Marouen Kachroudi
email: marouen.kachroudi@fst.rnu.tn

Gayo Diallo
email: gayo.diallo@isped.u-bordeaux2.fr

Ben -Sadok

Yahia

Initiating Cross-Lingual Ontology Alignment with Information Retrieval Techniques

Keywords: MOTS-CLÉS : Alignement d'ontologies, Inter-linguisme, Techniques de recherche d Ontology Alignment, Cross-lingualism, Information Retrieval Techniques

L'alignement d'ontologies est une tâche qui a attiré une attention considérable ces dernières années. La majorité des travaux dans ce contexte ont porté sur le développement d'algorithmes d'appariement monolingues. Comme de plus en plus de ressources sont disponibles dans plus d'une langue naturelle, de nouveaux algorithmes sont nécessaires. Ceux ci sont censés être capables d'aligner des ontologies qui partagent plus d'une langue. Pour relever ce défi, nous proposons dans ce papier une nouvelle méthode d'alignement inter-langues qui opère suivant une stratégie d'alignement basée sur l'utilisation des ressources externes et les techniques de recherche d'information. Les résultats obtenus montrent l'efficacité de la nouvelle méthode.

ABSTRACT. Ontology alignment is a task that has attracted considerable attention in recent years. The majority of works in this context focused on the development of monolingual matching algorithms. As more and more resources become available in more than one natural language, novel algorithms are required. The latter should be able to align ontologies which share more than one language. To meet this challenge, we propose in this paper a new cross-lingual alignment method that operates in a strategy based on the use of external resources and information retrieval techniques. The obtained results show the effectiveness of the new proposed method.

Introduction

Ontologies, as specifications of conceptualisations [START_REF] Gruber | Towards Principles for the Design of Ontologies used Knowledge Sharing[END_REF], are recognised as a "basic component of the Semantic Web" [START_REF] Berners-Lee | The semantic Web[END_REF] and have been widely used in knowledge management [START_REF] Jurisica | Ontologies for Knowledge Management : An Information Systems Perspective[END_REF]. Indeed, actors of this new Web generation publish their data sources in their own respective languages, in order to make this information interoperable and accessible to members of other linguistic communities [START_REF] Berners-Lee | Designing the web for an open society[END_REF]. As a solution, the ontology alignment process aims to provide semantic interoperable bridges between heterogeneous and distributed information systems. Indeed, the informative volume reachable via the Semantic Web stresses needs of techniques guaranteeing the share, reuse and interaction of all resources [START_REF] Suchanek | The Hidden Web, XML and Semantic Web : A Scientific Data Management Perspective[END_REF]. The explicitation of the associated concepts related to a particular domain of interest resorts to ontologies, considered as the kernel of the Semantic Web [START_REF] Berners-Lee | The semantic Web[END_REF]. In addition, the open and dynamic resources of the future Web endow it with a heterogeneous aspect, which reflects at once the formats or the language varieties of its description. Consequently, ontologies that are labelled in diverse natural languages are increasingly evident. Given ontologies that are likely to be authored by different actors using different terminologies, structures and natural languages, ontology alignment has emerged as a way to achieve semantic interoperability. Recent researches in ontology alignment field has largely focused on dealing with ontologies labelled in the same natural language. Still a few researches have focused on alignment scenarios where the involved ontologies are labelled in different natural languages. However, current monolingual alignment techniques often rely on lexical comparisons made between resource labels, which limits their deployment to ontologies in the same natural language or at least in comparable natural languages. Given the existing monolingual alignment tools limitations , there is a pressing need for the development of alignment techniques that can deal with ontologies in different natural languages. One approach is direct cross-lingual ontology mapping. In this paper, direct cross-lingual ontology alignment refers to the process of establishing relationships among ontological resources from two or more independent ontologies where each ontology is labelled in a different natural language.

A well known approach (Dos [START_REF] Santos | An API for Multi-lingual Ontology Matching[END_REF] to achieve cross-lingual ontology alignment is to use translation techniques with the goal of converting a direct cross-lingual alignment problem into a monolingual alignment problem, which can then be solved by monolingual ontology alignment tools. Various techniques [START_REF] Clark | Using Contextual Representations to Efficiently Learn Context-Free Languages[END_REF] such as statistical machine translation and rule-based machine translation have been developed with the aim of improving the translations quality through word sense disambiguation [START_REF] Navigli | Word sense disambiguation : A survey[END_REF]. Otherwise, machine translation tools are intended to assign an accurate meaning to a phrase in a specific natural language while limiting possible ambiguity, which is not necessarily a requirement in direct crosslingual ontology alignment however. Therefore, to achieve direct cross-lingual ontology alignment, translations should lead to the correct alignments generation, but it is not of interest whether these translations are the most accurate localisations in the specific natural language. Consequently, translating the ontology labels in the context of direct cross-lingual ontology alignment is not solely concerned with finding translated equivalents in the target natural language, but also is finding translations that can lead to correct mappings. There can be various ways to express the same concept in many natural languages. A simple example of this is : Ph.D. candidate and doctoral student both describe someone who is pursuing an academic degree of Doctor in any research axis. Envision this in the context of direct cross-lingual ontology alignment, assuming the target ontology is labelled in English and the source ontology is labelled in a natural language other than English. For a considered source ontology concept, its English translation can be Ph.D. candidate or doctoral student. Which one is more appropriate in the given alignment scenario ? To solve this problem, we would ideally like to know which candidate translation will lead to a correct mapping given that an equivalent concept is also presented in the target ontology. This translation selection process differs from traditional word sense disambiguation, as word sense disambiguation is the association of a given word in a text or discourse with a definition or meaning (sense) which is distinguishable from other meanings potentially attributable to that word [START_REF] Ide | Introduction to the Special Issue on Word Sense Disambiguation : The State of the Art[END_REF].

The remainder of this paper is organised as follows. A state of the art scrutinizes on direct cross-lingual ontology alignment approaches and related backgrounds are presented in section 2. To tackle this challenge, CLONA (Cross Lingual ONtology Alignment) is proposed in section 3. An overview of the evaluation is presented in section 4. Finally, section 5 concludes this paper with some outlooks for future work.

Context and literature review

This section presents the related knowledge background and a review on current approaches proposed in cross-lingual ontology alignment. Ontologies have gained increasing interest for structured meaning modeling among the Semantic Web community [START_REF] Sure | Handbook on Ontologies[END_REF]. However, in decentralised environments such as the Semantic Web, the heterogeneity issue occurs when ontologies are created by different authors. To address the heterogeneity issue arising from ontologies on the Semantic Web, ontology alignment has become an important research field [START_REF] Euzenat | Ontology Matching[END_REF].

Ontology alignment is considered as an evaluation of the resemblance degrees or the differences detected on them [START_REF] Ehrig | Ontology alignment : bridging the semantic gap[END_REF]. Besides, the alignment process can be defined as follows : being given two ontologies O 1 and O 2 , an alignment between O 1 and O 2 is a set of correspondences, (i.e., a quadruplet) : < e 1 , e 2 , r, Conf n >, with e 1 ∈ O 1 and e 2 ∈ O 2 , r is a relation between two given entities e 1 and e 2 , while Conf n represents the confidence level in this relation [START_REF] Euzenat | Ontology Matching (Second Edition)[END_REF].

Ontologies are widely used in knowledge-based systems and the ontological applications crosses several disciplines, discussed next. In agriculture, the Food and Agriculture Organization (FAO) 1 provides reference standards for defining and structuring agricultural terminologies. Since all FAO official documents must be made available in five official languages including Arabic, Chinese, English, French and Spanish. For this purpose, a large amount of research has been carried out on large multilingual agricultural thesauri translations, their alignment methodologies and a requirements definition to improve the interoperability between these multilingual resources [START_REF] Caracciolo | Requirements for the Treatment of Multilinguality in Ontologies within FAO[END_REF]. In higher education domain, the Bologna declaration 2 has introduced an ontology-based framework for qualification recognition [START_REF] Kismihók | Six Scenarios of Exploiting an Ontology Based, Mobilized Learning Environment[END_REF] across the European Union (EU). In an effort to better reconciliate labour markets with employment opportunities, an ontology is used to support the degrees and qualifications recognition within the EU. In e-learning, educational ontologies are used to enhance learning experiences and to empower system platforms with high adaptivity. In finance, ontologies are used to model knowledge in the stock market domain and portfolio management. In medicine, ontologies are used to improve knowledge sharing and reuse, such as work presented by Fang et al. [START_REF] Fang | Leveraging Ontology-Based Traditional Chinese Medicine Knowledge System : Using Formal Concept Analysis[END_REF] that focuses on a traditional Chinese medicine ontology creation. A key observation from ontology-based applications such as those mentioned above is that the ontology development is closely associated with natural languages. Given the diversity of natural languages and the different conceptual models of ontology authors, the heterogeneity issue is ineluctable with ontologies that are built on different conceptualisation models and natural languages. Ontologies vast existence in several natural languages provides an impetus to discover solutions to support semantic interoperability.

The recent evaluation campaign 3 for alignment systems was marked by the release of a new multilingual alignment systems which we are going to review the most relevant. AgreementMakerLight (AML) [START_REF] Faria | AML results for OAEI 2015[END_REF] is presented as an automated ontology matching system that heavily relies on element-level matching and on the use of some external resources as background knowledge. The AML workflow comprises nine steps, i.e., ontology loading and profiling, translation, baseline matching, background knowledge matching, word and string matching, structural matching, property matching, selection, and finally a repair module. AML employs the OWL API to read the input ontologies and extract the necessary information to populate its own data structures. The translation is achieved by querying Microsoft Translator for the full name in order to help provide the described context. AML employs an efficient, and generally precise weighted string-equivalence algorithm through the Lexical Matcher, to obtain a baseline class alignment between the input ontologies. AML has available four background knowledge sources used as mediators between the input ontologies. AML also employs the Multi-Word Matcher, which matches closely related multi-word names that have matching words and/or words with common WordNet synonyms or close hypernyms. For small and medium-sized ontologies, AML also employs a structural matching algorithm, called Neighbor Similarity Matcher. This algorithm computes similarity between two classes by propagating the similarity of their matched ancestors and descendants. AML employs a greedy selection algorithm, i.e., the Ranked Selector, to reduce the alignment cardinality. Depending on the size of the input ontologies, one of three selection strategies is used : strict, permissive, or 2. http ://eur-lex.europa.eu/legal-content/EN/TXT/ ?uri=uriserv :c11088 3. http ://oaei.ontologymatching.org/2015/ hybrid. AML employs a heuristic repair algorithm to ensure that the final alignment is coherent.

XMAP [START_REF] Djeddi | XMap : results for OAEI 2015[END_REF] is a highly scalable ontology alignment system, which is able to deal with voluminous ontologies in efficient computation time. It is a fast and effective highly precised system able to perform large ontologies alignment. A semantic similarity measure is defined by XMAP using UMLS and WordNet to provide a synonymy degree between two entities from different ontologies, by exploring both their lexical and structural contexts. XMAP relies on the Microsoft Translate API to translate ontologies into many languages. The terminological layer is in charge of carrying out the similarity computing process between the entity names, combining linguistic similarity with the semantic elements according to the entities context. The structural layer performs two key tasks related to the ontologies structure. The first is the similarity computation between the concepts taking into account the taxonomic hierarchy, the second is the the similarity computation using the internal concepts structure information, i.e., their properties, types and cardinality restrictions. The alignment layer is the final layer and its aim is to provide the final similarity matrix between the concepts taking into account the properties number influence and the similarity value that properties supply to the final similarity score.

LYAM++ alignment system [START_REF] Tigrine | LYAM++ results for OAEI 2015[END_REF] take as an input a source ontology, given in a natural language and a second target ontology. The overall processes consists of three main components : a terminological multilingual matcher, a mapping selection module and, finally, a structural matcher. One of the original contributions of LYAM++ is the choice orchestrating of the pre-described components.The multilingual terminological matching module, acts on the one hand as a preprocessing component and, on the other hand as a light-weight terminological matcher between cross-lingual labels. The Mapping selection is a module that transforms the initial many mappings to a 1 : 1 alignment based on the principle of iteratively retaining the concept pairs with maximal similarity value. Finally, the structural matcher component filters the trustworthy pairs of the aligned concepts by examining the similarity values produced for their parents and their descendants in the ontology hierarchies.

LOGMAP [START_REF] Jiménez-Ruiz | LogMap family results for OAEI[END_REF] is also a highly scalable ontology matching system that implements the consistency and locality principles. LOGMAP also supports real-time user interaction during the matching process, which is essential for use cases requiring very accurate mappings. LOGMAP is one of the few ontology matching system that can efficiently match semantically rich ontologies containing tens or even hundreds of thousands of classes. LOGMAP incorporates sophisticated reasoning and repair techniques to minimise the logical inconsistencies number, and provides support for user intervention during the matching process. LOGMAP extended a multilingual module to use both Google translate and Microsoft translator. Additionally, in order to split Chinese words, LOGMAP relies on the ICTCLAS4 library developed by the Institute of Computing Technology of the Chinese Academy of Sciences.

To sum up, a key observation from the literature review is that, using machine as a means to bridge the gap between natural languages is a feasible approach to achieve cross-lingual ontology alignment as shown in the literature. In addition, the alignment process can be enhanced by the integration of powerful treatments inherited from other related specialties, such as Information Retrieval (IR) [START_REF] Diallo | An effective method of large scale ontology matching[END_REF]. This paper presents the CLONA system and demonstrates its ability to align ontologies expressed in different natural languages using an external linguistic resource. CLONA as a few methods [START_REF] Groß | GOMMA Results for OAEI[END_REF][START_REF] Kachroudi | DAMO -Direct Alignment for Multilingual Ontologies[END_REF] meets challenges strictly bound at the linguistic level in the context of multilingual ontology alignment. The driven idea of our new method is to cross the natural language barrier. CLONA presents a novel view to improve the alignment accuracy that draws on the information retrieval techniques.

CLONA method

The CLONA workflow comprises six different complementary steps, as flagged by Figure 1 : (i) Parsing and Pretreatment, (ii) Translation, (iii) Indexation, (iv) Candidate Mappings Identification and (vi) Alignment Generation. CLONA is an alignment system which aims through specific techniques to identify the correspondences between two ontologies defined in two different natural languages. Indeed, it starts with a pretreatment stage to model the input ontologies in a well defined format for the rest of the process. The second phase is that of translation into a chosen pivot language achieved by the Microsoft Bing 5 translator. Thereafter, our method continues with an indexing phase over the considered ontologies. Then these indexes are queried to supply the candidate mappings list as a primary result. Before generating the alignment file, CLONA uses a filtering module for recovery and repair. Parsing and Pretreatment : This phase is crucial for ontologies pretreatment. It is performed using the OWL API. Indeed, it transforms the considered ontologies represented initially as two OWL files in an adequate format for the rest of the treatments. In our case, the goal is to substitute all the existing information in both OWL files so that each entity is represented by all its properties. Indeed, the parsing module begins by loading two ontologies to align described in OWL. This module allows to extract the ontological entities initially represented by a set of primitives. In other words, at the parsing stage, we seek primarily to transform an OWL ontology in a well defined structure that preserves and highlight all the information contained in this ontology. Furthermore, the resulting informative format, has a considerable impact on the similarity computation results thereafter. Thus, we get couples formed by the entity name and its associated label. In the next step we add an element to such couples to process these entities regardless of their native language.

Translation : The main goal of our approach is to solve the heterogeneity problem mainly due to multilingualism. This challenge brings us to choose between two alternatives, either we consider the translation path to one of the languages according to the two input ontologies, or we consider the translation path to a chosen pivot language. At this stage, we must have a foreseeable vision for the rest of our approach. Specifically, at the semantic alignment stage we use an external resource such as WordNet 6 . The latter is a lexical database for the English language. Therefore, the choice is governed by the use of WordNet, and we will prepare a translation of the two ontologies to the pivot language, which is English. To perform the translation phase we chose Bing Microsoft 7 tool.

Indexation : Whether on the Internet, with many search engine or local access, we need to find documents or simply sites. Such research is valuable to browse each file and the analysis thereafter. However, the full itinerary of all documents with the terms of a given query is expensive since there are too many documents and prohibitive response times. To enable faster searching, the idea is to execute the analysis in advance and store it in an optimized format for the search. Indexing is one of the novelties of our approach. It consists in reducing the search space through the use of effective search strategy on the built indexes. In fact, we no longer need the sequential scan because with the index structure, we can directly know what document contains a particular word. To ensure this indexing phase we use the Lucene 8 tool. Lucene is a Java API that allows developers to customize and deploy their own indexing and search engine. Lucene uses a suitable technology for all applications that require text search. Indeed, at the end of the indexing process, we get four different indexes to everyone of the two input ontologies depending on the type of the detected entities (i.e., concepts, data types, relationships, and instances). More precisely, we get a first index file I 1 6. https ://wordnet.princeton.edu/ 7. https ://www.bing.com/translator 8. https ://lucene.apache.org/ that corresponds to concepts index, a second one I 2 dedicated to relationships, a third one I 3 where we find a datatypes file index, and the last is I 4 a file indexing instances. The documents at the indexes represent semantic information about any ontological entity. These semantic information is enriched by means of the external resource (i.e., WordNet). Indeed, for each entity, CLONA keeps the entity name, the label, the translated label in English and its synonyms in English. So with Lucene, we created a set of indexes for the two ontologies, a search query is set up to return all the mapping candidates.

Candidate Mappings Identification : With Lucene, TermQuery is the most basic query type to search through an index. It can be built using one term. In our case, TermQuery's role is to find the entities in common between the indexes. Indeed, once the indexes are set up, the querying step of the latter is activated. Thus, the query implementation satisfies the terminology search and semantic aspects at once as we are querying documents that contain a given ontological entity and its synonyms obtained via WordNet. It is worthy to mention that indexes querying is done in both senses. In other words, if we have two indexes I 1 and I ′ 2 respectively belonging to O 1 and O 2 , querying is outlined as follows :

(I 1 → Query → I ′
2 ) and (I ′ 2 → Query → I 1 ). Indeed, this junction greatly increases the CLONA performance. The result of this process is a set of documents sorted by relevance according to the Lucene score assigned to each returned document. Thus, for each query, CLONA keep the first five documents returned and considers them as candidate mappings for the next phase.

Filtering and Recovery : The filtering module consists of two complementary submodules, each one is responsible of a specific task in order to refine the set of primarily aligned candidates. Indeed, once the list of candidates is ready, CLONA uses the first filter. Indeed, we should note that indexes querying may includes a set of redundant mappings. Doing so, this filter eliminates the redundancy. Indeed, it goes through the list of candidates and for each candidate, it checks if there are duplicates. If this is the case, it removes the redundant element(s). At the end of filtering phase, we have a candidates list without redundancy, however, there is always the concern of false positives, indeed, there was the need to establish a second filter. Once the redundant candidates are deleted, CLONA uses the second filter that eliminates false positives. This filter is applied to what we call partially redundant entities. An entity is considered as partially redundant if it belongs to two different mappings (i.e., being given three ontological entities e 1 , e 2 and e 3 . If on the one hand, e 1 is aligned to e 2 , and secondly, e 1 is aligned to e 3 , this last alignment is qualified as doubtful.

We note that CLONA generates (1 : 1) alignments. To overcome this challenge, CLONA compares the topology of the two suspicious entities (e 3 neighbors with e 1 neighbors, e 2 neighbors with e 1 neighbors ) with respect to the redundant entity e 1 , and retains the couple having the highest topological proximity value.

All candidates are subject of this filter, and as output we have the final alignment file.

Alignment Generation : The result of the alignment process provides a set of mappings, which are serialized in the RDF format.

Experimental Study

CLONA system has been developed with a unique focus on multilingual ontologies. The carried out experimental evaluation uses the battery of test files provided by the OAEI (Ontology Alignment Evaluation Initiative Campaign) 9 . The Multifarm dataset is composed of a subset of the Conference track, translated in nine different languages (i.e., Chinese (cn), Czech (cz), Dutch (nl), French (fr), German (de), Portuguese (pt), Russian (ru), Spanish (es) and Arabic (ar)). The considered ontologies are relatively small. With a special focus on multilingualism, it is possible to evaluate and compare the performance of alignment approaches through these test cases.

Since the CLONA method participated in the OAEI'2015 campaign, the evaluation process was conducted following a well established procedure. Therefore, CLONA was performed to determine 45 * 25 alignments. In this context, we distinguish two sub-cases, namely, "Same Ontologies" and "Different Ontologies". The first sub-case seeks alignment between same ontologies but expressed in two different natural languages. The second sub-case concerns alignment between all possible combinations of different ontologies pairs in different languages. By comparing the pioneering methods participating in the OAEI'2015 campaign and which developed specific techniques in multilingual processing, CLONA ranks second. It is anticipated by AML and succeeded by LogMap, LYAM ++ and XMAP. These values are an average of 1125 made alignments. The obtained values in the sub-case "Same Ontologies" are clearly better than those of "Different Ontologies" sub-cases. This is explained by the fact that ontologies have the same structure in the first scenario (i.e., "Same Ontologies"), which improves the process of the filtering and 9. http ://oaei.ontologymatching.org/ recovery module through enhancing similarity between the neighbor entities. Compared to the other participating methods, CLONA provides the second better values, especially in terms of Recall, i.e., 35% and 42% respectively for the first and second sub-cases. The F-Measure values flagged by In summary, and as shown in Figure 2, the CLONA method, and compared to other competing methods of the OAEI'2015 campaign, dominates 28.88% of tests occupying the first place. In addition, CLONA ranks second on 46.66% of the tests, third on 22.22% and fourth in 2.22% of tests. It should be emphasized also that in the subcase "Same Ontologies", and over 45 treated language pairs, CLONA ranked first out of 15 couples. The performance of the CLONA method is detailed on the link below as well as the alignment resulting files 10 . CLONA benefited from the power of the Lucene search and 10. http ://oaei.ontologymatching.org/2015/results/multifarm/index.html query engine, for detecting candidate alignments. This has been an asset for CLONA compared to other methods. Especially, the fact of bringing all ontological entities labels in a pivot language (i.e., English), endowed CLONA with a sharpness to cross the natural language barrier. In this context, it should be noted that the enrichment of the label entities translated lists with synonyms from WORDNET armed CLONA with a semantic layer to increase its ability to detect similar entities.

Conclusions

Multilingual ontology alignment is an important task in the ontology engineering field. In this paper, we introduced the CLONA method for cross-lingual ontology alignment. In addition, the results obtained by CLONA method are satisfactory. In this frame, it is important to highlight the external resource contribution. CLONA has implemented a technique for determining alignment candidates across the power of Lucene search engine. In addition, during the translation phase, we have set up a local translator that is built during the alignment process. This treatment reduces the translation time cost and access to the external resource. CLONA participation in OAEI'2015 campaign was encouraging, as it supplies good F-measure values in the two considered scenarios. Results reflects some strengths and some positive aspects. In addition, we aim to exploit more than one external resource during the alignment process and deploy CLONA in a semantic Web service.

Figure 1 .

 1 Figure 1. CLONA workflow.

Figure 2 .

 2 Figure 2. Percentages of occupied positions by CLONA over all tests

Table 1 .

 1 Table 1 summarizes the Precision (P), F-Measure (F-M) and Recall (R) metric values for all methods. Evaluation metrics average values for CLONA and other methods

		Different Ontologies	Same Ontologies
		P	F-M R	P	F-M R
	AML	0.53 0.51 0.50	0.93 0.64 0.50
	CLONA	0.46 0.39 0.35	0.91 0.58 0.42
	LOGMAP	0.75 0.41 0.29	0.95 0.45 0.30
	LYAM++	0.14 0.14 0.14	0.26 0.19 0.15
	XMAP	0.22 0.24 0.27	0.66 0.37 0.27

Table 2

 2 , confirm the CLONA performance which ranks second in the average metrics for all mixed sub-cases.

	Method F-Measure
	AML	0.56
	CLONA	0.43
	LOGMAP	0.40
	XMAP	0.38
	LYAM++	0.15

Table 2 .

 2 F-Measure average values for CLONA and other methods

http ://www.fao.org/home/fr/

https ://github.com/xylvxy/node-ictclas


Karel Pravda-Starov
email: karel@math.berkeley.edu

Contraction semigroups of elliptic quadratic differential operators

Keywords: contraction semigroups, elliptic quadratic differential operators, exponential decay, pseudospectrum, subellipticity. 2000 AMS Subject Classification. 47D06, 47F05

We study the contraction semigroups of elliptic quadratic differential operators. Elliptic quadratic differential operators are the non-selfadjoint operators defined in the Weyl quantization by complex-valued elliptic quadratic symbols. We establish in this paper that under the assumption of ellipticity, as soon as the real part of their Weyl symbols is a non-zero non-positive quadratic form, the norm of contraction semigroups generated by these operators decays exponentially in time.

Miscellaneous facts about elliptic quadratic differential operators

We study in this paper the class of elliptic quadratic differential operators. It is the class of pseudodifferential operators defined in the Weyl quantization (1.1) q(x, ξ) w u(x) = 1 (2π) n R 2n e i(x-y).ξ q x + y 2 , ξ u(y)dydξ, by some symbols q(x, ξ), where (x, ξ) ∈ R n × R n and n ∈ N * , which are some complexvalued elliptic quadratic forms i.e. complex-valued quadratic forms verifying (1.2) (x, ξ) ∈ R n × R n , q(x, ξ) = 0 ⇒ (x, ξ) = (0, 0).

Since the symbols of these operators are some quadratic forms, these are only differential operators. Indeed, the quadratic symbol x α ξ β , with (α, β) ∈ N 2n and |α + β| ≤ 2, is quantized in the differential operator

x α D β x + D β x x α 2 , D x = i -1 ∂ x .
Since their Weyl symbols are complex-valued, elliptic quadratic differential operators are a priori non-selfadjoint. We prove in this paper (Theorem 2.1) that under this assumption of ellipticity, as soon as the real part of their Weyl symbols is a non-zero non-positive quadratic form, the norm of contraction semigroups generated by these operators decays exponentially in time. Let us first begin by setting some notations and recalling known results about these operators.

Let q be a complex-valued elliptic quadratic form

q : R n x × R n ξ → C (x, ξ) → q(x, ξ),
with n ∈ N * , that is a complex-valued quadratic form verifying (1.2). The numerical range Σ(q) of q is defined by the subset in the complex plane of all values taken by this symbol

(1.3) Σ(q) = q(R n x × R n ξ ),

1
and the Hamilton map F ∈ M 2n (C) associated to the quadratic form q is uniquely defined by the identity (1.4) q (x, ξ); (y, η) = σ (x, ξ), F (y, η) , (x, ξ) ∈ R 2n , (y, η) ∈ R 2n , where q •; • stands for the polar form associated to the quadratic form q and σ is the symplectic form on R 2n , (1.5) σ (x, ξ), (y, η) = ξ.y -x.η, (x, ξ) ∈ R 2n , (y, η) ∈ R 2n .

Let us first notice from this definition that an Hamilton map is skew-symmetric with respect to σ. This is just a consequence of the properties of skew-symmetry of the symplectic form and symmetry of the polar form

(1.6) ∀X, Y ∈ R 2n , σ(X, F Y ) = q(X; Y ) = q(Y ; X) = σ(Y, F X) = -σ(F X, Y ).
Under this assumption of ellipticity, the numerical range of a quadratic form can only take very particular shapes. Indeed, J. Sjöstrand proves in the lemma 3.1 in [START_REF] Sjöstrand | Parametrices for pseudodifferential operators with multiple characteristics[END_REF] that if (1.7) q : R n x × R n ξ → C, n ≥ 2, is a complex-valued elliptic quadratic form then there exists z ∈ C * such that Re(zq) is a positive definite quadratic form. If n = 1, the same result is fulfilled if we assume besides that Σ(q) = C. This result shows that the numerical range of an elliptic quadratic form can only take two shapes. The first possible shape is when Σ(q) is equal to the whole complex plane. This case can only occur in dimension n = 1. The second possible shape is when Σ(q) is equal to a closed angular sector with a top in 0 and an opening strictly lower than π.

I Σ(zq)

O Indeed, if Σ(q) = C, using that the set Σ(q) is a semi-cone tq(x, ξ) = q( √ tx, √ tξ), t ∈ R + , (x, ξ) ∈ R 2n , because q is a quadratic form, we have

Σ(q) = R + z -1 I,
if z is the non-zero complex number mentioned above and I the compact interval

I = 1 + i Im(zq)(K),
where K is the following compact subset of R 2n , (x, ξ) ∈ R 2n : Re(zq)(x, ξ) = 1 .

The compactness of K is a direct consequence of the fact that Re(zq) is a positive definite quadratic form.

Elliptic quadratic differential operators define some Fredholm operators (see Lemma 3.1 in [START_REF] Hörmander | A Class of Hypoelliptic Pseudodifferential Operators with Double Characteristics[END_REF] or Theorem 3.5 in [START_REF] Sjöstrand | Parametrices for pseudodifferential operators with multiple characteristics[END_REF]), (1.8) q(x, ξ) w + z :

B → L 2 (R n ),
where B is the Hilbert space

(1.9) u ∈ L 2 (R n ) : x α D β x u ∈ L 2 (R n ) if |α + β| ≤ 2 , with the norm u 2 B = |α+β|≤2 x α D β x u 2 L 2 (R n ) .
The Fredholm index of the operator q(x, ξ) w + z is independent of z and is equal to 0 if n ≥ 2. In the case where n = 1, this index can take the values -2, 0 or 2. More precisely, this index is always equal to 0 if Σ(q) = C. If Σ(q) = C, J. Sjöstrand proves in the theorem 3.5 in [START_REF] Sjöstrand | Parametrices for pseudodifferential operators with multiple characteristics[END_REF] (see also Lemma 3.2 and Theorem 3.3 in [START_REF] Hörmander | A Class of Hypoelliptic Pseudodifferential Operators with Double Characteristics[END_REF]) that the spectrum of an elliptic quadratic differential operator

q(x, ξ) w : B → L 2 (R n ),
is only composed of eigenvalues with finite multiplicity

(1.10) σ q(x, ξ) w = λ∈σ(F ), -iλ∈Σ(q)\{0} r λ + 2k λ (-iλ) : k λ ∈ N ,
where F is the Hamilton map associated to the quadratic form q and r λ is the dimension of the space of generalized eigenvectors of F in C 2n belonging to the eigenvalue λ ∈ C. Let us notice that the spectra of these operators are always included in the numerical range of their Weyl symbols.

Contraction semigroups of elliptic quadratic differential operators

Let us consider a complex-valued elliptic quadratic form

q : R n x × R n ξ → C (x, ξ) → q(x, ξ), with n ∈ N * , verifying (2.1) Re q(x, ξ) ≤ 0, (x, ξ) ∈ R 2n .
Since the quadratic form Re q is non-positive, we deduce from the theorem 21.5.3 in [START_REF] Hörmander | The analysis of linear partial differential operators[END_REF] that there exists a real linear symplectic transformation χ of R 2n such that

(2.2) -Re(q • χ)(x, ξ) = k j=1 λ j (ξ 2 j + x 2 j ) + k+l j=k+1 x 2 j ,
with k, l ∈ N and λ j > 0 for all j = 1, ..., k. Then, it follows from the symplectic invariance of the Weyl quantization (Theorem 18.5.9 in [START_REF] Hörmander | The analysis of linear partial differential operators[END_REF]) that we can find a metaplectic operator U , which is a unitary transformation on L 2 (R n ) and an automorphism of the spaces S(R n ) and B, where B is the Hilbert space defined in (1.9) such that

(2.3) Re q(x, ξ) w = -U -1 k j=1 λ j (D 2 xj + x 2 j ) + k+l j=k+1 x 2 j U.
By using that U is a unitary operator on L 2 (R n ), we obtain that for all λ ∈ R and

u ∈ S(R n ), (2.4) Re(λu -q(x, ξ) w u, u) L 2 = λ u 2 L 2 -(Re q(x, ξ) w u, u) L 2 = λ u 2 L 2 + k j=1 λ j ( D xj U u 2 L 2 + x j U u 2 L 2 ) + k+l j=k+1 x j U u 2 L 2 ≥ λ u 2 L 2 ,
because the operator iIm q(x, ξ) w is formally skew-selfadjoint. By using the Cauchy-Schwarz inequality and the density of the Schwartz space S(R n ) in B, we deduce from (2.4) that

∀λ > 0, ∀m ∈ N * , q(x, ξ) w -λ -m ≤ λ -m ,
because we get from (1.10) and (2.1) that

σ q(x, ξ) w ⊂ {z ∈ C : Re z ≤ 0}.
This induces that the elliptic quadratic differential operator q(x, ξ) w generates a contraction semigroup e tq(x,ξ) w , t ≥ 0.

Indeed, this fact is a direct consequence of the following classical result, which can be found for example in the theorem 2.21 in [START_REF] Davies | One-Parameter Semigroups[END_REF]: let Q be a closed unbounded linear operator with a dense domain on L 2 (R n ), Q is the generator of a one-parameter semigroup e tQ verifying (2.5) ∃M > 0, ∃a ∈ R, ∀t ≥ 0, e tQ L(L 2 ) ≤ M e -at , if and only if

(2.6) ∀λ > -a, ∀m ∈ N * , λ ∈ σ(Q) and (Q -λ) -m ≤ M (λ + a) -m .
Remark. As mentioned in [START_REF] Hörmander | Symplectic classification of quadratic forms, and general Mehler formulas[END_REF] (p. 426), the assumption of ellipticity for a quadratic symbol q verifying (2.1) is not necessary to obtain that the operator defined in the Weyl quantization generates a contraction semigroup.

2.1. Statement of the main result. The main result contained in this paper is the following.

Theorem 2.1.

If q : R n x × R n ξ → C is a complex-valued elliptic quadratic form with n ∈ N * such that (2.7)
Re q ≤ 0, and ∃(x 0 , ξ 0 ) ∈ R 2n , Re q(x 0 , ξ 0 ) = 0, then the elliptic quadratic differential operator q(x, ξ) w generates a contraction semigroup with a norm decaying exponentially in time

(2.8) ∃M > 0, ∃a > 0, ∀t ≥ 0, e tq(x,ξ) w L(L 2 ) ≤ M e -at .
2.2. Some remarks about this result. Let us first notice that the assumption of ellipticity is essential. Indeed, let us consider the operator defined in the Weyl quantization by the following quadratic form

q(x, ξ) = -x 2 .
This operator is just the operator of multiplication by -x 2 , which generates the contraction semigroup

e tq(x,ξ) w u = e -tx 2 u, t ≥ 0, u ∈ L 2 (R n ).
We can easily check in this case that for all t ≥ 0,

e tq(x,ξ) w L(L 2 ) = 1. The second assumption in (2.7), ∃(x 0 , ξ 0 ) ∈ R 2n , Re q(x 0 , ξ 0 ) = 0,
is also important. Indeed, if the quadratic form Re q is identically equal to zero, the two elliptic quadratic differential operators q(x, ξ) w and -q(x, ξ) w generate some contraction semigroups verifying e tq(x,ξ) w -1 = e t(-q(x,ξ) w ) , t ≥ 0. This induces that we also have for all t ≥ 0, e tq(x,ξ) w L(L 2 ) = 1, in this case.

2.3.

Structure of the proof. We shall study two cases in the proof of the theorem 2.1. The first case is the case where the Hamilton map Re F associated to the real part of the operator's Weyl symbol Re q is non-nilpotent. This case corresponds to the case where the integer k appearing in (2.2) is non-zero. In this case, the assumption of ellipticity will not be needed and we shall get straightforwardly the result of the theorem 2.1 from (2.4).

The most interesting case is the case where the Hamilton map Re F is nilpotent. The assumption of ellipticity is then essential. In this case, we shall see that in a suitable new system of symplectic coordinates, we can tensor the variables

(x; ξ) = (x ′ , x ′′ ; ξ ′ , ξ ′′ ) ∈ R 2n ,
to write the Weyl symbol of the studied operator

q(x, ξ) = q(x ′ , ξ ′ ) + iε n ′′ j=1 µ j (ξ ′′2 j + x ′′2 j ),
with µ j > 0 for all j = 1, ..., n ′′ , ε ∈ {±1} and n = n ′ + n ′′ , as a sum of a complexvalued elliptic quadratic form q on R 2n ′ verifying Re q ≤ 0, such that its associated Hamilton map F has no real eigenvalue, and the symbol of an harmonic oscillator in dimension n ′′ . Thanks to this tensorization, it will be sufficient to prove the result of exponential decay in time for the norm of the contraction semigroup generated by the elliptic quadratic differential operator q(x ′ , ξ ′ ) w . In a second step, we shall see that the properties about its Hamilton map will induce that the symbol q(x ′ , ξ ′ ) is of finite order in every non-zero point of its numerical range. By using some results proved in [START_REF] Pravda-Starov | On the pseudospectrum of elliptic quadratic differential operators[END_REF] about the pseudospectrum of elliptic quadratic differential operators 1 , we will deduce from this fact some estimates for the resolvent's norm of the operator q(x ′ , ξ ′ ) w inside some particular regions of its numerical range Σ(q). These estimates will allow to give an integral formula for the contraction semigroup generated by the operator q(x ′ , ξ ′ ) w , which will permit to prove the result of exponential decay in time for its norm.

Proof of the theorem 2.1

Let us denote by F the Hamilton map associated to the quadratic form q. It follows from (1.4) that its real part Re F and its imaginary part Im F are respectively the Hamilton maps associated to the quadratic forms Re q and Im q. We deduce from (1.4) and (2.2) that we have in the system of symplectic coordinates (y 1 , η 1 , ..., y n , η n ), where (y, η) = χ(x, ξ),

(3.1) Re F = M             J 1 0 . . . 0
            M -1 , with (3.2) J j = 0 -λ j λ j 0 if 1 ≤ j ≤ k, and J j = 0 0 1 0 if k + 1 ≤ j ≤ k + l,
if M stands for the matrix of the real linear symplectic transformation χ. We deduce from (2.2), (3.1) and (3.2) that the Hamilton map Re F is nilpotent if and only if k = 0. In this case k = 0, the index of nilpotence is equal to 2. Let us begin by first considering the non-nilpotent case.

3.1. The non-nilpotent case. Let us consider the case where k ≥ 1. We deduce from (2.4) that for all λ ∈ R and u ∈ S(R n ),

Re(λu -q(x, ξ) w u, u) L 2 = λ u 2 L 2 -(Re q(x, ξ) w u, u) L 2 (3.3) = λ u 2 L 2 + k j=1 λ j ( D xj U u 2 L 2 + x j U u 2 L 2 ) + k+l j=k+1 x j U u 2 L 2 ≥ λ u 2 L 2 + λ 1 ( D x1 U u 2 L 2 + x 1 U u 2 L 2
), because λ j > 0 for all j = 1, ..., k. Since we get from the Cauchy-Schwarz inequality

D x1 U u 2 L 2 + x 1 U u 2 L 2 ≥ 2 D x1 U u L 2 x 1 U u L 2 ≥ 2Re(D x1 U u, ix 1 U u) L 2 = ([D x1 , ix 1 ]U u, U u) L 2 = U u 2 L 2 = u 2 L 2
, where [D x1 , ix 1 ] stands for the commutator of the operators D x1 and ix 1 , because U is unitary on L 2 (R n ), we deduce from (3.3) that for all λ ∈ R and u ∈ S(R n ), (3.4) λu -q(x, ξ)

w u L 2 ≥ (λ + λ 1 ) u L 2 .
Since from (1.10), the spectrum of the operator q(x, ξ) w is only composed of eigenvalues and that the Schwartz space S(R n ) is dense in B, we get from (3.4) that ∀λ > -λ 1 , ∀m ∈ N * , (q(x, ξ) w -λ) -m ≤ (λ + λ 1 ) -m , 1 These results are linked to some properties of subellipticity.

which induces in view of (2.5) and (2.6) that for all t ≥ 0, e tq(x,ξ) w L(L 2 ) ≤ e -λ1t . This proves the theorem 2.1 in the non-nilpotent case.

3.2. The nilpotent case. Let us now consider the nilpotent case, which is the most interesting one. The Hamilton map Re F is then nilpotent of index 2,

(3.5) (Re F ) 2 = 0.
We deduce from (2.2), (2.7), (3.1) and (3.2) that in this case

(3.6) k = 0 and l ≥ 1,
because the quadratic form Re q is non identically equal to zero.

Tensorization of variables.

Let us first notice that we can freely choose the symplectic coordinates, in which we express the symbol q of the studied operator. Indeed, by using the symplectic invariance of the Weyl quantization given by the theorem 18.5.9 in [START_REF] Hörmander | The analysis of linear partial differential operators[END_REF], we can find a metaplectic operator U , which is a unitary transformation on L 2 (R n ) and an automorphism of the spaces S(R n ) and B such that

(3.7) (q • χ)(x, ξ) w = U -1 q(x, ξ) w U, if χ is a real linear symplectic transformation of R 2n . Since (3.7) induces that e t(q•χ)(x,ξ) w = U -1 e tq(x,ξ) w U, t ≥ 0,
at the level of the generated semigroups, we actually get that for all t ≥ 0,

e t(q•χ)(x,ξ) w L(L 2 ) = e tq(x,ξ) w L(L 2 ) , because U is a unitary operator on L 2 (R n ).
This last identity justifies this first remark.

Let us now assume that we can find a decomposition of the phase space

T * R n = R n x × R n ξ , in a direct sum of symplectic sub-vector spaces S j ⊂ T * R n , 1 ≤ j ≤ N,
which are symplectically orthogonal and stable by the maps Re F and Im F ,

(3.8) T * R n = N j=1 σ⊥ S j , (Re F )S j ⊂ S j and (Im F )S j ⊂ S j ,
for all j ∈ {1, ..., N }. We consider a symplectic basis (e 1,j , ε 1,j , ..., e Nj,j , ε Nj,j ) of the symplectic space S j . By collecting these previous basis, we get a symplectic basis of T * R n . By using the stability and the orthogonality properties of the symplectic spaces S j , we obtain the following decomposition in this new system of symplectic coordinates

q(x, ξ) = σ 1≤j≤N, 1≤k≤Nj (x k,j e k,j + ξ k,j ε k,j ), F 1≤j≤N, 1≤k≤Nj (x k,j e k,j + ξ k,j ε k,j ) = 1≤j≤N σ 1≤k≤Nj (x k,j e k,j + ξ k,j ε k,j ), F 1≤k≤Nj (x k,j e k,j + ξ k,j ε k,j ) = 1≤j≤N
q j (x 1,j , ..., x Nj ,j , ξ 1,j , ..., ξ Nj ,j ), (3.9) with (3.10) q j (x 1,j , ..., x Nj ,j , ξ 1,j , ..., ξ Nj ,j ) = q(0, ..., 0, x 1,j , ..., x Nj ,j , 0, ..., 0, ξ 1,j , ..., ξ Nj ,j , 0, ..., 0), for all 1 ≤ j ≤ N . Since from (2.7), (3.9) and (3.10), q j is a complex-valued elliptic quadratic form on R 2Nj verifying Re q j ≤ 0, the operator q j (x 1,j , ..., x Nj ,j , ξ 1,j , ..., ξ Nj,j ) w generates of a contraction semigroup e tqj (x1,j ,...,xN j ,j ,ξ1,j ,...,ξN j ,j ) w , t ≥ 0, such that for all t ≥ 0,

(3.11) e tq(x,ξ) w = N j=1
e tqj (x1,j ,...,xN j ,j ,ξ1,j ,...,ξN j ,j ) w .

We deduce from (3.11) that it is sufficient for proving (2.8) to find an integer 1 ≤ j 0 ≤ N such that

∃M > 0, ∃a > 0, ∀t ≥ 0, e tq w j 0 L(L 2 (R N j 0 )) ≤ M e -at .
This second remark will allow us to reduce in the following our study to the case where the Hamilton map F associated to the quadratic form q has no real eigenvalue. Indeed, it will be a consequence of the following lemma. Lemma 3.1. If λ is a real eigenvalue of the Hamilton map F associated to a complexvalued elliptic quadratic form q verifying (2.7) then this eigenvalue λ is necessarily non-zero λ = 0, and the complex sub-vector space Ker(F +λ) is the sub-vector space complex conjugate of Ker(F -λ). If we consider

S λ = Ker(F -λ) ⊕ Ker(F + λ) ∩ T * R n ,
then the vector space Ker(F -λ) ⊕ Ker(F + λ) is a complexification of the space S λ . Moreover, S λ has a structure of symplectic vector space, which is stable by the Hamilton map Im F and verifies

Re F S λ = {0}.
Moreover, if µ is another real eigenvalue of F distinct from λ and -λ, then we have the following symplectically orthogonal direct sum

S λ ⊕ σ⊥ S µ ⊂ T * R n ,
where

S µ = Ker(F -µ) ⊕ Ker(F + µ) ∩ T * R n .
Proof of Lemma 3.1. Let λ be a real eigenvalue of F . The proposition 4.4 in [START_REF] Hörmander | Symplectic classification of quadratic forms, and general Mehler formulas[END_REF] shows that the sub-vector space of C 2n , Ker(F + λ),

is the complex conjugate of the complex sub-vector space Ker(F -λ). Moreover, we also get from this proposition that Re F Ker(F ± λ) = {0}

and that the vector space Ker(F -λ) ⊕ Ker(F + λ), if λ = 0, and Ker F are the complexifications of their intersections with T * R n . If λ = 0, then we could find

X 0 = X 1 + iX 2 = 0 such that X 1 , X 2 ∈ T * R n and X 0 , X 1 , X 2 ∈ Ker F .
Let us consider j ∈ {1, 2} such that X j = 0. Thus, we would get that q(X j ) = σ(X j , F X j ) = 0, which would contradict the ellipticity of the quadratic form q. This induces that the eigenvalue λ is non-zero. The fact that

(3.12) Re F Ker(F ± λ) = {0},
implies that Re F S λ = {0}. Moreover, since the space Ker(F -λ) ⊕ Ker(F + λ) is stable by F , it follows from (3.12) that it is also stable by Im F . The stability of the sub-space S λ by the matrix Im F then directly follows from the fact that Im F ∈ M 2n (R). Let us now consider

X 0 ∈ S λ such that for all Y ∈ S λ , σ(X 0 , Y ) = 0. It follows that for all Y and Z in S λ , σ(X 0 , Y + iZ) = 0, which induces that ∀X ∈ Ker(F -λ) ⊕ Ker(F + λ), σ(X 0 , X) = 0, because Ker(F -λ) ⊕ Ker(F + λ) is a complexification of S λ . On the other hand, since X 0 ∈ S λ , we have F X 0 ∈ Ker(F -λ) ⊕ Ker(F + λ), which implies that q(X 0 ) = σ(X 0 , F X 0 ) = 0.
We can then deduce from the ellipticity of the quadratic form q that X 0 = 0, which proves the symplectic structure of the space S λ .

Let us now assume that there exists a real eigenvalue µ of F distinct from λ and -λ. We already know that this eigenvalue µ is necessarily non-zero. Let

X ∈ Ker(F -ε 1 λ) and Y ∈ Ker(F -ε 2 µ), with ε 1 , ε 2 ∈ {±1}, we have σ(X, Y ) = σ(X, ε -1 2 µ -1 F Y ) = - 1 ε 2 µ σ(F X, Y ) = - ε 1 λ ε 2 µ σ(X, Y ),
according to the skew-symmetry of the Hamilton map F with respect to σ. Since

ε 1 λ ε 2 µ = 1,
because λ and µ are some real numbers such that µ ∈ {λ, -λ}, we finally deduce that σ(X, Y ) = 0, which proves that the sub-vector spaces S λ and S µ are symplectically orthogonal.

Let us consider the real eigenvalues of the Hamilton map F . In view of the previous lemma, we can write the two by two different elements of this set λ 1 , ..., λ r , -λ 1 , ..., -λ r .

By using the notations introduced above, let us consider the following space

(3.13) S = S λ1 ⊕ σ⊥ ... ⊕ σ⊥ S λr ⊂ T * R n ,
which is actually a direct sum because of the symplectic structure of sub-vector spaces S λj and their properties of two by two symplectic orthogonality. The lemma 3.1 shows that the symplectic vector space S is stable by the Hamilton maps Re F and Im F . Let us denote by S σ⊥ its symplectic orthogonal vector space in T * R n . This space S σ⊥ is then a symplectic vector space, which is also stable by Re F and Im F . Indeed, let us consider X ∈ S σ⊥ ⊂ T * R n . Since from the skew-symmetry of Hamilton maps with respect to σ, we get that for all

Y ∈ S ⊂ T * R n , σ(Y, Re F X) + iσ(Y, Im F X) = -σ(Re F Y, X) -iσ(Im F Y, X) = 0, because Re F Y ∈ S and Im F Y ∈ S. It follows that for all Y ∈ S, σ(Y, Re F X) = σ(Y, Im F X) = 0,
which proves that Re F X ∈ S σ⊥ and Im F X ∈ S σ⊥ . In view of our construction (3.13), the map Re F | S σ⊥ + iIm F | S σ⊥ has no real eigenvalue. If we now consider some symplectic coordinates (x, ξ),

x = (x ′ ; x ′′ ) ∈ R n ′ × R n ′′ , ξ = (ξ ′ ; ξ ′′ ) ∈ R n ′ × R n ′′ ,
such that (x ′ , ξ ′ ) and (x ′′ , ξ ′′ ) are respectively some symplectic coordinates in S and S σ⊥ , we deduce from the symplectic orthogonality of the spaces S and S σ⊥ , their stabilities by Hamilton maps Re F , Im F and the previous results that we can tensorize the variables in the symbol q,

(3.14) q(x, ξ) = q1 (x ′ , ξ ′ ) + q2 (x ′′ , ξ ′′ ), with (3.15) q1 (x ′ , ξ ′ ) = σ n ′ (x ′ , ξ ′ ), (Re F | S + iIm F | S )(x ′ , ξ ′ ) and (3.16) q2 (x ′′ , ξ ′′ ) = σ n ′′ (x ′′ , ξ ′′ ), (Re F | S σ⊥ + iIm F | S σ⊥ )(x ′′ , ξ ′′ ) ,
if σ n ′ and σ n ′′ stand respectively for the canonical symplectic forms on R 2n ′ and R 2n ′′ . We deduce from (2.7), (3.15) and (3.16) that q1 and q2 are some complexvalued elliptic quadratic forms respectively on R 2n ′ and R 2n ′′ verifying Re qj ≤ 0, j = 1, 2.

Since from (3.13) and the lemma 3.

, ξ ′′ 0 ) ∈ R 2n ′′ such that Re q2 (x ′′ 0 , ξ ′′ 0 ) = 0. Let us also notice that (3.5) implies that (Re F | S σ⊥ ) 2 = 0.
It follows in view of the reasoning mentioned above that we can reduce our study in the nilpotent case to the case where the Hamilton map F associated to the complexvalued quadratic form q defined in (2.7) has no real eigenvalue. In all the following, we make this additional licit assumption.

3.2.2. Some algebraic properties. Let us begin this paragraph by proving the following lemma.

Lemma 3.2. If q 1 and q 2 are two complex-valued quadratic forms on R 2n , then the Hamilton map associated to the complex-valued quadratic form defined by the Poisson bracket

{q 1 , q 2 } = ∂q 1 ∂ξ . ∂q 2 ∂x - ∂q 1 ∂x . ∂q 2 ∂ξ , is -2[F 1 , F 2 ] if [F 1 , F 2 ]
stands for the commutator of F 1 and F 2 the Hamilton maps of q 1 and q 2 .

Proof of Lemma 3.2. Let us first notice that we can write from (1.4) and (1.6) that for all X and Y in R 2n ,

q j (X; Y ) = σ(X, F j Y ) = -σ(F j X, Y ) = (-σF j )X, Y , for j ∈ {1, 2} if (3.17) σ = 0 I n -I n 0 and X, Y = 2n j=1 X j Y j .
It follows that for all X ∈ R 2n , (3.18) ∇q j (X) = -2σF j X.

Since we have

{q 1 , q 2 }(X) = σ ∇q 1 (X), ∇q 2 (X) , X = (x, ξ) ∈ R 2n ,
we deduce from (1.5), (1.6), (3.17) and (3.18) that for all X ∈ R 2n ,

(3.19) {q 1 , q 2 }(X) = σ(-2σF 1 X, -2σF 2 X) = 4 σ 2 F 1 X, σF 2 X = -4 F 1 X, σF 2 X = -4 σ t F 1 X, F 2 X = 4 σF 1 X, F 2 X = 4σ(F 1 X, F 2 X) = -4σ(X, F 1 F 2 X), which induces that (3.20) {q 1 , q 2 }(X) = 1 2 4σ(F 1 F 2 X, X) -4σ(X, F 1 F 2 X) = 2 σ(X, F 2 F 1 X) -σ(X, F 1 F 2 X) = -2σ(X, [F 1 , F 2 ]X),
by skew-symmetry of σ and skew-symmetry of Hamilton maps with respect to σ. Since for all X and Y in R 2n ,

σ(X, [F 1 , F 2 ]Y ) = σ(X, F 1 F 2 Y ) -σ(X, F 2 F 1 Y ) = σ(F 2 F 1 X, Y ) -σ(F 1 F 2 X, Y ) = -σ([F 1 , F 2 ]X, Y ) = σ(Y, [F 1 , F 2 ]X), we deduce that σ(X, -2[F 1 , F 2 ]Y
), is the polar form associated to the quadratic form {q 1 , q 2 } and that -2[F 1 , F 2 ] is its Hamilton map.

Let us now consider the following quadratic form

(3.21) r(X) = - 2n-1 j=0 Re q (Im F ) j X , X ∈ R 2n .
Since from (2.7), Re q is a non-positive quadratic form, we already know that r is a non-negative quadratic form. We are now going to see that r is actually a positive definite quadratic form.

Lemma 3.3. The quadratic form r is positive definite.

Proof of Lemma 3.3. Let us assume that there exists X 0 ∈ R 2n , X 0 = 0 such that r(X 0 ) = 0.

The non-positivity of the quadratic form Re q induces that for all j = 0, ..., 2n -1, (3.22) Re q (Im F ) j X 0 = 0.

If Re q(X; Y ) stands for the polar form associated to the quadratic form Re q, we deduce from the Cauchy-Schwarz inequality and (3.22) that for all j = 0, ..., 2n -1 and

Y ∈ R 2n , |Re q Y ; (Im F) j X 0 | 2 = |σ Y, Re F (Im F ) j X 0 | 2 ≤ [-Re q(Y )][-Re q (Im F ) j X 0 ] = 0.
It follows that for all j = 0, ..., 2n -1 and

Y ∈ R 2n , σ Y, Re F (Im F ) j X 0 = 0,
which implies that for all j = 0, ..., 2n -1,

(3.23) Re F (Im F ) j X 0 = 0,
because the symplectic form is non-degenerate. Let us set

(3.24)    e 1 = X 0 ε 1 = - 1 Im q(X 0 ) Im F X 0 .
This is licit. Indeed, since from (3.22), Re q(X 0 ) = 0, the ellipticity of the quadratic form q implies that Im q(X 0 ) = 0, because X 0 = 0. By using the skew-symmetry of the Hamilton map Im F , it follows that σ(ε 1 , e 1 ) = σ -(Im q(X 0 )) -1 Im F X 0 , X 0 = (Im q(X 0 )) -1 σ(X 0 , Im F X 0 ) = 1, which shows that the system (e 1 , ε 1 ) is symplectic.

If n = 1, we deduce that for all X ∈ R 2 , Re q(X) = σ(X, Re F X) = 0, because from (3.23) and (3.24),

Re F e 1 = Re F ε 1 = 0.

This contradicts (2.7) and ends the proof of the lemma 3.3 in the case where n = 1.

We can thus assume in the following that n ≥ 2.

In view of the proposition 21.1.3 in [START_REF] Hörmander | The analysis of linear partial differential operators[END_REF], this system can be completed in a symplectic basis (e 1 , ..., e n , ε 1 , ..., ε n ) of R 2n .

We now check that the vector space S generated by the vectors e 1 and ε 1 ,

(3.25) S = Vect(e 1 , ε 1 ),
is not stable by the Hamilton map Im F . Indeed, if the vector space S was stable by Im F , we could write in the symplectic coordinates associated to the symplectic basis (e 1 , ..., e n , ε 1 , ..., ε n ) the decomposition

(3.26) q(x, ξ) = q(x 1 , 0 ′ , ξ 1 , 0 ′ ) + 2q (x 1 , 0 ′ , ξ 1 , 0 ′ ); (0, x 2 , ..., x n , 0, ξ 2 , ..., ξ n )
+ q(0, x 2 , ..., x n , 0, ξ 2 , ..., ξ n ) = q(0, x 2 , ..., x n , 0, ξ 2 , ..., ξ n ) + iIm q(x 1 , 0 ′ , ξ 1 , 0 ′ ), because on one hand, we have

q(x 1 , 0 ′ , ξ 1 , 0 ′ ) = σ x 1 e 1 + ξ 1 ε 1 , F (x 1 e 1 + ξ 1 ε 1 ) = iσ x 1 e 1 + ξ 1 ε 1 , Im F (x 1 e 1 + ξ 1 ε 1 ) = iIm q(x 1 , 0 ′ , ξ 1 , 0 ′ ),
since from (3.23) and (3.24), (3.27) Re F e 1 = Re F X 0 = 0 and Re F ε 1 = -Im q(X 0 ) -1 Re F Im F X 0 = 0, and that on the other hand

q (x 1 , 0 ′ , ξ 1 , 0 ′ ); (0, x 2 , ..., x n , 0, ξ 2 , ..., ξ n ) = σ x 1 e 1 + ξ 1 ε 1 , F (x 2 e 2 + ... + x n e n + ξ 2 ε 2 + ... + ξ n ε n ) = -σ F (x 1 e 1 + ξ 1 ε 1 ), x 2 e 2 + ... + x n e n + ξ 2 ε 2 + ... + ξ n ε n ) = -iσ Im F (x 1 e 1 + ξ 1 ε 1 ), x 2 e 2 + ... + x n e n + ξ 2 ε 2 + ... + ξ n ε n ) = 0,
because we assume that the space S is stable by Im F . Since the quadratic form q is elliptic, we deduce from (3.26) that the quadratic form

(x 1 , ξ 1 ) → Im q(x 1 , 0 ′ , ξ 1 , 0 ′ ),
is also elliptic on R 2 . It follows from the result proved by J. Sjöstrand (Lemma 3.1 in [START_REF] Sjöstrand | Parametrices for pseudodifferential operators with multiple characteristics[END_REF]) and recalled in (1.7) that a real-valued elliptic quadratic form is necessarily positive definite or negative definite. Then, we deduce from (3.26) and the lemma 18.6.4 in [START_REF] Hörmander | The analysis of linear partial differential operators[END_REF] that we can find a new symplectic basis (ẽ 1 , ε1 ) of the space S such that in these new coordinates (3.28) q(x, ξ) = q(0, x 2 , ..., x n , 0, ξ 2 , ..., ξ n ) + iλ(x 2 1 + ξ 2 1 ), with λ ∈ R * . We can easily check that (3.28) implies that the Hamilton map F associated to the quadratic form q has some real eigenvalues, which contradicts our assumption. Indeed, by computing the Hamilton map F from (3.28) as in (3.1), we obtain that F X 1 = -λX 1 and F X 2 = λX 2 , if X 1 = ẽ1 + iε 1 and X 2 = ẽ1 -iε 1 . We conclude that the vector space S is actually not stable by the Hamilton map Im F .

We can now resume the symplectic coordinates associated to the symplectic basis (e 1 , ..., e n , ε 1 , ..., ε n ) defined just before (3.25). Since the vector space S is not stable by Im F , we can find by using the following symplectically orthogonal direct sum R 2n = S ⊕ σ⊥ S σ⊥ , and (3.24), an element X 3 ∈ S σ⊥ , X 3 = 0 and some real numbers (λ, µ) ∈ R 2 such that

(3.29) (Im F ) 2 X 0 = X 3 + λe 1 + µε 1 .
It follows from (3.23), (3.24) and (3.29) that

(3.30) Re F X 3 = Re F Im F X 3 = 0,
because n ≥ 2. We can then consider the following new vectors

(3.31)    e 2 = X 3 ε 2 = - 1 Im q(X 3 ) Im F X 3 ,
which is possible according to the fact that from (3.30), Re q(X 3 ) = σ(X 3 , Re F X 3 ) = 0, and to the ellipticity of q. It follows that (3.32) σ(ε 1 , e 1 ) = σ(ε 2 , e 2 ) = 1, σ(e 1 , e 2 ) = σ(ε 1 , e 2 ) = 0, because according to our construction e 1 , ε 1 ∈ S and e 2 ∈ S σ⊥ and that we get from (3.24), (3.31) and (3.32),

σ(ε 2 , e 1 ) = -Im q(X 3 ) -1 σ(Im F X 3 , X 0 ) = Im q(X 3 ) -1 σ(X 3 , Im F X 0 ) = -Im q(X 3 ) -1
Im q(X 0 )σ(e 2 , ε 1 ) = 0 and that from (3.24), (3.29), (3.31) and (3.32),

σ(ε 1 , ε 2 ) = Im q(X 0 )Im q(X 3 ) -1 σ(Im F X 0 , Im F X 3 ) = -Im q(X 0 )Im q(X 3 ) -1 σ (Im F ) 2 X 0 , X 3 = -Im q(X 0 )Im q(X 3 ) -1 σ(e 2 + λe 1 + µε 1 , e 2 ) = 0.
We deduce from (3.23), (3.24), (3.30) and (3.31) that

Re F e 1 = Re F e 2 = Re F ε 1 = Re F ε 2 = 0.

By checking as before that the vector space Vect(e 1 , e 2 , ε 1 , ε 2 ), cannot be stable by the Hamilton map Im F , because it would imply the existence of real eigenvalues for F , we can using (3.23) iterate this construction to build a symplectic basis (e 1 , ..., e n , ε 1 , ..., ε n ) of R 2n such that for all j = 1, ..., n,

Re F e j = Re F ε j = 0.

This induces that for all X ∈ R 2n , Re q(X) = σ(X, Re F X) = 0, which contradicts (2.7) and ends the proof of the lemma 3.3.

By using the two previous lemmas, we can now prove that the quadratic symbol q has a finite order τ verifying

(3.33) 1 ≤ τ ≤ 4n -2,
in every non-zero point of the numerical range

(3.34) Σ(q) \ {0} = q(R 2n ) \ {0}.
Let us first recall the classical definition of the order k(x 0 , ξ 0 ) of a symbol p(x, ξ) at a point (x 0 , ξ 0 ) ∈ R 2n (see section 27.2, chapter 27 in [START_REF] Hörmander | The analysis of linear partial differential operators[END_REF]). This order k(x 0 , ξ 0 ) is an element of the set N ∪ {+∞} defined by

(3.35) k(x 0 , ξ 0 ) = sup j ∈ Z : p I (x 0 , ξ 0 ) = 0, ∀ 1 ≤ |I| ≤ j ,
where I = (i 1 , i 2 , ..., i k ) ∈ {1, 2} k , |I| = k and p I stands for the iterated Poisson brackets

p I = H pi 1 H pi 2 ...H pi k-1 p i k ,
where p 1 and p 2 are respectively the real and the imaginary part of the symbol p, p = p 1 + ip 2 . The order of a symbol q at a point z is then defined as the maximal order of the symbol p = q -z at every point (x 0 , ξ 0 ) ∈ R 2n verifying p(x 0 , ξ 0 ) = q(x 0 , ξ 0 ) -z = 0.

Let X 0 be in R 2n , X 0 = 0. Let us denote by j 0 the lowest integer in {0, ..., 2n -1} such that (3.36) Re q (Im F ) j0 X 0 < 0, given the lemma 3.3. It follows that for all 0 ≤ j ≤ j 0 -1, (3.37) Re q (Im F ) j X 0 = 0.

Let us prove that (3.38) H 2j0 Im q Re q(X 0 ) = 0. Indeed, we deduce from the lemma 3.2 that the Hamilton map associated to the quadratic form H 2j0 Im q Re q is

(3.39) 4 j0 [Im F, [Im F, [..., [Im F, Re F ]...],
with exactly 2j 0 terms Im F appearing in the previous formula. We can write

(3.40) 4 j0 [Im F, [Im F, [..., [Im F, Re F ]...] = 2j0 j=0 c j (Im F ) j Re F (Im F ) 2j0-j ,
with c j ∈ R * for all j = 0, ..., 2j 0 . Indeed, by using the following identity

[P, [P, Q]] = P 2 Q -2P QP + QP 2 ,
we can prove by induction that for all n ∈ N * , there exist some positive constants d n,j , j = 0, ..., 2n such that

H 2j0 Im q Re q(X 0 ) = c j0 σ X 0 , (Im F ) j0 Re F(Im F ) j0 X 0 (3.41) + j0-1 j=0 c j σ X 0 , (Im F ) j Re F(Im F ) 2j0-j X 0 + j0-1 j=0 c 2j0-j σ X 0 , (Im F ) 2j0-j Re F(Im F ) j X 0 . Since on one hand σ X 0 , (Im F ) j0 Re F(Im F ) j0 X 0 = (-1) j0 σ (Im F ) j0 X 0 , Re F(Im F ) j0 X 0 = (-1) j0 Re q (Im F ) j0 X 0
by skew-symmetry of the Hamilton map Im F , and that on the other hand we get from (2.7), (3.37) and the Cauchy-Schwarz inequality

|σ X 0 , (Im F ) j Re F(Im F ) 2j0-j X 0 | = |σ (Im F ) j X 0 , Re F(Im F ) 2j0-j X 0 | = -Re q (Im F ) j X 0 ; (Im F ) 2j0-j X 0 ≤ [-Re q((Im F ) j X 0 )] 1 2 [-Re q((Im F ) 2j0-j X 0 )] 1 2 = 0 and |σ X 0 , (Im F ) 2j0-j Re F(Im F ) j X 0 | = |σ (Im F ) 2j0-j X 0 , Re F(Im F ) j X 0 | = -Re q (Im F ) 2j0-j X 0 ; (Im F ) j X 0 ≤ [-Re q((Im F ) 2j0-j X 0 )] 1 2 [-Re q((Im F ) j X 0 )]

H 2j0

Im q Re q(X 0 ) = (-1) j0 c j0 Re q (Im F ) j0 X 0 = 0, because c j0 ∈ R * . We conclude from (3.35), (3.36) and (3.38) that the quadratic symbol q has a finite order lower or equal than 4n -2 in every non-zero point of the numerical range Σ(q) \ {0}.

3.2.3. Some estimates for the resolvent of elliptic quadratic differential operators. To obtain the result of exponential decay in time for the norm of the contraction semigroup given by the theorem 2.1, we need to estimate the resolvent of its generator in some particular regions of the resolvent set. The first estimate is given by the following proposition, which shows that the resolvent of an elliptic quadratic differential operator cannot blow up in norm far from the numerical range of its symbol. This result is proved in [START_REF] Pravda-Starov | On the pseudospectrum of elliptic quadratic differential operators[END_REF]. For the sake of completeness of this paper, we recall its proof. Proposition 3.1. Let q : R n ×R n → C, n ∈ N * , be a complex-valued elliptic quadratic form. We have

∀z ∈ Σ(q), q(x, ξ) w -z -1 ≤ 1 d z, Σ(q)
, where d z, Σ(q) stands for the distance from z to the numerical range Σ(q).

Proof of Proposition 3.1. If the numerical range is equal to the whole complex plane, there is nothing to prove. If Σ(q) = C, we have seen after (1.7) that the numerical range is necessarily a closed angular sector with a top in 0 and an opening strictly lower than π. Let us consider z ∈ Σ(q) and denote by z 0 its orthogonal projection on the non-empty closed convex set Σ(q). According to the shape of the numerical range, it follows that z 0 belongs to its boundary and that we can find a complex number

z 1 ∈ C * , |z 1 | = 1 such that Σ(z 1 q) ⊂ z ∈ C : Re z ≥ 0 and (3.42) z 1 z ∈ z ∈ C : Re z < 0 , d z, Σ(q) = d(z 1 z, iR).
Using now that the operator i[Im(z 1 q)] w is formally skew-selfadjoint, we obtain that for all u ∈ S(R n ),

(3.43) Re z 1 q(x, ξ) w u -z 1 zu, u L 2 = d(z 1 z, iR) u 2 L 2 + Re z 1 q(x, ξ) w u, u L 2 .
Then, since the quadratic form Re(z 1 q) is non-negative, we deduce from the symplectic invariance of the Weyl quantization (Theorem 18.5.9 in [START_REF] Hörmander | The analysis of linear partial differential operators[END_REF]) and the theorem 21.5.3 in [START_REF] Hörmander | The analysis of linear partial differential operators[END_REF] that there exists a metaplectic operator U , which is a unitary transformation of L 2 (R n ) and an automorphism of the spaces S(R n ) and B, such that Re z 1 q(x, ξ)

w = U -1 k j=1 λ j (D 2 xj + x 2 j ) + k+ l j= k+1
x 2 j U, with k, l ∈ N and λ j > 0 for all j = 1, ..., k. By using that U is a unitary operator on L 2 (R n ), we obtain that the quantity

Re z 1 q(x, ξ) w u, u L 2 = k j=1 λ j D xj U u 2 L 2 + x j U u 2 L 2 + k+ l j= k+1 x j U u 2 L 2 ,
is non-negative. Then, we can deduce from the Cauchy-Schwarz inequality, (3.42) and (3.43) that for all u ∈ S(R n ),

d z, Σ(q) u L 2 ≤ |z 1 | q(x, ξ) w u -zu L 2 .
Finally, using the density of the Schwartz space S(R n ) in B and the fact that

|z 1 | = 1, we obtain that ∀z ∈ Σ(q), q(x, ξ) w -z -1 ≤ 1 d z, Σ(q)
, since according to (1.10), σ q(x, ξ) w ⊂ Σ(q).

The second estimate that we will use is a consequence of the semiclassical result given by the following proposition. This result deals with a result of absence of semiclassical pseudospectrum for elliptic quadratic differential operators at the boundary of their numerical ranges. It is linked to some properties of subellipticity of these operators. This result is proved in [START_REF] Pravda-Starov | On the pseudospectrum of elliptic quadratic differential operators[END_REF]. We also recall its proof for the sake of completeness of this paper.

Let us consider a complex-valued quadratic form q : R n x × R n ξ → C and ∆ an open half-line with a top in 0 included in its numerical range ∆ ⊂ Σ(q). We need to define a notion of order for the symbol q(x, ξ) on this half-line ∆. Since q is a quadratic form, all the iterated Poisson brackets are also some quadratic forms. The property of degree two homogeneity of these Poisson brackets induces that the symbol q has the same order at every point of the open half-line ∆. This allows to define the order of the symbol q on the half-line ∆ by defining this order with this common value. As underlined in [START_REF] Pravda-Starov | On the pseudospectrum of elliptic quadratic differential operators[END_REF], this order can be finite or infinite. In the case where the Weyl symbol of an elliptic quadratic differential operator is of finite order on an open half-line belonging to the boundary of its numerical range, we have the following result. Proposition 3.2. Let us consider a complex-valued elliptic quadratic form q : R n x × R n ξ → C, with a numerical range Σ(q) distinct from the whole complex plane

Σ(q) = C,
which is a closed angular sector with a top in 0 and a positive opening. If this symbol q(x, ξ) is of finite order k on an open half-line belonging to the boundary of its numerical range ∆ ⊂ ∂Σ(q) \ {0}, then this order is necessarily even and there is no semiclassical pseudospectrum of index k/(k + 1) on ∆ for the associated semiclassical operator, that is that for all z ∈ ∆,

(3.44) ∃C > 0, ∃h 0 > 0, ∀ 0 < h < h 0 , q(x, hξ) w -z -1 < Ch -k k+1 .
Proof of Proposition 3.2. Let us consider the following symbol belonging to the C ∞ b (R 2n , C) space composed of bounded complex-valued functions on R 2n with all derivatives bounded

(3.45) r(x, ξ) = q(x, ξ) -z 1 + x 2 + ξ 2 ,
with z ∈ ∆. Setting Σ(r) = r(R 2n ), we can first notice that

z ∈ ∂Σ(q) \ {0} ⇒ 0 ∈ ∂ Σ(r).
Let us also notice that the symbol r fulfills the principal-type condition in 0. Indeed, if (x 0 , ξ 0 ) ∈ R 2n was such that r(x 0 , ξ 0 ) = 0 and dr(x 0 , ξ 0 ) = 0, we would get from (3.45) that (3.46) dq(x 0 , ξ 0 ) = 0.

Since by using the lemma 3.1 in [START_REF] Sjöstrand | Parametrices for pseudodifferential operators with multiple characteristics[END_REF] and the lemma 18.6.4 in [START_REF] Hörmander | The analysis of linear partial differential operators[END_REF], we can find a non-zero complex number α ∈ C * and a real linear symplectic transformation χ of R 2n such that

(3.47) Re(αq) • χ(x, ξ) = n j=1 λ j (ξ 2 j + x 2 j ),
with λ j > 0 for all j = 1, ..., n, we deduce from (3.46) that

dRe(αq)(x 0 , ξ 0 ) = 2 n j=1 λ j (x 0 ) j dx j + ( ξ0 ) j dξ j • χ -1 = 0,
with (x 0 , ξ0 ) = χ -1 (x 0 , ξ 0 ). This would first imply that (x 0 , ξ0 ) = (0, 0) and (x 0 , ξ 0 ) = (0, 0), because λ j > 0 for all j = 1, ..., n. It would then follow that q(x 0 , ξ 0 ) = 0 because q is a quadratic form. However, since r(x 0 , ξ 0 ) = 0, we get from (3.45) that q(x 0 , ξ 0 ) = z = 0 because z ∈ ∆ ⊂ ∂Σ(q) \ {0}, which induces a contradiction. It follows that the symbol r actually fulfills the principal-type condition in 0. Let us now notice that, since symbol q is of finite order k in z, this induces in view of (3.45) that the symbol r is also of finite order k in 0. On the other hand, we deduce from (3.45) and (3.47) that the set {(x, ξ) ∈ R 2n : r(x, ξ) = 0} = {(x, ξ) ∈ R 2n : q(x, ξ) = z}, is compact. Under these conditions, we can apply the theorem 1.4 in [START_REF] Dencker | Pseudospectra of Semiclassical (Pseudo-)Differential Operators[END_REF], which proves that the integer k is even and gives the existence of positive constants h 0 and C 1 such that

(3.48) ∀ 0 < h < h 0 , ∀u ∈ S(R n ), r(x, hξ) w u L 2 ≥ C 1 h k k+1 u L 2 .
Remark. We did not check the dynamical condition (1.7) in [START_REF] Dencker | Pseudospectra of Semiclassical (Pseudo-)Differential Operators[END_REF], because this assumption is not necessary for the proof of Theorem 1.4. Indeed, this proof only use a part of the proof of Lemma 4.1 in [START_REF] Dencker | Pseudospectra of Semiclassical (Pseudo-)Differential Operators[END_REF] (a part of the second paragraph), where this condition (1.7) is not needed.

By using some results of symbolic calculus given by the theorem 18.5.4 in [START_REF] Hörmander | The analysis of linear partial differential operators[END_REF]

r 2 (x, ξ) = - 1 2 ∂ 2 r ∂x 2 (x, ξ) - 1 2 ∂ 2 r ∂ξ 2 (x, ξ).
We can easily check from (3.45) that these functions r 1 and r 2 belong to the space C ∞ b (R 2n , C), and we deduce from the Calderón-Vaillancourt theorem that there exists a positive constant C 2 such that for all u ∈ S(R n ) and 0 < h ≤ 1,

(3.52) r 1 (x, hξ) w u L 2 ≤ C 2 u L 2 and r 2 (x, hξ) w u L 2 ≤ C 2 u L 2 .
It follows from (3.48), (3.49), (3.52) and the triangular inequality that for all u ∈ S(R n ) and 0 < h < h 0 ,

C 1 h k k+1 (1 + x 2 + h 2 ξ 2 ) w u L 2 ≤ r(x, hξ) w (1 + x 2 + h 2 ξ 2 ) w u L 2 ≤ q(x, hξ) w u -zu L 2 + C 2 h(1 + h) u L 2 .
Since from the Cauchy-Schwarz inequality, we have for all u ∈ S(R n ) and 0 < h ≤ 1,

u 2 L 2 ≤ u 2 L 2 + xu 2 L 2 + hD x u 2 L 2 = (1 + x 2 + h 2 ξ 2 ) w u, u L 2 ≤ (1 + x 2 + h 2 ξ 2 ) w u L 2 u L 2 ,
we obtain that for all u ∈ S(R n ) and 0 < h < h 0 , (3.53) 2 , we deduce from (3.53) that there exist some positive constants h ′ 0 and C 3 such that for all 0 < h < h ′ 0 and u ∈ S(R n ), q(x, hξ) w u -zu

C 1 h k k+1 u L 2 ≤ q(x, hξ) w u -zu L 2 + C 2 h(1 + h) u L 2 . Since k ≥ 1 because z ∈ Σ(q)
L 2 ≥ C 3 h k k+1 u L 2 .
Using that the Schwartz space S(R n ) is dense in B and that the operator q(x, hξ) w + z, is a Fredholm operator of index 0, we obtain that for all 0 < h < h ′ 0 , q(x, hξ) w -z -1 ≤ C -1 3 h -k k+1 , which ends the proof of the proposition 3.2.

By using that q is a quadratic form, we obtain from the change of variables y = h 1/2 x, with h > 0, the following identity between the quantum operator q(x, ξ) w and its associated semiclassical operator (q(x, hξ) w ) 0<h≤1 , (3.54) q(x, ξ) wz h = 1 h q(y, hη) w -z , if z ∈ C. It follows that when (3.44) is fulfilled, we get from (3.54) the following estimate for the resolvent of the elliptic quadratic differential operator q(x, ξ) w , (3.55) ∃C > 0, ∃η 0 ≥ 1, ∀η ≥ η 0 , q(x, ξ) w -ηe iargz -1 < Cη -1 k+1 .

3.2.4. Exponential decay in time for the norm of contraction semigroups. Let q : R n x × R n ξ → C, n ∈ N * , be a complex-valued elliptic quadratic form verifying (3.56) Re q ≤ 0 and ∃(x 0 , ξ 0 ) ∈ R 2n , Re q(x 0 , ξ 0 ) = 0.

We consider the nilpotent case

(3.57) (Re F ) 2 = 0,
where F stands for the Hamilton map associated to q, and we assume that this Hamilton map F has no real eigenvalue. This reduction is licit according to section 3.2.1.

As mentioned after (1.7), the numerical range Σ(q) is necessarily a closed angular sector with a top in 0 and an opening strictly lower than π when the assumption (3.56) is fulfilled. We deduce from (2.2), (3.6), (3.56) and the ellipticity of q that the numerical range Σ(q) is actually a closed angular sector with a positive opening such that one of the two half-lines iR + or iR -belongs to its boundary. We will consider in the following the case where

(3.58) iR + ⊂ ∂Σ(q).
The proof of the theorem 2.1 in the second case iR -⊂ ∂Σ(q), will be a straightforward adaptation of the one given when (3.58) is fulfilled. We know from (1.10) that the spectrum of the elliptic quadratic differential operator q(x, ξ) w is only composed of eigenvalues and is given by

σ q(x, ξ) w = λ∈σ(F ), -iλ∈Σ(q)\{0} r λ + 2k λ (-iλ) : k λ ∈ N .
Since from (3.56), Σ(q) \ {0} ⊂ {z ∈ C : Re z ≤ 0}, it follows that Re(-iλ) = Im λ ≤ 0, if -iλ ∈ Σ(q)\ {0}. Since we assume that the Hamilton map F has no real eigenvalue, we obtain that Re(-iλ) < 0, if λ ∈ σ(F ) and -iλ ∈ Σ(q) \ {0}. This implies that

(3.59) σ q(x, ξ) w ⊂ {z ∈ C : Re z ≤ -a 0 }, with (3.60) a 0 = inf λ∈σ(F ), -iλ∈Σ(q)\{0} r λ + 2k λ -Re(-iλ) : k λ ∈ N > 0.
Let us prove that there exists a curve γ ∈ C ∞ (R, C) in the complex plane such that there exist some positive constants t 0 , b, c 1 and c 2 such that

(3.61) ∀t ∈ R, Re γ(t) ≤ -b < 0, (3.62) ∀t ∈ R, |t| ≥ t 0 , γ(t) = -c 1 |t| + it 4n-1 , (3.63) σ q(x, ξ) w ⊂ Γ 1 and (3.64) ∀z ∈ Γ 2 ∪ γ(R), q(x, ξ) w -z -1 ≤ c 2 (1 + |z|) -1 4n-1
, where Γ 1 and Γ 2 are the subsets appearing on the following figure in the cutoff of the complex plane

(3.65) Γ 1 ∪ Γ 2 ∪ γ(R) = C. Figure 3: Cutoff Γ 1 ∪ Γ 2 ∪ γ(R) = C. 0 Γ 2 Γ 1 γ
To justify this fact, let us begin by noticing that since we have proved in section 3.2.2 that the quadratic symbol q has always a finite order lower or equal than 4n -2 in every non-zero point of the numerical range Σ(q) \ {0}, we can deduce from the proposition 3.2 and (3.55) that (3.66) ∃C > 0, ∃η 0 ≥ 1, ∀η ≥ η 0 , q(x, ξ) w -iη -1 < Cη -1 4n-1 . This induces that for all u ∈ B and η ≥ η 0 , (3.67)

q(x, ξ) w u -iηu L 2 ≥ C -1 η 1 4n-1 u L 2 . It follows from (3.67) that for all u ∈ B and z ∈ {z ∈ C : Im z ≥ η 0 , |Re z| ≤ 2 -1 C -1 |Im z| 1 4n-1 }, we have q(x, ξ) w u -zu L 2 ≥ q(x, ξ) w u -iIm zu L 2 -|Re z| u L 2 ≥ 1 2C |Im z| 1 4n-1 u L 2 ,
which induces that q(x, ξ) w -z -1 ≤ 2C|Im z| -1 4n-1 , because the spectrum of the operator q(x, ξ) w is only composed of eigenvalues. Since

1 + |z| ≤ 1 + |Im z| + |Re z| ≤ 1 + |Im z| + 1 2C |Im z| 1 4n-1 ≤ 2|Im z|,
if the positive constant η 0 is sufficiently large and

z ∈ {z ∈ C : Im z ≥ η 0 , |Re z| ≤ 2 -1 C -1 |Im z| 1 4n-1 }, it follows that (3.68) ∃η 0 > 0, ∀z ∈ C, Im z ≥ η 0 , |Re z| ≤ 1 2C |Im z| 1 4n-1 , q(x, ξ) w -z -1 ≤ 2 4n 4n-1 C(1 + |z|) -1 4n-1 .
According to the shape of the numerical range (see the figure 2), we can assume that

{z ∈ C : Im z ≤ -η 0 , |Re z| ≤ 2 -1 C -1 |Im z| 1 4n-1 } ⊂ C \ Σ(q),
if the positive constant η 0 is chosen sufficiently large. We can then obtain from the proposition 3.1 that there exists a positive constant c 3 such that for all

z ∈ {z ∈ C : Im z ≤ -η 0 , |Re z| ≤ 2 -1 C -1 |Im z| 1 4n-1 }, (3.69) q(x, ξ) w -z -1 ≤ 1 d z, Σ(q) ≤ c 3 1 + |z| ,
if the positive constant η 0 is chosen sufficiently large. We can also deduce from the proposition 3.1 that there exists a positive constant c 4 such that for all

(3.70) z ∈ {z ∈ C : |Im z| ≥ η 0 , Re z ≥ 2 -1 C -1 |Im z| 1 4n-1 } ∪ z ∈ C : |Im z| ≤ η 0 , Re z ≥ 1 2 , (3.71) q(x, ξ) w -z -1 ≤ 1 d z, Σ(q) ≤ c 4 (1 + |z|) -1 4n-1 ,
since we can find a positive constant c 5 such that for all z belonging to the set (3.70), d z, Σ(q) ≥ Re z ≥ c 5 (1 + |z|) Using this curve γ, we can now give an integral formula for the contraction semigroup generated by the operator q(x, ξ) w , which allows to prove the result of exponential decay in time for its norm. In the following, we follow a method used by F. Hérau and F. Nier in [START_REF] Hérau | Isotropic hypoellipticity and trend to the equilibrium for the Fokker-Planck equation with high degree potential[END_REF].

Let us set

(3.72) T (s) = 1 2iπ γ e sz z -q(x, ξ) w -1 dz,
for all s > 0. Since from (3.64),

(3.73) ∀z ∈ γ(R), e sz z -q(x, ξ) w -1 ≤ c 2 e sRe z (1 + |z|) -1 4n-1 ,
we deduce from (3.62) that

T (s) ∈ L L 2 (R n ) ,
for all s > 0. Moreover if u 0 ∈ B, we can write that for all z ∈ σ q(x, ξ) w ,

(3.74) z-q(x, ξ) w -1 u 0 = 1 z -1 z-q(x, ξ) w -1 + q(x, ξ) w -1 -1 q(x, ξ) w -1 u 0 , because from (3.59), 1 ∈ σ q(x, ξ) w . Let us set (3.75) u(s) = 1 2iπ γ e sz z -q(x, ξ) w -1 u 0 dz,
for all s > 0. Since we can easily check from (3.62) that for all s > 0, e sz z -1 z -q(x, ξ) w -1 q(x, ξ) w -1 u 0 dz.

By using (3.62) and (3.73), we can check that the right-hand side of (3.77) is continuous with respect to s ≥ 0 and that for all s > 0, du(s) ds = 1 2iπ γ e sz z z -1 z -q(x, ξ) w -1 q(x, ξ) w -1 u 0 dz = 1 2iπ γ e sz z -1 q(x, ξ) w z -q(x, ξ) w -1 q(x, ξ) w -1 u 0 dz, (3.78) because z z -q(x, ξ) w -1 q(x, ξ) w -1 u 0 = z -q(x, ξ) w + q(x, ξ) w z -q(x, ξ) w -1 q(x, ξ) w -1 u 0 = q(x, ξ) w -1 u 0 + q(x, ξ) w z -q(x, ξ) w -1 q(x, ξ) w -1 u 0 and that from (3.76), 1 2iπ γ e sz z -1 dz = 0. Since 1 2iπ γ e sz z -1 q(x, ξ) w z -q(x, ξ) w -1 q(x, ξ) w -1 u 0 dz = q(x, ξ) w 1 2iπ γ e sz z -1 z -q(x, ξ) w -1 q(x, ξ) w -1 u 0 dz , because q(x, ξ) w ∈ L B, L 2 (R n ) , we deduce from (3.77) and (3.78) that for all s > 0, (3.79) du(s) ds = q(x, ξ) w u(s), in L 2 (R n ). On the other hand, it follows from (3.77) that (3.80) lim s→0 + u(s) = 1 2iπ γ z -q(x, ξ) w -1 dz z -1 q(x, ξ) w -1 u 0 .

We need the following lemma to conclude.

Lemma 3.4. We have 1 2iπ γ z -q(x, ξ) w -1 dz z -1 = -1 -q(x, ξ) w -1 .

By using this lemma, we get from (3.75), (3.79) and (3.80) that for all u 0 ∈ B and s > 0, e sq(x,ξ) w u 0 = 1 2iπ γ e sz z -q(x, ξ) w -1 u 0 dz.

By using that T (s) ∈ L L 2 (R n ) if s > 0, and the density of the space B in L 2 (R n ), we obtain that for all u ∈ L 2 (R n ) and s > 0, (3.81) e sq(x,ξ) w u = 1 2iπ γ e sz z -q(x, ξ) w -1 udz.

Let us notice from (3.61) and (3.64) that for all z ∈ γ(R), u ∈ L 2 (R n ) and s > 0, we obtain from (3.83) that there exists a positive constant c 6 such that for all s ≥ 1, (3.84) e sq(x,ξ) w L(L 2 ) ≤ c 6 e -b 2 s . Finally, by using that the elliptic quadratic differential operator q(x, ξ) w generates a contraction semigroup, we deduce from (3.84) that ∃M, a > 0, ∀t ≥ 0, e tq(x,ξ) w L(L 2 ) ≤ M e -at . It follows that it only remains to prove the lemma 3.4 to end our proof of the theorem 2.1.

To prove the lemma 3.4, it is sufficient to prove that for all u and v in L 2 (R n ),

(3.85) 1 2iπ γ (z -q(x, ξ) w ) -1 u, v L 2 dz z -1 = -(1 -q(x, ξ) w ) -1 u, v L 2 .
Since the function z → (z -q(x, ξ) w ) -1 u, v L 2 , is holomorphic in an open neighbourhood of the set Γ 2 ∪ γ(R), the Cauchy Theorem proves that

(3.86) 1 2iπ γR (z -q(x, ξ) w ) -1 u, v L 2 dz z -1 = -(1 -q(x, ξ) w ) -1 u, v L 2 ,
if R > 1 and γ R stands for the oriented contour appearing on the following figure

(3.87) γ R = γ R,1 ∪ γ R,2 ∪ γ R,3 ∪ γ R,4 ,
defined by (z -q(x, ξ) w ) -1 u, v L 2 dz z -1 (3.92) = 1 2iπ γ (z -q(x, ξ) w ) -1 u, v L 2 dz z -1 .

For j ≥ 2, we get from (3.64), (3.89), (3.90) and (3.91) that 1 2iπ γR,j (z -q(x, ξ) w ) -1 u, v L 2 dz z -1 (3.93)

≤ 1 2π sup z∈γR,j | (z -q(x, ξ) w ) -1 u, v L 2 | |z -1| Length(γ R,j ) ≤ c 2 2π max(2R, R + c 1 R 1 4n-1 ) sup z∈γR,j 1
|z -1|(1 + |z|) 1 4n-1 u L 2 v L 2 → 0,
when R → +∞. Finally, we deduce (3.85) from (3.92) and (3.93) by considering the limit when R → +∞ in (3.86). This ends the proof of the lemma 3.4 and also the proof of the theorem 2.1.

Figure 1 :

 1 Figure 1: Shape of the numerical range Σ(q) when Σ(q) = C.

[(- 1 )

 1 P, [P, [..., [P, Q]...] = 2n j=0 j d n,j P j QP 2n-j , if there are exactly 2n terms P in the left-hand side of the previous identity. It follows from (3.39) and (3.40) that

  j = 0, ..., j 0 -1, we deduce (3.38) from (3.36) and (3.41) that

Figure 2 :

 2 Figure 2: Numerical range Σ(q).

( 3 .82) e b 2 s

 32 e sz z -q(x, ξ) w -1 u L 2 ≤ c 2 (1 + |z|) -1 4n-1 e s(Re z+ b 2 ) u L 2 ≤ c 2 u L 2 .We deduce from (3.62), (3.81) and (3.82) that for all u ∈ L 2 (R n ) and s > 0,(3.83) e b 2 s e sq(x,ξ) w u L 2 ≤ c 2 2π t0 -t0 |γ ′ (t)|dt u L 2

Figure 4 : 2 -R γ R, 4 ( 3 1 4n- 1 ≤ t ≤ R 1 4n- 1 , 1 4n- 1 ≤ 1 4n- 1 .

 424311111111 Figure 4: Contour γ R .

  (1 + x 2 + h 2 ξ 2 ) w = q(x, hξ) w -z + hr 1 (x, hξ) w + h 2 r 2 (x, hξ) w ,

	(3.45), we can write (3.50) and (3.49) r(x, hξ) with (3.51)	r 1 (x, ξ) = -ix	∂r ∂ξ	(x, ξ) + iξ	∂r ∂x	(x, ξ)	and

w

We recall that Σ(q) is a closed angular sector.