Magnetic Wells in Dimension

Transcription

1 Magnetic Wells in Dimension Tree Bernard Helffer, Yuri Kordyukov, Nicolas Raymond, San Vu Ngoc To cite tis version: Bernard Helffer, Yuri Kordyukov, Nicolas Raymond, San Vu Ngoc. Tree. Analysis Magnetic Wells in Dimension PDE, Matematical Sciences Publisers, 2016, 9 7, pp < /apde >. <al > HAL Id: al ttps://al.arcives-ouvertes.fr/al Submitted on 6 May 2015 HAL is a multi-disciplinary open access arcive for te deposit and dissemination of scientific researc documents, weter tey are publised or not. Te documents may come from teacing and researc institutions in France or abroad, or from public or private researc centers. L arcive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recerce, publiés ou non, émanant des établissements d enseignement et de recerce français ou étrangers, des laboratoires publics ou privés.

2 MAGNETIC WELLS IN DIMENSION THREE B. HELFFER, Y. KORDYUKOV, N. RAYMOND, AND S. VŨ NGỌC Abstract. Tis paper deals wit semiclassical asymptotics of te treedimensional magnetic Laplacian in presence of magnetic confinement. Using generic assumptions on te geometry of te confinement, we exibit tree semiclassical scales and teir corresponding effective quantum Hamiltonians, by means of tree microlocal normal forms à la Birkoff. As a consequence, wen te magnetic field admits a unique and non degenerate minimum, we are able to reduce te spectral analysis of te low-lying eigenvalues to a one-dimensional -pseudo-differential operator wose Weyl s symbol admits an asymptotic expansion in powers of Introduction 1.1. Motivation and context. Te analysis of te magnetic Laplacian i A 2 in te semiclassical limit 0 as been te object of many developments in te last twenty years. Te existence of discrete spectrum for tis operator, togeter wit te analysis of te eigenvalues, is related to te notion of magnetic bottle, or quantum confinement by a pure magnetic field, and as important applications in pysics. Moreover, motivated by investigations of te tird critical field in Ginzburg-Landau teory for superconductivity, tere as been a great attention focused on estimates of te lowest eigenvalue. In te last decade, it appears tat te spectral analysis of te magnetic Laplacian as acquired a life on its own. For a story and discussions about te subject, te reader is referred to te recent reviews [11, 14, 24]. In contrast to te wealt of studies exploring te semiclassical approximations of te Scrödinger operator 2 + V, te classical picture associated wit te Hamiltonian p Aq 2 as almost never been investigated to describe te semiclassical bound states i.e. te eigenfunctions of low energy of te magnetic Laplacian. Te paper by Raymond and Vũ Ngọc [25] is to our knowledge te first rigorous work in tis direction. In tat paper, wic deals wit te two-dimensional case, te notion of magnetic drift, well known to pysicists, is cast in a symplectic framework, and using a semiclassical Birkoff normal form see for instance [27, 5, 28] it becomes possible to describe all te eigenvalues of order O. Independently, te asymptotic expansion of a smaller set of eigenvalues was establised in [12, 15] troug different metods wic act directly on te quantum side: explicit 1

3 2 B. HELFFER, Y. KORDYUKOV, N. RAYMOND, AND S. VŨ NGỌC unitary transforms and a Grusin like reduction are used to reduce te twodimensional operator to an effective one-dimensional operator. Te tree-dimensional case appens to be muc arder. Te only known results in tis case tat provide a full asymptotic expansion of a given eigenvalue concern toy models were te confinement is obtained by a boundary carrying a Neumann condition on an alf space in [23] or on a wedge in [22]. In te case of smoot confinement witout boundary, a construction of quasimodes by Helffer and Kordyukov in [13] suggests wat te expansions of te low lying eigenvalues could be. But, as was expected by Colin de Verdière in is list of open questions in [7], extending te symplectic and microlocal tecniques to te tree-dimensional case contains an intrinsic difficulty in te fact tat te symplectic form cannot be nondegenerate on te caracteristic ypersurface. Te goal of our paper is to answer tis question by fully carrying out tis strategy. After averaging te cyclotron motion, te effect of te degeneracy of te symplectic form can be observed on te fact tat te reduced operator is only partially elliptic. Hence, te key ingredient will be a separation of scales via te introduction of a new semiclassical parameter for only one part of te variables. Tese semiclassical scales are reminiscent of te tree scales tat ave been exibited in te classical picture in te large field limit, see [2, 6]. Tey are also related to te Born-Oppeneimer type of approximation in quantum mecanics see for instance [4, 20]. In fact, in a partially semiclassical context and under generic assumptions, a full asymptotic expansion of te first magnetic eigenvalues and te corresponding WKB expansions as been recently establised in any dimension in te paper by Bonnaillie-Noël Hérau Raymond [3] Magnetic geometry. Let us now describe te geometry of te problem. Te configuration space is R 3 = {q 1 e 1 + q 2 e 2 + q 3 e 3, q j R, j = 1, 2, 3}, were e j j=1,2,3 is te canonical basis of R 3. Te pase space is R 6 = {q, p R 3 R 3 } and we endow it wit te canonical 2-form 1.1 ω 0 = dp 1 dq 1 + dp 2 dq 2 + dp 3 dq 3. We will use te standard Euclidean scalar product, on R 3 and te associated norm. In particular, we can rewrite ω 0 as ω 0 u 1, u 2, v 1, v 2 = v 1, u 2 v 2, u 1, u 1, u 2, v 1, v 2 R 3. Te main object of tis paper is te magnetic Hamiltonian, defined for all q, p R 6 by 1.2 Hq, p = p Aq 2, were A C R 3, R 3.

4 MAGNETIC WELLS IN DIMENSION THREE 3 Let us now introduce te magnetic field. Te vector field A = A 1, A 2, A 3 is associated via te Euclidean structure wit te following 1-form α = A 1 dq 1 + A 2 dq 2 + A 3 dq 3 and its exterior derivative is a 2-form, called magnetic 2-form and expressed as dα = 1 A 2 2 A 1 dq 1 dq A 3 3 A 1 dq 1 dq A 3 3 A 2 dq 2 dq 3. Te form dα may be identified wit a vector field. If we let: B = A = 2 A 3 3 A 2, 3 A 1 1 A 3, 1 A 2 2 A 1 = B 1, B 2, B 3, ten, we can write 1.3 dα = B 3 dq 1 dq 2 B 2 dq 1 dq 3 + B 1 dq 2 dq 3. Te vector field B is called te magnetic field. Let us notice tat we can express te 2-form dα tanks to te magnetic matrix M B = 0 B 3 B 2 B 3 0 B 1. B 2 B 1 0 Indeed we ave 1.4 dαu, V = U, M B V = U, V B = [U, V, B], U, V R 3 R 3, were [,, ] is te canonical mixed product on R 3. We note tat B belongs to te kernels of M B and dα. An important role will be played by te caracteristic ypersurface Σ = H 1 0, wic is te submanifold defined by te parametrization: R 3 q jq := q, Aq R 3 R 3. We may notice te relation between Σ, te symplectic structure and te magnetic field in te following relation 1.5 j ω 0 = dα, were dα is defined in Confinement assumptions and discrete spectrum. Tis paper is devoted to te semiclassical analysis of te discrete spectrum of te magnetic Laplacian L,A := i q Aq 2, wic is te semiclassical Weyl quantization of H see 2.1. Tis means tat we will consider tat belongs to 0, 0 wit 0 small enoug. Let us recall te assumptions under wic discrete spectrum actually exist. In two dimensions, wit a non vanising magnetic field, a standard estimate see [1, 8] gives Bq uq 2 dq L,A u u, u C0 R 2. R 2

5 4 B. HELFFER, Y. KORDYUKOV, N. RAYMOND, AND S. VŨ NGỌC Except in special cases wen some components of te magnetic field ave constant sign, tis is no more te case in iger dimension see [10]. We sould impose a control of te oscillations of B at infinity. Under tis condition, we get a similar estimate at te price of a small loss. Tis kind of estimate actually follows from an analysis developed in [16]. Let us define bq := Bq. Let us now state te confining assumptions under wic we will constantly work in tis paper. Assumption 1.1. We consider te case of R 3 and assume 1.6 bq b 0 := inf q R 3 bq > 0, and te existence of a constant C > 0 suc tat 1.7 Bq C 1 + bq, q R 3. Under Assumption 1.1, it is proven in [16, Teorem 3.1] tat tere exist 0 > 0 and C 0 > 0 suc tat, for all 0, 0, C R 3 bq uq 2 dq L,A u u, u C 0 R 3. As a corollary, using Persson s teorem see [21], we obtain tat te bottom of te essential spectrum is asymptotically above b 1, were b 1 := lim inf q + bq. More precisely, under Assumption 1.1, tere exist 0 > 0 and C 0 > 0 suc tat, for all 0, 0, 1.9 s ess L,A [b 1 1 C 0 1 4, +. Assumption 1.2. We assume tat < b 0 < b 1. Moreover we will assume tat tere exists a point q 0 R 3 and ε > 0, β 0 b 0, b 1 suc tat 1.11 {bq β 0 } Dq 0, ε, were Dq 0, ε is te Euclidean ball centered at q 0 and of radius ε. For te rest of te article we let β 0 b 0, β 0. Witout loss of generality, we can assume tat q 0 = 0 and tat A0 = 0 wic can be obtained wit a cange of gauge. Note tat Assumption 1.2 implies tat te minimal value of b is attained inside Dq 0, ε. All along tis paper, we will strengten te assumptions on te nature of te point q 0. At some stage of our investigation, q 0 will be te unique minimum of b. Note in particular tat 1.11 is satisfied as soon as b admits a unique and non degenerate minimum.

6 MAGNETIC WELLS IN DIMENSION THREE Informal description of te results. Let us now informally walk troug te main results of tis paper. We will assume as precisely formulated in tat te magnetic field does not vanis and is confining. Of course, for eigenvalues of order O, te corresponding eigenfunctions are microlocalized in te semi-classical sense near te caracteristic manifold Σ see for instance [26, 31]. Moreover te confinement assumption implies tat te eigenfunctions of L,A associated wit eigenvalues less tat β 0 enjoy localization estimates à la Agmon. Terefore we will be reduced to investigate te magnetic geometry locally in space near a point q 0 = 0 R 3 belonging to te confinement region and wic, for notational simplicity, we may assume to be te origin. Ten, in a neigborood of 0, A0 Σ, tere exist symplectic coordinates x 1, ξ 1, x 2, ξ 2, x 3, ξ 3 suc tat Σ = {x 1 = ξ 1 = ξ 3 = 0} and 0, A0 as coordinates 0 R 6. Hence Σ is parametrized by x 2, ξ 2, x First Birkoff form. In tese coordinates suited for te magnetic geometry, it is possible to perform a semiclassical Birkoff normal form and microlocally unitarily conjugate L,A to a first normal form N = Op w N wit an operator valued symbol N depending on x 2, ξ 2, x 3, ξ 3 in te form N = ξ bx 2, ξ 2, x 3 I + f, I, x 2, ξ 2, x 3, ξ 3 + O I, ξ 3. were I = 2 Dx 2 1 +x 2 1 is te first encountered armonic oscillator and were, I, x 2, ξ 2, x 3, ξ 3 f, I, x 2, ξ 2, x 3, ξ 3 satisfies, for I 0, I 0, f, I, x 2, ξ 2, x 3, ξ 3 C I ξ Since we wis to describe te spectrum in a spectral window containing at least te lowest eigenvalues, we are led to replace I by its lowest eigenvalue and tus, we are reduced to te two-dimensional pseudo-differential operator N [1] = Op w N [1] were N [1] = ξ bx 2, ξ 2, x 3 + f,, x 2, ξ 2, x 3, ξ 3 + O, ξ Second Birkoff form. If we want to continue te normalization, we sall assume a new non-degeneracy condition te first one was te positivity of b. Now we assume tat, for any x 2, ξ 2 in a neigborood of 0, 0, te function x 3 bx 2, ξ 2, x 3 admits a unique and non-degenerate minimum denoted by sx 2, ξ 2. Ten, by using a new symplectic transformation in order to center te analysis at te partial minimum sx 2, ξ 2, we get a new operator N [1] wose Weyl symbol is in te form wit N [1] = ν2 x 2, ξ 2 ξ x bx 2, ξ 2, sx 2, ξ 2 + remainders, 1.12 νx 2, ξ 2 = bx 2, ξ 2, sx 2, ξ 2 1/4

7 6 B. HELFFER, Y. KORDYUKOV, N. RAYMOND, AND S. VŨ NGỌC and were te remainders ave been properly normalized to be at least formal perturbations of te second armonic oscillator ξ3 2 + x2 3. Since te frequency of tis oscillator is 1 2 in te classical picture, we are naturally led to introduce te new semiclassical parameter = 1 2 and te new impulsion so tat Op w ξ = 1 2 ξ ξ x 2 3 = 2 Op w ξ2 3 + x 2 3. We terefore get te -symbol of N [1] : N [1] = 2 ν 2 x 2, ξ 2 ξ x bx 2, ξ 2, sx 2, ξ 2 + remainders. We can again perform a Birkoff analysis in te space of formal series given by E = F [x 3, ξ 3, ] were F is a space of symbols in te form c, x 2, ξ 2. We get te new operator M = Op w M, wit M = 2 bx 2, ξ 2, sx 2, ξ J Op w ν2 x 2, ξ g, J, x 2, ξ 2 + remainders, were J = Op w ξ2 3 + x3 2 and g, J, x 2, ξ 2 is of order tree wit respect to J 1 2, 1 2. Motivated again by te perspective of describing te low lying eigenvalues, we replace J by and rewrite te symbol wit te old semiclassical parameter to get te operator M [1] = Opw M [1] = Op w M [1], wit 1.13 M [1] = bx 2, ξ 2, sx 2, ξ ν 2 x 2, ξ 2 + g 1 1 2, 2, x2, ξ 2 + remainders Tird Birkoff form. Te last generic assumption is te uniqueness and non-degeneracy of te minimum of te new principal symbol x 2, ξ 2 bx 2, ξ 2, sx 2, ξ 2 tat implies tat b admits a unique and non-degenerate minimum at 0, 0, 0. Up to an 1 2 -dependent translation in te pase space and a rotation, we are essentially reduced to a standard Birkoff normal form wit respect to te tird armonic oscillator K = 2 D 2 x 2 + x 2 2. Note tat all our normal forms may be used to describe te classical dynamics of a carged particle in a confining magnetic field see Figure 1.

8 MAGNETIC WELLS IN DIMENSION THREE 7 Figure 1. Te dased line represents te integral curve of te confining magnetic field B = curla troug q0 = 0.5, 0.6, 0.7 for Bx, y, z = y2, z2, 1 + x2 and te full line represents te projection in te q-space of te Hamiltonian trajectory wit initial condition q0, p0 wit p0 = 0.6, 0.01, 0.2 ending at q1, p Microlocalization. Of course, at eac step, we will ave to provide accurate microlocal estimates of te eigenfunctions of te different operators to get a good control of te different remainders. In a first approximation, we will get localizations at te following scales x1, ξ1, ξ3 ~δ δ > 0 is small enoug and x2, ξ2, x3 1. In a second approximation, we will get x3, ξ 3 ~δ. In te final step, we will refine te localization by x2, ξ2 ~δ A semiclassical eigenvalue estimate. Let us already state one of te consequences of our investigation. It will follow from te tird normal form tat we ave a complete description of te spectrum below te tresold 3 b0 ~ + 3ν 2 0, 0~ 2. Tis description is reminiscent of te results à la BorSommerfeld of [17] and [18, Appendix B] see also [15, Remark 1.4] obtained in te case of one dimensional semiclassical operators.

9 8 B. HELFFER, Y. KORDYUKOV, N. RAYMOND, AND S. VŨ NGỌC Teorem 1.3. Assume tat b admits a unique and non degenerate minimum at q 0. Denote 1.14 σ = Hess q 0 b B, B det Hess q0 b 2b 2, θ = 0 Hess q0 b B, B. Tere exists a function k C0 R2 wit arbitrarily small compact support, and k 1 2, Z = O + Z 3 2 wen, Z 0, 0, suc tat te following olds. For all c 0, 3, te spectrum of L,A below b 0 +cσ coincides modulo O wit te spectrum of te operator F acting on L 2 R x given by F = b 0 + σ 1 3 ζ θ 2 2 2θ K + k 1 2, K, K = 2 Dx 2 + x 2, wit some constant ζ. Remark 1.4. Te constant ζ in Teorem 1.3 is given by te formula ζ = ν 2 0, 0 2, were te function ν is given in Observe also tat σ = ν 4 0, 0. Corollary 1.5. Under te ypotesis of Teorem 1.3, let λ m m1 be te non decreasing sequence of te eigenvalues of L,A. For any c 0, 3, let N,c := {m N ; λ m b 0 + cσ }. Ten te cardinal of N,c is of order 1 2, and tere exist υ 1, υ 2 R and 0 > 0 suc tat [ λ m =b 0 +σ θm 1 2 ζ ] 2 +υ 1 m 1 2θ υ2 m O 5 2, uniformly for 0, 0 and m N,c. In particular, te splitting between two consecutive eigenvalues satisfies λ m+1 λ m = θ 2 + O 5 2. Proof. If te support of k is small enoug, te ypotesis k 1 2, Z = O + Z 3 2 implies tat, wen is small enoug, 1 + ηk K + 2 θ k 1 2, K 1 ηk, for some small η > 0. Terefore, since te eigenvalues of K are 2m 1, m N, te variational principle implies tat te number of eigenvalues of K + 2 θ k 1 2, K below a tresold C belongs to [ 1 2 C 1+η +1, 1 2 C 1 η +1]. Taking C = 2 θ c 1σ1/2 1/2 + ζ, and applying te teorem, we obtain θ 2 te estimate for te cardinal of N,c. Te corresponding eigenvalues of L,A are of te form λ m = b 0 + σ 1 3 ζ ] 2 2 2θ 2 + [θm k 1 2, 2m 1 + O,

10 MAGNETIC WELLS IN DIMENSION THREE 9 wit 2m 1 C 1 η. Terefore tere exists a constant C > 0, independent of, suc tat all m N,c satisfy te inequality 2m 1 C 1/2. Writing k 1 2, Z = c0 3/2 + υ 1 1/2 Z/2 + c υ 2 Z/2 2 + υ 3 Z + 1/2 O + Z 2 + OZ 3, we see tat, for m N,c, k 1 2, 2m 1 = υ1 3/2 m 1 + υ 2 2 m O 2 2 3/2, wic gives te result. Remark 1.6. An upper bound of λ m for fixed -independent m wit remainder in O 9 4 was obtained in [13] troug a quasimodes construction involving powers of 1 4. To te autors knowledge, Corollary 1.5 gives te most accurate description of magnetic eigenvalues in tree dimensions, in suc a large spectral window. Note also tat te non-degeneracy assumption on te norm of B is not purely tecnical. Indeed, at te quantum level, it appears troug microlocal reductions matcing wit te splitting of te Hamiltonian dynamics into tree scales: te cyclotron motion around field lines, te center-guide oscillation along te field lines, and te oscillation witin te space of field lines Organization of te paper. Te paper is organized as follows. In Section 2, we state our main results. Section 3 is devoted to te investigation of te first normal form see Teorem 2.1 and Corollary 2.4. In Section 4 we analyze te second normal form see Teorems 2.8 and 2.11 and Corollaries 2.9 and Section 5 is devoted to te tird normal form see Teorem 2.15 and Corollary Statements of te main results We recall see [9, Capter 7] tat a function m : R d [0, is an order function if tere exist constants N 0, C 0 > 0 suc tat mx C 0 X Y N 0 my for any X, Y R d. Te symbol class Sm is te space of smoot dependent functions a : R d C suc tat α N d, α x a x C α mx, 0, 1]. Trougout tis paper, we assume tat te components of te vector potential A belong to a symbol class Sm. Note tat tis implies tat B Sm, and conversely, if B Sm, ten tere exist a potential A and anoter order function m suc tat A Sm. Moreover, te magnetic Hamiltonian Hx, ξ = ξ Ax 2 belongs to Sm for an order function m on R 6.

11 10 B. HELFFER, Y. KORDYUKOV, N. RAYMOND, AND S. VŨ NGỌC We will work wit te Weyl quantization; for a classical symbol a = ax, ξ; Sm, it is defined as: 2.1 Op w a ψx = 1 2π 3 x + y e i x y,ξ / a R 6 2, ξ ψydydξ, ψ SR 3. Te Weyl quantization of H is te magnetic Laplacian L,A = i A Normal forms and spectral reductions. Let us introduce our first Birkoff normal form N. Teorem 2.1. If B0 0, tere exists a neigborood of 0, A0 endowed wit symplectic coordinates x 1, ξ 1, x 2, ξ 2, x 3, ξ 3 in wic Σ = {x 1 = ξ 1 = ξ 3 = 0} and 0, A0 as coordinates 0 R 6, and tere exist an associated unitary Fourier integral operator U and a smoot function, compactly supported wit respect to Z and ξ 3, f, Z, x 2, ξ 2, x 3, ξ 3 wose Taylor series wit respect to Z, ξ 3, is l c l,m,β x 2, ξ 2, x 3 Z m ξ β 3 suc tat k3 2l+2m+β=k 2.2 U L,AU = N + R, wit and were N = 2 D 2 x 3 + I Op w b + Opw f, I, x 2, ξ 2, x 3, ξ 3, a we ave I = 2 D 2 x 1 + x 2 1, b te operator Op w f, I, x 2, ξ 2, x 3, ξ 3 as to be understood as te Weyl quantization of an operator valued symbol, c te remainder R is a pseudo-differential operator suc tat, in a neigborood of te origin, te Taylor series of its symbol wit respect to x 1, ξ 1, ξ 3, is 0. Remark 2.2. In Teorem 2.1, te direction of B considered as a vector field on Σ is x 3 and te function b C R 6 stands for b j 1 Σ π were π : R 6 Σ : πx 1, ξ 1, x 2, ξ 2, x 3, ξ 3 = 0, 0, x 2, ξ 2, x 3, 0. In addition, note tat te support of f in Z and ξ 3 may be cosen as small as we want. Remark 2.3. In te context of Weyl s asymptotics, a close version of tis teorem appears in [19, Capter 6]. In order to investigate te spectrum of L,A near te low lying energies, we introduce te following pseudo-differential operator N [1] = 2 D 2 x 3 + Op w b + Opw f,, x 2, ξ 2, x 3, ξ 3, obtained by replacing I by.

12 MAGNETIC WELLS IN DIMENSION THREE 11 Corollary 2.4. We introduce 2.3 N = Opw N, wit N = ξ2 3 + I bx 2, ξ 2, x 3 + f,, I, x 2, ξ 2, x 3, ξ 3 and were b is a smoot extension of b away from D0, ε suc tat 1.11 still olds and were f, = χx 2, ξ 2, x 3 f, wit χ is a smoot cutoff function being 1 in a neigborood of D0, ε. We also define te operator attaced to te first eigenvalue of I 2.4 N [1], = Op w N [1], were N [1], = ξ bx 2, ξ 2, x 3 + f,,, x 2, ξ 2, x 3, ξ 3. If ε and te support of f are small enoug, ten we ave a Te spectra of L,A and N below β 0 coincide modulo O. b For all c 0, min3b 0, β 0, te spectra of L,A and N [1], below c coincide modulo O. Let us now state our results concerning te normal form of N [1] under te following assumption., or N [1], Notation 2.5. If f = fz is a differentiable function, we denote by T z f its tangent map at te point z. Moreover, if f is twice differentiable, te second derivative of f is denoted by T 2 z f,. Assumption 2.6. We assume tat T0 2 bb0, B0 > 0. Remark 2.7. If te function b admits a unique and positive minimum at 0 and tat it is non degenerate, ten Assumption 2.6 is satisfied. Under Assumption 2.6, we ave 3 b0, 0, 0 = 0 and, in te coordinates x 2, ξ 2, x 3 given in Teorem 2.1, b0, 0, 0 > 0. It follows from 2.5 and te implicit function teorem tat, for small x 2, tere exists a smoot function x 2, ξ 2 sx 2, ξ 2, s0, 0 = 0, suc tat bx 2, ξ 2, sx 2, ξ 2 = 0. Te point sx 2, ξ 2 is te unique in a neigborood of 0, 0, 0 minimum of x 3 bx 2, ξ 2, x 3. We define νx 2, ξ 2 := bx 2, ξ 2, sx 2, ξ 2 1/4. Teorem 2.8. Under Assumption 2.6, tere exists a neigborood V 0 of 0 and a Fourier integral operator V wic is microlocally unitary near V 0 and suc tat V N [1] V =: N [1] = Opw N [1],

13 12 B. HELFFER, Y. KORDYUKOV, N. RAYMOND, AND S. VŨ NGỌC were N [1] = ν 2 x 2, ξ 2 ξ x2 3 + bx2, ξ 2, sx 2, ξ 2 + r and r is a semiclassical symbol suc tat r = Ox Oξ2 3 + Oξ3 3 + O2. Corollary 2.9. Let us introduce N [1], = Op w N [1], were N [1], = ν 2 x 2, ξ 2 ξ x2 + bx2, ξ 2, sx 2, ξ 2 + r, wit r = χx 2, ξ 2, x 3, ξ 3 r, and were ν denotes a smoot and constant wit a positive constant extension of te function ν. Tere exists a constant c > 0 suc tat, for any cut-off function χ equal to 1 on D0, ε wit support in D0, 2ε, we ave: a Te spectra of N [1], and N [1], below b 0 + cε 2 coincide modulo O. b For all c 0, min3b 0, b 0 + cε 2, te spectra of L,A and N [1], below c coincide modulo O. Notation 2.10 Cange of semiclassical parameter. We let = 1 2 and, if A is a semiclassical symbol on T R 2, admitting a semiclassical expansion in 1 2, we write A := Op w A = Op w A =: A, wit A x 2, ξ 2, x 3, ξ 3 = A 2x 2, ξ 2, x 3, ξ 3. Tus, A and A represent te same operator wen = 1 2, but te former is viewed as an -quantization of te symbol A, wile te latter is an - pseudo-differential operator wit symbol A. Notice tat, if A belongs to some class Sm, ten A Sm as well. Tis is of course not true te oter way around. Teorem Under Assumption 2.6, tere exist a unitary operator W and a smoot function g, Z, x 2, ξ 2, wit compact support as small as we want wit respect to Z and wit compact support in x 2, ξ 2, wose Taylor series wit respect to Z, is c m,l x 2, ξ 2 Z m l, suc tat wit 2m+2l3, W N[1], W =: M = Op w M, M = 2 bx 2, ξ 2, sx 2, ξ J Op w ν2 x 2, ξ g, J, x 2, ξ R + S1. were a te operator N [1], is N [1], but written in te -quantization, b we ave let J = Op w ξ2 3 + x3 2,

14 MAGNETIC WELLS IN DIMENSION THREE 13 c te function R satisfies R x 2, ξ 2, x 3, ξ 3 = Ox 3, ξ 3. Remark Note tat te support of g wit respect to Z may be cosen as small as we want. Note also tat we ave used N [1], instead of N [1] : Since W is exactly unitary, we get a direct comparison of te spectra. Corollary We introduce wit M = Opw M, M = 2 bx 2, ξ 2, sx 2, ξ J ν 2 x 2, ξ g, J, x 2, ξ 2. We also define wit M [1], = Op w M [1], M [1], = 2 bx 2, ξ 2, sx 2, ξ ν 2 x 2, ξ g,, x 2, ξ 2. If ε and te support of g are small enoug, we ave a For all η > 0, te spectra of N [1], and M below b O 2+η coincide modulo O. b For c 0, 3, te spectra of M and M[1], below b cσ coincide modulo O. c If c 0, 3, te spectra of L,A and M [1], = M [1], below b 0 + cσ coincide modulo O. Finally, we can perform a last Birkoff normal form for te operator M [1], as soon as x 2, ξ 2 bx 2, ξ 2, sx 2, ξ 2 admits a unique and non degenerate minimum at 0, 0. Under tis additional assumption, b admits a unique and non degenerate minimum at 0, 0, 0. Terefore we will use te following stronger assumption. Assumption Te function b admits a unique and positive minimum at 0 and it is non degenerate. Teorem Under Assumption 2.14, tere exist a unitary -Fourier Integral Operator Q 1 2 wose pase admits an expansion in powers of 1 2 suc tat, Q M [1], 1 2 Q 1 2 = F + G, were a F is defined in Teorem 1.3, b te remainder is in te form G = Op w G, wit G = O z 2. Corollary If ε and te support of k are small enoug, we ave a For all η 0, 1 [1], 2, te spectra of M and F below b 0 + O 1+η coincide modulo O.

15 14 B. HELFFER, Y. KORDYUKOV, N. RAYMOND, AND S. VŨ NGỌC b For all c 0, 3, te spectra of L,A and F below b 0 + cσ coincide modulo O. Remark Since te spectral analysis of F is straigtforward, Item b of Corollary 2.16 implies Teorem 1.3. Te next sections are devoted to te proofs of our main results. Teorem 2.1 Corollary 2.4b N N [1] Teorem 2.8 L,A N [1] cange of semiclassical parameter Teorem 1.3 Teorem 2.11 N [1] F Teorem 2.15 M [1] cange of semiclassical parameter M [1] Corollary 2.13b M 3. First Birkoff normal form We assume tat B0 0 so tat in some neigborood Ω of 0 te magnetic field does not vanis. Up to a rotation in R 3 extended to a symplectic transformation in R 6 we may assume tat B0 = B0 e 3. In tis neigborood, we may defined te unit vector: 3.1 b = B B and find vectors c and d depending smootly on q suc tat b, c, d is a direct ortonormal basis Symplectic coordinates Straigtening te magnetic vector field. We consider te form dα and we would like to find a diffeomorpism, in a neigborood of 0, χ suc tat χˆq = q and χ dα = dˆq 1 dˆq 2. First, tis is easy to find a local diffeomorpism ϕ suc tat 3 ϕ q = bϕ q and ϕ q 1, q 2, 0 = q 1, q 2, 0. Tis is just te standard straigtening-out lemma for te non-vanising vector field b. Te vector e 3 is in te kernel of ϕ dα, wic implies tat we ave ϕ dα = f qd q 1 d q 2, for some smoot function f.

16 MAGNETIC WELLS IN DIMENSION THREE 15 But since te form ϕ dα is closed, f does not depend on q 3. Tis is ten easy to find anoter diffeomorpism ψ, corresponding to te cange of variables ˆq = ψ q = ψ 1 q 1, q 2, ψ 2 q 1, q 2, q 3, suc tat ψ ϕ dα = dˆq 1 dˆq 2. We let χ = ϕ ψ and we notice tat 3.2 χ dα = dˆq 1 dˆq 2 3 χˆq = bχˆq, Remark 3.1. It follows from 3.2 and 1.4 tat det T χ = B Symplectic coordinates. Let us consider te new parametrization of Σ given by ι : ˆΩ Σ wic gives a basis f 1, f 2, f 3 of T Σ : ˆq χˆq, A 1 χˆq, A 2 χˆq, A 3 χˆq, f j = T χe j, T A T χe j, j = 1, 2, 3. Using 1.5, and te fact tat f 3 is in te kernel of dα, we find ω 0 f j, f 3 = 0, j = 1, 2. Finally, ω 0 f 1, f 2 = dαt χe 1, T χe 2 = χ dαe 1, e 2 = 1. Te following vectors of R 3 R 3 form a basis of te symplectic ortogonal of T ιˆq Σ: 3.3 f 4 = B 1/2 c, t T χˆq Ac, f 5 = B 1/2 d, t T χˆq Ad, so tat ω 0 f 4, f 5 = 1. We let f 6 = 0, b + ρ 1 f 1 + ρ 2 f 2 were ρ 1 and ρ 2 are determined so tat ω 0 f j, f 6 = 0 for j = 1, 2. We notice tat ω 0 f j, f 6 = 0 for j = 4, 5 and ω 0 f 3, f 6 = Diagonalizing te Hessian. We recall tat Hq, p = p Aq 2 so tat, at a critical point p = Aq, te Hessian is T 2 HU 1, V 1, U 2, V 2 = 2 V 1 T q AU 1, V 2 T q AU 2. Let us notice tat T 2 Hf 4, f 5 = 2 B 1 B c, B d = 0, T 2 Hf 4, f 6 = 2 B c, b = 0, T 2 Hf 5, f 6 = 2 B d, b = 0. Te Hessian is diagonal in te basis f 4, f 5, f 6. Moreover we ave T 2 Hf 4, f 4 = d 2 Hf 5, f 5 = 2 B 1 B c 2 = 2 B 1 B d 2 = 2 B. Finally we ave: T 2 Hf 6, f 6 = 2.

17 16 B. HELFFER, Y. KORDYUKOV, N. RAYMOND, AND S. VŨ NGỌC Now we consider te local diffeomorpism: x, ξ ιx 2, ξ 2, x 3 + x 1 f 4 x 2, ξ 2, x 3 + ξ 1 f 5 x 2, ξ 2, x 3 + ξ 3 f 6 x 2, ξ 2, x 3. Te Jacobian of tis map is a symplectic matrix on Σ. We may apply te Moser-Weinstein argument see [29] to make tis map locally symplectic near Σ modulo a cange of variable wic is tangent to te identity. Near Σ, in tese new coordinates, te Hamiltonian H admits te expansion 3.4 Ĥ = H 0 + O x ξ ξ 3 3, were Ĥ denotes H in te coordinates x 1, x 2, x 3, ξ 1, ξ 2, ξ 3, and wit 3.5 H 0 = ξ bx 2, ξ 2, x 3 x ξ 2 1, b = Bx 2, ξ 2, x Semiclassical Birkoff normal form Birkoff procedure in formal series. Let us consider te space E of formal power series in x 1, ξ 1, ξ 3, wit coefficients smootly depending on x = x 2, ξ 2, x 3 : E = C x 2,ξ 2,x 3 [x 1, ξ 1, ξ 3, ]. We endow E wit te semiclassical Moyal product wit respect to all variables x 1, x 2, x 3, ξ 1, ξ 2, ξ 3 denoted by and te commutator of two series κ 1 and κ 2 is defined as [κ 1, κ 2 ] = κ 1 κ 2 κ 2 κ 1. Te degree of x α 1 1 ξα 2 1 ξβ 3 l = z1 αξβ 3 l is α 1 + α 2 + β + 2l = α + β + 2l. D N denotes te space of monomials of degree N. O N is te space of formal series wit valuation at least N. For any τ, γ E, we denote ad τ γ = [τ, γ]. Proposition 3.2. Given γ O 3, tere exist formal power series τ, κ O 3 suc tat e i 1 ad τ H 0 + γ = H 0 + κ, wit [κ, z 1 2 ] = 0. Proof. Let N 1. Assume tat we ave, for τ N O 3, e i 1 ad τn H 0 + γ = H 0 + K K N+1 + R N+2 + O N+3, wit K i D i, [K i, z 1 2 ] = 0 and R N+2 D N+2. Let τ D N+2. Ten we ave e i 1 ad τn +τ H 0 + γ = H 0 + K K N+1 + K N+2 + O N+3, wit K N+2 D N+2 suc tat K N+2 = R N+2 + i 1 ad τ H 0 + O N+3. Lemma 3.3. For τ D N+2, we ave i 1 ad τ H 0 = i 1 b ad τ z O N+3.

18 MAGNETIC WELLS IN DIMENSION THREE 17 To prove tis lemma, we observe tat Let us write i 1 ad τ H 0 = i 1 ad τ ξ i 1 ad τ b x z 1 2. τ = Ten, for te first term, we ave We also ave i 1 ad τ ξ 2 3 ={τ, ξ 2 3} α +β+2l=n+2 τ = 2ξ 3 x 3 = 2 α +β+2l=n+2 i 1 ad τ b x ={τ, b} + 2 O N = τ b + τ ξ 3 x 3 ξ 2 = α +β+2l=n+2 Terefore, for te second term, we get a α,β,l xz α 1 ξ β 3 l. a α,β,l x 3 xz α 1 ξ β+1 3 l O N+3. b τ b + O N+1 x 2 x 2 ξ 2 βa x b x 3 z α 1 z 1 2 ξ β 1 3 l + O N+1 O N+1. i 1 ad τ b x z 1 2 =i 1 ad τ b x z i 1 b x ad τ z 1 2 tat completes te proof of te lemma. By te lemma, we obtain tat tat we rewrite as =i 1 b x ad τ z O N+3, K N+2 = R N+2 + b ad τ z 1 2, R N+2 = K N+2 + i 1 b ad z1 2 τ = K N+2 + b{ z 1 2, τ }. Since b x 0, we deduce te existence of τ and K N+2 suc tat K N+2 commutes wit z Quantizing te formal procedure. Let us now prove Teorem 2.1. Using 3.4 and applying te Egorov teorem see [26, 31] or Teorem A.2, we can find a unitary Fourier Integral Operator U, and suc tat U L,AU = C 0 + Op w H0 + Op w r, were te Taylor series wit respect to x 1, ξ 1, ξ 3, of r satisfies r T = γ O 3 and C 0 is te value at te origin of te sub-principal symbol of U L,AU. One can coose U suc tat te subprincipal symbol is preserved

19 18 B. HELFFER, Y. KORDYUKOV, N. RAYMOND, AND S. VŨ NGỌC by conjugation 1, wic implies C 0 = 0. Applying Proposition 3.2, we obtain τ and κ in O 3 suc tat e i 1 ad τ H 0 + γ = H 0 + κ, wit [κ, z 1 2 ] = 0. We can introduce a smoot symbol a wit compact support suc tat we ave a T = τ in a neigborood of te origin. By Proposition 3.2 and Teorem A.4, we obtain tat te operator e i 1 Op w a Op w H0 + Op w r e i 1 Op w a is a pseudodifferential operator suc tat te formal Taylor series of its symbol is H 0 + κ. In tis application of Teorem A.4, we ave used te filtration O j defined in Section Since κ commutes wit z 1 2, we can write it as a formal series in z 1 2 : κ = l c l,m x 2, ξ 2, x 3 z 1 2m ξ β 3. k3 2l+2m+β=k Tis formal series can be reordered by using monomials z 1 2 m : κ = l c l,m x 2, ξ 2, x 3 z 1 2 m ξ β 3. k3 2l+2m+β=k Tanks to te Borel lemma, we may find a smoot function, wit a compact support as small as we want wit respect to, I and ξ 3, f, I, x 2, ξ 2, x 3, ξ 3 suc tat its Taylor series wit respect to, I, ξ 3 is l c l,m x 2, ξ 2, x 3 I m ξ β 3. k3 2l+2m+β=k Tis acieves te proof of Teorem Spectral reduction to te first normal form. Tis section is devoted to te proof of Corollary Numbers of eigenvalues. Lemma 3.4. Under Assumption 1.2, tere exists 0 > 0 and ε 0 > 0 suc tat for all 0, 0, inf s ess N β 0 + ε 0. Proof. By using te assumption we may consider a smoot function χ wit compact support and ε 0 > 0 suc tat ξ bx 2, ξ 2, x 3 + χx 2, x 3, ξ 2, ξ 3 β 0 + 2ε 0. 1 Tis is sometimes called te Improved Egorov Teorem. It was first discovered by Weinstein in [30], in te omogeneous setting. For te semiclassical case, see for instance [18, Appendix A].

20 MAGNETIC WELLS IN DIMENSION THREE 19 Ten, given η 0, 1 and estimating te second term in 2.3 by using tat te support of f is cosen small enoug and te semiclassical Calderon- Vaillancourt teorem, we notice tat, for small enoug, 3.6 N 1 η Opw ξ z 1 2 bx 2, ξ 2, x 3. Since te essential spectrum is invariant by relatively compact perturbations, we ave s ess N + 1 η Opw χx 2, x 3, ξ 2, ξ 3 = s ess. N Hence inf s ess N inf s N + 1 η Opw χx 2, x 3, ξ 2, ξ 3. In order to bound te r..s. from below, we write N + 1 η Opw χx 2, x 3, ξ 2, ξ 3 1 η Op w ξ z 1 2 bx 2, ξ 2, x η Op w χx 2, x 3, ξ 2, ξ 3 1 η Op w ξ bx 2, ξ 2, x 3 + χx 2, x 3, ξ 2, ξ 3 1 ηβ 0 + 2ε 0 C, were we ave used te semiclassical Gårding inequality. Taking η and ten small enoug, tis concludes te proof. By using te Hilbertian decomposition given by te Hermite functions e k, k1 associated wit I, we notice tat were N = k1 N [k],, 3.7 N [k], = 2 D 2 x 3 +2k 1 Op w b+opw f,, 2k 1, x 2, ξ 2, x 3, ξ 3, acting on L 2 R 2. Lemma 3.5. For all η 0, 1, tere exist C > 0 and 0 > 0 suc tat for all k 1 and 0, 0, we ave s 1 N [k], 1 2ηb 0 2k 1. Proof. Applying 3.6 to ψx 1, x 2, x 3 = ϕx 2, x 3 e k, x 1, we infer tat N [k], ϕ, ϕ 2k 11 η Op w bϕ, ϕ. Wit te Gårding inequality, we get Op w bϕ, ϕ b 0 C ϕ 2, and te conclusion follows by te min-max principle. We immediately deduce te following proposition. Proposition 3.6. We ave te following descriptions of te low lying spectrum of N.

21 20 B. HELFFER, Y. KORDYUKOV, N. RAYMOND, AND S. VŨ NGỌC a Tere exist 0 > 0 and K N suc tat, for 0, 0, te spectrum of N lying below β 0 is contained in te union K k=1 sp N [k],. b If c 0, min3b 0, β 0, ten tere exists 0 > 0 suc tat for all 0, 0 te eigenvalues of N lying below c coincide wit te eigenvalues of N [1], below c. We deduce te following proposition. Corollary 3.7. Under Assumption 1.10, we ave N L,A, β 0 = O 3/2, N N, β 0 = O 2. Proof. To get te first estimate, we use te Lieb-Tirring inequalities wic provide an upper bound of te number of eigenvalues in dimension tree and te diamagnetic inequality see [25] and Proposition 1.8. To get te second estimate, we use te first point in Proposition 3.6. Moreover, given η 0, 1, by using 0, 1 we infer ξ bx 2, ξ 2, x 3 ψ, ψ. N [k], ψ, ψ 1 η Op w Note tat te last inequality is very roug. By te min-max principle, we deduce tat N N [k],, β 0 N Op w ξ bx 2, ξ 2, x 3, 1 η 1 β 0. Ten, we conclude by using te Weyl asymptotics and our confinement assumption: N Op w ξ bx 2, ξ 2, x 3, 1 η 1 β 0 = O 2. Since N commutes wit I, we also deduce te following corollary. Corollary 3.8. For any eigenvalue λ of N suc tat λ β 0 we may consider an ortonormal eigenbasis of te space ker N λ formed wit functions in te form e k, x 1 ϕ x 2, x 3 wit k {1,... K}. Moreover we ave 1,β0 N = O 2 and eac eigenfunction associated wit λ β 0 is a linear combination of at most O 2 suc tensor products Microlocalization estimates. Te following proposition follows from te same lines as in dimension two see [16, Teorem 2.1]. Proposition 3.9. Under Assumptions 1.1 and 1.2, for any ɛ > 0, tere exist Cɛ > 0 and 0 ɛ > 0 suc tat for any eigenpair λ, ψ of L,A wit λ β 0 we ave for 0, 0 ɛ: 1 2 ψ 2 dq Cɛ expɛ 1 2 ψ 2, R 3 e 21 ɛφq/ Q,A e 1 ɛφq/ 12 ψ Cɛ expɛ 1 2 ψ 2,

22 MAGNETIC WELLS IN DIMENSION THREE 21 were φ is te distance to te bounded set { Bq β 0 } for te Agmon metric Bq β 0 + g, wit g te standard metric. Proposition Under Assumptions 1.1 and 1.2, we consider 0 < b 0 < β 0 < b 1 and tere exist C > 0 and 0 > 0 suc tat for any eigenpair λ, ψ of L,A wit λ β 0 we ave for 0, 0 and δ 0, 1 2 : ψ = χ 0 2δ L,A χ 1 qψ + O ψ, were χ 0 is a cutoff function compactly supported in te ball of center 0 and radius 1 and were χ 1 is a compactly supported smoot cutoff function being 1 in an open neigborood of { Bq β 0 }. Let us now investigate te microlocalization of te eigenfunctions of N. Proposition Let χ be a smoot cutoff function being 0 on {b β 0 } and 1 on te set {b β 0 + ε}. If λ is an eigenvalue of N suc tat λ β 0 and if ψ is an associated eigenfunction, ten we ave Op w χx 2, ξ 2, x 3 ψ = O ψ. Proof. Due to Corollary 3.8, it is sufficient to prove te estimate for a function in te form ψx 1, x 2, x 3 = e k, x 1 ϕx 2, x 3 were k lies in {1,..., K} and we ave N [k], ψ = λψ, or equivalently N ϕ = λϕ, were we recall 3.7. Ten, we write [ ] N [k], Op w χ ϕ = λ Opw χ ϕ + N [k],, Op w χ ϕ and it follows tat 3.8 N [k], Op w χ ϕ, Opw χ ϕ = λ Op w χ ϕ 2 [ ] + N [k],, Op w χ ϕ, Op w χ ϕ. Roug pseudo-differential estimates imply tat tere exist C > 0, 0 > 0 suc tat for all 0, 0, 3.9 [ ] N [k],, Op w χ ϕ, Op w χ ϕ C 2 Op w Combining 3.9 and 3.8, we get χ ϕ 2 +C Op w χ ϕ 2 + C Op w 3χ ϕ, Op w ξ 3 Op w χ ϕ Op w ξ 3 Op w χ ϕ C 1 2 Op w χ ϕ, were χ is a smoot cutoff function living on a sligtly larger support tan χ. By using 3.10, we can improve te commutator estimate [ ] N [k],, Op w χ ϕ, Op w χ ϕ 3 C 2 Op w χ ϕ 2.

23 22 B. HELFFER, Y. KORDYUKOV, N. RAYMOND, AND S. VŨ NGỌC We infer tat, tere exist C > 0, 0 > 0 suc tat for 0, 0, N [k], Op w χ ϕ, Opw χ ϕ β 0 Op w χ ϕ 2 + C 3 2 Op w χ ϕ 2. By using te semiclassical Gårding inequality and te support of χ, we get N [k], Op w χ ϕ, Opw χ ϕ β 0 + ε 0 Op w χ ϕ 2 and we deduce Op w χ ϕ 2 C 1 2 Op w χ ϕ 2. Te conclusion follows by a standard iteration argument. Te following proposition is concerned by te microlocalization wit respect to ξ 3. Proposition Let χ 0 be a smoot cutoff function being 0 in a neigborood of 0 and let δ 0, 1 2. If λ is an eigenvalue of N suc tat λ β 0 and if ψ is an associated eigenfunction, ten we ave χ 0 δ ξ 3 ψ = O ψ. Op w Proof. We write again ψx 1, x 2, x 3 = e k, x 1 ϕx 2, x 3 wit k {1,..., K} and we ave N [k], ϕ = λϕ. We use again te formula 3.8 wit χ 0 δ ξ 3. We get te commutator estimate [ ] N [k],, Op w χ 0 δ ξ 3 ϕ, Op w χ 0 δ ξ 3 ϕ C 3 2 δ Op w χ 0 δ ξ 3 ϕ 2. We ave Op w δ ξ 3 2 χ 2 0 δ ξ 3 = Op w ξ 2 1 δ 3 χ 2 0 ξ 3, so tat, wit te Gårding inequality, Op w δ ξ 3 2 χ 2 0 δ ξ 3 ϕ, ϕ We infer 2δ 1 C 1 δ β 0 Op w χ 0 δ ξ 3 ϕ 1 C 1 δ ϕ 2. 2 C 3 2 δ Op w χ 0 δ ξ 3 ϕ 2. Using Op w f, I, x 2, ξ 2, x 3, ξ 3 = Op w f, z 1 2, x 2, ξ 2, x 3, ξ 3, we deduce te following in te same way. Proposition Let χ 1 be a smoot cutoff function being 0 in a neigborood of 0 and let δ 0, 1 2. If λ is an eigenvalue of N suc tat λ β 0 and if ψ is an associated eigenfunction, ten we ave χ 1 δ x 1, ξ 1 ψ = O ψ. Op w

24 MAGNETIC WELLS IN DIMENSION THREE 23 Proposition Te spectra of L,A and N below β 0 coincide modulo O. Proof. We refer to [25, Section 4.3] wic contains similar arguments. Tis proposition provides te point a in Corollary 2.4. Wit Proposition 3.6, we deduce te point b. 4. Second Birkoff normal form 4.1. Birkoff analysis of te first level. Tis section is devoted to te proofs of Teorems 2.8 and Te goal now is to normalize a -pseudo-differential operator N [1] on R 2 wose Weyl symbol as te form N [1] = ξ bx 2, ξ 2, x 3 + r x 2, ξ 2, x 3, ξ 3, were r is a classical symbol wit te following asymptotic expansion: r = r 0 + r r 2 + in te symbol class topology, were eac r l as a formal expansion in ξ 3 of te form 4.1 r l x 2, ξ 2, x 3, ξ 3 c l,β x 2, ξ 2, x 3 ξ β 3. Te leading terms of N [1] are: 2l+β3 4.2 N [1] = ξ 2 3 +bx 2, ξ 2, x 3 +c 1,1 x 2, ξ 2, x 3 ξ 3 +Oξ 2 3+Oξ 3 3+O First normalization of te symbol. We consider te following local cange of variables ˆϕx 2, ξ 2, x 3, ξ 3 = ˆx 2, ˆξ 2, ˆx 3, ˆξ 3 : ˆx 2 := x 2 + ξ 3 2 sx 2, ξ 2, ˆξ 2 := ξ 2 + ξ 3 1 sx 2, ξ 2, 4.3 ˆx 3 := x 3 sx 2, ξ 2, ˆξ 3 := ξ 3. It is easy to ceck tat te differential of ˆϕ is invertible as soon as ξ 3 is small enoug. Moreover, we ave ˆϕ ω 0 ω 0 = O ξ 3. By te Darboux-Weinstein teorem see for instance [25, Lemma 2.4], tere exists a local diffeomorpism ψ suc tat 4.4 ψ = Id + Oξ 2 3 and ψ ˆϕ ω 0 = ω 0. Using te improved Egorov teorem, one can find a unitary Fourier Integral Operator V suc tat te Weyl symbol of V N [1] V is ˆN := N [1] ˆϕ ψ + O 2. From 4.4, and 4.3, we see tat ˆr := r ˆϕ ψ is still of te

25 24 B. HELFFER, Y. KORDYUKOV, N. RAYMOND, AND S. VŨ NGỌC form 4.1, wit modified coefficients c l,β. Tus, using te new variables and a Taylor expansion in ξ 3, we get ˆN = ˆξ 2 3 +bˆx 2 +Oˆξ 3, ˆξ 2 +Oξ 3, ˆx 3 +sˆx 2 +Oˆξ 3, ˆξ 2 +Oˆξ 3 +Oˆξ 2 3 and tus 4.5 ˆN = ˆξ bˆx 2, ˆξ 2, ˆx 3 + sˆx 2, ˆξ 2 + ˆξ 3 gˆx 2, ˆξ 2, ˆx 3 for some smoot function gˆx 2, ˆξ 2, ˆx 3. Terefore ˆN as te following form: + Oˆξ ˆr + O 2 + Oˆξ ˆr + Oˆξ O 2, ˆN = ˆξ 2 3+bˆx 2, ˆξ 2, ˆx 3 +sˆx 2, ˆξ 2 +ĉ 1,1 x 2, ˆξ 2, ˆx 3 ˆξ 3 +Oˆξ 2 3+Oˆξ 3 3+O Were te second armonic oscillator appears. We now drop all te ats off te variables. We use a Taylor expansion wit respect to x 3, wic, in view of 2.6, yields: bx 2, ξ 2, x 3 +sx 2, ξ 2 = bx 2, ξ 2, sx 2, ξ 2 + x bx 2, ξ 2, sx 2, ξ 2 +Ox 3 3. We let: 4.6 ν = bx 2, ξ 2, sx 2, ξ 2 1/4 and γ = ln ν. We introduce te cange of coordinates ˇx 2, ˇx 3, ˇξ 2, ˇξ 3 = Cx 2, x 3, ξ 2, ξ 3 defined by: ˇx 3 = νx 3, ˇξ 3 = ν 1 ξ 3, 4.7 ˇx 2 = x 2 + γ ξ 2 x 3 ξ 3, ˇξ 2 = ξ 2 γ x 2 x 3 ξ 3, for wic one can ceck tat C ω 0 ω 0 = Ox 3 ξ 3 = Oξ 3. As before, we can make tis local diffeomorpism symplectic by te Darboux-Weinstein teorem, wic modifies 4.7 by Oξ3 2. In te new variables wic we call x 2, x 3, ξ 2, ξ 3 again, te symbol Ň as te form: Ň = ν 2 x 2, ξ 2 ξ x bx2, ξ 2, sx 2, ξ 2 + č 1,1 x 2, ξ 2, x 3 ξ 3 + Ox Oξ Oξ O 2, for some smoot function č 1,1 x 2, ξ 2, x 3.

26 MAGNETIC WELLS IN DIMENSION THREE Normalizing te remainder. Te next step is to get rid of te term č 1,1 x 2, ξ 2, x 3 ξ 3. Let ax 2, ξ 2, x 3 := 1 2 x3 0 č 1,1 x 2, ξ 2, tdt. Since č 1,1 is compactly supported, a is bounded, and one can form te unitary pseudo-differential operator expia, A = Op w a. We ave exp ia Op w Ň expia = Op w Ň + exp ia[op w Ň, expia]. Te symbol of [exp ia Op w Ň, expia] is i e ia {N, e ia } + O 2 = {Ň, a} + O 2 = {Ň0, a} + O 2, were Ň0 is te principal symbol of Ň, wic satisfies: Ň 0 = ξ Oξ 3 3. Terefore {Ň, a} = {ξ3 2, a} + Oξ2 3. Since {ξ 2 3, a} = 2ξ 3 a x 3 = ξ 3 č 1,1, we get exp ia Op w Ň expia = Op w Ň ξ 3 č 1,1 + Oξ3 2 + O 2, wic sows tat we can remove te coefficient of ξ 3. Te new operator given by te conjugation formula N [1] = exp ia Op w Ň expia as a symbol of te form 4.8 N [1] = ν2 x 2, ξ 2 ξ x bx2, ξ 2, sx 2, ξ 2 + r, were r = Ox Oξ2 3 + Oξ3 3 + O2. Tis proves Teorem Te second Birkoff normal form. We now want to perform a Birkoff normal form for N [1], relative to te second armonic oscillator ν 2 x 2, ξ 2 ξ x 2 3. Using Notation 2.10, we introduce te new semiclassical parameter = 1 2, and use te relation Op w [1], N = Op w N[1],. Tus, let ξ j := 1/2 ξ j. Te new symbol N [1], as te form: N [1], x 2, ξ 2, x 3, ξ 3 = 2 ν 2 x 2, ξ 2 ξ x bx 2, ξ 2, sx 2, ξ r x 2 2, ξ 2, x 3, ξ 3.

27 26 B. HELFFER, Y. KORDYUKOV, N. RAYMOND, AND S. VŨ NGỌC We introduce momentarily a new parameter µ and define N [1], x 2, ξ 2, x 3, ξ 3 ; µ := ν 2 x 2, µ ξ 2 ξ x bx 2, µ ξ 2, sx 2, µ ξ 2 Notice tat N [1], + 2 r 2 x 2, µ ξ 2, x 3, ξ 3. x 2, ξ 2, x 3, ξ 3 ; = 2 N [1], x 2, ξ 2, x 3, ξ 3. We define now a space of functions suitable for te Birkoff normal form in x 3, ξ 3,. Let us now use te notation of te Appendix introduced in A.4 in te case wen te family of smoot linear maps R 2 R 2 is given by Let ϕ µ,r 2x 2, ξ 2 = x 2, µ ξ 2. F := C1 R 2, were te index R 2 means tat we consider symbols on R 2. More explicitly, we ave F = {d s. t. c S1; [0, 1] 0, 1] R 2 : dx 2, ξ 2 ; µ, = cϕ µ,r 2x 2, ξ 2 ; µ, }. Ten we define E := F [x 3, ξ 3, ], endowed wit te full Poisson bracket E E f, g {f, g} = j=2,3 f ξ j g x j g ξ j f x j E, and te corresponding Moyal bracket [f, g]. We remark tat te formal Taylor series of te symbol N [1], x 2, ξ 2, x 3, ξ 3 ; µ wit respect to x 3, ξ 3, belongs to E. We may apply te semiclassical Birkoff normal form relative to te main term ν 2 x 2, µ ξ 2 ξ x2 3 exactly as in Section and also [25, Proposition 2.7], were we use te fact tat te function x 2, ξ 2, x 3, ξ 3 ; µ, ν 2 x 2, µ ξ 2 1 belongs to E because ν 2 > C > 0 uniformly wit respect to µ. Let us consider γ E te formal Taylor expansion of 2 r 2 x 2, µ ξ 2, x 3, ξ 3 wit respect to x 3, ξ 3,. Te series γ is of valuation 3 and we obtain two formal series κ, τ E of valuation at least 3 suc tat and [κ, x ξ 2 3] = 0 e i 1 ad τ ν 2 x 2, µ ξ 2 ξ x γ = ν 2 x 2, µ ξ 2 ξ x κ. Te coefficients of τ are in S1 and one can find a smoot function τ S1 wit compact support wit respect to x 3, ξ 3, and wose Taylor series in x 3, ξ 3, is τ. By te Borel summation, τ will actually lie in Sm wit m x 2, ξ 2, x 3, ξ 3 = x 3, ξ 3 k for any k > 0, uniformly for small > 0 and µ [0, 1]. Notice tat N [1], Cm wit m = x 3, ξ 3 2 1, and

28 MAGNETIC WELLS IN DIMENSION THREE 27 tat mm = O1. Ten, we can apply Teorem A.3 wit te family of endomorpisms of R 4 defined Tus, te new operator ϕ µ,r 4x 2, ξ 2, x 3, ξ 3 = x 2, µ ξ 2, x 3, ξ 3. M = e i 1 Op w τ N [1], e i 1 Op w τ is a pseudo-differential operator wose Weyl symbol belongs to te class Cm modulo S1 see te notations of Teorem Moreover, tanks to Teorem A.4, its symbol M admits te following Taylor expansion wit respect to x 3, ξ 3, bx2, µ ξ 2, sx 2, µ ξ 2 + ν 2 x 2, µ ξ 2 ξ x κ. We write κ = m+2l3 c m,lx 2, µ ξ 2 z 3 2m l and we may find a smoot function g x 2, µ ξ 2, Z, suc tat its Taylor series wit respect to Z, is c m,l x 2, µ ξ 2 Z m l. 2m+2l3 We may now replace µ by, wic acieves te proof of Teorem Spectral reduction to te second normal form. Tis section is devoted to te proof of Corollary From N [1], Lemma 4.1. We ave N, β 0 N [1], to N [1],. In tis section, we prove Corollary 2.9. = O 2, N N [1],, β 0 = O 2. Proof. Te first estimate comes from Proposition 3.6 and Corollary 3.7. Te second estimate can be obtained by te same metod as in te proof of Corollary 3.7. Let us now summarize te microlocalization properties of te eigenfunctions of N [1], in te following proposition. Proposition 4.2. Let χ 0 be a smoot cutoff function on R being 0 in a neigborood of 0 and let δ 0, 1 2. Let χ be a smoot cutoff function being 0 on te bounded set {x bx 2, ξ 2, sx 2, ξ 2 β 0 } and 1 on te set {x bx 2, ξ 2, sx 2, ξ 2 β 0 + ε}, wit ε > 0. If λ is an eigenvalue of N [1], suc tat λ β 0 and if ψ is an associated eigenfunction, ten we ave and Op w χx 2, ξ 2, x 3 ψ = O ψ, Op w χ 0 δ ξ 3 ψ = O ψ. Proof. Te proof follows exactly te same lines as for Propositions 3.11 and 3.12.

29 28 B. HELFFER, Y. KORDYUKOV, N. RAYMOND, AND S. VŨ NGỌC Lemma 4.1 and Proposition 4.2 on te one and and Propositions 3.11 and 3.12 on te oter and are enoug to deduce from Teorem 2.8 te point a in Corollary 2.9. Te point b easily follows from Corollary From N [1], to M. Let us now prove te point a in Corollary We get te following roug estimate of te number of eigenvalues. Lemma 4.3. We ave 4.9 N N [1],, β 0 2 = N M, β 0 2 = O 4, 4.10 N M, β 0 2 = O 4. Proof. First, we notice tat N [1], and M are unitarily equivalent so tat 4.9 olds. Ten, given η > 0 and small enoug and up to srinking te support of g and by using te Calderon-Vaillancourt teorem as in te proof of Lemma 3.4, M M in te sense of quadratic forms, wit M = Opw 2 bx 2, ξ 2, sx 2, ξ J Op w ν 2 x 2, ξ 2 η. Since ν 2 c > 0, we get Op w 2 bx 2, ξ 2, sx 2, ξ J Op w Op w ν 2 x 2, ξ 2 2 bx 2, ξ 2, sx 2, ξ 2 η + c 2 2 J. We deduce te upper bound 4.10 by separation of variables and te minmax principle. Te following proposition deals wit te microlocal properties of te eigenfunctions of N [1],. Proposition 4.4. Let η 0, 1, δ 0, η 2, C > 0. Let χ be a smoot cutoff function being 0 on {bx 2, ξ 2, sx 2, ξ 2 β 0 } and being 1 on te set {bx 2, ξ 2, sx 2, ξ 2 β 0 + ε}, wit ε > 0. Let also χ 1 be a smoot cutoff function on R 2, being 0 in a neigborood of 0. If λ is an eigenvalue of N [1], suc tat λ β 0 2 and if ψ is an associated eigenfunction, we ave 4.11 Op w χx 2, ξ 2 ψ = O ψ and if λ is an eigenvalue of N [1], suc tat λ b C 2+η and if ψ is an associated eigenfunction, we ave 4.12 Op w χ 1 δ x 3, ξ 3 ψ = O ψ.

30 MAGNETIC WELLS IN DIMENSION THREE 29 Proof. Te estimate 4.11 is a consequence of Proposition 4.2. Ten, let us write te symbol of N [1], : N [1], = 2 ν 2 x 2, ξ 2 ξ2 3 + x 2 3 We write N [1], We get [ N [1], Op w [ + χ 1 δ x 3, ξ 3 ψ, Op w N [1], = λ Op w, Op w + 2 bx 2, ξ 2, sx 2, ξ 2 +R 2 x 2, ξ 2, x 3, ξ 3. χ 1 δ x 3, ξ 3 χ 1 δ x 3, ξ 3 ψ 2 ], Op w χ 1 δ x 3, ξ 3 χ 1 δ x 3, ξ 3 ψ., Op w χ 1 δ x 3, ξ ] 3, Op w χ 1 δ x 3, ξ 3 ψ C 3 Op w χ 1 δ x 3, ξ 3 ψ 2, were we ave used Ten, we use tat bx 2, ξ 2, sx 2, ξ 2 b 0, ν 2 x 2, ξ 2 c 0 > 0, λ b C 2+η, and te Gårding inequality to deduce 2 C 2δ C η Op w χ 1 δ x 3, ξ 3 ψ 2 C 3 Op w χ 1 δ x 3, ξ 3 ψ 2. Te desired estimate follows by an iteration argument. In te same way we can deal wit M. Proposition 4.5. Let η 0, 1, δ 0, η 2, C > 0. Let χ be a smoot cutoff function being 0 on {bx 2, ξ 2, sx 2, ξ 2 β 0 } and being 1 on te set {bx 2, ξ 2, sx 2, ξ 2 β 0 + ε}, wit ε > 0. If λ is an eigenvalue of M suc tat λ β 0 2 and if ψ is an associated eigenfunction, we ave 4.13 Op w χx 2, ξ 2 ψ = O ψ and if λ is an eigenvalue of M suc tat λ b C 2+η and if ψ is an associated eigenfunction, we ave 4.14 Op w χ 1 δ x 3, ξ 3 ψ = O ψ. Proof. In order to get 4.13, it is enoug to go back to te representation wit semiclassical, tat is M = M. Indeed te microlocal estimate follows by te same arguments as in Propositions 3.11 and Ten, 4.14 follows as in Proposition 4.4.