Julie Movellan

Françoise Rocher

Zohra Chikh

Cécile Marivingt-Mounir

Jean-Louis Bonnemain

Jean-François Chollet
email: jean.franc.chollet@univ-poitiers.fr

Synthesis and Evaluation as Biodegradable Herbicides of Halogenated Analogues of L-meta-Tyrosine

Keywords: Allelochemicals, Allelopathy, m-Tyrosine, Halogenated Analogues of L-metatyrosine, Herbicide, Sustainable agriculture

L-meta-tyrosine is a herbicidal non protein amino acid isolated some years ago from fine fescue grasses and characterized by its almost immediate microbial degradation in soil (half life less than 24 hours). Nine mono-or dihalogenated analogs of this allelochemical have been obtained through a seven-step stereoselective synthesis from commercial halogenated phenols. Bioassays showed a large range of biological responses, from a growth root inhibition of lettuce seedling similar to that noted with m-tyrosine [2-amino-3-(2chloro-5-hydroxyphenyl)propanoic acid or compound 8b] to an increase of the primary root growth concomitant with a delay of secondary root initiation [2-amino-3-[2-fluoro-5-hydroxy-3-(trifluoromethyl)phenyl]propanoic acid or compound 8h]. Compound 8b was slightly less degraded than m-tyrosine in the non-sterilized nutritive solution used for lettuce development while the concentration of compound 8h remained unchanged for at least two weeks. These data indicate that it is possible to manipulate both biological properties and degradation of mtyrosine by halogen addition.

Introduction

Protection of crops against pests probably arose at the same time as agriculture itself. Modern crop protection developed quickly after the Second World War with the advent of organic chemistry and numerous active ingredients became available in various domains: organophosphate and organochlorine insecticides, dithiocarbamate fungicides or auxinic herbicides also called phytohormones. These compounds were widely used to meet the high demand for food after the war without adequate assessment of the risks that could result from the use of some molecules [START_REF] Zadoks | A hundred and more years of plant-protection in the Netherlands[END_REF]. However, after many years, adverse effects on health or environment were often highlighted. Herbicides of the triazine family are leached and found in groundwater (Felding 1992b[START_REF] Hall | Leaching and runoff losses of herbicides in a tilled and untilled field[END_REF]) while organochlorine insecticides may accumulate in the environment or in lipophilic parts of some organisms [START_REF] Caro | Accumulation by plants of organochlorine insecticides from soil[END_REF][START_REF] Mueller | Persistent organochlorine pesticides in human milk samples from Australia[END_REF][START_REF] Ntow | Accumulation of persistent organochlorine contaminants in milk and serum of farmers from Ghana[END_REF][START_REF] Smith | Organochlorine chemicals in seafood: occurrence and health concerns[END_REF]. Thus, intensive agriculture, as practiced for several decades in developed countries, is changing and in particular, the part concerning the protection of cultivated plants [START_REF] Rabbinge | Scenario studies for future agriculture and crop protection[END_REF].

Many phytopharmaceutical compounds were withdrawn from the market after adverse effects were demonstrated [START_REF] Karabelas | Impact of European legislation on marketed pesticides -A view from the standpoint of health impact assessment studies[END_REF]). The arsenal of agrochemical products is restricted and no major mode of herbicide action has been introduced to the market place for about two decades [START_REF] Duke | Why have no new herbicide modes of action appeared in recent years?[END_REF]. Therefore new strategies must emerge to meet the specific requirements for crop protection in a context of sustainable agriculture [START_REF] Clark | A perspective on the role of quantitative structure-activity and structureproperty relationships in herbicide discovery[END_REF][START_REF] Dayan | Rationale for a natural products approach to herbicide discovery[END_REF][START_REF] Duke | Why have no new herbicide modes of action appeared in recent years?[END_REF][START_REF] Epstein | Patterns of pesticide use in California and the implications for strategies for reduction of pesticides[END_REF][START_REF] Gosme | Intensive versus low-input cropping systems: What is the optimal partitioning of agricultural area in order to reduce pesticide use while maintaining productivity[END_REF][START_REF] Jacobsen | Role of plant pathology in integrated pest management[END_REF][START_REF] Tesio | Allelopathy, a chance for sustainable weed management[END_REF][START_REF] Van Lenteren | A greenhouse without pesticides: fact or fantasy?[END_REF].

Thus, in order to identify new molecules able to control weed growth, allelopathy may be a promising way and in particular, to develop new synthetic herbicide families [START_REF] Bais | The role of root exudates in rhizosphere interations with plants and other organisms[END_REF][START_REF] Birkett | Does allelopathy offer real promise for practical weed management and for explaining rhizosphere interactions involving higher plants?[END_REF][START_REF] Duke | Natural-products with potential use as herbicides[END_REF][START_REF] Duke | New herbicide target sites from natural compounds[END_REF][START_REF] Duke | Natural phytotoxins as herbicides[END_REF][START_REF] Farooq | The role of allelopathy in agricultural pest management[END_REF][START_REF] John | Role of Allelopathy in vegetables crops production[END_REF][START_REF] Macias | Allelopathy in the search for natural herbicide models[END_REF][START_REF] Niemeyer | Hydroxamic acids derived from 2-hydroxy-2H-1,4-benzoxazin-3(4H)one: key defense chemicals of cereals[END_REF][START_REF] Petroski | Natural compounds for pest and weed control[END_REF][START_REF] Vyvyan | Allelochemicals as leads for new herbicides and agrochemicals[END_REF].

The definition of allelopathy has evolved over the years [START_REF] Duke | Allelopathy: Current status of research and future of the discipline: A Commentary[END_REF][START_REF] Molisch | Der Einfluss einer Pflanze auf die andere-Allelopathie[END_REF][START_REF] Rice | Allelopathy[END_REF][START_REF] Weston | Allelochemics: chemical interactions between species[END_REF] and currently, allelopathy is defined by the International Allelopathy Society as any process involving secondary metabolites produced by plants, algae, bacteria and fungi that influences the growth and development of agriculture and biological systems [START_REF] Chou | Introduction to allelopathy[END_REF]). More restrictively, allelochemicals are substances that are produced by one plant. Phenolics compounds are an important and common family of plant allelochemicals in the ecosystem [START_REF] Li | Phenolics and Plant Allelopathy[END_REF][START_REF] Wu | Allelochemicals in wheat (Triticum aestivum L.): cultivar difference in the exudation of phenolic acids[END_REF]. Recently, Bertin et al have isolated L-meta-tyrosine, a non protein amino acid exuded by the roots of fine fescue grasses, especially Festuca rubra spp commutata [START_REF] Bertin | Fescue ecology, physiology, and allelopathy -A case study[END_REF][START_REF] Bertin | Grass roots chemistry: meta-tyrosine, an herbicidal nonprotein amino acid[END_REF][START_REF] Weston | Allelochemics: chemical interactions between species[END_REF]. This compound which is synthesized by hydroxylation of phenylalanine in F. rubra [START_REF] Huang | meta-Tyrosine in Festuca rubra ssp commutata (Chewings fescue) is synthesized by hydroxylation of phenylalanine[END_REF]) exhibits herbicidal properties for a wide range of plant species [START_REF] Bertin | Grass roots chemistry: meta-tyrosine, an herbicidal nonprotein amino acid[END_REF]). More specifically m-tyrosine is a root growth inhibitor, contrary to its position isomers, tyrosine and ortho-tyrosine [START_REF] Bertin | Grass roots chemistry: meta-tyrosine, an herbicidal nonprotein amino acid[END_REF]. The mechanisms of m-tyrosine herbicidal activity are not known [START_REF] Bertin | Grass roots chemistry: meta-tyrosine, an herbicidal nonprotein amino acid[END_REF]. Furthermore, the concentrations required for significant root growth inhibition in the same plant species may vary considerably when filter paper bioassays are used [START_REF] Bertin | Grass roots chemistry: meta-tyrosine, an herbicidal nonprotein amino acid[END_REF][START_REF] Kaur | Taking ecological function seriously: soil microbial communities can obviate allelopathic effects of released metabolites[END_REF] and may be higher than what would normally be found in Festuca rubra spp commutata rhizosphere [START_REF] Kaur | Taking ecological function seriously: soil microbial communities can obviate allelopathic effects of released metabolites[END_REF]. This latter data and the rapid microbially loss of m-tyrosine in soil have raised a recent debate about the actual role of m-tyrosine in allelopathy [START_REF] Bertin | Assessment of the Phytotoxic Potential of m-Tyrosine in Laboratory Soil Bioassays[END_REF][START_REF] Duke | Allelopathy: Current status of research and future of the discipline: A Commentary[END_REF][START_REF] Kaur | Taking ecological function seriously: soil microbial communities can obviate allelopathic effects of released metabolites[END_REF]).

Here we report the synthesis in a stereo specific manner of nine mono-or polyhalogenated analogues of L-meta-tyrosine (Table 1). Using two bioassays methods, the phytotoxic activity of these compounds was then compared to that of m-tyrosine. Our data 1-support that mtyrosine is an efficient allelopathic compound, 2-demonstrate for the first time that manipulation of both biological properties and degradation of m-tyrosine by halogen addition is possible and 3-point out two derivatives that may be used as tools to elucidate the mechanisms of the biological activity of m-tyrosine.

Material and methods

Synthesis 1 H and 13 C FT-NMR spectra were measured with a Varian Model XL 400-MHZ spectrometer. Chemical shifts are reported as δ in units of parts per million (ppm) relative to chloroform ( 1 H 7.26 ppm, 13 C 77ppm) or water ( 1 H 4.79 ppm). Multiplicities are reported as follows: s (singlet), d (doublet), t (triplet), qd (quadruplet), dd (doublet of doublets), m (multiplet). Coupling constants are reported as a J value in Hertz (Hz). The number of protons (n) for a given resonance is indicated as nH, and is based on spectral integration values.

Melting points were determined on an Electrothermal IA9200 apparatus and are uncorrected.

Mass spectra were recorded using a Waters 3100 spectrometer in electro spray mode.

In vitro bioassays

In a first approach, biological effect of each halogenated compound was assessed in comparison to m-tyrosine using filter paper bioassays with lettuce [START_REF] Bertin | Grass roots chemistry: meta-tyrosine, an herbicidal nonprotein amino acid[END_REF]). Ten non-sterilized seeds (Lactuca sativa Var. Blonde de Paris) were placed in 9 cm Petri dishes on a filter paper impregnated with 1.5 mL of aqueous solutions of the tested compound at 0, 40, 80, 160, 320 or 640 µM concentrations. To avoid evaporation, Petri dishes were sealed with Parafilm ® . Two replicates of each assay were maintained under dim daylight conditions in a controlled environment (21 ± 0.5°C, HR 90% ± 5%). Radicle length was measured 4 days later.

The biological activities of m-tyrosine, 8b and 8h were also studied under non-sterilized hydroponic conditions. Shortened Pasteur pipettes were immersed in 15 mL centrifuge tubes filled with a nutrient solution composed of 2 mmol.l -1 of KNO 3 , 1 mmol.l -1 of KH 2 PO 4 , 1 mmol.l -1 of Ca(NO 3 ) 2 , 1 mmol.l -1 of MgSO 4 .7H 2 O, 0.3 µmol.l -1 (molybdenum equivalent) of (NH 4 ) 6 Mo 7 O 24 , 1 µmol.l -1 of H 3 BO 3 , 0.09 µmol.l -1 of MnSO 4 , H 2 O, 0.12 µmol.l -1 of NH 4 NO 3 , 0.035 µmol.l -1 of ZnSO 4 .7H 2 O and 18 µmol.l -1 of iron-EDDHA chelate 6% Fe. In these experiments, germination of lettuce seeds (Lactuca sativa Var. Blonde de Paris) occurs on an agar layer (4 mm high) at the top of each Pasteur pipette. Under these conditions, the radicle had to cross this layer to reach the nutritive solution. Unless stated otherwise, plants were grown at 24 ± 0.5°C and 60 ± 5% RH during the photoperiod (14 h, 250 µmol photons.m -2 .s -1 ).

Results and discussion

Synthesis of halogenated analogues of m-tyrosine

Several methods of synthesis for halogenated meta-tyrosine or tyrosine were previously proposed [START_REF] Bovonsombat | Facile syntheses of 3halo and mixed 3,5-dihalo analogues of N-acetyl-L-tyrosine via sulfonic acidcatalysed regioselective monohalogenation[END_REF][START_REF] Drain | Phenylalanine derivatives and their preparation[END_REF][START_REF] Kirk | Synthesis and adrenergic activity of ring-fluorinated phenylephrines[END_REF][START_REF] Konkel | Syntheses of 6-fluoro-meta-tyrosine and of its metabolites[END_REF]. Fluorinated analogues and particularly 6-fluoro-meta-tyrosine (Table 1, 8d) was synthesized to use it as a tracer for medical imaging [START_REF] Konkel | Syntheses of 6-fluoro-meta-tyrosine and of its metabolites[END_REF][START_REF] Vanbrocklin | A new precursor for the preparation of 6-[18F]Fluoro-L-m-tyrosine ([18F]FMT): efficient synthesis and comparison of radiolabeling[END_REF][START_REF] Vasdev | Selectivity of elemental fluorine towards L-tyrosine and L-alpha-methyltyrosine in acidic media and the syntheses of their F-18 3-fluoro and F-18 3,5-difluoro derivatives[END_REF]. Halogenated m-tyrosine derivatives were found to lower the blood pressure of mammals [START_REF] Drain | Phenylalanine derivatives and their preparation[END_REF]. Finally, we chose the synthesis in seven steps proposed by Konkel et al [START_REF] Kirk | Synthesis and adrenergic activity of ring-fluorinated phenylephrines[END_REF][START_REF] Konkel | Syntheses of 6-fluoro-meta-tyrosine and of its metabolites[END_REF] and among the nine products that were synthesized (Table 1), six were new (compounds 8a, 8b, 8e, 8f, 8h and 8i). The first six steps were performed with good to very good yields, whose median is between 65% (steps 2 and 4) and 95% (step 6). Only the last step (deprotection of the alpha-amino acid function) deserves to be optimized for some products, the yields ranging from 5% (8g) to 64% (8c) (median 42%). An interesting aspect of the proposed method is also the possibility of using crude products obtained at the end of reaction for the next step without further purifications.

Protection of the hydroxyl group of the phenolic derivatives (Fig. 1, 2a-i)

To a solution of 2-chlorophenol (1a, 6.43 g, 50 mmol) in dimethylformamide (25 mL), were added imidazole (3.75 g, 55 mmol) and ter-butyldimethylsilyl chloride (TBDMSCl, 8.29 g, 55 mmol). The mildly exothermic reaction was placed in an iced-bath. After the solution was stirred overnight under nitrogen atmosphere, water (125 mL) was added and the mixture was extracted with pentane (3 x 50 mL). The organic layer was washed successively with water (50 mL), 10% sodium carbonate aqueous solution (3 x 50 mL), water (3 x 50 mL) and dried over anhydrous magnesium sulphate. Removal of solvent under reduced pressure gave 2a (9.04 g, 37 mmol) which was used without further purification. The products 2b-i were obtained following the same procedure. Yields and 1 H NMR spectra for compounds 2a-i are given in the online resource 1 in the ESM.

Preparation of aldehydes (Fig. 1, 3a-i)

To a solution of tert-butyl(2-chlorophenoxy)dimethylsilane (2a, 8.9 g, 36.7 mmol) in dry THF (25 mL) stirred under argon atmosphere and cooled to -78°C, was added dropwise a solution of sec-butyllithium in hexane (1.4 M, 29 mL, 40 mmol). After stirring for 1 hour, an excess (3 mL, 40 mmol) of distillated DMF was added dropwise at -78 °C. After 30 min, tetrabutylammonium fluoride (TBAF, 1 M in THF, 44.5 mL, 44.5 mmol) was added dropwise at the same temperature. After 1.5 hour, the reaction was allowed to warm to room temperature, and water (50 mL) was added. After removal of most of the THF under reduced pressure, aqueous sodium hydroxide solution (1 N, 25 mL) was added and the basic solution was washed with diethyl ether (3 x 50 mL). The organic layer was washed 3 times with aqueous sodium hydroxide solution (1 N, 25 mL). The combined aqueous layers were acidified with hydrochloric acid solution (3 M), and the product was extracted with diethyl ether (4 x 50 mL). After drying over anhydrous magnesium sulphate, removal of solvent under reduced pressure gave 3a (2.25 g, 14.4 mmol) that was used without further purification. The products 3b-i were obtained following the same procedure. Yields and 1 H NMR spectra for compounds 3a-i are given in the online resource 2 in the ESM.

Benzylation of the hydroxyl group (Fig. 1, 4a-i)

To a solution of 2-chloro-3-hydroxybenzaldehyde (3a, 2.25 g, 16 mmol) in DMF (20 mL), potassium carbonate was added (2.30 g, 23 mmol). After 5 minutes of stirring, benzyl bromide (2.15 mL, 18 mmol) was added dropwise. The reaction mixture was stirred at room temperature for 20 hours. Diethyl ether (30 mL) was added and the mixture was washed with water (3 x 20 mL). The organic layer was dried over anhydrous magnesium sulphate and the removal of solvent under reduced pressure gave 4a (3.33 g, 13.5 mmol) that was used without further purification. The products 4b-i were obtained following the same procedure. Yields and 1 H NMR spectra for compounds 4a-i are given in the online resource 3 in the ESM.

Preparation of azalactones (Fig. 1, 5a-i)

To a solution of 3-(benzyloxy)-2-chlorobenzaldehyde (4a, 3.2 g, 13 mmol) in acetic anhydride (25 mL) were added sodium acetate (1.23 g, 15 mmol) and hippuric acid (2.69 g, 15 mmol). The reaction mixture was stirred at 80 °C for 2 hours and cooled at room temperature and with an iced-bath. Cold ethanol (30 mL) was added and after 15 min, iced water (15 mL) was added. The orange precipitate was collected by filtration on Büchner and dried under vacuum to give 5a (3.32 g, 8.53 mmol) that was used without further purification.

The products 5b-i were obtained following the same procedure. Yields and 1 H NMR spectra for compounds 5a-i are given in the online resource 4 in the ESM.

Ring opening of the azalactones (Fig. 1, 6a-i

) (Z)-4-(3-(benzyloxy)-2-chlorobenzylidene)-2-phenyloxazol-5(4H)-one (5a, 3.23 g, 8.3 mmol)
and sodium acetate (730 mg, 8.9 mmol) in MeOH (400 mL) were stirred at room temperature.

After 1 hour, the solvent was removed under reduced pressure. The residue was dissolved in ethylacetate (200 mL) and washed with water (3 x 50 mL). The organic phase was dried over anhydrous magnesium sulphate and the solvent was evaporated to give 6a (3.30 g, 7.82 mmol) that was used without further purification. The products 6b-i were obtained following the same procedure. Yields and 1 H NMR spectra for compounds 6a-i are given in the online resource 5 in the ESM.

Debenzylation and double bond saturation (Fig. 1, 7a-i)

A solution of (E)-methyl-2-benzamido-3-(3-(benzyloxy)-2-chlorophenyl) acrylate (6a, 1.0 g, 2.
37 mmol) in MeOH (100 mL) was hydrogenated over 10% Pd/C (205 mg) at 40 psi for 20 h. The palladium was then filtered on celite and the methanol was evaporated to give 7a (0.79 g, 2.37 mmol) which was used without further purification. The products 7b-e and 7g-i were obtained following the same procedure. 7f was obtained by hydrogenation at 60 °C under 80 psi during 20 h. Yields and 1 H NMR spectra for compounds 7a-i are given in the online resource 6 in the ESM.

Deprotection of the alpha-amino acid function to obtain L-m-tyrosine analogues (Table 1, Fig. 1, 8a-i)

A solution of methyl 2-benzamido-3-(2-chloro-3-hydroxyphenyl) propanoate (7a, 2.89 g, 8.66 mmol) in hydrochloric acid (3 M, 125 mL) was refluxed for 24 h. The solution was concentrated to dryness and the residue was dissolved in water (60 mL) and washed with diethyl ether (3 x 20 mL). The aqueous phase was then filtrated and neutralized to pH 6 with an aqueous solution of sodium hydroxide 1M. The water was evaporated until the formation of a precipitate. The precipitate was filtrated and product was dried under reduced pressure to give 8a (741 mg, 3.43 mmol). The products 8b-i were obtained following the same procedure.

Biological activity of compounds 8a-i

Filter paper bioassays

After 4 days of treatment with m-tyrosine used at 640 µM concentration, the lettuce root length was reduced by about 75% (Table 2). Adding a chlorine atom in the para position with respect to the hydroxyl group (compound 8b) did not affect significantly this inhibition (Table 2). By contrast, addition of a fluorine atom on the same carbon (compounds 8d and 8g) led to a lack of biological activity of these xenobiotics and further addition of a trifluoromethyl group on the free adjacent position (compound 8h) induced a clear increase of root growth under our experimental conditions (Table 2, Fig. 2). This increase was significant at concentrations ≥ 160 µM while the root growth inhibition induced by m-tyrosine and compound 8b were clearly significant at concentrations ≥ 40 and ≥ 80 µM respectively (Fig. 2). According to previous filter bioassays, the concentration of m-tyrosine required to achieve 50% reduction of lettuce root growth vary from 10 -20 µM [START_REF] Bertin | Assessment of the Phytotoxic Potential of m-Tyrosine in Laboratory Soil Bioassays[END_REF][START_REF] Bertin | Grass roots chemistry: meta-tyrosine, an herbicidal nonprotein amino acid[END_REF]) to about 150 µM [START_REF] Kaur | Taking ecological function seriously: soil microbial communities can obviate allelopathic effects of released metabolites[END_REF]. This variability may be due to the fact that m-tyrosine is not equally phytotoxic towards the cultivars used or / and to some microbial degradation of this molecule around the seeds when they are not sterilized.

Hydroponic bioassays

It is well known that m-tyrosine microbial degradation is high in soil, the half-life being estimated as less than 24 hours [START_REF] Kaur | Taking ecological function seriously: soil microbial communities can obviate allelopathic effects of released metabolites[END_REF]. By contrast, in a medium unsuitable for bacterial growth such as ultrapure water, the degradation of m-tyrosine as well as compound 8b was extremely low (21 and 17% respectively after 1.5 months in solution). In the nutritive medium used for lettuce growth, m-tyrosine and compound 8b concentrations remained unchanged during four days then they dropped sharply, especially one week after the beginning of the experiment (Fig. 3). m-tyrosine and compound 8b could not be detected at day 11 and day 14 respectively. If chloramphenicol -a broad spectrum antibiotic -was added at 0.25 g.L -1 to the medium, the degradation of both compounds was considerably reduced. At day 11, m-tyrosine was detected at 129 µM concentration (81% of the initial concentration) and 8b was detected at 144 µM concentration (90% of the initial concentration). These data indicate that bacterial degradation is slightly less efficient for compound 8b than for mtyrosine under our experimental conditions. Compound 8h concentration remained unchanged during the whole experiment (Fig. 3). This does not mean that the latter is not degraded in soil taking into account the diversity and the high content of microorganisms in this compartment.

Under our experimental conditions, the lettuce root growth was dramatically and similarly inhibited by both m-tyrosine and compound 8b at 40 µM (initial concentration) and was reduced by about 65% in presence of these two compounds at 10 µM (initial concentration) (Fig. 4). Our data support those of [START_REF] Bertin | Grass roots chemistry: meta-tyrosine, an herbicidal nonprotein amino acid[END_REF] [START_REF] Bertin | Grass roots chemistry: meta-tyrosine, an herbicidal nonprotein amino acid[END_REF], which clearly suggest that m-tyrosine is a potent inhibitor of plant development. In addition they show that compound 8b, a less soluble (Table 1) and less degradable molecule, can exhibit the same deleterious properties on root development at these two low concentrations. The root growth was not reduced by both m-tyrosine and compound 8b at the lowest concentration used (2.5 µM) (Fig. 4). However, it is likely that this initial concentration dropped quickly due to root uptake in addition to microbial degradation, and this may concern all the experiments conducted with higher concentrations. By contrast, in soil, the m-tyrosine flux occurs in a dynamic system with exudation from fescue roots on the one hand, microbial degradation, adsorption on soil constituents and receiving plant uptake on the other hand [START_REF] Duke | Allelopathy: Current status of research and future of the discipline: A Commentary[END_REF]. The mechanism of m-tyrosine uptake by plant tissues is not known but it is possible that a pHdependent carrier system is involved in addition to diffusion taking into account the low specificity of several amino-acid carrier systems [START_REF] Chen | ANT1, an aromatic and neutral amino acid transporter in Arabidopsis[END_REF][START_REF] Chollet | Synthesis and structure-activity relationships of some pesticides with an alpha-amino acid function[END_REF][START_REF] Deletage-Grandon | Carrier-mediated uptake and phloem systemy of a 350-Dalton chlorinated xenobiotic with an alpha-amino acid function[END_REF][START_REF] Fischer | Substrate specificity and expression profile of amino acid transporters (AAPs) in Arabidopsis[END_REF].

Long-term experiments suggest that the deleterious effects induced by high concentrations of m-tyrosine and compound 8b are irreversible. After 14 days of post-germination, the growth of the roots exposed initially to 640 µM m-tyrosine or compound 8b remained completely inhibited despite the microbial degradation occurring in the nutritive solution (Fig. 5 A, B andC). The brownish coloration of root tips probably due to oxidation of phenolic compounds by extracellular peroxidases suggests a necrotic state of the apical meristem. Similarly, the shoot development was stopped. Such a herbicidal effect is not necessary in field. In this regard, the dwarf germinations, which are induced by concentrations as low as 10 µM (Fig. 4), are not competitive for light and must be dramatically affected by soil drought taking into account the poor development of their root system. The complementary data with compound 8h from these long-term experiments support and extend those from filter paper bioassays (Table 2, Fig. 2). Treatments with this moderate hydrophilic (Table 1) and stable analogue under our experimental conditions (Fig. 3) led to an increase of primary root growth (Fig. 5D).

Furthermore, this response was concomitant with a clear delay of the secondary root emergence and a shoot growth inhibition (Fig. 5, compare A and D). By contrast, m-tyrosine promotes lateral root elongation in Arabidopsis and some lettuce isolates [START_REF] Bertin | Grass roots chemistry: meta-tyrosine, an herbicidal nonprotein amino acid[END_REF].

Conclusion

Nine halogenated m-tyrosine derivatives, among which six are new, have been obtained through a seven-step stereoselective synthesis from commercial halogenated phenol. Filter paper bioassays are an easy method to get preliminary information to evaluate the putative allelochemical properties of these compounds but they need to be completed by more suitable approaches such as hydroponic bioassays. Our experiments support previous data [START_REF] Bertin | Grass roots chemistry: meta-tyrosine, an herbicidal nonprotein amino acid[END_REF]. m-tyrosine is an efficient allelopathic agent but cannot be used in field because of its high microbial degradation [START_REF] Bertin | Assessment of the Phytotoxic Potential of m-Tyrosine in Laboratory Soil Bioassays[END_REF][START_REF] Kaur | Taking ecological function seriously: soil microbial communities can obviate allelopathic effects of released metabolites[END_REF]. Our data indicate that it is possible, by halogen addition, to manipulate: i/ the biological properties of m-tyrosine, from similar (compound 8b) to contrary (compound 8h) properties on root growth, ii/ the stability of these compounds in non-sterilized conditions. Compound 8b is slightly less degraded than m-tyrosine while compound 8h remained stable for at least two weeks under our experimental conditions. Finally, these two halogenated derivatives, which induced deleterious but contrary effects on seedling development, may be tools to elucidate the mechanisms of the biological activity of m-tyrosine. Our investigation is a complementary approach to opposite strategies consisting to confer resistance to exogenously added m-tyrosine [START_REF] Huang | Pleiotropic physiological consequences of feedback-insensitive phenylalanine biosynthesis in Arabidopsis thaliana[END_REF]. Radicle and shoot length were measured 6 days after sowing. The Kruskal-Wallis test was used to assess statistically significant differences in comparison to control. (*** P < 0.001 ; NS, not significant). For box plots, n = 25 for control and n = 9 for the other experiments Fig. 5 Long-term effect (14 days) of m-tyrosine (B), 8b (C) and 8h (D) used at 640 µM concentration on lettuce seedling growth. Seedlings were grown under dim daylight conditions in a controlled environment (21±0.5°C, HR 90±5%) for the first week after sowing and then at 24 ± 0.5°C and 60% RH during the photoperiod (14 h, 250 µmol photons.m -2 .s -1 ).

Figure legends

HO R 1 R 2 R 3 R 4 O R 1 R 2 R 3 R 4 Si HO R 1 R 2 R 3 R 4 CHO O R 1 R 2 R 3 R 4 CHO O R 1 R 2 R 3 R 4 N O O O R 1 R 2 R 3 R 4 HN COOCH 3 O HO R 1 R 2 R 3 R 4 HN COOCH 3 O HO R 1 R 2 R 3

8a, 2

 2 -amino-3-(2-chloro-3-hydroxyphenyl)propanoic acid, yield: 40%; brown solid; 1 H NMR δ (ppm): 3.27-3.56 (ABX system, J AB = 14 Hz, J AX = 9 Hz, J BX = 5.2 Hz, 2H, CH 2 ), 4.44 (m, 1H, CH), 6.97 (d, 3 J = 7.5 Hz, 1H, ArH), 7.04 (d, 3 J = 8.1 Hz, 1H, ArH), 7.33 (t, 3 J = 7.7 Hz, 1H, ArH);13 C NMR δ (ppm): (D 2 O/DCl): 34.72 (CH 2 ), 54.5 (CH), 116.1 (Ar CH ), 122.9 (C-Cl), 127.9 (2 Ar CH ), 134.5 (Ar Cq ), 173.7 (C-OH), 185.8 (COOH); Calculated [M] + (C 9 H 10 ClNO 3 ) m/z = 215.03. Found: ESI+: [M+H] + m/z = 216.08 8b, 2-amino-3-(2-chloro-5-hydroxyphenyl)propanoic acid; yield: 16%; grey solid; 1 H NMR (D 2 O/DCl) δ (ppm): 3.11 (m, 1H, CH 2 ), 3.33 (ddd, J =19.7 Hz, J =14.7 Hz, J = 5.6 Hz, 1H, CH 2 ), 4.04-4.06 (m, 1H, CH), 6.85-6.87 (m, 2H, ArH), 7.32 (dd, J = 17.6 Hz, J = 8.3 Hz, 1H, ArH); 13 C NMR (D 2 O) δ (ppm): 33.6 (CH 2 ), 53.7 (CH), 114.6 (Ar CH ), 116.1 (Ar CH ), 121.5 (Ar CH ), 130.5 (Ar Cq ), 150.2 (C-OH), 158.5 (C-Cl), 170.2 (COOH); Calculated [M] + (C 9 H 10 ClNO 3 ) m/z = 215.03. Found: ESI+: [M+H] + m/z = 216.08 8c, 2-amino-3-(2-fluoro-3-hydroxyphenyl)propanoic acid; yield: 64%; pink crystals; 1 H NMR (D 2 O) δ (ppm): 3.15-3.36 (ABX syst., J AB = 14.4 Hz, J AX = 7.6 Hz, J BX = 4.8 Hz, 2H, CH 2 ) , 4.11 (t, 1H, 3 J = 5.6 Hz, CH), 6.84 (t, 1H, 3 J = 6.2 Hz, 1H, ArH), 6,98 (dt, 1H, 4 J = 1.7 Hz, 3 J = 8.3 Hz, 1H, ArH), 7.05 (t, 1H, 3 J = 7.8 Hz, 1H, ArH); 13 C NMR (D 2 O) δ (ppm): 33.1 (CH 2 ), 57.9 (CH), 120.1 (Ar CH ), 125.3 (Ar CH ), 126.4 (Ar Cq ), 127.7 (Ar CH ), 146.2 (C-F), 146.3 (C-OH), 176.4 (COOH); Calculated [M] + (C 9 H 10 FNO 3 ) m/z = 199.06. Found: ESI+: [M+H] + m/z = 200.14 8d, 2-amino-3-(2-fluoro-5-hydroxyphenyl)propanoic acid; yield: 63%; beige solid; 1 H NMR (D 2 O) δ (ppm): 3.22 (ABX syst., J AB = 14.6 Hz, J AX = 7.2 Hz, J BX = 5.4 Hz, 2H, CH 2 ), 4.42 (m, 1H, CH), 6.86 (m, 2H, ArH), 7.07(m, 1H, ArH); 13 C NMR (D 2 O) δ (ppm): 33.3 (CH 2 ), 55 (CH), 119.3 (Ar CH ), 120.7 (Ar CH ), 124.3 (Ar CH ), 124.4 (Ar Cq ), 154.4 (C-OH), 157.1(C-F), 173.5 (COOH); Calculated [M] + (C 9 H 10 FNO 3 ) m/z = 199.06. Found: ESI+: [M+H] + m/z = 200.04 8e, 2-amino-3-(2,4-dichloro-3-hydroxyphenyl)propanoic acid; yield: 45%; grey solid; 1 H NMR (D 2 O) δ (ppm): 3.04-3.45 (ABX syst., J AB = 14 Hz, J AX = 7 Hz, J BX = 4.9 Hz, 2H, CH 2 ), 4.7-5.1 (m, 1H, CH), 6.8-6.92 (m, 1H, ArH), 7.10-7.30 (m, 1H, ArH); Calculated [M] + (C 9 H 9 Cl 2 NO 3 ) m/z = 249.00. Found: ESI+: [M+H] + m/z = 250.04 8f, 2-amino-3-(2,6-dichloro-3-hydroxyphenyl)propanoic acid; yield: 10%; grey solid; 1 H NMR (D 2 O) δ (ppm): 3.04-3.42 (ABX syst., J AB = 14 Hz, J AX = 6.9 Hz, J BX = 4.9 Hz, 2H, CH 2 ), 5.09-5.12 (m, 1H, CH), 6.82 (d, 3 J = 8.8 Hz, 1H, ArH), 6.93 (d, 3 J = 8.8 Hz, 1H, ArH), 13 C NMR (D 2 O) δ (ppm): 36.2 (CH 2 ), 55.9 (CH), 114.7 (Ar CH ), 121.49 (Ar CH ), 130.5 (Ar Cq ), 154.4 (C-OH), 158.0 (C-Cl), 173.5 (COOH); Calculated [M] + (C 9 H 9 Cl 2 NO 3 ) m/z = 249.00. Found: ESI+: [M+H] + m/z = 250.04 8g, 2-amino-3-(2,6-difluoro-3-hydroxyphenyl)propanoic acid; yield: 5%; grey solid; 1 H NMR (D 2 O) δ (ppm): 3.21-3.33 (ABX syst., J AB = 14.5 Hz, J AX = 7.9 Hz, J BX = 5.9 Hz, 2H, CH 2 ), 3.93-3.95 (m, 1H, CH), 6.85-7.01 (m, 2H, ArH); Calculated [M] + (C 9 H 9 F 2 NO 3 ) m/z = 217.06. Found: ESI+: [M+H] + m/z = 218.25 8h, 2-amino-3-[2-fluoro-5-hydroxy-3-(trifluoromethyl)phenyl]propanoic acid; yield: 90%; white solid; 1 H NMR (D 2 0) δ (ppm): 3.19-3.42 (syst. ABX, J AB = 14.5 Hz, J AX = 7.9 Hz, J BX = 5.9 Hz, 2H, CH 2 ), 4.06-4.09 (m, 1H, CH), 7.09-7.11 (m, 1H, ArH), 7.17-7.19 (m, 1H, ArH); 13 C NMR (D 2 0) δ (ppm) : 31.5 (CH 2 ), 55.9 (CH), 119.4 (Ar CH ), 122.5 (Ar CH ), 133.6 (Ar Cq ), 155.0 (C-OH), 149.1 (C-CF 3 ), 157.2 (C-F), 173.5 (COOH); Calculated [M] + (C 10 H 9 F 4 NO 3 ) m/z = 267.05. Found: ESI+: [M+H] + m/z = 268.18 8i, 2-amino-3-(6-chloro-2-fluoro-3-hydroxyphenyl)propanoic acid; yield: 32%; grey solid; 1 H NMR (D 2 O) δ (ppm): 2.67-2.73 (dd, 3 J = 9.5 Hz, 4 J = 9.45 Hz, 1H, CH), 3.21-3.25 (dd, 4 J = 4 Hz, 3 J = 12.1 Hz, 2H, CH 2 ), 6.68-6.72 (m, 1H, ArH), 6.81-7.01 (m, 2H, ArH, OH) ;. 13 C NMR (D 2 O) δ (ppm): 30.55 (CH 2 ), 54.19 (CH), 116.27(Ar CH ), 120.66 (Ar CH ), 123.70 (C-Cl), 125.18 (d, 2 J C-F = 12.6 Hz, ArCq), 145.05 (d, 2 J C-F = 12.3 Hz, C-OH), 149.25 (d, 1 J C-F = 240.7 Hz, C-F), 170.08 (COOH); Calculated [M] + (C 9 H 9 ClFNO 3 ) m/z = 233.03 Found: ESI+: [M+H] + m/z = 234.25

Fig. 1 Fig. 3 Fig. 4

 134 Fig. 1 General reaction schemeFig.2Effect of m-Tyr, 8b and 8h at various concentrations on lettuce (L. sativa var Bonde Parisienne) seedling root growth (filter paper bioassays). Radicle and shoot length were measured 4 days after sowing. The Kruskal-Wallis test was used to assess statistically significant differences in comparison to control. (*** P < 0.001 ; ** P < 0.01 ; NS, not significant). For box plots, n = 20 Fig. 3 Time-course changes over an-18-day period of m-Tyr, 8b and 8h concentrations in the nutrient solution used for lettuce growth. The initial concentration of the products was 160 µM. Dark conditions, temperature 20 ± 1°C. Mean of 3 assays Fig. 4 Effect of m-Tyr and 8b at various concentrations on lettuce (L. sativa var Bonde Parisienne) seedling root growth (hydroponic experiments). Radicle and shoot length were measured 6 days after sowing. The Kruskal-Wallis test was used to assess statistically significant differences in comparison to control. (*** P < 0.001 ; NS, not significant). For box plots, n = 25 for control and n = 9 for the other experiments

Figure 1 :Figure 2 :Figure 3 :Figure 4 :Figure 5 :

 12345 Figure 1: General reaction scheme

Table 1 .

 1 Structure and physicochemical properties of m-tyrosine and halogenated analogs. For all products, 2.1<pKa 1 <2.2, Polar Surface Area (PSA) = 83.55 Å 2 and number of Hydogen Bond Donors (HBD) = 4. MW = Molecular Weight. Mp = Melting point, decomp. = decomposition. All properties were computed using ACD Log D Sol Suite v.12.02 software except melting points that were experimentally determined.

	Two	arrows	localize	each	Pasteur	pipette.	A:	control

Table 2 .

 2 Short time effect of halogenated products at 640 µM concentration on lettuce root elongation in filter paper bioassays. For each experiment (A-F), one or two halogenated compounds were tested simultaneously with m-Tyr and control (without any product). Root length of seedlings was measured 4 days after placing seeds in control, m-tyrosine and m-tyrosine derivatives treatments. Results are expressed as the percentage of root growth inhibition in comparison to the control of the same experiment (A,B,C,D,E or F), taking into account the median of the main root length of 20 seedlings. The Kruskal-Wallis test was used to assess statistically significant differences in comparison to control (*** p<0.001, ** p<0.01, * p<0.05, (NS) non significant).

	Product	Experiment	Inhibition of root growth (%)
	m-Tyr	D	76.7 ***
	8a		34.9 **
	m-Tyr	A	82.9 ***
	8b		75.6 ***
	m-Tyr	F	78.3 ***
	8b		65.2 ***
	m-Tyr	A	82.9 ***
	8c		61.0 ***
	m-Tyr	D	76.7 ***
	8d		2.3 (NS)
	m-Tyr	C	75.8 ***
	8e		33.3 *
	m-Tyr	E	70.5 ***
	8f		53.8 ***
	m-Tyr	B	90.7 ***
	8g		7.0 (NS)
	m-Tyr	C	75.8 ***
	8h		-51.5 **
	m-Tyr	F	78.3 ***
	8h		-58.7 ***
	m-Tyr	E	70.5 ***
	8i		48.7 ***


Jean Zinn-Justin
email: jean.zinn-justin@cea.fr

Ulrich D Jentschura

Order-dependent mappings: strong coupling behaviour from weak coupling expansions in non-Hermitian theories

Keywords: numbers: 11.10.Jj, 11.15.Bt, 11.25.Db, 12.38.Cy, 03.65.Db

A long time ago, it has been conjectured that a Hamiltonian with a potential of the form x 2 +ivx 3 , v real, has a real spectrum. This conjecture has been generalized to a class of so-called PT symmetric Hamiltonians and some proofs have been given. Here, we show by numerical investigation that the divergent perturbation series can be summed efficiently by an order-dependent mapping (ODM) in the whole complex plane of the coupling parameter v 2 , and that some information about the location of level crossing singularities can be obtained in this way. Furthermore, we discuss to which accuracy the strong-coupling limit can be obtained from the initially weak-coupling perturbative expansion, by the ODM summation method. The basic idea of the ODM summation method is the notion of order-dependent "local" disk of convergence and analytic continuation by an order-dependent mapping of the domain of analyticity augmented by the local disk of convergence onto a circle. In the limit of vanishing local radius of convergence, which is the limit of high transformation order, convergence is demonstrated both by numerical evidence as well as by analytic estimates.

I. INTRODUCTION

The order-dependent mapping (ODM) summation method has been introduced in Ref. [START_REF] Seznec | Summation of divergent series by order dependent mappings: Application to the anharmonic oscillator and critical exponents in field theory[END_REF] and initially applied to series expansions of an integral whose perturbative expansion counts the number of Feynman diagrams with four-point vertices, to the quartic anharmonic oscillator, and to renormalization group (RG) functions in the three-dimensional φ 4 quantum field theory [START_REF] Seznec | Summation of divergent series by order dependent mappings: Application to the anharmonic oscillator and critical exponents in field theory[END_REF][START_REF] Zinn-Justin | Summation of divergent series: Orderdependent mapping[END_REF]. Later, other examples of quantum mechanics and field theory type have been studied [START_REF] Le Guillou | The hydrogen atom in strong magnetic fields: summation of the weak field series expansion[END_REF][START_REF] Guida | 3D Ising model: the scaling equation of state[END_REF][START_REF] Kneur | Asymptotically improved convergence of optimized perturbation theory in the Bose-Einstein condensation problem[END_REF][START_REF] Kneur | Critical and tricritical points for the massless 2D Gross-Neveu model beyond large N[END_REF][START_REF] Kneur | Emergence of tricritical point and liquid-gas phase in the massless 2+1 dimensional Gross-Neveu model[END_REF]. Some convergence proofs have been given in Refs. [START_REF] Duncan | Convergence proof for optimized δ expansion: Anharmonic oscillator[END_REF][START_REF] Guida | Improved Convergence Proof of the Delta Expansion and Order Dependent Mappings[END_REF]. However, these examples have in common, as a starting point, perturbations of Hermitian quantum Hamiltonians. In most of them the ODM summation method appears as a higher order extension of some variational calculation [START_REF] Seznec | Summation of divergent series by order dependent mappings: Application to the anharmonic oscillator and critical exponents in field theory[END_REF][START_REF] Caswell | Accurate energy levels for the anharmonic oscillator and a summable series for the double-well potential in perturbation theory[END_REF]. Therefore, we may wonder whether Hermiticity plays a role in explaining some specific convergence properties of the method.

Here, we thus consider a non-Hermitian example, the expansion generated by an i x 3 addition to the harmonic potential. The corresponding Hamiltonian is PT symmetric, that is, symmetric under a simultaneous complex conjugation and space reflection. Even though this is not quite obvious, such a Hamiltonian has a real spectrum, as first conjectured by Bessis and Zinn-Justin (1992) and proven in [START_REF] Dorey | Spectral equivalences, Bethe ansatz equations, and reality properties in PT-symmetric quantum mechanics[END_REF][START_REF] Shin | On the reality of eigenvalues for a class of PT-symmetric oscillators[END_REF]. In this note, we show numerically that the ODM method, suitably adapted to the problem, allows a precise determination of the ground state energy. The convergence properties also give some information about the analyticity as a function of the strength of the i x 3 perturbation, in particular excluding level crossing in some region of the Riemann surface. * Electronic mail: jean.zinn-justin@.cea.fr Another potential application of the method is the iφ 3 quantum field theory in two and three space dimensions, whose renormalization group functions describe the Lee-Yang edge singularity of Ising-like statistical models [START_REF] Fisher | Yang-Lee Edge Singularity and φ 3 Field Theory[END_REF][START_REF] Bender | Scalar Quantum Field Theory with Cubic Interaction[END_REF], should longer perturbative series become available.

II. THE IMAGINARY CUBIC POTENTIAL

We here consider the spectrum of the simplest example of a PT symmetric Hamiltonian,

H = - 1 2 d 2 dx 2 + 1 2 x 2 + i 6 √ g x 3 , (1)
where g is a real positive parameter. Indeed, the Hamiltonian is invariant under simultaneous complex conjugation and parity transformation x → -x. In this convention, the resonance energies of the cubic potential have their branch cut along the negative real g axis, where the quantity i √ g becomes real and the particle may tunnel through the "cubic wall" of the potential. One then verifies that the perturbative expansion of the energy eigenvalues E n for g → 0 contains only integer powers of g:

E n (g) = ∞ k=0 E n,k g k (2)
with real coefficients. Moreover, a steepest descent calculation of the path integral representation of the corresponding quantum partition function [START_REF] Lipatov | Divergence of the perturbation-theory series and the quasi-classical theory[END_REF][START_REF] Brézin | Perturbation theory at large order. I. The ϕ N interaction[END_REF][START_REF] Zinn-Justin | Perturbation series at large orders in quantum mechanics and field theories: application to the problem of resummation[END_REF] yields a large order behaviour of the form

E n,k ∼ k→∞ C (-1) k k b A -k k! , (3)
where A = 24/5 is the instanton action. The constant C and the half-integer b depend on n. The series is divergent for all g and when the expansion parameter is not small, a summation of the perturbative expansion is indispensable. Note that the sign oscillation with k and some additional considerations already suggest that the series is Borel summable (a property proved in [START_REF] Caliceti | Perturbation theory of odd anharmonic oscillators[END_REF]) and thus that the spectrum, beyond perturbation theory, is real. This was initially the origin of the conjecture of Bessis and Zinn-Justin, which was generalized to other PT symmetric Hamiltonians [START_REF] Bender | Real spectra in nonhermitian Hamiltonian having PT symmetry[END_REF][START_REF] Bender | Complex Extension of Quantum Mechanics[END_REF][START_REF] Bender | Making sense of non-Hermitian Hamiltonians[END_REF]. More recently, Padé summability was also rigorously established [START_REF] Grecchi | Pad summability of the cubic oscillator[END_REF]. This proof confirms numerical investigations based on the summation of the perturbative expansion by Padé approximants [START_REF] Bender | Numerical evidence that the perturbation expansion for a non-Hermitian PTsymmetric Hamiltonian is Stieltjes[END_REF]. Distributional Borel summability of the perturbation series to the complex resonance energies for negative coupling g was proved in [START_REF] Caliceti | Distributional Borel summability of odd anharmonic oscillators[END_REF].

In this note, we show by a combination of numerical and analytic arguments that the ODM method is especially well adapted to summing the perturbative series, converging for all values of the expansion parameter in the complex plane and, in addition, in a region of the second sheet of the Riemann surface. It thus also provides additional information about the spectrum for g complex on the Riemann surface [START_REF] Kleinert | Convergence behavior of variational perturbation expansion -A method for locating Bender-Wu singularities[END_REF].

However, before dealing with the quantum mechanics example, as a warm-up exercise, we first consider the simple integral

Z(g) = 1 √ 2π +∞ -∞ dx exp -1 2 x 2 -i 6 √ gx 3 , (4)
whose expansion coefficients count the number of Feynman diagrams that appear in the expansion of the partition function tre -βH corresponding to the Hamiltonian (1).

FIG. 1. Mapping: example of a function analytic in a cutplane. The cut in the g-plane (l.h.s.) is mapped onto the contour Γ of the λ-plane (r.h.s.).

III. ORDER-DEPENDENT MAPPING SUMMATION METHOD

A. Method

The ODM summation method [START_REF] Seznec | Summation of divergent series by order dependent mappings: Application to the anharmonic oscillator and critical exponents in field theory[END_REF] is based on some a priori knowledge, or educated guess, of the analytic properties of the function that is expanded. It applies both to convergent and divergent series, although it is mainly useful in the latter case [START_REF] Seznec | Summation of divergent series by order dependent mappings: Application to the anharmonic oscillator and critical exponents in field theory[END_REF].

Let E(g) be an analytic function that has the Taylor series expansion

E(g) = ∞ ℓ=0 E ℓ g ℓ , (5)
where the equality has to be understood in the sense of a (possibly asymptotic) series expansion. In what follows, we consider two functions analytic at least in a sector (as in the example of Fig. 1) and only mappings g → λ of the form

g = ρ λ (1 -λ) α , α > 1 , (6)
where α has to be chosen in accordance with the analytic properties of the function E (a rational number in our examples) and ρ is an adjustable parameter. A general discussion of the method can be found in Refs. [START_REF] Seznec | Summation of divergent series by order dependent mappings: Application to the anharmonic oscillator and critical exponents in field theory[END_REF][START_REF] Zinn-Justin | Summation of divergent series: Orderdependent mapping[END_REF]. An important property that singles out such a mapping is the following: inverting the mapping, one finds that, for g → ∞, λ approaches unity, and g -1/α has a regular expansion in powers of 1 -λ (see the Appendix).

After the mapping ( 6), E is given by a Taylor series in λ of the form

E g(λ) = ∞ ℓ=0 P ℓ (ρ) λ ℓ , (7)
where the coefficients P ℓ (ρ) are polynomials of degree ℓ in ρ. Since the result is formally independent of the parameter ρ, the parameter can be chosen freely. At ρ fixed, the series in λ is still divergent, but it has been verified on a number of examples (all Borel summable), and proved in certain cases [START_REF] Duncan | Convergence proof for optimized δ expansion: Anharmonic oscillator[END_REF][START_REF] Guida | Improved Convergence Proof of the Delta Expansion and Order Dependent Mappings[END_REF] that, by adjusting ρ order by order, one can devise a convergent algorithm. The basic idea is that ρ characterizes a "local radius of convergence" of the divergent input series. As the order of the mapping is increased, ρ → 0. However, the mapping g = g(ρ, λ) is constructed so that the circle of convergence of orderdependent radius ρ is always mapped onto a finite domain of the λ complex plane. At a risk of some oversimplification, we can describe the general paradigm of the ODM summation as follows: One "pretends" that the divergent input series is in fact a convergent series, but with a circle of convergence whose radius gradually vanishes as the order of the perturbative expansion is increased. Since both ρ → 0 and also |λ| < 1, the direct expansion in λ, with variable ρ, may then lead to a convergent output series even for large g.

The k-th approximant E (k) (g) is constructed in the following way: one truncates the expansion (7) at order k and chooses ρ = ρ k as to cancel the last term, i.e. P k (ρ k ) = 0. Because P k (ρ) has k roots (real or complex), in general one chooses for ρ the largest possible root (in modulus) ρ k for which P ′ k (ρ) is small (but see the more detailed discussion of Sec. III B).

This leads to a sequence of approximants

E (k) (g) = k ℓ=0 P ℓ (ρ k ) λ ℓ (g, ρ k ) with P k (ρ k ) = 0 , (8)
where λ = λ(g, ρ) is obtained by inverting the relation [START_REF] Kneur | Critical and tricritical points for the massless 2D Gross-Neveu model beyond large N[END_REF]. In the case of convergent series, it is expected that ρ k has a non-vanishing limit for k → ∞. By contrast, for divergent series it is expected that ρ k goes to zero for large k as

ρ k = O E -1/k k . ( 9
)
The intuitive idea here is that ρ k is proportional to a "local" radius of convergence.

The following remarks are in order. (i) Alternatively, one can choose the largest roots ρ k of the polynomials P ′ k (ρ) for which P k is small. Other mixed criteria involving a combination of P k and P ′ k can also be used. Indeed, the approximant is not very sensitive to the precise value of ρ k , within errors. Finally, P k+1 (ρ k )λ k+1 , as well as the values of P k (ρ)λ k in the neighbourhood of ρ k , give an order of magnitude of the error. (ii) In the ODM method, the main task is to determine the sequence of ρ k . Indeed, once the ρ k are known, for each value of g the calculation reduces to inverting the mapping ( 6) and simply summing the Taylor series in λ to the relevant order.

B. Optimal convergence analysis

We here give a heuristic analysis [START_REF] Seznec | Summation of divergent series by order dependent mappings: Application to the anharmonic oscillator and critical exponents in field theory[END_REF] of the convergence of the ODM method that shows how the convergence can be optimized. This will justify the choice of the class of zeros of the polynomials P k or P ′ k and provide a quantitative analysis of the corresponding convergence.

In this note, we consider only real functions analytic in a cut plane with a cut along the real negative axis (see Fig. 1) and a Cauchy representation of the form

E(g) = 1 π 0- -∞ dg ′ ImE(g ′ + i0) g ′ -g , (10)
up to possible subtractions to ensure the convergence of the integral for g → -∞. For later purpose, we introduce the notation

∆(g) = Im E(g + i0). (11)
Moreover, we assume that the function ∆ asymptotically fulfils

∆(g) ∝ g→0- (-g) -b-1 e A/g , A > 0 . ( 12
)
For the two examples discussed below, this exponential asymptotic form can be derived by steepest descent calculations [START_REF] Lipatov | Divergence of the perturbation-theory series and the quasi-classical theory[END_REF][START_REF] Brézin | Perturbation theory at large order. I. The ϕ N interaction[END_REF][START_REF] Zinn-Justin | Perturbation series at large orders in quantum mechanics and field theories: application to the problem of resummation[END_REF]. The function E(g) can be expanded in powers of g:

E(g) = ∞ ℓ=0 E ℓ g ℓ with E ℓ = 1 π 0- -∞ dg g ℓ+1 ∆(g) . (13)
The assumption [START_REF] Shin | On the reality of eigenvalues for a class of PT-symmetric oscillators[END_REF] then leads to a large order behaviour of the form

E k ∝ k→∞ (-A) -k Γ(k + b + 1) ∼ (-A) -k k b k! (14)
as assumed in Sec. II. We now introduce the mapping (6),

g = ρ λ (1 -λ) α . ( 15
)
The cut on the real negative axis of the g-plane is then mapped onto the contour Γ (Fig. 1).

The Cauchy representation then can be written as

E g(λ) = 1 2iπ Γ dλ ′ E g(λ ′ ) λ ′ -λ = 1 π Γ dλ ′ ∆ g(λ ′ ) λ -λ ′ , ( 16
)
where Γ is the image of the upper-part of the cut on the real negative axis under the mapping λ = λ(g) at fixed ρ, with a segment of the real λ negative axis, 1/(1 -α) < λ ′ < 0, and the complex contour Im( Γ) > 0 ending at λ = 1. The contours can be deformed if the function E(g) has analyticity properties beyond the first Riemann sheet.

We expand, omitting the dependence on ρ in λ,

E g(λ) = ∞ ℓ=0 P ℓ (ρ)[λ(g)] ℓ (17)
where the approximants are obtained by the truncated expansion

E g(λ) ≈ k ℓ=0 P ℓ (ρ k )[λ(g)] ℓ , (18)
and ρ k is chosen so that P k (ρ k ) ≈ 0. Moreover,

P ℓ (ρ) = 1 π Γ dλ λ ℓ+1 ∆ g(λ) . (19)
For k → ∞, the factor λ -k-1 favours small values of |λ|, but for too small values of λ, the exponential decay of ∆ g(λ) takes over. Thus, the remainder value of the polynomial P k (ρ k ) can be evaluated by the steepest descent method. The ansatz

ρ k ∼ R/k , R > 0 , (20)
implies that the saddle-point values of λ are independent of k and g(λ) → 0. Thus, ∆(g) can be replaced by its asymptotic form [START_REF] Shin | On the reality of eigenvalues for a class of PT-symmetric oscillators[END_REF] for g → 0 -, except for g of order 1 and thus λ close to 1. In what follows, we set

µ = R A , ( 21
)
since this is the only parameter (and it is independent of the normalization of g). The behaviour of P k (ρ k ), with ρ k being given by Eq. ( 20), is then given by the leading saddle point contributions:

P k (ρ k ) = O(e kσ ) with σ = 1 λµ (1 -λ) α -ln λ . (22)
At a zero of P k (ρ k ), several leading contributions cancel but each contribution is expected to give an order of magnitude of the error.

The saddle point equation is

σ ′ (λ) = - 1 µλ 2 (1 -λ) α-1 1 -(1 -α)λ + λµ = 0 . ( 23
) For µ large, the equation has a unique solution (for µ → 0, λ ∼ -1/µ), which is real negative and also exists for all values of µ with 1/(1 -α) < λ < 0. Then we note that at this saddle point, as a function of µ,

dσ dµ = - (1 -λ) α λµ 2 > 0 , (24)
since the saddle point value of λ is negative. This suggests decreasing µ as much as possible to improve the convergence, and thus, taking the zero ρ k with the smallest modulus. The exponential rate σ corresponding to the saddle point vanishes for

1 λµ (1 -λ) α -ln(-λ) = 0 , (25)
and this defines a special value µ c of the parameter µ. Some numerical values, obtained by combining Eqs. ( 23) and ( 25) are displayed in Table I. For µ > µ c , this contribution to P k (ρ k ) grows exponentially while for µ < µ c it decreases exponentially. The further analysis then somewhat depends on whether α is smaller or larger than 2. If α > 2, there exists a region on Γ where σ is positive and convergence is only possible if the initial integration contours Γ or Γ can be deformed in what corresponds to the second sheet of the function E(g). More generally, the contribution of the contour from the neighbourhood of λ = 1 plays an important role except, again, if it is possible to deform Γ in the first expression [START_REF] Brézin | Perturbation theory at large order. I. The ϕ N interaction[END_REF] in such a way that λ = 1 is no longer on the contour but inside it.

Two cases need to be distinguished: (i) One can decrease µ until other, complex, saddle points or maxima of the modulus on the contour give contributions with the same modulus. Relevant zeros of the polynomials P k and P ′ k then correspond to the cancellation between the different saddle point contributions. If then µ > µ c , σ is positive and the new approximants eventually also diverge and just yield a better behaved asymptotic expansion. If µ < µ c , σ is negative and the contributions to P k (ρ k ) decrease exponentially with k (this implies the possibility of contour deformation), and, as we show later, the method converges for all values of the parameter g. This also implies that the mapping ( 6) removes all singularities of E(g), obviously an exceptional situation but illustrated by the example of Sec. IV.

(ii) Generically, the mapping [START_REF] Kneur | Critical and tricritical points for the massless 2D Gross-Neveu model beyond large N[END_REF] does not cancel all singularities of the function E(g) and if it is possible to decrease µ up to µ c , σ vanishes and then other contributions of order at most exp C k 1-1/α (in particular coming from the integration near λ = 1) appear and the relevant zeros of P k and P ′ k correspond to the cancellation of these contributions. One can further decrease µ but the convergence properties are unlikely to improve since contributions of order exp C k 1-1/α survive and can no longer be reduced.

Returning to the expansion [START_REF] Zinn-Justin | Perturbation series at large orders in quantum mechanics and field theories: application to the problem of resummation[END_REF], at g fixed, from the behaviour of ρ k one infers

1 -λ ∼ R kg 1/α (26)
and so

λ k ∼ exp -k 1-1/α R g 1/α . ( 27
)
If the saddle point with λ < 0 is a leading saddle point and µ = µ c [case (i)], convergence is dominated by the behaviour of e kσ and this irrespective of the value of g.

If the integral is dominated by contributions of order 1 in the sense of case (ii), µ must be chosen in the range µ < µ c to avoid exponential divergence, but then to decrease the factor λ k one should choose R, and thus µ, as large as possible. This implies choosing µ = µ c .

In a generic situation, we then expect the behaviour of the contributions to P k (ρ k ) to be dominated by a factor of the form

exp C k 1-1/α , (28)
and the domain of convergence depends on the sign of the constant C. For C > 0, the domain of convergence is

|g| < R C -α [cos(arg(g)/α)] α . ( 29
)
For α > 2, this domain extends beyond the first Riemann sheet and requires analyticity of the function E(g) in the corresponding domain.

For C < 0, the domain of convergence is the union of the sector |arg(g)| < πα/2 and the domain

|g| > R |C| -α [-cos(arg(g)/α)] α . ( 30
)
Again for α > 2, this domain extends beyond the first Riemann sheet.

IV. EXAMPLE OF THE i x 3 INTEGRAL

We now study, as an example, the integral The function has a divergent series expansion in powers of g,

Z(g) = 1 √ 2π +∞ -∞ dx exp -1 2 x 2 -i 6 √ g x 3 . ( 31
)
Z(g) = 1 - 5 24 g + 385 1152 g 2 + O(g 3 ) , (32)
and the coefficient A in ( 3) is given by the non-trivial saddle point

x + i 2 √ g x 2 = 0 ⇒ x = 2i √ g , (33)
and so the saddle-point value of the integrand is

e A/g with A = 2 3 . (34)
Moreover, the function Z(g) has a convergent large g expansion of the form

Z(g) = g -1/6 ∞ ℓ=0 z ℓ g -ℓ/3 , (35)
where the evaluation of the integral yields

z 0 = 6 1/3 √ 2π 3 Γ( 2 3 ) = 1.1212331717419689582 . . . ( 36
)
This suggests the mapping (see the Appendix)

g = ρ λ (1 -λ) 3 (37)
and the introduction of the function

φ(λ, ρ) = (1 -λ) -1/2 Z g(λ) . (38)
Setting in the integral

x = s √ 1 -λ, one obtains φ(λ, ρ) = 1 √ 2π ∞ -∞ ds exp -1 2 s 2 + 1 2 λs 2 -i 6 λρs 3 = ∞ ℓ=0 φ ℓ λ ℓ (39)
where

φ k = 1 √ 2π 1 2iπ Γ dλ λ k+1 × ∞ -∞ ds exp -1 2 s 2 + 1 2 λs 2 -i 6 λρs 3 . ( 40
)
The first terms are

φ(λ, ρ) = 1 + 1 2 - 5 24 ρ λ + 3 8 - 35 48 ρ + 385 1152 ρ 2 λ 2 + O(λ 3 ) . ( 41
)
The asymptotic behaviour of Z for g → ∞,

Z(g) ∼ z 0 g -1/6 (42) is obtained for λ → 1, Z(g) ∼ g -1/6 ρ 1/6 φ(1, ρ). ( 43
) It follows φ(1, ρ) = z 0 ρ -1/6 . (44)
We use the following values of ρ k in order to calculate the first ODM approximants φ (k) (1, ρ) to φ(1, ρ) according to Eq. ( 8), but with the condition P ′ k (ρ k ) = 0 (zeros of the derivative) except for k = 1 where we choose the condition P 1 (ρ 1 ) = 0. These read (for k = 1, 2, 3, 4) 2.4 , 1.0909 , 0.7058 ± 0.1866 i 0.5894 ± 0.2633 i .

(45) The approximants to z 0 defined in Eq. ( 36) yield (we give only the real parts when ρ k is complex)

ρ 1/6 φ (1) (1, ρ) = 1.15709373 , (46a)
ρ 1/6 φ (2) (1, ρ) = 1.26825944 , (46b)
ρ 1/6 φ (3) (1, ρ) = 1.18358984 , (46c)
ρ 1/6 φ (4) (1, ρ) = 1.08048484 . (46d)
Surprisingly, using only a minimum number of terms from the input series, a rather good approximation to the strong-coupling limit is obtained. Figures 2,3, 4 and 5 show that for P k (ρ), in a rescaled variable τ = kρ/A, there is an essentially k-independent range where P k (ρ) and P ′ k (ρ) are small. This is also the region in the neighbourhood of which the zeros of P k and P ′ k are located. Quite generally, we choose for ρ k the complex zeros of P ′ k (ρ) with the largest real part in this range because they are easy to systematically identify.

The expected theoretical value, obtained by expressing that the real negative and the two complex conjugate solutions of the saddle point equation ( 23), λ = -0.1897168 . . . , (47a) λ = 0.8448584 . . . ± i 1.386261 . . . , yield contributions that have the same modulus (which allows for compensation) is µ = R/A ≈ 4.62987613 . . . and then the rate of convergence is predicted to be of the form 0.775 k from the corresponding estimate [START_REF] Grecchi | Pad summability of the cubic oscillator[END_REF].

The numerical investigation for g → ∞ extrapolated 31). We demonstrate that the favourable region where both P k (ρ) and P ′ k (ρ) are small persists in higher transformations orders.

to k = ∞ leads to (see Table II and Fig. 6)

lim k→∞ k Re(ρ k )/A = 4.65 ± 0.03 , (48)
yielding a value of µ significantly smaller than the value 5.31 . . . of Table I in section III B and consistent with the theoretical expectation.

To calculate the ODM approximants, we fit ρ k combining the theoretical asymptotic form and the calculated values of ρ k in the range k ≤ 60. This is achieved by adding a slightly ad hoc small correction term to the asymptotic formula.

For g → ∞, we define the following quantity which estimates how well the asymptotic strong-coupling limit is approximated by the ODM approximants of order k,

δ k = Re(Z(g)) g 1/6 ODM,k Z(g) g 1/6 exact -1 . ( 49
)
Some data are then displayed in Table II and Fig. 7.

From a fit to the numerical data in Fig. 7, One infers a geometric convergence of the form (0.78 ± 0.02) k quite consistent with the theoretical prediction. When g is finite and for k → ∞, the additional factor coming from λ k has the form (see Sec. III B)

λ k ∼ exp - k 3/5 R 2/5 g 2/5 . (50)
This factor can never cancel the geometric convergence factor and, thus, the ODM approximants converge on the whole Riemann surface.

FIG. 6. k Re(ρ k )/A, odd-even averaged, as a function of 1/k, for the model problem (31). The plot illustrates how the asymptotic limit given in Eq. ( 48) is approached for → ∞. 19) as a function of 1/k. We illustrate the asymptotic rate of convergence of the ODM approximants to the strong-coupling limit for the model problem (31), in higher transformation orders beyond k = 19.

FIG. 7. The plot shows (δ

k /δ19) 1/(k-

V. QUANTUM MECHANICS A. Strong coupling limit of the cubic potential

We now consider the spectrum of the Hamiltonian (1), which can be obtained by solving the time-independent Schrödinger equation

-1 2 ψ ′′ (x) + 1 2 x 2 + i 6 √ g x 3 ψ(x) = E ψ(x) (51)
with appropriate boundary conditions. The path integral representation of the corresponding partition function has a structure that bears some analogy with the integral (31) discussed in Sec. IV.

The spectrum has an expansion in integer powers of g with real coefficients. For example, for the ground state

E 0 (g) = 1 2 + 11 216 g - 155 384 g 2 + O(g 3 ). (52)
Instanton calculus, based on a steepest descent evaluation of the corresponding path integral, allows us to calculate the classical action relevant for the large order behaviour (3). One finds

A = 24 5 . (53)
For g large, from the scaling properties of Eq. ( 51), one infers that the ground state energy E 0 = E 0 (g), as a function of g, for large g, has an expansion of the form

E 0 = g 1/5 ∞ ℓ=0 ǫ ℓ g -2ℓ/5 . (54)
This suggests the mapping (see the Appendix)

g = ρ λ (1 -λ) 5/2 , E 0 = (1 -λ) -1/2 φ(λ, ρ) . (55)
In what follows, we again first concentrate on the behaviour of E 0 (g) for g → ∞, which is related to φ(1, ρ). More precisely, taking the strong coupling limit λ → 1, one finds

φ(1, ρ) = ρ 1/5 ǫ 0 . (56)
We now proceed as for the model problem discussed in Sec. IV. Figures 8 and9 show in a rescaled variable, that there is a range, which is k-independent up to corrections of order k -2/5 , where P k (ρ) and its derivative are small. This is also the region in the neighbourhood of which the zeros of P k and P ′ k are located. We have first determined the complex zeros ρ k of P ′ k (ρ) with the largest real part in this range. Again, these zeros have been chosen because it is easy to identify them systematically.

A fit of the numerical data for orders k ≤ 55 and an extrapolation to k → ∞ then yields

kRe(ρ k ) A = 5.5 ± 0.2 - 6.0 ± 0.5 k 2/5 , (57)
and thus

µ = R A = 5.5 ± 0.2 , (58)
to be compared with the expected maximum value µ = 4.895 . . . of Table I.

The numerical relative rate of convergence is compatible with an exponential of the form 0.05 × 0.94 k and yields the estimate However, the value of µ would lead eventually to a divergence of order 1.2 k . This indicates that the form (57) cannot be used asymptotically. If we take into account the expected value µ = 4.895 . . . and again fit Re(ρ k ), we find

kρ k A = 4.895690188 - 3.02 k 2/5 , (60)
a form we use in the remaining calculations. With this 11. E 0,k , even-odd averaged, as a function of 1/k 2/5 . As k → ∞, the energy converges-under oscillations-to the point indicated as 1/k 2/5 → 0. The plot illustrates the findings of Eqs. ( 61) and (62).

form, we indeed find a smoother convergence with

E 0 g 1/5 → g→∞ 0.37255 ± 0.00004 , (61)
at order 55 to be compared with the value obtained by solving the Schrödinger equation E 0 = 0.372545791 [START_REF] Jentschura | Calculation of the Characteristic Functions of Anharmonic Oscillators[END_REF].

The errors can be fitted by the form

1 2 (E 0,k + E 0,k-1 ) = E 0 as. (62)
-0.00092 exp[-0.292 k 3/5 ] cos(4.02 (k 3/5 -1.20)),

where the odd-even averaging improves the fit of the error (and slightly the convergence) by cancelling some subleading oscillatory correction. The form of the oscillatory factor in (62) is just a simple guess (see Fig. 11).

B. Finite value of the coupling parameter

For g finite, the additional factor [START_REF] Jentschura | Calculation of the Characteristic Functions of Anharmonic Oscillators[END_REF] governing the convergence of the ODM approximants in transformation order k is

λ k ∼ exp - R 2/5 k 3/5 g 2/5 (63) with R 2/5 ≈ 3.535 . . . . (64)
As a check, we have summed the series for g = 5. The result is

E 0 (5) = 0.60168393320519(6), (65)
and the convergence rate of the order of e -2.10 k 3/5 . Comparing with the convergence for g → ∞, one finds a variation of the coefficient of k 3/5 of 1.81 to be compared with 3.5/g 2/5 = 1.86. For g = 1, the result obtained from a numerical solution of the Schrödinger equation yields

E 0 (1) = 0.53078175930417667113556 , (66)
with all digits being significant. The ODM at order 55 yields (we give its explicit numerical value for illustration)

E 0 (1) = 0.53078175930417667113565 . (67)
The analysis of the convergence is consistent with the theoretical form with a coefficient of k 3/5 equal to 3.85 compared to the theoretical expectation of 3.82. Finally, to compare with the Padé summation of Ref. [START_REF] Bender | Numerical evidence that the perturbation expansion for a non-Hermitian PTsymmetric Hamiltonian is Stieltjes[END_REF], we give results for g = 288/49, which corresponds to λ 2 = 1/7 in Ref. [

where ∆E 0 = 49 (E 0 -1 2 ). At order 54, in Ref. [START_REF] Bender | Numerical evidence that the perturbation expansion for a non-Hermitian PTsymmetric Hamiltonian is Stieltjes[END_REF] one finds ∆E 0 = 5.5241683 and, at order 192, ∆E 0 = 5.52416721306031. The theoretical formula for the ODM methods predicts an error for ∆E of the order of a few times 10 -22 for k = 192, confirming the improved convergence achievable by use of the ODM. For completeness, we have verified that ∆E 0 = 5.5241672130602221133 . . . by numerical solution of the Schrödinger equation.

C. The imaginary part on the real negative axis

The ODM method converges on the real negative axis and thus the Stieltjes property [START_REF] Grecchi | Pad summability of the cubic oscillator[END_REF], that is, the positivity of the discontinuity, can be verified. For g → 0 -with Im(g) > 0, the imaginary part is known analytically from semi-classical considerations (instantons) [START_REF] Jentschura | Calculation of the Characteristic Functions of Anharmonic Oscillators[END_REF]:

ImE 0 (g) = g→0- 6 √ π e 24/5g √ -g 1 + 169 576 g + O(g 2 ) , (70)
where we recall that we are using the cubic Hamiltonian here in the convention (1). For |g| → ∞ and Im(g) = 0 + , one finds E 0 (g → -∞ + i 0) ∼ (0.301396 + i 0.218977) |g| 1/5 . (71)

Above the negative real axis, the antiresonance energy has a positive imaginary part. For g = -21.6, at order 55 we find E 0 = 0.5540544 + 0.351399i to be compared with the value E 0 = 0.554053519 + 0.351401778i given in [START_REF] Jentschura | Calculation of the Characteristic Functions of Anharmonic Oscillators[END_REF].

Similarly, for g = -5 + i0, E 0 = 0.4338906 + 0.1838582i, for g = -1 + i0, E 0 = 0.4425200442 + 0.015517927i. In fact, the imaginary part is a simple positive decreasing function providing a smooth interpolation between the two asymptotic forms (see Fig. 12).

D. Convergence and singularities

Combining Eqs. ( 62) and (63), one finds that the convergence domain is given by Re(g -2/5 ) > -0.0826 .

(72)

FIG.

12. The functionln ImE0(g) as a function of -g.

This contains the sector |arg g| < 5π/4, that is, the whole Riemann sheet and a sector of the second sheet. Moreover, in the neighbourhood of g = ∞, one finds the additional domain

|g| > 510.0 [-cos(2arg g/5)] 5/2 . (73)
In the union of these two domains, the function E 0 (g) is analytic and thus free of level-crossing singularities.

In particular, the expansion (54) has about 0.0826 as a radius of convergence.

VI. CONCLUSIONS

In this article, we have explored the properties of order-dependent mapping (ODM) for the summation of the divergent series originating from non-Hermitian Hamiltonians. In these cases, the resulting resonance energies may be complex, and their behaviour may be studied as a function of the complex coupling parameter g. In Sec. II, we discuss the motivation for the current study and the two physical problems treated here, which are the cubic anharmonic oscillator in quantum mechanics, and a model problem that counts the number of Feynman diagrams that occur in the perturbative expansion of the partition function of the cubic anharmonic oscillator.

In Sec. III, we recall some important properties of the ODM summation method, and in particular, we describe how the saddle-point approximation helps in determining asymptotic estimates for the parameter ρ k that are needed in order to construct the ODM in higher orders, for the convergence rate of the resulting approximants in the strong coupling limit, and for the analyticity domains of the approximants as the complex phase of the coupling parameter is varied.

In Sec. IV, we discuss the model problem defined in Eq. (IV) which is a simple integral that counts Feynman diagrams in a cubic theory and gives rise to a divergent series. We demonstrate that good approximations to the strong coupling limit can be obtained on the basis of few perturbative terms [see Eq. ( 46)]. In Sec. V, the analysis is first carried over to the PT symmetric cubic potential, with similarly encouraging numerical results, before demonstrating the applicability of the method to manifestly complex resonance energies (see Sec. V C).

The basic idea of the ODM is that one pretends that the divergent input series, which is to be summed, is in fact a convergent series, but with a circle of convergence whose radius gradually vanishes as the order of the perturbative expansion is increased. This algorithm, together with the representation of the coupling parameter g given in Eq. ( 6), leads to a very rapid convergence rate of the ODM transforms, in part because the double expansion in ρ and λ implied by Eq. ( 6) represents g as a function of two variables ρ and λ whose modulus is bounded by unity for approximants constructed in the strong coupling limit. If one additionally makes a suitable change of variable (and function), in order to incorporate the information about the strong coupling asymptotic expansion (see the Appendix), then one may approximate the strong coupling behaviour using only very few perturbative input data.

3 .

 3 The plot shows AP ′ k (ρ)/k as a function of τ = kρ/A for k = 30, for the model problem (31).

FIG. 4 .

 4 FIG.[START_REF] Guida | 3D Ising model: the scaling equation of state[END_REF]. The plot shows P k (ρ) as a function of τ = kρ/A for k = 60, for the model problem (31), demonstrating that a range where P k (ρ) is small persists in higher transformations orders.

FIG. 8 .FIG. 9 .

 89 FIG.8. We show P k (ρ) as a function of τ = kρ/A for the ODM approximant of order k = 55 to the ground state energy of the cubic Hamiltonian (51).

TABLE I .

 I Values of µc and λc.

	α	3/2	2	5/2	3	4
	µc	4.031233504	4.466846120	4.895690188	5.3168634291	6.1359656420
	-λc	0.2429640300	0.2136524524	0.1896450439	0.1699396648	0.14003129119

TABLE II .

 II Values of ρ k and errors in the ODM approximants in the limit g → ∞. For the estimated convergence rate of the output series obtained by the ODM, (δ k ) 1/k should approach a constant as k → ∞.

	k	5	10	15	20	25	30
	k Re(ρ k ) A	3.3612 4.3300 4.3450 4.4210 4.4954 4.5365
	k Im(ρ k ) A	0.7187	0	0	0.4544 0.2984 0.2260
	(δ k ) 1/k 0.4910 0.6149 0.6480 0.6526 0.4972 0.6819
	k	35	40	45	50	55	60
	k Re(ρ k ) A	4.54721 4.5509 4.5580 4.5698 4.5840 4.5884
	k Im(ρ k ) A	0.1800 0.4522 0.3706 0.5774	0	0.4390
	(δ k ) 1/k 0.7008 0.7064 0.7004 0.7019 0.7232 0.7323
	FIG. 2. The ODM polynomial P

k (ρ) as a function of the scaled variable τ = kρ/A for k = 30. The model problem (31) is being studied.

ACKNOWLEDGMENTS

U.D.J. acknowledges support by a Grant from the Missouri Research Board and by the National Science Foundation (Grant PHY-8555454). J. Z.-J. also gratefully acknowledges CERN's hospitality, where this work was completed.

Appendix A: Parameters in the ODM Let the strong-coupling asymptotic expansion for the quantity E(g) be known,

a property shared by the two examples we have discussed here (Eqs. ( 35),(54)) but also by the quartic anharmonic oscillator [START_REF] Seznec | Summation of divergent series by order dependent mappings: Application to the anharmonic oscillator and critical exponents in field theory[END_REF] and all x N perturbations to the quantum harmonic oscillator. We consider the conformal mapping g = ρλ (1 -λ) -α . This transformation maps the real positive g-axis onto the finite λ interval [0, 1].

For g → ∞, λ → 1 and g has an expansion at λ = 1 of the form

with Λ 0 = ρ -1/α . The function

(A3) has then a Taylor series expansion at λ = 0,

as well as at λ = 1,

with ϕ 0 = ǫ 0 ρ β , where ǫ 0 is the coefficient defined in (A1). This last property explains, to a large extent, the good convergence of the method even for g → ∞.