Ilaria Parenti

Cristina Gervasini

Jelena Pozojevic

Luitgard Graul-Neumann

Jacopo Azzollini

Diana Braunholz

Erwan Watrin

Kerstin S Wendt

Anna Cereda

Davide Cittaro

Gabriele Gillessen-Kaesbach

Dejan Lazarevic

Milena Mariani

Silvia Russo

Ralf Werner

Peter Krawitz

Lidia Larizza

Angelo Selicorni

Frank J Kaiser
email: frank.kaiser@uksh.de

Broadening of cohesinopathies: Exome sequencing identifies mutations in ANKRD11 in two patients with Cornelia de Lange-overlapping phenotype

Keywords: ANKRD11, cohesin, Cornelia de Lange syndrome, KBG syndrome, mosaicism, whole exome sequencing

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Cornelia de Lange syndrome (CdLS, OMIM #122470, 300590, 610759, 300882, and 614701) and KBG syndrome (OMIM #148050) are two distinct rare multisystem developmental disorders.

This article is protected by copyright. All rights reserved CdLS is characterized by clinical and genetic heterogeneity [START_REF] Kline | Cornelia de Lange syndrome: clinical review, diagnostic and scoring systems, and anticipatory guidance[END_REF]. The main features of the syndrome are pre-and post-natal growth retardation, intellectual disability, psychomotor delay, limb anomalies and a typical facies. These distinctive facial features include a low anterior hairline, arched eyebrows, synophrys, depressed nasal bridge, anteverted nares, long and smooth philtrum and thin upper lip with down-turned corners of the mouth. The cardiac, gastro-intestinal and genitourinary systems are also frequently involved [START_REF] Kline | Cornelia de Lange syndrome: clinical review, diagnostic and scoring systems, and anticipatory guidance[END_REF]. Heterozygous mutations in five different genes have been identified in patients with CdLS: NIPBL (5p13.2, OMIM #608667), SMC1A (Xp11.22, #300040), SMC3 (10q25.2, #606062), HDAC8 (Xq13.1, #300269) and RAD21 (8q24.11, #606462). All five genes are associated with the cohesin complex and its regulation. While the proteins SMC1A, SMC3 and RAD21 represent structural components of the cohesin complex, NIPBL and HDAC8 are associated regulators: NIPBL mediates the cohesin loading onto DNA and HDAC8 promotes SMC3 deacetylation to regulate proper cohesin dissociation from chromatin [START_REF] Tonkin | NIPBL, encoding a homolog of fungal Scc2-type sister chromatid cohesion proteins and fly Nipped-B, is mutated in Cornelia de Lange syndrome[END_REF][START_REF] Krantz | Cornelia de Lange syndrome is caused by mutations in NIPBL, the human homolog of Drosophila melanogaster Nipped-B[END_REF][START_REF] Musio | X-linked Cornelia de Lange syndrome owing to SMC1L1 mutations[END_REF][START_REF] Deardorff | Mutations in cohesin complex members SMC3 and SMC1A cause a mild variant of Cornelia de Lange syndrome with predominant mental retardation[END_REF][START_REF] Deardorff | RAD21 Mutations Cause a Human Cohesinopathy[END_REF][START_REF] Deardorff | HDAC8 mutations in Cornelia de Lange syndrome affect the cohesin acetylation cycle[END_REF]. The cohesin complex is involved in a large spectrum of functions including sister chromatid cohesion, DNA damage response, chromatin modification, transcriptional regualtion and long-range interactions between distant genomic regions [START_REF] Mehta | Cohesin: functions beyond sister chromatid cohesion[END_REF].

NIPBL is the major gene of the syndrome and alterations can be identified in more than half of the patients [START_REF] Selicorni | Clinical score of 62 Italian patients with Cornelia de Lange syndrome and correlations with the presence and type of NIPBL mutation[END_REF]. Mutations in NIPBL are usually associated with a severe phenotype and with a high frequency of limb anomalies [START_REF] Selicorni | Clinical score of 62 Italian patients with Cornelia de Lange syndrome and correlations with the presence and type of NIPBL mutation[END_REF]. Recently, a high proportion of mosaic mutations have been identified in patients with CdLS, thus partially accounting for the wide variability of expression: more than 20% of patients who were mutation negative in peripheral blood have been shown to carry mutations in NIPBL on DNA extracted from buccal mucosa or fibroblasts [START_REF] Huisman | High rate of mosaicism in individuals with Cornelia de Lange syndrome[END_REF][START_REF] Braunholz | Hidden Mutations in CdLS -Limitations of Sanger Sequencing in Molecular Diagnostics[END_REF].

This article is protected by copyright. All rights reserved Besides NIPBL, the remaining four genes account for about 15% of the patients, who are usually characterized by a more moderate or milder phenotype as those with mutations in NIPBL [START_REF] Deardorff | RAD21 Mutations Cause a Human Cohesinopathy[END_REF][START_REF] Deardorff | HDAC8 mutations in Cornelia de Lange syndrome affect the cohesin acetylation cycle[END_REF][START_REF] Gervasini | Cornelia de Lange individuals with new and recurrent SMC1A mutations enhance delineation of mutation repertoire and phenotypic spectrum[END_REF][START_REF] Kaiser | Loss-of-function HDAC8 mutations cause a phenotypic spectrum of Cornelia de Lange syndrome-like features, ocular hypertelorism, large fontanelle and X-linked inheritance[END_REF]. KBG syndrome is a very rare genetic disorder characterized by macrodontia of the upper central incisors, short stature, skeletal anomalies, psychomotor delay, intellectual disability, seizures, and craniofacial abnormalities which include a round facies, brachycephaly, hypertelorism, a bulbous nasal tip, long philtrum and broad and arched eyebrows [START_REF] Herrmann | The KBG syndrome-a syndrome of short stature, characteristic facies, mental retardation, macrodontia, and skeletal anomalies[END_REF][START_REF] Ockeloen | Further delineation of the KBG syndrome phenotype caused by ANKRD11 aberrations[END_REF]. KBG has been associated with loss of function mutations in the ANKRD11 gene (16q24.3, #611192), which codes for the ankyrin repeat-containing protein 11, also known as ankyrin repeat-containing cofactor 1 (ANCO-1) ( 16). This protein is known to inhibit the transcriptional activation of nuclear receptors target genes through the recruitment of the histone deacetylases HDAC3, HDAC4 and HDAC5 [START_REF] Zhang | Identification of a novel family of ankyrin repeats containing cofactors for p160 nuclear receptor coactivators[END_REF]. Sirmaci et al. [START_REF] Sirmaci | Mutations in ANKRD11 cause KBG syndrome, characterized by intellectual disability, skeletal malformations, and macrodontia[END_REF] demonstrated that ANKRD11 localizes within the nuclei of neurons and accumulates in discrete inclusions when neurons are depolarized, suggesting a role in neural plasticity. The protein contains five ankyrin repeats involved in protein-protein interactions, two repression domains (at the Nand C-terminus) and one activation domain capable of stimulating transcription [START_REF] Zhang | Characterization of transcriptional regulatory domains of ankyrin repeat cofactor 1[END_REF].

Herein we report on two patients with a tentative clinical diagnosis of CdLS who were found to be negative for mutations in the five known CdLS genes but carry heterozygous loss-offunction mutations in ANKRD11 identified by exome-sequencing approaches.

Patients and methods

Patients

Patient A (Fig. 1 a-h) is a German female, the fourth offspring of healthy and nonconsanguineous parents. The patient was born at 37 weeks of gestation after caesarian section. The Apgar scores were 9 and 10 at 1 and 5 minutes, respectively. Birth weight was This article is protected by copyright. All rights reserved 2350 g (-1.6 SD), birth length was 46 cm (1.75 SD) and the Occipital Frontal Circumference (OFC) was 31.5 cm (-2 SD). The patient showed early feeding difficulties resulting in a poor increase of weight. She also showed general muscular hypertonia.

The patient was re-evaluated at the age of four months. At this age, weight was 5600 g (+0.3 SD), height was 59 cm (-1.75 SD) and OFC was 38.3 cm (-0.3 SD). She showed a small brachycephalic head, open anterior fontanelle, synophrys, downslanting palpebral fissures, anteverted nares, depressed nasal bridge, long philtrum, thin lips with downturned corners of the mouth, short neck, mild brachydactyly, proximally set thumbs and mild micrognathia.

She presented with impressive hypertonia. She was able to sit at the age of 16 months, and to walk without support at the age of 20 months.

Last evaluation was performed at the age of four years. At this age, the patient was not able to eat without assistance. She presented with speech delay, being able to speak about 50 words and showed an aggressive and self-injurious behavior associated with autism. She showed a moderate intellectual disability. Her gait was wide based. Hearing or vision impairment were not observed.

Based on these clinical features the patient was considered to have CdLS.

Patient B (Fig. 1 i-k) is an Italian male and the third child of healthy and non-consanguineous parents. Birth weight was 3150 g (-1.7 SD), length was 53 cm (0.3 SD) and the OFC was 33 cm (-2.9 SD). A bilateral vesico-ureteral reflux was surgically treated in the first year of life.

Moreover, ultrasound kidney evaluation showed kidney dysplasia. Subsequent analysis of the renal function indicated the presence of a mild non nephrotic proteinuria without impairment of kidney function. He presented gastro-esophageal reflux with esophagitis of first degree.

For this reason, he received proton pump inhibitors therapy.

At the age of four he presented with moderate psychomotor and cognitive delay. He was able to walk independently, to speak in sentences and was completely able to manage himself in This article is protected by copyright. All rights reserved daily life. At the age of nine he started to show severe behavioral problems, characterized by auto-and hetero-aggression. He also showed generalized epilepsy.

At the age of 15 years the boy was in a good health condition. His weight was 41.2 kg (-0.5 SD), height was 161 cm (-1.6 SD) and the OFC was 54 cm (-1.2 SD).

He showed the following facial dysmorphisms: Low anterior hairline, broad and arched eyebrows with synophrys, long eyelashes, high nasal bridge, large nasal tip, smooth philtrum, large mouth with thin lips, macrodontia of superior central incisors, mild micrognathia. He also showed mild lumbar and forearms hirsutism, myopia and astigmatism, proximally placed thumbs and clinodactyly of the fifth finger. He was clinically diagnosed as having CdLS during childhood.

Methods

DNA extraction and sequencing

Genomic DNA was isolated from peripheral blood, buccal mucosa and fibroblasts. Extraction from blood and fibroblasts was performed with the QIAamp DNA Mini Kit (Qiagen, Milan, Italy), while the PureGene Buccal Cell Core Kit A (Qiagen) was used for gDNA isolation from buccal mucosa cells. PCR products were sequenced using the Big Dye terminator v3.1

Sequencing Kit and run on the ABI PRISM 3130xl sequencer (Applied Biosystems, Foster City, CA, USA). Electropherograms were analyzed with ChromasPro Software version 1.7.6 (Technelysium Pty Ltd, Tewantin QLD, Australia) and aligned with the wild type ANKRD11 sequence (RefSeq NM_013275).

Gene panel

Targeted gene panel analysis was performed on the fibroblast DNA of both patients with the Personal Genome Machine (Ion Torrent PGM, Life Technologies, Darmstadt, Germany) as described by Braunholz et al. [START_REF] Braunholz | Hidden Mutations in CdLS -Limitations of Sanger Sequencing in Molecular Diagnostics[END_REF].

Exome sequencing

This article is protected by copyright. All rights reserved Targeted enrichment was performed via Sure Select All Exon Kit V2 (Agilent, Waldbronn, Germany) for patient A, and through the TruSeq Exome Enrichment preparation kit (Illumina) for patient B. Sequencing was performed with Illumina's HiSeq 2500 system.

For bioinformatics analysis, read tags were aligned on human reference genome (GRCh37.5) using bwa 0.6.1 [START_REF] Li | Fast and accurate long-read alignment with Burrows-Wheeler transform[END_REF] or Novoalign (Novocraft Technologies, Selangor, Malesia). Single nucleotide variants (SNVs) and short insertions and deletions (indels) were called with GATK according to the best practice guidelines. The variant annotation on a functional level was performed using Jannovar or snpEff [START_REF] Jäger | Jannovar: a java library for exome annotation[END_REF] and filtered in GeneTalk for somatic variants [START_REF] Kamphans | GeneTalk: an expert exchange platform for assessing rare sequence variants in personal genomes[END_REF].

Pyrosequencing

The allelic dosage was quantified with an assay specific for each patient. The analysis was performed with the Pyro Gold Reagent Kit on the Pyro MarkID Q24 (Qiagen).

Results

Initial Sanger sequencing as well as high coverage target-enriched gene panel sequencing could not detect any pathogenic mutation in the CdLS-associated genes in either of the patients. Because of the high frequency of mosaic mutations in patients with CdLS that cannot be detected in DNA derived from blood, buccal mucosa and fibroblasts were also analyzed, as suggested by Braunholz et al. [START_REF] Braunholz | Hidden Mutations in CdLS -Limitations of Sanger Sequencing in Molecular Diagnostics[END_REF].

Since no obvious disease-causing mutation could be identified in any of the DNA samples, both patients were selected for a whole exome-sequencing analysis, which was performed on blood DNA samples.

For patient A four possible de-novo variants were considered as valid candidates: one splicing mutation in the Bone Morphogenetic Protein Receptor 2 (BMPR2) gene, two missense mutations in the ACAP3 (ArfGAP with Coiled-coil, Ankyrin repeat and PH This article is protected by copyright. All rights reserved domains 3) and in the ERC2 genes (ELKS/RAB6-interacting/CAST family member 2) as well as a nonsense mutation in ANKRD11. No obvious functional correlation was found between the first three genes and the cohesin complex. Moreover, three different bioinformatic tools (PolyPhen-2, SIFT and Mutation Taster) predicted the first three variants to be not pathogenic. Besides, mutations in ANKRD11 were recently identified in three patients with CdLS-overlapping phenotypes [START_REF] Ansari | Genetic heterogeneity in Cornelia de Lange syndrome (CdLS) and CdLS-like phenotypes with observed and predicted levels of mosaicism[END_REF]. The ANKRD11 mutation of patient A, c.5483G>T; p.S1828X, was found in 31% (836) of the 2664 sequencing reads, indicating a possible mosaic state.

For patient B four variants were detected. Three genes, namely ZW10 (Centromere/Kinetocore protein zw10 homolog), SMC1B (Structural Maintenance of Chromosome protein 1B) and ASPM (Abnormal Spindle-like Microcephaly-associated Protein), carried a missense mutation, while a four base-pairs deletion resulting in a frameshift was identified in the ANKRD11 gene. All three missense variants were found to be inherited; therefore, we focused on the frame-shift deletion affecting exon 9 of ANKRD11 (c.2297_2300delAGAA, p.K766_K767fsX9). Similar to the analysis of patient A, an uneven distribution of the wild type to the mutant allele (47/13 reads) was detected, thus raising the suspicion of a mosaicism. Both ANKRD11 mutations were confirmed and proven as de novo by Sanger sequencing on different tissues (Fig. 2a). While the electropherograms of patient A further indicated an unequal ratio between the wild type and the mutant allele, no significant difference in the distribution of the two alleles could be observed in patient B. To further investigate the role of the putative mosaicism for both mutations we performed pyrosequencing analyses. By this, we could confirm the presence of a mosaic state for patient A while the mutation in patient B was excluded to be mosaic. In detail, the quantification of the two alleles in patient A showed a 30:70 ratio between the mutant and the wild type allele in blood DNA and a 50:50 ratio on This article is protected by copyright. All rights reserved fibroblasts DNA, whereas equal distribution of wild type and mutant allele was observed in blood, buccal mucosa as well as fibroblast DNA of patient B (Fig. 2b).

Discussion

Although CdLS and KBG are rare distinct syndromes, both show overlapping features such as cognitive impairment, growth retardation and some craniofacial abnormalities including brachycephaly, broad arched eyebrows and anteverted nostrils. Limb anomalies, specifically small hands and feet, clinodactyly of the fifth finger and syndactyly of the second and third toes, are also described in both syndromes. In this context, it is worth to be noted that both ANKRD11 and the cohesin complex are involved in gene expression regulation. The main function of ANKRD11 is to repress the transcriptional activation of target genes of nuclear receptors by recruiting deacetylases to different promoters [START_REF] Zhang | Identification of a novel family of ankyrin repeats containing cofactors for p160 nuclear receptor coactivators[END_REF]. Similarly, the cohesin complex is involved in the regulation of gene expression, mediating both transcriptional activation and repression [START_REF] Horsfield | Cohesin-dependent regulation of Runx genes[END_REF][START_REF] Wendt | Cohesin mediates transcriptional insulation by CCCTC-binding factor[END_REF]. It is therefore tempting to speculate that the dysregulation of functionally interconnected sets of genes due to deficiency of the cohesin complex or ANKRD11 might result in overlapping phenotypical features.

Despite some shared phenotypical features, both syndromes must be considered as two distinct entities. In this regard, the head circumference and the teeth conformation may be used as main distinguishing features: CdLS patients are normally characterized by a marked microcephaly, which is not described for patients with KBG. Teeth anomalies including macrodontia of the central incisors, fused incisors, malposition, oligodontia and overcrowded teeth are instead frequent in patients with KBG, and not typical for CdLS (Table 1).

Here we report on two CdLS patients with a mutation in ANKRD11. Patient A shows some clinical signs common to both CdLS and KBG, specifically brachycephaly, short neck, arched eyebrows, anteverted nares, a long philtrum, intellectual disability and behavioral This article is protected by copyright. All rights reserved problems. Nevertheless, the small head circumference, synophrys, long eyelashes, a depressed nasal bridge and the absence of macrodontia or fused incisors were suggestive for the clinical diagnosis of CdLS. Patient B also shows features present in CdLS and KBG, such as broad and arched eyebrows, long philtrum, clinodactyly, intellectual disability and behavioral problems. Typical for CdLS are the low anterior hairline, synophrys, long eyelashes, smooth philtrum, thin lips with downturned corners of the mouth and a small head circumference. Despite the presence of a fifth finger clinodactyly, the hands of patient B show long fingers which is rather atypical for CdLS (Fig. 1k). Suggestive for KBG are the large nasal tip and macrodontia of the central upper incisors. Although patient B showed typical clinical features reminiscent for CdLS during early childhood, at the age of 15 years his phenotype was more compatible with KGB.

Interestingly, patient A carries the mutation in a mosaic state. In the last years, mosaicism has been proved to play an important role in CdLS [START_REF] Huisman | High rate of mosaicism in individuals with Cornelia de Lange syndrome[END_REF].

So far, only one mosaic mutation in the ANKRD11 gene has been reported in the mildly affected mother of a KBG proband [START_REF] Khalifa | Partial deletion of ANKRD11 results in the KBG phenotype distinct from the 16q24.3 microdeletion syndrome[END_REF]. In our patient A it might be possible that the presence of mosaicism with a variable ratio of the wild type and mutant gene product in different tissues can contribute to the mitigation of the KBG features and may explain the more CdLS-overlapping phenotype.

In summary, mutations in ANKRD11 can result in a CdLS-overlapping phenotype which seems to be most distinct in early childhood. Therefore, a detailed clinical follow up is strongly recommended. Sequencing analysis of ANKRD11 has to be considered for those patients with the suspected clinical diagnoses of CdLS during childhood who were tested as mutation negative in the five known CdLS genes.

Fig. 1 Phenotype

 1 Fig.1

Fig. 2 Analysis

 2 Fig.2

Fig. 1 Phenotype

 1 Fig. 1

  This article is protected by copyright. All rights reserved This article is protected by copyright. All rights reserved

Fig. 2 Analysis

 2 Fig. 2

This article is protected by copyright. All rights reserved

Acknowledgements

We thank Mariani Foundation, Milan, for the support of the clinical activity of UOS Pediatric Genetic Clinic of MBBM Foundation Monza. The study was supported under the frame of E-Rare-2 (TARGET-CdLS) by the German Federal Ministry of Education and Research (BMBF) (to FJK), by the by the Netherlands Organization for Health Research and Development (ZonMW) (to KSW), by the Agence Nationale de la Recherche (ANR) (to EW) and the "Schwerpunktprogramm (SPP) für medizinische Genetik" of the University of Lübeck (to DB, RW and FJK).

This article is protected by copyright. All rights reserved


Bernard Helffer
email: bernard.helffer@math.u-psud.fr

Yuri Kordyukov

AND S. V Ũ NGO . C Nicolas Raymond
email: nicolas.raymond@univ-rennes1.fr

San Vu

S Vũ

Magnetic Wells in Dimension Three

come   L'archive ouverte pluridisciplinaire

1. Introduction 1.1. Motivation and context. The analysis of the magnetic Laplacian (-i ∇ -A) 2 in the semiclassical limit → 0 has been the object of many developments in the last twenty years. The existence of discrete spectrum for this operator, together with the analysis of the eigenvalues, is related to the notion of "magnetic bottle", or quantum confinement by a pure magnetic field, and has important applications in physics. Moreover, motivated by investigations of the third critical field in Ginzburg-Landau theory for superconductivity, there has been a great attention focused on estimates of the lowest eigenvalue. In the last decade, it appears that the spectral analysis of the magnetic Laplacian has acquired a life on its own. For a story and discussions about the subject, the reader is referred to the recent reviews [START_REF] Fournais | Spectral methods in surface superconductivity[END_REF][START_REF] Helffer | Semiclassical spectral asymptotics for a magnetic schrödinger operator with non-vanishing magnetic field[END_REF][START_REF] Raymond | Little Magnetic Book[END_REF].

In contrast to the wealth of studies exploring the semiclassical approximations of the Schrödinger operator -2 ∆ + V , the classical picture associated with the Hamiltonian p -A(q) 2 has almost never been investigated to describe the semiclassical bound states (i.e. the eigenfunctions of low energy) of the magnetic Laplacian. The paper by Raymond and Vũ Ngo . c [START_REF] Raymond | Geometry and Spectrum in 2D Magnetic Wells[END_REF] is to our knowledge the first rigorous work in this direction. In that paper, which deals with the two-dimensional case, the notion of magnetic drift, well known to physicists, is cast in a symplectic framework, and using a semiclassical Birkhoff normal form (see for instance [START_REF] Ngo | Systèmes intégrables semi-classiques: du local au global, volume 22 of Panoramas et Synthèses [Panoramas and Syntheses[END_REF][START_REF] Charles | Spectral asymptotics via the semiclassical Birkhoff normal form[END_REF][START_REF] Ngo | Quantum Birkhoff normal forms and semiclassical analysis[END_REF]) it becomes possible to describe all the eigenvalues of order O( ). Independently, the asymptotic expansion of a smaller set of eigenvalues was established in [START_REF] Helffer | Semiclassical spectral asymptotics for a twodimensional magnetic Schrödinger operator: the case of discrete wells[END_REF][START_REF] Helffer | Accurate semiclassical spectral asymptotics for a twodimensional magnetic Schrödinger operator[END_REF] through different methods which act directly on the quantum side: explicit unitary transforms and a Grushin like reduction are used to reduce the twodimensional operator to an effective one-dimensional operator.

The three-dimensional case happens to be much harder. The only known results in this case that provide a full asymptotic expansion of a given eigenvalue concern toy models where the confinement is obtained by a boundary carrying a Neumann condition on an half space in [START_REF] Raymond | Semiclassical 3D Neumann Laplacian with variable magnetic field: a toy model[END_REF] or on a wedge in [START_REF] Popoff | When the 3D magnetic laplacian meets a curved edge in the semiclassical limit[END_REF]. In the case of smooth confinement without boundary, a construction of quasimodes by Helffer and Kordyukov in [START_REF] Helffer | Eigenvalue estimates for a three-dimensional magnetic Schrödinger operator[END_REF] suggests what the expansions of the low lying eigenvalues could be. But, as was expected by Colin de Verdière in his list of open questions in [START_REF] De Verdière | La méthode de moyennisation en mécanique semi-classique[END_REF], extending the symplectic and microlocal techniques to the three-dimensional case contains an intrinsic difficulty in the fact that the symplectic form cannot be nondegenerate on the characteristic hypersurface. The goal of our paper is to answer this question by fully carrying out this strategy. After averaging the cyclotron motion, the effect of the degeneracy of the symplectic form can be observed on the fact that the reduced operator is only partially elliptic. Hence, the key ingredient will be a separation of scales via the introduction of a new semiclassical parameter for only one part of the variables. These semiclassical scales are reminiscent of the three scales that have been exhibited in the classical picture in the large field limit, see [START_REF] Benettin | Adiabatic invariants and trapping of a point charge in a strong nonuniform magnetic field[END_REF][START_REF] Cheverry | Can one hear whistler waves ?[END_REF]. They are also related to the Born-Oppenheimer type of approximation in quantum mechanics (see for instance [START_REF] Born | Zur Quantentheorie der Molekeln[END_REF][START_REF] Martinez | A general effective Hamiltonian method[END_REF]). In fact, in a partially semiclassical context and under generic assumptions, a full asymptotic expansion of the first magnetic eigenvalues (and the corresponding WKB expansions) has been recently established in any dimension in the paper by Bonnaillie-Noël-Hérau-Raymond [START_REF] Bonnaillie-Noël | Magnetic WKB constructions[END_REF].

Magnetic geometry.

Let us now describe the geometry of the problem. The configuration space is R 3 = {q 1 e 1 + q 2 e 2 + q 3 e 3 , q j ∈ R, j = 1, 2, 3}, where (e j ) j=1,2,3 is the canonical basis of R 3 . The phase space is

R 6 = {(q, p) ∈ R 3 × R 3 }
and we endow it with the canonical 2-form (1.1) ω 0 = dp 1 ∧ dq 1 + dp 2 ∧ dq 2 + dp 3 ∧ dq 3 .

We will use the standard Euclidean scalar product •, • on R 3 and • the 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 .
The main object of this paper is the magnetic Hamiltonian, defined for all (q, p) ∈ R 6 by (1.2) H(q, p) = p -A(q) 2 ,

where A ∈ C ∞ (R 3 , R 3 ).

Let us now introduce the magnetic field. The vector field A = (A 1 , A 2 , A 3 ) is associated (via the Euclidean structure) with the 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 2 +(∂ 1 A 3 -∂ 3 A 1 )dq 1 ∧dq 3 +(∂ 2 A 3 -∂ 3 A 2 )dq 2 ∧dq 3 .
The form dα may be identified with 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 ),
then, we can write

(1.3) dα = B 3 dq 1 ∧ dq 2 -B 2 dq 1 ∧ dq 3 + B 1 dq 2 ∧ dq 3 .
The vector field B is called the magnetic field. Let us notice that we can express the 2-form dα thanks to the magnetic matrix

M B =   0 B 3 -B 2 -B 3 0 B 1 B 2 -B 1 0   .
Indeed we have

(1.4) dα(U, V ) = U, M B V = U, V ×B = [U, V, B], ∀(U, V ) ∈ R 3 ×R 3 ,
where [•, •, •] is the canonical mixed product on R 3 . We note that B belongs to the kernels of M B and dα.

An important role will be played by the characteristic hypersurface

Σ = H -1 (0),
which is the submanifold defined by the parametrization: R 3 q → j(q) := (q, A(q)) ∈ R 3 × R 3 . We may notice the relation between Σ, the symplectic structure and the magnetic field in the following relation

(1.5) j * ω 0 = dα ,
where dα is defined in (1.3).

1.3. Confinement assumptions and discrete spectrum. This paper is devoted to the semiclassical analysis of the discrete spectrum of the magnetic Laplacian L ,A := (-i ∇ q -A(q)) 2 , which is the semiclassical Weyl quantization of H (see (2.1)). This means that we will consider that belongs to (0, 0 ) with 0 small enough. Let us recall the assumptions under which discrete spectrum actually exist. In two dimensions, with a non vanishing magnetic field, a standard estimate (see [START_REF] Avron | Schrödinger operators with magnetic fields. I. General interactions[END_REF][START_REF] Cycon | Schrödinger operators with application to quantum mechanics and global geometry[END_REF]) gives

R 2 |B(q)||u(q)| 2 dq L ,A u | u , ∀u ∈ C ∞ 0 (R 2 ) .
Except in special cases when some components of the magnetic field have constant sign, this is no more the case in higher dimension (see [START_REF] Dufresnoy | Un exemple de champ magnétique dans R ν[END_REF]). We should impose a control of the oscillations of B at infinity. Under this condition, we get a similar estimate at the price of a small loss. This kind of estimate actually follows from an analysis developed in [START_REF] Helffer | Semiclassical analysis for the ground state energy of a Schrödinger operator with magnetic wells[END_REF]. Let us define b(q) := B(q) .

Let us now state the confining assumptions under which we will constantly work in this paper.

Assumption 1.1. We consider the case of R 3 and assume

(1.6) b(q) b 0 := inf q∈R 3 b(q) > 0 ,
and the existence of a constant C > 0 such that

(1.7) ∇B(q) C (1 + b(q)) , ∀q ∈ R 3 .
Under Assumption 1.1, it is proven in [16, Theorem 3.1] that there exist h 0 > 0 and C 0 > 0 such that, for all ∈ (0, h 0 ),

(1.8) (1 -C 0 1 4 ) R 3 b(q)|u(q)| 2 dq L ,A u | u , ∀u ∈ C ∞ 0 (R 3 ) .
As a corollary, using Persson's theorem (see [START_REF] Persson | Bounds for the discrete part of the spectrum of a semi-bounded Schrödinger operator[END_REF]), we obtain that the bottom of the essential spectrum is asymptotically above b 1 , where

b 1 := lim inf |q|→+∞ b(q).
More precisely, under Assumption 1.1, there exist h 0 > 0 and C 0 > 0 such that, for all ∈ (0, h 0 ),

(1.9) s ess (L ,A ) ⊂ [ b 1 (1 -C 0 1 4 ), +∞).
Assumption 1.2. We assume that

(1.10) 0 < b 0 < b 1 .
Moreover we will assume that there exists a point q 0 ∈ R 3 and ε > 0, β0

∈ (b 0 , b 1 ) such that (1.11) {b(q) β0 } ⊂ D(q 0 , ε),
where D(q 0 , ε) is the Euclidean ball centered at q 0 and of radius ε. For the rest of the article we let β 0 ∈ (b 0 , β0 ). Without loss of generality, we can assume that q 0 = 0 and that A(0) = 0 (which can be obtained with a change of gauge).

Note that Assumption 1.2 implies that the minimal value of b is attained inside D(q 0 , ε).

All along this paper, we will strengthen the assumptions on the nature of the point q 0 . At some stage of our investigation, q 0 will be the unique minimum of b. Note in particular that (1.11) is satisfied as soon as b admits a unique and non degenerate minimum.

Informal description of the results.

Let us now informally walk through the main results of this paper. We will assume (as precisely formulated in (1.10)-(1.11)) that the magnetic field does not vanish and is confining.

Of course, for eigenvalues of order O( ), the corresponding eigenfunctions are microlocalized in the semi-classical sense near the characteristic manifold Σ (see for instance [START_REF] Robert | Autour de l'approximation semi-classique[END_REF][START_REF] Zworski | Semiclassical analysis[END_REF]). Moreover the confinement assumption implies that the eigenfunctions of L ,A associated with eigenvalues less that β 0 enjoy localization estimates à la Agmon. Therefore we will be reduced to investigate the magnetic geometry locally in space near a point q 0 = 0 ∈ R 3 belonging to the confinement region and which, for notational simplicity, we may assume to be the origin.

Then, in a neighborhood of (0, A(0)) ∈ Σ, there exist symplectic coordinates

(x 1 , ξ 1 , x 2 , ξ 2 , x 3 , ξ 3 ) such that Σ = {x 1 = ξ 1 = ξ 3 = 0} and (0, A(0)) has coordinates 0 ∈ R 6 . Hence Σ is parametrized by (x 2 , ξ 2 , x 3 ).
1.4.1. First Birkhoff form. In these coordinates suited for the magnetic geometry, it is possible to perform a semiclassical Birkhoff normal form and microlocally unitarily conjugate L ,A to a first normal form N = Op w (N ) with an operator valued symbol N depending on (x 2 , ξ 2 , x 3 , ξ 3 ) in the form

N = ξ 2 3 + b(x 2 , ξ 2 , x 3 )I + f ( , I , x 2 , ξ 2 , x 3 , ξ 3 ) + O(|I | ∞ , |ξ 3 | ∞ ). where I h = 2 D 2
x 1 +x 2 1 is the first encountered harmonic oscillator and where ( , 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| 3 2 + |ξ 3 | 3 + 3 2

.

Since we wish to describe the spectrum in a spectral window containing at least the lowest eigenvalues, we are led to replace I by its lowest eigenvalue and thus, we are reduced to the two-dimensional pseudo-differential operator N [1] = Op w N [1] where

N [1] = ξ 2 3 + b(x 2 , ξ 2 , x 3 ) + f ( , , x 2 , ξ 2 , x 3 , ξ 3 ) + O( ∞ , |ξ 3 | ∞ ).
1.4.2. Second Birkhoff form. If we want to continue the normalization, we shall assume a new non-degeneracy condition (the first one was the positivity of b). Now we assume that, for any (x 2 , ξ 2 ) in a neighborhood of (0, 0), the function x 3 → b(x 2 , ξ 2 , x 3 ) admits a unique and non-degenerate minimum denoted by s(x 2 , ξ 2 ). Then, by using a new symplectic transformation in order to center the analysis at the partial minimum s(x 2 , ξ 2 ), we get a new operator N [1] whose Weyl symbol is in the form

N [1] = ν 2 (x 2 , ξ 2 )(ξ 2 3 + x 2 3 ) + b(x 2 , ξ 2 , s(x 2 , ξ 2 )) + remainders, with (1.12) ν(x 2 , ξ 2 ) = ( 1 2 ∂ 2 3 b(x 2 , ξ 2 , s(x 2 , ξ 2 ))) 1/4
and where the remainders have been properly normalized to be at least formal perturbations of the second harmonic oscillator ξ 2 3 + x 2 3 . Since the frequency of this oscillator is -1 2 in the classical picture, we are naturally led to introduce the new semiclassical parameter

h = 1 2
and the new impulsion

ξ = 1 2 ξ so that Op w ξ 2 3 + x 2 3 = h 2 Op w h ξ2 3 + x 2 3 .
We therefore get the h-symbol of N [1] :

N [1] h = h 2 ν 2 (x 2 , h ξ2 )( ξ2 3 + x 2 3 ) + h 2 b(x 2 , h ξ2 , s(x 2 , h ξ2 )) + remainders.
We can again perform a Birkhoff analysis in the space of formal series given by

E = F [[x 3 , ξ3 , h]]
where F is a space of symbols in the form c(h, x 2 , h ξ2 ).

We get the new operator M h = Op w h (M h ), with

M h = h 2 b(x 2 , h ξ2 , s(x 2 , h ξ2 )) + h 2 J h Op w h ν 2 (x 2 , h ξ2 ) + h 2 g (h, J h , x 2 , h ξ2 ) + remainders,
where J h = Op w h ξ2 3 + x 2 3 and g (h, J, x 2 , ξ 2 ) is of order three with respect to (J

1 2 , h 1 2
). Motivated again by the perspective of describing the low lying eigenvalues, we replace J h by h and rewrite the symbol with the old semiclassical parameter to get the operator M [1] = Op w h M

[1] h = Op w M [1] , with (1.13)

M [1] = b(x 2 , ξ 2 , s(x 2 , ξ 2 )) + 3 2 ν 2 (x 2 , ξ 2 ) + g ( 1 2 , 1 2 , x 2 , ξ 2 ) + remainders.
1.4.3. Third Birkhoff form. The last generic assumption is the uniqueness and non-degeneracy of the minimum of the new "principal" symbol

(x 2 , ξ 2 ) → b(x 2 , ξ 2 , s(x 2 , ξ 2 ))
that implies that b admits a unique and non-degenerate minimum at (0, 0, 0). Up to an 1 2 -dependent translation in the phase space and a rotation, we are essentially reduced to a standard Birkhoff normal form with respect to the third harmonic oscillator K = 2 D 2

x 2 + x 2 2 . Note that all our normal forms may be used to describe the classical dynamics of a charged particle in a confining magnetic field (see Figure 1). √ 1 + x2 and the full line represents the projection in the q-space of the Hamiltonian trajectory with initial condition (q 0 , p 0 ) (with p 0 = (-0.6, 0.01, 0.2)) ending at (q 1 , p 1 ).

1.4.4. Microlocalization. Of course, at each step, we will have to provide accurate microlocal estimates of the eigenfunctions of the different operators to get a good control of the different remainders. In a first approximation, we will get localizations at the following scales x 1 , ξ 1 , ξ 3 ∼ δ (δ > 0 is small enough) and x 2 , ξ 2 , x 3 ∼ 1. In a second approximation, we will get x 3 , ξ3 ∼ δ . In the final step, we will refine the localization by x 2 , ξ 2 ∼ δ . 1.5. A semiclassical eigenvalue estimate. Let us already state one of the consequences of our investigation. It will follow from the third normal form that we have a complete description of the spectrum below the threshold b 0 + 3ν 2 (0, 0) Theorem 1.3. Assume that b admits a unique and non degenerate minimum at q 0 . Denote

(1.14) σ = Hess q 0 b (B, B) 2b 2 0 , θ = det Hess q 0 b Hess q 0 b (B, B)
.

There exists a function k ∈ C ∞ 0 (R 2 ) with arbitrarily small compact support, and k (

1 2 , Z) = O(( + |Z|) 3
2 ) when ( , Z) → (0, 0), such that the following holds.

For all c ∈ (0, 3), the spectrum of L ,A below b 0 +cσ

1 2 3 2 coincides modulo O( ∞ )
with the spectrum of the operator F acting on L 2 (R x ) given by

F = b 0 + σ 1 2 3 2 - ζ 2θ 2 + θ 2 K + k ( 1 2 , K ) , K = 2 D 2 x + x 2 ,
with some constant ζ.

Remark 1.4. The constant ζ in Theorem 1.3 is given by the formula

ζ = ∇ν 2 (0, 0) 2 ,
where the function ν is given in (1.12). Observe also that σ = ν 4 (0, 0).

Corollary 1.5. Under the hypothesis of Theorem 1.3, let (λ m ( )) m 1 be the non decreasing sequence of the eigenvalues of L ,A . For any c ∈ (0, 3), let

N ,c := {m ∈ N * ; λ m ( ) b 0 + cσ 1 2 3 2 }.
Then the cardinal of N ,c is of order -1 2 , and there exist υ 1 , υ 2 ∈ R and 0 > 0 such that

λ m ( ) = b 0 +σ 1 2 3 2 + θ(m -1 2 ) - ζ 2θ 2 +υ 1 (m-1 2 ) 5 2 +υ 2 (m-1 2 ) 2 3 +O( 5 2
) , uniformly for ∈ (0, 0 ) and m ∈ N ,c . In particular, the splitting between two consecutive eigenvalues satisfies

λ m+1 ( ) -λ m ( ) = θ 2 + O( 5 2
).

Proof. If the support of k is small enough, the hypothesis k (

1 2 , Z) = O(( + |Z|) 3
2 ) implies that, when is small enough,

(1 + η)K K + 2 θ k ( 1 2 , K ) (1 -η)K ,
for some small η > 0. Therefore, since the eigenvalues of K are (2m -1) , m ∈ N * , the variational principle implies that the number of eigenvalues of

K + 2 θ k ( 1 2 , K ) below a threshold C belongs to [ 1 2 ( C (1+η) +1), 1 2 ( C (1-η) +1)]. Taking C = 2 θ (c -1)σ 1/2 1/2 + ζ θ 2
, and applying the theorem, we obtain the estimate for the cardinal of N ,c . The corresponding eigenvalues of L ,A are of the form

λ m ( ) = b 0 + σ 1 2 3 2 - ζ 2θ 2 + θ(m -1 2 ) + k ( 1 2 , 2m -1) + O( ∞ ), with (2m -1) C 1-η .
Therefore there exists a constant C > 0, independent of , such that all m ∈ N ,c satisfy the inequality (2m-1)

C 1/2 . Writing k ( 1 2 , Z) = c 0 3/2 + υ 1 1/2 (Z/2) + c 1 2 + υ 2 (Z/2) 2 + υ 3 Z + 1/2 O(h + |Z|) 2 + O(Z 3 ), we see that, for m ∈ N ,c , k ( 1 2 , (2m -1) ) = υ 1 3/2 m - 1 2 + υ 2 2 m - 1 2 2 + O( 3/2 ),
which gives the result.

Remark 1.6. An upper bound of λ m ( ) for fixed -independent m with remainder in O( 4 ) was obtained in [START_REF] Helffer | Eigenvalue estimates for a three-dimensional magnetic Schrödinger operator[END_REF] through a quasimodes construction involving powers of 1 4 . To the authors' knowledge, Corollary 1.5 gives the most accurate description of magnetic eigenvalues in three dimensions, in such a large spectral window. Note also that the non-degeneracy assumption on the norm of B is not purely technical. Indeed, at the quantum level, it appears through microlocal reductions matching with the splitting of the Hamiltonian dynamics into three scales: the cyclotron motion around field lines, the center-guide oscillation along the field lines, and the oscillation within the space of field lines. 1.6. Organization of the paper. The paper is organized as follows. In Section 2, we state our main results. Section 3 is devoted to the investigation of the first normal form (see Theorem 2.1 and Corollary 2.4). In Section 4 we analyze the second normal form (see Theorems 2.8 and 2.11 and Corollaries 2.9 and 2.13). Section 5 is devoted to the third normal form (see Theorem 2.15 and Corollary 2.16).

Statements of the main results

We recall (see [START_REF] Dimassi | Spectral asymptotics in the semi-classical limit[END_REF]Chapter 7]) that a function m :

R d → [0, ∞) is an order function if there exist constants N 0 , C 0 > 0 such that m(X) C 0 X -Y N 0 m(Y ) for any X, Y ∈ R d . The symbol class S(m) is the space of smooth - dependent functions a : R d → C such that ∀α ∈ N d , |∂ α x a (x)| C α m(x), ∀h ∈ (0, 1].
Throughout this paper, we assume that the components of the vector potential A belong to a symbol class S(m). Note that this implies that B ∈ S(m), and conversely, if B ∈ S(m), then there exist a potential A and another order function m such that A ∈ S(m ). Moreover, the magnetic Hamiltonian H(x, ξ) = ξ -A(x) 2 belongs to S(m ) for an order function m on R 6 .

We will work with the Weyl quantization; for a classical symbol a = a(x, ξ; ) ∈ S(m), it is defined as: (2.1)

Op w a ψ(x) = 1 (2π ) 3 R 6 e i x-y,ξ / a x + y 2 , ξ ψ(y)dydξ, ∀ψ ∈ S(R 3 ).
The Weyl quantization of H is the magnetic Laplacian L ,A = (-i ∇ -A) 2 .

Normal forms and spectral reductions. Let us introduce our first

Birkhoff normal form N .

Theorem 2.1. If B(0) = 0, there exists a neighborhood of (0, A(0)) endowed with symplectic coordinates

(x 1 , ξ 1 , x 2 , ξ 2 , x 3 , ξ 3 ) in which Σ = {x 1 = ξ 1 = ξ 3 = 0} and (0, A(0)) has coordinates 0 ∈ R 6
, and there exist an associated unitary Fourier integral operator U and a smooth function, compactly supported with respect to Z and ξ 3 , f ( , Z, x 2 , ξ 2 , x 3 , ξ 3 ) whose Taylor series with respect to Z, ξ 3 , is

k 3 2 +2m+β=k c ,m,β (x 2 , ξ 2 , x 3 )Z m ξ β 3 such that (2.2) U * L ,A U = N + R , with N = 2 D 2 x 3 + I Op w b + Op w f ( , I , x 2 , ξ 2 , x 3 , ξ 3 )
, and where (a) we have I = 2 D 2

x 1 + x 2 1 , (b) the operator Op w f ( , I , x 2 , ξ 2 , x 3 , ξ 3 ) has to be understood as the Weyl quantization of an operator valued symbol, (c) the remainder R is a pseudo-differential operator such that, in a neighborhood of the origin, the Taylor series of its symbol with respect to

(x 1 , ξ 1 , ξ 3 , ) is 0. Remark 2.2. In Theorem 2.1, the direction of B considered as a vector field on Σ is ∂ ∂x 3 and the function b ∈ C ∞ (R 6 ) stands for b • j -1 |Σ • π where π : R 6 → Σ : π(x 1 , ξ 1 , x 2 , ξ 2 , x 3 , ξ 3 ) = (0, 0, x 2 , ξ 2 , x 3 , 0).
In addition, note that the support of f in Z and ξ 3 may be chosen as small as we want.

Remark 2.3. In the context of Weyl's asymptotics, a close version of this theorem appears in [START_REF] Ivrii | Microlocal analysis and precise spectral asymptotics[END_REF]Chapter 6].

In order to investigate the spectrum of L ,A near the low lying energies, we introduce the following pseudo-differential operator

N [1] = 2 D 2 x 3 + Op w b + Op w f ( , , x 2 , ξ 2 , x 3 , ξ 3
), obtained by replacing I by .

Corollary 2.4. We introduce

(2.3) N = Op w N , with N = ξ 2 3 + I b(x 2 , ξ 2 , x 3 ) + f , ( , I , x 2 , ξ 2 , x 3 , ξ 3
) and where b is a smooth extension of b away from D(0, ε) such that (1.11) still holds and where f , = χ(x 2 , ξ 2 , x 3 )f , with χ is a smooth cutoff function being 1 in a neighborhood of D(0, ε). We also define the operator attached to the first eigenvalue of I (2.4)

N [1], = Op w N [1],
,

where

N [1], = ξ 2 3 + b(x 2 , ξ 2 , x 3 ) + f , ( , , x 2 , ξ 2 , x 3 , ξ 3
). If ε and the support of f are small enough, then we have (a) The spectra of L ,A and N below β 0 coincide modulo O( ∞ ). (b) For all c ∈ (0, min(3b 0 , β 0 )), the spectra of L ,A and N [1], below c coincide modulo O( ∞ ).

Let us now state our results concerning the normal form of N [1] (or N [1], ) under the following assumption.

Notation 2.5. If f = f (z) is a differentiable function, we denote by T z f (•)
its tangent map at the point z. Moreover, if f is twice differentiable, the second derivative of f is denoted by T 2 z f (•, •). Assumption 2.6. We assume that T 2 0 b(B(0), B(0)) > 0. Remark 2.7. If the function b admits a unique and positive minimum at 0 and that it is non degenerate, then Assumption 2.6 is satisfied. Under Assumption 2.6, we have ∂ 3 b(0, 0, 0) = 0 and, in the coordinates (x 2 , ξ 2 , x 3 ) given in Theorem 2.1, (2.5) ∂ 2 3 b(0, 0, 0) > 0 . It follows from (2.5) and the implicit function theorem that, for small x 2 , there exists a smooth function

(x 2 , ξ 2 ) → s(x 2 , ξ 2 ), s(0, 0) = 0, such that (2.6) ∂ 3 b(x 2 , ξ 2 , s(x 2 , ξ 2 )) = 0 .
The point s(x 2 , ξ 2 ) is the unique (in a neighborhood of (0, 0, 0)) minimum of

x 3 → b(x 2 , ξ 2 , x 3 ). We define ν(x 2 , ξ 2 ) := ( 1 2 ∂ 2 3 b(x 2 , ξ 2 , s(x 2 , ξ 2 ))) 1/4
. Theorem 2.8. Under Assumption 2.6, there exists a neighborhood V 0 of 0 and a Fourier integral operator V which is microlocally unitary near V 0 and such that V * N [1] V =: N [1] = Op w N [1] ,

where N

[1] = ν 2 (x 2 , ξ 2 ) ξ 2 3 + x 2 3 + b(x 2 , ξ 2 , s(x 2 , ξ 2 )) + r and r is a semiclassical symbol such that r = O( x 3 3 ) + O( ξ 2 3 ) + O(ξ 3 3 ) + O( 2 ). Corollary 2.9. Let us introduce N [1], = Op w N [1],
,

where

N [1], = ν 2 (x 2 , ξ 2 ) ξ 2 3 + x 2 3 + b(x 2 , ξ 2 , s(x 2 , ξ 2 )) + r , with r = χ(x 2 , ξ 2 , x 3 , ξ 3 )

r , and where ν denotes a smooth and constant (with a positive constant) extension of the function ν.

There exists a constant c > 0 such that, for any cut-off function χ equal to 1 on D(0, ε) with support in D(0, 2ε), we have:

(a) The spectra of N [1], and N [1], below (b 0 + cε 2 ) coincide modulo O( ∞ ).

(b) For all c ∈ (0, min(3b 0 , b 0 + cε 2 )), the spectra of L ,A and N [1], below c coincide modulo O( ∞ ).

Notation 2.10 (Change of semiclassical parameter). We let h = 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 h A h =: A h , with A h (x 2 , ξ2 , x 3 , ξ3 ) = A h 2 (x 2 , h ξ2 , x 3 , h ξ3 ).
Thus, A and A h represent the same operator when h = 1 2 , but the former is viewed as an -quantization of the symbol A , while the latter is an hpseudo-differential operator with symbol A h . Notice that, if A belongs to some class S(m), then A h ∈ S(m) as well. This is of course not true the other way around.

Theorem 2.11. Under Assumption 2.6, there exist a unitary operator W h and a smooth function g (h, Z, x 2 , ξ 2 ), with compact support as small as we want with respect to Z and with compact support in (x 2 , ξ 2 ), whose Taylor series with respect to Z, h is

2m+2 3 c m, (x 2 , ξ 2 )Z m h , such that W * h N [1], h W h =: M h = Op w h (M h ) , with M h = h 2 b(x 2 , h ξ2 , s(x 2 , h ξ2 )) + h 2 J h Op w h ν 2 (x 2 , h ξ2 ) + h 2 g (h, J h , x 2 , h ξ2 ) + h 2 R h + h ∞ S(1).
where (a) the operator N [START_REF] Avron | Schrödinger operators with magnetic fields. I. General interactions[END_REF], h is N [1], (but written in the h-quantization),

(b) we have let J h = Op w h ξ2 3 + x 2 3 , (c) the function R h satisfies R h (x 2 , h ξ2 , x 3 , ξ3 ) = O((x 3 , ξ3 ) ∞ ).
Remark 2.12. Note that the support of g with respect to Z may be chosen as small as we want. Note also that we have used N [START_REF] Avron | Schrödinger operators with magnetic fields. I. General interactions[END_REF], h instead of N [START_REF] Avron | Schrödinger operators with magnetic fields. I. General interactions[END_REF] h : Since W h is exactly unitary, we get a direct comparison of the spectra. Corollary 2.13. We introduce

M h = Op w h M h , with M h = h 2 b(x 2 , h ξ2 , s(x 2 , h ξ2 )) + h 2 J h ν 2 (x 2 , h ξ2 ) + h 2 g (h, J h , x 2 , h ξ2 ).
We also define

M [1], h = Op w h M [1], h , with M [1], h = h 2 b(x 2 , h ξ2 , s(x 2 , h ξ2 )) + h 3 ν 2 (x 2 , h ξ2 ) + h 2 g (h, h, x 2 , h ξ2 ).
If ε and the support of g are small enough, we have (a) For all η > 0, the spectra of

N [1], h and M h below b 0 h 2 + O(h 2+η ) coincide modulo O(h ∞ ). (b) For c ∈ (0, 3), the spectra of M h and M [1], h below b 0 h 2 + cσ 1 2 h 3 coincide modulo O(h ∞ ). (c) If c ∈ (0, 3), the spectra of L ,A and M [1], = M [1], h below b 0 + cσ 1 2 3 2 coincide modulo O( ∞ ).
Finally, we can perform a last Birkhoff normal form for the operator M

[1],

as soon as (x 2 , ξ 2 ) → b(x 2 , ξ 2 , s(x 2 , ξ 2 )) admits a unique and non degenerate minimum at (0, 0). Under this additional assumption, b admits a unique and non degenerate minimum at (0, 0, 0). Therefore we will use the following stronger assumption.

Assumption 2.14. The function b admits a unique and positive minimum at 0 and it is non degenerate.

Theorem 2.15. Under Assumption 2.14, there exist a unitary -Fourier Integral Operator Q 1 2 whose phase admits an expansion in powers of

1 2 such that Q * 1 2 M [1], Q 1 2 = F + G , where (a) F is defined in Theorem 1.3, (b) the remainder is in the form G = Op w (G ), with G = O(|z 2 | ∞ ).
Corollary 2.16. If ε and the support of k are small enough, we have (a) For all η ∈ 0, 1 2 , the spectra of M [1], and

F below b 0 + O( 1+η ) coincide modulo O( ∞ ).
(b) For all c ∈ (0, 3), the spectra of L ,A and F below b 0 + cσ

1 2 3 2 coincide modulo O( ∞ ).
Remark 2.17. Since the spectral analysis of F is straightforward, Item (b) of Corollary 2.16 implies Theorem 1.3.

The next sections are devoted to the proofs of our main results.

Theorem 1.3 N [1] N [1] h M h M [1] h M [1] L ,A N N [1]

First Birkhoff normal form

We assume that B(0) = 0 so that in some neighborhood Ω of 0 the magnetic field does not vanish. Up to a rotation in R 3 (extended to a symplectic transformation in R 6 ) we may assume that B(0) = B(0) e 3 . In this neighborhood, we may defined the unit vector:

(3.1) b = B B
and find vectors c and d depending smoothly on q such that (b, c, d) is a direct orthonormal basis.

3.1. Symplectic coordinates.

3.1.1. Straightening the magnetic vector field. We consider the form dα and we would like to find a diffeomorphism, in a neighborhood of 0, χ such that χ(q) = q and χ * (dα) = dq 1 ∧ dq 2 . First, this is easy to find a local diffeomorphism ϕ such that

∂ 3 ϕ(q) = b(ϕ(q))
and ϕ(q 1 , q2 , 0) = (q 1 , q2 , 0). This is just the standard straigthening-out lemma for the non-vanishing vector field b.

The vector e 3 is in the kernel of ϕ * (dα), which implies that we have ϕ * (dα) = f (q)dq 1 ∧ dq 2 , for some smooth function f . But since the form ϕ * (dα) is closed, f does not depend on q3 . This is then easy to find another diffeomorphism ψ, corresponding to the change of variables q = ψ(q) = (ψ 1 (q 1 , q2 ), ψ 2 (q 1 , q2 ), q3 ) , such that ψ * (ϕ * (dα)) = dq 1 ∧ dq 2 . We let χ = ϕ • ψ and we notice that

(3.2) χ * (dα) = dq 1 ∧ dq 2 ∂ 3 χ(q) = b(χ(q)) , Remark 3.1. It follows from (3.
2) and (1.4) that det T χ = B -1 .

Symplectic coordinates. Let us consider the new parametrization of Σ

given by ι : Ω -→ Σ q → (χ(q), A 1 (χ(q)) , A 2 (χ(q)), A 3 (χ(q))) , which gives a basis (f 1 , f 2 , f 3 ) of T Σ :

f j = (T χ(e j ), T A • T χ(e j )), j = 1, 2, 3 .
Using (1.5), and the fact that f 3 is in the kernel of dα, we find ω 0 (f j ,

f 3 ) = 0, j = 1, 2 . Finally, ω 0 (f 1 , f 2 ) = dα(T χe 1 , T χe 2 ) = χ * (dα)(e 1 , e 2 ) = 1 .
The following vectors of R 3 × R 3 form a basis of the symplectic orthogonal of T ι(q) Σ:

(3.3) f 4 = B -1/2 (c, ( t T χ(q) A)c), f 5 = B -1/2 (d, ( t T χ(q) A)d),
so that ω 0 (f 4 , f 5 ) = -1. We let f 6 = (0, b) + ρ 1 f 1 + ρ 2 f 2 where ρ 1 and ρ 2 are determined so that ω 0 (f j , f 6 ) = 0 for j = 1, 2. We notice that ω 0 (f j , f 6 ) = 0 for j = 4, 5 and ω 0 (f 3 , f 6 ) = -1.

3.1.3.

Diagonalizing the Hessian. We recall that H(q, p) = p -A(q) 2 so that, at a critical point p = A(q), the Hessian is

T 2 H((U 1 , V 1 ), (U 2 , V 2 )) = 2 V 1 -T q A(U 1 ), V 2 -T q A(U 2 ) .

Let us notice that

T 2 H(f 4 , f 5 ) = 2 B -1 B × c, B × d = 0, T 2 H(f 4 , f 6 ) = 2 B × c, b = 0, T 2 H(f 5 , f 6 ) = 2 B × d, b = 0.
The Hessian is diagonal in the basis (f 4 , f 5 , f 6 ). Moreover we have

T 2 H(f 4 , f 4 ) = d 2 H(f 5 , f 5 ) = 2 B -1 B × c 2 = 2 B -1 B × d 2 = 2 B .
Finally we have:

T 2 H(f 6 , f 6 ) = 2.
Now we consider the local diffeomorphism:

(x, ξ) → ι(x 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 ).
The Jacobian of this map is a symplectic matrix on Σ. We may apply the Moser-Weinstein argument (see [START_REF] Weinstein | Symplectic manifolds and their lagrangian submanifolds[END_REF]) to make this map locally symplectic near Σ modulo a change of variable which is tangent to the identity. Near Σ, in these new coordinates, the Hamiltonian H admits the expansion

(3.4) Ĥ = H 0 + O(|x 1 | 3 + |ξ 1 | 3 + |ξ 3 | 3 ),
where Ĥ denotes H in the coordinates (x 1 , x 2 , x 3 , ξ 1 , ξ 2 , ξ 3 ), and with

(3.5) H 0 = ξ 2 3 + b(x 2 , ξ 2 , x 3 )(x 2 1 + ξ 2 1 ), b = B(x 2 , ξ 2 , x 3 ) .

3.2.

Semiclassical Birkhoff normal form.

3.2.1.

Birkhoff procedure in formal series. Let us consider the space E of formal power series in (x 1 , ξ 1 , ξ 3 , ) with coefficients smoothly 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 with the semiclassical Moyal product (with respect to all variables (x 1 , x 2 , x 3 , ξ 1 , ξ 2 , ξ 3 )) denoted by and the commutator of two series κ 1 and κ 2 is defined as

[κ 1 , κ 2 ] = κ 1 κ 2 -κ 2 κ 1 .
The degree of

x α 1 1 ξ α 2 1 ξ β 3 = z α 1 ξ β 3 is α 1 + α 2 + β + 2 = |α| + β + 2 .
D N denotes the space of monomials of degree N . O N is the space of formal series with valuation at least N . For any τ, γ ∈ E, we denote ad τ γ = [τ, γ]. Proposition 3.2. Given γ ∈ O 3 , there exist formal power series τ, κ ∈ O 3 such that e i -1 adτ (H

0 + γ) = H 0 + κ , with [κ, |z 1 | 2 ] = 0 . Proof. Let N 1. Assume that we have, for τ N ∈ O 3 , e i -1 adτ N (H 0 + γ) = H 0 + K 3 + • • • + K N +1 + R N +2 + O N +3 , with K i ∈ D i , [K i , |z 1 | 2 ] = 0 and R N +2 ∈ D N +2 . Let τ ∈ D N +2 . Then we have e i -1 ad τ N +τ (H 0 + γ) = H 0 + K 3 + • • • + K N +1 + K N +2 + O N +3 , with K N +2 ∈ D N +2 such that K N +2 = R N +2 + i -1 ad τ H 0 + O N +3 . Lemma 3.3. For τ ∈ D N +2 , we have i -1 ad τ H 0 = i -1 b ad τ |z 1 | 2 + O N +3 .
To prove this lemma, we observe that

i -1 ad τ H 0 = i -1 ad τ ξ 2 3 + i -1 ad τ (b(x)|z 1 | 2 ). Let us write τ = |α|+β+2l=N +2 a α,β,l (x)z α 1 ξ β 3 l .
Then, for the first term, we have

i -1 ad τ ξ 2 3 ={τ , ξ 2 3 } = -2ξ 3 ∂τ ∂x 3 = -2 |α|+β+2 =N +2 ∂a α,β, ∂x 3 (x)z α 1 ξ β+1 3 ∈ O N +3 .
We also have

i -1 (ad τ b(x)) ={τ , b} + 2 O N = ∂τ ∂ξ 3 ∂b ∂x 3 + ∂τ ∂ξ 2 ∂b ∂x 2 - ∂τ ∂x 2 ∂b ∂ξ 2 + O N +1 = |α|+β+2 =N +2 βa(x) ∂b ∂x 3 z α 1 |z 1 | 2 ξ β-1 3 + O N +1 ∈ O N +1 .
Therefore, for the second term, we get

i -1 ad τ (b(x)|z 1 | 2 ) =i -1 (ad τ b(x))|z 1 | 2 + i -1 b(x) ad τ |z 1 | 2 =i -1 b(x) ad τ |z 1 | 2 + O N +3 ,
that completes the proof of the lemma. By the lemma, we obtain that

K N +2 = R N +2 + b ad τ |z 1 | 2 ,
that we rewrite as

R N +2 = K N +2 + i -1 b ad |z 1 | 2 τ = K N +2 + b{|z 1 | 2 , τ }.
Since b(x) = 0, we deduce the existence of τ and

K N +2 such that K N +2 commutes with |z 1 | 2 .

3.2.2.

Quantizing the formal procedure. Let us now prove Theorem 2.1. Using (3.4) and applying the Egorov theorem (see [START_REF] Robert | Autour de l'approximation semi-classique[END_REF][START_REF] Zworski | Semiclassical analysis[END_REF] or Theorem A.2), we can find a unitary Fourier Integral Operator U , and such that

U * L ,A U = C 0 + Op w (H 0 ) + Op w (r ),
where the Taylor series (with respect to x 1 , ξ 1 , ξ 3 , ) of r satisfies r T = γ ∈ O 3 and C 0 is the value at the origin of the sub-principal symbol of U * L ,A U . One can choose U such that the subprincipal symbol is preserved by conjugation 1 , which implies C 0 = 0. Applying Proposition 3.2, we obtain τ and κ in O 3 such that

e i -1 adτ (H 0 + γ) = H 0 + κ, with [κ, |z 1 | 2 ] = 0.
We can introduce a smooth symbol a with compact support such that we have a T = τ in a neighborhood of the origin. By Proposition 3.2 and Theorem A. [START_REF] Born | Zur Quantentheorie der Molekeln[END_REF], we obtain that the operator

e i -1 Op w (a ) (Op w (H 0 ) + Op w (r ))e -i -1 Op w (a )
is a pseudodifferential operator such that the formal Taylor series of its symbol is H 0 + κ. In this application of Theorem we have used the filtration O j defined in Section 3.2.1. Since κ commutes with |z 1 | 2 , we can write it as a formal series in |z 1 | 2 :

κ = k 3 2 +2m+β=k c ,m (x 2 , ξ 2 , x 3 )|z 1 | 2m ξ β 3 .
This formal series can be reordered by using monomials

(|z 1 | 2 ) m : κ = k 3 2 +2m+β=k c ,m (x 2 , ξ 2 , x 3 )(|z 1 | 2 ) m ξ β 3 .
Thanks to the Borel lemma, we may find a smooth function, with a compact support as small as we want with respect to , I and ξ 3 , f ( , I, x 2 , ξ 2 , x 3 , ξ 3 ) such that its Taylor series with respect to , I, ξ 3 is

k 3 2 +2m+β=k c ,m (x 2 , ξ 2 , x 3 )I m ξ β 3 .
This achieves the proof of Theorem 2.1.

3.3.

Spectral reduction to the first normal form. This section is devoted to the proof of Corollary 2.4.

Numbers of eigenvalues.

Lemma 3.4. Under Assumption 1.2, there exists h 0 > 0 and ε 0 > 0 such that for all ∈ (0, h 0 ), inf s ess (N ) (β 0 + ε 0 ) .

Proof. By using the assumption we may consider a smooth function χ with compact support and ε 0 > 0 such that

ξ 2 3 + b(x 2 , ξ 2 , x 3 ) + χ(x 2 , x 3 , ξ 2 , ξ 3 ) β 0 + 2ε 0 .
1 This is sometimes called the Improved Egorov Theorem. It was first discovered by Weinstein in [START_REF] Weinstein | Fourier integral operators, quantization, and the spectra of Riemannian manifolds[END_REF], in the homogeneous setting. For the semiclassical case, see for instance [18, Appendix A].

Then, given η ∈ (0, 1) and estimating the second term in (2.3) by using that the support of f is chosen small enough and the semiclassical Calderon-Vaillancourt theorem, we notice that, for small enough,

(3.6) N (1 -η) Op w ξ 2 3 + |z 1 | 2 b(x 2 , ξ 2 , x 3
) . Since the essential spectrum is invariant by (relatively) compact perturbations, we have

s ess N + (1 -η) Op w χ(x 2 , x 3 , ξ 2 , ξ 3 ) = s ess N . Hence inf s ess N inf s N + (1 -η) Op w χ(x 2 , x 3 , ξ 2 , ξ 3 ) .
In order to bound the r.h.s. from below, we write

N + (1 -η) Op w χ(x 2 , x 3 , ξ 2 , ξ 3 ) (1 -η) Op w ξ 2 3 + |z 1 | 2 b(x 2 , ξ 2 , x 3 ) + (1 -η) Op w χ(x 2 , x 3 , ξ 2 , ξ 3 ) (1 -η) Op w ξ 2 3 + b(x 2 , ξ 2 , x 3 ) + χ(x 2 , x 3 , ξ 2 , ξ 3 ) (1 -η)(β 0 + 2ε 0 -C ),
where we have used the semiclassical Gårding inequality. Taking η and then small enough, this concludes the proof.

By using the Hilbertian decomposition given by the Hermite functions (e k, ) k 1 associated with I , we notice that

N = k 1 N [k],
, where

(3.7) N [k], = 2 D 2 x 3 +(2k -1) Op w b+Op w 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), there exist C > 0 and h 0 > 0 such that for all k 1 and ∈ (0, h 0 ), we have

s 1 (N [k], ) (1 -2η)b 0 (2k -1) . Proof. Applying (3.6) to ψ(x 1 , x 2 , x 3 ) = ϕ(x 2 , x
3 )e k, (x 1 ), we infer that

N [k], ϕ, ϕ (2k -1) (1 -η) Op w (b)ϕ, ϕ .
With the Gårding inequality, we get

Op w (b)ϕ, ϕ (b 0 -C ) ϕ 2 ,
and the conclusion follows by the min-max principle.

We immediately deduce the following proposition.

Proposition 3.6. We have the following descriptions of the low lying spectrum of N .

where φ is the distance to the bounded set { B(q) β 0 } for the Agmon metric ( (B(q) -β 0 ) + g, with g the standard metric. Proposition 3.10. Under Assumptions 1.1 and 1.2, we consider 0 < b 0 < β 0 < b 1 and there exist C > 0 and 0 > 0 such that for any eigenpair (λ, ψ) of L ,A with λ β 0 we have for ∈ (0, 0 ) and δ ∈ (0, 1 2 ):

ψ = χ 0 -2δ L ,A χ 1 (q)ψ + O( ∞ ) ψ ,
where χ 0 is a cutoff function compactly supported in the ball of center 0 and radius 1 and where χ 1 is a compactly supported smooth cutoff function being

1 in an open neighborhood of { B(q) β 0 }.
Let us now investigate the microlocalization of the eigenfunctions of N .

Proposition 3.11. Let χ be a smooth cutoff function being 0 on {b β 0 } and 1 on the set {b β 0 + ε}. If λ is an eigenvalue of N such that λ β 0 and if ψ is an associated eigenfunction, then we have

Op w (χ(x 2 , ξ 2 , x 3 )) ψ = O( ∞ ) ψ .
Proof. Due to Corollary 3.8, it is sufficient to prove the estimate for a function in the form ψ(x 1 , x 2 , x 3 ) = e k, (x 1 )ϕ(x 2 , x 3 ) where k lies in {1, . . . , K} and we have

N ψ = λψ, or equivalently N [k], ϕ = λϕ,
where we recall (3.7). Then, we write

N [k], Op w (χ) ϕ = λ Op w (χ) ϕ + N [k], , Op w (χ) ϕ
and it follows that

(3.8) N [k], Op w (χ) ϕ, Op w (χ) ϕ = λ Op w (χ) ϕ 2 + N [k], , Op w (χ) ϕ, Op w (χ) ϕ .
Rough pseudo-differential estimates imply that there exist C > 0, 0 > 0 such that for all ∈ (0, 0 ),

N [k], , Op w (χ) ϕ, Op w (χ) ϕ C 2 Op w χ ϕ 2 +C Op w χ ϕ 2 + C Op w (∂ 3 χ) ϕ, Op w (ξ 3 ) Op w (χ) ϕ . (3.9)
Combining (3.9) and (3.8), we get

(3.10) Op w (ξ 3 ) Op w (χ) ϕ C 1 2
Op w χ ϕ , where χ is a smooth cutoff function living on a slightly larger support than χ. By using (3.10), we can improve the commutator estimate

N [k], , Op w (χ) ϕ, Op w (χ) ϕ C 3 2
Op w χ ϕ 2 .

We infer that, there exist C > 0, 0 > 0 such that for ∈ (0, 0 ),

N [k], Op w (χ) ϕ, Op w (χ) ϕ β 0 Op w (χ) ϕ 2 + C 3 2
Op w χ ϕ 2 .

By using the semiclassical Gårding inequality and the support of χ, we get

N [k], Op w (χ) ϕ, Op w (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 .

The conclusion follows by a standard iteration argument.

The following proposition is concerned by the microlocalization with respect to ξ 3 . Proposition 3.12. Let χ 0 be a smooth cutoff function being 0 in a neighborhood of 0 and let δ ∈ 0, 1 2 . If λ is an eigenvalue of N such that λ β 0 and if ψ is an associated eigenfunction, then we have

Op w χ 0 -δ ξ 3 ψ = O( ∞ ) ψ .
Proof. We write again ψ(x 1 , x 2 , x 3 ) = e k, (x 1 )ϕ(x 2 , x 3 ) with k ∈ {1, . . . , K} and we have N [k], ϕ = λϕ. We use again the formula (3.8) with χ 0 -δ ξ 3 . We get the 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 have

Op w ( -δ ξ 3 ) 2 χ 2 0 -δ ξ 3 = Op w 1-δ ξ 2 3 χ 2 0 (ξ 3
) , so that, with the Gårding inequality,

Op w ( -δ ξ 3 ) 2 χ 2 0 -δ ξ 3 ϕ, ϕ (1 -C 1-δ ) ϕ 2 .
We infer

( 2δ (1 -Ch 1-δ ) -β 0 ) Op w χ 0 -δ ξ 3 ϕ 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 the following in the same way. Proposition 3.13. Let χ 1 be a smooth cutoff function being 0 in a neighborhood of 0 and let δ ∈ 0, 1 2 . If λ is an eigenvalue of N such that λ β 0 and if ψ is an associated eigenfunction, then we have

Op w χ 1 -δ (x 1 , ξ 1 ) ψ = O( ∞ ) ψ .
Proposition 3.14. The spectra of L ,A and N below β 0 coincide modulo O( ∞ ).

Proof. We refer to [25, Section 4.3] which contains similar arguments.

This proposition provides the point (a) in Corollary 2.4. With Proposition 3.6, we deduce the point (b).

Second Birkhoff normal form

4.1. Birkhoff analysis of the first level. This section is devoted to the proofs of Theorems 2.8 and 2.11.

The goal now is to normalize a -pseudo-differential operator N [1] on R 2

whose Weyl symbol has the form

N [1] = ξ 2 3 + b(x 2 , ξ 2 , x 3 ) + r (x 2 , ξ 2 , x 3 , ξ 3 )
, where r is a classical symbol with the following asymptotic expansion:

r = r 0 + r 1 + 2 r 2 + • • •
(in the symbol class topology), where each r has a formal expansion in ξ 3 of the form

(4.1) r (x 2 , ξ 2 , x 3 , ξ 3 ) ∼ 2 +β 3 c ,β (x 2 , ξ 2 , x 3 )ξ β 3 .
The leading terms of N [1] are:

(4.2) N [1] = ξ 2 3 + b(x 2 , ξ 2 , x 3 )+c 1,1 (x 2 , ξ 2 , x 3 ) ξ 3 +O( ξ 2
3 )+O(ξ 3 3 )+O( 2 ).

4.1.1. First normalization of the symbol. We consider the following local change of variables φ(x 2 , ξ 2 , x 3 , ξ 3 ) = (x 2 , ξ2 , x3 , ξ3 ):

(4.3)            x2 := x 2 + ξ 3 ∂ 2 s(x 2 , ξ 2 ) , ξ2 := ξ 2 + ξ 3 ∂ 1 s(x 2 , ξ 2 ) , x3 := x 3 -s(x 2 , ξ 2 ) , ξ3 := ξ 3 .
It is easy to check that the differential of φ is invertible as soon as ξ 3 is small enough. Moreover, we have

φ * ω 0 -ω 0 = O(|ξ 3 |).
By the Darboux-Weinstein theorem (see for instance [START_REF] Raymond | Geometry and Spectrum in 2D Magnetic Wells[END_REF]Lemma 2.4]), there exists a local diffeomorphism ψ such that (4.4)

ψ = Id + O(ξ 2
3 ) and ψ * φ * ω 0 = ω 0 . Using the improved Egorov theorem, one can find a unitary Fourier Integral Operator V such that the Weyl symbol of V * N [1] V is N := N [1] • φ • ψ + O( 2 ). From (4.4), and (4.3), we see that r := r • φ • ψ is still of the 4.1.3. Normalizing the remainder. The next step is to get rid of the term č1,1 (x 2 , ξ 2 , x 3 ) ξ 3 . Let a(x 2 , ξ 2 , x 3 ) := -1 2

x 3 0 č1,1 (x 2 , ξ 2 , t)dt .
Since č1,1 is compactly supported, a is bounded, and one can form the unitary pseudo-differential operator exp(iA), A = Op w (a). We have

exp(-iA) Op w Ň exp(iA) = Op w Ň + exp(-iA)[Op w Ň , exp(iA)]. The symbol of [exp(-iA) Op w Ň , exp(iA)] is i e -ia {N, e ia } + O( 2 ) = { Ň , a} + O( 2 ) = { Ň0 , a} + O( 2 ),
where Ň0 is the principal symbol of Ň , which satisfies:

Ň0 = ξ 2 3 + O(ξ 3 3 ). Therefore { Ň , a} = {ξ 2 3 , a} + O(ξ 2 3 ). Since {ξ 2 3 , a} = 2ξ 3 ∂a ∂x 3 = -ξ 3 č1,1 ,
we get exp(-iA) Op w Ň exp(iA) = Op w ( Ň -ξ 3 č1,1 + O( ξ 2 3 ) + O( 2)), which shows that we can remove the coefficient of ξ 3 . The new operator given by the conjugation formula N [1] = exp(-iA) Op w Ň exp(iA) has a symbol of the form (4.8)

N [1] = ν 2 (x 2 , ξ 2 ) ξ 2 3 + x 2 3 + b(x 2 , ξ 2 , s(x 2 , ξ 2 )) + r , where r = O( x 3 3 ) + O( ξ 2 3 ) + O(ξ 3 3 ) + O( 2 )
. This proves Theorem 2.8. 4.1.4. The second Birkhoff normal form. We now want to perform a Birkhoff normal form for N [1], relative to the "second harmonic oscillator"

ν 2 (x 2 , ξ 2 ) ξ 2 3 + x 2 3 .
Using Notation 2.10, we introduce the new semiclassical parameter h = 1 2 , and use the relation

Op w (N [1], ) = Op w h (N [1],
h ). Thus, let ξj := -1/2 ξ j . The new symbol N [START_REF] Avron | Schrödinger operators with magnetic fields. I. General interactions[END_REF], h has the form:

N [1], h (x 2 , ξ2 , x 3 , ξ3 ) = h 2 ν 2 (x 2 , h ξ2 )( ξ2 3 + x 2 3 ) + b(x 2 , h ξ2 , s(x 2 , h ξ2 )) + h -2 r h 2 (x 2 , h ξ2 , x 3 , h ξ3 ) .
We introduce momentarily a new parameter µ and define

N [1], h (x 2 , ξ2 , x 3 , ξ3 ; µ) := ν 2 (x 2 , µ ξ2 )( ξ2 3 + x 2 3 ) + b(x 2 , µ ξ2 , s(x 2 , µ ξ2 )) + h -2 r h 2 (x 2 , µ ξ2 , x 3 , h ξ3 ). Notice that N [1], h (x 2 , ξ2 , x 3 , ξ3 ; h) = h -2 N [1],
h (x 2 , ξ2 , x 3 , ξ3 ). We define now a space of functions suitable for the Birkhoff normal form in (x 3 , ξ3 , h). Let us now use the notation of the Appendix introduced in (A.4) in the case when the family of smooth linear maps R 2 → R 2 is given by

ϕ µ,R 2 (x 2 , ξ2 ) = (x 2 , µ ξ2 ) . Let F := C(1) R 2 ,
where the index R 2 means that we consider symbols on R 2 . More explicitly, we have

F = {d s. t. ∃c ∈ S(1; [0, 1]×(0, 1]) R 2 : d(x 2 , ξ2 ; µ, h) = c(ϕ µ,R 2 (x 2 , ξ2 ); µ, h)} .
Then we define

E := F [[x 3 , ξ3 , h]] ,
endowed with the full Poisson bracket

E × E (f, g) → {f, g} = j=2,3 ∂f ∂ ξj ∂g ∂x j - ∂g ∂ ξj ∂f ∂x j ∈ E ,
and the corresponding Moyal bracket [f, g]. We remark that the formal Taylor series of the symbol N

[1],

h (x 2 , ξ2 , x 3 , ξ3 ; µ) with respect to (x 3 , ξ3 , h) belongs to E . We may apply the semiclassical Birkhoff normal form relative to the main term ν 2 (x 2 , µ ξ2 )( ξ2 3 + x 2 3 ) exactly as in Section 3.2.1 (and also [START_REF] Raymond | Geometry and Spectrum in 2D Magnetic Wells[END_REF]Proposition 2.7]), where we use the fact that the function

(x 2 , ξ2 , x 3 , ξ3 ; µ, h) → (ν 2 (x 2 , µ ξ2 )) -1
belongs to E because ν 2 > C > 0 uniformly with respect to µ. Let us consider γ ∈ E the formal Taylor expansion of h -2 r h 2 (x 2 , µ ξ2 , x 3 , h ξ3 ) with respect to (x 3 , ξ3 , h). The series γ is of valuation 3 and we obtain two formal series κ, τ ∈ E of valuation at least 3 such that

[κ, x 2 3 + ξ2 3 ] = 0 and e ih -1 adτ (ν 2 (x 2 , µ ξ2 )( ξ2 3 + x 2 3 ) + γ) = ν 2 (x 2 , µ ξ2 )( ξ2 3 + x 2
3 ) + κ. The coefficients of τ are in S(1) and one can find a smooth function τ h ∈ S(1) with compact support with respect to (x 3 , ξ3 , h) and whose Taylor series in (x 3 , ξ3 , h) is τ . By the Borel summation, τ h will actually lie in S(m ) with m (x 2 , ξ2 , x 3 , ξ3 ) = (x 3 , ξ3 ) -k for any k > 0, uniformly for small h > 0 and µ ∈ [0, 1]. Notice that N [START_REF] Avron | Schrödinger operators with magnetic fields. I. General interactions[END_REF], h ∈ C(m) with m = (x 3 , ξ3 ) 2 1, and Lemma 4.1 and Proposition 4.2 on the one hand and Propositions 3.11 and 3.12 on the other hand are enough to deduce from Theorem 2.8 the point (a) in Corollary 2.9. The point (b) easily follows from Corollary 2.4. 4.2.2. From N [START_REF] Avron | Schrödinger operators with magnetic fields. I. General interactions[END_REF], h to M h . Let us now prove the point (a) in Corollary 2.13. We get the following rough estimate of the number of eigenvalues. Lemma 4.3. We have

(4.9) N N [1], h , β 0 h 2 = N M h , β 0 h 2 = O(h -4 ) , (4.10) N M h , β 0 h 2 = O(h -4 ) .
Proof. First, we notice that N [START_REF] Avron | Schrödinger operators with magnetic fields. I. General interactions[END_REF], h and M h are unitarily equivalent so that (4.9) holds. Then, given η > 0 and h small enough and up to shrinking the support of g and by using the Calderon-Vaillancourt theorem (as in the proof of Lemma 3.4), M h M h in the sense of quadratic forms, with

M h = Op w h h 2 b(x 2 , h ξ2 , s(x 2 , h ξ2 )) + h 2 J h Op w h ν 2 (x 2 , h ξ2 ) -η .
Since ν 2 c > 0, we get

Op w h h 2 b(x 2 , h ξ2 , s(x 2 , h ξ2 )) + h 2 J h Op w h ν 2 (x 2 , h ξ2 ) -η Op w h h 2 b(x 2 , h ξ2 , s(x 2 , h ξ2 )) + c 2 h 2 J h .
We deduce the upper bound (4.10) by separation of variables and the minmax principle.

The following proposition deals with the microlocal properties of the eigenfunctions of N

[1], h . Proposition 4.4. Let η ∈ (0, 1), δ ∈ 0, η 2 , C > 0. Let χ be a smooth cutoff function being 0 on {b(x 2 , ξ 2 , s(x 2 , ξ 2 ))
β 0 } and being 1 on the set {b(x 2 , ξ 2 , s(x 2 , ξ 2 )) β 0 + ε}, with ε > 0. Let also χ 1 be a smooth cutoff function on R 2 , being 0 in a neighborhood of 0.

If λ is an eigenvalue of N [START_REF] Avron | Schrödinger operators with magnetic fields. I. General interactions[END_REF], h such that λ β 0 h 2 and if ψ is an associated eigenfunction, we have

(4.11) Op w h χ(x 2 , h ξ2 ) ψ = O(h ∞ ) ψ and if λ is an eigenvalue of N [1], h
such that λ b 0 h 2 + Ch 2+η and if ψ is an associated eigenfunction, we have

(4.12) Op w h χ 1 (h -δ (x 3 , ξ3 )) ψ = O(h ∞ ) ψ .
Proof. The estimate (4.11) is a consequence of Proposition 4.2. Then, let us write the symbol of N

[1], h : N [1], h = h 2 ν 2 (x 2 , h ξ2 ) ξ2 3 + x 2 3 +h 2 b(x 2 , h ξ2 , s(x 2 , h ξ2 ))+R h 2 (x 2 , h ξ2 , x 3 , h ξ3 ).
We write

N [1], h Op w h χ 1 (h -δ (x 3 , ξ3 )) ψ, Op w h χ 1 (h -δ (x 3 , ξ3 )) = λ Op w h χ 1 (h -δ (x 3 , ξ3 )) ψ 2 + N [1], h , Op w h χ 1 (h -δ (x 3 , ξ3 )) , Op w h χ 1 (h -δ (x 3 , ξ3 )) ψ .
We get

N [1], h , Op w h χ 1 (h -δ (x 3 , ξ3 )) , Op w h χ 1 (h -δ (x 3 , ξ3 )) ψ Ch 3 Op w h χ 1 (h -δ (x 3 , ξ3 )) ψ 2 ,
where we have used (4.11). Then, we use that

b(x 2 , h ξ2 , s(x 2 , h ξ2 )) b 0 , ν 2 (x 2 , h ξ2 ) c 0 > 0, λ b 0 h 2 + Ch 2+η ,
and the Gårding inequality to deduce

h 2 Ch 2δ -Ch η Op w h χ 1 (h -δ (x 3 , ξ3 )) ψ 2 Ch 3 Op w h χ 1 (h -δ (x 3 , ξ3 )) ψ 2 .
The desired estimate follows by an iteration argument.

In the same way we can deal with M h . Proposition 4.5. Let η ∈ (0, 1), δ ∈ 0, η 2 , C > 0. Let χ be a smooth cutoff function being 0 on {b(x 2 , ξ 2 , s(x 2 , ξ 2 )) β 0 } and being 1 on the set {b(x 2 , ξ 2 , s(x 2 , ξ 2 )) β 0 + ε}, with ε > 0. If λ is an eigenvalue of M h such that λ β 0 h 2 and if ψ is an associated eigenfunction, we have

(4.13) Op w h χ(x 2 , h ξ2 ) ψ = O(h ∞ ) ψ
and if λ is an eigenvalue of M h such that λ b 0 h 2 + Ch 2+η and if ψ is an associated eigenfunction, we have

(4.14) Op w h χ 1 (h -δ (x 3 , ξ3 )) ψ = O(h ∞ ) ψ .
Proof. In order to get (4.13), it is enough to go back to the representation with semiclassical , that is M h = M . Indeed the microlocal estimate follows by the same arguments as in Propositions 3.11 and 3.12. Then, (4.14) follows as in Proposition 4.4.

Then, there exists a linear symplectic change of variables that diagonalizes the Hessian, so that, if L is the associated unitary transform,

L * M [1], L = Op w M [1], , with
M [1], = b 0 + 2 θ(x 2 2 + ξ 2 2 ) + 3 2 ν 2 (0, 0) + ĉx 2 3 2 + dξ 2 3 2 + O(( 1 2 , z 2 ) 3 ), where θ = det Hess (0,0) b(x 2 , ξ 2 , s(x 2 , ξ 2 )) . Since (∂ x 3 b)(x 2 , ξ 2 , s(x 2 , ξ 2 )) = 0 and (0, 0) is a critical point of s, we notice that ∂ 2 x 2 x 3 b(0, 0, 0) = ∂ 2 ξ 2 x 3 b(0, 0, 0) = 0. Thus det Hess (0,0,0) b(0, 0, 0) = θ 2 ∂ 2
x 3 b(0, 0, 0). Using that b is identified with b • χ (see Remarks 2.2 and 3.1), this provides the expression given in (1.14).

Note that ĉ2 + d2 = (∇ x 2 ,ξ 2 ν 2 )(0, 0) 2 since the symplectic transform is in fact a rotation. Moreover we have )) . Here p is the Weyl symbol of P , and the canonical transformation κ is the time-1 Hamiltonian flow associated with principal symbol of Q.

Then the operator e

i h Q P e -i h Q is a pseudo-differential operator in Op w h (S(1)), and
e i h Q P e -i h Q -Op w h (p • κ) ∈ h Op w h (S( 1
From this classical version of Egorov's theorem, one can deduce the following refinement that is useful when p does not belong to S(1) (as it is the case in this paper). In order to quantize the formal Birkhoff procedure of Section 4.1.4, one needs to consider symbols in a class C stable under the Moyal product. For that purpose we first define the families of symbols S(m; [0, 1] × (0, 1]), that is of smooth functions a : R 2d × [0, 1] × (0, 1] → C such that, for any α ∈ N 2d , there exists C α such that, ∀(z; µ, h) ∈ R 2d × [0, 1] × (0, 1],

|∂ α z a(z; µ, h)| C α m(z) and where m is an order function on R 2d . The pair (µ, h) is considered as a parameter.

Then, let (ϕ µ ) µ∈[0,1] be a smooth family of linear maps R 2d → R 2d and define the following families of symbols on R 2d by (A.4) C(m) = a ∈ S(m; [0, 1] × (0, 1]); a(z; µ, h) = ã(ϕ µ (z); µ, h) with ã ∈ S(m; [0, 1] × (0, 1]) . Let κ be the canonical transformation associated with Q. Then, since m 1, we have p • κ ∈ C(m); indeed, if we write the Hamiltonian flow of Q in terms of the variable z = ϕ µ (z), we see from the linearity of ϕ µ that the components of the transformed vector field belong to C(m ). Therefore ϕ µ • κ is of the form κµ • ϕ µ , for some diffeomorphism κµ depending smoothly on µ.

Therefore, both terms in (A.2) belong to Op w h (C(1)). Applying this argument inductively in (A.3), we may write, for any k > 0, We conclude by letting P = Op w h (p • κ) + P . We will also need to examine how the Egorov theorem behaves with respect to taking formal power series of symbols. For this, it is convenient to introduce a filtration of S(m). Let P = Op w h p and Q = Op w h q be h-pseudo-differential operators on R d , with p ∈ S(m) and q ∈ S(m ), where m is an order function such that m and mm are bounded. Assume that: (A.5) i h ad q (O j ) ⊂ O j+1 ; ∀j 0.

Then for any k 0, the Weyl symbol of the pseudo-differential operator e i h Q P e -i h Qk j=0 1 j! ( i h ad Q ) j P belongs to Op w h (O k+1 ). In other words, the series of exp( i h ad Q )P converges to e i h Q P e -i h Q for the filtration (O j ) j∈N .

Proof. By the Taylor formula, we can write

e i h Q P e -i h Q = k j=0 1 j! (ad ih -1 Q ) j P + 1 k! (ad ih -1 Q ) k+1 1 0 (1 -t) k e it h Q P e -it h Q dt.
By Theorem A.2, we see that the integral belongs to Op w h (S(m)) = Op w h (O 0 ). Therefore, by Assumption (A.5), the remainder in the Taylor formula lies in Op w h (O k+1 ).

Figure 1 .

 1 Figure 1. The dashed line represents the integral curve of the confining magnetic field B = curlA through q 0 = (0.5, 0.6, 0.7) for B(x, y, z) = y 2 , z 2 ,

F c h a

  n g e o f s e m ic la s s ic a l p a r a m e t e

Theorem A. 1 ([ 31 ,

 131 Theorem 11.1, Remark (ii)]). Let P and Q be h-pseudodifferential operators on R d , with P ∈ Op w h (S(1)) and Q ∈ Op w h (S(1)).

Theorem A. 2 .

 2 Let P and Q be h-pseudo-differential operators on R d , with P ∈ Op w h (S(m)) and Q ∈ Op w h (S(m )), where m and m are order functions such that: (A.1) m = O(1); mm = O(1).

Theorem A. 3 .

 3 Let P and Q be h-pseudo-differential operators on R d , withP ∈ Op w h (C(m)) and Q ∈ Op w h (C(m )), where m and m are order functions such that:m 1; m = O(1); mm = O(1).Then e i h Q P e -i h Q = P + R, where P ∈ Op w h (C(m)), R ∈ h ∞ Op w h (S(1)), and with P -Op w h (p • κ) ∈ h Op w h (C(1)). Proof. Since ϕ µ is linear, one can see (using for instance [31, Theorem 4.17]) that C is stable under the formal Moyal product, i.e. for all order functions m 1 and m 2 , we have (C(m 1 )) (C(m 2 )) ⊂ C(m 1 m 2 ) + h ∞ S(1).

  e -i h Q -Op w h (p • κ) -(h P1 + h 2 P2 + • • • + h k Pk ) ∈ h k+1 Op w h (S(1)) , with Pj ∈ Op w h (C(1)). By a Borel summation in h, parametrized by z = ϕ µ (z), we can find a symbol P ∈ Op w h (C(1)) such that we have the asymptotic expansion in Op w h (S(1)):P ∼ h P1 + h 2 P2 + • • •

Theorem A. 4 .

 4 Let m be an order function on R 2d , and let (O j ) j∈N be a filtration of S(m), i.e.:O 0 = S(m), O j+1 ⊂ O j .

  Then the operator e i h Q P e -i h Q is a pseudo-differential whose symbol is in S(m), ande i h Q P e -i h Q -Op w h (p • κ) ∈ h Op w h (S(1)). Proof. The proof is based on the following observation. In order to compare Op w h (p • κ t ) and e it h Q P e -it h Q , we consider the derivative: moreover, if we denote by q 0 the principal symbol of Q, we have Op w h (S(1)), uniformly for s ∈ [0, t]. Applying Theorem A.1 to the integrand, we see that e it h Q P e -it h Q -Op w h (p • κ t ) ∈ h Op w h (S(1)).

	d dτ	e	iτ h Q Op w h (p • κ t-τ )e -iτ h Q
				= e	iτ h Q	i h	[Q, Op w
				d dτ	Op w
	which implies that this term is also in Op w h (S(1)). By symbolic calculus, we
	see that
	(A.2)		i h	[Q, Op w h (p • κ t-τ )] +	d dτ	Op w

h (p • κ t-τ )] + d dτ Op w h (p • κ t-τ ) e -iτ h Q . From Hypothesis (A.1), the term [Q, Op w h (p • κ t-τ )] belongs to Op w h (S(1)); h (p • κ t-τ ) = -Op w h ({q 0 , p • κ t-τ }), h (p • κ t-τ ) ∈ h Op w h (S(1)) ,

uniformly for t, τ in compact sets. It follows by integration from 0 to t that

(A.3) e it h Q P e -it h Q = Op w h (p • κ t ) + h t 0 e is h Q P 1 (s)e -is h Q ds,

for some P 1 (s) ∈

. This description is reminiscent of the results à la Bohr-Sommerfeld of[START_REF] Helffer | Puits de potentiel généralisés et asymptotique semi-classique[END_REF] and[18, Appendix B] (see also[START_REF] Helffer | Accurate semiclassical spectral asymptotics for a twodimensional magnetic Schrödinger operator[END_REF] Remark 1.4]) obtained in the case of one dimensional semiclassical operators.

Acknowledgments. The authors would like to thank Yves Colin de Verdière for stimulating discussions. This work was partially supported by the ANR (Agence Nationale de la Recherche), project Nosevol n o ANR-11-BS01-0019 and by the Centre Henri Lebesgue (program "Investissements d'avenir" -n o ANR-11-LABX-0020-01). Y. K. was partially supported by the Russian Foundation of Basic Research, project 13-01-91052-NCNI-a. During the completion of this work, B. H. was Simons foundation visiting fellow at the Isaac Newton Institute in Cambridge.

(a) There exist 0 > 0 and K ∈ N such that, for ∈ (0, 0 ), the spectrum of N lying below β 0 is contained in the union K k=1 sp N [k], .

(b) If c ∈ (0, min(3b 0 , β 0 )), then there exists 0 > 0 such that for all ∈ (0, 0 ) the eigenvalues of N lying below c coincide with the eigenvalues of N [1], below c .

We deduce the following proposition.

Corollary 3.7. Under Assumption 1.10, we have

Proof. To get the first estimate, we use the Lieb-Thirring inequalities (which provide an upper bound of the number of eigenvalues in dimension three) and the diamagnetic inequality (see [START_REF] Raymond | Geometry and Spectrum in 2D Magnetic Wells[END_REF] and Proposition 1.8). To get the second estimate, we use the first point in Proposition 3.6. Moreover, given η ∈ (0, 1), by using ∈ (0, 1) we infer

η) Op w ξ 2 3 + b(x 2 , ξ 2 , x 3 ) ψ, ψ . Note that the last inequality is very rough. By the min-max principle, we deduce that

Then, we conclude by using the Weyl asymptotics and our confinement assumption:

Since N commutes with I , we also deduce the following corollary.

Corollary 3.8. For any eigenvalue λ of N such that λ β 0 we may consider an orthonormal eigenbasis of the space ker N -λ formed with functions in the form e k, (x 1 )ϕ (x 2 , x 3 ) with k ∈ {1, . . . K}. Moreover we have

) and each eigenfunction associated with λ β 0 is a linear combination of at most O( -2 ) such tensor products.

Microlocalization estimates.

The following proposition follows from the same lines as in dimension two (see [START_REF] Helffer | Semiclassical analysis for the ground state energy of a Schrödinger operator with magnetic wells[END_REF]Theorem 2.1]).

Proposition 3.9. Under Assumptions 1.1 and 1.2, for any > 0, there exist C( ) > 0 and h 0 ( ) > 0 such that for any eigenpair (λ, ψ) of L ,A with λ β 0 we have for ∈ (0, h 0 ( )):

form (4.1), with modified coefficients c ,β . Thus, using the new variables and a Taylor expansion in ξ 3 , we get

and thus

for some smooth function g(x 2 , ξ2 , x3 ).

Therefore N has the following form:

4.1.2.

Where the second harmonic oscillator appears. We now drop all the hats off the variables. We use a Taylor expansion with respect to x 3 , which, in view of (2.6), yields:

3 ).

We let:

We introduce the change of coordinates (x 2 , x3 , ξ2 , ξ3 ) = C(x 2 , x 3 , ξ 2 , ξ 3 ) defined by: (4.7)

As before, we can make this local diffeomorphism symplectic by the Darboux-Weinstein theorem, which modifies (4.7) by O(ξ 2 3 ). In the new variables (which we call (x 2 , x 3 , ξ 2 , ξ 3 ) again), the symbol Ň has the form:

for some smooth function č1,1 (x 2 , ξ 2 , x 3 ).

that mm = O(1). Then, we can apply Theorem A.3 with the family of endomorphisms of R 4 defined

Thus, the new operator

h and we may find a smooth function g (x 2 , µ ξ2 , Z, h) such that its Taylor series with respect to Z, h is

We may now replace µ by h, which achieves the proof of Theorem 2.11. 4.2. Spectral reduction to the second normal form. This section is devoted to the proof of Corollary 2.13.

4.2.1.

From N [1], to N [1], . In this section, we prove Corollary 2.9.

Lemma 4.1. We have

Proof. The first estimate comes from Proposition 3.6 and Corollary 3.7. The second estimate can be obtained by the same method as in the proof of Corollary 3.7.

Let us now summarize the microlocalization properties of the eigenfunctions of N [1], in the following proposition.

Proposition 4.2. Let χ 0 be a smooth cutoff function on R being 0 in a neighborhood of 0 and let δ ∈ 0, 1 2 . Let χ be a smooth cutoff function being 0 on the bounded set {x 2 3 + b(x 2 , ξ 2 , s(x 2 , ξ 2 )) β 0 } and 1 on the set

such that λ β 0 and if ψ is an associated eigenfunction, then we have

Proof. The proof follows exactly the same lines as for Propositions 3.11 and 3.12. Let us consider the Hilbertian decomposition

h , where the symbol

h ξ2 ).

There exists h 0 > 0 such that for all k 1 and h ∈ (0, h 0 ),

Since each eigenfunction of M

[k], h associated with an eigenvalue less than β 0 h 2 provides an eigenfunction of M h , we infer that the eigenfunctions of M

[k], h are uniformly microlocalized in a (x 2 , ξ 2 )-neighborhood of (0, 0) as small as we want. Therefore, on the range of

h ), we have

and, with the Gårding inequality in the -quantization, we get

h ψ, ψ Op w h h 2 b 0 + (2k -1)h 3 (ν 2 (0, 0) -ε) -Ch 4 ψ, ψ . This implies the point (b) in Corollary 2.13.

Third Birkhoff normal form

5.1. Birkhoff analysis of the first level. In this section we prove Theorem 2.15.

We consider M [1], = Op w M [1], , with

By using a Taylor expansion, we get, (

where c = ∂ x 2 ν 2 (0, 0) and d = ∂ ξ 2 ν 2 (0, 0), and we have identified the Hessian with its quadratic form in (x 2 , ξ 2 ).

Thus, there exists a unitary transform Û 1 2 , which is in fact an -Fourier Integral Operator whose phase admits a Taylor expansion in powers of

, where 2 is + m + n and endowed with the Moyal product. Let F T be the full Taylor series of F . We find a formal series τ (x 2 , ξ 2 , 1 2 ) with a valuation at least 3 such that e i -1 adτ F T = F T , where F T is a formal series of the form

] (and that can be also written as a formal series in Moyal power of |z 2 | 2 , say (k T ) ).

Let τ (x 2 , ξ 2 , µ) be a compactly supported function whose Taylor expansion at (0, 0, 0) is equal to τ (x 2 , ξ 2 , µ). By the Egorov theorem A.2, uniformly with respect to the parameter µ, we obtain that e -i -1 Op w (τ ) Op w (F µ 2 )e i -1 Op w (τ ) =: Op w ( Fµ ) is an -pseudo-differential operator depending smoothly on µ. Expanding Fµ in powers of µ in the S(1) topology, and letting µ = √ , we see that F√ = F + G , where

with k a smooth function with a support as small as desired w.r. Proof. The first estimate follows from Lemma 4.3 and the second one from a comparison with the harmonic oscillator in x 2 .

The last proposition concerns the microlocalization of the eigenfunctions.

Proposition 5.2. Let η ∈ (0, 1), δ ∈ 0, η 2 , C > 0. Let χ be a smooth cutoff function being 0 in a bounded neighborhood of (0, 0) and 1 outside a bounded neighborhood of (0, 0). If λ is an eigenvalue of M [1], or of F such that λ b 0 + C 1+η and if ψ is an associated eigenfunction, we have

Proof. The proof is similar with the one of Proposition 4.4.

Appendix A. Egorov theorems

We start with the classical result (see for instance [START_REF] Zworski | Semiclassical analysis[END_REF]Theorem 11.1] and [START_REF] Robert | Autour de l'approximation semi-classique[END_REF]Théorème IV.10]).