HAL Id: hal-02570104
https://hal.archives-ouvertes.fr/hal-02570104v2
Preprint submitted on 2 Jun 2021
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magnetic fields
Léo Morin
To cite this version:
Léo Morin. A semiclassical Birkhoff normal form for constant-rank magnetic fields. 2021. �hal- 02570104v2�
LÉO MORIN
Abstract. We consider the semiclassical magnetic LaplacianL~on a Riemannian man- ifold, with a constant-rank and non-vanishing magnetic fieldB. Under the localization assumption thatB admits a unique and non-degenerate well, we construct a Birkhoff normal forms to describe the spectrum ofL~in the semiclassical limit~→0. We deduce an expansion of the eigenvalues of order~, in powers of~1/2.
1 – Introduction
1.1 – Context
We consider the semiclassical magnetic Laplacian with Dirichlet boundary conditions L~ = (i~d +A)∗(i~d +A)
on a d-dimensional oriented Riemannian manifold (M, g), which is either compact with boundary, or the EuclideanRd. Adenotes a smooth1-form on M, the magnetic potential.
The magnetic field is the 2-form B= dA.
The spectral theory of the magnetic Laplacian has given rise to many investigations, and appeared to have very various behaviours according to the variations of B and the geometry of M. We refer to the books and review [18, 12,40] for a description of these works. Here we focus on the Dirichlet realisation ofL~, and we give a description of semi- excited states, eigenvalues of order O(~), in the semiclassical limit ~ → 0. As explained in the above references, the magnetic intensity has a great influence on these eigenvalues, and one can define it in the following way.
Using the isomorphismTqM ≃TqM∗ given by the metric, one can define the following skew-symmetric operatorB(q) :TqM →TqM, by:
(1.1) Bq(X, Y) =gq(X,B(q)Y), ∀X, Y ∈TqM, ∀q ∈M .
The operator B(q) being skew-symmetric with respect to the scalar productgq, its eigen- values are purely imaginary and symmetric with respect to the real axis. We denote these repeated eigenvalues by
±iβj(q),· · · ,±iβs(q),0,
withβj(q) >0. In particular, the rank ofB(q) is 2sand may depend on q. However, we will focus on the constant-rank case. We denote bykthe dimension of the Kernel ofB(q), so that d= 2s+k. The magnetic intensity (or "Trace+") is the following scalar-valued function,
b(q) = Xs j=1
βj(q).
The function b is continuous on M, but non-smooth in general. We are interested in discrete magnetic wells and non-vanishing magnetic fields.
Key words and phrases. magnetic Laplacian, normal forms, spectral theory, semiclassical limit, pseudo differential operators, microlocal analysis, symplectic geometry.
1
Assumption 1. The magnetic intensity is non-vanishing, and admits a unique global mini- mum b0 >0atq0 ∈M\∂M, which is non-degenerate. Moreover, in the non-compact case M =Rd we assume that
b∞:= lim inf
|q|→+∞b(q)> b0 and the existence of C >0 such that
|∂ℓBij(q)| ≤C(1 +|B(q)|), ∀ℓ, i, j ∀q ∈Rd.
Assumption 2. The rank of B(q) is constant equal to 2s >0 on a neighborhoodΩ of q0. Under Assumption 1, the following useful inequality was proven in [20]. There is a C0 >0 such that, for~small enough,
(1.2) (1 +~1/4C0)hL~u, ui ≥ Z
M
~(b(q)−~1/4C0)|u(q)|2dq , ∀u∈Dom(L~). Remark 1.1. Actually, one has the better inequality obtained replacing~1/4 by~. This was proved by Guillemin-Uribe [13] in the case of a non-degenerate B, by Borthwick-Uribe [7]
in the constant rank case, and by Ma-Marinescu [32] in a more general setting.
Remark 1.2. Using this inequality, one can prove Agmon-like estimates for the eigenfunc- tions ofL~. Namely, the eigenfunctions associated to an eigenvalue < b1~are exponentially small outside Kb1 = {q , b(q) ≤ b1}. We will use this result to reduce our analysis to the neighborhood Ωof q0. In particular, the greater b1 is, the larger Ω must be.
Finally, we will assume the following, in order to get smooth functions βj on Ω.
Assumption 3. βi(q0)6=βj(q0) for every 1≤i < j ≤s.
Under Assumptions 1 and 2, estimates on the ground states of L~ in the semiclassical limit ~→0 were proven in several works, especially in dimension d= 2,3.
OnM =R2, asymptotics for the j-th eigenvalue ofL~
(1.3) λj(L~) =b0~+ (α(2j−1) +c1)~2+o(~2)
with explicit α, c1 ∈ R were proven by Helffer-Morame [21] (for j = 1) and Helffer- Kordyukov [15] (j ≥1). Actually, this second paper contains a description of some higher eigenvalues. They proved that, for any integers n, j ∈ N, there exist ~jn > 0 and for
~∈(0,~jn) an eigenvalueλn,j(~)∈sp(L~) such that
λn,j(~) = (2n−1)(b0~+ ((2j−1)α+cn)~2) +o(~2),
for another explicit constant cn. In particular, it gives a description of some semi-excited states (of order (2n−1)b0~). Finally, Raymond-V˜u Ngo.c [41] (and Helffer-Kordyukov [17]) got a description of the whole spectrum below b1~, for any fixedb1 ∈(b0, b∞). More precisely, they proved that this part of the spectrum is given by a familly of effective operators N~[n] (n∈N) modulo O(~∞). These effective operators are ~-pseudodifferential operators with principal symbol given by the function ~(2n−1)b. More interestingly, they explained why the two quantum oscillators
(2n−1)b0~, and (2j−1)α~2,
appearing in the eigenvalue asymptotics correspond to two oscillatory motions in classical dynamics : The cyclotron motion, and a rotation arround the minimum point of b. The results of Raymond-V˜u Ngo.c were generalized to an arbitrary d-dimensional Riemannian manifold in [36], under the assumption k = 0 (B(q) has full rank), proving in particular similar estimates (1.3) in a general setting. Actually, these eigenvalue estimates were proven simultaneously by Kordyukov [28] in the context of the Bochner Laplacian.
In this paper we are interested on the influence of the kernel of B (k >0). The rank of B being even, this kernel always exists in odd dimensions : if d = 3 the kernel direc- tions correspond to the usual field lines. On M =R3, Helffer-Kordyukov [16] proved the existence of λnmj(~)∈sp(L~) such that,
λnmj(~) =(2n−1)b0~+ (2n−1)1/2(2m−1)ν0~3/2 + ((2n−1)(2j−1)α+cnm)~2+O(~9/4),
for some ν0 > 0 and α, cnm ∈ R. Motivated by this result and the 2D case, Helffer- Kordyukov-Raymond-V˜u Ngo.c [19] gave a description of the whole spectrum below b1~, proving in particular the eigenvalue estimates
(1.4) λj(L~) =b0~+ν0~3/2+α(2j−1)~2+O(~5/2).
Their results exhibit a new classical oscillatory motion in the directions of the field lines, corresponding to the quantum oscillator (2m−1)ν0~3/2.
The aim of this paper is to generalize the results of [19] to an arbitrary Riemannian manifold M, under the assumptions 1 and 2. In particular we describe the influence of the kernel of B in a general geometric and dimensional setting. Their approach, which we adapt, is based on asemiclassical Birkhoff normal form. Theclassical Birkhoff normal form has a long story in physics, and goes back to Delaunay [10] and Lindstedt [31]. This formal normal form was the starting point of a lot of studies on stability near equilibrium, and KAM theory (after Kolmogorov [27], Arnold [1], Moser [38]). The works of Birkhoff [2] and Gustavson [14] gave its name to this normal form. We refer to the books [39] and [23] for precise statements. Our approach here relies on a quantization. Physicists and quantum chemists already noticed in the 1980’ that a quantum analogue of the Birkhoff normal form could be used to compute energies of molecules ([11],[25],[33],[43]). Joyeux and Sugny also used such techniques to describe the dynamics of excited states (see [26]
for example). In [44], Sjöstrand constructed a semi-classical Birkhoff normal form for a Schrödinger operator −~2∆ +V, using the Weyl quantization, to make a mathematical study of semi-excited states. In their paper [41], Raymond and V˜u Ngo.c had the idea to adapt this method forL~ onR2, and with Helffer and Kordyukov onR3 [19]. This method is reminiscent of Ivrii’s approach (in his book [24]).
1.2 – Main results
The first idea is to link the classical dynamics of a particle in the magnetic fieldB with the spectrum of L~ using pseudodifferential calculus. Indeed, L~ is a ~-pseudodifferential operator with symbol
H(q, p) =|p−Aq|2+O(~2), ∀p∈TqM∗,∀q∈M ,
and H is the classical Hamiltonian associated to the magnetic field B. One can use this property to prove that, in the phase spaceT∗M, the eigenfunctions (with eigenvalue< b1~) are microlocalized on an arbitrarily small neighborhood of
Σ =H−1(0)∩T∗Ω ={(q, p)∈T∗Ω, p=Aq}.
Hence, the second main idea is to find a normal form for H on a neighborhood of Σ.
Namely, we find canonical coordinates near Σ in which H has a "simple" form. The symplectic structure of Σ, as submanifold of T∗M is thus of great interest. One can see that the restriction of the canonical symplectic form dp∧dq on T∗M to Σ is given by B (Lemma 2.1); and when B has constant-rank, one can find Darboux coordinates ϕ: Ω′ ⊂R2s+k(y,η,t)→Ω such that
ϕ∗B = dη∧dy ,
up to reducing Ω. We will start from these coordinates to get the following normal form for H.
Theorem 1.3. Under Assumptions1, 2, and3, there exists a diffeomorphism Φ1:U1′ ⊂R4s+2k→U1 ⊂T∗M
between neighborhoods U1′ of 0 andU1 of Σ such that
H(x, ξ, y, η, t, τ) :=b H◦Φ1(x, ξ, y, η, t, τ) satisfies (with the notation βbj =βj◦ϕ),
Hb =hM(y, η, t)τ, τi+ Xs j=1
βbj(y, η, t) ξj2+x2j
+O((x, ξ, τ)3),
uniformly with respect to(y, η, t), for some(y, η, t)-dependant positive definite matrixM(y, η, t).
Moreover,
Φ∗1(dp∧dq) = dξ∧dx+ dη∧dy+ dτ∧dt .
Remark 1.4. We will use the following notation for our canonical coordinates:
z= (x, ξ)∈R2s, w= (y, η)∈R2s, τ = (t, τ)∈R2k.
This theorem gives the Tayor expansion of H on a neighborhood of Σ. In particular (x, ξ, τ) ∈Rd measures the distance to Σ whereas (y, η, t) ∈Rd are canonical coordinates on Σ.
Remark 1.5. This theorem exhibits the harmonic oscillatorξj2+x2j in the first-order expan- sion of H. This oscillator, which is due to the non-vanishing magnetic field, corresponds to the well-known cyclotron motion.
Actually, one can use the Birkhoff normal form algorithm to improve the remainder.
Using this algorithm, we can change the O((x, ξ)3) remainder into an explicit function of ξ2j +x2j, plus some smaller remaindersO((x, ξ)r). This remainder powerr is restricted by resonances between the coefficients βj. Thus, we take an integer r1∈Nsuch that
∀α∈Zs, 0<|α|< r1 ⇒ Xs j=1
αjβj(q0)6= 0. Here, |α| = P
j|αj|. Moreover, we can use the pseudodifferential calculus to apply the Birkhoff algorithm toL~, changing the classical oscillatorξ2j+x2j into the quantum harmonic oscillator
I~(j)=−~2∂x2j+x2j,
whose spectrum consists of the simple eigenvalues (2n−1)~, n ∈N. Following this idea we prove the following theorem.
Theorem 1.6. Let ε > 0. Under Assumptions 1, 2 and 3, there exist b1 ∈ (b0, b∞), an integer Nmax >0 and a compactly supported function f1⋆∈ C∞(R2s+2k×Rs×[0,1)) such that
|f1⋆(y, η, t, τ, I,~)|. (|I|+~)2+|τ|(|I|+~) +|τ|3 ,
satisfying the following properties. For n ∈ Ns, denote by N~[n] the ~-pseudodifferential operator in (y, t) with symbol
N~[n]=hM(y, η, t)τ, τi+ Xs j=1
βbj(y, η, t)(2nj −1)~+f1⋆(y, η, t, τ,(2n−1)~,~).
For ~<<1, there exists a bijection
Λ~ :sp(L~)∩(−∞, b1~)→ [
|n|≤Nmax
sp N~[n]
∩(−∞, b1~),
such that Λ~(λ) =λ+O(~r21−ε) uniformly with respect to λ.
Remark 1.7. In this theorem sp(A) denotes the repeated eigenvalues of an operator A, so that there might be some multiple eigenvalues, but Λ~ preserves this multiplicity. We only consider self-adjoint operators with discrete spectrum.
Remark 1.8. One should care of how large b1 can be. As mentionned above, the eigenfunc- tions of energy < b1~ are exponentially small outside Kb1 ={q ∈M , b(q)≤b1}. Thus, we will choseb1 such that Kb1 ⊂Ω, where Ω is some neighborhood of q0. Hence the larger Ω is, the greater b1 can be. However, there are three restrictions on the size of Ω:
• The rank of B(q) is constant onΩ,
• There exist canonical coordinates ϕ on Ω (i.e. such thatϕ∗B = dη∧dy),
• There is no resonance in Ω:
∀q∈Ω, ∀α∈Zs, 0<|α|< r1 ⇒ Xs j=1
αjβj(q)6= 0.
Remark 1.9. If k = 0 we recover the result of [36]. Here we want to study the influence of a non-zero kernel k >0. This result generalizes the result of [19], which corresponds to d= 3, s=k= 1, on the EuclideanR3. However, this generalization is not straightforward since the magnetic geometry is much more complicated in higher dimensions, in particular if k >1. Moreover, there is a new phenomena in higher dimensions : resonances between the functionsβj (as in[36]).
The spectrum ofL~ in(−∞, b1~)is reduced to the operators N~[n]. Actually if we chose b1 small enough, it is reduced to the first operator N~[1] (Here we denote the multi-integer 1 = (1,· · · ,1)∈Ns). Hence in the second part of this paper, we study the spectrum N~[1]
using a second Birkhoff normal form. Indeed, the symbol ofN~[1] is
N~[1](w, t, τ) =hM(w, t)τ, τi+~bb(w, t) +O(~2) +O(τ~) +O(τ3),
so if we denote bys(w) the minimum point oft7→bb(w, t) (which is unique on a neighbor- hood of0), we get the following expansion
N~[1](w, t, τ) =hM(w, s(w))τ, τi+~ 2h∂2bb
∂t2(w, s(w))·(t−s(w)), t−s(w)i+· · · and the principal part is a harmonic oscillator with frequences√
~νj(w)(1≤j≤k) where (νj2(w))1≤j≤k are the eigenvalues of the symmetric matrix:
M(w, s(w))1/2·1
2∂t2ˆb(w, s(w))·M(w, s(w))1/2.
These frequences are smooth non-vanishing functions ofwon a neighborhood of0, as soon as we assume that they are simple.
Assumption 4. νi(0)6=νj(0) for indices 1≤i < j ≤k.
We fix an integerr2∈Nsuch that
∀α∈Zk, 0<|α|< r2 ⇒ Xk j=1
αjνj(0)6= 0, and we prove the following reduction theorem forN~[1].
Theorem 1.10. Let c > 0 and δ ∈(0,12). Under assumptions 1, 2, 3 and 4, with k > 0, there exists a compactly supported function f2⋆ ∈ C∞(R2s×Rk×[0,1)) such that
|f2⋆(y, η, J,√
~)|.
|J|+√
~ 2
,
satisfying the following properties. For n ∈ Nk, denote by M[n]~ the ~-pseudodifferential operator in y with symbol
M~[n](y, η) =bb(y, η, s(y, η)) +√
~ Xk j=1
νj(y, η)(2nj −1) +f2⋆(y, η,(2n−1)√
~,√
~). For ~<<1, there exists a bijection
Λ~ :sp(N~[1])∩(−∞,(b0+c~δ)~)→ [
n∈Nk
sp(~M[n]~ )∩(−∞,(b0+c~δ)~), such that Λ~(λ) =λ+O(~1+δr2/2) uniformly with respect to λ.
Remark 1.11. The thresholdb0+c~δis needed to get microlocalization of the eigenfunctions of N~[1] in an arbitrarily small neighborhood of τ = 0.
Remark 1.12. This second harmonic oscillator (in variables (t, τ)) corresponds to a clas- sical oscillation in the directions of the field lines. We see that this new motion, due to the kernel of B, induces powers of √
~in the spectrum.
As a corollary, we get a description of the low-lying eigenvalues of L~ by the effective operator ~M[1]~ .
Corollary 1.13. Let ε > 0 and c ∈ (0,minjνj(0)). Denote by ν(0) = P
jνj(0) and r = min(2r1, r2+ 4). Under assumptions1, 2, 3 and 4, with k >0, there exists a bijection
Λ~ :sp(L~)∩(−∞,~b0+~3/2(ν(0) + 2c))→sp(~M[1]~ )∩(−∞,~b0+~3/2(ν(0) + 2c)) such that Λ~(λ) =λ+O(~r/4−ε) uniformly with respect toλ.
We deduce the following eigenvalue asymptotics.
Corollary 1.14. Under the assumptions of corollary 1.13, for j∈N, thej-th eigenvalue of L~ admits an expansion
λj(L~) =~
⌊r/2X⌋−2 ℓ=0
αjℓ~ℓ/2+O(~r/4−ε), with coefficients αjℓ∈R such that:
αj,0 =b0, αj,1 = Xk j=1
νj(0), αj,2=Ej+c0,
where c0 ∈R and ~Ej is the j-th eigenvalue of a s-dimensional harmonic oscillator.
Remark 1.15. ~Ej is the j-th eigenvalue of a harmonic oscillator whose symbol is given by the Hessian at w= 0 ofˆb(w, s(w)). Hence, it corresponds to a third classical oscillatory motion : a rotation in the space of field lines.
Remark 1.16. The asymptotics
λj(L~) =b0~+ν(0)~3/2+ (Ej +c0)~2+o(~2) were unknown before, except in the special 3d-case M =R3 in [19].
1.3 – Related questions and perspectives
In this paper, we are restricted to energies λ < b1~, and as mentionned in Remark1.8, the thresholdb1 > b0 is limited by three conditions, including the non-resonance one:
∀q∈Ω, ∀α∈Zs, 0<|α|< r1 ⇒ Xs j=1
αjβj(q)6= 0.
It would be interesting to study the influence of resonances between the functionsβj on the spectrum ofL~. Maybe a Grushin reduction method could help, as in [17] for instance. A Birkhoff normal form was given in [9] for a Schrödinger operator−~2∆+V with resonances, but the situation is somehow simpler, since the analogues ofβj(q) are independent of q in this context.
We are also restricted by the existence of Darboux coordinates ϕ on (Σ, B), such that ϕ∗B = dη∧dy. Indeed, the coordinates (y, η) on Σ are necessary to use the Weyl quan- tization. To study the influence of the global geometry of B, one should consider another quantization method for the presymplectic manifold (Σ, B). In the symplectic case, for instance in dimension d= 2, a Toeplitz quantization may be useful. This quantization is linked to the complex structure induced by B on Σ, and the operator L~ can be linked with this structure in the following way:
L~ = 4~2
∂+ i 2~A
∗
∂+ i 2~A
+~B = 4~2∂∗A∂A+~B, with
A=A1+iA2, B=∂1A2−∂2A1, 2∂=∂1+i∂2.
In [45], this is used to compute the spectrum of L~ on a bidimensional Riemann surface M with constant curvature and constant magnetic field. See also the recent papers [8,29]
where semi-excited states for constant magnetic fields in higher dimensions are considered.
If the 2-formB is not exact, we usually consider a Bochner Laplacian on thep-th tensor product of a complex line bundle L over M, with curvature B. This Bochner Laplacian
∆p, depends on p ∈ N, and the limit p → +∞ is interpreted as the semi-classical limit.
∆p is a good generalization of the magnetic Laplacian because locally it can be written
1
~2(i~∇+A)2, where the potential A is a local primitive of B, and ~=p−1. For details, we refer to the recent articles [30], [28], [34], and the references therein. In [28], Kordyukov constructed quasimodes for∆p in the case of a symplecticB and discrete wells. He proved expansions:
λj(∆p)∼X
ℓ≥0
αjℓp−ℓ/2.
Our work also gives such expansions for∆p as explained in [37].
In this paper, we only mentionned the study of the eigenvalues of L~: What about the eigenfunctions ? WKB expansions for the j-th eigenfunction were constructed on R2 in [5], and on a 2-dimensional Riemannian manifold in [4]. We do not know how to construct magnetic WKB solutions in higher dimensions. This article suggests that the directions corresponding to the kernel of B could play a specific role.
An other related question is the decreasing of the real eigenfunctions. Agmon estimates only give a O(e−c/√~) decay outside any neighborhood of q0, but 2D WKB suggest a O(e−c/~) decay. In the recent paper [6], Bonthonneau, Raymond and V˜u Ngo.c proved this onR2, using the FBI Transform to work on the phase spaceT∗R2. This kind of question is motivated by the study of the tunneling effect: The exponentially small interaction between two magnetic wells for example.
In this paper, we only have investigated 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 Euclidean 2D case by Boil and V˜u Ngo.c in [3].
Finally, it would be interesting to study higher Landau levels and the effect of resonances in our normal forms, as was done by Charles and V˜u Ngo.c in [9] for an electric Schrödinger operator −~2∆ +V.
1.4 – Structure of the paper
In section 2 we prove Theorem 1.3, reducing the symbol H of L~ on a neighborhood of Σ = H−1(0). In section 3 we construct the normal form, first in a space of formal series (section 3.2), and then the quantized version N~ (section 3.3). In section 4 we prove Theorem 1.6. For this we describe the spectrum of N~ (section 4.1), then we prove microlocalization properties on the eigenfunctions of L~ and N~ (section 4.2), and finally we compare the spectra ofL~ and N~ (section 4.3).
In section 5 we focus on Theorem 1.10 which describes the spectrum of the effective operator N~[1]. In5.1 we reduce its symbol, in 5.2 we construct a second formal Birkhoff normal form, and in 5.3the quantized versionM~. In5.4we compare the spectra of N~[1]
and M~.
Finally, sections 6and 7are dedicated to the proofs of Corollaries1.13and 1.14respec- tively.
2 – Reduction of the principal symbol H
2.1 – Notations
L~ is a~-pseudodifferential operator onM with principal symbol H:
H(q, p) =|p−Aq|2gq∗, p∈Tq∗M , q ∈M .
Here, T∗M denotes the cotangent bundle of M,p ∈ Tq∗M is a linear form on TqM. The scalar product gq on TqM induces a scalar product gq∗ on Tq∗M, and | · |gq∗ denotes the associated norm. In this section we prove Theorem 1.3, reducing H on a neighborhood of its minimum:
Σ ={(q, p) ∈T∗M , q∈Ω, p=Aq}.
Recall that Ω is a (small) neighborhood of q0 ∈ M \∂M. We will construct canonical coordinates (z, w, v) ∈R2d with:
z= (x, ξ) ∈R2s, w= (y, η)∈R2s, v= (t, τ)∈R2k. R2d is endowed with the canonical symplectic form
ω0 = dξ∧dx+ dη∧dy+ dτ∧dt . We will identify Σ with
Σ′ ={(x, ξ, y, η, t, τ)∈R2d, x=ξ= 0, τ = 0}=R2s+k(y,η,t)× {0}.
We will use several lemmas to prove Theorem 1.3. Before constructing the diffeomor- phismΦ−11 on a neighborhood U1 ofΣ, we will restrict toΣ. Thus we need to understand the structure of Σ induced by the symplectic structure on T∗M (Section 2.2). Then we will construct Φ1 and finally prove Theorem1.3 (Section2.3).
2.2 – Structure of Σ
Recall that onT∗M we have the Liouville 1-form α defined by
α(q,p)(V) =p((dπ)(q,p)V), ∀(q, p)∈T∗M , V ∈T(q,p)(T∗M),
whereπ:T∗M →M is the canonical projection : π(q, p) =q, anddπits differential. T∗M is endowed with the symplectic formω = dα. Σ is a d-dimensional submanifold of T∗M which can be identified withΩ using
j:q ∈Ω7→(q, Aq)∈Σ, and its inverse, which is π.
Lemma 2.1. The restriction of ω toΣ isωΣ=π∗B.
Proof. Fixq ∈Ωand Q∈TqM. Then
(j∗α)q(Q) =αj(q)((dj)Q) =Aq((dπ)◦(dj)Q) =Aq(Q),
becauseπ◦j=id. Thusj∗α=Aand αΣ=π∗j∗α=π∗A. Taking the exterior derivative we get
ωΣ= dαΣ=π∗(dA) =π∗B .
Since B is a closed 2-form with constant rank equal to2s, (Σ, π∗B) is a presymplectic manifold. It is equivalent to (Ω, B), usingj. We recall the Darboux Lemma, telling that such a manifold is locally equivalent to(R2s+k,dη∧dy).
Lemma 2.2. Up to reducing Ω, there exists an open subset Σ′ of R2s+k(y,η,t) and a diffeomor- phismϕ: Σ′ →Ω such that ϕ∗B= dη∧dy.
One can always take (any) coordinate system onΩ. Up to working in these coordinates, it is enough to consider the caseM =Rdwith
H(q, p) = Xd k,ℓ=1
gkℓ(q)(pk−Ak(q))(pℓ−Aℓ(q)), (q, p)∈T∗Rd≃R2d to prove Theorem1.3. This is what we will do. In coordinates, ω is given by
ω= dp∧dq = Xd j=1
dpj∧dqj and Σis the submanifold
Σ ={(q,A(q)), q ∈Ω} ⊂R2d, and j◦ϕ: Σ′→Σ.
In order to extendj◦ϕto a neighborhood ofΣ′inR2din a symplectic way, it is convenient to split the tangent spaceTj(q)(R2d)according to tangent and normal directions toΣ. This is the purpose of the following two lemmas.
Lemma 2.3. Fix j(q) = (q,A(q))∈Σ. Then the tangent space to Σis Tj(q)Σ ={(Q, P)∈R2d, P =∇qA·Q}. Moreover, the ω-orthogonal Tj(q)Σ⊥ is
Tj(q)Σ⊥={(Q, P)∈R2d, P = (∇qA)T ·Q}. Finally,
Tj(q)Σ∩Tj(q)Σ⊥=Ker(π∗B).
