Jessica Santos Da Silveira

Noël Durand

Stella Lacour

M-P Belleville

Ana Perez

Gérard Loiseau

Manuel Dornier
email: manuel.dornier@supagro.fr

Marie-Pierre Belleville

Solid-state fermentation as a sustainable method for coffee pulp treatment and production of an extract rich in chlorogenic acids

Keywords: coffee pulp, food waste treatment, alcoholic fermentation, phenolic compounds extraction, chlorogenic acids Highlights

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

Introduction

Worldwide, population growth has accelerated the agro-industrial residues generation causing major environmental, economic and societal problems (water/soil/air pollution, toxicity, waste treatment and disposal). The coffee production sector, one of the most important commodity in the world, generates huge amounts of residues, such as coffee pulp, husk, skin, among others [START_REF] Esquivel | Functional properties of coffee and coffee by-products[END_REF], which contribute about 50% of the total coffee fruit production. According to data from the International Coffee Organization ("ICO" 2017), global green coffee production in 2017 was estimated to be around 9.5 million t with over 90% of coffee production taking place in developing countries (Central and South America, Central and East Africa, Asia), whereas consumption is mainly in the industrialized economies. Coffee pulp is the main residue (40% mass of coffee fruit) obtained from the wet processing of coffee cherries and is considered as both ecotoxic and antinutritional to animals due to its polyphenols, tannins, and caffeine content [START_REF] Murthy | Sustainable management of coffee industry by-products and value addition-A review[END_REF]. Nevertheless, coffee pulp is rich in sugars, proteins, minerals, amino-acids and also contains functional molecules of high industrial interest (phenolic compounds with antioxidative, anti-inflammatory, antimutagenic, antibacterial, and anticancer properties; caffeine with well-known psychotropic and diuretic properties) [START_REF] Shahidi | Hydroxycinnamates and their in vitro and in vivo antioxidant activities[END_REF][START_REF] Arellano-González | Antioxidant Activity of Coffee Pulp Extracts[END_REF].

Studies have shown that the data on coffee pulp composition can vary due to the characteristics of coffee fruits (cultivar, place of production, culture conditions, maturity, etc.), the postharvest management (methods of depulping, drying, storage, etc.), or even the analytical methods used [START_REF] Duangjai | Comparison of antioxidant, antimicrobial activities and chemical profiles of three coffee (Coffea arabica L.) pulp aqueous extracts[END_REF][START_REF] Rios | Changes in composition, antioxidant content, and antioxidant capacity of coffee pulp during the ensiling process[END_REF][START_REF] Rodríguez-Durán | Soluble and Bound Hydroxycinnamates in Coffee Pulp (Coffea arabica) from Seven Cultivars at Three Ripening Stages[END_REF]. Four major classes of phenolic compounds were identified in recent studies: flavonols, flavan-3-ols, anthocyanidins, hydroxycinnamic acids and chlorogenic acids as the predominant phenolic compound (Heeger et al., 2016). These molecules find evident applications in pharmaco-cosmetic, food and medical industries [START_REF] Quideau | Plant Polyphenols: Chemical Properties, Biological Activities, and Synthesis[END_REF].

Moreover, hydroxycinnamic acids (HA) including chlorogenic acids (CGA) can be converted into products of even higher added value (vanillin hemisynthesis as an example) [START_REF] Di Gioia | Metabolic engineering of Pseudomonas fluorescens for the production of vanillin from ferulic acid[END_REF]. However, most of these molecules are found in coffee pulp conjugated to sugars as glycosides. Compounds such as chlorogenic acids are found in an ester form bonded to cell wall forming highly complex polysaccharide structures [START_REF] Asther | Feruloyl esterase from Aspergillus niger: a comparison of the production in solid state and submerged fermentation[END_REF].

Biotechnological processes such as solid-state fermentation can be used to enhance the phenolic content in plant extracts through the breakage of ester bonds between phenolics and the plant cell wall, increasing their concentration and consequently functional properties [START_REF] Palmieri | Enhancement of antioxidant properties from green coffee as promising ingredient for food and cosmetic industries[END_REF][START_REF] Arellano-González | Antioxidant Activity of Coffee Pulp Extracts[END_REF]. Solid-state fermentation is defined as any fermentation process performed on moist solid materials in the absence of free-flowing water that acts both as physical support and source of nutrients for microorganisms. Due to this low water availability, a limited number of microorganisms, mainly yeasts and fungi, can be used for solid-state fermentation [START_REF] Thomas | Current developments in solid-state fermentation[END_REF].

Saccharomyces cerevisiae is an important microorganism used worldwide for producing food and beverages. This yeast has several advantages for ethanol production from lignocellulosic biomass: efficient ethanol production from simple sugars, it does not require oxygenation, has a relatively high tolerance to ethanol and inhibitors, has low pH optimum and is generally recognized as safe (GRAS) as a food additive for human consumption. The production of ethanol during alcoholic fermentation also presents an advantage for the recovery of phenolic compounds from agri-food solid wastes, as mixtures of water/ethanol have been shown to enhance the solubilization of phenols [START_REF] Benmeziane | Optimization of extraction parameters of phenolic compounds from Algerian fresh table grapes (Vitis Vinifera)[END_REF].

Solid-state fermentation offers numerous advantages over other techniques such as lower energy requirements and the absence of sophisticated and complex machinery and control systems, providing a low-cost process in a sustainable framework. The main drawbacks of this method concern the process scale-up (heat transfer and culture homogeneity) and production yields. Several researchers have studied the application of ultrasound-assisted extraction to increase the conversion of starch materials to glucose as well as overall ethanol yield during fermentation processes by destroying plant cell wall and making it easier to microorganisms to access sugars from plant matrix [START_REF] Chemat | Ultrasound assisted extraction of food and natural products. Mechanisms, techniques, combinations, protocols and applications[END_REF][START_REF] Nikolić | Ultrasound-assisted production of bioethanol by simultaneous saccharification and fermentation of corn meal[END_REF]. Ultrasonication has been applied widely in various biological and chemical processes. However, the use of ultrasound-assisted extraction coupled to solid-state fermentation to treat food waste has not been widely investigated. The use of ultrasonics has the potential to break the pulp cell wall and to release sugars due to acoustic cavitation, enhancing ethanol production and phenolic compounds recovery.

A major concern expressed with regards to the extraction of polyphenols has been in relation to their stability. Phenolic compounds may undergo degradation due to temperature, light, oxygen, enzymes, and pH. Light and oxygen in the air are the two most important factors that facilitate degradation reactions. Enzymes (mainly oxidative enzymes) already present in the plant material can be released during the extraction process and promote such degradation reactions [START_REF] Mäkilä | Stability of Hydroxycinnamic Acid Derivatives, Flavonol Glycosides, and Anthocyanins in Black Currant Juice[END_REF]. Therefore, to build an efficient extraction method, it is crucial to keep the stability of phenolic compounds.

One way to avoid phenol degradation during extraction is by adding a stabilizing agent to the solution. In addition to presenting antifungal and antibacterial properties, sulfur dioxide (SO2) is commonly used as a reducing agent and as an inhibitor of endogenous oxidases in winemaking [START_REF] Blouin | Le SO2 en oenologie[END_REF][START_REF] Ribereau-Gayon | The Microbiology of Wine and Vinifications[END_REF]. Especially under its bisulfite form (HSO3 -), it can efficiently protect phenolics from chemical and enzymatic oxidations. Another well-known food preservative is ascorbic acid, a watersoluble antioxidant naturally found in many fruits and vegetables. In industry, ascorbic acid can be added to plant-based products to preserve its antioxidant capacity and provide chemical stability. Furthermore, in the presence of oxygen, ascorbic acid tends to oxidize, removing the environmental resources of oxygen [START_REF] Varvara | The use of the ascorbic acid as food additive and technical-legal issues[END_REF].

The acidification of the solution presents an alternative to prevent phenol degradation by adjusting the pH with acidity regulators. The pH optimum of enzymes, such as polyphenol oxidase (PPO), ranges around pH 6-7 and becomes inactive below pH 4. Hence, the role of acidity regulators is to maintain the pH well below that necessary for optimal catalytic activity. Moreover, it is well-known that hydroxycinnamic acids are stable at lower pH (3-5), unlike basic conditions where isomerization and oxidation reactions can occur (pH≥7.0) [START_REF] Friedman | Saccharomyces cerevisiae commits to a programmed cell death process in response to acetic acid[END_REF][START_REF] Ma | Rapid resolution liquid chromatography (RRLC) analysis and studies on the stability of Shuang-Huang-Lian preparations[END_REF][START_REF] Narita | Degradation Kinetics of Chlorogenic Acid at Various pH Values and Effects of Ascorbic Acid and Epigallocatechin Gallate on Its Stability under Alkaline Conditions[END_REF]. A typical acidity regulator is acetic acid, an organic acid that has traditionally been used to improve the shelf-life and microbiological safety of food products. Glacial acetic acid can potentially be used during fermentation processes to enhance phenolic compounds extraction and stability by acidification of the medium, but can cause yeast death at certain concentrations (Leão et al., 2001).

Based on these premises, the aim of this study was to investigate the extraction and stabilization of chlorogenic acids from coffee pulp during solid-state fermentation using commercial yeast strains. First, the process optimization was investigated at laboratory scale under well-controlled conditions. Then, the fermentation process was applied to larger scales in real condition of production in order to evaluate its interest for industrial application. The outcomes of this study are expected to contribute to the development of a sustainable and competitive method for coffee pulp treatment, applicable to all regions where coffee is produced by the wet processing method (around 50% of the worldwide production).

Materials and Methods

Raw material

Coffee pulp from the wet depulping and demucilaging process of coffee beans (Coffea arabica) was supplied by the Beneficio Coopeunión, a coffee-producer cooperative located in Trés Rios (Costa Rica), during the 2015 and 2017 harvests for assays at laboratory and semi-pilot scale, respectively. As soon as obtained, raw coffee pulp was split into lots of 3 kg, frozen at -20°C, and shipped to France by aircraft in a controlled temperature system. Upon arrival, the bags containing frozen coffee pulp were stored at -20°C. Prior to experiments, batches of 1 kg of coffee pulp were thawed at room temperature and ground using a mixer Thermomix TM31 (Vorwerk, Wuppertal, Germany) for 1 min at maximum power (level 10) and without heating. The coffee pulp was then split into hermetically sealed flasks and stored at -20°C.

For the fermentation at pilot scale, coffee pulp was supplied by the Beneficio San Diego, a coffee-producer cooperative located in San José (Costa Rica), during the 2018 harvest. The night before the experiment, coffee pulp was collected and stored at room temperature. Prior to the experiment, batches of 10 kg of coffee pulp were ground using a Bowl Cutter SM 45 (K+G Wetter, Biedenkopf, Germany) for 7 min at maximum power (level 2) and without heating. All batches of coffee pulp were gathered in a sterilized container and placed in a room at 28°C.

Chemicals

Methanol, sulfuric acid, and phosphoric acid were all of analytical grade from Honeywell (Seelze, Germany). Standard of chlorogenic acids containing a mix of 3-Caffeoylquinic acid (3-CQA), 5-Caffeoylquinic acid (5-CQA), 4-Caffeoylquinic acid (4-CQA), 4-Feruloylquinic acid (4-FQA), 5-Feruloylquinic acid (5-FQA), 3.4-Dicaffeoylquinic acid (3.4-diCQA), 3.5-Dicaffeoylquinic acid (3.5-diCQA), and 4.5-Dicaffeoylquinic acid (4.5-diCQA), was purchased from International Development and Manufacturing (New Jersey, USA). Standard of glucose, fructose and ethanol were purchased from Sigma Aldrich (Steinheim, Germany). L(+)-ascorbic acid, glacial acetic acid, and sodium metabisulfite, used to stabilize the chlorogenic acids during the assays, were all of analytical grade purchased from Sigma Aldrich (Steinheim, Germany).

Solid-state fermentation

Two Saccharomyces cerevisiae strains used for solid-state fermentation, with a population of 10 7 CFU g -1 , were kindly provided by Lallemand (Toulouse, France). The yeast strains were dried and stored at 4°C.

Fermentations at laboratory scale were carried out using a double-walled glass reactor (Legallais, Montferrier-sur-Lez, France) with a working volume of 400 mL. The day prior to experiments, samples of 400 g of coffee pulp were thawed at room temperature under darkness. The yeast strain was re-activated and multiplied using deionized water at 35°C for 30 min, at a ratio of 10 mL of water per g dried yeast. During this time, the coffee pulp was maintained at 28°C in the fermentation vessels. The temperature of the outer jacket was controlled with a thermostatic water bath. Coffee pulp was inoculated at a ratio of 2.5 g yeast/kg coffee pulp, and solid-state fermentation was carried out at a fixed temperature of 28°C, in darkness and without stirring to maintain anaerobic conditions during the fermentation. Prior to sampling, coffee pulp was gently mixed using a sterilized laboratory tool to ensure homogeneity of the mixture. Samples were clarified by centrifugation at 8603 x g for 15 min.

Semi-pilot and pilot scale fermentations were carried out respectively in France in a 20 L stainless steel tank (Artame, Baguim do Monte, Portugal) containing 12 kg of coffee pulp and in Costa Rica in a 200 L sterilized polypropylene drum (Lacoplast S.A., Guatemala) containing 90 kg of coffee pulp. The tanks were placed in climate rooms set and preheated at 28°C for temperature control. Fermentations were carried out at 28°C, without stirring and under darkness. Coffee pulp was inoculated, sampled and clarified as described for laboratory scale. The coffee pulp extract was separated from the mash by pressing using a hydraulic press (Stossier LI P MO, Simaco, Bouzonville, France or TC Y125, Owatonna Tool Company, Minnesota, USA) at 50-60 bar for 30 min.

Stabilization procedures for chlorogenic acid stability

In order to prevent chlorogenic acid (CGA) degradation during the fermentation process, the performance of different stabilizing agents was studied at laboratory scale.

Sulfur dioxide was used as sodium metabisulfite at 3 concentrations: 30 mg kg -1 according to data from the yeast supplier as the limit sulfite concentration supported by their Saccharomyces cerevisiae strains, 500 mg kg -1 a concentration used for sulfitic maceration of musts in winemaking that favors polyphenol extraction rate [START_REF] Blouin | Le SO2 en oenologie[END_REF], and 0.5 wt% a very high concentration which guarantees total protection against oxidation. Glacial acetic acid was chosen as an acidity regulator at two concentrations:

1 wt% to stabilize CGA and allow yeast growth, and at 10 wt% to stabilize CGA and inhibit yeast activity. Finally, ascorbic acid at 1 wt% was used as an antioxidant [START_REF] Narita | Degradation Kinetics of Chlorogenic Acid at Various pH Values and Effects of Ascorbic Acid and Epigallocatechin Gallate on Its Stability under Alkaline Conditions[END_REF]. The stabilizing agent was directly mixed to the coffee pulp before inoculation or at a certain time during the fermentation to study its effect on the stabilization of CGA and on yeast activity.

Ultrasound-assisted extraction

For particular experiments, the fermentation at laboratory scale was coupled to ultrasound-assisted extraction in order to evaluate the possible enhancement of chlorogenic acid extraction. The ultrasound treatment was carried out using a Bransonic model 3510E-MT 100 W ultrasound equipment (Danbury, USA) using 42 kHz as frequency. Ultrasounds were applied during 10 min (1) prior to inoculation or (2) at a selected time during fermentation where the maximum amount of CGA was extracted.

Total power used was 250 W/kg of coffee pulp.

Analytical methods

The moisture of the raw coffee pulp was determined by gravimetric method in triplicate at 100°C after 48 h. The pH and temperature were obtained using an ALMEMO measuring instrument (Ahlborn, Ilmenau, Germany). The concentration of sugars and ethanol was determined by High-Performance Liquid Chromatography (HPLC) using a Dionex Ultimate 3000 chromatograph (Thermo Scientific) that was equipped with a refractive index detector (RID-10A Schimadzu), a UV detector (210 nm), and an Aminex HPX87H column that was operated at a temperature of 35°C using 5 mM sulfuric acid as the eluent at a flow rate of de 0.6 mL min -1 .

The concentration of chlorogenic acid isomers was determined by HPLC on a system consisting of Shimadzu (Kyoto, Japan) Model LC 20AD pumping units, an automated sample injector (Shimadzu SIL 20 AXR), a variable-wavelength UV detector (Shimadzu SPD20A), column Uptisphere type ODB 5 (5 µm particle size, 250 x 4.6 mm), with identical pre-column, thermostatically controlled at 30°C. The elution program used two solvents, A and B. Solvent A was 4 mM phosphoric acid and solvent B was methanol.

The following elution program was used: A-B mixture 95/5 v/v from min 0 to 35, to A-B mixture 25/75 v/v from min 35 to 40, then pure solvent B from min 40 to 50, to A-B mixture 95/5 v/v from min 50 to 55. Flow rate was 1 mL min -1 . UV detection was at 327 nm, which corresponds to maximum CGA absorption.

Samples were filtered through 0.45 µm pore size filter before HPLC injection.

The compounds (sugars, ethanol, CGA) were identified by comparison of their retention times with the retention times of certified standards. The quantification of compounds was performed using calibration curves with 5 different concentrations of standard solutions (R 2 > 0.99).

Results and Discussion

Coffee pulp

This study explored the fermentation of coffee pulp at three different scales: laboratory, semi-pilot, and pilot. Different batches of coffee pulp with different pretreatment conditions were used during the experiments, which resulted in a variability of the raw material composition. A chemical characterization of coffee pulp was carried out to determine its initial composition in terms of fermentable sugars, ethanol and chlorogenic acid 5-CQA (Table 1).

Pretreatment steps Grinding

Pretreatment steps

Freezing, grinding and freezing

Laboratory scale

Semi-pilot scale

Pretreatment steps

Freezing, grinding and freezing

Pilot scale

The chlorogenic acid 5-CQA was reported as being the most abundant simple polyphenol present in fresh coffee pulp [START_REF] Martínez | Coffee Pulp Polyphenols: An Overview[END_REF] and its concentration was followed in all assays of this work. For laboratory scale experiments, eight samples of coffee pulp from the 2015 batch were used. Fermentation at semi-pilot and pilot scales was carried out only once due to limitations in coffee pulp supply.

According to results, we observed a difference in coffee pulp composition between samples of the same batch (2015) as well as significant differences between the three batches. Glucose and fructose contents varied by less than 10 % within the laboratory scale samples whereas changed by 41% (glucose) and 36% (fructose) between batches at different scales. Traces of ethanol were found in coffee pulp in concentrations inferior to 1 g L -1 , probably due to a natural (but very limited) fermentation of coffee pulp during storage. The biggest difference was found in terms of the amount of 5-CQA. We observed that the highest concentration of 5-CQA, 198 mg L -1 , was found in the 2018 batch that was only stored for one night prior to the experiment. Much lower concentrations were found in the coffee pulp batches of 2015 that were milled, frozen and stored for several months. An intermediate value was obtained for the batch of 2017 that was stored for a shorter period than the 2015 batch. This can be explained by the high sensitivity of phenolic compounds, such as chlorogenic acid 5-CQA, to isomerization and/or oxidation by external factors (temperature, light, oxygen) [START_REF] Narita | Degradation Kinetics of Chlorogenic Acid at Various pH Values and Effects of Ascorbic Acid and Epigallocatechin Gallate on Its Stability under Alkaline Conditions[END_REF][START_REF] Xie | Investigation of Isomeric Transformations of Chlorogenic Acid in Buffers and Biological Matrixes by Ultraperformance Liquid Chromatography Coupled with Hybrid Quadrupole/Ion Mobility/Orthogonal Acceleration Time-of-Flight Mass Spectrometry[END_REF] that can occur during the pretreatment and storage steps, leading to CGA degradation.

This finding illustrates the importance of studying the initial coffee pulp composition. Even though the same biomass is used as a substrate for fermentation, quality of the raw material may lead to different ethanol production and chlorogenic acid extraction during fermentation, and this can be a challenge for industrial scale productions.

Study of the fermentation process at laboratory-scale

Selection of yeast strains and kinetics of CGA extraction

A preliminary study was carried out to select a Saccharomyces cerevisiae strain for coffee pulp fermentation with the view to enhance chlorogenic acid extraction. Two strains were tested, Yeast A and Yeast B, which differed in terms of sulfite production during fermentation (Yeast A presenting lower SO2 production). Table 2 presents the relative concentration (ratio between the final and initial concentrations of a compound)

for the glucose, fructose, ethanol and 5-CQA after 24 h of fermentation.

Table 2

Relative concentration C/C0 (ratio between the final and initial concentrations) of glucose, fructose and ethanol after 24 h of fermentation and final concentration of 5-CQA obtained at laboratory scale for both yeast strains tested (Yeast A and Yeast B) and for a control of coffee pulp non-inoculated. Results from the control (non-inoculated coffee pulp) show that no alcoholic fermentation occurred within the 24 h period and that the final coffee pulp extract contained 8.0 mg L -1 of 5-CQA. Stability of the raw coffee pulp even with its endogenous flora is probably correlated with its high phenolic content that inhibits microorganism growth. Both yeasts were able to ferment coffee pulp and lead to very similar performances in terms of sugar consumption and ethanol production. Nevertheless, only one strain was capable of increasing the concentration of 5-CQA in the extract from coffee pulp. While the fermentation using Yeast A increased by almost 4 times the concentration of 5-CQA compared to the control, results with Yeast B showed the degradation of this molecule during the fermentation process. This result is probably correlated to a difference in the production of pectinases, β-glucanases and other hydrolytic enzymes that are able to facilitate the release of the 5-CQA out of plant cells [START_REF] Alimardani-Theuil | Yeasts: An attractive source of pectinases-From gene expression to potential applications: A review[END_REF][START_REF] Pinelo | Upgrading of grape skins: Significance of plant cellwall structural components and extraction techniques for phenol release[END_REF]. This shows that only specific strains can be used for the extraction of chlorogenic acids from coffee pulp. The Yeast A was then selected for all the experiments presented in this work.

Relative concentration (C/C0) after 24 h Final concentration of 5-CQA after 24 h [mg L -

To better understand the fermentation and extraction processes, a kinetic study was carried out with coffee pulp inoculated with Yeast A and the results were compared to a control experiment (non-inoculated coffee pulp). Figure 1 presents the evolution of glucose, fructose and ethanol concentrations during the solid-state fermentation for both samples. Results showed that glucose was the preferred carbon source, followed by fructose as usual. Both sugars were almost completely consumed after 12 h of fermentation. The production of ethanol was correlated to the consumption of sugars, increasing during fermentation and levelling off to a final concentration of 31.4 g L -1 (around 4% vol.) at t = 12 h, when sugar stocks reached a very low level (glucose < 1 g L -1 ; fructose < 6 g L - 1 ). No fermentation occurred in the control.

Figure 2 presents the evolution of chlorogenic acids (CGA) extraction in the inoculated coffee pulp compared to the control. The CGA identified were: 5-CQA, 4-CQA, 4-FQA, 5-FQA, and 4.5-diCQA. The evolution of the total amount of these five chlorogenic acids is also presented (total CGA).

Results from the inoculated pulp (Yeast A) show that 5-CQA was the only isomer that was significantly extracted during the fermentation process. Its concentration increased over time reaching a maximum at t = 8 h and decreasing afterward, probably due to a higher degradation rate of this compound compared to the extraction. The concentration of 4-CQA, 5-FQA, and 4.5-diCQA varied very little over time, while the concentration of 4-FQA slowly decreased. The control showed similar behavior for the 4-CQA, 5-FQA, and 4.5-diCQA, with their concentrations staying relatively stable over time. On the contrary the concentrations of 5-CQA and 4-FQA decreased over time.

During fermentation, different behaviors were observed. Surprisingly for 5-CQA the concentration increased over time reaching a maximum at 8 h and then decreased drastically. It was already reported that the hydroxycinnamic acids are mostly present in the coffee pulp under linked-form with cell walls whereas the soluble compounds could be mainly located inside the plant vacuoles [START_REF] Rodríguez-Durán | Soluble and Bound Hydroxycinnamates in Coffee Pulp (Coffea arabica) from Seven Cultivars at Three Ripening Stages[END_REF]. The enzymatic pool of the yeast which is responsible for the cell wall weakening could favor the wall-bonded 5-CQA release as well as the increasing ethanol concentration could favor the solubility of this compound and its extraction. Losses of 5-CQA become predominant after 8 h of fermentation through oxidation reactions by chemical, enzymatic and even fermentation pathways or through adsorption by the yeasts itselves [START_REF] Mazauric | Interactions between Yeast Lees and Wine Polyphenols during Simulation of Wine Aging: I. Analysis of Remnant Polyphenolic Compounds in the Resulting Wines[END_REF]. Furthermore, with lower sugar stocks in the media, chlorogenic acids could become the carbon source in its turn. Figure 3 summarizes the possible pathways of extraction and degradation of CGA during coffee pulp fermentation. Finally, chlorogenic acids may also undergo various isomerization reactions [START_REF] Liang | Role of Chlorogenic Acids in Controlling Oxidative and Inflammatory Stress Conditions[END_REF][START_REF] Moon | Role of Roasting Conditions in the Level of Chlorogenic Acid Content in Coffee Beans: Correlation with Coffee Acidity[END_REF] and could explain the surprising stability observed in Figure 2 for 4-CQA, 5-FQA and 4.5-diCQA, compared to 5-CQA and 4-CQA, during fermentation (as an example, 5-CQA could be partially converted into 4-CQA and 5-FQA into 4-FQA).

To conclude this part of the study, these results showed that it was possible to significantly extract the 5-CQA using solid-state fermentation. In fact, alcoholic fermentation with Yeast A allowed an increase of 3.5 times of 5-CQA initial concentration, from 4.3 mg L -1 to 15 mg L -1 , at only 8 h of fermentation. However, in order to preserve the 5-CQA extracted during the fermentation process, it is important to develop a stabilization method for chlorogenic acids.

Stabilization of chlorogenic acids

A study was carried out to compare different strategies for the stabilization of chlorogenic acids in the coffee pulp. Initially, two concentrations of sulfite were chosen to stabilize coffee pulp prior to inoculation: 30 and 500 mg kg -1 . It was found that when adding this agent at the beginning of the fermentation, there were either complete yeast inactivation resulting in no chlorogenic acid extraction (500 mg kg -1 of SO2) or coffee pulp fermentation but no chlorogenic acid stabilization (30 mg kg -1 of SO2). So, under those tested conditions, the stabilization step applied to the coffee pulp before the fermentation process was not suitable. A complementary study was then performed comparing 3 different stabilizing agents (sulfite 0.5 wt%, ascorbic acid 1 wt%, and acetic acid 1 wt% and 10 wt%) added into the culture medium at the optimal time of 5-CQA extraction, i.e. at 8 h of fermentation (as previously shown in Figure 2). This new strategy aimed to stabilize the 5-CQA at its highest concentration in the extract and to study the effect of the addition of stabilizing agent on the yeast activity.

Figure 4 presents the evolution of glucose and 5-CQA concentrations for the different assays. Since 5 coffee pulp samples from batch 2015 were used during the experiments, the concentrations were expressed in terms of the relative concentration of each compound (C/C0) due to raw material variability. Indeed, it is worth noting that, irrespective to the fermentation trials, fructose consumption and ethanol production showed the same evolution as previously described that confirmed the good reproducibility of the fermentation of the pulp by yeast. For this raison, the evolution of glucose concentration was used as the fermentation indicator in all the following results. From Fig. 4, the addition of ascorbic acid at 1 wt% or acetic acid at 1 wt% didn't affect yeast activity; the concentration of glucose continued to decrease over time as it was observed in the experiment done without any stabilizing agent. As expected, the addition of sulfite at 0.5 wt% or acetic acid at 10 wt% stopped the fermentation process, which indicates that, at the given concentration, sulfite and acetic acid inhibited yeast activity. Both of these stabilizing agents are known for acting through a strong acidification of intracellular content and, also by an intrinsic toxic activity like in the case of HSO3 -, by blocking many carbonyl functions that disrupt the cell metabolism [START_REF] Blouin | Le SO2 en oenologie[END_REF]. Concerning the extraction of 5-CQA, the addition of ascorbic acid tended to stabilize the 5-CQA until 24 h of fermentation. Afterward, due to a gradually oxidation of ascorbic acid, the 5-CQA started to degrade faster. The acetic acid, in both concentrations tested, was successful to stabilize the 5-CQA. Surprisingly, the addition of sulfite not only stabilized the 5-CQA, but also increased its concentration over time, reaching a maximum at 24 h of fermentation and representing an enhancement of 400%

of 5-CQA concentration compared to its initial concentration. This can be explained first by the fact that 5-CQA can undergo reversible electrochemical reactions [START_REF] Tomac | Electrochemical Properties of Chlorogenic Acids and Determination of Their Content in Coffee Using Differential Pulse Voltammetry[END_REF], that is to say, oxidized forms of 5-CQA can be reversibly reduced by the redox action of sulfite, unlike the other stabilizing agents tested in the study. Second, this result could also be due to a faster destruction of the plant cells resulting in higher amounts of 5-CQA being released. This well-known phenomenon in winemaking is often called the "dissolvent activity" of sulfites [START_REF] Ribereau-Gayon | The Microbiology of Wine and Vinifications[END_REF].

Combining the results in terms of glucose consumption and 5-CQA extraction, two stabilizing agents could be used for the fermentation of coffee pulp: acetic acid at 1 wt% that allows yeast growth while stabilizing the 5-CQA gradually extracted, and the sulfite at 0.5 wt% that stops fermentation, stabilizes and enhances 5-CQA extraction. For acetic acid, the advantage is a complete consumption of sugars during the fermentation.

So, the low sugar content of the final extract should considerably simplify the further purification steps during downstream processing. The use of sulfite will provide an extract richer in chlorogenic acids but less pure (with more sugars), and the use of acetic acid at 1wt% will provide a purer extract, but with lower antioxidant capacity. Therefore, a compromise must be found when choosing a stabilizing agent for this process.

Study of the process optimization using ultrasound-assisted extraction

Two strategies were chosen to combine ultrasound-assisted extraction to the fermentation process: prior to fermentation (i.e. before addition of yeasts) or after 8 h of fermentation when the highest concentration of 5-CQA was achieved (according to results in Section 3.2.2). Even if ultrasounds can disrupt also the yeast cells causing yeast death and thus stop the fermentation as it was recently reported [START_REF] Chemat | Ultrasound assisted extraction of food and natural products. Mechanisms, techniques, combinations, protocols and applications[END_REF], the fermentation during the first hours could potentiate ultrasound effects and thus favor CGAs' extraction. Since the main purpose of our process was to enhance the amount of CGA in the coffee pulp extract, the use of ultrasounds during the fermentation was an interesting strategy to explore. Figure 5 presents the results for glucose consumption and 5-CQA extraction during the fermentation process. In that case, coffee pulp was stabilized using acetic acid at 1 wt% in order to preserve the 5-CQA while allowing the fermentation process to continue.

Results showed that glucose consumption was similar for both systems.

Surprisingly, the glucose was not completely consumed even after 48 h of fermentation, compared to previous results that showed the glucose being mainly consumed at 24 h of fermentation (Figure 4). A moderate 5-CQA extraction optimization could be obtained with ultrasound-assisted extraction, as the content of 5-CQA at 8 h of fermentation was only increased by around 280% (US at 8 h) or 220% (US prior to fermentation) compared to 200% (fermentation without ultrasounds). Moreover, 5-CQA became unstable after 24 h of fermentation, which could be explained by the possible release of endogenous oxidative enzymes due to the ultrasound treatment. Another proposed explanation for the 5-CQA instability relies on the fact that the ultrasound treatment could release other compounds that could interact with the 5-CQA throughout chemical reactions causing its degradation. Additionally, ultrasound treatment could enhance the extraction of many phenolic compounds from the coffee pulp that in turn could create an inhibitory environment for yeast growth, leading to the decrease of fermentation activity. Under the experimental conditions applied in this work, the coupling of ultrasounds prior or during the fermentation process is not an interesting method to improve the extraction yields of 5-CQA from coffee pulp. It didn't improve the 5-CQA extraction yield as expected, and even slowed down the fermentation process. Therefore, no ultrasound treatment was applied on further studies.

Application of the fermentation process at larger scales

To evaluate the validity of the results obtained at laboratory scale, fermentation at semi-pilot scale was carried out and monitored during 24 h. Sulfite at 0.5 wt% was chosen as the best stabilizing agent for the coffee pulp stabilization after a first step of fermentation. As mentioned in Section 3.1, a different batch of coffee pulp was used for this assay (batch of 2017), which resulted in differences on the initial concentrations of sugars and chlorogenic acids compared to the coffee pulp used for laboratory scale experiments (batch of 2015). The batch of 2017, used for this study, contained lower concentrations of sugars and a higher initial amount of 5-CQA, as presented in the Table 1 and discussed in Section 3.1. Figure 6 presents the results for the glucose consumption and 5-CQA extraction during the fermentation process. The glucose was completely consumed after only 6 h of fermentation, which can be explained by the lower initial amount of sugars in the coffee pulp from batch 2017 (14.39 g L -1 of glucose and 21.42 g L -1 of fructose) compared to the coffee pulp from batch 2015 (31.73 g L -1 of glucose and 42.97 g L -1 of fructose). With most of the fermentable sugars being consumed before 8 h of fermentation, the addition of sulfite at t = 8 h did not disturb the yeast activity since the fermentation was already completed at this time. The concentration of 5-CQA increased over time as previously observed at laboratory scale, even after the addition of the sulfite (as explained in Section 3.3), up to a final concentration of 151 mg L -1 of 5-CQA at t 24 h, 275% higher than the initial concentration (55 mg L -1 of 5-CQA). When the fermentation was carried out at laboratory scale, using sulfite at 0.5 wt% at t = 8 h as the stabilizing agent, the content of 5-CQA in the coffee pulp extract was increased by 400% at t 24h, against 275% at semi-pilot scale.

Nevertheless, it was shown that the fermentation of coffee pulp and the extraction of chlorogenic acids were successful at semi-pilot scale, despite of the differences in coffee pulp composition (Table 1) and of the limitations encountered (reduced mass and heat transfers during the process scale-up).

Finally, to evaluate the process in a coffee producing country, fermentation was carried out at pilot-scale in Costa Rica with 90 kg of fresh coffee pulp nearby the green coffee production area. In order to compare the results from the semi-pilot fermentation, sulfite at a concentration of 0.5 wt% was used to stabilize the chlorogenic acids extracted during the fermentation. Table 3 summarizes the results for the fermentation performed at different scales. As mentioned in Section 3.1, the coffee pulp composition varied due to the year of harvest and the storage time, so the relative concentration (C/C0) of each compound (glucose, fructose, ethanol and CGA) was chosen instead of direct concentrations (C) values.

Table 3

Overview of the year of coffee pulp batches, the average temperature during the fermentation process (T) and the relative concentration (ratio between the final C and initial C0 concentrations) of glucose, fructose, ethanol and 5-CQA after 24 h of fermentation at laboratory, semi-pilot and pilot scales. For all three scales, sulfite at 0.5 wt% was used to stabilize the 5-CQA extracted, added at 8 h of fermentation. At laboratory scale, the fermentation was immediately stopped when sulfite was mixed to coffee pulp (Figure 4), which explains the higher relative concentrations of glucose and fructose in the product after 24 h. This was not the case for the 2017 and 2018 batches, which had lower initial concentrations of sugars, allowing the almost complete consumption of glucose and fructose prior to the addition of sulfite. In that cases, ethanol production yields reached a quite high level between 0.37 and 0.41 g ethanol / g sugar which corresponds to 72 and 80% of the maximal theoretical conversion yield (0.51 g ethanol / g sugar). Nevertheless, the extraction of 5-CQA was very similar for all three scales, with an augmentation of around 300% of its initial concentration in less than 1 day of fermentation.

Although the operating temperature was set at 28 °C, the average temperature of coffee pulp varied with the scale of the process. Obviously the bigger the amount of pulp used as substrate, the harder the temperature control of the coffee pulp. Since no stirring was applied during the fermentation, heat transfer became an important problem for the process. However, even at lower temperatures, solid-state fermentation using a commercial yeast (Yeast A) was able to ferment coffee pulp and to produce a rich phenolextract containing more than 600 mg of 5-CQA per kg of coffee pulp and purer in terms of the lower concentration of sugars (5 g L -1 of glucose and 12 g L -1 of fructose). We showed that the fermentation of coffee pulp at a pre-industrial (pilot) scale with very simple equipment can be used to obtain a valuable product. It was important to develop a process that could be easily implemented in the coffee producing countries, without big financial investment and operating costs, lower environmental impact and that could generate a new source of income for the small coffee producers.

Conclusions

Solid-state fermentation was used to produce a valuable product from coffee pulp at laboratory, semi-pilot and pilot scales. Saccharomyces cerevisiae consumed the sugars releasing chlorogenic acids that are found linked to the plant cell wall probably as glycosides. Investigation of the extract stability and the effect of ultrasounds revealed that higher extraction yields were obtained when fermentation was carried out without ultrasound treatment and by using sulfite at 0.5 wt% as the stabilizing agent. Irrespective of the scales, a phenol-rich extract was obtained containing 300-400% more chlorogenic acids than its initial concentration, within less than 24 h of fermentation. After fermentation and solid/liquid separation steps, the quantity of the remaining solid residue was considerably reduced. This final by-product will contain much less caffeine and phenolics (compounds considered as anti-nutritional and eco-toxic), which could facilitate its reuse as animal feeding, fertilizer or composting substrate.

For further studies, it could be of great interest to test other yeast strains and to optimize fermentation conditions for even increasing the extraction yield. In order to find the best stabilizing method, in terms of (1) CGA preservation, (2) effect on the fermentation activity and (3) cost and environmental impact, new stabilizing agents must be tested at larger scales. To obtain a ready-to-market product, the concentration and the purification of the extract need to be performed and investigated using sustainable technologies. Finally, the characterization of the final product will have to be carried out, e.g., in terms of antioxidant activity, to better determine its market added value and to enlighten about the economic feasibility of the process.

Fig. 1 .

 1 Fig. 1. Substrate consumption (glucose, fructose) and ethanol production during solid-state fermentation of coffee pulp carried out at lab scale with Yeast A, compared to a control noninoculated.

Fig. 2 .

 2 Fig. 2. Extraction of chlorogenic acids from coffee pulp during fermentation at lab scale with yeast A compared to control non-inoculated.

Fig. 3 .

 3 Fig. 3. Suggested competing pathways of the extraction and degradation of chlorogenic acids (CGA) during fermentation of coffee pulp.

Fig. 4 .

 4 Fig. 4. Effect of different stabilizing agents on the (A) glucose consumption and (B) 5-CQA extraction during solid-state fermentation at lab scale with yeast A (glucose: C0 = 32.5 ± 1.5 g L - 1 ; 5-CQA: C0 = 5 ± 2 mg L -1 ).

Fig. 5 .

 5 Fig. 5. Effect of ultrasounds on glucose consumption (A) and 5-CQA extraction (B) during fermentation at laboratory-scale. (Glucose: C0 = 31.9 ± 1.8 g L -1 ; 5-CQA: (C0 = 6 ± 2 mg L -1 ).

Fig. 6 .

 6 Fig. 6. Glucose consumption and 5-CQA extraction during fermentation at semi-pilot scale.

Table 1

 1 Characterization of raw coffee pulp from different batches in terms of origin, pretreatment steps, local and year of assays, pH, moisture, sugar, ethanol and 5-CQA content.

	s

L -1 ] Fructose [g L -1 ] Ethanol [g L -1 ] 5-CQA [mg L -1 ]

		14.39	21.42	0.6	54.91
	Harvest year	2018	Supplier	Beneficio San Diego
	Year of assay(s)	2018	Local of assay(s)	Costa Rica
			Moisture [%]	85 ± 0.8
			pH	4.3
		Glucose [g L -1 ] Fructose [g L -1 ]	Ethanol [g L -1 ]	5-CQA [mg L -1 ]
		19.47	28.44	0.65	198.66

Acknowledgements

This project is supported by Agropolis Fondation under the reference ID 1403-079 through the « Investissements d'avenir » programme (Labex Agro: ANR-10-LABX-0001-01) » and Eurodia Industrie SA (Pertuis, France). The authors acknowledge the support from Fabrice Vaillant, researcher, and Joël Grabulos, technician at CIRAD, for lending their facilities and for help with the HPLC analyses. We also thank Lallemand (Toulouse, France) for providing Saccharomyces cerevisiae strains.


A Ifa

H Louati

M Rouleux
email: rouleux@univ-tln.fr

On semi-classical spectral series for an atom in a periodic polarized electric field

In this report we present preliminary results about the tunneling problem for a magnetic Schrödinger operator. As a motivation we consider the 3-D time-dependent Schrödinger operator

where V is a radial potential and E(t) a circularly polarized field with uniform frequency ω.

The quantum monodromy operator (QMO) that takes the system through a complete period T = 2π/ω, turns out to be unitarily equivalent to e iT P A (x,hDx)/h , where P A (x, hDx)) identifies with a magnetic Schrödinger operator. When V is sufficiently confining, P A (x, hDx)) presents a double magnetic well. Then we construct its semi-classical ground state and examine the splitting between its two first eigenvalues.

Introduction

The general motivation is to describe some (semiclassical) spectrum for the quantum monodromy operator (QMO) associated with a periodic time dependent self-adjoint N -body Schrödinger operator H(t) on L 2 (R 3N ). Recall QMO is the unitary operator that evolves the physical system through one period. This spectrum is the same as for Floquet Hamiltonian H = D t +H(t) on L 2 (R 3N

x ×R t ), see e.g. [START_REF] Schach-Möller | 2-body short-range systems in a time-periodic electric field[END_REF]. Scattering theory for Floquet Hamiltonian leads to important properties, such as local decay (in time) for the probability density. Bound states are also of interest, e.g. regarding tunneling properties. A.Tip [START_REF] Tip | Atoms in circularly polarized fields: the dilation-analytic approach[END_REF] has investigated the case where H(t) is the Hamiltonian for an atom with N particles of charge e j and mass m j (e 0 = N being the charge of the nucleus of mass m 0 ) subject to the circularly polarized periodic electric field E 1 (t) = (sin ωt,cos ωt, 0) ∈ R 3 . Let also V (rotationally invariant in R 3N ) be the sum of all Coulomb potentials between the particles of charge e j . Neglecting spin effects, we have

H 1 (t) = N -1 j=0 (2m j ) -1 p 2 j + V (x) - N -1 j=0 e j x j • E 1 (t)
Time-evolution U 1 (t, s) corresponding to H 1 (t) is well defined, and QMO is U 1 (T +s, s) (independent of s ∈ R). An equivalent formulation can be given in terms of

H 2 (t) = N -1 j=0 (2m j ) -1 p j -e j A 2 (t) + V (x)
with E 1 (t) = -∂ t A 2 (t), and the corresponding time-evolutions U j (t, s), j = 1, 2 are related by a unitary operator X(t). One of the main observations in [START_REF] Tip | Atoms in circularly polarized fields: the dilation-analytic approach[END_REF] is that the group of unitary operators R(t) = e iωtL3 , L 3 being the vertical component of the angular momentum (tensor product over the N particles), takes A 2 (t) into a constant magnetic potential A 0 = ( 1 ω , 0, 0), i.e.

R(t)H

2 (t)R(t) -1 = j (2m j ) -1 (p j -e j A 0 ) 2 + V (x)
and U 2 (t, s) = e -iωtL3 e i(t-s)PA e iωsL3 where P A is the time-independent self-adjoint operator

P A = N -1 j=0 (2m j ) -1 (p j -e j A 0 ) 2 + V (x) -ωL 3
Quantum monodromy operator U 2 (s + T, s) over the period T = 2π/ω, is of the form

U 2 (s + T, s) = e -iωsL3 e iT PA e iωsL3
This relates the (discrete) spectra of U 2 (s + T, s) and P A (either real spectrum or resonances), in the sense that if E is an eigenvalue of P A , then F = e iT E is an eigenvalue of U 2 (s + T, s). For simplicity we assume henceforth there is only one heavy nucleus and one electron x ∈ R 3 , and introduce a semi-classical parameter h so that L 3 = x 1 hD x2 -x 2 hD x1 , and P A on L 2 (R 3 ) takes the form

P A (x, hD x ) = (hD x -νA 0 ) 2 + V (x) -ωL 3
, where ν is a coupling constant. In case V is Coulomb potential, the complex dilation theory of P A set up in [START_REF] Tip | Atoms in circularly polarized fields: the dilation-analytic approach[END_REF] gives an insight into the resonant spectrum of P A . Here we still assume V to be rotation invariant, but allow for non Coulomb (confining) potentials, so that P A has discrete, real spectrum.

Hamiltonians

The classical Hamiltonian is

p A (x, ξ) = (ξ 1 - ν ω ) 2 +ξ 2 2 +ξ 2 3 +V (|x|)-ω(x 1 ξ 2 -x 2 ξ 1 )
(1) and we may rewrite P A as a magnetic Schrödinger operator. Namely, after the affine change of coordinates x = 2ν ω 2 y + (0, 2ν ω 2 , 0), and the substitution

W (y) = 4ν 2 ω 4 V (|x(y)|) -λ 2 ω 2 (y 2 1 + y 2 2 ) , we get p A (x, ξ) = ω 4 4ν 2 p ′ A (y, η) + ν 2 ω 2 , where p ′ A (y, η) = (η 1 + 2ν 2 ω 3 y 2 ) 2 + (η 2 - 2ν 2 ω 3 y 1 ) 2 + η 2 3 + W (y) = (η -ω ′ A(y)) 2 + W (y) (2
) Note that the corresponding operator P ′ A (y, hD y ) commutes with symmetry y 3 → -y 3 . General spectral properties of magnetic Schrödinger operators are well-known, see e.g. [START_REF] Helffer | Semi-classical Analysis for the Schrödinger and applications[END_REF], in particular when W (y) → +∞ as y → ∞, P ′ A (y, hD y ) has only discrete spectrum.

Discrete semi-classical spectrum of P A near some energy level E is generally associated with local minima of the Hamiltonian. Among these minima it is easy to find those which are the critical points of r → V (r), namely

ρ ± 0 = (x 0 , ξ 0 ) ± = (0, 2νω -2 , ± r 2 0 -4ν 2 ω -4 ), 0 with V (r 0 ) = E -ω 2 ν -2 .
We will write as well ρ ± 0 = (y ± 0 , η ± 0 ) = (y ± 0 , 0) in the new coordinates y. So the y-projection of the critical points y ± 0 of p ′ A (y, η) are located on the y 3 -axis, symmetric with respect to the (y 1 , y 2 ) plane. At such a critical point, η ± 0 = 0.

For a suitable choice of parameters (r 0 , V ′′ (r 0 ), ω, ν), it may happen that ρ ± 0 is an elliptic point even if r 0 is not a minimum of V . However, to meet hypotheses of [START_REF] Martinez | Microlocal WKB expansions[END_REF], we assume that E is the bottom of the spectrum of P A , and thus r 0 is a global minimum of V . To avoid multiple tunneling, we assume also that there is no other critical points of p A than ρ ± 0 at energy E. Such elliptic fixed points ρ ± 0 of Hamiltonian p A , will be called the magnetic microlocal wells. We consider Dirichlet realization P M ± A (x, hD x ) localized in some (large enough) neighborhoods M ± ⊂ R 3 of x ± 0 (by symmetry P M ± A (x, hD x ) are unitarily equivalent) and study their semi-classical spectrum in a h-neighborhood of E, see [START_REF] Helffer | Multiple wells in the semi-classical limit I[END_REF]. Note that the spectral parameters for P M ± A and P ′ A M ± are related as follows:

P M ± A (x, hD x )u ± (x) = (E + λ A (h))u ± (x) ⇐⇒ P ′ A M ± (y, hD y )u ′ ± (y) = (E ′ + λ ′ A (h))u ′ ± (y) (3) where E ′ = 4ν 2 ω 4 V (r 0 ), λ ′ A (h) = 4ν 2 ω 4 λ A (h)
, and u ′ ± (y) = u ± (x) we will denote for short again by u ± (y). We know that the first eigenvalue λ A (h) of P M ± A (x, hD x ) -E is non degenerate. By tunneling, this will gives raise to splitted eigenvalues E 0 (h) < E 1 (h) for P A (x, hD x ) exponentially close to λ A (h). Thus our main goal consists in solving the rather standard problems: (1) Find semiclassical asymptotics of λ A (h), h → 0. (2) Estimate the splitting E 1 (h)-E 0 (h) (tunneling between ρ ± 0 ). Point (1) results directly from the general microlocal arguments of [START_REF] Martinez | Microlocal WKB expansions[END_REF] near ρ ± 0 applied to P A , that we recall here. Point (2) relies more specifically on the fact that P A (or P ′ A ) is a magnetic Schrödinger operator. Both use complexification of time t → it for Hamilton equations in the "classically forbidden region".

Microlocal approach: WKB expansions near the microlocal magnetic well

We work with P A rather than the rescaled operator P ′ A , and since our constructions will hold in a small neighborhood of x 0 = x + 0 , x 0 ∈ M = M + say, we shall write sometimes P A instead of P M + A (x, hD x ). At an elliptic fixed point Floquet exponents, i.e. eigenvalues of the fundamental matrix F pA = J Hess p A (ρ ± 0 ), are purely imaginary. Their square µ ′2 are the roots of a 3:rd degree polynomial with real coefficients. It turns out that we can find an open set of parameters (r 0 , V ′′ (r 0 ), ω, ν) such that r 0 ω 2 ν -1 ≥ 2 and µ ′ 2 1 < µ ′ 2 2 < µ ′ 2 3 < 0. When these conditions are met, applying a linear canonical transformation κ A , and setting the reference energy to E ′ = 0 to simplify notations, which amounts to set V (r 0 ) = 0, p A takes the form, with

µ ′ j = iµ j , µ j > 0 p A (x, ξ) = 3 j=1 µ j (x 2 j + ξ 2 j )/2 + O(|x, ξ| 3 ) (4)
in local coordinates (x, ξ) vanishing at ρ ± 0 (but not the same as the original coordinates of the problem). It is important to note that, although κ A changes drastically the geometry of the phasespace, we still get informations on u ′ ± (y), see [START_REF] Martinez | Microlocal WKB expansions[END_REF], Remark 5.2 and below. Namely, κ A does not generate additional caustics in the region of interest.

Examine first WKB constructions and decay properties of the normalized quasi-mode associated with λ A (h) by microlocal technics and the method of complex deformations [START_REF] Helffer | Multiple wells in the semi-classical limit I[END_REF], [START_REF] Helffer | Effet tunnel pour l'équation de Schrödinger avec champ magnétique[END_REF], [START_REF] Helffer | Semi-classical Analysis for the Schrödinger and applications[END_REF], [START_REF] Martinez | Estimations de l'effet tunnel pour le double puits II. Etats hautement excités[END_REF]. We follow here [START_REF] Martinez | Microlocal WKB expansions[END_REF].

Consider the global FBI transform (with c(h) a normalization constant)

T u(x, ξ; h) = c(h) R 3 e i(x-y)ξ/h-(x-y) 2 /2h u(y) dy (5) that maps isometrically L 2 (R 3 ) into L 2 (R 6 ). It is associated with the transformation κ T : T * R 3 → Λ = {ζ = i Im z}, (x, ξ) → (z, ζ) = (x -iξ, -iξ), such that -Im(dz ∧ dζ) = dx ∧ dξ, Re(dz ∧ dζ) = 0 on Λ. The pluri-subharmonic (pl.s.h) weight defining Λ, is -φ 0 (z, z) = (Im z) 2 , namely Λ = {ζ = ∂φ0
∂z }. Transformation T intertwines P A with P A (x, ξ, hD x , hD ξ ) whose Weyl symbol is given by p A (x, ξ; x * , ξ * ) = p w A (xξ * , x * ), so that the eigenvalue equation becomes, with u T = T u and u = u + as in (3)

( P A (x, ξ, hD x , hD ξ ) -λ A (h))u T (x, ξ; h) ∼ 0 (6)
which we will solve by WKB method, by looking for u T of the form u T (z; h) ∼ e -ξ 2 /2h-ϕ(z)/h a(z; h). The reduced operator Ψ(x, ξ, hD x , hD ξ ) = e ξ 2 /2h+ϕ(z)/h P A (x, ξ, hD x , hD ξ )e -ξ 2 /2h-ϕ(z)/h acting (formally) on symbols a

(z; h) is in- deed a h-PDO Ψ(z, hD z ; h) with symbol σ Ψ (z, ζ; h) ∼ σ 0 (z, ζ) + hσ 1 (z, ζ) + • • • , and σ 0 (z, ζ) = -p A (z -ζ, iζ).
Remark 1 We could also use the standard unitary Bargman transform [START_REF] Sjöstrand | Singularités analytiques microlocales[END_REF]

T 0 : L 2 (R 3 ) → H Φ (C 3 ),
space of holomorphic functions square integrable with respect to the weight Φ(z) = |x| 2 /4. It is somewhat more natural in this context, but we found it convenient to stick to notations of [START_REF] Martinez | Microlocal WKB expansions[END_REF], [START_REF] Martinez | An Introduction to Semiclassical and Microlocal Analysis[END_REF].

Performing another "absolute" metaplectic complex transformation T 2 (independent of all parameters), associated with a complex canonical transformation κ 2 , we can arrange so that σ 0 has the quadratic approximation σ 02 (z, ζ) = 3 j=1 2µ j z j ζ j near the magnetic microlocal well. We still denote by u T the corresponding quasi-mode. In fact T 2 could be avoided if we replace T by T 0 .

We solve Hamilton-Jacobi (HJ) equation σ 02 (z, ∂ z ϕ 0 (z) = 0 by ϕ 0 (z) = 1 4 z 2 . This gives the complex "outgoing" Lagrangian manifold Λ 0 + = {ζ = ∂ z ϕ 0 (z)}, and also the "incoming" one Λ 0 -= {ζ = -∂ z ϕ 0 (z)}. By the stable-unstable manifold Theorem this carries to σ 0 (z, ζ), thus we can solve HJ equation with holomorphic phase function ϕ(z) = ϕ 0 (z) + O(|z| 3 ) defining Λ ± . Continuing this way, we can also solve the transport equations with an analytic symbol a(z; h). Altogether we get a WKB solution of (Ψ(z,

hD z ; h) -λ A (h))a(z; h) ∼ 0 microlocally near ρ 0 = ρ + 0 , such that u T (z, h) ∼ e -ξ 2 /2h-ϕ(z)/h a(z; h) (7)
Here ∼ means modulo a remainder term O(e -1/Ch ). Applying classical variational principles, this is justified by Agmon type estimates, which say how close the microlocal solution approaches the actual eigenfunction.

To this end, we use the method of non characteristic deformations. Namely in some neighborhood Ω of (0,0) where microlocal solution (7) holds (i.e. before the occurence of focal points on Λ ± ) there is deformation of -φ 0 to a family of pl.s.h. weights -φ t (z, z), such that the R-Lagrangian manifold Λ t = exp tH q (Λ) is transverse to {z = 0} at (0,0), and hence of the form ζ = ∂φt ∂z . We check that φ t (z, z) → 2 Re ϕ(z) as t → ∞. Moreover σ 0 remains elliptic outside (0,0) along these deformations, in the sense -σ 0 (z, ζ)| Λt ≥ Const.(|z| 2 + |ζ| 2 ) for all t, and φ t satisfies the eikonal equation ∂ t φ t (z, z) = -2σ 0 (z, ∂ t φ t ). We then modify φ t to another weight ψ t such that ψ t < 2 Re ϕ near ∂Ω, where WKB constructions of type [START_REF] Helffer | Semi-classical Analysis for the Schrödinger and applications[END_REF] fail to exist, and get the following Agmon estimate: If u is the 1:st normalized eigenfunction of P M A then for any compact set K ⊂ Ω, there exists ε > 0 and h ε > 0, such that uniformly for z = xiξ ∈ K, 0 < h < h ε we have

e ξ 2 /2h+ϕ(x-iξ)/h (T u + (x, ξ; h) -u T (x, ξ; h)) =
O(e -ε /h ) (8) Altogether, this gives the asymptotics for the first eigenvalue λ A (h) for

P M + A (x, hD x ) λ A (h) = h Tr + (F pA ) + O(h 2 ) ( 9
)
where Tr + (F pA ) = 3 j=1 µ j . This gives also the asymptotics of T u + in Ω ⊂ M .

The main drawback is the poor control of the neighborhood of x 0 where this asymptotics holds for u = T -1 u T (left inverse), because κ A is not merely implied by a change of variables in the xspace. Nevertheless since κ -1 T ( Λ + ), Λ + = {ζ = i∂ z ϕ(z)} is a strictly positive Lagrangian manifold at (0,0) (also after applying κ 2 ), the same holds for

κ -1 A • κ -1
T ( Λ + ) (in the initial canonical variables), and so Λ + has no focal points near ρ 0 . So Agmon estimate (8) again justifies the asymptotics of u of

P M A (x, hD x ) in some (real) neighborhood Ω R of x 0 .
Of course by symmetry y 3 → -y 3 everything carries to M -. To proceed we need to extend WKB constructions, when ω ′ is small enough, near minimal geodesics for Agmon metric associated with the Hamiltonian without a magnetic field, .

Geometry and quantization of magnetic Hamiltonians

We work here with the rescaled operator P ′ A (y, hD y ) = (P ′ A ) M (y, hD y ) localized in M = M + . In Classical Mechanics the generalized momentum η of a particle is related to its velocity by v = η -A(y). The magnetic symplectic 2-form is

σ A = dv ∧ dy = dη ∧ dy + dA(y) = σ + dA(y)
Consider first the kinetic term K 0 (y, η) = |v| 2 = (η -A(y)) 2 . We know [START_REF] Ivrii | Microlocal Analysis[END_REF], [START_REF] Raymond | Geometry and Spectrum in 2D Magnetic Wells[END_REF] that if dA(y) = 0, then Σ = {K 0 = 0} = {dK 0 = 0} is a 3-D submanifold of T * R 3 , and Σ ∩ Σ σ (Σ σ denotes orthogonal complement with respect to the symplectic 2-form σ) is the Hamilton vector field for 3 j=1 F j (y)(η j -A j (y) where F j (y) are the components of the vector intensity of the magnetic field B(y). In the present case Σ ∩ Σ σ = R ∂ ∂y3 . (Note that in the 2-D case, Σ is just a 2-D symplectic manifold). Many examples of integrable magnetic Hamiltonians are provided in [START_REF] Ivrii | Magnetic Schrodinger operators: Geometry, Classical and Quantum Dynamics and Spectral Asymptotics[END_REF].

As in the case of Schrödinger operator without a magnetic field, we may define complex branches of the energy surface by considering imaginary times. They glue along Σ. For Hamiltonian (1) with a potential V sufficiently confining, the "magnetic classically allowed region" (MCAR) is a domain in R 3 with 2 connected components in ±y 3 > 0 bounded by the closed 2-D manifolds defined by π y Σ ∩ {p ′ A (y, η) = E ′ } , where π y : T * R 3 → R 3 is the natural projection, and the "magnetic classically forbidden region" (MCFR) its complement. For small ω ′ , the MCAR can be viewed as a deformation of the "classically allowed region" (CAR) for the Hamiltonian P ′ 0 (y, hD y ) = (hD y ) 2 + W (y) without a magnetic field.

We stress that in the MCFR we make the substitution v → iv instead of ξ → iξ as in [START_REF] Helffer | Effet tunnel pour l'équation de Schrödinger avec champ magnétique[END_REF]. By this transformation the Hamiltonian remains real and gauge invariant. This is consistent with the following example of operator on L 2 (R 2 ) H(y, hD y ) = (hD y1 ) 2 +(hD y2 -by 1 ) 2 +V 1 (y 1 )+ω 2 y 2 2 (10) which is unitarily equivalent (after partial Fourier transform with respect to x 2 ) to the Schrödinger operator without a magnetic field [START_REF] Brüning | Splitting of lower energy levels in a quantum double well in a magnetic field and tunneling of wave packets in Nanowires[END_REF], [START_REF] Dobrokhotov | Tunneling, Librations and Normal Forms in a Quantum Double Well with a Magnetic Field[END_REF]). It turns out that our computations lead (up to the present accuracy) to the same quantities as [START_REF] Helffer | Effet tunnel pour l'équation de Schrödinger avec champ magnétique[END_REF], but without factor i at some places, see e.g. Proposition 1 below.

H ′ (x, hD x ) = (hD x1 ) 2 + (hD x2 ) 2 + (ωx 2 -bx 1 ) 2 + V 1 (x 1 ) (see
Remark 2 Quantization of the kinetic term with a linear A(y) = (-y 1 , 0, 0) (Landau gauge) was considered first in [START_REF] Landau | Mécanique Quantique[END_REF], p.496, leading to Landau levels with infinite degeneracy. In this case, the yprojection of classical trajectories consist in a family of helices, with arbitrary center. The generalized wave-functions are expressed in term of a family of Hermite functions oscillating in the y 1direction around the center. This gives a hint at understanding the case of magnetic Schrödinger operator (MSO) P ′ A (y, hD y ) : degeneracy is lifted by the potential, that fixes somehow the center of the helix.

Notice that the potential well U E

′ (h) for p ′ 0 (y, η) = η 2 +W (y) at energy E ′ (h) = 4ν 2 ω 4 (V (r 0 )+ λ A (h)) with 2 connected components U ± E ′ (h) is given by U E ′ (h) ± = {y : V (|x(y)|) - ν 2 ω 2 (y 2 1 + y 2 2 ) ≤ V (r 0 ) + λ A (h), ±y 3 > 0} (11)
Thus y ± 0 (the "center" of each well) verifies y ± 0 ∈ U ± E ′ , and ∂U ± E ′ (h) ⊂ M ± if r → V (r) grows fast enough at infinity and near 0, so that in particular W | y3=0 > 0. Under these conditions, by the hyperplane symmetry, p ′ 0 has two potential wells

U ± E ′ (h)
localized near y 0 = y ± 0 . Conditions on Floquet exponents (see Sect.3) ensure that W has nondegenerate minima at y ± 0 . Taking Weyl quantization of p ′ A (x, ξ) = (ηω ′ A(y)) 2 + W (y), where A(y) = (-y 2 , y 1 , 0) (symmetric gauge) and ω ′ = 2ν 2 ω 3 , we get the magnetic Schrödinger Hamiltonian

P ′ A (y, hD y )u(y; h) = (2πh) -3 R 6 e i(y-z)η/h p ′ A ( y + z 2 , η)u(z) dz dη (12)
and by a change of variables

P ′ A (y, hD y )u(y; h) = (2πh) -3 R 6 e i(y-z) η+A y+z 2 /h p ′ 0 ( y + z 2 , ξ)u(z) dz dη (13)
Remark 3 For non linear magnetic potentials A(y) we need to modify this quantization, since it is no longer gauge invariant, see [START_REF] Mantoiu | The Magnetic Weyl Calculus[END_REF]. There are farreaching generalisation of such h-PDO's, in particular Berezin-Toeplitz operators, see [START_REF] Charles | Aspects semi-classiques de la quantification géometrique[END_REF], [START_REF] Charles | Symbolic calculus for Toeplitz operators with half-form[END_REF], [START_REF] Hitrik | Asymptotic Bergman kernels in the analytic case[END_REF].

5 WKB constructions for P ′ 0 and P ′

A

We will somehow relate the quasi-modes of P ′ 0 and P ′ A in M ± , outside U ± E ′ (h) . Since we are looking for rather rough estimates on the eigenvalue splitting, it is sufficient here to search for WKB solutions of P ′ 0 (y, hD y )λ ′ A (h) w(y; h) ∼ 0 in the classically forbidden region (CFR) W (y) ≥ E ′ (h), although P ′ 0 and P ′ A do not have the same ground state, see e.g. the discussion in [START_REF] Helffer | Semi-classical Analysis for the Schrödinger and applications[END_REF], Sect.7.2.

First we recall some known facts on tunneling for Schrödinger operators without a magnetic field.

Let S + E ′ (h) (y) be Agmon distance from U + E ′ (h) to y for the Riemannian conformal metric W (y) - E ′ (h)) + dy 2 , E ′ (h) = 4ν 2 ω 4 (V (r 0 ) + λ A (h)
) which vanishes on U E ′ . We know that the exponential decay of w from the well U + E ′ , say, is given roughly by e -S + E ′ (y)/h , see [START_REF] Helffer | Multiple wells in the semi-classical limit I[END_REF], [START_REF] Martinez | Estimations de l'effet tunnel pour le double puits II. Etats hautement excités[END_REF]. So we have to compute w near minimal geodesics γ E ′ (h) between ∂U ± E ′ (h) . Such (finitely many) minimal geodesics are also called librations [START_REF] Brüning | Splitting of lower energy levels in a quantum double well in a magnetic field and tunneling of wave packets in Nanowires[END_REF], [START_REF] Dobrokhotov | Tunneling, Librations and Normal Forms in a Quantum Double Well with a Magnetic Field[END_REF], [START_REF] Anikin | Multidimensional tunneling between potential wells at non degenerate minima[END_REF]. Within the required accuracy on tunneling rates, we could again replace E ′ (h) by E ′ , which amounts to replace the librations by the instanton between U ± E ′ = {y ± 0 }. Changing t to it we may assume that in

W (y) ≥ E ′ (h) (CFR) P ′ 0 -E ′ (h) takes the form Q ′ 0 (y, hD y )+E ′ (h) = (hD y ) 2 -W (y)+E ′ (h). Then the WKB solutions of (Q ′ 0 (y, hD y ) + E ′ )w(y, h) = λ A (h)w(y, h) are given by w(y; h) = e -φ(y,E ′ )/h b(y, E ′ ; h) (14)
In particular the phase function φ solves (locally) Hamilton-Jacobi equation (∇φ(y, E ′ )) 2 -W (y) + E ′ = 0 and the amplitude b(y, E ′ ; h) = b 0 (y, E ′ ; h) + hb 1 (y, E ′ ; h) + • • • the transport equations. These WKB expansions can be justified by Agmon estimates near the minimal geodesics γ E ′ as in [START_REF] Martinez | Estimations de l'effet tunnel pour le double puits II. Etats hautement excités[END_REF].

Let now

Q ′ A (y, hD y )+E ′ (h) = (hD y -ω ′ A(y)) 2 - W (y) + E ′ (h)
be the corresponding Hamiltonian in MCFR. Recall that MCFR (with the magnetic field of strength ω ′ ) is a deformation O(ω ′ ) of the CFR (without the magnetic field).

To solve (Q ′ A (y, hD y ) + E ′ (h))u(y; h) ∼ 0, we try u(y; h) = e -ψ(y,E ′ )/h b(y, E ′ ; h)w(y, E ′ ; h) = e -(φ(y,E ′ )+ψ(y;E ′ ))/h b(y, E ′ ; h)c(y, E ′ ; h) (15) and replace E ′ (h) by E ′ as stated above. We find (omitting parameter E ′ (h)), using [START_REF] Ivrii | Microlocal Analysis[END_REF] and ( 13)

(Q ′ A (y, hD y ) + E ′ )u(y; h) = (2πh) -3 exp i (y -z)(η + ω ′ A( y + z 2 )) + φ(z) + ψ(z) /h q ′ 0 ( y + z 2 , η) + E ′ b(z, E ′ ; h)c(z, E ′ ; h) dz dη (16)
(we have identified the operator with its Weyl symbol). Applying asymptotic stationary phase yields eikonal equation [START_REF] Martinez | Estimations de l'effet tunnel pour le double puits II. Etats hautement excités[END_REF] which is Hamilton-Jacobi equation for Hamiltonian (ηω ′ A(y) + φ ′ (y)) 2 + W (y) + E ′ . By a stability argument [START_REF] Martinez | Estimations de l'effet tunnel pour le double puits II. Etats hautement excités[END_REF] has again a unique solution near γ E ′ , and can be solved perturbatively in ω ′ > 0 small enough.

(-ω ′ A(y) + φ ′ (y) + ψ ′ (y)) 2 -W (y) + E ′ = 0
Namely, look for ψ(y) = ω ′ ψ 0 (y) + ω ′2 ψ 1 (y, ω ′ ). Substituting into [START_REF] Martinez | Estimations de l'effet tunnel pour le double puits II. Etats hautement excités[END_REF] we get at first order in ω ′ : 2∇φ(y)∇ψ 0 (y) = 2 ∇φ(y), A(y) which is a transport equation along the integral curves of Hamilton vector field H Q ′ 0 . Using Hamilton Eq. for Q ′ 0 it can be written as d dt ψ 0 (y(t)) = 2 -y 2 ∂φ ∂y1 + y 1 ∂φ ∂y2

or in cylindrical coordinates y 1 = r cos θ, y 2 = r sin θ, y 3 = y 3 , d dt ψ 0 (x(t)) = 2 ∂φ ∂θ (x(t)) which can be integrated as (see also [START_REF] Helffer | Effet tunnel pour l'équation de Schrödinger avec champ magnétique[END_REF], Eq.(2.27) 1 ) ψ 0 (t) = ψ 0 | H * + 2 t 0 ∂φ ∂θ (x(s)) ds. Higher order approximations in ω ′ are obtained similarly. In particular setting

ψ 1 = ψ 1 + ω ψ 2 , we find 2∇φ(y)∇ψ ′ 1 (y) = -(ω ′ y) 2 -ψ ′ 0 2 + 2 A(y), ψ ′ 0 = -|ψ ′ 0 -A(y)| 2
or (see also Eq. (2.27) 2 and Lemma 3.9 of [START_REF] Helffer | Effet tunnel pour l'équation de Schrödinger avec champ magnétique[END_REF]),

d dt ψ 1 (x(t)) = -|ψ ′ 0 (x(s))-A(x(s))| 2 ds
The same type of arguments holds for transport equations defining c(y, E ′ ; h).

Note that in dimension 1, the leading order term of the symbol b(y, E ′ ; h)c(y, E ′ ; h) would assume the familiar form b(y, E ′ ; h)c(y, E ′ ; h) = Const. exp -1 2 y (W (z) -E ′ ) -1/2 dz , and we expect this formula to hold in the 3-D case as well.

Remark 4

The computation above is based on representation [START_REF] Ivrii | Magnetic Schrodinger operators: Geometry, Classical and Quantum Dynamics and Spectral Asymptotics[END_REF] in the real domain, can actually be carried over MCFR where the phase becomes complex (possibly purely imaginary), if we choose suitable branches of solutions with positive imaginary part. This can e.g. be achieved after performing a FBI transform of type (8) which allows for complex phases.

This eventually gives

u = u WKB outside some ω ′ - neighborhood of CAR {y : W (y) ≤ E ′ (h)} of the form Ω + ω ′ ∪ Ω - ω ′ .
In fact CFR and MCFR may differ by O(ω ′ ). To ensure that our WKB solutions overlap with the microlocal solutions obtained in Sect.3, we need to assume that representation [START_REF] Helffer | Semi-classical Analysis for the Schrödinger and applications[END_REF] of the actual eigenfunction holds in a sufficiently large neighborhood (of size O(ω ′ )) of the CAR {y : W (y) ≤ E ′ } (without the magnetic field). This requires that we have a good control, sufficiently far away the microlocal wells, of the pl.s.h. weights -φ t constructed in Sect.3. It should be instructive to consider the model [START_REF] Helffer | Effet tunnel pour l'équation de Schrödinger avec champ magnétique[END_REF] from this viewpoint.

Let us summarize our constructions in the following:

Proposition 1 : Assume Ω + ω ′ ∪ Ω - ω ′ ⊂ Ω + R ∪ Ω - R ,
where Ω ± R have been defined at the end of Sect.2. Then there are WKB solutions u WKB as in [START_REF] Mantoiu | The Magnetic Weyl Calculus[END_REF] of

(P A (x, hD x ) -E -λ A (h))u WKB ∼ 0 extending the quasi-modes u ± of P M ± A constructed in (7) along the (finite set) of minimal geodesics γ E ′ between ∂U ± E ′ (h) .
As in the case of Schrödinger operator without a magnetic field, we can control the decay of eigenfunctions outside U ± E ′ by Agmon estimates. We start with a rough weighted energy estimate. Let E a = {y : [START_REF] Helffer | Semi-classical Analysis for the Schrödinger and applications[END_REF],p.43. Let also Φ be a Lipschitz function on ∂ Ω. Then for all u ∈ C 2 (Ω; C), u| ∂ Ω = 0, we recall from [START_REF] Helffer | Semi-classical Analysis for the Schrödinger and applications[END_REF],Prop.7.2.25

d E ′ (U + E ′ , y) + d E ′ (U - E ′ , y) ≤ S E ′ + a} ⊂ M + ∪ M -for suitable a > 0, and Ω ⊂ R 3 large enough with C 2 boundary, such that U + E ′ ⊂ Ω, U - E ′ ∩ Ω = ∅, Γ = ∂Ω ∩ E a ⊂ M + ∩ M -, see
Ω |(hD x -A(x))(e Φ/h u)| 2 dx+ Ω (W (x) -E ′ -|Φ ′ (x)| 2 )e 2Φ/h |u| 2 dx = Re Ω e 2Φ/h (P ′ A -E ′ )u(x)u(x) dx (18)
The weight Φ should of course be related with φ+ψ, but the term (W -E ′ -|Φ ′ (x)| 2 ) in the second integral refers rather to the eikonal equation verified by φ alone, so instead of the decay observed in the WKB solution, we end up with this we would get in case A(x) = 0. So sharper estimates would require either to assume analyticity of the potential as in [START_REF] Helffer | Effet tunnel pour l'équation de Schrödinger avec champ magnétique[END_REF], or to change formula [START_REF] Martinez | An Introduction to Semiclassical and Microlocal Analysis[END_REF], to a formula verified by the ratio φ A = u A /u 0 of ground states with and without a magnetic field. Note that in case λ A (h) = λ 0 (h), which is achieved in particular (Bohm-Aharonov vector potential) when rot A = 0 in Ω, necessarily non simply connected, and γ A(x) dx ∈ 2πhZ for all closed path γ ⊂ Ω, then (hD x -A(x))φ A = 0. This is due to an identity by R.Lavine and M.O'Carroll, see [START_REF] Helffer | Semi-classical Analysis for the Schrödinger and applications[END_REF],p.93 and references therein, and leads in particular that φ A is just a (x-dependent) phase factor. So in this case there would be no additionnal decay due to the magnetic field.

6 The gap formula and estimate of the splitting . As in the case of Schrödinger operator without a magnetic field, we relate the splitting to an integral of u + ∂u-∂y3 over Γ ⊂ {y 3 = 0} that bissects the classically forbidden region near the minimal geodesics.

The gap formula is given by the interaction matrix using Stokes formula with the magnetic potential

The second term vanish since A 3 (y) = 0, hence ( 19) is identical to the gap formula without a magnetic field. This altogether with ( 18) yields an estimate for the splitting of the form e -S E ′ +O(ω ′ ) /h ≤ E 1 (h)-E 0 (h) ≤ e -S E ′ -O(ω ′ ) /h (20) but getting a sharper exponent requires to modify (18) as suggested above. Precise Agmon estimates also depend on the number of these geodesics, and on the structure of the eigenfunction u A of P A (y, hD y ) near ∂U ± E ′ (h) , obtained in Sect.3.

Acknowledgements

We thank B.Helffer for interesting discussions on Agmon estimates in the presence of a magnetic field, and a referee for some useful references.