"Problèmes pour chercher" : Experience, Possible and Necessity in Pupils Reasonings

Problèmes pour chercher» are, literally, «problems to research» in mathematics at the primary school level. Their aim is for students to solve non classical problems, and to discuss and argue about them. The goal is not to build curricular knowledge. Hence, these problems are, by some aspects, close to the activity of mathematicians. But what is really the activity of the pupils during these classroom works? Are they working on proofs? What kind of difficulties do they have? This contribution deals with the attitude of pupils towards Experience, Possible, Necessity and proofs. To achieve this, we propose to analyze the productions of 8-9 year old pupils on the "No-Three-In-Line" problem.

Introduction

In France, official texts for the primary school level require teachers to propose "problème pour chercher", literally "problems to research", in mathematics. Their aim is for pupils to develop mathematical reasoning skills. Is it easy to develop these abilities at the end of primary school? What kind of reasoning can produce pupils? What kind of difficulties do they have? Can these "problème pour chercher" bring pupils to a mathematical activity?

In this communication, we are interested in tackling these questions. And to this means, we study the activity of 8-9 year old pupils while solving the "No-Three-In-Line" problem. In particular, we analyse their reasonings and propositions of proof, and their links with the Possible, the Necessity and proofs in mathematics. This analysis leads us to clarify the origins of the pupils' difficulties.

Presentation of the problem and of the sequence

In this communication, we focus on the activity of pupils about the "No-Three-In-Line" problem, introduced by Henry Dudeney in 1917. It asks for the maximum number of points that can be placed on the n × n grid so that no three points are collinear. This problem is still open, and for large n we only have conjectures (for more details, see http://wwwhomes.uni-bielefeld.de/achim/no3in/readme.html). For 2≤n≤32, we know that it is possible to put 2n points on the grid.

Scenario of the sequence

We propose the No-Three-In-Line problem for n equal to 4 or 5 to third grade pupils (Cours élémentaire 2, in France). The scenario of the sequence is the following. First, the pupils have to put the maximum number of points on a 4×4 grid. At this stage, our aim is for pupils to understand the problem and its constraints : they have to take into account not only horizontal or vertical lines, but any slanted "straight line" as well, whatever its slope. The teacher collect the pupils propositions, and pupils exclude the ones that do not obey the constraints. Then, the teacher proposes to solve a similar problem for a 5×5 grid. This time, after a long time researching and examinating the propositions, and after most of pupils have found a solution with 10 points, we ask pupils : is it worth trying to keep on researching? At this point, we expect pupils to propose some explanation. The sequence ends with a discussion about these explanations.

Characteristics of the problem

The No-Three-In-Line problem is not a classical problem for the pupils. Even though it may look like a training exercise to check alignments, the resolution needs more. The aim of this problem is not to build curricular knowledge but to take part in the construction of knowledge about reasoning and proof. This problem corresponds to a "problème pour chercher", as their characteristics are described in the official texts of the French Ministry of Education (see Appendix). It also has the main characteristics of a "problème ouvert" [START_REF] Arsac | Problème ouvert et situation-problème, IREM de Lyon, Villeurbanne Balacheff N[END_REF].

How to explain it at the end of the primary school?

At the primary school level, this problem, which is a discrete geometry problem, corresponds to an optimization problem because the 8-9 year old pupils do not have any knowledge about plane transformations and collinearity. Thus, what kind of explanation1 can they produce?

For 8-9 year olds, and for n equal to 4 or 5, the proof consists in three points2 :

1. No row or column can have more than two points on it, so one can at most put 8 (resp. 10) points on a 4×4 (resp. 5× 5) grid ;

2. On the other hand, one can indeed success in placing 8 (resp. 10) points ;

3. Consequently, the maximun number of points that one can place on a 4× 4 (resp. 5× 5) grid is attained.

The nature of these three elements is different. The first one is the product of a reasoning, considering the constraints of the problem and some knowledge about collinearity. The third one is the conclusion and also corresponds to a thought process. But, the second element strongly depends on the experience. Thus, this one is mostly related to the pragmatic register while the others are mostly related to the intellectual register. So, this proof rests both on pragmatic and intellectual registers.

Analysis of the productions of the pupils

The pupils are first asked to find correct and optimal configurations. Their activity at this stage is mostly related to the empirical register. When we ask them "is it worth trying to keep on researching?", we expect it will concern the axiomatical register. Does it correspond to the reality?

In this paragraph, we study the explanations produced in a 26 pupils classroom. These pupils worked on this problem during four one-hour-classroom sessions.

3.1 Are pupils' propositions related to some concept of proof? If yes, to which kind?

Afer some long time searching for a solution on a 5×5 grid, the teacher asked pupils : "who thinks that it is possible to put more than 10 points on this grid? Should we believe that 10 is the maximum?" Eight pupils strongly thought that one could do better than 10. The teacher proposed them to keep on searching. She asked the other pupils to explain why it was not worth trying to keep on searching?

We have translated below the main propositions given by the pupils to explain that it is impossible to put more than 10 points on a 5×5 grid:

1. "I think that there are no more than 10. Because I have checked for the rows, columns and vertically (sic) and could not find more."

2. "Because on every column and row there are two points."

3. "Because Jean said there are two points a row and a column."3 4. "Because there are 5 nodes (sic) and we cannot place more than the double." 4 5. "Because we need a 10× 10 grid to place more than 10 points."

6. "Because it is already very difficult to place 10 points. I think we cannot do better."

7. "Because we tried many times and we did not do succeed."

These explanations have been produced inside small groups and accepted inside them. So they can be consided as proof at this level (Balacheff, 1982). However, the third one has a special status : the original argument is the same as the second one, but the pupils grant credit to it, because they grant credit to Jean. Can we accept this proposition as a proof? Whatever it is, these pupils do not work on mathematical reasoning. Proposition 5 is also different: the pupils did not take into account the constraints of the problem (especially the fact that they have to work with a 5×5 grid) and proposed to change the problem.

According to Balacheff's classification [START_REF] Balacheff | Processus de preuve et situations de validation[END_REF], there are two main categories of proofs: the pragmatic ones, linked to the action, and the intellectual ones that suppose some remoteness from the action. This classification excludes the third propostion from the domain of proof. The explanations number 1, 6 and 7 are clearly pragmatic proofs, while number 2 and 4 may correspond to intellectual proofs, even if the second one can also be understood in the following way as: there are already 2 points on every line, if we add a new one there will be three points on the line. More precisely, according to Balacheff's classification of pragmatic and intellectual proofs once again, Proposition 1 matches a generic example. Indeed, the pupils consider a grid with 10 points, respecting the no-three in line constraint, and try to place another point on it. As they do not succeed, they conclude to the impossibility. So the grid which is the support of their reasoning corresponds to some generic example. In our opinion, Propositions 6 and 7 match naïve empirism, especially because of the lack of questioning about validation (it does not work because I do not succeed even with many tries). However, these two propositions seem different to us. Let us now try to analyze the pupils propositions with the help of the "problematisation" framework to make this difference precise.

Possible and Necessity in the pupils reasonings

This framework was developed by Orange [START_REF] Orange | Problème et problématisation dans l'enseignement scientifique[END_REF] in the context of didactic of sciences. It is a way to think the relationships beetween problem and solution, as Orange said (Orange, 2005, p. 70): « c'est une construction explicite du champ des possibles, en lien avec deux caractéristiques essentielles du travail scientifique : la relation dynamique entre savoirs et problèmes, et l'apodicticité (le caractère de nécessité) des savoirs produits » This concept is more and more used as a theoretical framework in France, in the domain of didactic of sciences. It is used to produce teaching sequences or to analyze students and teacher activities. It does take into account neither the «suffisance» question, wich is strongly linked with necessity in mathematics, nor the axiomatic step. However, it appears to us as a possible framework to study pupils' mathematical reasoning. Indeed, this framework takes into account epistemological aspects of knowledge building in mathematics: problems found mathematical research and teaching, their resolution generates mathematical knowledge. Moreover "Possible" and "Necessity" are two essential notions in mathematics.

Proposition 6 does not show any exploration of the possible (it only deals with the difficulty to place 10 points for many grids), whereas Proposition 7 is based on the exploration of many cases (we tried but . . . ). For this reason we think that these pupils did not build the problem the same way.

The first explanation shows that the pupils explore the possibles on a given grid, which has a generic status, with already 10 points put in place. It is a local exploration that has a generical value. The exploration leads pupils to construct a necessity: we cannot place another point, so we can only place 10 points on the grid.

In Propositions 2 and 4, a mathematical necessity appears. But, we do not have information about the way the pupils have built it.

The major difficulties of the problem

These propositions have been posted on the table and the whole class have to discuss about them. The question is especially to know if we accept them as a proof inside the classroom.

The discussion among the pupils shows the following elements. The pupils quickly set aside the third proposition: for them, it is neither an explanation, nor a proof. A girl said: "Jean can be wrong". They also quickly set aside the fifth one: this proposition changes the problem and does not give any explanation. But for the other propositions, the discussion was more difficult. If all the pupils agreed with the fact that it is already difficult to place 10 points, some kept on thinking that it was possible to do better. They were saying: "if we keep on searching, we will certainly find a better solution." So, for them, this explanation is not proof. However, none of them points to the fact that this pragmatic argument has no mathematical value; they only reject its conclusion. And many pupils who do not seem convinced do not manage to say why. Finally, the teacher said : "this is not a mathematical argument."

Many pupils also reject or do not understand Propositions 2 and 4. For them, these propositions use the constraint of the problem in the reasoning: saying that one cannot put more than 2 points on a line corresponds to the main constraint of the problem. They do not see the feature of necessity restrained in these propositions whereas, for them, this feature seems to be more clear in Propositions 1, 6 and 7. We believe that there lies the major difficulty for these pupils: it prevents them to have access to an intellectual proof.

However, some pupils accepted these intellectual proofs, but they did not consider the necessity to precise that, moreover, one succeeds in placing 10 points. For them, this return to a pragmatic aspect of the problem corresponds to some strong difficulty inherent to this problem. Indeed, they could not produce a proof of this problem.

Conclusion

Then, using in a complementary way the classification of proofs of Balacheff and the framework of "problematisation" [START_REF] Orange | Problème et problématisation dans l'enseignement scientifique[END_REF], we studied the explanations produced by 8-9 year old pupils on the "No-Three-In-Line" problem and pointed out the major difficulties that forbid pupils to reach an intellectual proof. Now, according to this analysis, let us give some elements of comparison between the pupils activity and the mathematician one [START_REF] Perrin | L'expérimentation en mathématiques[END_REF]. The experimental part of the pupils research is real; in particular, some of them use reasonings on many or generical examples. But, where the mathematician will easily distinguish between the constraints of the problem and the necessity, and more precisely, will change the constraints of the problem into necessities, the pupils get stuck. Moreover, the mathematician knows that 10 non-collinear points put on a 5×5 grid can contribute to a proof only because they prove the existence of a maximal solution, whereas this game on the status of the argument is difficult for the pupils.

In reference to the distinction ofBalacheff (1982), we speak about explanation rather than of proof.

We number these points to facilitate the recall but 1 and 2 can completely be reversed.

Jean said this before, about the

4× 4 grid. 4 In fact, these pupils confuse «nodes» for «lines».


Olivier Pitois
email: olivier.pitois@ifsttar.fr

Asmaa Kaddami

Vincent Langlois

Capillary imbibition in open-cell monodisperse foams

Keywords: porous media, fluid flows, permeability, foam

come

Introduction

Solid foams are widely used from years in many different fields thanks to their large specific surface area, low density, mechanical properties and thermal/acoustic insulation efficiency [START_REF] Gibson | Cellular Solids: Structure and Properties[END_REF]. Liquid capillary imbibition occurs in solid foams, either as wanted process or as unexpected event. For example, clean-up of oily pollution and oil-water separation can be performed by using solid foams.

In such a case, oil is sucked by the foam whose surface has been made hydrophobic [START_REF] Zhang | Bioinspired multifunctional foam with self-cleaning and oil/water separation[END_REF][START_REF] Luo | Bio-inspired modification of silicon carbide foams for oil/water separation and rapid power-free absorption towards highly viscous oils[END_REF]. Food industry uses increasingly dried foamed products which have to be imbibed and dissolved in water [START_REF] Hardy | Foam-mat drying technology: A review[END_REF]. Imbibition is required for impregnation purpose with simple or complex fluids in order to modify the properties of the foam surface. This increases drastically the spectrum of properties which can be given to foams, i.e. superhydrophobicity [START_REF] Zhang | One-step fabrication of robust superhydrophobic and superoleophilic surfaces with self-cleaning and oil / water separation function[END_REF][START_REF] Gao | Robust Superhydrophobic Foam : A Graphdiyne-Based Hierarchical Architecture for Oil / Water Separation[END_REF], antibacterial [START_REF] Jain | Potential of Silver Nanoparticle-Coated Polyurethane Foam As an Antibacterial Water Filter[END_REF], electrical [START_REF] Xinzhao | Electrically conductive graphene-coated polyurethane foam and its epoxy composites[END_REF] or fire-safe [START_REF] Xie | A green approach to fabricating nacre-inspired nanocoating for super-efficiently fire-safe polymers via one-step self-assembly[END_REF] properties. On a different note, durability of foamed construction materials can be jeopardized by successive imbibition/drying cycles. Therefore, controlling sorption properties of those materials is a crucial issue and water repellents can be used for preventing or decreasing water absorption [START_REF] Ma | Properties of foamed concrete containing water repellents[END_REF].

For each example given above, the rate at which the porous material can be filled due to capillary imbibition is a crucial aspect, and theoretical prediction of the imbibition kinetics is of great interest. Said in another way, designing materials with appropriate imbibition properties requires understanding how sorption behavior depends on the material's morphology. Theory of capillary imbibition has been intensively studied from the pioneering works of Bell & Cameron [START_REF] Bell | The flow of liquids through capillary spaces[END_REF], Lucas [START_REF] Lucas | Ueber das Zeitgesetz des kapillaren Aufstiegs von Flüssigkeiten[END_REF] and Washburn [START_REF] Washburn | The dynamics of capillary flow[END_REF]. They assumed that a wet front progresses through the sample under the action of capillary suction while saturating the material behind. Such a basic theory has to be modified to account for the specific microstructure encountered in various porous materials [START_REF] Dullien | Porous media : fluid transport and pore structure[END_REF]. It is to say that such a specific work has not been performed to date for solid foams.

We investigate the imbibition behavior of solid foams characterized by independent pore size and solid volume fractions. The results allow for a consistent description of the capillary imbibition process in foams. Theory of capillary imbibition is presented in section 2. Foam samples are produced and characterized in section 3. Imbibition results are presented and discussed in section 4.

Theory

Let us briefly recall the classical theory for capillary imbibition of liquids through porous media.

Here is considered the vertical capillary rise of liquid at height ℎ through a porous material with porous volume fraction . By introducing the mean curvature for the water interface in the pores, 1 ̅ ⁄ , the driving capillary pressure is given by the Laplace law: ∆ = 2 ̅ ⁄ . Note that in the following we consider the driving pressures as positive values, i.e. ∆ = |2 ̅ ⁄ |. The progression of the interface at height ℎ is restrained by gravity, so the effective driving pressure is ∆ = ∆ -ℎ .

The resulting interface velocity (= ℎ ⁄ ) can be estimated via the Darcy law: = ⁄ × ∆ ℎ ⁄ , where is the fluid dynamic velocity, the permeability, and (= ) the Darcy velocity.

The equation for the evolution of the interface height is therefore:

= ∆ ! " # - $% ! = & # -' (eq. 1)
where the parameters

( = ∆ ! et ' = $% !
have been introduced. Note that the ratio ( ' ⁄ corresponds to the Jurin height: ℎ ) = ∆ ⁄ . The solution of equation ( 1) is given by [START_REF] Washburn | The dynamics of capillary flow[END_REF]:

= - # * - & * + ,-.1 - *# & / (eq. 2)
Note also that equation ( 1) simplifies for ℎ ≪ ℎ ) and the Washburn solution can be conveniently used:

≅ & # = ∆ ! " # (eq. 3)
ℎ ≅ 2( " 2 ⁄ = .

2 ∆ ! / " 2 ⁄

(eq. 4)

Production and characterization of the solid foam samples

Studied solid foams are geopolymer foams produced from metakaolin (MK) particles by using a dedicated process presented in the following.

Materials

We used an activating solution prepared by mixing NaOH solution (mass concentration Cw = 0.35) with a solution containing Na2O (Cw = 0.08) and SiO2 (Cw = 0.27) provided by MERCK KGaA, and water. The density is ℓ = 1400 kg/m 3 and the chemical composition is given by molar ratios H2O/Na2O = 14.68 and SiO2/Na2O = 1.01. The typical setting time is about two hours.

Glucopon® 225 DK (alkyl polyglycoside provided by BASF) was used as surfactant for foaming purpose. Surface tension of Glucopon water solutions was measured (using a pendant drop tensiometer) to be = 30 mN/m for concentrations above the critical micellar concentration.

Dynamic viscosity of such water solutions is very close to that of pure water, i.e. ≅ 10 -3 Pa.s. Such a nonionic surfactant of molar mass equal to 420 g/mol was shown to be suitable to stabilize the foam morphology during the two hours preceding the geopolymerization stage.

Production

The method used for the production of controlled MK foams has been presented in a previous paper [START_REF] Kaddami | A physical approach towards controlling the microstructure of metakaolin-based geopolymer foams[END_REF]. Here we recall the main steps but the reader can refer to [START_REF] Kaddami | A physical approach towards controlling the microstructure of metakaolin-based geopolymer foams[END_REF] for more details.

In broad outline, metakaolin foams were produced by mixing precursor aqueous foam with the metakaolin (MK) suspension described previously. The production process consists of two steps: [START_REF] Gibson | Cellular Solids: Structure and Properties[END_REF] production of the precursor aqueous foam, (2) mixing of the precursor aqueous foam with the MK suspension and in-line filling of the resulting MK fresh foam in a vessel.

Step 1: Aqueous foam was generated by pushing both gas (nitrogen) and foaming liquid (glucopon water solution) through a T-junction. Tuning the gas/liquid flow rates ratio allowed producing monodisperse bubbles with chosen diameters ranging between 200 µm and few millimeters. Generated bubbles were continuously collected into a vertical glass column. The average bubble diameter ; * was measured using a camera focused at the wall of the column. For foam made with bubble size ; * ≳ 500 µm, ripening turned out to be insignificant over the duration of the production step, so highly monodisperse precursor aqueous foams were obtained. In contrast, smaller bubble sizes were observed to evolve during the production step because of the ripening process. This can be explained by both the increase of the ripening rate for smaller bubbles [START_REF] Cantat | Foams: Structure and Dynamics[END_REF] and the decrease of the production rate (larger time required to produce the same volume of foam) for smaller bubbles. Such an evolution was efficiently counteracted by using perfluorohexane as a saturating vapor in the bubbling gas [START_REF] Gandolfo | Interbubble Gas Diffusion and the Stability of Foams[END_REF].

Step 2: After production of the required precursor foam volume, the latter was pushed with the MK suspension through a static mixer (flow focusing device and classical helical geometries). The gas volume fraction within the resulting MK foam could be adjusted by tuning the ratio of flow rates for the precursor foam and the MK suspension. The resulting fresh foam was continuously pushed in a sample vessel of diameter 26 mm and height = > = 50 mm. Typical volume flow rates for the production of the studied samples were of the order of 10 mL/min. The vessel was filled to the top and closed. Note that the values for the solid volume fraction ? were also determined a posteriori by comparing the measured density of the fresh MK foam with the density of the MK suspension.

After 2 days in the closed vessels, samples were let for drying at room temperature for 2 days before unmolding.

Foam microstructure

Optical microscopy (see Figure 1c) was used to observe the foam structure and to measure the size of both pores and apertures that connect them. In such a case, mean values (over ≈ 100 measurements) will be reported.

In addition, X-ray tomography (EasyTom RX Solutions) was used to obtain size distributions.

Images of small samples (i.e. 5x5x10 mm) were obtained with a Ultratom scanner from RX solutions.

Measurement involved a Hamamatsu L10801 X-ray source (160 kV) and a Paxscan Varian 2520 V atpanel imager. All scans were performed at 60 kV and 85 µA. Frame rate was 3 images per second and 12 images were averaged to produce one projection (the resulting effective exposure time was therefore 4s). 3D tomographic reconstruction was performed with the X-Act commercial software developed by RX-Solutions. Voxel size for the obtained images was 5 µm. Pores appeared black on the reconstructed images and interstitial geopolymer solid was light grey. This allowed us to analyze the images with the freeware ImageJ program [START_REF] Rueden | ImageJ2: ImageJ for the next generation of scientific image data[END_REF] to compute the pore size distribution: First, a closing filter from MorphoLibJ plugin [START_REF] Legland | MorphoLibJ: integrated library and plugins for mathematical morphology with ImageJ[END_REF] with a 2 voxel-radius ball element was applied to reduce noise from the images. Then, image threshold was calculated using the Otsu method [START_REF] Otsu | A Threshold Selection Method from Gray-Level Histograms[END_REF]. 3D

Watershed from MorphoLibJ was applied, then 1-voxel dilatation filter. Finally, 3D Object Counter plugin returned the volume ( 4,A ) of each pore, from which the equivalent pore diameter B6 4,A D ⁄ E "/G was calculated.

The produced samples are shown in Fig. 1a-c. Highly monodisperse open cell foams are obtained with several pores sizes (Fig. 1d). The effect of solid volume fraction is shown in Fig. 1c:

foam can be described as interconnected struts at small ? values, spherical pores with small circular apertures are seen for larger ? values. The decrease of the mean aperture diameter H as a function of solid volume fraction is presented in Fig. 1e. Note that classical models proposed for packed beds of solid spheres overestimate significantly the foam permeability. For example, at ? ≈ 0.2, the Carman-Kozeny model [START_REF] Carman | Fluid flow through granular beds[END_REF][START_REF] Kozeny | Ueber kapillare Leitung des Wassers im Boden[END_REF] gives 2 10 -2 for the dimensionless permeability, while the measured value is 2 10 -3 (overestimation by factor 10).

Actually, available models do not capture the exponential decrease and most of them overestimate our permeability data [START_REF] Kumar | State-of-the-Art of Pressure Drop in Open-Cell Porous Foams: Review of Experiments and Correlations[END_REF]. In the following we use permeability values given by equation 5.

Imbibition experiment

Water imbibition was studied by measuring the capillary rise through foam samples. Samples were put in contact with a thin layer (3 mm) of Glucopon solution (1.2 g/L) in a glass container with large cross-section area (800 cm 2 ). The use of the surfactant solution provided good wetting conditions of the liquid at the walls of the foam skeleton, i.e. the solid/liquid contact angle ^ was expected to be such that _`a^≅ 1. The rising front ℎ was followed as a function of time using a camera (see the picture in figure 3). The liquid front was always easy to be distinguished due to appropriate contrast between filled and dry porosities. However, after the expected saturation height for imbibition of the foams pores (the so-called Jurin height), a further increase of the front height could be observed (see Figure 1b) due to the imbibition of the porous skeleton (i.e. geopolymer matrix). Note that this second regime is classical for porous media characterized by two pore sizes [START_REF] Bico | Wetting of textured surfaces[END_REF][START_REF] Lerouge | Synergistic actions of mixed small and large pores for capillary absorption through biporous polymeric materials[END_REF]. Here, the pores of geopolymer matrix are very small [START_REF] Davidovits | Geopolymers[END_REF] and the associate Washburn velocity is negligible compared to the one corresponding to the large pores. As we were mostly interested in the imbibition properties provided by the foam geometry of such foam material, rather than the intrinsic imbibition properties of the geopolymer matrix, in the following we consider only the first and rapid regime.

Results and discussion

Imbibition curves

Typical results for liquid imbibition are presented in figure 4a. Note that data are plotted as a function of " 2 ⁄ in order to identify the Washburn regime. Each curve is characterized by rapid increase which is, indeed, in reasonable agreement with the Washburn behavior. It appears that the velocity depends on both the pore size and solid volume fraction. As expected, the front height saturates at longer times to a value which increases as the pore size decreases. Note that inertial effects are expected to be limited to the very first instants (or heights) of the imbibition process [START_REF] Fries | The transition from inertial to viscous flow in capillary rise[END_REF], so in the following we restrict our analysis to the Washburn equations (i.e. eqs 1 and 2). Equation 2 is now compared against experimental values ℎ , as shown in Fig. 4b. Note that the value for the saturation height ℎ ) sets the value for ∆ , so fitting equation 2 to our data provides the value for the second parameter, that is, foam permeability . Following such a procedure, we obtain values for ∆ and , as presented in Figs 5a&b respectively. Note that the saturation height with ; 4 ≈ 200 µm samples, was found to be larger than the sample heights so the capillary pressure could not be measured. For such cases however, it was assumed that the values for the dimensionless capillary pressure are similar to values measured for other samples (within the spread observed for the data in Fig. 5a -see the red crosses), so the corresponding permeability was also deduced by fitting the theoretical curve to the data.

1) as a function of foam permeability, as

measured by Darcy experiment (Fig. 2). Each symbol corresponds to a given pore size, as indicated.

The solid line corresponds to = . The dashed line corresponds to = 10 ⁄ .

Capillary pressure

Let us first discuss values obtained for the capillary pressure ∆ . It is shown in Fig. 5a Note that the maximal value for ∆ , i.e. 4 ; 4 ⁄ , corresponds to |1 ̅ ⁄ | ≈ 2 ; 4 ⁄ , which means that the radius of curvature for the liquid-gas interface is approximately equal to the pore radius.

This can be understood from the spherical shape of the pores surface for such ? values (see Fig. 1c).

As ? decreases, the size of the apertures connecting the pores increases as well and the resulting pore surface is increasingly open. For the smallest ? values, the foam solid structure mainly consists in an assembly of struts. Such structure has some similarity with bundles of cylindrical fibers as studied by Princen [START_REF] Princen | Capillary phenomena in assemblies of parallel cylinders. II. Capillary rise in systems with more than two cylinders[END_REF]. Therefore, the decrease observed for the capillary pressure is expected to be related to the evolution of the pore geometry. In order to quantify this effect, we propose to consider the middle cross-section g of a single representative pore, such as the one shown in Fig. 6a.

Mechanical equilibrium for the liquid within the structure is reached when the liquid weight, ℎg , is supported exactly by the surface tension along the wetted perimeter ℓ of the pore surface, ℓ. As ℎ = Δ at equilibrium, the dimensionless capillary pressure writes: ∆ B ; 4 ⁄ E ⁄ = ℓ; 4 g ⁄ . and ? ? = 0.48. For smaller ? values, ℓ decrease from D; 4 to 0 whereas g decreases weakly. Note the particular behavior for the BCC structure around ? ≅ 0.11 due to the transition from 14 apertures (Kelvin cell) for ? ≲ 0.11 to 8 apertures for ? ≳ 0.11 [START_REF] Weaire | The Physics of Foams[END_REF]. Using values presented in Figs 6b&c, the dimensionless capillary pressure is calculated for the three structures and the results are plotted in Fig. 7 against experimental data. The FCC structure cannot be used to describe our results for ? Q 0.26 and the SC structure appears to underestimate significantly the data. In contrast, the BCC structure describes reasonably well the measured capillary pressures over the studied range of solid volume fraction. It is to say that the Kelvin structure is often used as ideal structure in order to evaluate foam properties such as drainage [START_REF] Cantat | Foams: Structure and Dynamics[END_REF], mechanics [START_REF] Jang | On the compressive strength of open-cell metal foams with Kelvin and random cell structures[END_REF] or acoustics [START_REF] Trinh | Tuning membrane content of sound absorbing cellular foams: Fabrication, experimental evidence and multiscale numerical simulations[END_REF]. Note that more general models have been proposed to describe the capillary pressure on porous media. For example, Masoodi & Pillai [START_REF] Masoodi | A general formula for capillary suction-pressure in porous media[END_REF] proposed the following expression:

Δ = ? 1 -? Γ ⁄
, where Γ is the volume-to-surface area of the solid matrix. The corresponding dimensionless capillary pressure thus writes ∆ B ; 4 ⁄ E ⁄ = ; 4 g ? 1 -? ⁄ where g ? is the specific surface area, i.e. the surface area of the solid matrix in contact with the fluid divided by the total volume of porous material. Such a parameter can be found elsewhere [START_REF] Cantat | Foams: Structure and Dynamics[END_REF][START_REF] Pitois | Specific surface area model for foam permeability[END_REF] for the BCC structure. The calculated values for ∆ B ; 4 ⁄ E ⁄ are plotted in Fig. 7 and show significant overestimation of our experimental data. , where g ? is the specific surface area for the BCC structure.

Apparent permeability

For an ideal porous material, it is expected that = = . Here, permeability values deduced from imbibition results (see Fig. 5b) were found to be one order of magnitude smaller than measured Darcy permeability . In other words, using the measured foam permeability values in equation 2 would give imbibition velocity 10 times larger than the measured velocity (see Fig. 4b). In order to understand this discrepancy we record movies from the magnified surface of the sample (at the scale of ~10 pores, see Fig. 3c). This reveals a halting pore-filling process: from pore to pore, the capillary ascending motion pauses regularly (see the space-time image). It is recognized that interface motions can be slowed down or even stopped by passing through pore junctions [START_REF] Sadjadi | Meniscus arrest during capillary rise in asymmetric microfluidic pore junctions[END_REF] or through capillaries with significant expansion of the cross-section [START_REF] Erickson | Numerical simulations of capillary-driven flows in nonuniform cross-sectional capillaries[END_REF][START_REF] Mehrabian | Wicking flow through microchannels[END_REF] and such intermittent front rise has been observed in similar cellular materials [START_REF] Lerouge | Synergistic actions of mixed small and large pores for capillary absorption through biporous polymeric materials[END_REF]. In order to reconcile and values, we introduce an additional time l corresponding to the time spent by the fluid to pass through the aperture between two neighbor pores. Note that the vertical motion in pores is still considered as mainly governed by the balance between capillary effects and Darcy's law (leading to eqs 1 and 4).

Therefore, the average time for the interface to travel a length ; 4 is equal to ; 4 ⁄ O l, and the modified imbibition velocity is: The physical mechanism involved in time l is not known so we turn to a phenomenological approach to identify a functional form that allows for data to be understood with eq. 7. Our results for imbibition show a Washburn behavior for ℎ ≪ ℎ ) , which suggests that l is proportional to ℎ: l = uB ? , ; 4 v w" ⁄ E × ℎ ⁄ , where v w" = x ⁄ is the capillary length and u is a dimensionless function. As ,m ≡ ∆ ℎ ⁄ , the apparent permeability writes as a function of uB ? , ; 4 v w" ⁄ E:

= z " S O ∆ U c u| w"
(eq. 8)

Note that the term ; 4 Δ u ⁄ in eq. 8 can be considered as an additional permeability associated in series with the Darcy permeability. We have sought for uB ? , ; 4 v w" ⁄ E as a power law function, that is, u = }B; 4 v w" ⁄ E ~ ?

• , where } is a numerical constant. As shown in Fig. 8 it is possible to obtain parameters }, Z, € such that eq. 8 is fulfilled for all the values and : u~B

; 4 v w" ⁄ E ? G 2 ⁄ .
Equivalently, the time l expresses as l B ; 4 ⁄ E ⁄ ~ ?

G 2 ⁄ ℎ v w" ⁄ and it is expected to be related to the size of the apertures as well as the geometry of the surrounding borders. The porosity of the geopolymer matrix may also influence the time l. A dedicated study of the interface breakthrough should be performed at the scale of the single aperture in order to understand and clarify the reported dependence for time l. ⁄ with } = 34 10 3 . Each symbol corresponds to a given pore size, as indicated.

Conclusion

Solid foam samples were produced with monodisperse pore sizes and controlled pore volume fractions. Vertical imbibition experiments were performed in order to measure both the imbibition rate and the saturation height. Saturation heights allowed us to deduce the driving capillary pressure in the foam structures, showing values inversely proportional to pore size. Moreover, the capillary pressure was found to decrease as a function of the pore volume fraction, which was understood by the evolution of the pore geometry, from spherical shape of the pore surface, for pore volume fractions close to 60%, to increasingly open pore surface for higher pore volume fractions. Moreover, these experimental values were found to be consistent with values calculated by adapting the approach proposed by Princen [START_REF] Princen | Capillary phenomena in assemblies of parallel cylinders. II. Capillary rise in systems with more than two cylinders[END_REF] for bundles of cylindrical fibers. Here, the calculation based on the ideal BCC (Kelvin) foam structure was found to provide values very close to experimental values.

Eventually, the dimensionless values reported in this paper for the capillary pressure can be used easily to estimate saturation of liquids in various foamed materials. We recall that experiments were conducted within good wetting conditions, i.e. ^≅ 0. For non-vanishing contact angle, one can anticipate that the driving capillary pressure is decreased by a factor _`a^, i.e. ∆ ≈ _`a^ 1.95 O 6.4 ? ; 4 ⁄ can be used in eq. 4 for example. This behavior is expected for contact angles below the critical value ^ , for which ∆ = 0. Further study could be focused on the the effect of contact angle, and more precisely, to the determination of ^ .

The measured imbibition rate was found to be qualitatively compatible with the Washburn theory [START_REF] Bell | The flow of liquids through capillary spaces[END_REF][START_REF] Lucas | Ueber das Zeitgesetz des kapillaren Aufstiegs von Flüssigkeiten[END_REF][START_REF] Washburn | The dynamics of capillary flow[END_REF], i.e. the imbibition length is proportional to the square root of time, but it is one order of magnitude smaller than the values predicted from the Darcy foam permeability measured on the same samples. Our result clearly shows that the use of the classical Washburn theory can induce overestimation of the imbibition rate by factor 10. Overestimation by factor 100 can even be reached if appropriate values for both foam permeability (eq. 5) and foam capillary pressures (eq. 6)

are not used.

In the last part of this work we tried to reconcile theory with measured imbibition rates. From video-microscopy performed at the pore scale, we identify an excess time spent by the liquid-gas interface at connections between pores and we introduced this excess time in the expression of the imbibition velocity. This excess time is believed to be related to the particular motions for the interface when it reaches pore junctions [START_REF] Sadjadi | Meniscus arrest during capillary rise in asymmetric microfluidic pore junctions[END_REF] or areas characterized by sharp changes of channel cross-section [START_REF] Erickson | Numerical simulations of capillary-driven flows in nonuniform cross-sectional capillaries[END_REF][START_REF] Mehrabian | Wicking flow through microchannels[END_REF]. A dedicated study should be performed at the scale of a single pore connection in order to clarify this effect. Here we turned to a phenomenological approach and an empirical expression was provided for the excess time. Additionally, studies should be performed at both pore scale and sample scale with several couples liquid/solid in order to quantify the effect of contact angle as it becomes close to the critical contact angle ^ .

Expressions provided for both foam permeability, foam capillary pressure and excess time, can be used to estimate imbibition rates in solid foams. For example, this would be helpful to control foam imbibition and impregnation processes involved in the design of foamed materials with improved functional properties, such as superhydrophobicity [START_REF] Zhang | One-step fabrication of robust superhydrophobic and superoleophilic surfaces with self-cleaning and oil / water separation function[END_REF][START_REF] Gao | Robust Superhydrophobic Foam : A Graphdiyne-Based Hierarchical Architecture for Oil / Water Separation[END_REF], antibacterial [START_REF] Jain | Potential of Silver Nanoparticle-Coated Polyurethane Foam As an Antibacterial Water Filter[END_REF], electrical [START_REF] Xinzhao | Electrically conductive graphene-coated polyurethane foam and its epoxy composites[END_REF] or fire-safe [START_REF] Xie | A green approach to fabricating nacre-inspired nanocoating for super-efficiently fire-safe polymers via one-step self-assembly[END_REF] properties.

Figure 1 :

 1 Figure 1: Microstructure of the solid MK foams. (a) Picture of a sample seen from above (; 4 ≈ 600 µm, ? = 0.2). (b) SEM images of foam microstructure (; 4 ≈ 800 µm, ? = 0.15). (c) Closeup of foam samples revealing the open-cell microstructure and showing pores (continuous lines) and pore apertures (dotted lines). Top: ; 4 ≈ 800 µm, ? = 0.08; bottom: ; 4 ≈ 800 µm, ? = 0.3. (d) Volume-weighted pore size distribution (probability density function) obtained from the X-ray tomography method, for three samples. The three Gaussian curves are plotted with mean and standard deviation equal to: 200 µm and 15 µm, 770 µm and 40 µm, 1070 µm and 40 µm. (e) Diameter of apertures H (measured using optical microscopy) divided by the pore diameter ; 4 as a function of solid volume fraction ? for different pore sizes as indicated. As indicated, the continuous lines correspond to values obtained by using the Surface Evolver software (see the main text for details) for the Kelvin/BCC structure, the FCC structure and the SC structure.

3. 4 .

 4 Fluid permeabilityGas-flow permeability was measured with a home-made setup. A gas flow controller was used to push dry nitrogen through the foam sample (length 9 and circular cross-section : = 5.3 cm 2 ) embedded with PTFE ribbon and fitting a cylindrical tube. The pressure difference ∆ across the sample was measured thanks to a manometer. Several measurements were performed as a function of the gas flow rate I and the foam permeability was deduced from the Darcy law:= J & 9 : ⁄ K J Δ I ⁄ K ⁄ ,where & = 1.8 10 -5 Pa.s is the dynamic viscosity of the gas. A typical example is presented in Fig. 3a. Most of the measurements were performed within the identified linear regime characterized by low Reynolds number MN = I : ⁄ ; 4 & ⁄ ≪ 1, where ≅ 1.2 kg/m 3 . For the largest pore sizes this condition was not fully satisfied so we used the Darcy-Forchheimer equation: ∆ 9 ⁄ = & ⁄ × I : ⁄ O P I : ⁄ 2 , where P is the non-Darcy flow coefficient. Darcian permeability was obtained from a quadratic regression performed for the pressure gradient ∆ 9 ⁄ as a function of I : ⁄ .

Figure 2 : 6 ⁄

 26 Figure 2: Darcy permeability of the studied foam samples divided by 4 2 G ⁄ = BD; 4 G 6 ⁄ E 2 G ⁄ . Each empty symbol corresponds to a given pore size, as indicated. The dashed line corresponds to eq. 5.Note that permeability vanishes for ? Q ? * ≅ 0.36, i.e. Random Close Packing of spheres.

Figure 3 :

 3 Figure 3: Imbibition experiment. (a) Image of the solid foam sample during imbibition. The sample is put in contact with water solution and the wet front height, ℎ, is followed as a function of time. The front rise is better highlighted through the space-time image shown in (b). The red points and line highlight the water saturation front for the foam pores (macropores). The dotted line highlights the water saturation front in the porous solid skeleton of the foam. The corresponding velocity is small, so the second front is better distinguished after the water front stops in the macropores. (c) Close-up of the pore filling process during imbibition (left) and corresponding space-time image (right). The blue line shows the positions of the water front.

Figure 4 :

 4 Figure 4: Typical imbibition curves, i.e. capillary rise height as a function of time. (a) Height ℎ as a function of " 2 ⁄ for samples. Squares: ; 4 = 1030 µm and ? = 0.33; Circles: ; 4 = 630 µm and ? = 0.28; Diamonds: ; 4 = 300 µm and ? = 0.17; Triangles: ; 4 = 195 µm and ? = 0.18. The dotted lines corresponds observed saturation heights. (b) Height ℎ as a function of time for sample characterized by ; 4 = 300 µm and ? = 0.17. Solid line corresponds to equation 2 with parameters: ∆ = 333 Pa (equivalently ℎ ) = 34 mm), = 6 10 -11 m 2 . The dashed line corresponds to equation 2 with parameters: ∆ = 333 Pa (equivalently ℎ ) = 34 mm), = = 22.6 10 -11 m 2 .

Figure 5 :

 5 Figure 5: (a) Capillary pressure b divided by ; 4 ⁄ as a function of solid volume fraction ? . Each symbol corresponds to a given pore size, as indicated (crosses correspond to samples with pore size close to 200 µm, for which the dimensionless capillary pressure could not be measured but was assumed to be similar to other samples). The dashed line corresponds to eq. 6. (b) Foam permeability as deduced from imbibition experiments (see eq. 1) as a function of foam permeability, as

Fig. 6 : 2 ⁄ = D 4 ⁄

 624 Fig. 6: (a) Sketch of single representative foam pore (cut at the middle cross-section for clarity). In each pore, surface tension acting at the solid surface (shown by the arrows) supports the column of liquid beneath the interface. (b) Wetted solid length ℓ divided by ; 4 as a function of solid volume fraction for three ideal foam structures: FCC, BCC (Kelvin) and SC. (c) Pore cross section g divided by ; 4 2 for three ideal foam structures: FCC, BCC (Kelvin) and SC.

Fig. 7 :

 7 Fig. 7: Capillary pressure b divided by ; 4 ⁄ as a function of solid volume fraction ? . Black diamonds correspond to experimental data presented in Fig. 5a. Solid lines correspond to equation ∆ B ; 4 ⁄ E ⁄ = ℓ; 4 g ⁄ with ℓ and g given by Fig. 6 for the three studied foam structures: FCC, BCC (Kelvin) and SC. The dotted line corresponds to equation ∆ B ; 4 ⁄ E ⁄ = ; 4 g ? 1 -? ⁄ , where g ? is

Figure 8 :

 8 Figure 8: Permeability values given by eq. 7 as a function of permeability values deduced from imbibition experiments. In eq. 8, values are given by eq. 5, ∆ values are presented in Fig. 5a, and u = }B; 4 v w" ⁄ E ? G 2

  Metakaolin (MK) particles were provided by AGS Minéraux (France): Argical M 1200s. Chemical compositions: SiO2 55.0%, Al2O3 39.0%, Fe2O3 1.8%, TiO2 1.5%, K2O+Na2O 1.0%, CaO+MgO 0.6%. As provided by the supplier: BET specific surface area is equal to 19 m 2 /g and mass average diameter is approximately 4 ≈ 2 µm. Their density is 67 = 2200 kg/m 3 .

	The activated MK suspensions were prepared by mixing a mass 8 ℓ of the activating solution with a mass 8 67 of metakaolin particles. The suspension is characterized by the ratio 9 : ⁄ = 8 ℓ 8 67 ⁄ . For the present study 9 : ⁄ was within the range 1.75-2. The chemical composition of those
	samples is given by Si/Al and Na2O/Al2O3 ratios, respectively within the ranges 1.8-2.5 and 1.2-2.7.

Acknowledgments

The authors thank P. Aimedieu (at Navier Laboratory) for technical support with the X-ray tomography. This work has benefited from French government Grant managed by the Agence Nationale de la Recherche [Grant number ANR-13-RMNP-0003-01].

The measured rise velocity of liquid in solid foam is one order of magnitude smaller than the predicted value. This deviation is attributed to the excess time spent by the liquid-gas interface at connections between pores.


Lech Zieliński

Semiclassical Distribution of Eigenvalues for Elliptic Operators with Hölder Continuous Coefficients, Part I: Non-critical Case

Keywords: Mathematics Subject Classification (2000): 35P20 spectral asymptotics, semiclassical approximation, Weyl formula, elliptic operators

We consider a version of the Weyl formula describing the asymptotic behaviour of the counting function of eigenvalues in the semiclassical approximation for self-adjoint elliptic differential operators under weak regularity hypotheses. Our aim is to treat Hölder continuous coefficients and to investigate the case of critical energy values as well.

Introduction

Since the papers of J. Chazarain [START_REF] Chazarain | Spectre d'un hamiltonien quantique et mécanique classique[END_REF] and B. Helffer, D. Robert [START_REF] Helffer | Comportement semi-classique du spectre des hamiltoniens quantiques elliptiques[END_REF], the semiclassical spectral asymptotics have been investigated in numerous works and we refer to the monographs [START_REF] Dimassi | Spectral asymptotics in the semiclassical limit[END_REF], [START_REF] Ivrii | Microlocal analysis and precise spectral asymptotics[END_REF], [START_REF] Levendorskii | Asymptotic distribution of eigenvalues of differential operators[END_REF] and [START_REF] Robert | Autour de l'approximation semi-classique[END_REF]. The main results have been obtained by using the tools of the microlocal analysis based on the approach of L. Hörmander [START_REF] Hörmander | The spectral function of an elliptic operator[END_REF]. However this approach works only for smooth problems and the semiclassical framework is usually considered for a non-critical energy value. Our aim is to present a method of obtaining semiclassical estimates for more general classes of differential operators. (A) Formulation of the results Let r ∈]0; 1] and denote by B r the set of bounded, Hölder continuous functions of exponent r on IR d , i.e. a ∈ B r means that a ∈ L ∞ (IR d ) and 1 there is C > 0 such that (1.1) |a(x) -a(y)| ≤ C|x -y| r (x, y ∈ IR d ).

Let m ∈ IN * and for ν, ν ∈ IN d , |ν|, |ν| ≤ m we consider real-valued a ν,ν ∈ B r such that a ν,ν = a ν,ν and

(1.2)

|ν|=|ν|=m a ν,ν (x)ξ ν+ν ≥ c|ξ| 2m (x, ξ ∈ IR d ),
holds for a certain constant c > 0.

For h > 0 let A h be the quadratic form defined for ϕ, ψ ∈ C m 0 (IR d ) by

(1.3) A h [ϕ, ψ] = |ν|,|ν|≤m
(a ν,ν (hD) ν ϕ, (hD) ν ψ),

where (•, •) is the scalar product of L 2 (IR d ) and (hD) ν = (-ih) |ν| ∂ ν /∂x ν . The ellipticity hypothesis (1.2) ensures the fact that A h is bounded from below and its closure defines a self-adjoint operator A h . We introduce

(1.4) a(x, ξ) = |ν|,|ν|≤m a ν,ν (x)ξ ν+ν ,
and for E ∈ IR we denote (1.5) Γ E = a -1 (] -∞; E[) = {(x, ξ) ∈ IR 2d : a(x, ξ) < E}.

We have Proposition 1.1 Let E, E 0 ∈ IR be such that E < E 0 and Γ E 0 is bounded. Then one can find h 0 > 0 such that for h ∈]0; h 0 ], the spectrum of A h is discrete in ] -∞; E].

Further on E 0 , E, h 0 are as in Proposition 1.1 and |Γ E | = a(x,ξ)<E dxdξ is the Lebesgue measure of Γ E . For h ∈]0; h 0 ] we define the counting function N (A h , E) as the number of eigenvalues (counted with their multiplicities) smaller than E. Our principal result is: Theorem 1.2 Let A h be as above with a ν,ν ∈ B r for a certain r ∈]0; 1]. If µ ∈]0; 2r 2+r [, then we have

(1.6) N (A h , E) = |Γ E |(2πh) -d + (|Γ E+h µ | -|Γ E-h µ |)O(h -d ).
Similarly as in [START_REF] Buzano | Weyl formula for hypoelliptic operators of Schrödinger type[END_REF] one can observe that some additional conditions on a are needed to obtain a good estimate of |Γ E+h µ | -|Γ E-h µ | as h → 0. In this paper we are interested in the following condition (1.7) a(x, ξ) = E ⇒ ∇ ξ a(x, ξ) = 0.

If (1.7) holds then E will be called a non-critical energy value and it is easy to see that this condition ensures

|Γ E+h µ | -|Γ E-h µ | = O(h µ ).
Moreover it is possible to obtain the following stronger estimates:

Theorem 1.3 We assume moreover (1.7). If µ ∈]0; r[, then we have (1.8) N (A h , E) = |Γ E |(2πh) -d + O(h µ-d ).

(B) Comments

The proof of Theorem 1.3 is presented below and a suitable development, which allows us to prove Theorem 1.2, will be described in [START_REF] Zielinski | Semiclassical distribution of eigenvalues for elliptic operators with Hölder continuous coefficients, Part II: critical case[END_REF]. The basic idea is to replace irregular coefficents by smooth ones and to investigate the corresponding smooth problem following some ideas of our earlier papers [START_REF] Zielinski | Asymptotic behaviour of eigenvalues of differential operators with irregular coefficients on a compact manifold[END_REF][START_REF] Zielinski | Asymptotic distribution of eigenvalues of some elliptic operators with intermediate remainder estimates[END_REF]. In the case of a non-critical energy value it is also possible to investigate the smooth problems adopting the theory developed in the book of V. Ivrii [START_REF] Ivrii | Microlocal analysis and precise spectral asymptotics[END_REF], deducing Theorem 1.3 according to [START_REF] Ivrii | Sharp spectral asymptotics for operators with irregular coefficients[END_REF] (and we have also (1.8) with the optimal value µ = 1 if the first order derivates of the coefficients are Hölder continuous, cf. [START_REF] Ivrii | Sharp spectral asymptotics for operators with irregular coefficients[END_REF] or [START_REF] Zielinski | Semiclassical Weyl formula for elliptic operators with non-smooth coefficients[END_REF]).

However the approach we present here is quite different from [START_REF] Ivrii | Sharp spectral asymptotics for operators with irregular coefficients[END_REF] or [START_REF] Zielinski | Semiclassical Weyl formula for elliptic operators with non-smooth coefficients[END_REF]. It seems to us that the most interesting feature of this approach is the possibility of investigating non-critical and critical energy values in a quite similar way and the fact that the analysis of the smooth problem is working for suitable classes of pseudodifferential operators as well.

A general plan is the following. In Section 2 we define the regularized operators P h and the Fourier transform allows us to express suitable functions fh (P h ) by means of the evolution group U t = e itP h /h µ . In Section 3 we describe an approximation of U t giving a pseudodifferential approximation of fh (P h ) with correct asymptotic properties. The correct asymptotic behaviour of the approximation is proved at the end of Section 3 by means of simple integrations by parts. In Section 4 we explain how to implement a similar idea to estimate the difference between fh (P h ) and the approximation. The final computations justifying this idea are presented in Section 5 and some standard supplementary details are given in Section 6.

(C) Developments 1. Let Ãh = A h + hA h,1
, where A h is like above and (1.9)

∃ C 0 > 0 ∀ϕ ∈ C m 0 (IR d ), |A h,1 [ϕ, ϕ]| ≤ A h [ϕ, ϕ] + C 0 ||ϕ|| 2 . Then ( Ãh [ϕ, ϕ]+C 0 ||ϕ|| 2 ) 1/2 and (A h [ϕ, ϕ]+C 0 ||ϕ|| 2 )
(M ) × C m (M ) satisfying supp φ ∩ supp ψ = ∅ ⇒ A M,h [ φ, ψ] = 0.
Assume that in local coordinates on U ⊂ IR d the form A M,h is acting on ϕ, ψ ∈ C m 0 (U) according to the formula (1.3) with all the hypotheses of Theorem 1.2 (or 1.3) satisfied. Then a standard reasoning can be applied to obtain analogical estimates for the counting function of A M,h , the associated self-adjoint operator in L 2 (M, dx). 3. For an operator A M,1 considered in the point 2, we can deduce the classical Weyl formula considering a semiclassical problem ÃM,h with h = λ -1/(2m) . We need to assume the Hölder continuity of top order coefficients (|ν| = |ν| = m) and we can consider the lower order coefficients belonging to L ∞ . Indeed, reasoning as in the point 1 we can modify lower order coefficients and since the principal symbol is ξ-homogeneous, the energy value 1 is not critical, allowing us to adapt the proof of Theorem 1.3 and to obtain (1.10)

N (A M,1 , λ) = N ( ÃM,λ -1/(2m) , 1) = cλ d 2m + O(λ d-µ 2m )
for every µ ∈ [0; r[. This result was described in [START_REF] Zielinski | Asymptotic behaviour of eigenvalues of differential operators with irregular coefficients on a compact manifold[END_REF][START_REF] Zielinski | Asymptotic distribution of eigenvalues of some elliptic operators with intermediate remainder estimates[END_REF] and we refer to [START_REF] Ivrii | Microlocal analysis and precise spectral asymptotics[END_REF][START_REF] Ivrii | Sharp spectral asymptotics for operators with irregular coefficients[END_REF][START_REF] Levendorskii | Asymptotic distribution of eigenvalues of differential operators[END_REF] and [START_REF] Zielinski | Asymptotic distribution of eigenvalues for elliptic boundary value problems[END_REF] for results concerning boundary value problems. [START_REF] Helffer | Comportement semi-classique du spectre des hamiltoniens quantiques elliptiques[END_REF]. The regularity hypothesis made on the coefficients a ν,ν are really essential only for x such that (x, ξ) ∈ Γ E 0 with a certain E 0 > E and the behaviour of coefficients for other values of x can be more general: the main requirement is the possibility of reducing the problem by adding an auxiliary cut-off supported in Γ E 0 as in Lemma 6.1. In particular we have assumed a ν,ν ∈ L ∞ (IR d ) for sake of simplicity, but it is possible to consider unbounded coefficients in a framework of tempered variation models on T * IR d (cf. e.g. [START_REF] Helffer | Calcul fonctionnel par la transformation de Mellin et opérateurs admissibles[END_REF]).

2 Regularized problem

(A) Definition of smooth operators Let γ ∈ C ∞ 0 (IR d ) satisfy γ(x) dx = 1 and let γ ε (x) = ε -d γ(x/ε) for ε > 0. We fix δ ∈]0; 1[ and define (2.1) a ν,ν,h (x) = (a ν,ν * γ h δ )(x) = a ν,ν (y)γ(h -δ (x -y)) h -δd dy.
As explained in Section 6, the hypothesis a ν,ν ∈ B r ensures the estimates

(2.2) |a ν,ν (x) -a ν,ν,h (x)| ≤ Ch δr , (2.3) |∂ α x a ν,ν,h (x)| ≤ C α (1 + h δ(r-|α|) ) (for every α ∈ IN d ). We define (2.4) p h (x, ξ) = |ν|,
P • h = |ν|,|ν|≤m (hD) ν a ν,ν,h (x)(hD) ν satisfies |((A h -P • h )ϕ, ϕ)| ≤ Ch µ ((I -h 2 ∆) m ϕ, ϕ
) and defining (2.7)

P ± h = P • h ± Ch µ (I -h 2 ∆) m (with C large enough) we ensure P - h ≤ A h ≤ P + h (in the sens of quadratic forms). If h ∈]0; h 0 ]
with h 0 as in Proposition 1.1, then the min-max principle (cf. [START_REF] Reed | Methods of modern mathematical physics[END_REF]) ensures

N (P + h , E) ≤ N (A h , E) ≤ N (P - h , E) and it is clear that it suffices to prove Theorem 2.1 The formula (1.8) holds with P ± h instead of A h . (B) Microlocal trace formula For E , E ∈ IR let 1 I [E ; E] : IR → {0, 1} be the characteristic function of [E ; E] and let 1 I [E ; E] (P ± h ) denote the spectral projector of P ± h on [E ; E]. If b h is a polynomially bounded smooth function of (x, ξ) ∈ IR 2d , then B h = b h (x, hD) denotes the pseudodifferential operator acting on ϕ ∈ S(IR d ) according to the formula (B h ϕ)(x) = dξ (2πh) d e ixξ/h b h (x, ξ) dy e -iyξ/h ϕ(y). Let s ∈ IR. We write b ∈ S s 0,δ if b = (b h ) h∈]0; 1] is a family of smooth functions satisfying the estimates (2.8) |∂ α ξ ∂ β x b h (x, ξ)| ≤ C α,β h -s-|β|δ
for every α, β ∈ IN d . In Section 6 we explain that Theorem 2.1 follows from

Theorem 2.2 Let Γ be a closed subset of Γ E 0 such that (2.9) (x, ξ) ∈ Γ ⇒ |∇ ξ p h (x, ξ)| ≥ c
holds for a certain constant c > 0. Let l = (l h ) h∈]0; 1] ∈ S 0 0,δ be such that l h is real-valued and supp l h ⊂ Γ for every h ∈]0

; 1]. If L h = l h (x, hD) and L * h denotes its adjoint in L 2 (IR d ), then (2.10) tr L h 1 I [E ; E] (P ± h )L * h = E <p h (x,ξ)<E dxdξ (2πh) d l h (x, ξ) 2 + O(h µ-d ).
(C) Plan of the proof of Theorem 2.2 Further on we drop the index h. In particular we write simply L, l, p instead of L h , l h , p h and we abbreviate

P ± h = P . Let γ ∈ C ∞ 0 (] -1 2 ; 1 2 [) be such that γ = 1 and γ ≥ 0. Then the convolution with γh µ (λ) = h -µ γ(λ/h µ ) allows us to replace 1 I [E ; E] by the approximations (2.11) f - h = 1 I [E +h µ /2; E-h µ /2] * γh µ , f + h = 1 I [E -h µ /2; E+h µ /2] * γh µ , satisfying 1 I [E +h µ ; E-h µ ] ≤ f - h ≤ 1 I [E ; E] ≤ f + h ≤ 1 I [E -h µ ; E+h µ ] , hence (2.12) tr L f - h (P )L * ≤ tr L1 I [E ; E] L * ≤ tr L f + h (P )L * .
Clearly it suffices to prove (2.10) with f ± h (P ) instead of 1 I [E ;E] . Further on we abbreviate f ± h = fh and observe the estimates of derivatives

(2.13) | f (k) h (λ)| ≤ C k h -kµ (k ∈ IN).
Next we introduce the h µ -Fourier transform of fh ,

(2.14) f h (t) = (F h µ fh )(t) = ∞ -∞ dλ e -iλt/h µ fh (λ)
and remark that for any k ∈ IN we have

(2.15) t k f h (t) = (-i) k h kµ (F h µ f (k) h )(t) = O(1). Since fh (λ) = (F -1 h µ f h )(λ) = ∞ -∞ dt 2πh µ f h (t) e itλ/h µ , we can write (2.16) tr L fh (P )L * = ∞ -∞ dt 2πh µ f h (t) tr LU t L *
where we have denoted U t = e itP/h µ . Then our principal task is to construct a sequence of operators (Q N ,t ) N ∈IN being a suitable approximation of LU t . More precisely: assuming Q N ,t|t=0 = L and defining (2.17)

Q N ,t = d dt Q N ,t -iQ N ,t P/h µ .
we can write formally

(2.18) LU t -Q N ,t = t 0 dτ d dτ Q N ,t-τ U τ = - t 0 dτ Q N ,t-τ U τ
and observe that due to (2.15-16) and (2.18), Theorem 2.2 follows from

Proposition 2.3 Let N ∈ IN. Then there is Q N ,t ∈ B(L 2 (IR d )) satisfying (2.19) ∞ -∞ dt 2πh µ f h (t) tr Q N ,t L * = E <p<E dxdξ (2πh) d l 2 + O(h µ-d ) and (2.20) tr Q N ,t-τ U τ L * ≤ h (1-δ) N -5d (2 + |t|) C N where Q N ,t is given by (2.17), C N > 0 is a constant large enough, (t, τ ) ∈ IR * × IR and τ /t ∈ [0; 1].
In Section 3 we describe the construction of (2.21)

Q N ,t = e itp/h µ 0≤n≤ N t n q • N ,n (x, hD),
as suitable pseudodifferential operators. At the end on Section 3 we check that (2.19) follows via integrations by parts and in Section 4 we describe a similar strategy to obtain (2.20), completing the proof in Section 5.

Description of the approximation

The operators (2.7) can be written in a standard form (3.1)

P ± h = |ν|≤2m p ± ν,h (x)(hD) ν = p ± h (x, hD)
and it easy to check that (2.3) still holds with p ± ν,h instead of a ν,ν,h and

(3.2) p ± h (x, ξ) = |ν|≤2m p ± ν,h (x)ξ ν = a(x, ξ) + O(h µ )(1 + |ξ|) 2m .
For a smooth function (x, ξ) → b t (x, ξ) ∈ C we denote

(3.3) PN b t = e -itp/h µ ∂ t (b t e itp/h µ ) - |α|≤ N h |α|-µ α! i |α|+1 ∂ α ξ (b t e itp/h µ ∂ α x p ± ) .
Further on l ∈ S 0 0,δ is as in Theorem 2.2. Proof. First of all we recall that b ∈ S s 0,δ , b ∈ S s 0,δ ⇒ b b ∈ S s+s 0,δ and it is easy to see that bp, bp ± belong to S s 0,δ . Since (2.5) still holds with p ± h instead of p h

Lemma 3.1 Let ρ = µ -1 + δ. Let b ∈ S s 0,
and |α| ≥ 1 ⇒ h |α| ∂ α x a ν,ν,h = O(h δr+(1-δ)|α| ), we obtain b(p-p ± ) ∈ S s-δr-(1-δ) 0,δ and h |α| b∂ α x p ± ∈ S s-δr-(1-δ)|α| 0,δ if |α| ≥ 1.
Using moreover δr ≥ µ, we obtain

(3.5) b 0 = ih -µ (p -p ± )b + 1≤|α|≤ N h |α|-µ α! i |α|+1 ∂ α ξ (b ∂ α x p ± ) ∈ S s 0,δ .
Next for n ∈ {1, ..., N } we obtain

b n = α=α 0 +...+αn |α|≤ N , α k =0 if k =0 c α 0 ,...,αn h |α|-(n+1)µ ∂ α 0 ξ (b ∂ α x p ± ) ∂ α 1 ξ p . . . ∂ αn ξ p ∈ S s+nρ 0,δ observing that 1 ≤ n ≤ |α| ⇒ (n + 1)µ -(1 -δ)|α| -δr ≤ nρ. Proposition 3.2 Assume that N ∈ {0, 1, ..., N } and ρ = µ -1 + δ ≥ 0.
Then we can find

(3.6(N )) q N ,N,t = 0≤n≤N t n q • N ,n , such that q N ,N,t|t=0 = q • N ,0 = l and (3.7(N )) P N q N ,N,t = N ≤n≤N + N t n q • N ,N,n with (3.8(N )) q • N ,n ∈ S (n-1)ρ 0,δ , supp q • N ,n ⊂ supp l (n ∈ {1, ..., N }), (3.9
(N )) q • N ,N,n ∈ S nρ 0,δ , supp q • N ,N,n ⊂ supp l (n ∈ {N, ..., N + N }).
Proof. If N = 0 then we take q N ,0,t = q • N ,0 = l ∈ S 0 0,δ and Lemma 3.1 with b = l gives the statement of Proposition 3.2 for N = 0. Next we assume that the statement holds for a given N ≤ N -1 and we prove that it still holds for N + 1 instead of N . Using the induction hypothesis (3.7(N )) to express P N q N ,N,t we find

P N q N ,N +1,t = P N (t N +1 q • N ,N +1 ) + P N q N ,N,t = t N (N + 1)q • N ,N +1 + q • N ,N,N + t N +1 P N q • N ,N +1 + N +1≤n≤N + N t n q • N ,N,n .
In order to obtain (3.7(N + 1)) it suffices to cancel the term with t N taking q • N ,N +1 = -q • N ,N,N /(N + 1).

Since q • N ,N,N ∈ S N ρ 0,δ and supp q • N ,N,N ⊂ supp l by the induction hypothesis, we obtain (3.8(N + 1)) and using Lemma 3.

1 with b = q • N ,N +1 we observe that ρ ≥ 0 ⇒ S (N +n)ρ 0,δ ⊂ S (N +1+n)ρ 0,δ
and (3.9(N + 1)) holds.

Lemma 3.3 If b ∈ C ∞ 0 (IR 2d
) then we denote

(3.10) J t (b) = dxdξ (2πh) d e itp(x,ξ)/h µ b(x, ξ). If b ∈ S s 0,δ satisfies supp b ⊂ supp l, then for every k ∈ IN one can find b k ∈ S s 0,δ such that supp b k ⊂ supp l and (3.11) h -kµ t k J t (b) = J t (b k ).
Proof. The hypothesis (2.9) ensures the existence of bj ∈ S s 0,δ such that b = d j=1 bj ∂ ξ j p and integrating by parts we find (3.12) h -µ tJ t ( bj ∂ ξ j p) = J t (i∂ ξ j bj ).

Thus the statement of Lemma 3.3 holds for k = 1 and we complete the proof reasoning by induction with respect to k ∈ IN.

Proof of the estimate (2.19). Taking Q N ,t given by (2.21) with q N , N ,t defined in Proposition 3.2, we observe that Q N ,t L * has the integral kernel

K N ,t (x, y) = dξ (2πh) d e i(x-y)ξ/h+itp(x,ξ)/h µ q N , N ,t (x, ξ)l(y, ξ), hence (3.13) tr Q N ,t L * = dx K N ,t (x, x) = dxdξ (2πh) d e itp/h µ lq N , N ,t .
Therefore using Lemma 3.3 with k = n -1 and b = h (n-1)µ q • N ,n l ∈ S 0 0,δ for n = 2, ..., N we can find q N ∈ S 0 0,δ such that supp q N ⊂ supp l and

tr Q N ,t L * = 0≤n≤ N t n J t (q • N ,n l) = J t (l 2 ) + tJ t (q N ).
Changing the order of integrals we find

(3.14) ∞ -∞ dt 2πh µ f h (t)J t (l 2 ) = dxdξ (2πh) d l(x, ξ) 2 fh (p(x, ξ))
and supp ( fh

-1 I ]E ; E[ ) ⊂ [E -h µ ; E + h µ ] ∪ [E -h µ ; E + h µ ] allows us to write (3.14) as E <p(x,ξ)<E dxdξ (2πh) d l(x, ξ) 2 + R(l, p -E, h µ ) + R(l, p -E , h µ ),
where

(3.15) R(l, p -E, h µ ) := h -d E-h µ ≤p(x,ξ)≤E+h µ dxdξ l(x, ξ) 2
can be estimated by O(h µ-d ) due to the hypohesis (2.9) and the same is true with E instead of E. Next we use tf

h (t) = -ih µ (F h µ f h )(t) to write (3.16) ∞ -∞ dt 2πh µ f h (t)tJ t (q N ) = dxdξ (2πh) d q N (x, ξ)(-i)h µ f h (p(x, ξ))
and we complete the proof of (2.19) observing that q

N = O(1), supp q N ⊂ supp l, h µ f h = O(1) and supp f h ⊂ [E -h µ ; E + h µ ] ∪ [E -h µ ; E + h µ ] allow us to estimate (3.16) as before by R(q N , p -E, h µ ) + R(q N , p -E , h µ ) = O(h µ-d ).
4 Auxiliary notations and properties

(A) Expression of Q N ,t-τ Further on we denote V = {(t, τ ) ∈ IR * × IR : τ /t ∈ [0; 1]}.
(4.2) (Op t,τ (b)ϕ)(x) = dydξ (2πh) d e i(x-y)ξ/h+i(t-τ )p(x,ξ)/h µ b h,t,τ (x, ξ, y)ϕ(y) defines Op t,τ (b) ∈ B(L 2 (IR d )) satisfying (4.3) ||Op t,τ (b)|| ≤ Ch -s (2 + |t|) C 0 (1 + |t|/h µ ) 2d
for a certain constant C > 0 due to Calderón-Vaillancourt theorem (cf. e.g. [START_REF] Hörmander | The analysis of linear partial differential operators[END_REF], Section 18.6). As explained at the end of Section 6, b ∈ Ss ensures also

(4.4) ||Op t,τ (b)|| tr ≤ h -s-5d (2 + |t|) C .
Using (2.21), (3.6( N )) and P = P * = p ± (x, hD) * , we find Q N ,t P = Op t,0 ( q N , N ,t (x, ξ)p ± (y, ξ) ), hence writing the Taylor's development of y → p ± (y, ξ) in x and using (y -x) α e i(x-y)ξ/h = (ih) |α| ∂ α ξ j (e i(x-y)ξ/h ) to perform standard integrations by parts in (4.2), we find

(4.5) Q N ,t = Op t,0 ( ( P N q N , N ,t )(x, ξ) + r N ,t ),
with the remainder term of the Taylor's development of order N , (4.6) r N ,t (x, ξ, y) = e -itp(x,ξ)/h µ ( N + 1)

1 0 dσ (1 -σ) N r N ,σ,t (x, ξ, y)
where r N ,σ,t (x, ξ, y) = (4.7)

|α|= N +1 h N +1-µ i N +2 α! ∂ α ξ (q N , N ,t e itp/h µ )(x, ξ) ∂ α x p ± (x + σ(y -x), ξ) .
Let q • N , N ,n be as in Proposition 3.2 and define

(4.8) q • N ,n,t,τ (x, ξ, y) = (1 -τ /t) n q • N , N ,n (x, ξ).
Then q • N ,n ∈ Snρ ⊂ Snµ-N (1-δ) and we can write (4.9)

P N q N , N ,t-τ (x, ξ) = N ≤n≤2 N t n q • N ,n,t,τ (x, ξ, y).
Using the form of q N , N ,t in (4.6-7), similarly as in the proof of Lemma 3.1 we find r

• N ,n ∈ Snµ-N (1-δ) such that (4.10) r N ,t-τ = 0≤n≤2 N t n r • N ,n,t,τ
and conclude Corollary 4.1 There exist q N ,n ∈ Snµ-N (1-δ) such that

(4.11) Q N ,t-τ = 0≤n≤2 N t n Op t,τ (q N ,n ).
(B) Auxiliary operator algebra Y It is easy to see that U t : S(IR d ) → S(IR d ) and for any B :

S(IR d ) → S(IR d ) we can define Y τ (B) := U * τ BU τ . We write Y ∈ Y 0 if there is N ∈ IN such that Y = (Y τ ) τ ∈IR has the form (4.12) Y τ = 1≤k≤N [0; 1] N dw s w,k,τ Y τ s w,k,1 (B k,1 ) • • • Y τ s w,k,N (B k,N ),
where

B k,k = b k,k (x, hD) with      b k,k ∈ S 0 0,δ for (k, k ) ∈ {1, ..., N } 2 , s w,k,k ∈ [-1; 1] for (w, k, k ) ∈ [0; 1] N ×{1, ..., N } 2 , s w,k,τ ∈ C and |s w,k,τ | ≤ (2 + |τ |) N for (w, k, τ ) ∈ [0; 1] N ×{1, ..., N }×IR We observe that taking s w,k,k = s w,k,τ = 1 we obtain Y τ (B k,1 ) • • • Y τ (B k,N ) ∈ Y 0
and it is not difficult to see that Y 0 is an algebra with the property Using this notation and (4.11) we can write

(4.15) tr Q N ,t-τ U τ L * = 0≤n≤2 N t n J t,τ (q N ,n , L * ).
In Section 5 we will prove

(4.16) h -nµ t n J t,τ (b 0 , Y ) = 1≤k≤Kn J t,τ (b n,k , Y n,k ).
It is easy to see that Proposition 4.2 implies the estimates (2.20). Indeed, Proposition 4.2 allows us to replace each term of (4.15) by

J t,τ (b N ,n,k , Y N ,n,k ) with b N ,n,k ∈ S-N (1-δ)
, hence (2.20) follows from (4.13) and (4.4).

(C) Commutators with x j /h

We denote by x j the operator of multiplication by the j-th coordinate.

Lemma 4.3 If Y ∈ Y and ρ = µ -1 + δ, then there exist Ỹ + , Ỹ -∈ Y such that (4.17) [Y τ , x j /h] = Ỹ + τ + h -ρ τ Ỹ - τ .
Proof. Let B = b(x, hD) with b ∈ Ŝ0 0,δ . We will show that (4.18) [Y τ (B k,k ),

x j /h] = Y + τ + h -ρ τ Y - τ holds with some Y + , Y -∈ Y. To start we write (4.19) [Y τ (B), x j /h] = U * τ [B, Y -τ (x j /h)]U τ and introduce (4.20) P j := [iP, x j /h] = ∂ ξ j p ± (x, hD).
Then we can write the Taylor formula

Y -τ (x j /h) = x j /h -τ (∂ τ Y τ )| τ =0 (x j /h) + τ 2 1 0 dσ (1 -σ)∂ 2 τ Y -στ (x j /h) (4.21) = x j /h + h -µ τ P j + h -2µ τ U τ Y τ ([P j , iP ])U * τ , where Y τ ( B) := τ 1 0 dσ (1 -σ)Y (1-σ)τ ( B)
for any B : S(IR d ) → S(IR d ). Using (4.21) we can express the commutator (4.19) in the form

(4.22) Y τ ([B, x j /h])+h -ρ τ Y τ (h δ-1 [B, P j ])+h -ρ τ [Y τ (B), Y τ (h δ-1-µ [P j , P ])]
and since [B, x j /h] = -i∂ ξ j b(x, hD), the first term of (4.22) belongs to Y. Then using Lemma 6.3 we find that h δ-1 [B, P j ] = b j (x, hD) with b j ∈ Ŝ0 0,δ and h δ-1-µ [P, P j ] = p j (x, hD) with (1 + |ξ| 2 ) -2m p j (x, ξ) belonging to S 0 0,δ . Thus choosing E 0 ∈ IR large enough we have P = P + E 0 I ≥ I and we can write Y τ (h δ-1-µ [P j , P ])] = Y τ (BP 2 )Y τ (h δ-1-µ P -2 [P j , P ])]. Then BP 2 = b(x, hD) with b ∈ Ŝ0 0,δ and the standard parametrix construction (cf. e.g. [START_REF] Hörmander | The analysis of linear partial differential operators[END_REF], Theorem 8.1.9) gives P -2 h δ-1-µ [P, P j ] = pj (x, hD) with pj ∈ S 0 0,δ , completing the proof of (4.18). If we assume only B = b(x, hD) with b ∈ S 0 0,δ , then reasoning as above we can find

Y + , Y -∈ Y 0 such that (4.18 ) [Y τ (B k,k ), x j /h] = Y + τ + h -ρ τ Y - τ P 2 .
It is clear that to obtain the general statement of Lemma 4.3 it remains to commute succesively

x j /h with Y τ s w,k,k (B k,k ), k ∈ {1, ..., N }. If b k,k ∈ S 0 0,δ , then we have still P 2 B k,k P -2 = bk,k (x, hD) with b k,k ∈ S 0 0,δ , hence P 2 from the last term of (4.18 ) can be put near b k,k (x, hD) with k such that b k,k ∈ Ŝ0 0,δ , i.e. Y ∈ Y ⇒ Y P 2 ∈ Y. Lemma 4.4 Let Y ∈ Y and P j = [iP, x j /h]. Then one can find Y + , Y -∈ Y such that (4.23) h -µ τ P j U τ Y τ = [U τ Y τ , x j /h] + U τ (Y + τ + h -ρ τ Y - τ ).
Proof. Using (4.21) to express [U τ , x j /h] = (Y -τ (x j /h) -x j /h)U τ and applying Lemma 4.3, we find

[U τ Y τ , x j /h] = [U τ , x j /h]Y τ + U τ [Y τ , x j /h] = h -µ τ P j U τ Y τ + h -ρ τ U τ Y τ (h δ-1-µ [P j , iP ])Y τ + U τ ( Ỹ + τ + h -ρ τ Ỹ - τ ), i.e. (4.23) holds with Y - τ = -Y τ (h δ-1-µ [P j , iP ])Y τ -Ỹ - τ , Y + τ = -Ỹ + τ .
5 End of the proof of Theorem 2.2

It remains to prove Proposition 4.2. We start by Lemma 5.1 Let b ∈ Ss . We consider b∂ ξ j p ∈ Ss defined by

(5.1) (b∂ ξ j p)(x, ξ, y) = b(x, ξ, y)∂ ξ j p(x, ξ). Then (5.2) [Op t,τ (b), x j /h] = h -µ (t -τ )Op t,τ (b∂ ξ j p) -Op t,τ (i∂ ξ j b).
Proof. Since the integral kernel of [Op t,τ (b), x j /h] is (5.3) (x, y) → dξ (2πh) d y j -x j h e i(x-y)ξ/h+i(t-τ )p(x,ξ)/h µ b(x, ξ, y), using y j -x j h e i(x-y)ξ/h = i∂ ξ j e i(x-y)ξ/h to integrate by parts, we can rewrite it as dξ (2πh) d e i(x-y)ξ/h+i(t-τ )p(x,ξ)/h µ (h -µ (t -τ )∂ ξ j pb -i∂ ξ j b)(x, ξ, y).

p (j),t,τ (x, ξ, y) = 1 - τ t ∂ ξ j p(x, ξ) + τ t ∂ ξ j p ± (y, ξ).
Then there exist b k ∈ Ss , Y k ∈ Y such that

(5.5) h -µ t J t,τ (p (j) b, Y ) = 1≤k≤2 J t,τ (b k , Y k ) + h -ρ tJ t,τ (b -k , Y -k ) .
Proof. Since P j = P * j = ∂ ξ j p ± (x, hD) * is a differential operator of order 2m -1, the standard composition formula gives (5.6) Op t,τ (b)P j = Op t,τ (b∂

ξ j p + h 1-δ b),
where (b∂ ξ j p )(x, ξ, y) = b(x, ξ, y)∂ ξ j p ± (y, ξ) and b ∈ Ss has the form b(x, ξ, y) = 1≤|α|≤2m-1

h |α|-1+δ i |α| α! ∂ α y b(x, ξ, y)∂ α ξ ∂ ξ j p ± (y, ξ).
By definition h -µ t J t,τ (p (j) b, Y ) can be expressed as (5.7) h -µ (t -τ ) tr Op t,τ (b∂ ξ j p)U τ Y τ + h -µ τ tr Op t,τ (b∂ ξ j p )U τ Y τ .

Due to (5.2) the first term of (5.7) can be written as (5.8) tr [Op t,τ (b), x j /h]U τ Y τ + J t,τ (i∂ ξ j b, Y ) and (5.6) allows us to write the second term of (5.7) in the form (5.9)

h -µ tr Op t,τ (b)τ P j U τ Y τ + h -ρ τ J t,τ ( b, Y ).
Then due to (4.23), the first term of (5.9) can be written as Proof. Let ε = min{1-δ, 1-µ}. We are going to show that for every n ∈ IN one can find b n ∈ Ss-nε such that Op t,τ (b-b χ ) = Op t,τ (b n ) and b n (x, ξ, y) = 0 for x -y belonging to a small neighbourhood of 0. Reasoning by induction we assume the existence of such b n for a given n ∈ IN. Then we can find b n,j ∈ Ss-nε such that b n = 1≤j≤d (x j -y j )b n,j and b n,j (x, ξ, y) = 0 for y -x belonging to a neighbourhood of 0. Next we observe that decomposing Op t,τ ((x j -y j )b n,j ) as a sum (5.11) Op t,τ ((x j -

(5.10) tr Op t,τ (b)[U τ Y τ , x j /h] + J t,τ (b, Y + ) + h -ρ τ J t,τ (b, Y -). Taking b 1 = i∂ ξ j b, Y 1 = Y , b 2 = b, Y 2 = Y + , b -1 = τ t b, Y -1 = Y , b -2 = τ t b, Y -2 = Y -we obtain (5.5) observing that tr [Op t,τ (b), x j /h]U τ Y τ + tr Op t,τ (b)[U τ Y τ , x j /h] = tr [Op t,τ (b)U τ Y τ , x j /h] = 0.
y j + h 1-µ (t -τ )∂ ξ j p)b n,j ) -h 1-µ (t -τ )Op t,τ (∂ ξ j pb n,j ),
we can write the first term of (5.11) as Op t,τ (ih∂ x j b n,j ). Thus (5.11) can be written as Op t,τ (b n+1,j ) with b n+1,j = ih∂ x j b n,j -(t-τ )h 1-µ ∂ ξ j pb n,j ∈ Ss-(n+1)ε and b n+1,j (x, ξ, y) = 0 for x -y belonging to a small neighbourhood of 0, i.e. the statement holds for n + 1. Proof of Proposition 4.2. Choosing h 0 > 0 small enough we ensure

h -µ t J t,τ (b 0 , Y ) = 1≤k≤K (J t,τ (b k , Y k ) + h 1-δ-µ tJ t,τ (b -k , Y -k )).
with some b k ∈ Ss , Y k ∈ Y. Applying the analogical reasoning to express h -µ t J t,τ (b -k , Y -k ), k = 1, ..., K, we can write a formula similar to (5.12) with h 1-δ-µ replaced by h 2(1-δ)-µ and after N iterations we can replace h

a h (x) -a(x) = (a(y) -a(x))γ h δ (x -y) dy, i.e. |a h (x) -a(x)| ≤ C |y -x| r |γ h δ (x -y)| dy = C r h rδ .
(B) We describe how to deduce Theorem 2.1 from Theorem 2.2. Let E ∈ IR be fixed such that p(x, ξ) ≥ E and the spectrum σ

(P ) ⊂ [E ; ∞[. Next we fix E 1 < E 0 and consider g ∈ C ∞ 0 (] -∞; E 1 [).
We observe that (2.5) allows us to find h g > 0 such that supp (g • p) ⊂ Γ E 1 ⊂ Γ E 0 for h ∈]0; h g] and we recall our hypothesis that Γ E 0 is bounded. Then reasoning as in the proof of Lemma 3.1 of [START_REF] Zielinski | Semiclassical Weyl formula for elliptic operators with non-smooth coefficients[END_REF] we obtain Proposition 6.1 Let g be as above. We fix l ∈ C ∞ 0 (IR 2d ) such that l = 1 on a neighbourhood of Γ E 1 and denote L = l(x, hD). and using (6.3) with g 1 , g 0 instead of g we find that (6.6) can be written as

  δ be independent of t and such that supp b ⊂ supp l. Then we can find b n ∈ S s+nρ 0,δ for n ∈ {0, ..., N } such that supp b n ⊂ supp l and (3.4) P N b = 0≤n≤ N t n b n .

For s ∈

 ∈ IR we will writeb ∈ Ss if b = (b h,t,τ ) h∈]0; 1[, (t,τ )∈V satisfies (4.1) |∂ α ξ ∂ β x,y b h,t,τ (x, ξ, y)| ≤ C α,β h -s-|β|δ (1 + |t|) C 0 for every (α, β) ∈ IN d × IN 2dand supp b h,t,τ ⊂ (supp l) × IR d . Further on || • || is the norm of bounded operators B(L 2 (IR d )) and ||B|| tr := tr (B * B) 1/2 is the trace class norm. If b ∈ Ss then the formula

(4. 13 )

 13 Y ∈ Y 0 ⇒ ∃ C > 0, ||Y τ || ≤ (2 + |τ |) C . We define Ŝs 0,δ as the class of symbols b such that b n (x, ξ) = b(x, ξ)(1 + |ξ| 2n ) belongs to b ∈ S s 0,δ for any n ∈ IN. Finally we define Y as the set of operators Y ∈ Y 0 of the form (4.12), where b k,k , s w,k,k , s w,k,τ are as before and for every k ∈ {1, ..., N } there is k ∈ {1, ..., N } such that b k,k ∈ Ŝ0,δ . For b ∈ Ss , Y ∈ Y we denote (4.14) J t,τ (b, Y ) := tr Op t,τ (b)U τ Y τ .

Proposition 4 . 2

 42 Let Y ∈ Y and b 0 ∈ Ss . Then for every n ∈ IN one can find K n ∈ IN * and b n,k ∈ Ss , Y n,k ∈ Y for k = 1, ..., K n such that

Proposition 5 . 2

 52 Let b ∈ Ss , Y ∈ Y, j ∈ {1, ..., d} and set(5.4)

Lemma 5 . 3

 53 Let χ ∈ C ∞ 0 (IR d ) be such that χ = 1 on a neighbourhood of 0. Let b ∈ Ss and b χ (x, ξ, y) = b(x, ξ, y)χ(x -y). Then one can find b ∈ Ss-1 such that Op t,τ (b) = Op t,τ (b χ + b ).

d

  j=1 |p (j) (x, ξ, x)| ≥ c/2 for (x, ξ) ∈ supp l and h < h 0 . Then using Lemma 5.3 we can modify b 0 to ensure d j=1 |p (j) (x, ξ, y)| ≥ c/3 for (x, ξ, y) ∈ supp b 0 . Therefore we can find b (j) ∈ Ss such that b 0 = d j=1 b (j) p (j) and applying Proposition 4.2 we can write(5.12)

(6. 4 )

 4 E 1 ≤ a(x, ξ) ≤ E 1 ⇒ ∇ ξ a(x, ξ) = 0and due to (2.5) we can find c > 0 and h 0 > 0 such that (6.5)E 1 ≤ p(x, ξ) ≤ E 1 ⇒ |∇ ξ p(x, ξ)| ≥ c for h ∈]0; h 0 ]. Let g 0 ∈ C ∞ 0 (]E 1 ; E 1 [), g 1 ∈ C ∞ 0 (] -∞; E[) be real-valued, g 0 =1 in a neighbourhood of E and g 1 + g 2 0 = 1 on [E ; E]. Then (6.6) N (P, E) = tr 1 I [E ; E] (P ) = tr g 1 (P ) + tr (g 2 0 1 I [E ; E] )(P )

(6. 7 ) 6 . 3

 763 tr (g 1 • p)(x, hD) + tr L1 I [E ; E] (P )L * + O(h µ-d ),where L = l(x, hD) with l = g 0 • p. Since (x, ξ) ∈ supp l ⇒ E 1 ≤ p(x, ξ) ≤ E 1 , (6.5) ensures (2.9) and we can use Theorem 2.2 to express (6.7) as(6.8) dxdξ (2πh) d g 1 (p(x, ξ)) + dxdξ (2πh) d (g 2 0 1 I [E ; E] )(p(x, ξ)) + O(h µ-d ).Sinceg 1 + g 2 0 1 I [E ; E] = 1 I [E ; E], the sum of two integrals from (6.8) gives 20 (2πh) -d p<E dxdξ. Finally (2.5) and (1.7) ensureΓ E -C h µ ≤ Γ E-Ch µ ≤ p<E dxdξ ≤ Γ E+Ch µ ≤ Γ E + C h µ .(C) Proof of Proposition 1.1 follows as described in Appendix of[START_REF] Zielinski | Semiclassical Weyl formula for elliptic operators with non-smooth coefficients[END_REF].(D) Some properties of pseudodifferential operators. Following well known composition formulas (cf. e.g.[START_REF] Hörmander | The analysis of linear partial differential operators[END_REF], Theorem 18.5.4 and 18.5.10) we haveLemma Let b ∈ S s 0,δ , b ∈ S s 0,δ . Then there is b b ∈ S s+s 0,δ such that b(x, hD) b(x, hD) = (b b)(x, hD).Let s , s , s , s ∈ IR be such that the conditions∂ x k b ∈ S s 0,δ , ∂ ξ k b ∈ S s 0,δ , ∂ x k b ∈ S s 0,δ , ∂ ξ k b ∈ S s 0,δ hold for k ∈ {1, . . . , d} and let s = max{s + s , s + s } -1. Then [b(x, hD), b(x, hD)] = b (x, hD) with b ∈ S s 0,δ . Proof of (4.4). Let l d (x, ξ) = (1 + |x| 2 ) d (1 + |ξ| 2 ) d . Then l d (x, hD) -1 is of trace class and ||Op t,τ (b)|| tr ≤ ||(l d (x, hD) * ) -1 || tr ||l d (x, hD) * Op t,τ (b)||, where l d (x, hD) * Op t,τ (b) = Op t,τ (b d ) with b d ∈ Ss+2d , i.e. (4.4) follows from (4.3).

  1/2 are equivalent norms if h < h 0 with h 0 small enough and we can define Ãh , the associated selfadjoint operator in L 2 (IR d ). Moreover the assertions of Proposition 1.1, Theorem 1.2 and 1.3 still hold with Ãh instead of A h .

2. Let M be a compact (boundaryless) manifold with a density dx of class C m and let A M,h be a quadratic form on C m

  -y)|h -δ|α| dy = Ch (r-|α|)δ |y| r |γ (α) (y)| dy completes the proof of (2.3). To obtain (2.2) we write

	Further on we assume |α| ≥ 1, hence γ h δ (x -y)dy = 0 and (6.1) still holds (α)
	if a(y) is replaced by a(y) -a(x). Therefore
		|∂ α x a h (x)| ≤ |a(y) -a(x)||γ h δ (x -y)|h -δ|α| dy ≤ (α)
		C |y -x| r |γ h δ (x (α)
	6 Appendix
	(A) Proof of (2.2) and (2.3). Dropping the indices ν, ν we can write
	(6.1)	∂ α x a h (x) = a(y)γ h δ (x -y)h -δ|α| dy. (α)

1-δ-µ

by h N (1-δ)-µ . Thus for N ≥ µ/(1 -δ) we obtain (4.16) with n = 1 and the general statement of Proposition 4.2 clearly follows by induction with respect to n ∈ IN.

  ) -(g • p)(x, hD)|| tr = O(h µ-d ).Proof. Let l be as in Proposition 6.1 andQ • t = (e itp l)(x, hD). Then it is easy to see that computing Q• t = d dt Q • t -iP Q • t we can apply (4.3) with s = -δr, µ = 0 and obtain || tr ≤ || Q• t || || L * || tr ≤ h δr-d (2 + |t|) C .Reasoning as in (2.18) we find||( Le itP -Q • t )L * || tr ≤ h δr-d (2 + |t|) C+1and introducing the Fourier transform F g = g ∈ S(IR) to write( Lg(P ) -(g • p)(x, hD)) L * = ( Le itP -Q • t ) L *, we obtain ||( Lg(P ) -(g • p)(x, hD)) L * || tr = O(h µ-d ), which implies (6.3) due to (6.2). . Due to (1.7) we have E ∈]E 1 ; E 1 [ with some E 1 , E 1 ∈ IR satisfying

	Proposition 6.2 If g is as above, then (6.3) ||g(P || Q•	∞ -∞	dt 2π	g(t)

Then for every N ∈ IN, (6.2) ||g(P )(I -L)|| tr = O(h N ). If E < E 1 < E 0 then we can find g as above satisfying g ≥ 1 I [E ; E] and ||1 I [E ; E] (P )|| tr ≤ ||g(P )|| tr ≤ ||g(P )|| || L|| tr + O(h N ) = O(h -d ) follows from (6.2) and from the well-known estimate || l(x, hD)|| tr = O(h -d ). t L *