H Guezi

Isabelle Boutet

Ann C Andersen

F Lallier

Arnaud Tanguy
email: atanguy@sb-roscoff.fr

Guezi

Tanguy Arnaud

Comparative analysis of symbiont ratios and gene expression in natural populations of two Bathymodiolus mussel species

Keywords: Hydrothermal vent, Cold seeps, Bathymodiolus, symbiont, FISH 47 48 DAPI: 4', 6-DiAmidino-2-Phenyl-Indole, double-stranded DNA staining Fluorescencent In Situ Hybridization 52 MOX: Methane OXidizing bacteria 53 pmoA: particulate methane oxygenase, subunit

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Several studies have been conducted that link symbiont ratios to the local environment Gwen mussels (a hydrothermal site rich in sulfide), which showed a greater abundance in 134 sulfur-oxidizing (SOX) bacteria and higher level of ATP sulfurylase gene expression (Boutet 135 et al. 2011). In another study, a lower abundance of MOX symbionts was detected in mussels 136 from Lucky Strike compared to Menez Gwen individuals using TEM micrographs analysis 137 (Fiala-Médioni et al. 2002). Using a fluorescence in situ hybridization protocol coupled with

Variation in MOX ratio

258 Samples collected from the three REGAB sites exhibited similar MOX ratio (Fig. 1) with RQ 259 values at 384.1±49.4 , 273.8±37.4 and 301.3±20.9 for NW, W01 and W02 respectively. These 260 relative quantities were comparable to those found for SOX ratio. No significant differences 261 were detected for MOX ratio between the two populations from LS (RQ = 38.2±4.4) and MG

262

(RQ = 33.6±3.8) (t=0.82, p-value=0.42) (Fig. 1) but these relative quantities were much lower 263 than those found for B. azoricus SOX ratio (t=6.05, p-value = 1.358e-07) (Fig. 1).

279

Samples collected from the REGAB sites had similar levels of pmoA expression: 0.0011, 280 0.0035, and 0.162 at NW, W01, and W02 respectively (Fig. 2). No significant differences 281 were found in the expression of the symbiont-specific pmoA gene between the two vent fields 282 MG (RQ = 2.00) and LS (1.21) (t=-1.53 and p-value=0.13) (Fig. 2).

Comparative analysis of symbiont ratios and gene expression in two mussel species

285

A comparative analysis of symbiont ratios revealed similar SOX ratio (t=-0.94 and p-286 value=0.35), but a significant difference in MOX ratio (t=14.22 and p-value<2.2e -16 ) between 287 the two mussels species. The comparatively larger B. aff. boomerang species (shell length 288 between 50 and 160 cm) had markedly higher MOX ratio (316.24±19.6) (Fig. 1), but lower 289 ATP-sulfurylase (3.04±0.74) and pmoA (0.07±0.05) gene expression, compared to the smaller 290 species B. azoricus (shell length between 53 and 96 cm), with values of (35.43±2.8, 46.42±7.9 and 1.69±0.29, for MOX ratio, ATP-sulfurylase and pmoA gene expression respectively (Fig. 292 2). In their study dedicated to the characterization of the SOX-symbiont population in the 337 bivalve Codakia orbicularis, [START_REF] Caro | Characterization of the Population 487 of the Sulfur-Oxidizing Symbiont of Codakia orbicularis (Bivalvia, Lucinidae) by 488 Single-Cell Analyses[END_REF] evidenced the existence of different sub-338 populations characterized by various numbers of genome-copies ranging from haploid to 339 tetraploid stages. The multigenomic state of bacteria has been considered to be a signature of 340 their ability for rapid cell division when nutrients become available (Thorsen et al. 1992). The 341 ploidy degree has never been addressed in the symbiont of any Bathymodiolus species, so it is 342 at present impossible to know if one or both symbiont exhibit variable numbers of genome 343 copies and in which proportion. Our PCR approach does not allow to address this question 344 since we do not have any way to quantify the gill cells, nor the bacterial cells and then 345 subsequently establish a correlation between C(t) values and cell numbers. The observation of 346 polyploidy made by [START_REF] Caro | Characterization of the Population 487 of the Sulfur-Oxidizing Symbiont of Codakia orbicularis (Bivalvia, Lucinidae) by 488 Single-Cell Analyses[END_REF] could have an impact on what we have considered as 347 the symbiont ratio, in such a way that the real number of bacterial cells could not be directly 348 correlated with the C(t) value. Without any additional information, we assume that even this 349 could happen in mussel bacteriocytes, as the medium ratio of cells showing the different 350 polyploidy status is quite similar in all samples. Our ratios show differences between 351 individuals, even if not strictly correlated to an exact cell number. In previous studies using 352 FISH as a tool for symbiont ratios estimates, the question of the number of genome copies is The short distance that separate W02 from W01 (around 100 meters) may partly 373 explain the similar SOX ratio and ATP-sulfurylase expression in mussels from those areas, 374 and may also support the hypothesis that mussels from W01 and W02 are exposed to similar 375 environmental conditions. This is in agreement with previous studies conducted on mussels 376 from the REGAB site that showed no significant difference in SOX abundance in gills of B. outward to the periphery. We also hypothesize that mussels from NW are exposed to more 387 transient sulfide fluxes than mussels from W02. According to [START_REF] Charlou | Physical and chemical characterization 499 of gas hydrates and associated methane plumes in the Congo-Angola Basin[END_REF], the 388 methane concentration (13.85 µl/l) observed in an area close to NW was much lower than 389 those in W01 (89.51µl/l) and W02 (128.9µl/l). This may indicate that mussels from this area 390 rely more on sulfide than methane as an energy and carbon source to compensate for the lower concentration of methane, Sulfide probably represents an important secondary energy 392 source for mussels from the NW. This implies that fluid flow is intense but different in the 393 three areas. The sulfide and methane concentrations were identified as important factors 394 influencing the symbiont bacteria quantities in gills of hydrothermal Bathymodiolus mussels 395 (Fiala-Médioni et al. 2002;Duperron et al. 2006), but the link between symbiont ratios and 396 environmental conditions is complex as evidenced by the fact that the SOX symbiont ratio is were measured in fluids at MG ranging from 0.6 to 3.3 mM, and at LS from 1.3 to1.82 mM 412 while methane concentrations of up to 0.52 mM and 1.7 µM were measured at LS and MG 413 respectively [START_REF] Douville | The rainbow vent fluids (36°14′N, MAR): the influence of 526 ultramafic rocks and phase separation on trace metal content in Mid-Atlantic Ridge 527 hydrothermal fluids[END_REF][START_REF] Charlou | Geochemistry of high H 2 and CH 4 vent fluids issuing from ultramafic rocks at the Rainbow hydrothermal field 496 (36°14′ N, MAR)[END_REF], Chralou et al. 2004). These findings

444

For both species, size dependent variations on symbiont ratios and gene expression 445 were not significant as found by Duperron et al (2011). Martins et al (2008) showed that small 446 individuals of B. azoricus mussels were strongly dependent on filter-feeding while larger 447 mussels obtained a significant portion of their energy from endosymbiosis, suggesting that 448 symbiont ratios could increase with age. We did not observe such correlation between 449 symbiont ratios and mussel size in our B. azoricus sample but this might be due to the fact 450 that very small specimens were not present in our collection.

451

To conclude, we have presented a comparative analysis of symbiont ratios and gene 452 expression in the gills of two Bathymodiolus species to improve the understanding of inter-453 and intra-site variability observed in mussels from contrasting environments. Symbiont ratios 454 in the gills of mussels reflect the environmental levels of methane and sulfide. B. aff.

455

boomerang from a methane rich environment (cold seeps) showed a significantly higher 456 relative abundance of MOX, and B. azoricus from a sulfide rich environment (hydrothermal 457 vent) showed a significantly higher relative abundance of SOX. We also evidenced that 458 symbiont ratios (based on 16S DNA qPCR assay) is not strictly correlated to gene expression TABLES Table1: Sulfide and methane concentrations recorded in fluids at Menez Gwen and Lucky Strike. (adapted from [START_REF] Douville | The rainbow vent fluids (36°14′N, MAR): the influence of 526 ultramafic rocks and phase separation on trace metal content in Mid-Atlantic Ridge 527 hydrothermal fluids[END_REF]Charlou et al. 2000[START_REF] Charlou | Physical and chemical characterization 499 of gas hydrates and associated methane plumes in the Congo-Angola Basin[END_REF] and methane concentration reported around mussels at three minor pockmarks within REGAB giant pockmark [START_REF] Charlou | Physical and chemical characterization 499 of gas hydrates and associated methane plumes in the Congo-Angola Basin[END_REF]

  127 parameters. The relative abundance of each symbiont is influenced by the hydrothermal fluid 128 characteristics, especially methane and sulfide concentrations. Recently, symbiont ratios and 129 metabolic gene expression were shown to vary significantly among specimens of B. azoricus 130 from two contrasted hydrothermal vent sites, Menez Gwen and Rainbow. Specimens of B. 131 azoricus from Rainbow (a hydrothermal site rich in methane) had a high concentration of 132 methane oxidizing (MOX) bacteria and a high level of pmoA expression compared to Menez133

  138 3-dimensional reconstruction of bacteriocyte sections (3D FISH), Halary et al (2008) also 139 demonstrated that specimens of B. azoricus from different aggregates of animals within a 140 single site varied in their symbiont abundances. On the contrary B. aff. boomerang collected 141 from different mussel-aggregates within the giant cold seep pockmark REGAB (800m in 142 diameter) showed a relatively homogenous symbiont ratios (Duperron et al. 2011). The two 143 latter studies aimed at understanding the mechanisms developed by endosymbiont bacteria to 144 adapt in response to changing environmental parameters. However, our understanding of the 145 inter-and intra-site variability of the mussel symbiont ratios in contrasting ecosystems is 146 poor. To explore this variability, we compared symbiont ratios and gene expression in two 147 Bathymodiolus species from different habitats (vents/seeps), and from different sites within 148 each habitat. We investigated the variability of their relative abundance and the gene 149 expression of both symbionts in the two mussel species using real time PCR and we applied 150 the FISH technique for a visual confirmation of the bacterial dominance in the mussel gills.

  individuals were sampled at two vent fields: Menez Gwen (MG: 37°50' N, 31°31' 155 W, 830m depth; n =33), and Lucky Strike (LS: 37°17' N, 32°28' W, 1700m depth; n =22) in 156 the Azores region of the Mid-Atlantic Ridge using the ROV (Remotely Operated Vehicle) 157 Victor 6000 operated from the ship N/O Pourquoi Pas? during the BIOBAZ cruise in July 158 and the tubes were slowly mixed by inversion for 3 min before a centrifugation at 14,000 rpm 180 at 4°C for 10 min. The supernatant was collected in a fresh tube, and DNA was precipitated 181 with 2/3 volume of cold isopropanol (1 h at -20°C). The DNA pellet was recovered by 182 centrifugation at 14,000 rpm at 4°C for 20 min, washed with 75% cold ethanol, air-dried and 183 re-suspended in 200 μl of sterile water. The relative ratio of symbionts in both mussel species 184 was estimated by real-time PCR amplification using 16S specific primers (Table 3) designed 185 according to the probes previously developed for FISH analysis (Duperron et al. 2006). All 186 experiments were carried out using a Chroma4 thermal cycler (Bio-Rad Laboratories, Inc. 187 Hercules, CA) and 1× ABsolute™QPCR SYBR® Green mix (ABgene, Epsom, UK), 70 nM 188 of each primer, and diluted DNA (2.5 ng in a final volume of 10 μl). A 120 bp-fragment of 189 cytosolic malate dehydrogenase gene (MDH) from the host was used as an internal PCR 190 control. The relative ratio of each symbiont type was estimated by the comparative Ct method 191 formula: RQ = 2 -ΔCt , with ΔCt = Ct16S-CtMDH. The expression of functional genes that are specific for a given metabolic pathway can be 193 considered as an appropriate tool for investigating the metabolic potential of an organism, and 194 can be an indicator of its physiological activity. In this study, we choose to analyze the 195 expression of two bacterial genes, the ATP-sulfurylase (specific of the SOX metabolism) and 196 the pmoA gene, which encodes for the particulate methane mono-oxygenase subunit-A 197 (specific of the MOX metabolism). Using gill tissue, total RNA for both mussels and bacteria 198 were extracted together using the Tri-Reagent (Invitrogen) according to the manufacturer's 199 instructions. Five μg of total RNA was reverse-transcripted using M-MLV reverse-200 transcriptase (Promega, Madison, WI), random hexamers (Promega) and an anchor-oligo(dT) 201 primer (5'-CGCTCTAGAACTAGTGGATCT-3'). The relative gene expression of symbionts 202 in both mussel species was estimated by real time PCR amplification using specific ATP-203 sulfurylase and pmoA-primers (Table 2; GenBank accession numbers AB178052 and 204 AY945761, respectively). A volume of 4 μl of each diluted reverse transcription product 205 (1:200 dilution) was subjected to real-time PCR in a final volume of 10 μl containing 70 nM 206 of primers and 1×ABsolute™ QPCR SYBR® Green Mix (ABgene). The amplification was 207 carried out as follows in triplicates: initial enzyme activation at 94°C for 15 min, then 40 208 cycles of 94°C for 15 sec, and then 60°C for 1 min. A fragment of ribosomal protein L15 gene 209 (RpL15) from the host was used as an internal PCR control (

  results of SOX and MOX ratios, we chose 3 samples showing a high symbiont 222 ratio, and three samples showing a low symbiont ratio for each species previously fixed for 223 FISH. The gills embedded in paraffin were cut at 6 µm on a Leica, RM2245 microtome 224 (Germany). The sections were mounted on Super Frost slides (Euromedex, France) pre-coated with Biobond (Euromedex, France). The slides were dewaxed and the gill tissue rehydrated in 226 a decreasing ethanol series. Hybridization was performed in a buffer composed of (0.18M 227 NaCl, 0.02 M Tris-HCl, 0.05 % SDS). To stain the symbionts, two probes were used: the 228 MOX-specific probe ImedM-138 (5'-ACCATGTTGTC CCCCACTAA-3') labeled with Cy3 229 (green), and the SOX-specific probe BangT-642 (5'-CCTATACTCTAGCTTGCCAG-3') 230 labeled with Cy5 (red) (Amann et al. 1990; Duperron et al. 2005, Duperron et al. 2008, 231 Duperron et al. 2011). The sections were incubated with 50 ng of each probe for hybridization 232 in a 30% formamide buffer, for 3 h at 46°C in pre-heated moisture chambers. After 233 hybridization, the samples were washed twice in a solution containing (0.04M NaCl, 0.04M 234 Tris/HCl, 0.1% SDS and 0.02M EDTA) at 48°C for 15 min, then rinsed with MilliQ water, air 235 dried, and mounted with VectaShield containing nuclear DAPI-staining and covered with a 236 coverslip sealed with nail-varnish. The hybridized sections were photographed under a 237 confocal laser-scanning microscope (Leica TCS SP5II, Germany).

  that the relative abundance of symbiont ratios and gene expression were 243 normally distributed for each sample tested in this study. The SOX and MOX ratios 244 determination in the two mussel species from the various sites at vents or at seeps are shown t-test showed no significant difference between SOX symbiont ratio quantified in 249 specimens from W01 (RQ = 312.0±41.7) compared to specimens from W02 and NW, (t=1.93, 250 p-value=0.059 and t=-0.8, p-value=0.40, respectively), but there was a significant difference 251 in SOX ratio between specimen from W02 and NW (t=-3.42, p-value=0.001). Mussels 252 collected at NW had a significantly higher SOX level (RQ = 355.6±32.7) compared to 253 mussels collected at W02 (RQ = 222.0±22.5) (Fig.1). No significant differences were detected 254 for SOX ratio between B. azoricus populations from the two vent sites, LS (RQ = 301.2±34.8) 255 and MG (RQ = 355.7±80.4) (t=-0.63, p-value=0.53) (Fig.1).256 257

  in ATP sulfurylase gene expression 267Using real-time PCR, no significant difference was detected (t=0.53 and p-value=0.59) in 268 symbiont-specific gene expression of ATP-sulfurylase between W01 and W02, but significant 269 differences were found between, and between W02 270 and NW (t=3.14 and p-value=0.003). Mussels collected at NW had a lower ATP-sulfurylase 271 gene expression than the populations from W01 and W02 (Fig.2). There was no significant 272 difference of symbiont gene expression of ATP-sulfurylase between the two vent fields MG 273 and LS (t=-0.90 and p-value=0.36) (Fig. 2). in pmoA gene expression 276 The pmoA gene encodes the active site of the pMMO (membrane-bound methane 277 monooxygenase) enzyme. The expression level of pmoA was generally very low in both 278 species (Fig. 2, note scale difference) with higher values for B. azoricus vs B. aff. boomerang.

Fluorescence

  in situ hybridization of oligonucleotide probes to 16S RNA was used to identify 296 both MOX and SOX symbiont bacteria in gills of Bathymodiolus mussels (Fig.3). MOX 297 dominated the gills of B. aff. boomerang and colonized the basal regions of the bacteriocytes 298 (Fig.3 A, B), whereas they appeared faintly in B. azoricus (Fig.3 C, D). In both 299 Bathymodiolus species, SOX were always found in the apical parts of the bacteriocytes and 300 were the most abundant symbiont in the gills of B. azoricus (Fig.3 C, D). The evidence from 301 in situ hybridization shows that B. aff. boomerang from the deeper REGAB site had obviously 302 a greater symbiont ratio compared to B. azoricus (Fig.3).

  of real time PCR to estimate symbiont ratios 306 Relative or absolute quantification of symbiotic bacteria is necessary to understand the 307 dynamics of the symbiotic associations between chemoautotrophic bacteria (methanotrophs 308 and thiotrophs) and Bathymodiolus mussels. Previous microscopic studies have attempted to 309 quantify the relative abundance of each bacterial type (Salerno et al. 2005). Fiala-Médioni et 310 al (2002) qualitatively identified two symbiotic bacteria morphotypes in the gills of B. 311 azoricus and B. heckerae (a species living in cold seeps) using transmission electron 312 microscopy. Halary et al (2008) and Duperron et al (2011) used 3D-FISH to quantify the 313 volume occupied by different symbionts in bacteriocytes from B. azoricus and B. aff. 314 boomerang. The 3D-FISH technique is based on fluorescence in situ hybridization and image 315 analysis. The main limitation of microscopy techniques in quantification is the fact that they 316 do not estimate the total volume of symbionts in the entire gill of a specimen, but only in a 317 few thin section of this gill (Halary et al., 2008). Evidence for the activity of both sulfur-318 oxidizing and methane-oxidizing metabolic pathways has also been reported in B. azoricus 319 (Fiala-Médioni et al. 2002). Regarding the stability of bacterial mRNA, a previous study 320 conducted on B. puteoserpentis (a vent species very close to B. azoricus) evidenced that the 321 half-life of pmoA mRNA was long, ranging from a few hours to several days (Wendeberg et 322 al. 2012). Deana and Belasco (2005) have shown that while bacterial mRNAs are actively translated, they are very stable in comparison with their non-translated counterparts. In our 324 study, mussels are brought on board and dissected within 2 to 6 hours after sampling on the 325 bottom, optimizing the quality of mRNA. Indeed, using qPCR on DNA and RNA extracted 326 from a whole piece of gill, we are not able to take into account the local heterogeneity of the 327 symbiont distribution within each gill filament, but we generate a better estimation of the 328 relative ratio between both symbionts. In their study, Wendeberg et al (2012) localized pmoA 329 and aprA (encoding subunit A of the dissimilatory adenosine-5'-phosphosulfate-reductase 330 which is involved in the metabolism of the SOX symbiont). These authors evidenced gene 331 expression by FISH and showed that symbionts exhibited a stronger gene expression in 332 bacteriocytes close to the frontal ciliated surface at the posterior ends of the infra-branchial 333 chamber compared to the anterior gill ends. These authors also observed a high variability in 334 mRNA expression between samples that could reflect local modifications of the 335 environmental conditions.

  336

353

  not addressed either although it might affect fluorescence intensity. Overall, FISH techniques 354 give a snapshot of a gill section, with precise and accurate spatial information but inadequate 355 quantitative estimates of relative abundance in symbiont ratio. Moreover FISH or TEM 356 techniques are time-consuming and only a limited number of samples/individuals can be analyzed. Applying qPCR is considerably faster and cheaper, and allows taking into account a 358 large number of individuals and their natural variation. Therefore, the use of q-PCR approach 359 to estimate the relative abundance of symbionts in the mussel gills has to be considered as 360 complementary to other methods such as 2D or 3D FISH and TEM micrographs.361Furthermore, the expression of additional metabolic genes has to be considered to better 362 characterize the physiological status of symbionts. Genomes of both B. azoricus symbionts 363 genomes are presently being sequenced and annotated and this new information should help in SOX ratio and ATP sulfurylase gene expression 367 Mussels at the REGAB site were mostly concentrated in two areas: W01 and W02 368 (corresponding to M1 and M2 as described by Marcon et al. 2013). In our study, we focused 369 on mussels from three areas W01 and W02 described by Duperron et al (2005) and Olu-Leroy 370 et al (2007) and NW (NorthWest). We report the first data for symbiont ratios and gene 371 expression in Bathymodiolus specimens from the NW habitat.

  372

  377aff. boomerang collected in three areas separated by less than thirty meters (Duperron et al. 378 2011). The NW area is located 140 meters from W01 and at 190 meters from W02. Mussels 379 from NW seem to have a high SOX ratio and a low ATP-sulfurylase expression compared to 380 mussels from W02. These differences are likely the result of local differences in 381 environmental conditions specific to NW, as well as overall varying conditions at the REGAB 382 pockmark. Ondréas et al (2005), Charlou et al (2004) and Olu-Le Roy et al (2007) reported 383 that methane concentrations in the giant REGAB pockmark decrease from the center to the 384 periphery. Our results on the MOX and SOX symbionts in the mussels lead us to suggest that, 385 contrary to methane, sulfide concentrations increase in the center of REGAB pockmark 386

397

  similar in B. azoricus inhabiting two physically and chemically contrasting vent fields, LS and 398 MG. Our results suggest that the variation in sulfide availability in the cold seep areas 399 W01/W02 and NW impacted the SOX symbiont ratio in B. aff. boomerang at a short spatial in MOX ratio and pmoA gene expression 403 Mussels from W01, W02 and NW have statistically similar MOX ratio and pmoA expression 404 levels indicating that methane concentrations could be similar in these areas. The three 405 (smaller pockmark) sites sampled are located in the center of REGAB giant pockmark where 406 the methane flux is the highest and methane fluxes have been observed in all mussel beds 407 (Ondréas et al. 2005; Olu-Le Roy et al. 2007; Marcon et al. 2013). MOX ratio and pmoA 408 expression were also similar for B. azoricus specimens from two hydrothermal sites, LS and 409 MG. We have no specific measurements of sulfide and methane in the vicinity of the B. 410 azoricus mussel beds sampled, but as previously reported, reduced concentrations of sulfide 411

  414 are in agreement with our results and could explain the predominance of SOX thiotrophic 415 symbionts in the gills of all specimens of B. azoricus. analysis of symbiont ratios and gene expression between species 418 In the present study, we report the first attempt to compare symbiont ratios and gene 419 expression in the gills of two Bathymodiolus species from contrasting environments: 420 hydrothermal vents and cold seeps. Our results indicate that the MOX ratio is higher in the 421 gills of B. aff. boomerang compared to B. azoricus. The dominance of MOX is not surprising 422 since methane concentrations are significantly higher in cold seeps. Mussel beds from the 423 three areas within REGAB pockmark showed obvious pulses of free methane gas into the 424 water column, which could result in an appreciable variability in the concentration of gas and both indicate that chemosynthetic bacteria make a substantial contribution to the nutrition 443 of the B. aff. boomerang.

FIGURESFig. 1

 1 FIGURES

FigureFig. 2

 2 Figure

Fig. 3

 3 Fig. 3 Localization of MOX (green, FISH probeImedM-138) and SOX (orange, FISH probeBangT193) symbionts in gill cross-sections from B. aff. boomerang with high (A), and low (B) MOX symbiont content and from B. azoricus with high (C) and low (D) SOX symbiont content. Host cell nuclei are blue (DAPI).

Table 3

 3

	). Relative expression of

.3 Statistical Analysis 214 Data from quantitative real-time PCR experiments were tested for normality using the Shapiro 215 test, and differences in bacteria ratios and gene expression level were analyzed for

	216
	217	significance with a Welch t-test in the R Studio software (RStudio 0.96.122 Version, 2012).
	218	Boston, http://www.rstudio.org/. Differences were considered statistically significant when
		p≤0.05.

Table 2 :

 2 . ND: not determined. Primers used in real-time PCR amplification of bacteria and host gene.

	SITES	DEPTH (m)	H2S concentration	CH4 concentration
	LS	1700	0.6-3.3 (mM)	0.52 (mM)
	MG	850	1.3-1.82 (mM)	1.7 (mM)
	W01	3156	ND	89.5 µl/l
	W02	3156	ND	128.9 µl/l
	NW	3156	ND	13.8 µl/l

Acknowledgements

 [START_REF] Boetius | Hydrate Ridge: a natural laboratory for the study of microbial life 482 fueled by methane from near-surface gas hydrates[END_REF][START_REF] Beer | In situ fluxes and zonation of microbial activity in surface sediments of the 509 Haakon Mosby Mud Volcano[END_REF][START_REF] Beer | In situ fluxes and zonation of microbial activity in surface sediments of the 509 Haakon Mosby Mud Volcano[END_REF]). These observations 426 support the hypothesis that relative abundance of symbiont bacteria could be significantly 427 influenced by environmental conditions (Fiala-Médioni et al. 2002;Duperron et al. 2006).

428

The average ratio and gene expression for each symbiont was significantly different between 429 species. B. aff. boomerang from deeper sites (3170 m) exhibits a higher symbiont ratio, but 430 associated with the lowest gene expression. Generally, mussels collected from shallow water 431 benefit from a higher particulate flux (Riou et al. 2010) which is the case for MG and LS 432 populations that are located at 830 and 1700 meters depth respectively. The presence of 433 particle flux suggests that these mussels could rely less on symbionts to meet their carbon 434 needs. At the contrary, in the deeper sites, particles are less accessible to mussels increasing 435 the dependence of mussels on their symbionts. Surprisingly, we observed that gene expression 436 of both ATP-sulfurylase and pmoA are significantly higher in B. azoricus individuals 437 suggesting that gene expression in both species can be modulated by various factors, but also 438 that there is no evident correlation between symbiont ratios and gene expression at least for


Anne-Sophie Bonnet-Ben Dhia
email: sophie.bonnet-bendhia@ensta-paristech.fr

Lucas Chesnel
email: lucas.chesnel@cmap.polytechnique.fr

Sergei A Nazarov
email: srgnazarov@yahoo.co.uk

Anne-Sophie Bonnet-Ben Dhia

Non-scattering wavenumbers and far field

Keywords: Non-scattering wavenumbers, interior transmission problem, invisibility, energy identities, asymptotic analysis, relative scattering matrix

Introduction

Consider a reference acoustic medium, say R d , d = 2, 3, presenting a defect (penetrable inclusion) localized in the bounded domain D. Generating incident plane waves and measuring the resulting far field pattern of the scattered fields (the only available information far from D), one can try to reconstruct the features of the defect in the reference medium. In particular, the classical inverse scattering problem is to determine the support of the inclusion. In view of this aim, many methods have been developed [START_REF] Potthast | A survey on sampling and probe methods for inverse problems[END_REF][START_REF] Cakoni | Qualitative methods in inverse scattering theory. An introduction[END_REF], as for example, the Linear Sampling Method (LSM) [START_REF] Colton | A simple method for solving inverse scattering problems in the resonance region[END_REF][START_REF] Colton | A simple method using morozov's discrepancy principle for solving inverse scattering problems[END_REF]. It seems that the LSM works, without modifications and a priori knowledge, only at wavenumbers which are not transmission eigenvalues. Transmission eigenvalues correspond to wavenumbers k > 0 such that there are incident fields, generalized1 combination of plane waves, which produce arbitrarily small scattered fields outside the inclusion. This, and the fact that nice questions of spectral theory appear in its study, explains why the interior transmission eigenvalue problems has been so intensely investigated for now more than 25 years (see [START_REF] Kirsch | The denseness of the far field patterns for the transmission problem[END_REF][START_REF] Colton | The inverse scattering problem for time-harmonic acoustic waves in an inhomogeneous medium[END_REF][START_REF] Rynne | The interior transmission problem and inverse scattering from inhomogeneous media[END_REF] and the recent review paper [START_REF] Cakoni | Transmission eigenvalues in inverse scattering theory inverse problems and applications[END_REF]).

All the aforementioned theory supposed that we can produce incident plane waves in all directions θ i ∈ S d-1 and measure the far field pattern of the associated scattered fields for all θ s ∈ S d- 1 , where S d-1 denotes the unit sphere in R d . However, this is not quite realistic. Some works have been devoted to the study of the LSM in partial aperture (see [START_REF] Cakoni | Recent developments in the qualitative approach to inverse electromagnetic scattering theory[END_REF] and [START_REF] Collino | Numerical and analytical studies of the linear sampling method in electromagnetic inverse scattering problems[END_REF][START_REF] Colton | The linear sampling method in inverse electromagnetic scattering theory[END_REF][START_REF] Colton | Target identification of coated objects[END_REF] for the corresponding numerical experiments), that is when the observer knows only the restriction of the far field operator on a non empty open set of S d-1 . Thanks to the analytic dependence with respect to the wavenumber, the theory can be developed as in the case of full aperture. But again, this is not completely satisfactory for applications because in practice, one cannot access to this continuum of information. Very often, for example in the numerical implementation of the LSM, one can produce incident plane waves and measure the far field pattern of the associated scattered fields only in a finite set of directions. To study such a configuration, let us consider given N distinct incident directions θ 1 , . . . , θ N ∈ S d-1 . We shall assume that the emitters and the receivers are located at the same positions so that measurements can be made only in the directions -θ 1 , . . . , -θ N (backscattering directions). Then, the question of the injectivity of the far field operator amounts to the question of the injectivity of a relative scattering matrix of size N × N denoted A (k). In this context, we shall say that k > 0 is a non-scattering wavenumber if A (k) is not injective, equivalently, if there is a non trivial incident field, combination of the plane waves of directions of propagation θ 1 , . . . , θ N , such that the far field pattern of the associated scattered field vanishes in the directions -θ 1 , . . . , -θ N . Let us emphasize that in this case, unlike in the continuous setting, the scattered field has no reason to be identically null outside the defect. The first objective of the present article is to find criteria on the physical parameters of the inclusion to guarantee that the non-scattering wavenumbers, defined by means of the relative scattering matrix, form a (possibly empty) discrete set. This kind of results seems an important first step to justify the practical implementation of reconstruction methods such as the LSM mentioned above.

On the other hand, all techniques are not equally sensitive to transmission eigenvalues. Thus, it has been shown in [START_REF] Lechleiter | The factorization method is independent of transmission eigenvalues[END_REF] that the Factorization Method (FM) [START_REF] Kirsch | The factorization method for inverse problems[END_REF] is stable at transmission eigenvalues. Again, this property has been demonstrated in the continuous framework. Can we hope to justify the FM in the setting with a finite set of emitters and receivers for all wavenumbers k > 0? This is the second question we investigate in this work. More precisely, under certain assumptions on the directions θ 1 , . . . , θ N , we show that the answer is negative proving constructively that for any k > 0 and any domain D, there are penetrable inclusions supported in D for which the matrix A (k) is null (that is equivalent to say that A (k) has a kernel of dimension N ). Let us underline that we consider only non dissipative isotropic inclusions, the result being simpler to establish for dissipative or anisotropic inclusions. The above proposition indicates in particular that the data of the relative scattering matrix does not uniquely determine the position of the defect. Bearing in mind that A (k) belongs to a space of finite dimension while there is an infinite number of degrees of freedom for the definition of the defect, this result is not completely surprising. However, we do not know any existing proof in the literature. Note that in the continuous setting, this construction is impossible. Indeed, the knowledge of the far field operator on S d-1 × S d-1 uniquely determines the parameter of the inclusion (see [START_REF] Colton | Inverse acoustic and electromagnetic scattering theory[END_REF]Theorem 10.5] and the references therein: [START_REF] Nachman | Reconstructions from boundary measurements[END_REF][START_REF] Sylvester | A global uniqueness theorem for an inverse boundary value problem[END_REF] in 3D, [START_REF] Bukhgeim | Recovering a potential from Cauchy data in the two-dimensional case[END_REF][START_REF] Yu | Inverse boundary value problem for Schrödinger equation in two dimensions[END_REF] in 2D). This text is organized as follows. In Section 2, we formulate the problem and introduce the notations. Section 3 is dedicated to the proof of discreteness of non-scattering wavenumbers in the setting with a finite set of emitters and receivers. We first give a sense to the relative scattering matrix for complex wavenumbers. Then, we establish some energy identities for purely imaginary wavenumbers k. Using the analytic dependence on k, these equalities allow us to prove that the real wavenumbers k for which A (k) is not injective form a (possibly empty) discrete set. In the process, we also consider sound hard obstacles for which the analysis is slightly simpler. In Section 4, we adopt a different point of view. For a given wavenumber k > 0 and a given domain D, we present a constructive technique to demonstrate that there are non trivial inclusions supported in D for which A (k) is the null matrix. To implement the approach, which was developed in [START_REF] Nazarov | Asymptotic expansions of eigenvalues in the continuous spectrum of a regularly perturbed quantum waveguide[END_REF][START_REF] Bonnet-Ben Dhia | Obstacles in acoustic waveguides becoming "invisible" at given frequencies[END_REF] in a context related to waveguides problems, we need in particular to assume that the scattering directions are different from the incident directions. In the second part of Section 4, we show that it is much more complicated (it might be impossible but we are not able to prove it) to impose far field invisibility in the incident direction. Finally in the last part of Section 4, we derive a numerical algorithm to provide examples of (approximated) invisible inclusions.

Setting

We assume that the propagation of acoustic waves in time harmonic regime in the reference medium R d , d = 2, 3, is governed by the Helmholtz equation ∆u + k 2 u = 0. Here, u denotes the pressure, k > 0 corresponds to the wavenumber proportional to the frequency of harmonic oscillations and ∆ is the Laplace operator. The localized perturbation in the reference medium is modelled by some bounded open set D ⊂ R d with Lipschitz boundary Γ := ∂D. To represent the physical properties of the penetrable inclusion D, we introduce

A ∈ L ∞ (R d , R d×d ) and ρ ∈ L ∞ (R d , R), two parameters such that A = Id and ρ = 1 in R d \ D. We assume that A(x) is symmetric for all x ∈ D, satisfies A(x)ζ • ζ ≥ C|ζ| 2 for all x ∈ R d , ζ ∈ R d and that ρ verifies ρ(x) ≥ C for all x ∈ R d , for some constant C > 0. The scattering of the incident plane wave u i := e ikθ i •x , of direction of propagation θ i ∈ S d-1 , by D, is described by the problem Find u such that -div(A∇u) = k 2 ρ u in R d , u = u i + u s in R d , lim r→+∞ r d-1 2 ∂u s ∂r -iku s = 0. (1)
We emphasize that in (1), u i is given. On the other hand, the last line of [START_REF] Blåsten | Corners always scatter[END_REF], where r = |x|, is the Sommerfeld radiation condition which ensures that the scattered field u s is outgoing. It is known (see for example [START_REF] Colton | Inverse acoustic and electromagnetic scattering theory[END_REF]) that for all k > 0, Problem (1) has a unique solution in H 1 loc (R d ), the Sobolev space of measurable functions whose H 1 -norm is finite on each bounded subset of R d . We shall denote u s (•, θ i ) the associated scattered field. Using Green's representation formula and computing explicitly the asymptotic behaviour of the Green's function for the Helmholtz equation far from D, one proves (see [START_REF] Colton | Inverse acoustic and electromagnetic scattering theory[END_REF]Theorem 2.6]) the expansion

u s (x, θ i ) = e ikr r -d-1 2 u ∞ s (θ s , θ i ) + O(1/r) , (2)
as r → +∞, uniformly in θ s ∈ S d-1 . Here θ s is the direction of observation. The function u ∞ s (•, θ i ) : S d-1 → C, is called the far field pattern associated with u i := e ikθ i •x . In other words, at infinity, u s (•, θ i ) behaves at the first order like a cylindrical wave in 2D or like a spherical wave in 3D. The far field pattern is given by the following integral representation

u ∞ s (θ s , θ i ) := c d ∂ ν u + s (•, θ i ), e -ikθs•x Γ -c d ∂ ν (e -ikθs•x ), u s (•, θ i ) Γ . (3)
In this expression, •, • Γ stands for the duality pairing (without complex conjugation) between H -1/2 (Γ) and H 1/2 (Γ). On the other hand, ν denotes the unit normal vector to Γ orientated to the interior of D and

∂ ν u + s (•, θ i ) refers to the normal trace of u s (•, θ i )| Ω , with Ω := R d \ D.
Finally, the constant c d verifies c 2 = e iπ/4 / √ 8πk and c 3 = 1/(4π). In particular when A = Id in R d , we deduce from (3) that

u ∞ s (θ s , θ i ) = c d k 2 D (ρ -1)(u ∞ s (•, θ i ) + e ikθ i •x ) e -ikθs•x dx. ( 4
)
As mentioned in the introduction, we shall assume in this article that we have a finite set of emitters and receivers located at the same positions so that we can produce incident plane waves in some given directions θ 1 , . . . , θ N ∈ S d-1 and measure the far field pattern of the resulting scattered field only in the directions -θ 1 , . . . , -θ N . This corresponds to knowing all elements of the relative scattering matrix

A (k) ∈ C N ×N such that A mn (k) = u ∞ s (-θ m , θ n ). (5)
In the next section, we prove that for given physical parameters A and ρ, the values of k > 0 such that A (k) is not injective form a (possibly empty) discrete set. Then in Section 4, imposing A = Id in R d , for a given k > 0 and a given domain D, we construct a ρ supported in D such that A (k) is the null matrix.

Discreteness of non-scattering wavenumbers

We remind the reader that the wavenumber k > 0 is called non-scattering wavenumber if A (k) is not injective. Before working on the relative scattering matrix A (k), we investigate first the case where D is a sound hard obstacle (rather than a penetrable inclusion). This study is convenient for pedagogical purposes because the analysis for the sound hard obstacle is (slightly) simpler than the one for the penetrable inclusion. On the other hand, it yields a result which is interesting by itself. Indeed, while the discreteness of transmission eigenvalues for the sound hard obstacle in the continuous setting is obtained for free (they correspond to the eigenvalues for the Neumann Laplacian in D), the equivalent result is more delicate to show in the discrete framework (note that transmission eigenvalues are then called non-scattering wavenumbers).

Sound hard obstacle

Denote Ω = R d \ D and consider the following scattering problem Find u such that

-∆u = k 2 u in Ω u = u i + u s in Ω ∂ ν u = 0 on Γ lim r→+∞ r d-1 2 ∂u s ∂r -iku s = 0, (6)
with u i = e ikθ i •x . In [START_REF] Bukhgeim | Recovering a potential from Cauchy data in the two-dimensional case[END_REF], the third equation on the boundary Γ models the sound hard obstacle. According e.g. to [START_REF] Mclean | Strongly elliptic systems and boundary integral equations[END_REF]Theorem 9.11], we know that Problem (6) has a unique solution in H 1 loc (R d ) for all k > 0. We call u(•, θ i ) and u s (•, θ i ) the corresponding total and scattered fields. Formula (3), which is also valid for this problem, and a simple integration by parts on D provide the equalities

u ∞ s (θ s , θ i ) = c d ∂ ν u s (•, θ i ), e -ikθs•x Γ -c d ∂ ν (e -ikθs•x ), u s (•, θ i ) Γ 0 = c d ∂ ν u i , e -ikθs•x Γ -c d ∂ ν (e -ikθs•x ), u i Γ . ( 7
)
Summing the two equations of [START_REF] Cakoni | Recent developments in the qualitative approach to inverse electromagnetic scattering theory[END_REF] and remembering that

∂ ν (u i + u s (•, θ i )) = ∂ ν u(•, θ i ) = 0 on Γ, we deduce that u ∞ s (θ s , θ i ) = -c d ∂ ν (e -ikθs•x ), u(•, θ i ) Γ . ( 8
)
We denote B(k) ∈ C N ×N the relative scattering matrix for this problem. It is defined elementwise via

B mn (k) = u ∞ s (-θ m , θ n ) = -c d ∂ ν (e ikθm•x ), u(•, θ n ) Γ . ( 9
)
As for the case of the penetrable inclusion, we shall say that k ∈ R * is a non-scattering wavenumber when the kernel of B(k) contains a non-zero element. In the sequel, we wish to prove that the set of non-scattering wavenumbers is either empty or discrete.

We start by giving a sense to B(k) for non real k.

For k ∈ C \ R, f ∈ L 2 c (Ω) (the set of functions of L 2 (Ω) which are compactly supported) and g ∈ H -1/2 (Γ), the Lax-Milgram theorem ensures there is a unique solution u ∈ H 1 (Ω) to the following problem Find u such that -∆u -k 2 u = f in Ω ∂ ν u = g on Γ. (10)
In particular, this allows us to define, for k verifying ℑm k > 0, the resolvent R(k

) : L 2 c (Ω) × H -1/2 (Γ) → H 1 loc (Ω) such that R(k)(f, g) = u.
The map R can be meromorphically continued to Λ where Λ is equal to C or to some Riemann surface according to the parity of the space dimension (the interested reader can find more details for example in [START_REF] Melrose | Geometric scattering theory[END_REF]). Moreover, R has no pole on (0; +∞) and, according to the limiting absorption principle, for k ∈ (0; +∞), we have R(k)(0, -∂ ν u i ) = u s where u s is defined in [START_REF] Bukhgeim | Recovering a potential from Cauchy data in the two-dimensional case[END_REF]. We deduce that the matrix valued map k → B(k) introduced in ( 9) is analytic on Λ\P, P denoting the set of poles of R. As a consequence, k → det B(k) is analytic on Λ \ P and, according to the principle of isolated zeros, to show that the set of non-scattering wavenumbers is either empty or discrete, it is sufficient to exhibit some k ∈ Λ \ P such that B(k) is injective. We will prove that this is true when k is purely imaginary with ℑm k > 0 (note that {k ∈ C | ℑm k > 0} is indeed included in Λ \ P).

Set k = iκ with κ > 0. In this case, if the source terms in [START_REF] Cakoni | Transmission eigenvalues in inverse scattering theory inverse problems and applications[END_REF] are real valued, then the solution in H 1 (Ω) of Problem ( 10) is also real-valued. Since the functions x → e ikθn•x are real valued, it is sufficient to prove that B(iκ) is invertible as a matrix of R N ×N . Consider the incident field

u i = N n=1 α n e ikθn•x = N n=1 α n e -κθn•x , (α 1 , . . . , α N ) ⊤ ∈ R N . ( 11
)
We call u the unique solution to the problem

Find u such that -∆u = -κ 2 u in Ω ∂ ν u = 0 on Γ u s := u -u i ∈ H 1 (Ω). ( 12
)
If u(•, θ n ) denotes the solution of ( 12) with u i = e -κθn•x , then by linearity, we have u

= N n=1 α n u(•, θ n ). Since u = u s + u i in Ω and ∂ ν u = 0 on Γ, we have u s = u -u i and ∂ ν u s = -∂ ν u i on Γ. This allows us to write Ω |∇u s | 2 + κ 2 u 2 s dx = Ω |∇u s | 2 + u s ∆u s dx = ∂ ν u s , u s Γ = -∂ ν u i , (u -u i ) Γ = - D |∇u i | 2 + κ 2 u 2 i dx -∂ ν u i , u Γ . ( 13
)
The "-" in front of the first term of the right-hand side of the above equation appears because ν is orientated to the interior of D. On the other hand, using [START_REF] Cakoni | The existence of an infinite discrete set of transmission eigenvalues[END_REF], which also holds for k such that ℑm k > 02 , we find

∂ ν u i , u Γ = N m=1 N n=1 α m α n ∂ ν e -κθm•x , u(•, θ n ) Γ = -c -1 d α ⊤ B(iκ) α, ( 14
)
where α = (α 1 , . . . , α N ) ⊤ . Gathering ( 13) and ( 14), we obtain the energy identity

c -1 d α ⊤ B(iκ) α = Ω |∇u s | 2 + κ 2 u 2 s dx + D |∇u i | 2 + κ 2 u 2 i dx. ( 15
)
Assume that D = ∅. If α = (α 1 , . . . , α N ) ⊤ is an element of ker B(iκ), then, according to [START_REF] Colton | A simple method for solving inverse scattering problems in the resonance region[END_REF], the field

u i = N n=1 α n e -κθn•x must satisfy u i = 0 on D.
This implies that α is the null vector and proves that B(iκ) is injective (or equivalently det B(iκ) = 0). Since the zeros of the analytic function k → det B(k) are isolated, we deduce the following result: Proposition 3.1. (Sound hard obstacle) -The set of non-scattering wavenumbers for Problem [START_REF] Bukhgeim | Recovering a potential from Cauchy data in the two-dimensional case[END_REF] is either empty or discrete.

Remark 3.1. The case of the sound soft obstacle, for which we replace, in [START_REF] Bukhgeim | Recovering a potential from Cauchy data in the two-dimensional case[END_REF], the homogeneous Neumann boundary condition by a homogeneous Dirichlet boundary condition, can be treated in a similar way.

Penetrable inclusion

to Λ, with Λ equal to C or to some Riemann surface according to the parity of the space dimension. Therefore, again, to prove that the set of non-scattering wavenumbers is either empty or discrete, it is sufficient to exhibit some k = iκ, with κ > 0, such that A (k) is injective. As explained above, it is sufficient to prove that A (iκ) is invertible as an element of R N ×N . Consider an incident field like in [START_REF] Cardone | The localization effect for eigenfunctions of the mixed boundary value problem in a thin cylinder with distorted ends[END_REF] (we use the same notation). We call u the unique solution to the problem

Find u such that -div(A∇u) = k 2 ρ u in R d u s := u -u i ∈ H 1 (Ω).
As in the previous subsection, we wish to establish energy identities to show that A (iκ) ∈ R N ×N is invertible. Using [START_REF] Cakoni | Recent developments in the qualitative approach to inverse electromagnetic scattering theory[END_REF] and working as in [START_REF] Colton | The linear sampling method in inverse electromagnetic scattering theory[END_REF], we derive

c -1 d α ⊤ A (iκ) α = ∂ ν u + s , u i Γ -∂ ν u + i , u s Γ (16)
In the right hand side of ( 16), ∂ ν u + s and ∂ ν u + i refer, respectively, to the normal trace of u s | Ω and u i | Ω on Γ. We shall denote ∂ ν u - i the normal trace of u i | D on Γ. We define similarly A∂ ν u -and ∂ ν u + . Notice that on Γ, we have

A∂ ν u -= ∂ ν u + , ∂ ν u - i = ∂ ν u + i and ∂ ν u + = ∂ ν u + i + ∂ ν u + s . Since u i = u -u s , we obtain ∂ ν u + s , u i Γ -∂ ν u + i , u s Γ = ∂ ν u + , u i Γ -∂ ν u + i , u i Γ + ∂ ν u + s , u s Γ -∂ ν u + , u s Γ = A∂ ν u -, u i -u s Γ -∂ ν u - i , u i Γ + ∂ ν u + s , u s Γ . ( 17
)
In the following arguments, we shall write F |∇ϕ| 2 instead of F ∇ϕ • ∇ϕ when F is a matrix and ϕ is a function. Integrating by parts, we find from ( 16), ( 17)

c -1 d α ⊤ A (iκ) α = - D A∇u • ∇(u i -u s ) + κ 2 ρ u (u i -u s ) dx + D |∇u i | 2 + κ 2 u 2 i dx + Ω |∇u s | 2 + κ 2 u 2 s dx = R d A|∇u s | 2 + κ 2 ρ u 2 s dx + D (Id -A)|∇u i | 2 + κ 2 (1 -ρ)u 2 i dx. ( 18
)
The above identity will allow us to conclude when Id -A is nonnegative definite and 1ρ is nonnegative.

To handle the case where A -Id is nonnegative definite and ρ -1 is nonnegative, now we prove a second energy identity. Starting like in [START_REF] Colton | The inverse scattering problem for time-harmonic acoustic waves in an inhomogeneous medium[END_REF], we write

∂ ν u + s , u i Γ -∂ ν u + i , u s Γ = ∂ ν u + s , u Γ -∂ ν u + s , u s Γ -∂ ν u + i , u s Γ = A∂ ν u -, u Γ -∂ ν u - i , u + u s Γ -∂ ν u + s , u s Γ . ( 19
)
From this expression, using ( 16), we deduce

c -1 d α ⊤ A (iκ) α = - D A|∇u| 2 + κ 2 ρ u 2 dx + D ∇u i • ∇(u + u s ) + κ 2 u i (u + u s ) dx - Ω |∇u s | 2 + κ 2 u 2 s dx = - R d |∇u s | 2 + κ 2 u 2 s dx - D (A -Id)|∇u| 2 + κ 2 (ρ -1)u 2 dx. ( 20
)
The previous analysis leads us to formulate the two assumptions:

< C|ζ| 2 ≤ (A -Id)ζ • ζ for all ζ ∈ R d or 0 < C ≤ ρ -1.
From relations [START_REF] Colton | Target identification of coated objects[END_REF] and [START_REF] Elschner | Corners and edges always scatter[END_REF], we obtain the following result. Using Equalities ( 18) and ( 20), we can provide other criteria, analogous to the ones derived in the study of the continuous interior transmission eigenvalue problem (see [START_REF] Bonnet-Ben Dhia | On the use of T -coercivity to study the Interior Transmission Eigenvalue Problem[END_REF][START_REF] Lakshtanov | Ellipticity in the interior transmission problem in anisotropic media[END_REF][START_REF] Bonnet-Ben Dhia | Strongly oscillating singularities for the interior transmission eigenvalue problem[END_REF]), which only involve the sign of A -Id and ρ -1 in D ∩ V where V is a neighbourhood of Γ = ∂D. To derive such criteria, we first prove a lemma which is very close to [43, Proposition 2.1]. For δ > 0, define

D δ := {x ∈ D | dist(x, ∂D) < δ}. Lemma 3.1. Let à ∈ L ∞ (D, R d×d ) and ρ ∈ L ∞ (D, R) be two parameters that satisfy Ã(x)ζ • ζ ≥ C|ζ| 2 for all x ∈ D, ζ ∈ R d and ρ(x) ≥ C for all x ∈ D, for some constant C > 0. Assume that Ã(x) is symmetric for all x ∈ D. Consider some δ > 0. If v ∈ H 1 (D) verifies -div( Ã∇v) + κ 2 ρ v = 0, ( 21
)
then there exists a constant c > 0 independent of v, κ such that, for κ > 0 large enough,

e cκ D\D δ |∇v| 2 + κ 2 v 2 dx ≤ D δ v 2 dx. ( 22
)
Remark 3.2. In particular, the result of this lemma ensures that for large values of κ, functions satisfying ( 21) are more and more localized in a neighbourhood of Γ.

Proof. We adopt an approach which is used for example in [11, §3.2].

For δ > 0, set Φ δ (x) = min(dist(x, ∂D), δ). Introduce c 1 , c 2 two positive constants such that Ã(x)ζ • ζ ≤ c 1 |ζ| 2 for all x ∈ D δ , ζ ∈ R d and ρ(x) ≥ c 2 for all x ∈ D δ . Define the function E such that E(x) = e εκΦ δ (x) -1 with ε = c 2 /c 1 (
this ad hoc value for ε will be needed in ( 26)). It is known that for a Lipschitz boundary

∂D, Φ δ is an element of L ∞ (D) such that ∇Φ δ ∈ (L ∞ (D)) 2 . Moreover, there holds |∇Φ δ | ≤ 1 on D.
Multiplying ( 21) by E 2 v, integrating by parts and noticing that E 2 v = 0 on ∂D, we find

D Ã∇v • ∇(E 2 v) + κ 2 ρ (Ev) 2 dx = 0. ( 23
) Writing Ã∇v • ∇(E 2 v) = E 2 Ã∇v • ∇v + 2Ev Ã∇v • ∇E and Ã∇(Ev) • ∇(Ev) = E 2 Ã∇v • ∇v + v 2 Ã∇E • ∇E + 2Ev Ã∇v • ∇E, we deduce Ã∇v • ∇(E 2 v) = Ã∇(Ev) • ∇(Ev) -v 2 Ã∇E • ∇E.
Plugging this equality in [START_REF] Kato | Perturbation Theory For Linear Operators[END_REF], we get the identity

D Ã∇(Ev) • ∇(Ev) + κ 2 ρ(Ev) 2 dx = D v 2 Ã∇E • ∇E dx. ( 24
)
Since E is constant on D \ D δ , from [START_REF] Kirsch | The denseness of the far field patterns for the transmission problem[END_REF] we obtain

D\D δ Ã∇(Ev) • ∇(Ev) + κ 2 ρ(Ev) 2 dx ≤ D δ v 2 ( Ã∇E • ∇E -κ 2 ρE 2 ) dx. ( 25
)
With our particular choice for ε, on D δ we can write

( Ã∇E • ∇E -κ 2 ρE 2 ) ≤ κ 2 (ε 2 c 1 e 2εκΦ δ (x) -c 2 (e εκΦ δ (x) -1) 2 ) ≤ 2 c 2 κ 2 e εκδ . ( 26
)
Using [START_REF] Lakshtanov | Ellipticity in the interior transmission problem in anisotropic media[END_REF] in [START_REF] Kirsch | The factorization method for inverse problems[END_REF] yields

(e εκδ -1) 2 2 c 2 κ 2 e εκδ D\D δ Ã∇v • ∇v + κ 2 ρv 2 dx ≤ D δ v 2 dx,
which leads to [START_REF] Yu | Inverse boundary value problem for Schrödinger equation in two dimensions[END_REF] for κ large enough. Now, we can use Lemma 3.1 to localize to Γ the assumptions made on A and ρ.

Assumption 3.3.

There is a neighbourhood of Γ, denoted V, such that Id -A is nonnegative definite on D ∩ V. Moreover, there exists a constant C such that there holds 0 < C ≤ 1ρ on D ∩ V.

Assumption 3.4.

There is a neighbourhood of Γ, denoted V, such that A -Id is nonnegative definite on D ∩ V. Moreover, there exists a constant C such that there holds 0 < C ≤ ρ -1 on D ∩ V. Proof. Assume that A and ρ verify Assumption 3.3. Introduce δ > 0 small enough so that the set

D δ = {x ∈ D | dist(x, ∂D) < δ} verifies D δ ⊂ V. If α = (α 1 , . . . , α N )
⊤ is an element of ker A (iκ) then, according to [START_REF] Colton | Target identification of coated objects[END_REF], the field u i = N n=1 α n e -κθn•x and the associated scattered field u s must verify

0 = R d A|∇u s | 2 + κ 2 ρ u 2 s dx + D (Id -A)|∇u i | 2 + κ 2 (1 -ρ)u 2 i dx. ( 27
)
The boundedness of A, ρ and Lemma 3.1 applied to u i (with à = Id and ρ = 1) allow to write

D (Id -A)|∇u i | 2 + κ 2 (1 -ρ)u 2 i dx ≥ D δ (Id -A)|∇u i | 2 + κ 2 (1 -ρ)u 2 i dx -c D\D δ |∇u i | 2 + κ 2 u 2 i dx ≥ D δ (Id -A)|∇u i | 2 + κ 2 (1 -ρ)u 2 i dx -c D δ u 2 i dx , ( 28
)
where c > 0 is a constant independent of κ and α which may change from one line to another. Plugging [START_REF] Lechleiter | The factorization method is independent of transmission eigenvalues[END_REF] in [START_REF] Lakshtanov | Applications of elliptic operator theory to the isotropic interior transmission eigenvalue problem[END_REF] and using that 0

< C ≤ 1 -ρ on D δ , we deduce 0 ≥ c κ 2 D δ u 2 i dx
for κ large enough. This implies u i = 0 on D δ , which is possible if and only if α is the null vector. Therefore, A (iκ) is injective for κ large enough and the analyticity of the map k → det A (k) leads to the conclusion of the proposition. The case where A and ρ verify Assumption 3.4 can be treated similarly using Lemma 3.1 applied to u with à = A and ρ = ρ.

Remark 3.3. As mentioned previously, the technique to prove Proposition 3.3 is directly inspired by

what has been done to consider the continuous interior transmission eigenvalue problem. However, in the discrete setting, Assumption 3.3 and Assumption 3.4 may probably be relaxed. Thus, one could imagine that imposing conditions on the physical parameters A and ρ only in some particular regions of Γ, associated with the directions of the incident waves, is enough to obtain the result of Proposition 3.3. This is obvious when there is only one incident direction. And when there are more and more incident directions, one would retrieve the conditions of Assumption 3.3 and Assumption 3.4. But obtaining such a criterion for a given finite number of incident directions is still an open problem.

Open questions

Around these questions, we can formulate a series of problems we do not know how to solve. Can we prove that the set of non-scattering wavenumbers is discrete or empty when the scattering directions are not equal to the opposite of the incident directions? Can we show that the non-scattering wavenumbers for Problem (6) (resp. (1)) do not accumulate at 0? Can we relax Assumptions 3.1, 3.2, 3.3 and 3.4? Do the non-scattering wavenumbers in this discrete setting (if they exist) converge to the transmission eigenvalues of the continuous framework when the number of incident and scattering directions tend to +∞?

Far field invisibility

In this section, we change the point of view. Let us consider k > 0 a given wavenumber and D a given domain. We want to construct an inclusion supported in D for which the relative scattering matrix A (k) defined in ( 5) is null. We assume that the physical coefficient A verifies A = Id in R d so that the material of the inclusion is characterized only by the real valued parameter ρ. Then, the scattering of the incident field u i = N n=1 α n e ikθn•x is described by the problem

Find u ∈ H 1 loc (R d ) such that -∆u = k 2 ρ u in R d , u = u i + u s in R d , lim r→+∞ r d-1 2 ∂u s ∂r -iku s = 0. ( 29
)
We search for ρ under the form ρ = 1 + εµ where ε > 0 is a parameter small with respect to 1 and where µ ∈ L ∞ (R d , R) is a function supported in D. We emphasize that the inclusion we wish to create is a small perturbation of the reference material. This is a key element of the approach we will follow.

The technique we will use has been introduced in [START_REF] Nazarov | Asymptotic expansions of eigenvalues in the continuous spectrum of a regularly perturbed quantum waveguide[END_REF][START_REF] Nazarov | Eigenvalues of the laplace operator with the Neumann conditions at regular perturbed walls of a waveguide[END_REF][START_REF] Nazarov | Enforced stability of an eigenvalue in the continuous spectrum of a waveguide with an obstacle[END_REF][START_REF] Nazarov | Enforced stability of a simple eigenvalue in the continuous spectrum of a waveguide[END_REF][START_REF] Cardone | Asymptotic behaviour of an eigenvalue in the continuous spectrum of a narrowed waveguide[END_REF][START_REF] Nazarov | Trapped waves in a cranked waveguide with hard walls[END_REF] with the concept of "enforced stability for embedded eigenvalues". In these works, the authors develop a method for constructing small regular and singular perturbations of a waveguide that preserve the multiplicity of the point spectrum on a given interval of the continuous spectrum. The approach has been adapted in [START_REF] Bonnet-Ben Dhia | Obstacles in acoustic waveguides becoming "invisible" at given frequencies[END_REF] (see also [START_REF] Bonnet-Ben Dhia | Underwater topography "invisible" for surface waves at given frequencies[END_REF] for an application to a water wave problem) to prove the existence of regular perturbations of a waveguide, for which several waves at given frequencies pass through without any distortion or with only a phase shift.

One incident direction and N scattering directions

To simplify the presentation of the method, we first assume that there is only one incident direction θ i (i.e. in [START_REF] Maz'ya | Asymptotic theory of elliptic boundary value problems in singularly perturbed domains[END_REF], we take u i = e ikθ i •x ) and N scattering directions θ 1 , . . . , θ N . For a given ε > 0, we denote u ε the solution of Problem (29) with ρ = 1 + εµ. We proceed to a formal asymptotic expansion of u ε . This function admits the decomposition u ε = u i + u ε s where u ε s corresponds to the field scattered by the inclusion. We take the simplest ansatz for a regular perturbation of a partial differential equation [START_REF] Kato | Perturbation Theory For Linear Operators[END_REF][START_REF] Maz'ya | Asymptotic theory of elliptic boundary value problems in singularly perturbed domains[END_REF] u ε s = 0 + εû + ε 2 ũ + . . . , where the correction terms û, ũ have to be determined and where the dots stand for higher order terms insignificant in our asymptotic analysis. Let us point out that we choose an ansatz for the scattered field which vanishes at the zero order because ρ = 1 + εµ is a perturbation of order ε of the parameter of the reference material (which does not produce any scattered field). Plugging the expression of u ε and the expansion of u ε s in (29), we find that û and ũ are uniquely determined as the solutions of the problems

Find û ∈ H 1 loc (R d ) such that -∆û -k 2 û = k 2 µu i in R d , lim r→+∞ r d-1 2 ∂ û ∂r -ikû = 0 Find ũ ∈ H 1 loc (R d ) such that -∆ũ -k 2 ũ = k 2 µû in R d , lim r→+∞ r d-1 2 ∂ ũ ∂r -ikũ = 0. (30)
From (4), we deduce that the far field pattern of u ε s in the direction of observation θ s , denoted u ε ∞ s (θ s , θ i ), admits the asymptotic expansion

u ε ∞ s (θ s , θ i ) = 0 + ε c d k 2 D µ e ik(θ i -θs)•x dx + ε 2 c d k 2 D µ û e -ikθs•x dx + . . . . (31)
Observing [START_REF] Melrose | Geometric scattering theory[END_REF], we see it is easy to find functions µ such that u ε ∞ s (θ n , θ i ) vanishes at order ε for n = 1, . . . , N . However, this is not sufficient since we want to impose u ε ∞ s (θ n , θ i ) = 0 (at any order in ε). To control the higher order terms in ε 2 , ε 3 ,. . . whose dependence with respect to µ is less simple than for the first term of the asymptotics, we will use the fixed point theorem. To obtain a fixed point formulation, we look for µ under the form

µ = µ 0 + N n=1 τ 1,n µ 1,n + N n=1 τ 2,n µ 2,n . ( 32
)
In this expression, τ 1,n , τ 2,n are real parameters that we will tune to achieve invisibility in the directions θ 1 , . . . , θ N . We need 2N real parameters because we want to cancel N complex coefficients. Moreover, in (32) µ 0 , µ 1,n , µ 2,n are given real valued functions, supported in D, verifying

D µ 0 cos(k(θ i -θ n ′ ) • x)) dx = 0, D µ 0 sin(k(θ i -θ n ′ ) • x)) dx = 0 D µ 1,n cos(k(θ i -θ n ′ ) • x)) dx = δ nn ′ , D µ 1,n sin(k(θ i -θ n ′ ) • x)) dx = 0 D µ 2,n cos(k(θ i -θ n ′ ) • x)) dx = 0, D µ 2,n sin(k(θ i -θ n ′ ) • x)) dx = δ nn ′ ( 33
)
for all n, n ′ = 1, . . . , N . In [START_REF] Nazarov | Asymptotic expansions of eigenvalues in the continuous spectrum of a regularly perturbed quantum waveguide[END_REF], δ nn ′ denotes the Kronecker delta such that δ nn ′ = 1 if n = n ′ and δ nn ′ = 0 else. At this stage, it is important to assume that there holds θ iθ n ′ = 0 for n ′ = 1, . . . , N so that we can indeed find functions µ 2,n which satisfy the last equation of [START_REF] Nazarov | Asymptotic expansions of eigenvalues in the continuous spectrum of a regularly perturbed quantum waveguide[END_REF]. In §4.4, dedicated to numerical experiments, we will explain how to build explicit functions µ 1,n , µ 2,n verifying [START_REF] Nazarov | Asymptotic expansions of eigenvalues in the continuous spectrum of a regularly perturbed quantum waveguide[END_REF]. With this particular choice for µ, plugging ( 32) in (31) and using [START_REF] Nazarov | Asymptotic expansions of eigenvalues in the continuous spectrum of a regularly perturbed quantum waveguide[END_REF], we obtain, for n = 1, . . . , N , the expansion

u ε ∞ s (θ n , θ i ) = ε c d k 2 (τ 1,n + iτ 2,n ) + ε 2 c d k 2 (F ε 1,n (τ ) + iF ε 2,n (τ )), (34)
where F ε 1,n , F ε 2,n denote some functions (whose precise definition can be found in ( 38)) of ε and τ := (τ 1,1 , . . . , τ 1,N , τ 2,1 , . . . , τ 2,N ) ⊤ . Now, to impose u ε ∞ s (θ n , θ i ) = 0 for n = 1, . . . , N , we see from ( 34) that we have to solve the problem Find τ ∈ R 2N such that τ = F ε (τ ), [START_REF] Nazarov | Trapped waves in a cranked waveguide with hard walls[END_REF] with

F ε (τ ) := -ε (F ε 1,1 (τ ), . . . , F ε 1,N (τ ), F ε 2,1 (τ ), . . . , F ε 2,N (τ )) ⊤ . ( 36
)
Lemma 4.1 hereafter ensures that for any given parameter γ > 0, the map F ε is a contraction of B γ := {τ ∈ R 2N |τ | ≤ γ} for ε small enough. Therefore, the Banach fixed-point theorem guarantees the existence of some ε 0 > 0 such that for all ε ∈ (0; ε 0 ], Problem [START_REF] Nazarov | Trapped waves in a cranked waveguide with hard walls[END_REF] has a unique solution in B γ .

It is important to observe that because of the orthogonality conditions [START_REF] Nazarov | Asymptotic expansions of eigenvalues in the continuous spectrum of a regularly perturbed quantum waveguide[END_REF], we are sure that the constructed function µ defined by [START_REF] Nachman | Reconstructions from boundary measurements[END_REF] verifies µ ≡ 0 when µ 0 ≡ 0. As a consequence, we indeed defined a non trivial perturbation of the reference medium, supported in D, which is such that the far field pattern of the scattered field associated with the incident plane wave u i = e ikθ i •x , vanishes in the directions θ 1 , . . . , θ N .

Let us summarize this result in the following statement.

Proposition 4.1. Let k > 0 be a given wavenumber and D refer to a given Lipschitz domain. Consider θ i , θ 1 , . . . , θ N N + 1 elements of S d-1 such that θ n = θ i for n = 1, . . . , N . Define the incident plane wave u i := e ikθ i •x . Then, there exists a non trivial parameter ρ, with ρ -1 supported in D, such that the far field pattern of the scattered field defined by [START_REF] Maz'ya | Asymptotic theory of elliptic boundary value problems in singularly perturbed domains[END_REF] vanishes in the directions θ 1 , . . . , θ N .

In the following lemma, we show some properties of the operator F ε that we used in the previous analysis.

Lemma 4.1. Let γ > 0 be a given parameter. Then, there exists ε 0 > 0 such that for all ε ∈ (0;

ε 0 ], the map F ε is a contraction of B γ = {τ ∈ R 2N |τ | ≤ γ}.
Proof. We assume that γ > 0 is given. Decomposing u ε s as u ε s = 0 + εû + ε 2 ǔε , where û is defined in [START_REF] Mclean | Strongly elliptic systems and boundary integral equations[END_REF], we obtain that ǔε must be the solution of the problem

Find ǔε ∈ H 1 loc (R d ) such that -∆ǔ ε -k 2 ρ ǔε = k 2 µû in R d , lim r→+∞ r d-1 2 ∂ ǔε ∂r -ikǔ ε = 0. ( 37
)
Therefore, using (4), we find that the functions F ε 1,n , F ε 2,n introduced in (34) verify

F ε 1,n (τ ) = ℜe D µ(û + εǔ ε ) e -ikθn•x dx and F ε 2,n (τ ) = ℑm D µ (û + εǔ ε ) e -ikθn•x dx . ( 38
)
General results of perturbations theory for linear operators (see [START_REF] Kato | Perturbation Theory For Linear Operators[END_REF]Chap. 7], [START_REF] Hille | Functional analysis and semi-groups[END_REF]Chap. 4]) yield, for ε small enough, the continuity estimate

û -û′ D + ǔε -ǔε′ D ≤ C |τ -τ ′ |, ∀τ, τ ′ ∈ B γ . ( 39
)
Here, C > 0 is a constant (which can change from one line to another) independent of ε while û′ , ǔε′ denote respectively the solutions of ( 30), [START_REF] Nazarov | Enforced stability of a simple eigenvalue in the continuous spectrum of a waveguide[END_REF] with τ replaced by τ ′ in the definition of µ [START_REF] Nachman | Reconstructions from boundary measurements[END_REF]. Using [START_REF] Päivärinta | Transmission eigenvalues[END_REF] in [START_REF] Päivärinta | Strictly convex corners scatter[END_REF], we deduce

|F ε 1,n (τ ) -F ε 1,n (τ ′ )| + |F ε 2,n (τ ) -F ε 2,n (τ ′ )| ≤ C |τ -τ ′ |.
This result and definition (36) then imply

|F ε (τ ) -F ε (τ ′ )| ≤ C ε |τ -τ ′ |, ∀τ, τ ′ ∈ B γ . ( 40
)
Taking τ ′ = 0 in [START_REF] Potthast | A survey on sampling and probe methods for inverse problems[END_REF] and remarking that |F ε (0)| ≤ C ε (use [START_REF] Päivärinta | Strictly convex corners scatter[END_REF] and the conditions (33) imposed on µ 0 to show the latter inequality), we find [START_REF] Potthast | A survey on sampling and probe methods for inverse problems[END_REF], this finally allows to conclude that the map F ε is a contraction of B γ for ε small enough.

|F ε (τ )| ≤ C ε for all τ ∈ B γ . With
Let us denote τ sol ∈ B γ the unique solution of Problem [START_REF] Nazarov | Trapped waves in a cranked waveguide with hard walls[END_REF]. The end of the previous proof ensures that there exists a constant c 0 > 0 independent of ε such that

|τ sol | = |F ε (τ sol )| ≤ c 0 ε, ∀ε ∈ (0; ε 0 ]. (41)
Estimate [START_REF] Robbiano | Spectral analysis of the interior transmission eigenvalue problem[END_REF] tells us that the function µ is equal to µ 0 at first order.

Backscattering measurements

We come back to the study of the relative scattering matrix A (k) ∈ C N ×N associated with [START_REF] Maz'ya | Asymptotic theory of elliptic boundary value problems in singularly perturbed domains[END_REF]. We remind the reader that in the definition of A (k), we assumed that the scattering directions are -θ 1 , . . . , -θ N where θ 1 , . . . , θ N denote the incident directions. We wish to construct an inclusion of material for which A (k) = 0 N ×N . It is well-known (see [START_REF] Colton | Inverse acoustic and electromagnetic scattering theory[END_REF]Theorem 8.8]) that the far field pattern introduced in (2) satisfies the reciprocity relation

u ∞ s (θ s , θ i ) = u ∞ s (-θ i , -θ s ), ∀θ s , θ i ∈ S d-1 .
From the definition of A (k) (see [START_REF] Bonnet-Ben Dhia | Underwater topography "invisible" for surface waves at given frequencies[END_REF]) and this property, we deduce that

A mn (k) = u ∞ s (-θ m , θ n ) = u ∞ s (-θ n , θ m ) = A nm (k).
Therefore, the matrix A (k) is symmetric and we need to cancel only N (N + 1)/2 complex terms. Following the approach of the previous section, we search for ρ under the form ρ = 1 + εµ with

µ = µ 0 + N n=1 n m=1 τ 1,m,n µ 1,m,n + N n=1 n m=1 τ 2,m,n µ 2,m,n . (42)
In this expression, τ 1,m,n , τ 1,m,n denote some real parameters to fix while µ 0 , µ 1,m,n , µ 2,m,n refer to given real valued functions, supported in D, verifying

D µ 0 cos(k(θ n ′ + θ m ′ ) • x)) dx = 0, D µ 0 sin(k(θ n ′ + θ m ′ ) • x)) dx = 0, D µ 1,m,n cos(k(θ n ′ + θ m ′ ) • x)) dx = 1 if (m, n) = (m ′ , n ′ ) or (m, n) = (n ′ , m ′ ) 0 else, D µ 1,m,n sin(k(θ n ′ + θ m ′ ) • x)) dx = 0, D µ 2,m,n cos(k(θ n ′ + θ m ′ ) • x)) dx = 0, D µ 2,m,n sin(k(θ n ′ + θ m ′ ) • x)) dx = 1 if (m, n) = (m ′ , n ′ ) or (m, n) = (n ′ , m ′ ) 0 else. (43)
The existence of such functions µ 0 , µ 1,m,n , µ 2,m,n can be shown working as in §4.4 provided that the N (N + 1)/2 elements of the family {θ n ′ + θ m ′ } 1≤m ′ ≤n ′ ≤N are all non null and all different. With such a µ, using (4) and working as in [START_REF] Nazarov | Eigenvalues of the laplace operator with the Neumann conditions at regular perturbed walls of a waveguide[END_REF], we obtain, for m, n = 1, . . . , N , the expansion

u ε ∞ s (-θ m , θ n ) = ε c d k 2 (τ 1,m,n + iτ 2,m,n ) + ε 2 c d k 2 (G ε 1,m,n (τ ) + iG ε 2,m,n (τ )). (44)
Here, G ε 1,m,n , G ε 2,m,n denote some functions, defined as in [START_REF] Päivärinta | Strictly convex corners scatter[END_REF], of ε and τ := (τ 1 , τ 2 ), where

τ 1 = (τ 1,m,n ) N ×N , τ 2 = (τ 2,m,n ) N ×N . According to (44), to impose u ε ∞ s (-θ m , θ n ) = 0 for m, n = 1, . . . , N , it just remains to solve the problem Find τ ∈ S N × S N such that τ = G ε (τ ), (45)
where S N denotes the space of symmetric matrices. In this expression, the map τ → G ε is defined via

G ε (τ ) := -ε (G ε 1 (τ ), G ε 2 (τ )), (46)
where G ε 1 (τ ), G ε 2 (τ ) stand for the N × N symmetric matrices made of the terms G ε 1,m,n (τ ), G ε 1,m,n (τ ) respectively. A simple adaptation of the proof of Lemma 4.1 allows to demonstrate that for any γ > 0, there exists ε 0 > 0 such that for all ε ∈ (0; ε 0 ], the map G ε is a contraction of {τ ∈ S N × S N |τ | ≤ γ}. This final result leads to the following statement. Proposition 4.2. Let k > 0 be a given wavenumber and D refer to a given Lipschitz domain. Consider θ 1 , . . . , θ N N vectors of S d-1 such that the N (N + 1)/2 elements of the family {θ n ′ + θ m ′ } 1≤m ′ ≤n ′ ≤N are all non null and all different. Then, there exists a non trivial parameter ρ, with ρ-1 supported in D, such that the relative scattering matrix A (k) associated with [START_REF] Maz'ya | Asymptotic theory of elliptic boundary value problems in singularly perturbed domains[END_REF] verifies A (k) = 0 N ×N . As a consequence, for such inclusions, for any incident field combination of the plane waves e ikθ 1 •x , . . . , e ikθ N •x , the far field pattern of the scattered field vanishes in the directions -θ 1 , . . . , -θ N . Remark 4.1. In dimension two, the assumptions of this proposition can be slightly simplified. Indeed, we can see that if θ 1 , . . . , θ N denote N vectors of S 1 , then the elements of the family {θ n ′ + θ m ′ } 1≤m ′ ≤n ′ ≤N are all different as soon as they are all non null. Remark 4.2. We have focused our attention only on the case where the propagation of acoustic waves is governed by the equation ∆u + k 2 ρ u = 0. Let us mention that we can play similarly with the coefficients A or (A, ρ) to obtain the same kind of results when the acoustic field verifies div(A∇u) + k 2 u = 0 or div(A∇u) + k 2 ρ u = 0.

The case of the scattering direction coinciding with the incident direction

In the previous constructions (see e.g. ( 33)), we needed to assume that there holds θ s = θ i to control, in the imaginary part of u ε ∞ s (θ s , θ i ), the terms of orders ε 2 , ε 3 , . . . by the term of order ε. When θ s = θ i (see Figure 1), the approach we proposed cannot be implemented. To cope with this problem, one could consider a complex valued parameter ρ or an anisotropic material characterized by some matrix valued coefficient A. But imagine that we want to use non dissipative isotropic materials with A = Id only. For a given wavenumber k > 0, can we find a real valued parameter ρ for the inclusion D such that the far field pattern associated with the incident plane wave u i := e ikθ i •x , vanishes in the direction θ i ? In the sequel, we shall assume there is only one scattering direction (N = 1).

D Incident direction θ i

Scattering direction θs = θ i According to (4), we know that the far field pattern in the direction θ s of the scattered field solution of the problem

-∆u = k 2 ρ u in R d , u = u i + u s in R d , lim r→+∞ r d-1 2 ∂u s ∂r -iku s = 0, (47)
is given by

u ∞ s (θ s , θ i ) = c d k 2 D (ρ -1) u e -ikθs•x dx. ( 48
)
When θ s = θ i , we find, using the relation

-∆u s -k 2 ρ u s = k 2 (ρ -1) u i , u ∞ s (θ i , θ i ) = c d k 2 D (ρ -1) (u i + u s ) u i dx = c d k 2 D (ρ -1) |u i | 2 dx -c d D u s ∆u s + k 2 ρ u s dx = c d k 2 D (ρ -1) |u i | 2 dx + c d D |∇u s | 2 -k 2 ρ|u s | 2 dx + c d ∂ ν u s , u s Γ . ( 49
)
Let us work on the last term of the right hand side of the previous equation as in the proof of [START_REF] Colton | Inverse acoustic and electromagnetic scattering theory[END_REF]Theorem 2.5]. We take R large enough such that D is contained in the ball B R := {x ∈ R d |x| < R} and apply Green formula in B R \ D to obtain

∂ ν u s , u s Γ = -∂ ν u s , u s ∂B R + B R \D |∇u s | 2 -k 2 |u s | 2 dx.
On ∂B R , ν refers to the unit outward normal to B R . But u s satisfy the radiation condition of (47) and admits the decomposition (2). Therefore, there holds

ℑm ∂ ν u s , u s Γ = k S d-1 |u ∞ s (θ, θ i )| 2 dθ. ( 50
)
Using (50) in (49), we conclude

ℑm (c -1 d u ∞ s (θ i , θ i )) = k S d-1 |u ∞ s (θ, θ i )| 2 dθ.
According to the Rellich's lemma (see, e.g., [START_REF] Cakoni | Qualitative methods in inverse scattering theory. An introduction[END_REF]Theorem 4.1]

= 0 in R d \ D.
This shows that it is much more complicated to impose far field invisibility in the direction θ i than in other directions. In particular, if k > 0 and ρ are such that the the far field pattern of u s vanishes in the direction θ i , according to Proposition 4.3, we must have u s = uu i = 0 and ∂ ν u s = ∂ ν (uu i ) = 0 on Γ. This implies that the pair (u,

u i ) = (u i + u s , u i ) verify ∆u + k 2 ρ u = 0 in D, ∆u i + k 2 u i = 0 in D, u -u i = 0 on Γ, ∂ ν (u -u i ) = 0 on Γ.
(51) This is nothing else than a transmission eigenvalue problem with a strong hypothesis on u i . Compared to the classical transmission eigenvalue problem where the restriction of the incident field to D belongs to L 2 (D), here we additionally impose u i = e ikθ i •x . As a consequence, for a given ρ which verifies 0 < c ≤ ρ ≤ C < in a neighbourhood of Γ or 1 < c ≤ ρ in a neighbourhood of Γ, we know according to [START_REF] Sylvester | Discreteness of transmission eigenvalues via upper triangular compact operators[END_REF] (see also [START_REF] Lakshtanov | Applications of elliptic operator theory to the isotropic interior transmission eigenvalue problem[END_REF][START_REF] Robbiano | Spectral analysis of the interior transmission eigenvalue problem[END_REF] in the case of a smooth ρ) that the set of wavenumbers for which (51) has a non trivial solution is either empty or discrete. However, we cannot use the results of existence of transmission eigenvalues (see [START_REF] Päivärinta | Transmission eigenvalues[END_REF][START_REF] Cakoni | The existence of an infinite discrete set of transmission eigenvalues[END_REF]) to conclude to the existence of non trivial solutions to (51) because here, we impose a very restrictive condition for u i . More interesting for our configuration are the recent results of [START_REF] Blåsten | Corners always scatter[END_REF][START_REF] Päivärinta | Strictly convex corners scatter[END_REF][START_REF] Elschner | Corners and edges always scatter[END_REF]. In these articles, situations where the support of ρ -1 has corners or edges are considered. In this context, the authors provide sufficient criteria on ρ which ensure that all non trivial incident Herglotz functions (i.e. all non trivial generalized combinations of plane waves), for all wavenumbers k > 0, produce non trivial scattered field. For such coefficients ρ, the only solution of (51) is null. For more general ρ, for example such that the support of ρ -1 is smooth, even assuming u i = e ikθ i •x , we do not know any result showing that (51) has only the trivial solution. Nevertheless, in any case, and this is the important message of this discussion, it seems particularly delicate to impose far field invisibility in the direction θ i .

Numerical experiments

We want to implement numerically the approach developed in §4.1. For a given wavenumber k > 0, we consider the 2D scattering problem

Find u ∈ H 1 loc (R 2 ) such that -∆u = k 2 ρ u in R 2 , u = u i + u s in R 2 , lim r→+∞ √ r ∂u s ∂r -iku s = 0, (52)
with u i = e ikθ i •x (one incident direction θ i = (cos(ψ i ), sin(ψ i ))). Our goal is to build ρ such that the far field pattern of u s vanishes in the three scattering directions θ 1 = (cos(ψ 1 ), sin(ψ 1 )), θ 2 = (cos(ψ 2 ), sin(ψ 2 )) and θ 3 = (cos(ψ 3 ), sin(ψ 3 )). In other words, we take N = 3. Consider D a given Lipschitz domain. Following [START_REF] Nachman | Reconstructions from boundary measurements[END_REF], we search for ρ of the form ρ = 1 + εµ, where µ is a function supported in D such that

µ = µ 0 + 3 n=1 τ 1,n µ 1,n + 3 n=1 τ 2,n µ 2,n . ( 53
)
To define functions µ 1,n , µ 2,n that satisfy the conditions of (33), we start by computing the matrix

B =   B 11 B 12 B 21 B 22  
where B 21 = (B 12 ) ⊤ and

B 11 = (B 11 n,n ′ ) 3×3 with B 11 n,n ′ = D cos(k(θ i -θ n ) • x) cos(k(θ i -θ n ′ ) • x) dx B 12 = (B 12 n,n ′ ) 3×3 with B 12 n,n ′ = D sin(k(θ i -θ n ) • x) cos(k(θ i -θ n ′ ) • x) dx B 22 = (B 22 n,n ′ ) 3×3 with B 22 n,n ′ = D sin(k(θ i -θ n ) • x) sin(k(θ i -θ n ′ ) • x) dx. Observing that {cos(k(θ i -θ 1 ) • x)), . . . , cos(k(θ i -θ 3 ) • x)), sin(k(θ i -θ 1 ) • x)), . . . , sin(k(θ i -θ 3 ) • x))}
is a family of linearly independent functions on D (when there holds θ i = θ n for n = 1 . . . 3), we can prove that B is invertible. We denote D its inverse. Finally, we take

µ 1,n = 3 n ′ =1 D n ′ ,n cos(k(θ i -θ n ′ ) • x) + 3 n ′ =1 D 3+n ′ ,n sin(k(θ i -θ n ′ ) • x) µ 2,n = 3 n ′ =1 D n ′ ,3+n cos(k(θ i -θ n ′ ) • x) + 3 n ′ =1 D 3+n ′ ,3+n sin(k(θ i -θ n ′ ) • x).
Then, we construct a µ 0 that verifies the six orthogonality conditions of (33) taking

µ 0 = µ # 0 - 3 n=1 D µ 1,n µ # 0 dx µ 1,n - 3 n=1 D µ 2,n µ # 0 dx µ 2,n (54)
where µ # 0 is an arbitrary function such that µ # 0 / ∈ span{µ 1,1 , . . . , µ 1,3 , µ 2,1 , . . . , µ 2,3 }. Here, we choose µ # 0 such that µ # 0 (x) = 1 + x + y.

For the numerical experiments, we take

ψ i = 0 • , ψ 1 = 90 • , ψ 2 = 180 • (backscattering direction), ψ 3 = 225
• and D = B 1 (so that the inclusion in contained in the unit disk). The wavenumber k is set to k = 4. Now, we describe the procedure to solve the fixed point problem (35) by induction.

We denote τ j = (τ j 1,1 , . . . , τ j 1,3 , τ j 2,1 , . . . , τ j 2,3 ) ⊤ (resp. µ j ) the value of τ = (τ 1,1 , . . . , τ 1,3 , τ 2,1 , . . . , τ 2,3 ) ⊤ (resp. µ) at iteration j ≥ 0 (we remind the reader that τ 1,n , τ 2,n are the parameters appearing in (53)). Using formulas [START_REF] Nazarov | Eigenvalues of the laplace operator with the Neumann conditions at regular perturbed walls of a waveguide[END_REF], [START_REF] Nazarov | Trapped waves in a cranked waveguide with hard walls[END_REF], for j ≥ 0, n = 1, . . . , 3, we define τ j+1

1,n = τ j 1,n -ℜe ( (ε c d k 2 ) -1 u ε j ∞ s (θ n , θ i ) )

τ j+1 2,n = τ j 2,n -ℑm ( (ε c d k 2 ) -1 u ε j ∞ s (θ n , θ i ) ). (55)
In the above definition, u ε j ∞ s (θ n , θ i ) denotes the far field pattern in the direction θ n of the function u ε j s satisfying the problem

Find u ε j s ∈ H 1 loc (R 2 ) such that -∆u ε j s -k 2 (1 + ε µ j ) u ε j s = k 2 ε µ j u i in R 2 , lim r→+∞ √ r ∂u ε j s ∂r -iku ε j s = 0. ( 56
)
According to formula (48), we know that

u ε j ∞ s (θ n , θ i ) = c d k 2 D ε µ j (u i + u ε j s ) e -ikθn dx. ( 57
)
We approximate the solution of Problem (56) with a P2 finite element method set on the ball B 8 (8 is the radius). On ∂B 8 , a truncated Dirichlet-to-Neumann map with 13 harmonics serves as a transparent boundary condition. We choose τ 0 = (0, . . . , 0). For the simulations of Figures 2345, we stop the procedure when3 n=1 |τ j+1 1,nτ j 1,n | + |τ j+1 2,nτ j 2,n | ≤ 10 -13 (corresponding to 37 iterations) and we take ε = 0.15. To obtain the results of Figures 67, we perform 10 iterations and we try several values of ε. For the computations, we use the FreeFem++ 3 software while we display the results with Matlab The dashed curve represents the approximated far field pattern at iteration 0 (taking ρ = 1 + εµ 0 = 1 + εµ 0 ). The solid curve corresponds to the approximated far field pattern computed for the parameter ρ obtained at the end of the fixed point procedure. The dotted lines indicate the directions where we want u ε ∞ s to vanish. From these results, and in accordance with estimate (41), we infer that the term µ 0 in the expression of µ (see (53)) determines the shape of u ε ∞ s . But, we also notice that the fixed point procedure allows to correct significantly u ε ∞ s in the directions θ 1 , θ 2 and θ 3 . The results we obtain are in agreement with this estimate. This shows that the convergence of the fixed point procedure gets quicker as ε goes to zero. But in this case, the perturbation of the reference medium becomes smaller and smaller, as indicated by the results of Figure 7.

Assumption 3 . 1 .

 31 In D, Id -A is nonnegative definite and 1ρ is nonnegative. Moreover, there exists a constant C and a non empty open set ̟ ⊂ D on which there hold 0 < C|ζ| 2 ≤ (Id -A)ζ • ζ for all ζ ∈ R d or 0 < C ≤ 1ρ. Assumption 3.2. In D, A -Id is nonnegative definite and ρ -1 is nonnegative. Moreover, there exists a constant C and a non empty open set ̟ ⊂ D on which there hold 0

Proposition 3 . 2 .

 32 (Penetrable inclusion) -Assume that A and ρ verify Assumption 3.1 or Assumption 3.2. Then the set of non-scattering wavenumbers for Problem (1) is either empty or discrete.

Proposition 3 . 3 .

 33 (Penetrable inclusion) -Assume that A and ρ verify Assumption 3.3 or Assumption 3.4. Then the set of non-scattering wavenumbers for Problem (1) is either empty or discrete.

Figure 1 :

 1 Figure 1: Scattering problem with θ s = θ i .

  4 and Paraview 5 .

Figure 2 :

 2 Figure 2: Coefficient ρ. Interestingly, the fixed point procedure converges though the coefficient ρ is not a very small perturbation of the parameter (uniformly equal to one) of the reference medium. The domain represented here is equal to B 8 and the inclusion D is located in B 1 .

Figure 3 :

 3 Figure 3: Restriction of the coefficient ρ to D. In this particular case, with our choice for the functions defining µ, we have ρ| D ∈ C ∞ (D). The domain represented here is B 1 .

Figure 4 :Far

 4 Figure 4: Absolute value of the approximation of the scattered field u ε s at the end of the fixed point procedure in logarithmic scale. As desired, we see it is very small far from D in the directions corresponding to the angles 90 • , 180 • and 225 • . The domain represented here is equal to B 8 and the inclusion D is located in B 1 .

Figure 5 :

 5 Figure 5: The dashed curve represents the approximated far field pattern at iteration 0 (taking ρ = 1 + εµ 0 = 1 + εµ 0 ). The solid curve corresponds to the approximated far field pattern computed for the parameter ρ obtained at the end of the fixed point procedure. The dotted lines indicate the directions where we want u ε ∞

Figure 6 :

 6 Figure 6: Curves ln 3 n=1 |u ε j ∞ s

Figure 7 :

 7 Figure 7: Minima and maxima of the parameter ρ obtained after 10 iterations for several values of ε.

  Let u s refer to the scattered field, defined by Problem (47), associated with the incident plane wave u i = e ikθ i •x . Then the far field pattern of u s vanishes in the direction θ i if and only if there holds u s

	in dimension 2 and [16, Theorem 2.14] in
	dimension 3), this proves the
	Proposition 4.3.

Here, the adjective "generalized" means that the incident field can be the combination of an infinite number of plane waves. In the literature, such a field is referred to as a Herglotz wave function.

Throughout the paper, the complex square root is chosen so that if ξ = re iγ for r ≥ 0 and γ ∈ [0; 2π), then √ ξ = √ re iγ/2 . With this choice, there holds ℑm √ ξ ≥ 0 for all ξ ∈ C.

FreeFem++, http://www.freefem.org/ff++/.

Matlab, http://www.mathworks.se/.

Paraview, http://www.paraview.org/.

Acknowledgments

The work of the first author is supported by the Academy of Finland (decision 140998) and by the FMJH through the grant ANR-10-CAMP-0151-02 in the "Programme des Investissements d'Avenir". The research of the third author is supported by the Russian Foundation of Fundamental Investigations, grant No. 15-01-02175.