n |k u k + h Op w ~ (ˆ b ) u, u i ) ≥ ~ (1 − ζ )(2 c | n | + b − ˜ c ~ ) k u k . This proves (5.4) for a new c > . Moreover, if you take any eigenpair ( λ, ψ ) of N ~ with λ ≤ b ~ , it is an eigenpair of some N ( n ) ~ , with ψ = u ⊗ h n , and: ~ (1 − ζ )(2 c | n | + b − ˜ c ~ ) k u k ≤ hN ( n ) ~ u, u i = hN ~ ψ, ψ i ≤ b ~ k ψ k . Thus, there is a n max > independent of ~ , λ, ψ such that | n | ≤ n max . We deduce (5.3). (cid:3) Using the previous Lemma and the well-known expansion of the first eigenvaluesof Op w ~ (ˆ b ) , we deduce an expansion of the first eigenvalues of N ~ . Theorem 5.1. Let ε > and N ≥ . There exist ~ > and c > such that,for ~ ∈ (0 , ~ ] , the N first eigenvalues of N ~ : ( λ j ( ~ )) ≤ j ≤ N admit an expansion inpowers of ~ / of the form: λ j ( ~ ) = ~ b + ~ ( E j + c ) + ~ / c j, + ~ c j, + ..., where ~ E j is the j -th eigenvalue of the d/ -dimensional harmonic oscillator associ-ated to the Hessian of ˆ b at , counted with multiplicity.Proof. The smallest eigenvalues of N ~ are those of N (0) ~ , which has the symbol ~ ˆ b ( w ) + f ⋆ ( w, ~ , ..., ~ ) = ~ (ˆ b ( w ) + ~ c + O ( ~ )) . The first eigenvalues of a semiclassical pseudodifferential operator with principalsymbol ˆ b (which admits a unique and non-degenerate minimum) have an expansionof the form: µ j ( ~ ) = b + ~ E j + ~ / X m ≥ a j,m ~ m/ , (5.6)where ~ E j is the j -th eigenvalue of the d/ -dimensional harmonic oscillator associ-ated to the Hessian of ˆ b at the minimum. Let us recall the idea of the proof of thisresult. Since the minimum of ˆ b is non degenerate, we can write ˆ b ( w ) = b + 12 Hess ˆ b ( w, w ) + O ( | w | ) . A linear symplectic change of coordinates changes Hess ˆ b into d/ X j =1 ν j ( y j + η j ) , for some positive numbers ( ν j ) ≤ j ≤ d/ . In these coordinates the symbol becomes ˆ b ( y, η ) = b + d/ X j =1 ν j ( y j + η j ) + O ( | w | ) + O ( ~ ) , SEMICLASSICAL BIRKHOFF NORMAL FORM 21 and Helffer-Sjöstrand proved in [11] that the first eigenvalues of a pseudo-differentialoperator with such a symbol admits an expansion in powers of ~ / . 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]. (cid:3) 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 − (0) since the rest R ~ can belarge 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 K = { b ( q ) ≤ ˜ b } ⊂ Ω . For ε > , we denote K ε = { q : d( q, K ) ≤ ε } . (6.1)For ε > small enough, K ε ⊂ Ω .The following Theorem states the well-known Agmon estimates (see Agmon’spaper [1]), which gives exponential decay of the eigenfunctions of the magneticLaplacian L ~ outside the minimum q of the magnetic intensity b . In particular,these eigenfunctions are localized in Ω . Theorem 6.1 (Agmon estimates) . Let α ∈ (0 , / and b < b < ˜ b . There exist C, ~ > such that for all ~ ∈ (0 , ~ ] and for all eigenpair ( λ, ψ ) of L ~ with λ ≤ ~ b ,we have: Z M | e d ( q,K ) ~ − α ψ | d q ≤ C k ψ k . In particular, if χ : M → [0 , is a smooth function being on K ε , ψ = χ ψ + O ( ~ ∞ ) in L ( M ) . Proof. If Φ : M → R is a Lipschitz function such that e Φ ψ belongs to the domainof q ~ , the Agmon formula (Theorem A.3 in Appendix), q ~ ( e Φ ψ ) = λ k e Φ ψ k + ~ k dΦ e Φ ψ k , together with the Assumption 1, (1 + ~ / C ) q ~ ( e Φ ψ ) ≥ Z ~ ( b ( q ) − ~ / C ) | e Φ ψ | d q g , yields to: Z (cid:2) ~ ( b ( q ) − ~ / C ) − (1 + ~ / C )( λ + ~ | dΦ | ) (cid:3) | e Φ ψ | d q g ≤ . We split this integral into two parts: Z K c (cid:2) ~ ( b ( q ) − ~ / C ) − (1 + ~ / C )( λ + ~ | dΦ | ) (cid:3) | e Φ ψ | d q g ≤ Z K (cid:2) − ~ ( b ( q ) − ~ / C ) + (1 + ~ / C )( λ + ~ | dΦ | ) (cid:3) | e Φ ψ | d q g . We choose Φ : Φ m ( q ) = χ m ( d ( q, K )) ~ − α for m > , where χ m ( t ) = t for t < m , χ m ( t ) = 0 for t > m , and χ ′ m uniformly bounded withrespect to m . Since Φ m ( q ) = 0 on K and b ( q ) − ~ / C ≥ , we have: Z K c (cid:2) ~ ( b ( q ) − ~ / C ) − (1 + ~ / C )( λ + ~ | dΦ m | ) (cid:3) | e Φ m ψ | d q g ≤ C ~ k ψ k . Morever, λ ≤ b ~ and | dΦ m | ≤ ˜ C ~ − α : Z K c h ~ ( b ( q ) − ~ / C ) − (1 + ~ / C )( b ~ + ˜ C ~ − α ) i | e Φ m ψ | d q g ≤ C ~ k ψ k . Thus, up to changing the constant C : Z K c ~ (˜ b − b − ~ / C − ˜ C ~ − α ) | e Φ m ψ | d q ≤ C ~ k ψ k . Since ˜ b > b , we have ˜ b − b − ~ / C − ˜ C ~ − α > for ~ small enough. Hence Z K c | e Φ m ψ | d q ≤ C k ψ k , and since Φ m = 0 on K : Z | e Φ m ψ | d q ≤ ( C + 1) k ψ k . By Fatou’s lemma in the limit m → + ∞ , Z | e d ( q,K ) ~ − α ψ | d q ≤ ( C + 1) k ψ k . To prove the second result, notice that k ψ − χ ψ k = Z χ =1 | (1 − χ ) ψ | d q ≤ Z χ =1 | ψ | d q ≤ Z K cε | ψ | d q ≤ e − ε ~ − α Z K cε | e d ( q,K ) ~ − α ψ | d q ≤ Ce − ε ~ − α k ψ k = O ( ~ ∞ ) . (cid:3) Now we prove the microlocalization of the eigenfunctions of L ~ near Σ . Theorem 6.2. Let ε > , δ ∈ (0 , ) , and < b < ˜ b . Let χ : M → [0 , be asmooth function being on K ε . Let χ : R → [0 , be a smooth compactly supportedcutoff function being near . Then for any normalized eigenpair ( λ, ψ ) of L ~ suchthat λ ≤ ~ b we have: ψ = χ ( ~ − δ L ~ ) χ ( q ) ψ + O ( ~ ∞ ) in L ( M ) . SEMICLASSICAL BIRKHOFF NORMAL FORM 23 Proof. Using Theorem 6.1, we have ψ = χ ψ + O ( ~ ∞ ) in L ( M ) . Since χ ( ~ − δ L ~ ) is a bounded operator, we get: χ ( ~ − δ L ~ ) ψ = χ ( ~ − δ L ~ ) χ ψ + O ( ~ ∞ ) in L ( M ) . In fact, ψ = χ ( ~ − δ L ~ ) ψ. Indeed, there exists a C > such that χ ( ~ − δ . ) = 1 on B (0 , C ~ δ ) , and for ~ ∈ (0 , ~ ) small enough, λ ∈ B (0 , b ~ ) ⊂ B (0 , C ~ δ ) . Thus, χ ( ~ − δ L ~ ) ψ = χ ( ~ − δ λ ) ψ = ψ. (cid:3) Microlocalization of the eigenfunctions of N ~ . The next two theoremsstates the microlocalization of the eigenfunctions of the normal form. We recall thatif ϕ is defined by Theorem 1.1, we have: ϕ ( K ) = { w ∈ V : ˆ b ( w ) ≤ ˜ b } , with ˆ b ( w ) = b ◦ ϕ − ( w ) . We also recall the definition (6.1) of K ε . This first lemmagives a microlocalization result on the w variable. Lemma 6.1. Let ~ ∈ (0 , ~ ] and b ∈ (0 , ˜ b ) . Let χ be a smooth cutoff function on R dw supported on V such that χ = 1 on ϕ ( K ε ) . Then for any normalized eigenpair ( λ, ψ ) of N ~ such that λ ≤ ~ b , we have: ψ = Op w ~ ( χ ) ψ + O ( ~ ∞ ) in L ( R dx,y ) . Proof. Let χ = 1 − χ , which is supported in ϕ ( K ε ) c . The eigenvalue equation yieldsto hN ~ Op w ~ ( χ ) ψ, Op w ~ ( χ ) ψ i ≤ b ~ k Op w ~ ( χ ) ψ k + h [ N ~ , Op w ~ ( χ )] ψ, Op w ~ ( χ ) ψ i . (6.2)Using Lemma 5.1, we can write ψ = u ⊗ h n for some n ∈ N d/ , u ∈ L ( R d/ w ) , with ≤ | n | ≤ n max . Then [ N ~ , Op w ~ ( χ )] ψ = [ N ( n ) ~ , Op w ~ ( χ )] u ⊗ h n = ~ d/ X j =1 (2 n j + 1) Op w ~ ( ˆ β j ) , Op w ~ ( χ ) ψ + O ( ~ ) , because the principal symbol of N ( n ) ~ is P d/ j =1 ~ (2 n j + 1) ˆ β j . Since the symbol of thecommutator is of order ~ and supported in supp χ , we have h [ N ~ , Op w ~ ( χ )] ψ, Op w ~ ( χ ) ψ i ≤ C ~ k Op w ~ ( ¯ χ ) ψ k , (6.3)where ¯ χ is a small extension of χ , with value on supp χ and on a neighborhoodof ϕ ( K ε ) . Moreover using Proposition 4.1, hN ~ Op w ~ ( χ ) ψ, Op w ~ ( χ ) ψ i ≥ (1 − ζ ) hL ~ Op w ~ ( χ ) ψ, Op w ~ ( χ ) ψ i≥ (1 − ζ ) ~ ˜ b k Op w ~ ( χ ) ψ k , where we used the G ◦ arding inequality because, the symbol of L ~ is greater than ˜ b on supp χ . Together with (6.2) and (6.3), we get ~ (cid:16) (1 − ζ )˜ b − b (cid:17) k Op w ~ ( χ ) ψ k ≤ C ~ k Op w ~ ( ¯ χ ) ψ k . For η small enough, (1 − ζ )˜ b > b . Hence, dividing by ~ and iterating with ¯ χ insteadof χ , we get k Op w ~ ( χ ) ψ k = O ( ~ ∞ ) . (cid:3) Now we prove the microlocalization of the eigenfunctions of N ~ on a neighborhoodof ϕ (Σ) = { ( z, w ) : z = 0 } . Theorem 6.3. Let ~ ∈ (0 , ~ ] , b ∈ (0 , ˜ b ) , and δ ∈ (0 , / . Let χ be a smoothcutoff function on R d/ w supported on V such that χ = 1 on ϕ ( K ε ) and χ a realcutoff function being near . Then for any normalized eigenpair ( λ, ψ ) of N ~ suchthat λ ≤ ~ b , we have: ψ = χ ( ~ − δ I (1) ~ ) ...χ ( ~ − δ I ( d/ ~ ) Op w ~ ( χ ( w )) ψ + O ( ~ ∞ ) in L ( R d ) . Proof. According to Lemma 6.1, ψ = Op w ~ ( χ ) ψ + O ( ~ ∞ ) . Since χ d/ ( ~ − δ I ~ ) := χ ( ~ − δ I (1) ~ ) ...χ ( ~ − δ I ( d/ ~ ) is a bounded operator, we have χ d/ ( ~ − δ I ~ ) ψ = χ d/ ( ~ − δ I ~ ) Op w ~ ( χ ) ψ + O ( ~ ∞ ) . It remains to prove that ψ = χ d/ ( ~ − δ I ~ ) ψ for ~ small enough. Using Lemma 5.1, ψ = u ⊗ h n for some u ∈ L ( R d/ y ) , n ∈ N d/ with ≤ | n | ≤ n max , and so χ d/ ( ~ − δ I ~ ) ψ = χ ( ~ − δ (2 n + 1)) ...χ ( ~ − δ (2 n d/ + 1)) ψ. But χ = 1 on a neighborhood of , so there is ~ > such that, for any ~ ∈ (0 , ~ ] and any ≤ | n | ≤ n max , χ ( ~ − δ (2 n + 1)) ...χ ( ~ − δ (2 n d/ + 1)) = 1 . Thus, ψ = χ d/ ( ~ − δ I ~ ) ψ. (cid:3) Rank of the spectral projections. We want the microlocalization Theorems6.2 and 6.3 to be uniform with respect to λ ∈ ( −∞ , b ~ ] . That is why we need therank of the spectral projections to be bounded by some finite power of ~ − . If A is a bounded from below self-adjoint operator, and α ∈ R , we denote N ( A , α ) thenumber of eigenvalues of A smaller than α , counted with multiplicities. It is therank of the spectral projection ] −∞ ,α ] ( A ) .The proof of the following estimate is inspired by the proof of Lemma A.4 inAppendix, adapted from [9]. The idea is to locally approximate the magnetic fieldto a constant. SEMICLASSICAL BIRKHOFF NORMAL FORM 25 Lemma 6.2. Let b < b < ˜ b . There exists C > and ~ > such that for all ~ ∈ (0 , ~ ] , we have: N ( L ~ , ~ b ) ≤ C ~ − d/ . Proof. Take ( χ m ) m ≥ a smooth partition of unity, such that: X m ≥ χ m ( q ) = 1 and X m ≥ | d χ m ( q ) | ≤ C, ∀ q ∈ M, with supp ( χ m ) ⊂ V m a local chart. Then, by Lemma A.5 (in Appendix), for any ψ ∈ D ( q ~ ) , q ~ ( ψ ) = X m ≥ q ~ ( χ m ψ ) − ~ X m ≥ k ψ d χ m k ≥ X m ≥ q ~ ( χ m ψ ) − C ~ k ψ k . Since K = { b ( q ) ≤ ˜ b } is compact, there is a m > such that, for m > m : q ~ ( χ m ψ ) ≥ ~ Z (cid:0) b ( q ) − ~ / C (cid:1) | χ m ψ | d q g (6.4) q ~ ( χ m ψ ) ≥ ~ (˜ b − ~ / C ) k χ m ψ k ≥ ~ b k χ m ψ k , (6.5)for ~ small enough. For ≤ m ≤ m , we can work like in R d using the charts, andwe can find a new partition of unity on V m such that J X j =0 | χ ~ m,j | = 1 , and J X j =0 | d χ ~ m,j ( x ) | ≤ C ~ − α , (6.6)where C > does not depend on m , and with supp ( χ ~ m,j ) ⊂ B m,j := { x : | x − z m,j | ≤ ~ α } . Thus we have for ≤ m ≤ m : q ~ ( χ m ψ ) ≥ J X j =0 q ~ ( χ m,j ψ ) − C ~ − α k χ m ψ k . (6.7)On each B m,j , we will approximate the magnetic field by a constant. Up to a gaugetransformation, we can assume that the vector potential vanishes at z m,j . In otherwords, we can find a smooth function ϕ m,j on B m,j such that ˜ A ( z m,j ) = 0 , where ˜ A = A + ∇ ϕ m,j . The potential ˜ A defines the same magnetic field B as A .Let us define A lin ( x ) = B ( z m,j ) . ( x − z m,j ) , so that | ˜ A ( x ) − A lin ( x ) | ≤ C | x − z m,j | on B m,j . (6.8)Then if ˜ q ~ denotes the quadratic form for the new potential ˜ A , for v ∈ C ∞ ( B m,j ) , ˜ q ~ ( v ) = q lin ~ ( v ) + k ( ˜ A − A lin ) v k + 2 ℜh ( ˜ A − A lin ) v, ( i ~ ∇ + A lin ) v i , and using (6.8) and the Cauchy-Schwarz inequality, ˜ q ~ ( v ) ≥ q lin ~ ( v ) − C k| x − z m,j | v k q q lin ~ ( v ) . We use | ab | ≤ ε a + ε − b to get: ˜ q ~ ( v ) ≥ (cid:0) − C ~ β (cid:1) q lin ~ ( v ) − C ~ − β k| x − z m,j | v k and so ˜ q ~ ( v ) ≥ (cid:0) − C ~ β (cid:1) q lin ~ ( v ) − C ~ α − β k v k . (6.9)Changing A into ˜ A amounts to conjugate the magnetic Laplacian by e i ~ − ϕ j,m , so: ˜ q ~ ( v ) = q ~ ( e i ~ − ϕ j,m v ) . Hence, for any v ∈ C ∞ ( B m,j ) , q ~ ( v ) ≥ (cid:0) − C ~ β (cid:1) q lin ~ ( e − i ~ − ϕ j,m v ) − C ~ α − β k v k . (6.10) q lin ~ ( v ) is the quadratic form associated to a constant magnetic field operator. Now,we approximate the metric with a flat one: q lin ~ ( v ) = X k,l Z | g ( x ) | / g kl ( x )( i ~ ∂ k v + A link v )( i ~ ∂ l v + A linl v )d x (6.11) ≥ (1 − C ~ α ) X k,l Z | g ( z m,j ) | / g kl ( z m,j )( i ~ ∂ k v + A link v )( i ~ ∂ l v + A linl v )d x, (6.12) = (1 − C ~ α ) q flat ~ ( v ) . (6.13)Hence, from (6.7) and (6.5) we get: q ~ ( ψ ) ≥ m X m =0 J X j =0 q ~ ( χ ~ m,j ψ ) − m X m =0 C ~ − α k χ m ψ k − C ~ k ψ k + X m>m ~ (˜ b − C ~ / ) k χ m ψ k , and using (6.10) and (6.13): q ~ ( ψ ) ≥ (1 − C ~ β )(1 − C ~ α ) m X m =0 J X j =0 q flat ~ ( e − i ~ − ϕ j,m χ ~ m,j ψ ) − C ~ α − β m X m =0 J X j =0 k χ ~ m,j ψ k − ( C ~ − α + C ~ + C ~ / ) k ψ k + X m>m ~ ˜ b k χ m ψ k ≥ (1 − C ~ β )(1 − C ~ α ) m X m =0 J X j =0 q flat ~ ( e − i ~ − ϕ j,m χ ~ m,j ψ ) + X m>m ~ ˜ b k χ m ψ k − ( C ~ α − β + C ~ − α + C ~ + C ~ / ) k ψ k . SEMICLASSICAL BIRKHOFF NORMAL FORM 27 Then q ~ ( ψ ) ≥ (1 − C ~ β )(1 − C ~ α ) q L (( χ j,m ψ ) j,m , ( χ m ψ ) m>m ) − K ( ~ ) k ψ k , (6.14)where K ( ~ ) = C ~ α − β + C ~ − α + C ~ + C ~ / and q L (( ψ j,m ) j,m , ( ψ m ) m>m ):= m X m =0 J X j =0 q flat ~ ( e i ~ − ϕ j,m ψ j,m ) + (1 − C ~ β ) − (1 − C ~ α ) − X m>m ~ ˜ b k ψ m k is the quadratic form associated to L = m M m =0 J M j =0 L m,j ! ⊕ M m>m L m ! , where L m,j is a Schrödinger operator with constant magnetic field acting on L ( B m,j ) ,and L m is the multiplication by (1 − C ~ β ) − (1 − C ~ α ) − ~ ˜ b acting on L ( V m ) .We test inequality (6.14) on the N ( L ~ , b ~ ) -dimensional space V spanned by the N ( L ~ , b ~ ) first eigenfunctions of L ~ (Corresponding to eigenvalues ≤ ~ b ). For ψ ∈ V , (1 − C ~ β )(1 − C ~ α ) q L (( χ j,m ψ ) j,m , ( χ m ψ ) m>m ) ≤ ( b ~ + K ( ~ )) k ψ k . Then, since ψ (( χ j,m ψ ) ≤ m ≤ m , ≤ j ≤ J , ( χ m ψ ) m>m ) is one-to-one, the space ( (( χ j,m ψ ) j,m , ( χ m ψ ) m ) ∈ M j,m L ( B m,j ) ! ⊕ M m>m L ( V m ) ! ; ψ ∈ V ) is N ( L ~ , b ~ ) -dimensional, and the min-max principle yields to: N ( L ~ , b ~ ) ≤ N (cid:16) L , ( ~ b + K ( ~ ))(1 − C ~ β ) − (1 − C ~ α ) − (cid:17) . Since L m,j is a magnetic Laplacian with constant magnetic field, we know that, for ~ small enough: N ( L m,j , O ( ~ )) = O ( ~ − d/ ) , ≤ m ≤ m , ≤ j ≤ J, and N ( L m , ~ b + o ( ~ )) = 0 , m > m . With α = 3 / and β = 1 / , K ( ~ ) = o ( ~ ) , so we deduce: N ( L ~ , ~ b ) = O ( ~ − d/ ) . (cid:3) The same result holds for N ~ : Lemma 6.3. Let b ∈ (0 , ˜ b ) . There exists C > and ~ > such thatfor all ~ ∈ (0 , ~ ) , N ( N ~ , ~ b ) ≤ C ~ − d/ . Proof. By Lemma 4.1, we have: hN ~ ψ, ψ i ≥ (1 − ζ ) hL ~ ψ, ψ i ≥ (1 − ζ ) ~ hB ~ ψ, ψ i , with B ~ = Op w ~ (ˆ b ) . Using the min-max principle, it follows that N ( N ~ , ~ b ) ≤ N ( B ~ , (1 − ζ ) − b ) , and using Weyl estimates ([6] Chapter 9, or [13]), we get N ( B ~ , (1 − ζ ) − b ) = O ( ~ − d/ ) . (cid:3) Comparison of the spectra of L ~ and N ~ Proof of Theorem 1.3. We denote λ ( ~ ) ≤ λ ( ~ ) ≤ ... the smallest eigenvalues of L ~ and ν ( ~ ) ≤ ν ( ~ ) ≤ ... the smallest eigenvalues of N ~ . The goal of this section is to prove the followingtheorem, using the results of section 6. Theorem 7.1. If b < ˜ b and δ ∈ (0 , / , then λ n ( ~ ) = ν n ( ~ ) + O ( ~ δr ) , uniformly in n such that λ n ( ~ ) ≤ ~ b and ν n ( ~ ) ≤ ~ b . Together with Theorem 5.1, this theorem concludes the proofs of Theorems 1.3and 1.5. Proof. We will prove that ν n ( ~ ) ≤ λ n ( ~ ) + O ( ~ δr ) , the other inequality being similar.Let ≤ n ≤ N ( L ~ , ~ b ) , and let us denote ψ , ~ , ..., ψ n, ~ the normalized eigenfunctionsassociated to the first eigenvalues of L ~ . We also denote V n, ~ = span { χ ( ~ − δ L ~ ) χ ( q ) ψ j, ~ : 1 ≤ j ≤ n } , where χ and χ are defined in Theorem 6.2. We have the normal form: ˜ U ∗ ~ L ~ ˜ U ~ = N ~ + R ~ , (7.1)where ˜ U ~ = V ~ U ~ , is given by (4.1) and Theorem 4.1.We will use the min-max principle. For ψ ∈ span ≤ j ≤ n ψ j, ~ , we denote ˜ ψ = χ ( ~ − δ L ~ ) χ ( q ) ψ ∈ V n, ~ Such a ˜ ψ is microlocalized on Ω ~ ⊂ U ⊂ T ∗ M, where Ω ~ = { ( q, p ) ∈ T ∗ M : | p − A ( q ) | < c ~ δ , q ∈ Ω } . (Indeed, the symbol of χ ( ~ − δ L ~ ) is O ( ~ ∞ ) where χ ( ~ − δ | p − A ( q ) | ) ≡ ). Thus,since V ~ V ∗ ~ = I microlocally on U (4.3) and U ~ is unitary, we deduce from (7.1) that: hN ~ ˜ U ∗ ~ ˜ ψ, ˜ U ∗ ~ ˜ ψ i = hL ~ ˜ ψ, ˜ ψ i − h R ~ ˜ U ∗ ~ ˜ ψ, ˜ U ∗ ~ ˜ ψ i + O ( ~ ∞ ) k ˜ ψ k , (7.2) SEMICLASSICAL BIRKHOFF NORMAL FORM 29 On the first hand, by Theorem 6.2, we can change ˜ ψ into ψ up to an error of order ~ ∞ . Indeed, by Lemma 6.2, the estimates of Theorem 6.2 remain true for ψ . Weget: hL ~ ˜ ψ, ˜ ψ i = hL ~ ψ, ψ i + O ( ~ ∞ ) k ψ k ≤ ( λ n ( ~ ) + O ( ~ ∞ )) k ψ k . On the other hand, the remainder is: h R ~ ˜ U ∗ ~ ˜ ψ, ˜ U ∗ ~ ˜ ψ i = h U ~ R ~ U ∗ ~ V ∗ ~ ˜ ψ, V ∗ ~ ˜ ψ i . The function V ∗ ~ ˜ ψ is microlocalized in V ~ = { ( w, z ) : w ∈ V, | z | ≤ c ~ δ } , because V ~ is a Fourier integral operator with phase function associated to the canon-ical transformation Φ , which is sending Ω ~ (where ˜ ψ is microlocalized) on V ~ . More-over, the symbol of the pseudo-differential operator U ~ R ~ U ∗ ~ on V is O (( ~ + | z | ) r/ ) (Theorem 4.1), so we get: U ~ R ~ U ∗ ~ V ∗ ~ ˜ ψ = O ( ~ δr ) . Thus equation (7.2) yields to: hN ~ ˜ U ∗ ~ ˜ ψ, ˜ U ∗ ~ ˜ ψ i ≤ ( λ n ( ~ ) + O ( ~ δr )) k ˜ U ∗ ~ ˜ ψ k , for all ˜ ψ ∈ V n, ~ . Since V n, ~ is n -dimensional, the min-max principle gives ν n ( ~ ) ≤ λ n ( ~ ) + O ( ~ δr ) . The same arguments give the opposite inequality, replacing Theorem 6.2 and Lemma6.2 by Theorem 6.3 and Lemma 6.3. (cid:3) Proof of Corollary 1.1. Let us prove the Weyl estimates stated in Corollary1.1. The proof relies on the classical Weyl asymptotics for pseudo-differential op-erators with elliptic principal symbol ([6] Chapter 9, [13] Appendix). Let us firstprove the Weyl estimates for the Normal form. For any n ∈ N d/ , N ( n ) ~ is a pseudo-differential operator with principal symbol ~ ˆ b [ n ] ( w ) = ~ d/ X j =1 (2 n j + 1) b β j ( w ) . Note that V n := { ˆ b [ n ] ( w ) ≤ b } is empty for all but finitely many n . For these n , the G ◦ arding inequality gives hN ( n ) ~ ψ, ψ i ≥ ~ ( b − c ~ ) k ψ k , ∀ ψ ∈ S ( R d/ ) , so that N ( N ( n ) ~ , b ~ ) = N ( 1 ~ N ( n ) ~ , [ b − c ~ , b ]) which is o ( ~ − d/ ) by the classical Weyl asymptotics. For the other finitely many n , V n ⊂ { ˆ b ( w ) ≤ b } is a compact set with positive volume and thus the classical Weyl asymptotics gives N (cid:16) N ( n ) ~ , b ~ (cid:17) = N (cid:18) ~ N ( n ) ~ , b (cid:19) ∼ π ~ ) d/ Vol ( V n ) . Using sp ( N ~ ) = [ n sp ( N ( n ) ~ ) , we deduce that N ( N ~ , b ~ ) ∼ π ~ ) d/ X n Vol ( V n ) . Moreover, Vol( V n ) = Z V n d y d η = Z ϕ − ( V n ) ϕ ∗ (d y d η ) , where ϕ is defined in Theorem 1.1. Since ϕ is a symplectomorphism, we have B = ϕ ∗ (d η ∧ d y ) and thus B d/ ( d/ d/ ϕ ∗ ((d η ∧ d y ) d/ ) = ϕ ∗ (d y d η ) . Hence Vol( V n ) = Z b [ n ] ( q ) ≤ b B d/ ( d/ , so that N ( N ~ , b ~ ) ∼ π ~ ) d/ X n ∈ N d/ Z b [ n ] ( q ) ≤ b B d/ ( d/ , where the sum is finite. It remains to compare N := N ( N ~ , b ~ ) and N := N ( L ~ , b ~ ) . If we apply Theorem 1.3 with some b + δ > b , we get a c > such that for ~ smallenough, N ( N ~ , ~ b − c ~ r/ − ε ) ≤ N ≤ N ( N ~ , ~ b + c ~ r/ − ε ) , so: | N − N | ≤ N ( N ~ , [ ~ b − c ~ r/ − ε , ~ b + c ~ r/ − ε ]) . Classical Weyl asymptotics gives N ( N ( n ) ~ , [ ~ b − c ~ r/ − ε , ~ b + c ~ r/ − ε ]) = o ( ~ − d/ ) , for any n ∈ N d/ , so | N − N | = o ( ~ − d/ ) , and the proof is complete.8. The case r = ∞ If r = ∞ (where r is defined in (1.9)), there is no resonances: d/ X j =1 α j β j ( q ) = 0 , ∀ α ∈ N d/ , α = 0 . (8.1)Of course, we can take any finite r ≥ , and construct the corresponding normalform. From Theorem 1.5 we deduce that ∀ r ≥ , ∀ j ≥ , λ j ( ~ ) = ~ b + r − X k =4 c j,k ~ k/ + O ( ~ r/ ) , so we get a complete expansion of λ j ( ~ ) in powers of ~ / . However, the normalform depends on r . A natural question is : Could we construct a normal form which SEMICLASSICAL BIRKHOFF NORMAL FORM 31 does not depend on r ? The answer is yes, but we need to restrict to lower energies.Let us describe this construction.The reduction of the classical Hamiltonian does not depend on r , so there isnothing to change. The first problem appear with the formal normal form (Theorem3.1). The problem is that the neighborhood V on which the normal form is validmust reduce as r goes to infinity. So we slightly change our definition of the spaceof formal series O N ( N ≥ . Since the degree of a formal series τ ∈ E = C ∞ ( R dw )[[ x, ξ, ~ ]] depend on w , we define O N to be the set of formal series with valuation at least N on a neighborhood of 0. Then this neighborhood might go to zero as N grows.Then the proof of Theorem (3.1) remains true for r = ∞ , and we get: Theorem 8.1. If γ ∈ O , there exist τ, κ ∈ O such that: • e i ~ ad τ ( H + γ ) = H + κ, • [ κ, | z j | ] = 0 for ≤ j ≤ d/ . Then we can quantize this result exactly as in Theorem 4.1, and we get: Theorem 8.2. For ~ ∈ (0 , ~ ] small enough, there exist a unitary operator U ~ : L ( R d ) → L ( R d ) , a smooth function f ⋆ ( w, I , ..., I d/ , ~ ) , and a pseudodifferential operator R ~ suchthat: ( i ) U ∗ ~ b L ~ U ~ = L ~ + Op w ~ f ⋆ ( w, I (1) ~ , ..., I ( d/ ~ , ~ ) + R ~ , ( ii ) f ⋆ has an arbitrarily small compact ( I , ..., I d/ , ~ ) -support (containing 0), ( iii ) ∀ N ≥ , σ w, T ~ ( R ~ ) ∈ O N and σ w, T ~ ( U ~ R ~ U ∗ ~ ) ∈ O N . with I ( j ) ~ = Op w ~ ( | z j | ) and L ~ = Op w ~ ( H ) . We call N ~ = L ~ + Op w ~ f ⋆ ( w, I (1) ~ , ..., I ( d/ ~ , ~ ) the normal form, and R ~ the remainder. Moreover, up to replacing f ⋆ by χ ( ~ − . ) f ⋆ , (which does not change the propertiesof the normal form because f ⋆ is defined by its Taylor series), we can adapt the proofof Proposition 4.1 to get Lemma 8.1. We can construct the normal form N ~ such that, for ~ ∈ (0 , ~ ] smallenough and some C > : (1 − C ~ ) hL ~ ψ, ψ i ≤ hN ~ ψ, ψ i ≤ (1 + C ~ ) hL ~ ψ, ψ i , ∀ ψ ∈ S ( R d ) . It remains to prove the analog of Theorem 1.3. For this, we need the followingmicrolocalization results. Their proofs follow the same lines as in section 6. Notethe retriction to energies λ ≤ ~ ( b + c ~ η ) , necessary to localize in a neighborhood of q of decreasing size as ~ → . We define, for any fixed c > : K ~ := { q ∈ M : b ( q ) ≤ b + 2 c ~ η } , (8.2) and its small neighborhood K , ~ := { q ∈ M : d( q, K ~ ) ≤ ~ η } . (8.3) Theorem 8.3. Let δ ∈ (0 , ) , c > , and η ∈ (0 , / . Let χ ~ : M → [0 , bea smooth cutoff function being on K , ~ . Let χ : R → [0 , be a smooth cutofffunction being near . Then for any normalized eigenpair ( λ, ψ ) of L ~ such that λ ≤ ~ ( b + c ~ η ) we have: ψ = χ ( ~ − δ L ~ ) χ ~ ( q ) ψ + O ( ~ ∞ ) in L ( M ) , uniformly with respect to ( λ, ψ ) . The proof follows the same lines as Theorem 6.1, with α = 1 / , K replaced by K ~ , K ε replaced by K , ~ , and Theorem 6.2 with no change. The uniformity withrespect to ( λ, ψ ) follows from Lemma 6.2.Similarly, we have the microlocalization Theorem for the normal form N ~ . Wedenote V ~ := { w ∈ R d , d( w, ϕ ( K , ~ )) < ~ } . Theorem 8.4. Let ~ ∈ (0 , ~ ] , c > , η ∈ (0 , / and δ ∈ (0 , η/ . Let χ be asmooth cutoff function on R d/ w supported on V such that χ = 1 near and χ asmooth cutoff function being near . Then for any normalized eigenpair ( λ, ψ ) of N ~ such that λ ≤ ~ ( b + c ~ η ) , we have: ψ = χ ( ~ − δ I (1) ~ ) ...χ ( ~ − δ I ( d/ ~ ) Op w ~ ( χ ( ~ − δ w )) ψ + O ( ~ ∞ ) in L ( R d ) , uniformly with respect to ( λ, ψ ) .Proof. We follow the proof of Lemma 6.1. With χ ( w ) = 1 − χ ( ~ − δ w ) , Inequality(6.2) becomes hN ~ Op w ~ ( χ ) ψ, Op w ~ ( χ ) ψ i ≤ ~ ( b + c ~ η ) k Op w ~ ( χ ) ψ k + h [ N ~ , Op w ~ ( χ )] ψ, Op w ~ ( χ ) ψ i , And the estimate (6.3) on the commutator becomes h [ N ~ , Op w ~ ( χ )] ψ, Op w ~ ( χ ) ψ i ≤ ~ − δ k Op w ~ ( ¯ χ ) ψ k , because the commutator is of order ~ − δ . The lower bound becomes hN ~ Op w ~ ( χ ) ψ, Op w ~ ( χ ) ψ i ≥ (1 − C ~ ) hL ~ Op w ~ ( χ ) ψ, Op w ~ ( χ ) ψ i≥ (1 − C ~ ) ~ ( b + ˜ C ~ δ ) k Op w ~ ( χ ) ψ k . Hence we get h (1 − C ~ )( b + ˜ C ~ δ ) − ( b + c ~ η ) i k Op w ~ ( χ ) ψ k ≤ ~ − δ k Op w ~ ( ¯ χ ) ψ k . Since δ < η , we get a new C > such that for ~ small enough: C ~ δ k Op w ~ ( χ ) ψ k ≤ ~ − δ k Op w ~ ( ¯ χ ) ψ k . Iterating with ¯ χ instead of χ , for δ < / we get Op w ~ ( χ ) ψ = O ( ~ ∞ ) . The end of the proof is the same as the proof of Theorem 6.3. The uniformity withrespect to ( λ, ψ ) comes from Lemma 6.3. (cid:3) SEMICLASSICAL BIRKHOFF NORMAL FORM 33 Since the eigenfunctions of N ~ and L ~ are microlocalized on a neighborhood ofthe minimum of diameter going to as ~ → , we can follow the proof of Theorem1.3 (section 7) to get: Theorem 8.5. Let c > and η ∈ (0 , / . We denote λ ( ~ ) ≤ λ ( ~ ) ≤ ... and ν ( ~ ) ≤ ν ( ~ ) ≤ ... the first eigenvalues of L ~ and N ~ . Then λ n ( ~ ) = ν n ( ~ ) + O ( ~ ∞ ) , uniformly in n such that λ n ( ~ ) ≤ ~ ( b + c ~ η ) and ν n ( ~ ) ≤ ~ ( b + c ~ η ) . Appendix A. Lemma A.1. The principal and subprincipal symbols of the operator L ~ = ( i ~ d + A ) ∗ ( i ~ d + A ) are σ ( L ~ ) = | p − A ( q ) | g ∗ ( q ) , and σ ( L ~ ) = 0 . Proof. We will compute these symbols in coordinates, in which L ~ acts as: L coord ~ = X kℓ | g | − / ( i ~ ∂ k + A k ) g kℓ | g | / ( i ~ ∂ ℓ + A ℓ ) . The principal symbol is always well-defined. The subprincipal symbol is well-definedif we restrict the changes of coordinates to be volume-preserving. This amounts toconjugating L coord ~ by | g | / . Thus the subprincipal symbol is defined in coordinatesby: σ ( L ~ ) = σ ( | g | / L coord ~ | g | − / ) . The total symbol of − i ~ ∂ k − A k is σ ( − i ~ ∂ k − A k ) = p k − A k , so we can use the star product ⋆ on symbols to compute the symbol of L ~ : σ ( | g | / L coord ~ | g | − / ) = X kℓ | g | / ⋆ | g | − / ⋆ ( p k − A k ) ⋆ g kℓ | g | / ⋆ ( p ℓ − A ℓ ) ⋆ | g | − / . Now we will use the formula σ ( f ⋆ g ) = f g + ~ i { f, g } + O ( ~ ) several times to compute the symbol, where { f, g } denotes the Poisson brackets. Ofcourse, we directly deduce the principal symbol: σ ( | g | / L coord ~ | g | − / ) = X kℓ g kℓ ( p k − A k )( p ℓ − A ℓ ) so that σ ( L ~ ) = | p − A ( q ) | g ∗ ( q ) . To compute the subprincipal symbol, we will use: σ ( | g | / L coord ~ | g | − / ) = X kℓ h | g | − / ⋆ ( p k − A k ) ⋆ | g | / i ⋆g kℓ ⋆ h | g | / ⋆ ( p ℓ − A ℓ ) ⋆ | g | − / i . Let us compute a k = | g | − / ⋆ ( p k − A k ) ⋆ | g | / . a k = ( p k − A k ) + ~ i (cid:2) {| g | − / ( p k − A k ) , | g | / } + {| g | − / , p k − A k }| g | / (cid:3) + O ( ~ )= ( p k − A k ) + ~ i (cid:20) | g | − / ∂ | g | / ∂q k − ∂ | g | − / ∂q k | g | / (cid:21) + O ( ~ )= ( p k − A k ) + ~ i | g | − / ∂ | g | / ∂q k + O ( ~ ) . We also get the similar result for b ℓ = | g | / ⋆ ( p ℓ − A ℓ ) ⋆ | g | − / : b ℓ = ( p ℓ − A ℓ ) − ~ i | g | − / ∂ | g | / ∂q ℓ + O ( ~ ) Thus we can compute a k ⋆ g kℓ = g kℓ ( p k − A k ) + ~ i { p k − A k , g kℓ } + ~ i | g | − / ∂ | g | / ∂q k g kℓ + O ( ~ )= g kℓ ( p k − A k ) + ~ i ∂g kℓ ∂q k + ~ i | g | − / ∂ | g | / ∂q k g kℓ + O ( ~ ) , and a k ⋆ g kℓ ⋆ b ℓ = g kℓ ( p k − A k )( p l − A l ) + ~ i { g kℓ ( p k − A k ) , p ℓ − A ℓ } − ~ i g kℓ ( p k − A k ) | g | − / ∂ | g | / ∂q ℓ + ~ i ∂g kℓ ∂q k ( p ℓ − A ℓ ) + ~ i | g | − / ∂ | g | / ∂q k ( p ℓ − A ℓ ) + O ( ~ ) . Summing over k, ℓ , we get X kℓ a k ⋆ g kℓ ⋆ b ℓ = X kℓ g kℓ ( p k − A k )( p l − A l ) + ~ i { g kℓ ( p k − A k ) , p ℓ − A ℓ } + ~ i ∂g kℓ ∂q k ( p ℓ − A ℓ ) + O ( ~ )= X kℓ g kℓ ( p k − A k )( p ℓ − A ℓ ) + ~ i g kℓ ∂ ( p ℓ − A ℓ ) ∂q k − ~ i ∂g kℓ ( p k − A k ) ∂q ℓ + ~ i ∂g kℓ ∂q k ( p ℓ − A ℓ ) + O ( ~ )= X kℓ g kℓ ( p k − A k )( p ℓ − A ℓ ) + O ( ~ ) . Since σ ( | g | / L coord ~ | g | − / ) = X kℓ a k ⋆ g kℓ ⋆ b ℓ , we deduce that: σ ( | g | / L coord ~ | g | − / ) = 0 , and σ ( L ~ ) = 0 . (cid:3) SEMICLASSICAL BIRKHOFF NORMAL FORM 35 The following Lemma due to Weinstein [23] tells that, if two 2-forms coincide ona submanifold, they are equal up to a transformation tangent to the identity. Lemma A.2 (Relative Darboux lemma) . Let ω and ω be two -forms on Ω × R dz which are closed and non degenerate. Assume that ω | z =0 = ω | z =0 . Then there existsa change of coordinates S on a neighborhood of Ω × { } such that S ∗ ω = ω and S = Id + O ( | z | ) . For a proof, see for example [19] and the references therein. The next Lemmastates the Agmon formula (see [1]). Lemma A.3 (Agmon formula) . Let ψ be an eigenfunction of L ~ associated to λ ,and Φ : M → R is a Lipschitz function such that e Φ ψ be in the domain of q ~ , then dΦ is defined almost everywhere and: q ~ ( e Φ ψ ) = λ k e Φ ψ k + ~ k e Φ ψ dΦ k . Proof. First note that: q ~ ( e Φ ψ ) = hL ~ e Φ ψ, e Φ ψ i L ( M ) = λ k e Φ ψ k + h [ L ~ , e Φ ] ψ, e Φ ψ i L ( M ) , so we need to compute the bracket. h [ L ~ , e Φ ] ψ, e Φ ψ i = Z M h ( i ~ d + A ) ∗ ( i ~ d + A ) e Φ ψ, e Φ ψ i d q − Z M h e Φ ( i ~ d + A ) ∗ ( i ~ d + A ) ψ, e Φ ψ i d q = Z M | ( i ~ d + A ) e Φ ψ | d q − Z M h ( i ~ d + A ) ψ, ( i ~ d + A ) e ψ i d q On the one hand, Z M h ( i ~ d + A ) ψ, ( i ~ d + A ) e ψ i d q = Z M (cid:0) | e Φ ( i ~ d + A ) ψ | + 2 e h ( i ~ d + A ) ψ, i ~ ψ dΦ i (cid:1) d q, and taking the real part: Z M h ( i ~ d + A ) ψ, ( i ~ d + A ) e ψ i d q = Z M (cid:0) | e Φ ( i ~ d + A ) ψ | + 2 ℜ e h ( i ~ d + A ) ψ, i ~ ψ dΦ i (cid:1) d q. On the other hand, Z M | ( i ~ d + A ) e Φ ψ | d q = Z M | e Φ ( i ~ d + A ) ψ | + | i ~ ψe Φ dΦ | + 2 ℜh e Φ ( i ~ d + A ) ψ, i ~ ψe Φ dΦ i d q, so we finally get: h [ L ~ , e Φ ] ψ, e Φ ψ i = ~ k e Φ ψ dΦ k . (cid:3) In [9], the following theorem is proved in the case M is compact or the Euclidean R d . Here we just adapted their proof for non-compact manifolds, with a possibleboundary. Lemma A.4. Assume that ( M, g ) is either compact or with bounded variations inthe following sense : There exists a compact subset K ⊂ M and finitely many charts Ψ n : U n → V ⊂ R d , ≤ n ≤ n , with M \ K = n [ n =1 U n , under which the Riemannian metric satisfies ∂g ij ∂x k is bounded for ≤ i, j, k ≤ d, and ∂ | g | / ∂x k is bounded for ≤ k ≤ d. Then if B is such that |∇ B ( q ) | ≤ C (1 + | B ( q ) | ) , (A.1) there exists ~ > and C > such that, for ~ ∈ (0 , ~ ] , ∀ u ∈ D ( q ~ ) , (1 + ~ / C ) q ~ ( u ) ≥ Z M ~ ( b ( q ) − ~ / C ) | u ( q ) | d q. Proof. Take ( χ m ) m ≥ a smooth partition of unity on M , such that: X m ≥ χ m = 1 and X m ≥ | d χ m ( q ) | ≤ C, ∀ q ∈ M, with supp ( χ m ) ⊂ V m a bounded local chart. Then by Lemma A.5 (below), for any u ∈ D ( q ~ ) , q ~ ( u ) = X m ≥ q ~ ( χ m u ) − ~ X m ≥ k u d χ m k ≥ X m ≥ q ~ ( χ m u ) − C ~ k u k , and we can deal with every q ~ ( χ m u ) in local coordinates x = ( x , ..., x d ) : we canwrite q ~ ( χ m u ) = q coord ~ ( ˜ χ m ˜ u ) , where ˜ u stands for u written in coordinates. We denote h B ( x ) i = (1 + | B ( x ) | ) / . Under assumption (A.1), up to taking V m small enough,we can find z m ∈ M and C > such that: C − h B ( x ) i ≤ h B ( z m ) i ≤ C h B ( x ) i , ∀ x ∈ V m . (A.2)Indeed, by (A.1), if we denote M ( ε ) = sup | x − y |≤ ε h B ( x ) ih B ( y ) i and m ( ε ) = inf | x − y |≤ ε h B ( x ) ih B ( y ) i ,we have: ∀| x − y | ≤ ε, ∃ c xy ∈ [ x, y ] , h B ( x ) i ≤ h B ( y ) i + C h B ( c xy ) i| x − y | , which implies M ( ε ) ≤ CM ( ε ) ε, and for ε < / C , M ( ε ) ≤ . Similarily, we have h B ( x ) ih B ( y ) i ≥ − C h B ( c cy ) ih B ( y ) i | x − y | ≥ − CM ( ε ) | x − y | ≥ − Cε. SEMICLASSICAL BIRKHOFF NORMAL FORM 37 Rescaling a standard partition of unity on R d , we can find a new partition of unity ( χ ~ m,j ) j ≥ on V m such that: X j ≥ | χ ~ m,j | = 1 , and X j ≥ | d χ ~ m,j ( x ) | ≤ C h B ( z m ) i ~ − α , (A.3)where C > does not depend on m , and with supp ( χ ~ m,j ) ⊂ B m,j := { x : | x − y m,j | ≤ h B ( z m ) i − / ~ α } . (A.4)Then for any u ∈ C ∞ ( V m ) , q ~ ( u ) ≥ X j q ~ ( χ m,j u ) − C h B ( z m ) i ~ − α k u k (A.5) ≥ X j q ~ ( χ m,j u ) − C ~ − α Z ( b ( x ) + 1) | u | d x g , (A.6)because h B ( z m ) i ≤ C h B ( x ) i ≤ C ′ ( b ( x ) + 1) . Since b is continuous, on each B m,j wecan choose z m,j such that b ( z m,j ) ≥ b ( x ) , ∀ x ∈ B m,j . (A.7)On each B m,j , we will approximate the magnetic field by a constant. Up to a gaugetransformation, we can assume that the vector potential vanishes at z m,j . In otherwords, we can find a smooth function ϕ m,j on B m,j such that ˜ A ( z m,j ) = 0 , where ˜ A = A + ∇ ϕ m,j . The potential ˜ A defines the same magnetic field B as A .Let us define A lin ( x ) = B ( z m,j ) . ( x − z m,j ) , so that | ˜ A ( x ) − A lin ( x ) | ≤ k∇ B k B j,m | x − z m,j | , on B m,j , and using (A.1) and (A.2), | ˜ A ( x ) − A lin ( x ) | ≤ C h B ( z m,j ) i| x − z m,j | , on B m,j . (A.8)Then if ˜ q ~ denotes the quadratic form for the new potential ˜ A , for v ∈ C ∞ ( B m,j ) , ˜ q ~ ( v ) = q lin ~ ( v ) + k ( ˜ A − A lin ) v k + 2 ℜh ( ˜ A − A lin ) v, ( i ~ ∇ + A lin ) v i , and using (A.8) and the Cauchy-Schwarz inequality, ˜ q ~ ( v ) ≥ q lin ~ ( v ) − C kh B ( z m,j ) i| x − z m,j | v k q q lin ~ ( v ) . We use | ab | ≤ ε a + ε − b to get: ˜ q ~ ( v ) ≥ (cid:0) − C ~ β (cid:1) q lin ~ ( v ) − C ~ − β kh B ( z m,j ) i| x − z m,j | v k ≥ (cid:0) − C ~ β (cid:1) q lin ~ ( v ) − ˜ C ~ α − β k v k by (A.2) and (A.4).Changing A into ˜ A amounts to conjugate the magnetic Laplacian by e i ~ − ϕ j,m , so: ˜ q ~ ( v ) = q ~ ( e i ~ − ϕ j,m v ) . We get for v ∈ C ∞ ( B m,j ) : q ~ ( v ) ≥ (cid:0) − C ~ β (cid:1) q lin ~ ( e − i ~ − ϕ j,m v ) − ˜ C ~ α − β k e i ~ − ϕ j,m v k ≥ (cid:0) − C ~ β (cid:1) q lin ~ ( e − i ~ − ϕ j,m v ) − ˜ C ~ α − β k v k . It remains to estimate q lin ~ ( v ) . Using the assumptions on M , q lin ~ ( v ) = X k,l Z | g ( x ) | / g kl ( x )( i ~ ∂ k v + A link v )( i ~ ∂ l v + A linl v )d x ≥ (1 − C ~ α ) X k,l Z | g ( z m,j ) | / g kl ( z m,j )( i ~ ∂ k v + A link v )( i ~ ∂ l v + A linl v )d x. For this new Schrödinger operator with constant magnetic field on a flat metric, thedesired inequality is well known: q lin ~ ( v ) ≥ (1 − C ~ α ) ~ Z b ( z m,j ) | v | | g ( z m,j ) | / d x ≥ (1 − C ~ α ) ~ Z b ( x ) | v | | g ( z m,j ) | / d x ≥ (1 − C ~ α ) ~ Z b ( x ) | v | | g ( x ) | / d x. because of (A.7) and the assumptions on M . Thus, q lin ~ ( e − i ~ − ϕ m,j v ) ≥ (1 − C ~ α ) ~ Z b ( x ) | v | | g ( x ) | / d x. Finally, we get a C > such that, for ~ small enough, (1 + C ~ β + C ~ α ) q ~ ( u ) ≥ ~ Z M b ( x ) | u | d x g − ˜ C ~ α − β k u k − C ~ k u k − C ~ − α ( Z M b ( x ) | u | d x g + k u k ) , where the last part comes from (A.6). The desired inequality follows if we choose β = 1 / and α = 3 / . (cid:3) Lemma A.5. If ( χ m ) m ≥ is a smooth partition of unity on M , such that X m ≥ χ m = 1 , then for any u ∈ D ( q ~ ) : q ~ ( u ) = X m ≥ q ~ ( χ m u ) − ~ X m ≥ k u d χ m k . SEMICLASSICAL BIRKHOFF NORMAL FORM 39 Proof. q ~ ( u ) = X m Z | χ m ( i ~ d + A ) u | d q g = X m Z | ( i ~ d + A )( χ m u ) − [ i ~ d + A, χ m ] u | d q g = X m Z | ( i ~ d + A )( χ m u ) − i ~ u (d χ m ) | d q g = X m Z | ( i ~ d + A )( χ m u ) | + ~ | u d χ m | − ℜh ( i ~ d + A )( χ m u ) , i ~ u d χ m i d q g = X m q ~ ( χ m u ) + ~ k u d χ m k − Z ℜh ( i ~ d + A )( χ m u ) , i ~ u d χ m i d q g . Moreover, h ( i ~ d + A )( χ m u ) , i ~ u d χ m i = h i ~ u d χ m + i ~ χ m d u + χ m uA, i ~ u d χ m i = ~ | u d χ m | + ~ h χ m d u, u d χ m i − i ~ | u | h A, d χ m i | {z } real . Thus, q ~ ( u ) = X m (cid:0) q ~ ( χ m u ) − ~ k u d χ m k (cid:1) + 2 ~ ℜ Z X m h χ m d u, u d χ m i d q g = X m (cid:0) q ~ ( χ m u ) − ~ k u d χ m k (cid:1) + ~ ℜ Z X m h ¯ u d u, χ m d χ m i d q g = X m (cid:0) q ~ ( χ m u ) − ~ k u d χ m k (cid:1) + ~ ℜ Z h ¯ u d u, X m χ m !| {z } =0 i d q g = X m (cid:0) q ~ ( χ m u ) − ~ k u d χ m k (cid:1) (cid:3) Acknowledgement I would like to thank Nicolas Raymond and San Vu Ngoc for many stimulatingdiscussions, helpful advices, and their readings of the preliminary drafts of thisversion. References [1] S. Agmon, Lectures on exponential decay of solutions of second-order elliptic equations , Math-ematical Notes 29, Princeton University Press, 1982.[2] G. Boil, S. 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