A semiclassical Birkhoff normal form for constant-rank magnetic fields

We consider the semiclassical magnetic Laplacian $\mathcal{L}_h$ on a Riemannian manifold, with a constant-rank and non-vanishing magnetic field $B$. Under the localization assumption that $B$ admits a unique and non-degenerate well, we construct three successive Birkhoff normal forms to describe the spectrum of $\mathcal{L}_h$ in the semiclassical limit $\hbar \rightarrow 0$. We deduce an expansion of all the eigenvalues under a threshold, in powers of $\hbar^{1/2}$.

1. Introduction 1A.Context.We consider the semiclassical magnetic Laplacian with Dirichlet boundary conditions on a d-dimensional oriented Riemannian manifold (M, g), which is either compact with boundary, or the Euclidean ‫ޒ‬ d .A denotes a smooth 1-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 behaviors according to the variations of B and the geometry of M. We refer to the books and review [Helffer and Kordyukov 2014;Fournais and Helffer 2010;Raymond 2017] for a description of these works.Here we focus on the Dirichlet realization of L − h , and we give a description of semiexcited states, eigenvalues of order O( − h) in the semiclassical limit − h → 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 isomorphism T q M ≃ T q M * given by the metric, one can define the following skew-symmetric operator B(q) : T q M → T q M by Since the operator B(q) is skew-symmetric with respect to the scalar product g q , its eigenvalues are purely imaginary and symmetric with respect to the real axis.We denote these repeated eigenvalues by ±iβ 1 (q), . . ., ±iβ s (q), 0, with β j (q) > 0. In particular, the rank of B(q) is 2s and may depend on q.However, we will focus on the constant-rank case.We denote by k the dimension of the kernel of B(q), so that d = 2s + k.The magnetic intensity (or "trace+") is the scalar-valued function b(q) = s j=1 The function b is continuous on M, but nonsmooth in general.We are interested in discrete magnetic wells and nonvanishing magnetic fields.
Assumption 1.We assume that: • The magnetic intensity is nonvanishing and admits a unique global minimum b 0 > 0 at q 0 ∈ M \ ∂ M.
• The rank of B(q) is constant equal to 2s > 0 on a neighborhood of q 0 .
Remark 1.1.Since the nonzero eigenvalues of B are simple at q 0 , the function b is smooth on a neighborhood of q 0 .In particular, it is meaningful to say that the minimum of b is nondegenerate.
Under Assumption 1, the following useful inequality was proven in [Helffer and Mohamed 1996].There is a C 0 > 0 such that, for − h small enough, (1-2) Remark 1.2.Actually, one has the better inequality obtained replacing − h 1/4 by − h.This was proved in [ Guillemin and Uribe 1988] in the case of a nondegenerate B, in [Borthwick and Uribe 1996] in the constant rank case, and in [Ma and Marinescu 2002] in a more general setting.
Remark 1.3.Using this inequality, one can prove Agmon-like estimates for the eigenfunctions of L − h .Namely, the eigenfunctions associated to an eigenvalue < b 1 − h are exponentially small outside K b 1 = {q : b(q) ≤ b 1 }.We will use this result to localize our analysis to the neighborhood of q 0 .In particular, the greater b 1 is, the larger must be.
Under Assumption 1, estimates on the ground states of L − h in the semiclassical limit − h → 0 were proven in several works, especially in dimensions d = 2, 3.
On M = ‫ޒ‬ 2 , asymptotics for the j-th eigenvalue of with explicit α, c 1 ∈ ‫ޒ‬ were proven in [Helffer and Morame 2001] (for j = 1) and [Helffer and Kordyukov 2011] ( j ≥ 1).Actually, this second paper contains a description of some higher eigenvalues.They proved that, for any integers n, j ∈ ‫,ގ‬ there exist − h jn > 0 and for − h ∈ (0, − h jn ) an eigenvalue λ n, j for another explicit constant c n .In particular, it gives a description of some semiexcited states (of order (2n − 1)b 0 − h).Finally, [Raymond andVũ Ngo . c 2015] (and[Helffer andKordyukov 2015]) gives a description of the whole spectrum below b 1 − h, for any fixed b 1 ∈ (b 0 , b ∞ ).More precisely, they proved that this part of the spectrum is given by a family of effective operators N [n]   − h (n ∈ ‫)ގ‬ modulo O( − h ∞ ).These effective operators are − h-pseudodifferential operators with principal symbol given by the function − h(2n − 1)b.More interestingly, they explained why the two quantum oscillators appearing in the eigenvalue asymptotics correspond to two oscillatory motions in classical dynamics: the cyclotron motion and a rotation around the minimum point of b.The results of Raymond and Vũ Ngo .c were generalized to an arbitrary d-dimensional Riemannian manifold in [Morin 2022b], 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 in [Kordyukov 2019] in the context of the Bochner Laplacian.
We are interested on the influence of the kernel of B (k > 0).Since the rank of B is even, this kernel always exists in odd dimensions: if d = 3, the kernel directions correspond to the usual field lines.On M = ‫ޒ‬ 3 , Helffer and Kordyukov [2013] proved the existence of λ nm j ( − h) ∈ sp(L − h ) such that for some ν 0 > 0 and α, c nm ∈ ‫.ޒ‬ Motivated by this result and the 2-dimensional case, Helffer, Kordyukov, Raymond and Vũ Ngo . c [Helffer et al. 2016] gave a description of the whole spectrum below b 1 − h, proving in particular the eigenvalue estimates (1-4) Their results exhibit a new classical oscillatory motion in the directions of the field lines, corresponding to the quantum oscillator (2m − 1)ν 0 − h 3/2 .The aim of this paper is to generalize the results of [Helffer et al. 2016] to an arbitrary Riemannian manifold M, under Assumption 1.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 a semiclassical Birkhoff normal form.The classical Birkhoff normal form has a long story in physics and goes back to [Delaunay 1860;Lindstedt 1883].This formal normal form was the starting point of a lot of studies on stability near equilibrium, and KAM theory (after [Kolmogorov 1954;Arnold 1963;Moser 1962]).The name of this normal form comes from [Birkhoff 1927;Gustavson 1966].We refer to the books [Moser 1968;Hofer and Zehnder 1994] for precise statements.Our approach here relies on a quantization.Physicists and quantum chemists already noticed in the 1980s that a quantum analogue of the Birkhoff normal form could be used to compute energies of molecules [Delos et al. 1983;Jaffé and Reinhardt 1982;Marcus 1985;Shirts and Reinhardt 1982].Joyeux and Sugny [2002] also used such techniques to describe the dynamics of excited states.Sjöstrand [1992] constructed a semiclassical Birkhoff normal form for a Schrödinger operator − − h 2 + V using the Weyl quantization, to make a mathematical study of semiexcited states.Raymond and Vũ Ngo . c [2015] had the idea to adapt this method for L − h on ‫ޒ‬ 2 , and with Helffer and Kordyukov on ‫ޒ‬ 3 [Helffer et al. 2016].This method is reminiscent of Ivrii's approach [2019].
1B. Main results.The first idea is to link the classical dynamics of a particle in the magnetic field B with the spectrum of L − h using pseudodifferential calculus.Indeed, L − h is an − h-pseudodifferential operator with principal symbol for all p ∈ T q M * , for all 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 space T * M, the eigenfunctions (with eigenvalue < b 1 − h) are microlocalized on an arbitrarily small neighborhood of Hence, the second main idea is to find a normal form for H on a neighborhood of .Namely, we find canonical coordinates near in which H has a "simple" form.The symplectic structure of as a submanifold of T * M is thus of great interest.One can see that the restriction of the canonical symplectic form d p ∧ dq on T * M to is given by B (Lemma 2.1), and when B has constant rank, one can find Darboux coordinates ϕ : ′ ⊂ ‫ޒ‬ 2s+k (y,η,t) → such that ϕ * B = dη ∧ dy, up to shrinking .We will start from these coordinates to get the following normal form for H.
Theorem 1.4.Under Assumption 1, there exists a diffeomorphism satisfies (with the notation βj = β j • ϕ) uniformly with respect to (y, η, t) for some (y, η, t)-dependent positive definite matrix M(y, η, t).Moreover, Remark 1.5.We will use the following notation for our canonical coordinates: This theorem gives the Taylor expansion of H on a neighborhood of .In particular (x, ξ, τ ) ∈ ‫ޒ‬ d measures the distance to , whereas (y, η, t) ∈ ‫ޒ‬ d are canonical coordinates on .
Remark 1.6.This theorem exhibits the harmonic oscillator ξ 2 j + x 2 j in the expansion of H.This oscillator, which is due to the nonvanishing 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 ξ 2 j + x 2 j , plus some smaller remainders O((x, ξ ) r ).This remainder power r is restricted by resonances between the coefficients β j .Thus, we take an integer r 1 ∈ ‫ގ‬ such that, (1-5) Here, |α| = j |α j |.Moreover, we can use the pseudodifferential calculus to apply the Birkhoff algorithm to L − h , changing the classical oscillator ξ 2 j + x 2 j into the quantum harmonic oscillator ގ‬ Following this idea we construct a normal form for L − h in Theorem 3.4.We also deduce a description of its spectrum.Theorem 1.7.Let ε > 0. Under Assumption 1, there exist b 1 ∈ (b 0 , b ∞ ), an integer N max > 0 and a compactly supported function f satisfying the following properties.For n ∈ ‫ގ‬ s , denote by For − h ≪ 1, there exists a bijection uniformly with respect to λ.
Remark 1.8.In this theorem sp(A) denotes the repeated eigenvalues of an operator A, so that there might be some multiple eigenvalues, but − h preserves this multiplicity.We only consider self-adjoint operators with discrete spectrum.
Remark 1.9.One should care of how large b 1 can be.As mentioned above, the eigenfunctions of energy < b 1 − h are exponentially small outside K b 1 = {q ∈ M : b(q) ≤ b 1 }.Thus, we will chose b 1 such that K b 1 ⊂ , where is some neighborhood of q 0 .Hence the larger is, the greater b 1 can be.However, there are three restrictions on the size of : • The rank of B(q) is constant on .
• There is no resonance in : for all q ∈ , for all α ∈ ‫ޚ‬ s , 0 Of course the last condition is the most restrictive.However, if we forget the second condition, which is of global geometric nature, given a magnetic field and an r 1 one can estimate an associated b 1 satisfying the third condition.In particular we can construct simple examples on ‫ޒ‬ d such that the threshold b 1 − h includes several Landau levels.
Remark 1.10.If k = 0 we recover the result of [Morin 2022b].Here we want to study the influence of a nonzero kernel k > 0. This result generalizes the result of [Helffer et al. 2016], which corresponds to d = 3, s = k = 1 on the Euclidean ‫ޒ‬ 3 .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 functions β j (as in [Morin 2022b]).

The spectrum of L
Actually if we choose b 1 small enough, it is only given by the first operator N [1]   − h (here we denote the multi-integer 1 = (1, . . ., 1) ∈ ‫ގ‬ s ).Hence in the second part of this paper, we study the spectrum N [1]   − h using a second Birkhoff normal form.Indeed, the symbol of so if we denote by s(w) the minimum point of t → b(w, t) (which is unique on a neighborhood of 0), we get the expansion where we will show that the remaining terms are only perturbations.As explained in Section 5, in (1-6) we can recognize a harmonic oscillator with frequencies , where (ν 2 j (w)) 1≤ j≤k are the eigenvalues of the symmetric matrix . These frequencies are smooth nonvanishing functions of w on a neighborhood of 0, as soon as we assume that they are simple.
We fix an integer r 2 ∈ ‫ގ‬ such that, and we construct a normal form for N [1]   − h in Theorem 5.4.Again, we deduce a description of its spectrum.
Theorem 1.11.Let c > 0 and δ ∈ 0, 1 2 .Under Assumptions 1 and 2, with k > 0, there exists a compactly supported function f satisfying the following properties.For n ∈ ‫ގ‬ k , denote by M [n]   − h the − h-pseudodifferential operator in y with symbol For − h ≪ 1, there exists a bijection ) uniformly with respect to λ.
Remark 1.12.The threshold b 0 + c − h δ is needed to get microlocalization of the eigenfunctions of N [1]   − h in an arbitrarily small neighborhood of τ = 0.
Remark 1.13.This second harmonic oscillator (in variables (t, τ )) corresponds to a classical oscillation in the directions of the field lines.We see that this new motion, due to the kernel of B, induces powers of √ − h in the spectrum.
We deduce the following eigenvalue asymptotics.
Corollary 1.15.Under the assumptions of Corollary 1.14, for j ∈ ‫,ގ‬ the j-th eigenvalue of L − h admits an expansion with coefficients α jℓ ∈ ‫ޒ‬ such that where c 0 ∈ ‫ޒ‬ and − h E j is the j-th eigenvalue of an s-dimensional harmonic oscillator.
Remark 1.16.Note − h E j is the j-th eigenvalue of a harmonic oscillator whose symbol is given by the Hessian at w = 0 of b(w, s(w)).Hence, it corresponds to a third classical oscillatory motion: a rotation in the space of field lines.
1C. Related questions and perspectives.In this paper, we are restricted to energies λ < b 1 − h, and as mentioned in Remark 1.9, the threshold b 1 > b 0 is limited by three conditions, including the nonresonance one: It would be interesting to study the influence of resonances between the functions β j on the spectrum of L − h .Maybe the Grushin techniques could help, as in [Helffer and Kordyukov 2015] for instance.A Birkhoff normal form was given in [Charles and Vũ Ngo . c 2008] for a Schrödinger operator − − h 2 +V with resonances, but the situation is somehow simpler, since the analogues of β j (q) are independent of q in this context.
We are also restricted by the existence of Darboux coordinates ϕ on ( , B) such that ϕ * B = dη ∧ dy.Indeed, the coordinates (y, η) on are necessary to use the Weyl quantization.To study the influence of the global geometry of B, one should consider another quantization method for the presymplectic manifold ( , 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 , and the operator L − h can be linked with this structure in the following way: In [Tejero Prieto 2006], this is used to compute the spectrum of L − h on a bidimensional Riemann surface M with constant curvature and constant magnetic field.See also [Charles 2020;Kordyukov 2022], where semiexcited states for constant magnetic fields in higher dimensions are considered.
If the 2-form B is not exact, we usually consider a Bochner Laplacian on the p-th tensor product of a complex line bundle L over M, with curvature B. This Bochner Laplacian p depends on p ∈ ‫,ގ‬ and the limit p → +∞ is interpreted as the semiclassical limit.The Bochner Laplacian p is a good generalization of the magnetic Laplacian because locally it can be written (1/ − h 2 )(i − h∇ + A) 2 , where the potential A is a local primitive of B, and − h = p −1 .For details, we refer to [Kordyukov 2019;2020;Marinescu and Savale 2018].Kordyukov [2019] constructed quasimodes for p in the case of a symplectic B and discrete wells.He proved expansions Our work also gives such expansions for p as explained in [Morin 2022a].

In this paper, we only mention the study of the eigenvalues of L −
h : what about the eigenfunctions?WKB expansions for the j-th eigenfunction were constructed on ‫ޒ‬ 2 in [Bonthonneau and Raymond 2020] and on a 2-dimensional Riemannian manifold in [Bonthonneau et al. 2021a].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.
Another related question is the decreasing of the real eigenfunctions.Agmon estimates only give a O(e −c/ √ − h ) decay outside any neighborhood of q 0 , but the 2-dimensional WKB suggests a O(e −c/ − h ) decay.
Recently Bonthonneau, Raymond and Vũ Ngo . c [Bonthonneau et al. 2021b] proved this on ‫ޒ‬ 2 using the FBI transform to work on the phase space T * ‫ޒ‬ 2 .This kind of question is motivated by the study of the tunneling effect: the exponentially small interaction between two magnetic wells for example.
Finally, 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 2-dimensional case in [Boil and Vũ Ngo . c 2021].
1D. Structure of the paper.In Section 2 we prove Theorem 1.4, describing the symbol H of L − h on a neighborhood of = H −1 (0).In Section 3 we construct the normal form, first in a space of formal series (Section 3B) and then the quantized version N − h (Section 3C).In Section 4 we prove Theorem 1.7.For this we describe the spectrum of N − h (Section 4A), then we prove microlocalization properties on the eigenfunctions of L − h and N − h (Section 4B), and finally we compare the spectra of L − h and N − h (Section 4C).In Section 5 we focus on Theorem 1.11 which describes the spectrum of the effective operator N [1]   − h .In Section 5A we study its symbol, in Section 5B we construct a second formal Birkhoff normal form, and in Section 5C the quantized version M − h .In Section 5D we compare the spectra of N [1]   − h and M − h .Finally, Sections 6 and 7 are dedicated to the proofs of Corollaries 1.14 and 1.15 respectively.

2A. Notation. L −
h is an − h-pseudodifferential operator on M with principal symbol H : Here, T * M denotes the cotangent bundle of M, and p ∈ T * q M is a linear form on T q M. The scalar product g q on T q M induces a scalar product g * q on T * q M, and | • | g * q denotes the associated norm.In this section we prove Theorem 1.4, thus describing H on a neighborhood of its minimum: Recall that is a small neighborhood of q 0 ∈ M \ ∂ M. We will construct canonical coordinates (z, w, v) ∈ ‫ޒ‬ 2d on , with ‫ޒ‬ 2d is endowed with the canonical symplectic form We will identify with We will use several lemmas to prove Theorem 1.4.Before constructing the diffeomorphism −1 1 on a neighborhood U 1 of , we will first define it on .Thus we need to understand the structure of induced by the symplectic structure on T * M (Section 2B).Then we will construct 1 and finally prove Theorem 1.4 (Section 2C).
2B. Structure of .Recall that on T * M we have the Liouville 1-form α defined by where π : T * M → M is the canonical projection: π(q, p) = q, and dπ is its differential.T * M is endowed with the symplectic form ω = dα. is a d-dimensional submanifold of T * M which can be identified with using j : q ∈ → (q, A q ) ∈ and its inverse, which is π .
Taking the exterior derivative we get Since B is a closed 2-form with constant rank equal to 2s, ( , π * B) is a presymplectic manifold.It is equivalent to ( , B), using j.We recall the Darboux lemma, which states that such a manifold is locally equivalent to ‫ޒ(‬ 2s+k , dη ∧ dy).
One can always take any coordinate system on .Up to working in these coordinates, it is enough to consider the case M = ‫ޒ‬ d with to prove Theorem 1.4.This is what we will do.In coordinates, ω is given by and is the submanifold In order to extend j • ϕ to a neighborhood of ′ in ‫ޒ‬ 2d in a symplectic way, it is convenient to split the tangent space T j (q) ‫ޒ(‬ 2d ) according to tangent and normal directions to .This is the purpose of the following two lemmas.
Lemma 2.3.Fix j (q) = (q, A(q)) ∈ .Then the tangent space to is Moreover, the ω-orthogonal Proof.Since is the graph of q → A(q), its tangent space is the graph of the differential Q → (∇ q A) • Q.
In order to characterize T ⊥ , note that the symplectic form ω = d p ∧ dq is defined by where ⟨ • , • ⟩ denotes the Euclidean scalar product on ‫ޒ‬ d .Thus, Finally, with Lemma 2.1 we know that the restriction of ω to T is given by π * B. Hence, T j (q) ∩T j (q) ⊥ is the set of (Q, P) ∈ T j (q) such that It is the kernel of π * B. □ Now we define specific basis of T j (q) and its orthogonal.Since B(q) is skew-symmetric with respect to g, there exist orthonormal vectors (2-2) These vectors are smooth functions of q because the nonzero eigenvalues ±iβ j (q) are simple.They define a basis of ‫ޒ‬ d .Define the following ω-orthogonal vectors to T : These vectors are linearly independent and Similarly, the tangent space T j (q) admits a decomposition Thus its differential d( j • ϕ) maps the kernel of dη ∧ dy on the kernel of π * B: (2-4) A complementary space of K in T is given by From all these considerations we deduce: Lemma 2.4.Fix j (q) = (q, A(q)) ∈ .Then we have the decomposition where L is any Lagrangian complement of K in (E ⊕ F) ⊥ .
Proof.We have T + T ⊥ = E ⊕ K ⊕ F, and the restriction of ω = d p ∧ dq to this space has kernel K = T ∩T ⊥ .Hence, the restriction ω E⊕F of ω to E ⊕ F is nondegenerate and its orthogonal (E ⊕ F) ⊥ as well.Moreover (E ⊕ F) ⊥ has dimension 2d − 4s = 2k, and we have K is a Lagrangian subspace of (E ⊕ F) ⊥ .Therefore it admits a complementary Lagrangian: a subspace L of (E ⊕ F) ⊥ with dimension k such that ω L = 0 and Remark 2.5.From now on, we fix any choice of Lagrangian complement L. With this choice, we define a basis (ℓ j ) of L as follows.First note that the decomposition (E ⊕ F) ⊥ = K ⊕ L yields a bijection between L and the dual K * , which is ℓ → ω(ℓ, • ).We emphasize that this bijection depends on the choice of L. Using this bijection, we define ℓ j to be the unique vector in L satisfying (2-6) Using the canonical coordinates (w, t, τ, z), we identify with ′ .
Lemma 2.8.The Hamiltonian H = H • 1 has the Taylor expansion Proof.Let us compute the differential and Hessian of at a point (q, A(q)) ∈ .First, and at p = A(q) the Hessian is We can deduce a Taylor expansion of H (w, t, τ, z) with respect to (τ, z) (with fixed q = ϕ(w, t)).First, Then we can compute the partial differential using Lemma 2.6, . The Taylor expansion of H is thus where ∂ 2 τ,z H is the partial Hessian with respect to (τ, z).We have and computing the Hessian matrix amounts to computing ∇ 2 j (q) H on the vectors g j , f j , and In the special case (Q, P) = f j we have and similarly Finally, it remains to prove Actually, (2-10) follows from the identity where ⊥ H denotes the orthogonal with respect to the quadratic form ∇ 2 H (which is different from the symplectic orthogonal ⊥).Indeed, to prove (2-11) note that }, because the vectors u j , v j span the range of B. Hence we find We construct the Birkhoff normal form in the space It is a space of formal series in (x, ξ, τ, − h) with coefficients smoothly depending on (w, t).We see these formal series as Taylor series of symbols, which we quantize using the Weyl quantization.Given an h (with symbol a − h admitting an expansion in powers of − h in some standard class), we denote by [a − h ] or σ T (A − h ) the Taylor series of a − h with respect to (x, ξ, τ ) at (x, ξ, τ ) = 0. Conversely, given a formal series ρ ∈ E 1 , we can find a bounded symbol a − h such that [a − h ] = ρ.This symbol is not uniquely defined, but any two such symbols differ by O((x, ξ, − h) ∞ ), uniformly with respect to (w, t) ∈ U.
• In order to make operations on Taylor series compatible with the Weyl quantization, we endow E 1 with the Weyl-Moyal product ⋆, defined by Op where Note that to define such a product it is necessary to assume that our formal series depend smoothly on (w, t).
• The degree of a monomial is We denote by D N the C ∞ (U )-module spanned by monomials of degree N, and which satisfies If ρ 1 , ρ 2 ∈ E 1 , we denote their commutator by , and we have the formula In particular, and ).The Birkhoff normal form algorithm is based on the following lemma.We recall the definition (1-5) of r 1 .
(1) Every series ρ ∈ E 1 satisfies i ( Proof.The first statement is a simple computation.For the second and the third, it suffices to consider monomials so that R N commutes with every |z j | 2 (1 ≤ j ≤ s) if and only if α = α ′ , which amounts to saying that R N is a function of |z j | 2 and proves (3).Moreover, where ⟨γ , β⟩ = s j=1 γ j βj (w, t).Under the assumption |α| + |α ′ | + |α ′′ | + 2ℓ < r 1 , we have |α − α ′ | < r 1 and by the definition of r 1 the function ⟨α ′ − α, β(w, t)⟩ cannot vanish for (w, t) ∈ U, and this proves (2).□ 3B.Formal Birkhoff normal form.In this section we construct the Birkhoff normal form at a formal level.We will work with the Taylor series of the symbol H of L − h , in the new coordinates 1 .According to Theorem 1.4, H = H • 1 defines a formal series (3-4) At a formal level, the normal form can be stated as follows.
Theorem 3.3.For every γ ∈ O 3 , there are κ, ρ ∈ O 3 such that where κ is a function of harmonic oscillators: Moreover, if γ has real-valued coefficients, then so do ρ, κ and the remainder O r 1 .
Proof.We prove this by induction on an integer N ≥ 3. Assume that we found Rewriting the remainder as R N + O N +1 , with R N ∈ D N , we have We are looking for a ρ ′ ∈ O N .For such a ρ ′ we apply e (i/ − h) ad ρ ′ : The new term because a derivation with respect to (y, η, t) does not decrease the degree.Similarly, and thus (3-6) becomes i Using this formula in (3-5) we get Thus, we are looking for K N , ρ ′ ∈ D N such that with [K N , |z j | 2 ] = 0.By Lemma 3.2, we can solve this equation provided N < r 1 , and this concludes the proof.Moreover, (i/ − h) ad |z j | 2 is a real endomorphism, so we can solve this equation on ‫.ޒ‬ □ 3C.Quantizing the normal form.We now construct the normal form N − h , quantizing Theorems 1.4 and 3.3.We denote by I ( j) − h the harmonic oscillator with respect to x j , defined by We prove the following theorem.
Theorem 3.4.There exist (1) a microlocally unitary operator Remark 3.5.U − h is a Fourier integral operator quantizing the symplectomorphism ˜ 1 ; see [Martinez 2002;Zworski 2012].In particular, if h is a pseudodifferential operator on ‫ޒ‬ d with symbol Remark 3.6.Due to the parameters (y, η, t, τ ) in the formal normal form, an additional quantization is needed, hence the Op It is a quantization with respect to (y, η, t, τ ) of an operator-valued symbol f ⋆ 1 (y, η, t, τ, I (1) − h , . . ., I (s) − h ).Actually, this operator symbol is simple since one can diagonalize it explicitly.Denoting by h j n j (x j ) the n j -th eigenfunction of Proof.In order to prove Theorem 3.4, we first quantize Theorem 1.4.Using the Egorov theorem, there exists a microlocally unitary operator Then we use the following lemma to quantize the formal normal form and conclude.□ Lemma 3.7.There exists a bounded pseudodifferential operator Q − h with compactly supported symbol such that where N − h and R − h satisfy the properties stated in Theorem 3.4.
Remark 3.8.As explained below, the principal symbol Q of Q − h is O((x, ξ, τ ) 3 ).Thus, the symplectic flow ϕ t associated to the Hamiltonian Q is ϕ t (x, ξ, τ ) = (x, ξ, τ ) + O((x, ξ, τ ) 2 ).Moreover, the Egorov theorem implies that e −(i/ − h)Q− h quantizes the symplectomorphism ϕ 1 .Hence, Proof.The proof of this lemma follows the exact same lines as in the case k = 0 [Morin 2022b, Theorem 4.1].Let us recall the main arguments.The symbol σ − h is equal to H + O( − h 2 ) on U ′ 1 .Thus, its associated formal series is [σ − h ] = H 2 + γ for some γ ∈ O 3 .Using the Birkhoff normal form algorithm (Theorem 3.3), we get κ, ρ ∈ O 3 such that If Q − h is a smooth compactly supported symbol with Taylor series [Q − h ] = ρ, then by the Egorov theorem the operator has a symbol with Taylor series H 2 + κ + O r 1 .Since κ commutes with the oscillator |z j | 2 , it can be written as We can reorder this formal series using the monomials If f ⋆ 1 is a smooth compactly supported function with Taylor series then the operator (3-7) is equal to In this section we describe the spectral properties of N − h .We can use the properties of harmonic oscillators to diagonalize it in the following way.For 1 ≤ j ≤ s and n j ≥ 1, we recall that the n j -th Hermite function h j n j (x j ) is an eigenfunction of h(2n j − 1)h j n j , and the functions (h n ) n∈‫ގ‬ s defined by form a Hilbertian basis of L 2 ‫ޒ(‬ s x ).Thus, we can use this basis to decompose the space L 2 ‫ޒ(‬ 2s+k x,y,t ) on which N − h acts: N − h preserves this decomposition and where N [n]   − h is the pseudodifferential operator with symbol In particular, the spectrum of Moreover, as in the k = 0 case, for any b 1 > 0 there is an The reason is that the symbol N [n]   − h is greater than b 1 − h for n large enough.Finally, to prove our main result, Theorem 1.7, it remains to compare the spectra of L − h and N − h .
4B. Microlocalization of the eigenfunctions.Here we prove microlocalization results for the eigenfunctions of L − h and N − h .These results are needed to show that the remainders O((x, ξ, τ ) r 1 ) we got are small.More precisely, for each operator we need to prove that the eigenfunctions are microlocalized • inside (space localization), • where |(x, ξ, τ )| ≲ − h δ for δ ∈ 0, 1 2 (i.e., close to ).Fix b1 such that Proof.This follows from the Agmon estimates, as in the k = 0 case (in [Morin 2022b]).Indeed, from (4-2) we deduce where the O( − h ∞ ) is in the space of bounded operators L(L 2 , L 2 ) and independent of

Then the eigenfunction ψ
With the notation χ = 1 − χ 1 , we will prove that We will bound from above the right-hand side, and from below the left-hand side.First, since g − h (λ) is supported where λ ≤ b1 − h, we have Moreover, the commutator [L − h , χ w ] is a pseudodifferential operator of order − h, with symbol supported on supp χ .Hence, if χ is a cutoff function having the same general properties of χ, such that χ = 1 on supp χ , we have Finally, the symbol of χ w is equal to 0 on an − h δ -neighborhood of , and thus the symbol on the support of χ w .Hence the Gårding inequality yields Using this last inequality in (4-4), and bounding the right-hand side with (4-5) and (4-6) we find and we deduce that Iterating with χ instead of χ, we finally get, for arbitrarily large N > 0, This is true for every ψ, with ϕ = g − h (L − h )ψ, and thus x,ξ,τ ) be a cutoff function equal to 1 on a neighborhood of 0. Then every eigenfunction ψ − h of N − h associated with an eigenvalue λ Proof.Just as in the previous lemma, it is enough to show that ).We prove this using the same method.
The right-hand side can be bounded from above as before.On the left-hand side we find ε > 0 such that The symbol of χ w vanishes on an − h δ -neighborhood of x = ξ = τ = 0. Thus we can bound from below the symbol of H 2 and use the Gårding inequality: We conclude the proof as in Lemma 4.2.□ Lemma 4.4 (space localization for N − h ).Let b 1 ∈ (b 0 , b1 ) and χ 0 ∈ C ∞ 0 ‫ޒ(‬ 2s+k y,η,t ) be a cutoff function equal to 1 on a neighborhood of { b(y, η, t) ≤ b1 }.Then every eigenfunction ψ − h of N − h associated with an eigenvalue λ Proof.Every eigenfunction of N − h is given by ψ − h (x, y, t) = u − h (y, t)h n (x) for some Hermite function h n with |n| ≤ N max and some eigenfunction u − h of N [n]   − h .Thus, it is enough to prove the lemma for the eigenfunctions of We will prove that ∥χ We first have the bound ] is a pseudodifferential operator of order − h with symbol supported on supp χ.Moreover, its principal symbol is {N where χ has the same general properties as χ, and is equal to 1 on supp χ.By Lemma 4.3, we can find a cutoff where |τ | ≲ − h δ and we get Finally for ε > 0 small enough we have the lower bound and χ vanishes on a neighborhood of { b(w, t) ≤ b1 }.Using this lower bound in (4-10), and bounding the right-hand side with (4-11) and (4-12) we get and we can iterate with χ instead of χ to conclude.□ 4C.Proof of Theorem 1.7.To conclude the proof of Theorem 1.7, it remains to show that Here λ n (A) denotes the n-th eigenvalue of the self-adjoint operator A, repeated with multiplicities.
Lemma 4.5.One has where U − h is given by Theorem 3.4.We will use ϕ According to Lemmas 4.3 and 4.4, ψ is microlocalized, where (w, t) ∈ { b(w, t) ≤ b1 } ⊂ U and for suitable δ ∈ 0, 1 2 .By (4-14) and (4-15) we have The reversed inequality is proved in the same way: we take the eigenfunctions of L − h as quasimodes for N − h , and we use the microlocalization lemma, Lemma 4.2.□

5.
A second normal form in the case k > 0 In the previous sections, we compared the spectrum of L − h and the spectrum of the normal form (5-1) In this section, we will construct a Birkhoff normal form again, to describe the spectrum of N [1]   − h by an effective operator M − h on ‫ޒ‬ s y .For that purpose, in Section 5A we will find new canonical variables ( t, τ ) in which N [1]   − h is the perturbation of a harmonic oscillator.In Sections 5B and 5C we will construct the semiclassical Birkhoff normal form M − h .In Section 5D we will prove that the spectrum of N [1]   − h is given by the spectrum of M − h .Under Assumption 1 we know that t → b(w, t) admits a nondegenerate minimum at s(w) for w in a neighborhood of 0, and we denote by (ν 2 1 (w), . . ., ν 2 k (w)) the eigenvalues of the positive symmetric matrix The maps ν 1 , . . ., ν k are smooth nonvanishing functions in a neighborhood of w = 0.
5A. Geometry of the symbol N [1]   − h .We prove the following lemma.Lemma 5.1.There exists a canonical (symplectic) Proof.We want to expand N [1]   − h near its minimum with respect to the variables v = (t, τ ).First, from the Taylor expansion of f ⋆ 1 we deduce We will Taylor-expand t → b(w, t) on a neighborhood of its minimum point s(w).For that purpose, we define new variables ( ỹ, η, t, τ ) = φ(y, η, t, τ ) by Then φ * ω 0 = ω 0 + O(τ ).Using Theorem B.2, we can make φ symplectic on a neighborhood of 0, up to a change of order O(τ 2 ).In these new variables, the symbol Then we remove the tildes and expand this symbol in powers of t, τ , − h.We find Now, we want to diagonalize the positive quadratic forms M(w, s(w)) The diagonalization of quadratic forms in orthonormal coordinates implies that there exists a matrix P(w) such that We define the new coordinates ( y, η, ť, τ ) = φ(y, η, t, τ ) by . Again, we can make it symplectic up to a change of order O(|t| 3 +|τ | 2 ) by Theorem B.2.In these new variables, the symbol becomes (after removing the "checks") The last change of coordinates ( ŷ, η, t, τ ) = φ(y, η, t, τ ), defined by , so it can be corrected modulo O(|τ | 2 ) to be symplectic, and we get the new symbol which concludes the proof.□ 5B.Second formal normal form.The harmonic oscillators appearing in N − h are the symbol of J ( j) − h for the h-quantization is τ 2 j + t 2 j .This is why we use the mixed quantization It is related to the − h-quantization by the relation In other words, if a is a symbol in some standard class S(m), and if we define a(h, y, η, t, τ ) = a(h 2 , y, η, t, h τ ), However, if we take a ∈ S(m), then Op w ♯ (a) is not necessarily an − h-pseudodifferential operator, since the associated a may not be bounded with respect to − h, and thus it does not belong to any standard class.For instance, we have But still Op w ♯ (a) is an h-pseudodifferential operator, with symbol a(h, y, η, t, τ ) = a(h, y, h η, t, τ ).

With this notation
Op w ♯ (a) = Op w h (a).
Thus, in this sense, we can use the properties of − h-pseudodifferential and h-pseudodifferential operators to deal with our mixed quantization.
Remark 5.2.Operators of the form (5-2) are just special cases of the usual h-pseudodifferential operators for which the reader can refer to [Martinez 2002;Zworski 2012].Moreover, our mixed quantization could be interpreted as a where we first quantize with respect to (y, η) so that Op w − h a is an operator-valued symbol which depends on (t, τ ).In the following we could have used this formalism, thus dealing with operator-valued symbols in (t, τ ) instead of real-valued symbols and mixed quantization.
In our case, we have Let us construct a semiclassical Birkhoff normal form with respect to this quantization.We will work in the space of formal series (5-4) where U = U 2 ‫ޒ∩‬ 2s w ×{0}.This space is endowed with the star product ⋆ adapted to our mixed quantization.In other words The change of variable τ = h τ between the usual − h-quantization and our mixed quantization yields the following formula for the star product: (5-5) with The degree function on E 2 is defined by We denote by D N the C ∞ (U )-module spanned by monomials of degree N, and For τ 1 , τ 2 ∈ E 2 , we define We define with the notation ṽj = t j + i τj , so that 1 Now we construct the following normal form.Recall that r 2 is an integer chosen such that, for all α ∈ ‫ޚ‬ k , 0 < |α| < r 2 , s j=1 α j ν j (0) ̸ = 0.
Moreover, this nonresonance relation at w = 0 can be extended to a small neighborhood of 0.
Theorem 5.4.There exist Proof.Lemma 5.1 provides us with a symplectomorphism 2 such that We can apply the Egorov theorem to get a Fourier integral operator V 2, − h such that and following the notation of Section 5B, we have the associated formal series We apply Lemma 5.3 and we get formal series κ, ρ such that We take a compactly supported symbol a(h, w, t, τ ) with Taylor series ρ.Then the operator (5-9) has a symbol with Taylor series If we take f ⋆ 2 (h, w, J 1 , . . ., J k ) a smooth compactly supported function with Taylor series then the operator (5-9) is equal to Multiplying by h 2 , and getting back to the − h-quantization, we get Lemma 5.5.Let δ ∈ 0, 1 2 and c > 0.
Proof.Using the mixed quantization and h The principal part of N [1]  h is of order h 2 , and implies a microlocalization of the eigenfunctions, where Since b admits a unique and nondegenerate minimum b 0 at 0, this implies that w lies in an arbitrarily small neighborhood of 0, and that The technical details follow the same ideas of Lemmas 4.2, 4.3 and 4.4.Now we can focus on M − h , whose principal symbol with respect to the Op w ♯ -quantization is Hence its eigenfunctions are microlocalized where b(y, η, s(y, η)) which implies again that w lies in an arbitrarily small neighborhood of 0 and that Using the same method as before, we deduce from Theorem 5.4 and Lemma 5.5 a comparison of the spectra of N [1]   − h and M − h .With the notation the following lemma concludes the proof of Theorem 1.11.
Lemma 5.6.Let δ ∈ 0, 1 2 and c > 0. We have In this section we prove that the spectrum of We recall that c ∈ (0, min j ν j (0)) and r = min(2r 1 , r 2 + 4).We can apply Theorem 1.7 for b 1 > b 0 arbitrarily close to b 0 .Thus the spectrum of L and we deduce from the Gårding inequality that Then, we apply Theorem 1.11 for δ close enough to 1 2 , and we see that the spectrum of N and the eigenfunctions of M [n]   − h are microlocalized in an arbitrarily small neighborhood of (y, η) = 0 (Lemma 5.5), and M [n]   − h satisfies in this neighborhood Using the Gårding inequality, the spectrum of M [n]   − h (n ̸ = 1) is thus ≥ b 0 + − h 1/2 (ν(0) + 2c) for ε and − h small enough.It follows that the spectrum of N [1]   − h below − hb 0 + − h 3/2 (ν(0) + 2c) is given by the spectrum of − hM [1]   − h .

Proof of Corollary 1.15
We explain here where the asymptotics for λ j (L − h ) come from.First we use Corollary 1.14 so that the spectrum of L − h below − hb 0 + − h 3/2 (ν(0) + 2c) is given by M with ν(w) = k j=1 ν j (w).The principal part admits a unique minimum at 0, which is nondegenerate.The asymptotics of the first eigenvalues of such an operator are well known.First one can make a linear change of canonical coordinates diagonalizing the Hessian of b and get a symbol of the form One can factor the ∇ν(0) • w term to get µ j (η 2 j + y 2 j ) + − h 1/2 ν(0) + − hc 0 + O(w 3 + − hw + − h 3/2 + − h 1/2 w 2 ).
For an operator with such symbol (i.e., harmonic oscillator + remainders) one can apply the results of [Charles and Vũ Ngo . c 2008, Theorem 4.7] or [Helffer and Sjöstrand 1984] and deduce that the j-th eigenvalue λ j (M [1]   − h ) admits an asymptotic expansion in powers of − h 1/2 such that where − h E j is the j-th repeated eigenvalue of the harmonic oscillator with symbol s j=1 µ j (η 2 j + y 2 j ).
For By the nondegeneracy of ω t , this determines Y t .We know ψ t exists until time t = 1 on a small enough neighborhood of x = 0, and ψ * t ω t = ω 0 .Thus ψ = ψ 1 is the desired diffeomorphism.If M is a compact manifold, a pseudodifferential operator A − h on L 2 (M) is an operator acting as a pseudodifferential operator in coordinates.Then the principal symbol of A − h (and its Kohn-Nirenberg class) does not depend on the coordinates, and we denote it by σ 0 (A − h ).The subprincipal symbol σ 1 (A − h ) is also well-defined, up to imposing that the charts be volume-preserving (in other words, if we see A − h as acting on half-densities, its subprincipal symbol is well-defined).In the case where M is a compact manifold, L − h is a pseudodifferential operator, and its principal and subprincipal symbols are In this case, we assume that B belongs to some standard class.This is equivalent to assuming that H belongs to some (other) standard class.Then, L − h is a pseudodifferential operator with total symbol H.
with operator-valued symbols for which we refer to [Keraval 2018; Martinez 2007].Indeed we can write Op w ♯ (a) = Op w h (Op w − h a), (5-3) operator with symbol O r 2 .Note that M − h is an − h-pseudodifferential operator whose symbol admits an expansion in powers of Proof of Theorem 1.11.In order to prove Theorem 1.11, we need the following microlocalization lemma.