Joran Van Turnhout

Domenico Aceto

Arnaud Travert

Philippe Bazin

Frédéric Thibault-Starzyk

Annemie Bogaerts

Federico Azzolina

Frédéric Thibault- Starzyk

Federico Azzolina-Jury

Observation of surface species in plasma-catalytic dry reforming of methane in a novel atmospheric pressure dielectric barrier discharge in situ IR cell

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

Introduction

In the fight against climate change, the dry reforming of methane (DRM) has gained significant interest for the conversion of greenhouse gases (GHGs) into value-added chemicals. Indeed, this reaction involves the conversion of CO2 and CH4, the two main greenhouse gases, into syngas. The latter is a valuable feedstock for gas-to-liquid (GTL) technology via the Fischer-Tropsch synthesis, and can also be used for methanol synthesis 1 . However, despite its rich history, DRM remains a significant challenge in chemical engineering. Indeed, high operating temperatures (400-1000 °C) and significant coking of the used catalysts (typically containing Ni, Co or precious metals) form grave limitations for the thermo-catalytic DRM, so that it is not yet widely used on an industrial scale today. A possible alternative is the combination of a catalyst with nonthermal plasma (NTP). In such a plasma-catalytic system, inelastic collisions of high-energy electrons with reactant molecules lead to the production of reactive and excited species, thus enabling the activation of CO2 and CH4 near room temperature [2][3][4] . Concurrently, reactive species formed in the plasma can interact with the catalyst and thus change its properties, and vice versa, the presence of a catalyst in the plasma influences the plasma properties 6 ; this can give rise to synergetic effects, which can yield an enhanced conversion 7 , improved selectivity towards targeted reaction products [3][4][5] , improvement of the energy efficiency 6,8 , and may enhance the catalyst stability by reducing coke formation and sintering 9,10 . Among the many types of NTP that are being investigated, a dielectric barrier discharge (DBD) is the most suitable for plasma catalysis, because its simple design allows for easy implementation of catalyst material into the discharge zone, it operates near room temperature, is easily switched on and off, is simple to operate and scale up, and operates under atmospheric pressure 11 . Currently, for DRM, packed bed DBDs can provide conversions of up to 80% with an energy cost in the range of 15-100 eV/molec 12 , which is much higher than the set benchmark of 4.27 eV/molec (i.e. energy efficiency of 60%) as proposed by Snoeckx and Bogaerts 12 , in order for plasmacatalytic syngas formation to be a competitive alternative to thermal catalysis. Clearly, significant improvement is required in order for plasma-catalytic syngas production to be a viable approach. For this, a better understanding of the plasmacatalytic synergy is paramount. Plasma-catalytic synergy can be partly attributed to plasmainduced changes in catalyst surface reaction pathways. For example, highly reactive plasma species (like radicals) may have a larger sticking probability on the catalyst surface than their molecular counterparts, paving the way for new reactive pathways on the surface 6 . Additionally, reactive plasma species are more likely to react with adsorbed species via Eley-Rideal mechanisms 9 , whereas in thermal catalysis surface reactions overwhelmingly occur via the Langmuir-Hinshelwood mechanism 13,14 . Moreover, high-energy electrons in the plasma may influence the reactions on the catalyst surface 9,15 . Finally, vibrationally excited species may play an important role in the plasma-catalytic synergy [16][17][18] . All of the above implies that for a certain reaction, the optimal plasma catalyst may differ significantly from the optimal thermal catalyst. Indeed, this was already shown computationally for other plasma-catalytic systems 16,18 .

Clearly, a thorough understanding of the effects of NTP on catalyst surface chemistry is paramount for the rational design of a plasma-catalytic system. While microkinetic models [16][17][18][19] and DFT studies 8 can provide useful insights in the possible effect of plasma on the catalytic surface chemistry, their accuracy is inherently limited by approximations and therefore, experimental validation to those models is imperative. For this, in-situ (i.e. inside the plasma, during operation) IR spectroscopy analysis of catalyst surface species is particularly interesting. In previous work, in-situ observation of species at the catalyst surface in plasma catalysis has mostly been performed by diffuse reflectance infrared Fourier transform spectroscopy (DRIFTS) 10,[20][21][22][23] , a sampling technique widely used in thermal catalysis. Alternatively, transmission IR can be applied. Advantages of the latter include (i) the possibility for relatively straightforward quantitative studies, as transmission FTIR obeys the Beer-Lambert law, in contrast to DRIFTS, which follows the Kubelka Munk theory, (ii) lower temperature dependence, (iii) the elimination of grain size dependence and (iv) a smaller temperature gradient in the catalyst material [START_REF] Thibault-Starzyk | Infrared Spectroscopy: Classical Methods[END_REF]26 . While previously developed transmission cells allowed for the analysis of surface species in glow discharge plasma [START_REF] Azzolina-Jury | [END_REF] or in a closed system at atmospheric pressure 27 , to date, no transmission IR cell has allowed the study of DBD plasma catalysis under continuous flow. To the best of our knowledge, to this day, only one publication 22 reported an in-situ IR study of plasma-catalytic DRM using DRIFTS, and the influence of NTP on catalyst surface species in DRM is largely unexplored. Therefore, in the present work, we describe a novel in-situ DBD IR cell that allows for investigation of plasma catalysis in a DBD in transmission mode, atmospheric pressure, flow conditions (WHSV ~ 0 -50,000 mL g -1 h -1 ), relevant discharge voltages (~ 0 -50 kV) and frequencies (~ 0 -5 kHz) and temperatures up to 450 °C. We applied it for the first time to study the surface species formed during plasmacatalytic DRM over SiO2 and 3 wt% Ru/SiO2. Ru has shown promising results for thermo-catalytic DRM 28 , and SiO2 was chosen as an inert support in an attempt to minimize the contribution of the support to the catalytic activity of the metalloaded material. Moreover, we also tested both SiO2 and Ru/SiO2 in a packed bed DBD reactor, and measured the CO2 and CH4 conversion, as well as the product yields and selectivities, in order to link the observations of our in-situ experiments to the catalytic activity. Finally, it is crucial to know the catalyst surface temperature during plasma catalysis, to better understand the underlying mechanism, i.e., whether the formation of surface species is due to plasma-induced heating of the catalyst surface or due to other plasma effects, such as plasma-generated radicals. However, in-situ measurement of the catalyst surface temperature inside a DBD plasma is particularly challenging. To our knowledge, it has only been reported twice in literature, and for other reactions than DRM 23,29 . In this work, we provide a novel technique to estimate the temperature of the catalyst wafer under DBD plasma using FTIR spectroscopy in the in-situ DBD IR cell.

Experimental

Catalyst preparation and characterization

Ru loading of the SiO2 support (Solvay) was carried out by wet impregnation (3 wt% Ru) using RuCl3.xH2O (Sigma Aldrich, 99.98%) as precursor salt dissolved in H2O. Note that the provider did not specify the hydrate of the salt, and therefore we estimated the metal loading using EDS-SEM (see section 3.1). The water was evaporated under continuous stirring using a rotary evaporator at 70 °C under vacuum (0.3 Torr) until complete dryness. Afterwards, the catalyst powder was calcined at 450 °C for 4 h and subsequently reduced at its reduction temperature as determined by TPR (see section 3.1) for 4 h in a gas mixture of 20 vol% H2 and 80 vol% Ar with a total gas flow rate of 16 mL min -1 (STP). Finally, for the transmission IR measurements, the catalyst powder was pressed (1 ton, 10 s) into wafers (2 cm 2 , ~ 20 mg). For the packed bed experiments, the catalyst powder was shaped into irregular extrudates (> 1 mm in every dimension). For this, 50 mg of the catalyst material was pressed (2 tons, ~ 5 seconds) in a pellet with height > 1 mm. The pellet was crushed and the pieces were sieved to select only those having each dimension > 1 mm, to guarantee gaps in the catalyst bed that are larger than the Debye length of the plasma (typically 0.5-1 µm for DBD). The shaping of SiO2 for experiments on the support alone was done similarly to the shaping of the metal-loaded SiO2. H2 temperature programmed reduction (H2-TPR) was applied to the catalyst pre thermal reduction in order to determine the reduction temperature of the deposited metal oxide particles on the support. For this, 0.05 g of a catalyst sample was heated to 1000 °C in a 5 vol% H2 and 95 vol% Ar gas flow (total flow rate of 40 mL min -1 (STP)) at a heating rate of 10 °C/min using a Micromeritics Autochem II chemisorption analyser equipped with a thermal conductivity detector and subsequently cooled down to room temperature. X-ray diffraction patterns of the Ru/SiO2 catalyst were recorded post-reduction with a PANalytical X'pert PRO diffractometer with CuKα radiation (λ = 0.15418 nm, 40 mA, 45 kV). The sample was scanned from 5 to 80° at a step size of 0.0083556° and scanning rate of 0.026526 °/min. The specific surface area (SA) and pore volume of the catalyst and SiO2 support were determined by N2-physisorption measurements at -196 °C using a Micromeritics ASAP 2020 Plus adsorption analyser. Before N2-adsorption, the sample (0.2 g) was degassed at 200 °C overnight. The specific SA of the sample was calculated using the Brunauer, Emmett and Teller (BET) method. The total pore volume was measured at P/P 0 = 0.99. Energy-dispersive X-ray spectroscopy (EDS) coupled to scanning electron microscopy (SEM) was used for the elemental analysis of Ru/SiO2 catalyst using an EDAX XM2-30T apparatus. Two separate regions of the catalyst were analysed.

In-situ IR

Experimental setup

Please do not adjust margins Please do not adjust margins A schematic representation of the novel in-situ DBD IR cell used for transmission IR measurements is given in Figure 1. A more detailed scheme and photographs of the cell can be found in the supporting information (Section S1; Figures S1-S3). The H2, CH4, CO2 and Ar gas inlet is controlled by four separate Brooks Delta Smart II mass flow controllers. The gas inflow is mixed and introduced in the glass DBD cell. The gas outlet is placed higher up in the cell and is connected to a Pfeiffer Vacuum Prisma Pro quadrupole mass spectrometer to monitor the outlet gas composition. Additionally, the cell is connected to a rotary pump (~ 0.3 Torr, Edwards nXDS6iC) and turbopump (10 -7 Torr, Pfeiffer vacuum HiCube 80) to allow for easy purging of the cell.

The DBD IR cell is positioned inside the sample compartment of a Bruker Vertex 80v spectrometer with a liquid nitrogen cooled mercury-cadmium-telluride (MCT) detector. The glass cell is sealed with two KBr windows for the IR beam to pass through, ensuring an IR transmission window in the 4000-400 cm -1 range.

The catalyst is introduced as a wafer in the sample holder, which is attached to a grid electrode. Parallel to this electrode, a similar grid electrode is placed behind one of the aforementioned KBr windows; the latter functions as a dielectric barrier. This electrode is electrically isolated with a third KBr window. The gap between both electrodes is ~ 5 mm. The inner grid electrode is grounded, whereas the outer grid electrode is connected to a signal generator (Générateur de fonctions arbitraires 50 MHz, Française d'Intrumentation) which is amplified (Trek 20/20C). A voltage probe (Testec HVP-2739-1000:1) is placed on the connection between the outer high voltage (HV) electrode and the amplifier. Additionally, a voltage probe is positioned on the connection between the inner ground electrode and the ground, and a current monitor (Hioki CT6710) is placed on this same connection. The voltage and current probes are connected to a Tektronix MDO3024 oscilloscope. The latter illustrates a significant advantage of our cell, being that our setup allows for continuous measurement of plasma parameters during an experiment.

2.2.2.

In-situ observation of surface species

After being introduced into the cell, the Ru/SiO2 wafer was activated at 350 °C for 1.5 h under secondary vacuum (10 -7 Torr) after which the catalyst was kept at the same pressure for 16 h. The cell was subsequently filled with Ar to a pressure of 1 atm and an Ar plasma was ignited at a voltage of ~ 16 kV, a frequency of 3 kHz and an Ar gas flow rate of 20 mL min -1 (STP). Next, the gas flow was changed to 20 vol% H2 and 80 vol% Ar (20 mL min - 1 total, STP) and the voltage was increased to ~ 28 kV to sustain the plasma. In this way, the catalyst was reduced in-situ for 20 min. Afterwards, the plasma was extinguished and the cell was purged under vacuum (0.3 Torr) in order to remove the water formed during the reduction step. After the purging that followed the in-situ reduction, the cell was filled with a gas flow of 20 mL min -1 (STP) consisting of 35 vol% CH4, 35 vol% CO2 and 30 vol% Ar to a pressure of 1 atm. Then, plasma was again ignited at a voltage of ~ 30 kV and a frequency of 3 kHz. The Ar in the system was necessary to sustain the discharge. After reaching a stable conversion (measured with the mass spectrometer), the plasma was extinguished and the outlet gas flow was again stabilized. Subsequently, the system was put under vacuum (0.3 Torr) to purge the cell and to monitor the adsorbed species on the catalyst surface. Then, we heated the catalyst pellet in-situ to set temperatures to monitor the desorption behaviour of strongly adsorbed species as a function of temperature. Note that the catalyst surface (and gas around the surface) was continuously monitored during both the in-situ reduction and the DRM reaction using transmission FTIR (resolution of 4 cm -1 , 64 scans per spectrum and 1 spectrum recorded per 9s) as described in section 2.2.1 above.

In addition to the tests with the Ru/SiO2-loaded sample holder, a test was performed with the SiO2 support and in the empty (i.e. no wafer in the sample holder) cell. For the former, no insitu reduction was required, while for the latter neither activation, nor in-situ reduction was required. The rest of the protocol remained identical. Note that while, in principle, the presence of a mass spectrometer at the gas outlet should allow for operando operation (i.e. activity measurements in parallel to surface characterization), the operation under plasma conditions requires spatial separation of the electrodes from any metallic parts of the spectrometer, leading to large dead volumes in the current cell. Measured conversions are therefore very low (see table S2), so that we proceed with activity testing in a packed bed DBD reactor (see section 2.3). However, it is noteworthy that, while the observed conversions in our in-situ cell are much lower, the observed selectivity towards CO and H2 on both SiO2 and Ru/SiO2 show similar values to the ones obtained in the packed bed reactor below (see Table S2).

In-situ temperature measurements

BaSO4 was used as an internal standard for temperature measurements. This salt presents absorption bands of low intensity near 2000 cm -1 , which can be assigned to overtone and combination bands of low wavenumber stretching and bending S-O vibrations 35,[START_REF] Nakamoto | Infrared and Raman Spectra of Inorganic and Coordination Compounds: Part A: Theory and Applications in Inorganic Chemistry[END_REF] . These bands were observed to experience a downward frequency shift (bathochromic shift) with the Please do not adjust margins Please do not adjust margins increase of the sample temperature. A band located at 1965.9 cm -1 at room temperature and not overlapping with vicinal bands was used to establish a temperature scale based on the position of its maximum 𝜈 ̅ 𝑚𝑎𝑥 measured in a temperaturecalibrated IR cell. The calibration curve yielded the following equation (R 2 = 0.997, SD = 5°C):

𝑡 °C ⁄ = -35.30985 × 𝜈 ̅ 𝑚𝑎𝑥 cm -1 ⁄ + 69 442.7 (1)
Experiments were performed with either a wafer of 40 mg of pure BaSO4 in a H2/Ar plasma or wafer of a mixture of 10 mg of Ru/SiO2 powder with 30 mg of BaSO4 (Alfa Aesar, 97%). The determination of the position of band maxima for calibration and measures was carried out by quadric interpolation with SpectroChempy 38,39 .

Thermal In-situ operando IR "sandwich" cell

To distinguish the effect of plasma-induced heating from other plasma effects, the Ru/SiO2 wafer was also tested in a thermal in-situ operando IR "sandwich" cell at 100 and 150 °C (i.e. around the observed temperature range of the catalyst wafer in the plasma, as described in section 3.2.3). More information on the setup and experimental procedure is available in the supporting information (section S6).

Packed bed DBD

Experimental setup

The packed bed reactor consists of a cylindrical inner electrode (stainless steel, Ø = 5 mm) positioned at the centre of a glass cylinder (Øinner = 10 mm) which is surrounded by the outer electrode (Cu coil, L = 100 mm). Thus, a discharge gap of 2.5 mm is obtained. Catalyst material is positioned in the discharge zone (i.e. within the gap between the electrodes) and is held in place with silicon wool, which is positioned right underneath the discharge zone. A scheme of the DBD packed bed reactor is given in the supporting information (Figure S4). The outer coil electrode is grounded, whereas the inner rod electrode is connected to a signal generator (Générateur de fonctions arbitraires 50 MHz, Française d'Intrumentation) which is amplified (Trek 20/20C). The voltage and current during the experiment are monitored similarly to the in-situ setup as described above.

Likewise, the H2, CH4, CO2 and Ar gas inlet control, as well as the analysis of the outlet gas, is executed similarly to the in-situ setup described above.

In one experiment for both SiO2 and Ru/SiO2, IR spectroscopy is used to detect the presence of higher hydrocarbons or oxygenates such as methanol in the outlet gas. Thus another line is added between the mass spectrometer and the gas exhaust, leading to a glass cell positioned inside the sample compartment of a Bruker Vertex 80v spectrometer with a liquid nitrogen cooled mercury-cadmium-telluride (MCT) detector.

The glass cell is sealed with two KBr windows for the IR beam to pass through, ensuring an IR transmission window in the 4000-400 cm -1 range.

Experimental procedure

First, the discharge gap was packed with reduced 3 wt% Ru/SiO2 extrudates (1.5 g). Then, the setup was purged under vacuum (~ 0.3 Torr), and the DBD packed bed reactor was subsequently filled with Ar to 1 atm. The catalyst was reduced in-situ at 350 °C for 1.5 h under a gas flow of 20 vol% H2 in Ar (total of 45 mL min -1 , STP) after which the system was kept under vacuum (~ 0.3 Torr) for 2 h. The catalyst was again treated in-situ using H2/Ar plasma as described for the in-situ cell, with the only difference being the applied voltage (20 kV) and duration (10 min) to limit the strain on our reactor. Likewise, the continuation of the experiment was similar to the procedure described for the in-situ cell described above. However, for the packed bed reactor, we used a total gas flow rate of 22 mL min -1 (STP) and applied a voltage of 25 kV, as further increasing the voltage led to arc discharges, making safe operation impossible. Moreover, to allow for easier discharging, plasma was ignited in pure Ar first, after which the reactant gases were introduced. The outlet gas was continuously monitored using mass spectrometry. More specifically, the following mass-to-charge ratios (m/z) were monitored using PV MassSpec software: 2 (H2), 15 (CH3, as a proxy for CH4, to avoid interference with O), 28 (CO), 40 (Ar) and 44 (CO2). The contribution of CO2 to the m/z = 28 signal was accounted for during quantitative data processing to avoid an underestimation of CO production. The experiments were executed three times. In one experiment, the outlet gas was also monitored using FTIR to detect the presence of higher hydrocarbons or oxygenates such as methanol.

Another experiment was run (again three times) with SiO2 support alone following the same procedure, but without in-situ reduction.

The formulas used for the calculation of CO2 and CH4 conversion, and for the product yields and selectivities, are given in section S2 of the supporting information. The calibration curves for these calculations are given in section S3 of the supporting information (Figures S5-S10). Detailed catalyst characterization is presented in section S4 of the supporting information. The H2-TPR profile of the Ru-loaded catalyst shows a peak at 166 °C corresponding to the reduction of RuO2 to metallic Ru (see Figure S11). A smaller peak was found at 417 °C, possibly indicating the reduction of Ru clusters on the SiO2 support. The X-ray diffractogram of the amorphous SiO2 support shows a broad peak at 2θ = 23°. The 3 wt% Ru/SiO2 sample shows a minor peak at 2θ = 44° corresponding to Ru(101) (see Figure S12). A complete reduction of RuO2 to metallic Ru is indicated by the absence of peaks at 2θ = 28°, 35° and 54°, corresponding to RuO2(110), RuO2(101) and RuO2(211), respectively.

Results and discussion

Catalyst characterization

Please do not adjust margins

Please do not adjust margins

The SiO2 support and 3 wt% Ru/SiO2 catalyst exhibit a BET surface area of 151 and 148 m 2 g -1 , respectively. The weight loss upon degassing, which is a measure for the water holding capacity of the samples, is 9 and 5 wt% for the SiO2 support and the 3 wt% Ru/SiO2 sample, respectively. The isotherms (see Figure S13 and S14), being of type IVa in the IUPAC classification of physisorption isotherms, suggest a mesoporous adsorbent with pores wider than 4 nm. The hysteresis loop of the samples can be classified as type H2b, suggesting the presence of small pore necks with a large size distribution. The presence of Ru was confirmed using EDS-SEM. Figure 2 shows the Ru distribution in two analysed regions of the Ru/SiO2 catalyst. Clearly, the Ru distribution over the SiO2 support is mostly homogeneous, although some Ru clusters of ~1 µm have been observed on the SiO2 surface. Additionally, trace amounts of Al and Na appear to be present, possibly due to impurities in the SiO2 support (See Table S1 and Figures S15-S16). Finally, we estimate a Ru loading of ~ 5 wt% (See Figure S17).

In-situ IR

SiO2 support

In order to disentangle the effects of the individual reactants of DRM (i.e. CO2 and CH4), we performed experiments with CO2/Ar and CH4/Ar plasma in addition to experiments with a mixture of the reactants. The results of these experiments are presented in section S5 of the supporting information (Figures S18-S24).

The FTIR difference spectra (3800-2400 cm -1 ), recorded during DRM in plasma on the SiO2 support are shown in Figure 3.

A characteristic IR band of gas phase CH4 can be observed at 3015 cm -1 and is ascribed to the ν3 mode of CH4. Moreover, the presence of gas phase CO2 is evidenced by the combination bands at 3727 and 3703 cm -1 (ν1 + ν3) and those at 3627 and 3603 cm -1 (2 ν2 + ν3). Putting the system under vacuum reveals IR bands corresponding to (strongly) adsorbed species on the SiO2 support in the 3000-2800 cm -1 region (see inset in Figure 3). Considering the wavenumbers of these bands, they may indicate the formation of CHx species onto the SiO2 support. This assignment is further supported by the existence of similar bands in a CH4/Ar system under plasma (see Figure S23 and S24), and the lack thereof in a CO2/Ar system under plasma (see Figure S20 and S21). Interestingly, as gas phase adsorption of CH4 onto SiO2 occurs through weak physisorption and is fully reversible upon evacuation 30 , the adsorption behavior of CH4 appears to be strongly modified under plasma. Indeed, it seems that new chemical bonds could be formed between CH4 and the thermally inert SiO2 support, likely due to the formation of CHx Please do not adjust margins Please do not adjust margins radicals in plasma. Indeed, even upon exposure of the sample to temperatures up to 145 °C, these species remain adsorbed on the surface (see Figure S26). This already points to a first difference between plasma-catalytic and thermo-catalytic DRM, which may affect the catalytic surface reactions.

Figure 4 shows the FTIR difference spectra at lower wavenumbers (2230-1300 cm -1 ) recorded during DRM in plasma on the SiO2 support. Apart from bands corresponding to gas phase CO2 (2357 and 2336 cm -1 , ν3, not shown in Figure 4) and CO (2165 and 2120 cm -1 , ν1), another, strong IR band appears at 1722 cm -1 . For the assignment of this band, it is important to note that (i) the band shows a similar behaviour upon increasing the temperature as the bands representing physisorbed formaldehyde (see Figures S26 andS40), and (ii) the band is not (or to much smaller extents) formed upon exposure of the sample to CO2/Ar or CH4/Ar plasma alone (see SI, section S5). On one hand, the band could indicate the formation of chemisorbed formaldehyde species (Si-O-CH=O) on the SiO2 support. A possible pathway towards these species on SiO2 could be the direct adsorption of CHO, which is formed during DRM in DBD plasma, even without the presence of a catalyst, hence formed by plasma-chemical reactions 3 . However, the rather low temperature at which this band decreases, similar to the desorption temperature of formaldehyde (see (i) above), may contradict this assignment. In this regard, assignment of the 1722 cm -1 band to physisorbed formic acid may be more realistic; indeed, a band at 1720 cm -1 has been observed for the physisorption of formic acid on SiO2, and the simultaneous development of bands in the 1500-1400 cm -1 region is consistent with this assignment 31,32 . Moreover, formic acid was previously observed as a reaction product of plasma-catalytic DRM 33 , albeit in trace amounts, and the band at 1760 cm -1 , which disappears under vacuum, can be assigned to gas phase formic acid.

Furthermore, during the reaction, a band at 1631 cm -1 develops, which remains under vacuum (see Figure S25), showing the formation of adsorbed water in the plasma. Bands at 1684 and 1381 cm -1 , which persist under vacuum, may evidence the presence of bicarbonate like species, whereas bands at 1596 and 1466 cm -1 remain under vacuum and may indicate the presence of monodentate carbonates on SiO2, and the appearance of bands at 1668 and 1450 cm -1 upon switching off the plasma may evidence the formation of bidentate carbonates on SiO2. While the latter observation may suggest that bidentate carbonates are exclusively formed in the afterglow of the plasma, alternatively, it may indicate that their destruction is discontinued upon extinguishing the plasma. It is worth noting that neither monodentate, nor bidentate carbonates are observed on SiO2 during thermo-catalytic DRM 34 , and thus, their formation must also be attributed to plasma effects, such as Eley-Rideal reactions involving e.g. CO.

For an alternative representation of the evolution of the FTIR difference spectra over time, we refer the reader to waterfall plots in the supporting information (see Figure S27).

3 wt% Ru/SiO2

A H2/Ar plasma is first applied for in-situ reduction of Ru on the catalyst wafer. Two minutes after plasma ignition, the formation of H2O is observed, indicating the reduction of RuO to metallic Ru by the H2 plasma (see Figure S28). Reduction is Please do not adjust margins Please do not adjust margins completed after 6 minutes. Afterwards, negative IR bands at 3800-3500 and ~1625 cm -1 are recorded, indicating that the initially adsorbed water at the catalyst surface is also desorbed. Again, in order to disentangle the effects of the individual reactants of DRM, we performed experiments with CO2/Ar and CH4/Ar plasma in addition to experiments with a mixture of the reactants, the results of which are presented in Figures S29-S35. While the spectra in the 3800-2000 cm -1 range recorded during DRM on Ru/SiO2 are similar to the corresponding spectra recorded on SiO2 alone (see Figures S36 andS37), the 2000-1300 cm -1 region shows a remarkable difference (see Figures 6 andS36). More specifically, the band at 1722 cm -1 that was previously observed on the SiO2 support and assigned to adsorbed formic acid exhibits a much lower intensity, and the amount of adsorbed water on the catalyst surface is much smaller as well, as evidenced by the vast decrease in intensity of the 1631 cm -1 band on Ru/SiO2 compared to SiO2. Indeed, on Ru/SiO2, both bands exhibit about half of the intensity they show on SiO2. Interestingly, when looking at the development of both bands over time on SiO2 alone (see Figure 5), water is immediately formed on the surface upon switching the plasma on, whereas formic acid is formed with some delay. This could indicate that the water on the catalyst surface serves as a source of OH groups for the formation of formic acid from, for example, CHO, which is a known reactive species in DRM in plasma 3 . We observe a similar delay on Ru/SiO2, although the intensity of the bands is much lower. Note that the 1760 cm -1 band, illustrative of gas phase formic acid, initially shows a low intensity, which increases as the reaction progresses, potentially showing desorption of physisorbed formic acid from the SiO2 surface. We hypothesize that the difference in adsorbed water between both samples can be attributed to the surface temperature. Indeed, the presence of water vapour in the plasma with Ru/SiO2 catalyst is clearly visible in Figure 6, whereas the same bands are not observed in the plasma with SiO2. This is a good indication that while on SiO2, the plasma remains at a low temperature, the temperature of the plasma on Ru/SiO2 is elevated above 100 °C, which is indeed confirmed in section 3.2.3. At these elevated temperatures, the amount of surface adsorbed water is low, so that the formation of formic acid is limited. It should be mentioned that temperature-induced desorption of formic acid cannot be excluded at these temperatures (see Figure S26), which would contribute to a lower intensity of the 1722 cm -1 band. While it is not yet clear to what extent formic acid adsorbed on SiO2 exactly contributes to the plasma-catalytic reaction, its Please do not adjust margins Please do not adjust margins formation is remarkable, and has, to the best of our knowledge, not yet been considered in plasma-catalytic DRM.

Clearly, this type of interaction between the plasma species and the catalyst differs from the classical interpretation of catalysis in a thermal system. Indeed, in the latter, catalytic activity is determined by the available pathways in which reactants can be converted to reaction products on the catalytic surface, with the Langmuir-Hinshelwood mechanism being dominant. While these surface reactions can also play a role in plasma catalysis, clearly, the role of a plasma catalyst goes beyond that. In the present work, we have illustrated that the plasma-induced heating of the catalyst surface is dependent on the catalyst material, which in its turn affects the plasma-induced adsorption on the catalyst surface.

Estimation of catalyst temperature under DBD plasma

In order to better understand whether the formation of surface species is due to plasma-induced surface heating or other plasma effects, such as plasma-generated radicals, we need to know the catalyst surface temperature, determined with the method explained in section 2.2.3 above. Figure 7 shows the evolution of the wavenumber of the BaSO4 band 𝝂 ̅ 𝒎𝒂𝒙 (see Equation 1 in section 2.2.3), used to probe the temperature of the pure BaSO4 sample, along with the corresponding temperature, as a function of time in a H2/Ar DBD plasma. We select this as a simple system with limited chemistry to first illustrate our novel technique in a comprehensible way. While it clearly shows that switching on the plasma leads to a roughly continuous temperature increase up to ~90 °C, with a stabilization after ~15-20 min, there is no clear relation between applied voltage and observed temperature. Moreover, the applied voltage does not greatly influence the heating rate of the sample, and upon plasma extinction, the sample temperature again drops to its initial temperature at rates similar to those observed for the temperature increase.

The position of the maximum of the BaSO4 band used to probe the temperature of the Ru/SiO2/BaSO4 system as a function of time in a CO2/CH4/Ar plasma, along with the corresponding temperature, is shown in Figure 8. In this case, while the data Please do not adjust margins Please do not adjust margins are noisier than for pure BaSO4 due to dilution, the trends are still clear and we observed a temperature increase in the range of 70-120 °C, with peaks up to 150 °C. Hence, the maximum temperature of the Ru/SiO2 wafer during the plasma-catalytic DRM can be estimated to be lower than 150 °C. We also performed experiments in a thermal in-situ operando IR "sandwich cell", as described in section 2.2.4 and in section S6 of the supporting information (see Figure S38). Our spectra (see Figure S39) reveal that this temperature increase is insufficient for the formation of surface species on the catalyst sample, as well as for the conversion of CO2 and CH4 (i.e. our samples exhibit no activity at this temperature), which was not observed in our thermal in-situ operando cell, even at 450 °C. Hence, the formation of the species at the catalyst surface and the conversion of CO2 and CH4 must be attributed to effects other than plasma-induced surface heating, such as the adsorption of plasma-generated radicals or molecules, or the occurrence of Eley-Rideal reactions. Note that the field of plasma catalysis is greatly hindered by a lack of understanding of the actual role of plasma-induced surface heating in possible synergism [START_REF] Lovell | [END_REF] , and therefore, these observations provide a valuable contribution towards a better understanding of plasma-catalytic synergism by the decoupling of thermal and plasma effects.

Packed bed DBD experiments

Table 1 shows the CO2 and CH4 conversion, as well as the product yields and selectivities, obtained in the packed bed DBD reactor, for both the SiO2 support and the 3 wt% Ru/SiO2 catalyst. Besides H2 and CO, which are the major products of DRM (i.e. syngas), C2H4 is produced in non-negligible amounts. Note that the yield and selectivity are H-based for H2, and Cbased for CO and C2H4. The CO2 and CH4 conversions are lower for the Ru/SiO2 catalyst than for the bare SiO2 support, as well as the selectivity towards syngas (both to H2 and CO). Moreover, the C2H4 selectivity is significantly higher on Ru/SiO2, i.e., 3.1% vs. 1.2% for the bare SiO2 support. We hypothesize that these results can indeed (at least partly) be understood with the findings in our in-situ cell.

Indeed, the conversions of CO2 and CH4 are to a great extent determined by backreactions of plasma species, i.e. reactions in which the initial reactants (CO2 and CH4) are formed again. These reactions involve CHO and OH 3 , the concentrations of which are reduced by the formation of formic acid on the SiO2 surface, which could reduce the contribution of these backreactions, and thus increase the CO2 and CH4 conversion. Note that during plasma operation, condensate is visibly formed at the reactor outlet, so that an accurate detection of liquid reaction products like CH3OH or formic acid during steady-state plasma operation was not possible.

Conclusions

In this work, we present an in-situ study (i.e. inside the plasma and during operation) of the infrared-active species formed on the surface of (Ru-loaded) silica during plasma-catalytic DRM using a novel in-situ transmission IR DBD cell. Moreover, to link the observations of the in-situ experiments to catalytic activity, we also tested the catalysts in a packed bed DBD reactor, and we measured the CO2 and CH4 conversion and product yields and selectivities. We found that in the plasma-catalytic DRM on SiO2, adsorbed water is immediately formed, while adsorbed formic acid is formed with a delay, possibly indicating that water provides OH for the formation of formic acid. We hypothesize that, as Ru/SiO2 is subject to greater plasma-induced surface heating, the amount of adsorbed water is lower, leading to less formation of formic acid on this catalyst. In addition, we present a novel technique to estimate the temperature of the catalyst inside our plasma-catalytic system in-situ, and found a maximum catalyst temperature increase in the range of 70-120 °C, with peaks up to 150 °C for Ru/SiO2. This temperature increase is insufficient for the formation of surface-adsorbed species on the catalyst, as revealed by our experiments in a thermal in-situ operando IR "sandwich" cell. Hence, the formation of these species cannot be attributed to plasma-induced heating, but must be due to other plasma effects, such as the adsorption of plasma-generated radicals, molecules or the occurrence of Eley-Rideal reactions. This novel Please do not adjust margins Please do not adjust margins technique thus allows to decouple thermal and plasma effects, and therefore, it contributes greatly towards a better understanding of plasma-catalytic synergism.

In general, our study provides valuable insights in the occurrence of surface-adsorbed species that may play a role in plasma-catalytic DRM. Nevertheless, more work is needed to obtain a comprehensive picture of the entire mechanism. A better understanding of the surface reactions could be obtained by for example Steady-State Isotopic Transient Kinetic Analysis (SSITKA) and step-scan FTIR analysis. Additionally, combining the obtained experimental results with plasma-chemical and microkinetic surface chemistry models, in which the fundamental reaction steps on the catalyst surface and in the plasma are modelled, could lead to a better understanding of the observed experimental results, and thus provide an overall picture of the underlying interactions between the plasma and a catalyst in plasma-catalytic DRM.

Figure 1 :

 1 Figure 1: Schematic representation of the novel in-situ DBD IR cell

Figure 2 :

 2 Figure 2: EDS elemental mapping of two spots on the Ru/SiO2 catalyst, showing a homogeneous Ru loading on the SiO2 support with additional presence of larger (~1 µm) Ru clusters

Figure 3 :

 3 Figure 3: FTIR difference spectra as a function of time of the SiO2 support during DRM in plasma in the 3800-2400 cm -1 region with detail of the 3100-2800 cm -1 region. Spectra are shown with a time interval of 36 s up to steady state.

Figure 2 :

 2 Figure 2: FTIR difference spectra as a function of time (36 s between spectra up to steady state) of the SiO2 support during DRM in plasma in the 2230-1300 cm -1 region with detail of the 1800-1550 cm -1 and 1500-1300 cm -1 regions. Spectra are shown with a time interval of 36 s up to steady state.

Figure 5 :

 5 Figure 5: Evolution of the intensity of bands observed in the FTIR spectra over time during plasma-catalytic DRM over SiO2 at 1722 (blue), 1632 (purple) and 1760 cm -1 (green)

Figure 6 :

 6 Figure 6: FTIR difference spectra as a function of time (up to steady state) of the 3 wt% Ru/SiO2 catalyst during DRM in plasma in the 2000-1300 cm -1 region with detail of the 1250-1950 cm -1 region. Spectra are shown with a time interval of 36 s up to steady state.

Figure 7 :Figure 8 :

 78 Figure 7: Wavenumber of the BaSO4 band used to probe the BaSO4 temperature in a H2/Ar DBD plasma as a function of time. The intensity of the background color is indicative of the applied voltage, going from grey (0 kV, plasma off) to dark red (50 kV)

Table 1 :

 1 CO2 and CH4 conversion, and H2, CO and C2H4 yield and selectivity obtained for the plasma-catalytic DRM over SiO2 and 3 wt% Ru/SiO2 in a packed bed DBD reactor.The H2 yield and selectivity are H-based, while the CO and C2H4 yield and selectivity are C-based (see formulas in SI, section S2). Error bars for C2H4 yield and selectivity are not available, since C2H4 production was only monitored with FTIR during one experiment. This journal is © The Royal Society of Chemistry 20xx

			SiO2 extrudates	3 wt% Ru/SiO2 extrudates
	CO2 conversion (%)	17 ± 2	10 ± 2
	CH4 conversion (%)	25.4 ± 0.5	19 ± 2
	H2	Yield (%) Selectivity (%)	22.3 ± 0.4 87.9 ± 0.1	11.6 ± 0.9 61 ± 4
	CO	Yield (%) Selectivity (%)	17.1 ± 0.6 81 ± 3	7.2 ± 0.9 50 ± 6
	C2H4 (ethylene)	Yield (%) Selectivity (%)	0.46 1.2	0.65 3.1

This journal is © The Royal Society of Chemistry 20xx J. Name., 2013, 00, 1-3 | 3

This journal is © The Royal Society of Chemistry 20xx J. Name., 2013, 00, 1-3 | 5

This journal is © The Royal Society of Chemistry 20xx J. Name., 2013, 00, 1-3 | 7

Acknowledgements

This research was supported by the European Research Council (ERC) under the European Union's Horizon 2020 Research and Innovation programme (grant agreement No 810182 -SCOPE ERC Synergy project) and the European Union's Horizon 2020 Research and Innovation programme under Marie Sklodowska-Curie grant agreement No 813393 (PIONEER).

Conflicts of interest

There are no conflicts to declare.


Fanny Bocquentin

Lior Weissbrod

Hamoudi

Hamoudi Khalaily

Eynan


Fanny Bocquentin

Lior Weissbrod

Hamoudi

published or not. The documents may come


V Bonnaillie-Noël

B Helffer

Numerical analysis of nodal sets for eigenvalues of Aharonov-Bohm Hamiltonians on the square and application to minimal partitions

Numerical analysis of nodal sets for eigenvalues of Aharonov-Bohm Hamiltonians on the square and application to minimal partitions

Virginie Bonnaillie-Noël, Bernard Helffer

Introduction

The recent papers [START_REF] Helffer | Nodal domains and spectral minimal partitions[END_REF] and [START_REF] Bonnaillie-Noël | Aharonov-bohm hamiltonians, isospectrality and minimal partitions[END_REF] have shown the strong links between the analysis of nodal sets of some eigenfunctions of Aharonov-Bohm Hamiltonians and the question of minimal partitions. In connection with recent papers of B. Noris and S. Terracini [START_REF] Noris | Nodal sets of magnetic Schrödinger operators of Aharonov-Bohm type and energy minimizing partitions[END_REF][START_REF] Noris | [END_REF], we analyze the dependence of the nodal picture of the eigenvalues of the Aharonov-Bohm Hamiltonian on the square as a function of the pole and propose new candidates for minimal k-partitions of a specific topological type for the square.

Aharonov-Bohm Hamiltonian

Let us recall some definitions and results about the Aharonov-Bohm Hamiltonian (for short ABX-Hamiltonian) with a singularity at X introduced in [START_REF] Bonnaillie-Noël | Aharonov-bohm hamiltonians, isospectrality and minimal partitions[END_REF][START_REF] Helffer | Nodal sets for groundstates of Schrödinger operators with zero magnetic field in nonsimply connected domains[END_REF] and motivated by the work of Berger-Rubinstein [START_REF] Berger | On the zero set of the wave function in superconductivity[END_REF]. We denote by X = (x 0 , y 0 ) the coordinates of the pole and consider the magnetic potential with flux Φ = 1/2 :

A X (x, y) = (A X 1 (x, y), A X 2 (x, y)) = 1 2 - y -y 0 r 2 ,
x -x 0 r 2 .

(1.1)

Minimal partitions

where λ(D i ) is the groundstate energy of the Dirichlet Laplacian on D i . We denote the infimum of Λ over all k-partitions of Ω by

L k (Ω) = inf D∈O k Λ(D) . (1.4)
We look for minimal k-partitions, that are partitions D = (D 1 , . . . , D k ), such that

L k (Ω) = Λ(D) .
We recall that these minimal k-partitions, whose existence was proven in [START_REF] Conti | An optimal partition problem related to nonlinear eigenvalues[END_REF][START_REF] Conti | A variational problem for the spatial segregation of reaction-diffusion systems[END_REF][START_REF] Conti | On a class of optimal partition problems related to the Fučík spectrum and to the monotonicity formulae[END_REF], share with nodal domains many properties of regularity, except that the number of half-lines meeting (with equal angle) at critical points of their boundary set can be odd [START_REF] Helffer | Nodal domains and spectral minimal partitions[END_REF]. Here by critical points we mean points which are at the intersection of at least three ∂D i 's.

Main goals

In [START_REF] Bonnaillie-Noël | Numerical simulations for nodal domains and spectral minimal partitions[END_REF], we have combined results of [START_REF] Helffer | Nodal domains and spectral minimal partitions[END_REF] and [START_REF] Helffer | On spectral minimal partitions : the case of the sphere[END_REF] with efficient numerical computations to exhibit some candidates to be minimal 3-partitions for the square, the disk, . . . . This approach was based on an assumption of symmetry permitting the reduction to a more standard spectral analysis and consequently only gave symmetric candidates. Using two different symmetries of the square, we get the surprise of finding two candidates D 1 and D 2 with Λ(D 1 ) = Λ(D 2 ) (≃ 66.581) and give numerical evidence that the critical point for these partitions is at the center of the square. These candidates are represented in Figure 1. This leads naturally to questions of isospectrality which were solved using the Figure 1: Two candidates with different symmetries for the minimal 3-partition of the square.

ABX-Hamiltonians in [START_REF] Bonnaillie-Noël | Aharonov-bohm hamiltonians, isospectrality and minimal partitions[END_REF] with X = C, the center of the square. It was also sketched there an alternative proof using the construction of a double covering of the punctured square; this approach seems more adequate for performing the numerical analysis because of possible numerical difficulties in treating the singularities due to the Aharonov-Bohm potential and the fact that the eigenfunctions would have been complex-valued. This kind of arguments also appears in a similar context in [START_REF] Levitin | Isospectral domains with mixed boundary conditions[END_REF] and [START_REF] Jakobson | Spectral problems with mixed Dirichlet-Neumann boundary conditions: isospectrality and beyond[END_REF]. We will push this numerical analysis with several goals:

• Illustrate the fact that these two symmetric candidates for minimal 3-partitions on the square belong actually to a continuous family of non necessarily symmetric candidates (see Figure 2 and Figure 9). • Check, by moving the pole X of the ABX-Hamiltonian, the conjecture that the singular point of the minimal 3-partition is at the center.

• Understand and illustrate the mechanism of deformation of the nodal set, and hence extend or guess, in connection with recent papers of B. Noris and S. Terracini [START_REF] Noris | Nodal sets of magnetic Schrödinger operators of Aharonov-Bohm type and energy minimizing partitions[END_REF][START_REF] Noris | [END_REF], some of the properties described in Berger-Rubinstein [START_REF] Berger | On the zero set of the wave function in superconductivity[END_REF] and Helffer-M.-and-T. Hoffmann-Ostenhof-Owen [START_REF] Helffer | Nodal sets for groundstates of Schrödinger operators with zero magnetic field in nonsimply connected domains[END_REF] for the ground state energy (see also [START_REF] Alziary | Eigenfunctions and Hardy inequalities for a magnetic Schrödinger operator in R 2[END_REF]).

Organization of the paper

In Section 2, we explain how we implement the computations on the double covering of the punctured square. In Section 3, we apply Courant's theory for comparing the Aharonov-Bohm eigenvalues associated to this puncturing to the eigenvalues of the square.

Section 4 analyzes the dependence of the eigenfunctions of the Dirichlet Laplacian on the puncturing point appearing in the construction of the double covering.

Sections 5, 6 are more specifically devoted to the analysis of the behavior of the nodal sets and eigenvalues when the poles belong respectively to the perpendicular bisector y = 1 2 , the diagonal y = x, which correspond to cases when some symmetry of the square is respected in the puncturing. We treat also the case of the axis y = 1 4 + x 2 as an example of a generic situation.

Section 7 describes the possible applications of our analysis of nodal sets to the research of minimal partitions with a given topological type.

Numerical implementation

The ABX-Hamiltonian has a singularity at the pole X and the eigenfunctions are complexvalued. For these reasons, we prefer to deal with the Dirichlet Laplacian on the doublecovering ΩR X whose eigenfunctions are real-valued. Some of these eigenfunctions, which will be described below, are directly related with the K X -real eigenfunctions of the ABX-Hamiltonian.

The numerical results were realized by using the Finite Element Library Mélina (see [START_REF] Martin | Mélina, bibliothèque de calculs éléments finis[END_REF]). The computations consist only of the determination of the eigenfunctions of a Dirichlet Laplacian. Nevertheless, since we are interested in the nodal lines of these eigenfunctions, computations have to be quite accurate and we choose the package Mélina permitting the implementation of high order elements. In our computations, we use for the approximation polynomials of degree 6 in each triangle of the discretization.

The main point of the numerical part consists in meshing the double covering ΩR X of the punctured domain Ω \ {X}. To do this, we use the two-dimensional mesh generator Triangle (see [START_REF] Shewchuk | Triangle: A two-dimensional quality mesh generator and Delaunay triangulator[END_REF]). Let us explain in more details how we proceed.

Let Ω be the square [0 , 1] × [0 , 1] and X be a point in [0 , 1 2 ] × [0 , 1 2 ]. We start with meshing the Ω so that (see Fig. 3):

• the segment joining (0, 0) to the pole X does not go through any element of the mesh,

• the segment [(0, 0); X] is the union of edges of an even number of triangles,

• the pole X is the vertex of some triangles. This first mesh is essentially done for the first sheet and we repeat this mesh for the second sheet. To obtain a mesh of the double covering ΩR X , we have to exchange the vertex along the segment [(0, 0); X] between the first and second sheet: we choose an orientation and we decide, for example to exchange the vertices between the first and second sheet of the triangles which are at the left of the segment. Then we remove the point X of the second sheet by equaling it to the vertex X of the first sheet.

In this paper X is an element of the lattice of the first quarter of the square [0 , 1 2 ] × [0 , 1 2 ] with step size 1 100 . Theoretically, the eigenvalues and eigenfunctions depend only on the pole and are independent of the cut chosen for our construction. The introduction of the segment [(0, 0); X] is only a technical point and we have verified that the numerical computations of the eigenfunctions and eigenvalues are (with a rather good accuracy ∼ 10 -3 ) independent of the choice of the line joining the pole to the boundary, that is the line between the first and second sheet. This is illustrated in Table 1 where we give the first fourteen eigenvalues of the Dirichlet Laplacian on the double covering of the square ΩR C realized with two different cutting lines: the half-diagonal and the half-perpendicular bisector. Many computations are available on the web page (see [START_REF] Bonnaillie-Noël | Computations for nodal domains and spectral minimal partitions[END_REF]):

http://w3.bretagne.ens-cachan.fr/math/simulations/MinimalPartitions The eigenvalues given all along this paper provide from our computations with an approximation at 5 • 10 -4 of the computed eigenvalues.

We observe that the eigenvalues of the square are numerically computed with a good accuracy whereas it is more difficult to be accurate for the ABX-eigenvalues, particularly in the case of multiplicity 2. In the following, we use a P 6 approximation with at least 6000 elements.

To detect the nodal lines, we use a program realized by G. Vial. The idea is that it is very easy to compute the zero set of linear functions. In our case, we deal with a function which is piecewise P k and given by a finite element method. We know the values of this function at some points. As soon as we have these values, we can replace this function by a new function which is piecewise linear. For this, we introduce some new points by an interpolation method. Then we detect the zero set of this new function. 3 A few theoretical comparison theorems

Notation

We denote by Ω the square [0 , 1] × [0 , 1] and by C = (1 2 , 1 2 ) the center of the square. We compute the eigenfunctions of the Laplacian on the double covering ΩR X of ΩX = Ω \ {X}. By a symmetry argument, it is enough to consider X = (x 0 , y 0 ) in the quarter square [0 , 1 2 ] × [0 , 1 2 ].

There are two ways of labelling the eigenvalues.

We can label it in the standard way and this is the way we get the eigenvalues in our numerical program. With this in mind, we denote by λ k ( ΩR X ) the k-th eigenvalue of the Dirichlet Laplacian on ΩR X .

We can also take account of the symmetry relative to the deck map D R X associating to a given point in the covering the distinct point with same projection by the covering map π R X of ΩR X onto ΩX . This splits the spectrum into two independent spectra relative to two orthogonal spaces in the L 2 space of our covering square. The eigenvalues correspond

• either to eigenfunctions lifted from the eigenfunctions (of the Dirichlet Laplacian) 1 on the square by the covering map (sometimes called D R X -symmetric because they are symmetric with respect to the deck map),

• or to eigenfunctions which are D R X -antisymmetric with respect to the deck map.

We also call the second ones ABX-eigenvalues because they can be seen as eigenvalues of an AB-Hamiltonian with a pole X creating a (renormalized) flux equal to 1 2 . We shortly write ABX-Hamiltonian, if we want to make explicit the reference to the pole. We denote by

λ ABX k = λ AB k ( ΩX ) = λ AS k ( ΩR X )
the k-th eigenvalue of the AB-Hamiltonian with pole at X. We denote by λ S k ( ΩR X ) = λ k (Ω) the k-th eigenvalue corresponding to the D R Xsymmetric eigenvalues.

We will also observe that, when puncturing on the perpendicular bisectors or the diagonals of the square, additional symmetries will be important for explaining effective crossings of eigenvalues.

Eigenvalues of the square

The eigenvalues of the square are well known and given by the double sequence π 2 (m 2 +n 2 ) with m ∈ N \ {0}, n ∈ N \ {0}, with corresponding basis of eigenfunctions given by Ω ∋ (x, y) → φ mn (x, y) := sin(mπx) sin(nπy) .

Labelling the eigenvalues in increasing order leads to the sequence denoted by λ k (Ω) , k ∈ N * . Table 2 gives the first thirteen eigenvalues and the nodal set of the associated eigenfunctions belonging to the above basis. The second, fifth, seventh, ninth and twelfth eigenvalues are double and consequently, it is also natural to look at the nodal sets of linear combinations in order to determine all the possible nodal configurations associated with this eigenvalue.

Eigenvalues of the square (m, n)-labelling Nodal sets for φ mn We notice that the D R X -symmetric spectrum of the Dirichlet Laplacian on the double covering ΩR X is the spectrum of the square and is independent of the pole. This is a consequence of the fact that the spectra of the Dirichlet Laplacian in Ω and ΩX are the same, the puncturing point being of capacity 0. So it is more the ABX-spectrum which is of interest because depending on the position of the pole. Nevertheless, the standard labelling of all the eigenvalues on ΩR X can play a role when applying Courant's theorem. Of course we have λ 1 ( ΩR X ) = λ 1 (Ω).

λ 1 (Ω) = 2π 2 ≃ 19.739 (1, 1) λ 2 (Ω) = λ 3 (Ω) = 5π 2 ≃ 49.348 (2, 1) , (1, 2) λ 4 (Ω) = 8π 2 ≃ 78.957 (2, 2) λ 5 (Ω) = λ 6 (Ω) = 10π 2 ≃ 98.696 (3, 1) , (1, 3) λ 7 (Ω) = λ 8 (Ω) = 13π 2 ≃ 128.305 (3, 2) , (2, 3) λ 9 (Ω) = λ 10 (Ω) = 17π

Theoretical estimates of the eigenvalues

This subsection is concerned with the comparison between the spectrum on the square, the spectrum on the double covering ΩR X and the ABX-spectrum. We propose some equalities and upper-bounds between the eigenvalues essentially based on the minimax principle and on the Courant's nodal theorem which is recalled now. Theorem 3.1 Let k ≥ 1, λ k (Ω) be the k-th eigenvalue for the Dirichlet Laplacian on Ω. Then the number of nodal domains of any associated eigenfunction is bounded from above by k.

We would also apply this theorem for the K X -real eigenfunctions of the ABX-Hamiltonian on ΩX . Equivalently, this corresponds to a Courant nodal theorem for the D R X -antisymmetric eigenfunctions on ΩR X , already discussed in [START_REF] Bonnaillie-Noël | Aharonov-bohm hamiltonians, isospectrality and minimal partitions[END_REF]. Combination of the Courant nodal theorem and Max-Min principle for the ABX-Hamiltonian leads to the following proposition.

Proposition 3.2 Let X ∈ [0, 1 2 ] × [0, 1 2 ], then λ 1 (Ω) = λ 1 ( ΩR X ) , λ ABX 1 = λ 2 ( ΩR X ) , λ ABX 2 = λ 3 ( ΩR X ) , (3.1)
and

λ ABX 2 < λ 2 (Ω) = λ ℓ 2 ( ΩR X ) with ℓ 2 ≥ 4 (with multiplicity at least 2) , (3.2)
λ ABX 3 ≤ λ 4 (Ω) = λ ℓ 4 ( ΩR X ) with ℓ 4 ≥ 7 , (3.3)
λ 5 (Ω) = λ ℓ 5 ( ΩR X ) with ℓ 5 ≥ 8, (3.4)
λ ABX 5 ≤ λ 7 (Ω) = λ ℓ 7 ( ΩR X ) with ℓ 7 ≥ 12 , (3.5)
λ 9 (Ω) = λ ℓ 9 ( ΩR X ) with ℓ 9 ≥ 14 , (3.6)
λ ABX 8 ≤ λ 11 (Ω) = λ ℓ 11 ( ΩR X ) with ℓ 11 ≥ 19 . (3.7)
If X belongs to the perpendicular bisectors of the square, we have more accurately:

ℓ 4 = ℓ 4 (X) ≥ 8 , (3.8)
λ ABX 6 ≤ λ 7 (Ω) . (3.9) Remark 3.3
The multiplicity of λ ℓ (Ω) as eigenvalue of the Dirichlet Laplacian on ΩR X is of course larger or equal to its multiplicity on Ω. This could permit to improve some inequalities above when we can find for a given pole an eigenfunction u of the Dirichlet Laplacian on Ω vanishing at the pole. The number of nodal domains of the lifted symmetric function on the covering is then two times the number µ(u) of nodal domains of u instead of 2µ(u) -1.

To find this eigenfunction could be easier when the eigenspace is of higher dimension. This appears for example for λ 2 (Ω).

Proof :

Let us first prove (3.2). We first observe that, for any X ∈ [0 , 1 2 ]×[0 , 1 2 ], there exists an eigenfunction u X of the Dirichlet Laplacian associated with λ 2 (Ω) and u X (X) = 0. We have just to look for u X of the form u

X = αφ 1,2 +βφ 2,1 with (α, β) = (0, 0) satisfying αφ 1,2 (X) + βφ 2,1 (X) = 0 . By lifting on ΩR X , this gives a D R X -symmetric eigenfunction u X • π R
X on ΩR X with four nodal domains and associated with λ 2 (Ω). Hence by Courant's theorem, λ 2 (Ω) = λ ℓ 2 ( ΩR X ) with ℓ 2 ≥ 4. To establish the first inequality in (3.2), we consider the functions max(u X , 0) and max(-u X , 0), which span a two-dimensional space in the form domain of the ABX-Hamiltonian for which the energy is less than λ 2 (Ω). We can conclude by the minimax principle. It is easy to see that the inequality is strict. Hence at this stage we also get (3.1).

Let us now prove (3.3). Using the three functions obtained by restriction to one nodal domain of the function φ 2,2 (then extended by 0) which does not contain X, we obtain a 3-dimensional space of functions in the form domain of the ABX-Hamiltonian for which the energy is less than λ 4 (Ω) (or a 4-dimension space if X is on the perpendicular bisector to the side of the square because we can in this case get a 4-dimensional space, see Remark 3.3). We then conclude by the minimax principle. The relation with λ ℓ 4 ( ΩR X ) is an application of the Courant's nodal theorem using the function

φ 2,2 • π R X . Relation (3.4) is a consequence of (3.3).
For (3.5), we can this time use the function φ 3,2 which has at least 5 nodal domains not containing X. For X on the perpendicular bisector, we get (3.9).

The function φ 4,1 has at least 3 nodal domains not containing X. Using (3.5) and the multiplicity of λ 9 (Ω), we obtain (3.6).

Using the function φ 3,3 which has at least 8 nodal domains not containing X, we deduce (3.7).

The lower bound for ℓ 7 , ℓ 9 and ℓ 11 results immediately of the upper bounds of λ ABX The nodal set of the second K X -real eigenfunction u ABX 2 consists of one line joining the pole X to the boundary.

Proof :

We know from [START_REF] Helffer | Nodal sets for groundstates of Schrödinger operators with zero magnetic field in nonsimply connected domains[END_REF] that a piecewise regular line in the nodal set should join the pole X to the boundary. Another piece in the nodal set should necessary create an additional nodal domain which will lead to

λ 2 ≤ λ ABX 2 in contradiction to (3.2). ✷ Let X = (x, y) ∈ [0 , 1 2 ] × [0 , 1 2 ], we denote X = ( x 2 , y 2
). Using a dilation argument, we can establish in complement of (3.3), that

λ ABX 4 ≤ 4λ ABX 1 .
The property that λ ABX 1 ≥ λ 1 (Ω) is of course a particular case of the diamagnetic inequality. We will observe on the pictures that the situation is much more complicate for the excited states. Except in the case of additional symmetries where some monotonicity will be proven, we have no theoretical results.

Remark 3.5

We will see in Figures 18,22 and 26, that the upper-bounds (3.2), (3.3) and (3.5) in Proposition 3.2 for the ABX-eigenvalues are optimal in the sense that we can find a pole such that the upper-bound is false with a smaller eigenvalue of the square.

4 Behavior of the eigenvalues on the double covering of the punctured square, when moving the pole

In this section, we start to discuss the influence of the location of the puncturing point X (or pole) on the topological structure of the nodal set of the first eigenfunctions.

Behavior when the pole tends to the boundary

It has been announced by B. Noris and S. Terracini [START_REF] Noris | Nodal sets of magnetic Schrödinger operators of Aharonov-Bohm type and energy minimizing partitions[END_REF][START_REF] Noris | [END_REF] that the k-th ABX-eigenvalue of the punctured square tends to the k-th eigenvalue of the Dirichlet Laplacian on the square as the pole tends to the boundary (see also [START_REF] Hillairet | The eigenvalues of the Laplacian on domains with small slits[END_REF] for connected results). They also establish in [START_REF] Noris | Nodal sets of magnetic Schrödinger operators of Aharonov-Bohm type and energy minimizing partitions[END_REF] the continuity with respect to a pole and prove that

X → λ ABX k is of class C 1 if λ ABX k is simple.
Because after a translation by X, we get a fixed operator with moving regular boundary and fixed pole at (0, 0), the regularity is actually easy. These results are illustrated in Figures 4-14 which represent the eigenvalues λ k ( ΩR X ), k = 2, 3, 6, 7, 8, 9, 11, 12, 13, according to the location of the pole X ∈ P and Table 3

Eigenvalues 2 and 3

We observe numerically, see Figure 4, that for X ∈ P, the function X → λ 2 ( ΩR X ) has a global maximum, denoted by λ max 2 for X = C and is minimal when X belongs to the boundary x = 0 or y = 0. This minimum equals λ 2 (Ω). Moreover we do not observe other critical points in P. Looking at Figure 5, we observe numerically that the function X → λ 3 ( ΩR X ) behaves conversely : it has a global minimum, denoted by λ min 3 , for X = C and the maximum is reached at the boundary x = 0 or y = 0 and equals λ 3 (Ω). We have monotonicity along lines joining a point of the boundary to the center C. Furthermore, we notice that λ max Figure 6 gives the eigenvalues and the nodal lines of the eigenfunctions associated with the second and third eigenvalues of the Dirichlet Laplacian on ΩR X on the first and second lines respectively. The j-th column corresponds to the domain ΩR X j with X j = ( 1 5 , j 10 ), j = 1, . . . , 5. These figures are illustration of the theory of Berger-Rubinstein [START_REF] Berger | On the zero set of the wave function in superconductivity[END_REF] and Helffer-Hoffmann-Ostenhof-Hoffmann-Ostenhof-Owen [START_REF] Helffer | Nodal sets for groundstates of Schrödinger operators with zero magnetic field in nonsimply connected domains[END_REF] (see also [START_REF] Alziary | Eigenfunctions and Hardy inequalities for a magnetic Schrödinger operator in R 2[END_REF]). For the ground state energy, we recover the theorem of these authors that the nodal set is composed of a line joining the pole to the boundary. We observe that the nodal line in the first case is choosing a kind of minimal distance between the pole and the boundary whereas the nodal line in the second case seems to choose a kind of maximal distance. We do not have a rigorous explanation for this property except that it should be related to the theorem proved in [START_REF] Berger | On the zero set of the wave function in superconductivity[END_REF][START_REF] Helffer | Nodal sets for groundstates of Schrödinger operators with zero magnetic field in nonsimply connected domains[END_REF] that λ ABX 1 is the infimum over the Dirichlet eigenvalue of the Laplacian in Ω \ γ where γ is a regular path joining the pole X to the boundary.

( ΩR X ), k = 2, 3, for poles X = ( 1 5 , j 10 ), 1 ≤ j ≤ 5 .
We also recover the two last equations in (3.1).

Eigenvalues 4 and 5

Using (3.2), we have proved that λ 2 (Ω) ≥ λ 5 ( ΩR X ). We observe numerically (see also ahead Figures 18 and22 for poles along a symmetry axis and Figure 26) that, for any X ∈ P, we have Numerics shows that the 6-th eigenvalue λ 6 ( ΩR X ) has a unique maximum λ max 6 ≃ 66.581 which is attained when the pole is at the center and is minimal at the boundary. We do not observe other critical points. The 7-th eigenvalue λ 7 ( ΩR X ) is minimal when the pole is at the center and its minimum λ min 7 is equal to λ max

λ 4 ( ΩR X ) = λ 5 ( ΩR X ) = λ 2 (Ω) . ( 4

6

. When the pole is at the center, the zero set of the 6-th eigenfunction provides, by projection, a candidate for a 3-partition and λ max 6 is the conjectured value for L 3 (Ω). We observe that the 7-th eigenvalue becomes constant equal to λ max 7 = λ 4 (Ω) = 8π 2 as a function of the pole when the pole is close to the boundary. Looking at the nodal set for the eigenfunctions associated with λ 7 ( ΩR X ), we notice that when X is close to the boundary, λ 7 ( ΩR X ) = λ 4 (Ω). This corresponds to a crossing when X approaches the boundary between fixed eigenvalues belonging to the spectrum of the square (i.e. the D R X -symmetric spectrum on the covering) and the Xdependent ABX-spectrum (i.e. the D R X -antisymmetric spectrum on the covering ΩR X ). Applying relation (3.3), we have proved theoretically that λ 4 (Ω) ≥ λ 7 ( ΩR X ). We observe numerically that this relation is optimal in the sense that we have equality for X close to the boundary.

Considering the linear combination of the eigenfunctions u 6 and u 7 associated respectively with λ 6 ( ΩR C ) and

λ 7 ( ΩR C ), with C = ( 1 2 , 1
2 ), we can construct a family of 3-partitions with the same energy. Using the double covering approach, it is very easy, for a given eigenvalue with multiplicity ≥ 2, to compute linear combinations of the two eigenfunctions given by the program, because their approximations are given at the same points. If we wanted to construct a family of 3-partitions with the same energy without this approach, we should, for example, construct linear combinations of the two candidates presented in Figure 1. Unfortunately the approximation of the eigenfunctions are not defined at the same points and we should defined new approximations at some new common points. This approach is more technical and requires a longer computation time than the double covering approach.

It is interesting to discuss if we can prove the numerically observed inequality We note indeed that if the conjecture is true, we will get (4.2) by the minimax principle. This conjecture is illustrated in Fig. A1, having in mind that λ 6 ( ΩR X ) = λ ABX 3 .

λ ABX 3 ≥ λ 2 (Ω) .

Eigenvalues 8 and 9

Figures 10-11 represent the numerical computations of λ 8 ( ΩR X ) and λ 9 ( ΩR X ) for X ∈ P. We observe numerically that the function X → λ 8 ( ΩR X ) has a unique maximum denoted by λ max 8 at a point C 1 on the diagonal and X → λ 9 ( ΩR X ) reaches its unique minimum, λ min 9 , at this point. We can recover this behavior on Figure 18 where are drawn the eigenvalues for poles on the diagonal. Numerically, λ max According to (3.8), we have proved that λ 8 ( ΩR X ) ≤ λ 7 (Ω). This theoretical upperbound is coarse and the numerics suggests that we have in fact the better bound λ 8 ( ΩR X ) ≤ λ 5 (Ω). Moreover C 1 is singular for the maps X → λ 8 ( ΩR X ) and λ 9 ( ΩR X ).

Eigenvalue 10

We observe numerically that, for any X ∈ P, we have As for the 7-th eigenvalue λ 7 ( ΩR X ), the 11-th eigenvalue λ 11 ( ΩR X ) becomes constant as X is far enough from the center and equal to λ 5 (Ω). It seems that λ 11 ( ΩR X ) has a unique global maximum when the pole is at the center (see Figure 12). On the contrary, see Figure 13, it seems that λ 12 ( ΩR X ) has only a local minimum at the center and admits local maxima at the points C 2 on the diagonal (see Figure 18) and A 1 = (a 1 , 1 2 ) and A ′ 1 = ( 1 2 , a 1 ) on the perpendicular bisector (see Figure 22). These local maxima can also be observed in Figures 192021222324where is given the 12-th eigenvalue around the points A 1 and C 2 . With a finer grid for X around A 1 = (a 1 , 1

λ 10 ( ΩR X ) = λ 5 (Ω). ( 4
2 ) and C 2 = (c 2 , c 2 ), we get, thanks to the change of symmetry in the nodal domains, the estimates: 0.4255 ≤ a 1 ≤ 0.4256 and 112.05 ≤ λ 12 ( ΩR A 1 ) ≤ 112.06, 0.3689 ≤ c 2 ≤ 0.3690 and 112.01 ≤ λ 12 ( ΩR C 2 ) ≤ 112.02.

It seems that X → λ 12 ( ΩR X ) presents a global maximum for X = C 2 and local maxima at X = A 2 = (a 2 , 1 2 ) and X = ( 1 2 , a 2 ) but the accuracy of our computation, although we used a refined grid for X, does not allow us to be more affirmative. Theoretically, relation (3.5) gives λ 11 ( ΩR X ) ≤ λ 7 (Ω). This estimate is optimal since there exist poles for which λ 11 ( ΩR X ) > λ 5 (Ω) = λ 6 (Ω) (see also for poles along the diagonal or the perpendicular bisector). We recall that λ 12 ( ΩR X ) = λ AB,X We observe that X → λ 13 ( ΩR X ) tends to λ 7 (Ω) when X tends to the boundary and admits a local minimum for X = C 2 along the diagonal. It seems also that this function admits a local minimum for X on one perpendicular bisector at points A 1 and A ′ 1 (see Figures 18 and22) and a local maximum, λ C 13 at X = C. Looking at Figure 15, we observe numerically that for X sufficiently far from C, we have λ 14 ( ΩR X ) = λ 7 (Ω) . It seems that the function X → λ 14 ( ΩR X ) reaches its unique minimum λ min 14 at X = C and we have λ min 14 = λ C 13 , as observed in Figure 22. We observe numerically that, for any X ∈ P, we have

λ 15 ( ΩR X ) = λ 7 (Ω) . (4.4)

Eigenvalues 16 and 17

Looking at Figure 16, we observe that X → λ 16 ( ΩR X ) admits three local maxima : two on the perpendicular bisectors reaching at points A 2 and A ′ 2 and one on the diagonal at point C 3 , see Figures 18 and22. We also observe that λ 16 ( ΩR X ) tends to λ 7 (Ω) as X tends to the boundary or X tends to the center X = C.

Let us now deal with λ 17 ( ΩR X ) represented in Figure 17. The behavior is in reverse order: the function X → λ 17 ( ΩR X ) seems to have three local minima at the same points A 2 , A ′ 2 and C 3 and these minima equal the maxima of λ 16 ( ΩR X ). With finer discretization around the points A 2 = (a 2 , 1 2 ) and C 3 = (c 3 , c 3 ), we estimate numerically according to the change of symmetry in the nodal domains: 5 Moving the pole along the axis y = 1/2

0.2389 ≤ c 3 ≤ 0.

Analysis of the symmetries

Let us begin with some considerations about the ABX-Hamiltonian on the X-punctured square, using the symmetry along the perpendicular bisector y = 1 2 . We refer to [START_REF] Bonnaillie-Noël | Aharonov-bohm hamiltonians, isospectrality and minimal partitions[END_REF] for more details. We fix a point X = (x 0 , 1 2 ) along this line. The square is invariant under the symmetry σ 1 (x, y) = (x, 1 -y).

We consider the antilinear operator

Σ c 1 = ΓΣ 1 ,
where Γ is the complex conjugation (Γu = u) and Σ 1 is associated with σ 1 by the relation

Σ 1 u(x, y) = u(x, 1 -y).
We use the symmetry of the X-punctured square to give an orthogonal decomposition of

L 2 K = L 2 K X : L 2 K = L 2 K,Σ 1 ⊕ L 2 K,a Σ 1 , (5.1)
where

L 2 K,Σ 1 = {u ∈ L 2 K , Σ c 1 u = u }, (5.2)
and

L 2 K,a Σ 1 = {u ∈ L 2 K , Σ c 1 u = -u }. (5.3)
As established in [START_REF] Bonnaillie-Noël | Aharonov-bohm hamiltonians, isospectrality and minimal partitions[END_REF]Lemma 5.6], we can prove that:

• if u ∈ C ∞ ( ΩX ) ∩ L 2 K,Σ 1 , then its nodal set contains [0, x 0 ] × { 1 2 }, • if u ∈ C ∞ ( ΩX ) ∩ L 2 K,aΣ 1 , then its nodal set contains [x 0 , 1] × { 1 2 }.
Dealing with a mixed Dirichlet-Neumann condition on the half-domain, we deduce that the eigenvalues for which the eigenfunctions are symmetric are increasing with respect to x 0 , whereas the eigenvalues for which the eigenfunctions are antisymmetric are decreasing with respect to x 0 .

Spectral variation

Figure 18 gives the eigenvalues for poles along the axis y = 1/2 and 0 < x ≤ 1/2. We observe that λ ABX k is increasing with the first coordinate of X for k = 1, 3, 5, 6, whereas it is decreasing for k = 2, 4.

We denote by X(x) = (x, 1 2 ) the pole and introduce A j = X(a j ), B j = X(b j ) specific points which can be seen on the Figure 18. The below mentioned symmetry (resp. antisymmetry) is in this section with respect to Σ c 1 (see (5.2) and (5.3)) and denoted by Σ 1 (resp. aΣ 1 ) on the figure. Then, we observe numerically:

A 1 A 2 B 1 B 2 ! 1 (")=2# 2 ! 2 (")=! 3 (")=5# 2 ! 4 (")=8# 2 ! 5 (")=! 6 (")=10# 2 ! 7 (")=! 8 (")=13# 2 ! 9 (")=! 10 (")=17# 2 ! 1 AB ! 2 AB ! 3 AB ! 4 AB ! 5 AB ! 6 AB ! 7 AB ! 8 AB ! 9 AB $ 1 a$ 1 $ 1 a$ 1 $ 1 $ 1 a$ 1 $ 1 a$ 1 a$ 1 $ 1 $ 1 ! 1 (" R X ) ! 2 (" R X ) ! 3 (" R X ) ! 4 (" R X ) ! 5 (" R X ) ! 6 (" R X ) ! 7 (" R X ) ! 8 (" R X ) ! 9 (" R X ) ! 10 (" R X ) ! 11 (" R X ) ! 12 (" R X ) ! 13 (" R X ) ! 14 (" R X ) ! 15 (" R X ) ! 16 (" R X ) ! 17 (" R X ) !
is strictly increasing from [0, 1 2 ] onto [λ 5 (Ω) , λ ABC , λ 9 (Ω)] with antisymmetric eigenfunctions and strictly increasing from [a 2 , b 2 ] onto [λ ABA 2 9 , λ 9 (Ω)] with symmetric eigenfunctions.

Exchange of symmetry and crossing points

When moving the pole on one bisector, and for each eigenvalue of multiplicity 1, the corresponding K X -real eigenfunction should be either symmetric or antisymmetric with respect to Σ c 1 . Figure 18 suggests that there exists two poles A 1 = (a 1 , 1 2 ) and A 2 = (a 2 , 1 2 ) on the perpendicular bisector such that λ 12 ( ΩR A 1 ) and λ 16 ( ΩR A 2 ) are eigenvalues of multiplicity 2. Taking the Aharonov-Bohm point of view, this corresponds to a crossing between λ ) and the corresponding nodal set for X along the perpendicular bisector and close to A 1 (resp. A 2 ). We can verify on each line of the two figures, that there exists an exchange of symmetry3 between the third column and the fourth column predicting the existence of the points A 1 and A 2 . We observe in the second line and fifth column of Figure 19 one nodal line with no selfcrossing. This is an artefact produced by our algorithm to detect the nodal lines and the accuracy of our computation. There is actually a crossing between a closed line and the horizontal segment joining the pole to the boundary of the square. , k = 6, 7 for poles X = ( i 100 , , k = 8, 9 for poles X = ( i 100 , 1 2 ), 26 ≤ i ≤ 31.

Nodal deformation : an example

The last column of Figure A3 gives the nodal set for the eigenfunction associated with the fifth eigenvalue of the ABX-Hamiltonian for poles X = ( i 20 , 1 2 ), i = 1, . . . , 10, on the perpendicular bisector of one side of the square. The deformation mechanism for the nodal set is represented in Figure 21. Between the fourth and fifth figures, we have a nodal structure where there are two double points at the boundary. 6 Moving the pole on the diagonal

Analysis of the symmetries

As for the case of poles along the perpendicular bisector, we can use the symmetry along the diagonal to have additional information about the spectrum. The square is invariant under the symmetry

σ 2 (x, y) = (y, x).
We consider the antilinear operator Σ c 2 = ΓΣ 2 where Σ 2 is associated with σ 2 by the relation Σ 2 u(x, y) = u(y, x). We use this symmetry to give an orthogonal decomposition of

L 2 K = L 2 K X : L 2 K = L 2 K,Σ 2 ⊕ L 2 K,a Σ 2 , (6.1)
where L 2 K,Σ 2 and L 2 K,a Σ 2 are respectively symmetric and antisymmetric with respect to Σ c 2 . As in [3, Lemma 5.6], we can prove that, with X = (x 0 , x 0 ),

• if u ∈ C ∞ ( ΩX ) ∩ L 2 K,Σ 2 , then the nodal set of u contains {(x, x), 0 < x < x 0 } ; • if u ∈ C ∞ ( ΩX ) ∩ L 2 K,aΣ 2 , then the nodal set of u contains {(x, x), x 0 < x < 1} .
Dealing with a mixed Dirichlet-Neumann condition on the half-domain, we deduce that the eigenvalues for which the eigenfunctions are symmetric are increasing with respect to x 0 whereas the eigenvalues for which the eigenfunctions are antisymmetric are decreasing with respect to x 0 .

Spectral variation

Figure 22 gives the eigenvalues for poles along the diagonal line of the square x = y with 0 ≤ x ≤ 1/2. We observe that the eigenvalues λ AB X(x) k with X(x) = (x, x) are increasing with x ∈ [0, 1 2 ] for k = 1, 3 and decreasing for k = 2. We introduce C j = X(c j ), D j = X(d j ) specific crossing points appearing on the figure. The below mentioned symmetry (resp. antisymmetry) is in this section with respect to Σ c 2 and denoted by Σ 2 (resp. aΣ 2 ) on the Figure 22. Then, we observe:

(a) x → λ 1 ( ΩR X(x) ) equals λ 1 (Ω), in adequation with (3.1). (b) x → λ 2 ( ΩR X(x) ) = λ AB X(x) 1
is strictly increasing from [0 , 1 2 ] onto [λ 1 (Ω) , λ ABC

1

] and the eigenfunctions are symmetric. The equality between λ 2 ( ΩR X(x) ) and λ AB X(x) 1

observed numerically was rigourously proved in (3.1).

(c) x → λ 3 ( ΩR X(x) ) = λ AB X(x) 2
is strictly decreasing from [0 , 1 2 ] onto [λ ABC

! 1 (")=2# 2 ! 2 (")=! 3 (")=5# 2 ! 4 (")=8# 2 ! 5 (")=! 6 (")=10# 2 ! 7 (")=! 8 (")=13# 2 ! 9 (")=! 10 (")=17# 2 C 1 C 2 C 3 D 1 D 2 D 3 ! 1 AB ! 2 AB ! 3 AB ! 4 AB ! 5 AB ! 6 AB ! 7 AB ! 8 AB ! 9 AB $ 2 a$ 2 $ 2 $ 2 a$ 2 a$ 2 $ 2 a$ 2 $ 2 $ 2 a$ 2 $ 2 $ 2 a$ 2 a$ 2 $ 2 ! 1 (" R X ) ! 2 (" R X ) ! 3 (" R X ) ! 4 (" R X ) ! 5 (" R X ) ! 6 (" R X ) ! 7 (" R X ) ! 8 (" R X ) ! 9 (" R X ) ! 10 (" R X ) ! 11 (" R X ) ! 12 (" R X ) ! 13 (" R X ) ! 14 (" R X ) ! 15 (" R X ) ! 16 (" R X ) ! 17 (" R X ) ! on [d 2 , 1
2 ] where it is strictly increasing onto [λ 5 (Ω) , λ ABC 5 ] and the eigenfunctions are symmetric. This illustrates the fact that relation (3.5) is optimal.

, λ 9 (Ω)] with symmetric eigenfunctions. We observe that λ 17 ( ΩR X(x) ) ≤ λ 9 (Ω) whereas we have proved in (3.7) the upper-bound by λ 11 (Ω).

Exchange of symmetry

As in the case of the symmetry with respect to the bisector, when moving the pole on one diagonal, and for each eigenvalue of multiplicity 1, the corresponding K X -real eigenfunction should be either symmetric or antisymmetric with respect to the quantized symmetry with respect to the diagonal (see [START_REF] Bonnaillie-Noël | Aharonov-bohm hamiltonians, isospectrality and minimal partitions[END_REF]). Figure 22 The nodal lines have to respect the symmetry according to the diagonal line. Consequently, in each case, there exists a nodal line touching the point (0, 0) or [START_REF] Alziary | Eigenfunctions and Hardy inequalities for a magnetic Schrödinger operator in R 2[END_REF][START_REF] Alziary | Eigenfunctions and Hardy inequalities for a magnetic Schrödinger operator in R 2[END_REF]. In some picture, the touching point is not exactly one of these points but it is quite difficult to detect accurately these lines near the boundary because the eigenfunction vanishes at the boundary.

87

Moving the pole without respecting the symmetries of the square

Figure 26 gives the eigenvalues λ k ( ΩR X ) for 1 ≤ k ≤ 12 when the pole X belongs to the line y = 1 4 + x 2 . We choose this axis to exhibit a case without symmetry and we notice that the 25 AB-eigenvalues λ ABX k are no longer monotone with respect to x when X = (x, 1 4 + x 2 ). The result should be the same for any arbitrary line (except the perpendicular bisector and the diagonal). We choose to present the simulations for this line because this line contains enough points in P to use the previous numerical computations.

! 1 (")=2# 2 ! 2 (")=! 3 (")=5# 2 ! 4 (")=8# 2 ! 5 (")=! 6 (")=10# 2 ! 7 (")=! 8 (")=13# 2 ! 9 (")=! 10 (")=17# 2 ! 1 AB ! 2 AB ! 3 AB ! 4 AB ! 5 AB ! 6 AB ! 7 AB ! 8 AB ! 9 AB ! 1 (" R X ) ! 2 (" R X ) ! 3 (" R X ) ! 4 (" R X ) ! 5 (" R X ) ! 6 (" R X ) ! 7 (" R X ) ! 8 (" R X ) ! 9 (" R X ) ! 10 (" R X ) ! 11 (" R X ) ! 12 (" R X ) ! 13 (" R X ) ! 14 (" R X ) ! 15 (" R X ) ! 16 (" R X ) ! 17 (" R X ) ! 18 (" R X )
Figure 26: Moving the pole along the axis y =

1 4 + x 2 , 0 ≤ x ≤ 1 2 .
It would be interesting to make computations for a finer grid of X = (x, 1 4 + x 2 ) for x around 0.44 to detect possible crossings between λ AB 5 , λ AB 6 and λ AB 7 .

Nodal sets and minimal partitions

The analysis of λ ABX The eigenfunctions computed with the double covering approach do not produce a 5-partition actually a small perturbation of it with only three nodal sets in ΩC , but we can have a better insight by taking the Dirichlet-Neumann approach along the bisector orthogonal to one side: we compute the 5-th eigenfunction of the Dirichlet Laplacian on the square with Dirichlet condition on {(x, 1 2 ), 1 2 < x < 1}, see Figure 28. This time, neither numerics nor theory is giving the existence of a continuous family of 5-partitions. Actually, one knows from elementary results on the perturbation of harmonic polynomials of order 2 that the perpendicular crossing of two lines will generically disappear by perturbation. In the 2D eigenspace of λ 11 ( ΩR C ) only four eigenfunctions seem to lead to a configuration on one sheet with five nodal domains. The other eigenfunctions seem to have four symmetric (for the deck map) pairs of domains. These eigenfunctions are symmetric or antisymmetric with respect of one of the four bisectors of the square (see Figure 27). Looking at the linear combination tu 11 + (1 -t)u 12 of the eigenfunction associated with λ 11 ( ΩR C ) and λ 12 ( ΩR C ), we observe in Figure 27 Of course it is interesting to compare with what can be obtained by looking at other topological types for the 5-minimal partitions. We recall that these types can be classified by using Euler formula (see [START_REF] Helffer | On spectral minimal partitions : new properties and applications to the disk[END_REF] for the case of 3-partitions). Inspired [START_REF] Cybulski | Minimization of the Renyi entropy production in the space-partitioning process[END_REF], we look for a partition which has the symmetries of the square and four critical points. We get two types of models that we can reduce to a Dirichlet-Neumann problem on a triangle corresponding to the eighth of the square. Moving the Neumann boundary on one side like in [START_REF] Bonnaillie-Noël | Numerical simulations for nodal domains and spectral minimal partitions[END_REF] leads to two candidates. Numerical computations demonstrate a lower energy in one case which coincides with one of the pictures in [START_REF] Cybulski | Minimization of the Renyi entropy production in the space-partitioning process[END_REF] (see Figure 28). Note that in the case of the disk a similar analysis leads to a different answer. The partition of the disk by five halfrays with equal angle has a lower energy than the minimal 5-partition with four singular points (see Figure 29). We note that, on the basis of standard computations (see for example (A1) and (A5) in [START_REF] Helffer | Nodal domains and spectral minimal partitions[END_REF], Appendix A) this energy corresponds to the eleventh eigenvalue of the Dirichlet problem on the double covering on the punctured disk (hence is not Courant sharp) but corresponds to the fifth of the Aharonov-Bohm spectrum of the disk. Hence it is Courant sharp in the sense developed in [START_REF] Helffer | On spectral minimal partitions : the case of the sphere[END_REF] (for the sphere) and shows the minimality of this 5-partition inside the minimal 5-partitions of the disk having a unique critical point at the center.

Conclusion

We have explored rather systematically how minimal partitions could be obtained by looking at nodal domains of a problem on the double covering of a punctured square. We have analyzed the behavior of the nodal set when moving the pole in the square. This has permitted to confirm the status of "main" candidate for some 3-partitions in the case of the square. This has also permitted to exhibit a natural candidate for a minimal 5-partition which finally appears to be less favorable than another partition with four critical points. This is a starting point for a program which can be developed in at least two directions :

• analyze other domains,

• do the same work by considering the double covering of a multi-punctured domain and moving the poles.

For a given

  partition D of an open set Ω by k disjoint open subsets D i , we consider Λ(D) = max i=1,...,k λ(D i ) ,

Figure 2 :

 2 Figure 2: An asymmetric candidate for the minimal 3-partition of the square.

Figure 3 :

 3 Figure 3: Mesh of the double covering ΩR X .

5 by λ 7

 57 (Ω) (hence by λ 9 (Ω)) and of λ ABX 8 by λ 11 (Ω) established in (3.5) and (3.7). ✷ Lemma 3.4

Figure 4 :

 4 Figure 4: X → λ 2 ( ΩR X ) for X ∈ P.

Figure 5 :

 5 Figure 5: X → λ 3 ( ΩR X ) for X ∈ P.

Figure 7 :

 7 Figure 7: λ 6 ( ΩR X ), in function of the pole X ∈ P.

Figure 8 :

 8 Figure 8: λ 7 ( ΩR X ), in function of the pole X ∈ P.

  Figures 9 give the projection by π R C of the nodal set for the functions tu 6 + (1 -t)u 7 with t = k/8, k = 0, . . . , 8.

Figure 9 :

 9 Figure 9: Continuous family of 3-partitions with the same energy.

(4. 2 )

 2 This is directly related to the conjecture proposed by S. Terracini 2 :Conjecture 4.1 Except at the center X = C = ( 1 2 , 1 2 ), λ ABX 3is simple and the corresponding nodal set of 2 Personal communication the K X -real eigenfunction is the union of a line joining the pole to the boundary and of another line joining two points of the boundary.

8 = λ min 9 andFigure 10 :

 8910 Figure 10: X → λ 8 ( ΩR X ) for X ∈ P.

Figure 11 :

 11 Figure 11: X → λ 9 ( ΩR X ) for X ∈ P.

. 3 )Figure 12 :

 312 Figure 12: λ 11 ( ΩR X ), according to the location of the pole X ∈ P.

Figure 13 :

 13 Figure 13: λ 12 ( ΩR X ) according to X ∈ P with additional zoom.

6 .Figure 14 :

 614 Figure 14: λ 13 ( ΩR X ) according to X ∈ P.

Figure 15 :

 15 Figure 15: λ 14 ( ΩR X ) according to X ∈ P.

Figure 16 :Figure 17 :

 1617 Figure 16: λ 16 ( ΩR X ) according to X ∈ P.

  (a) x → λ 1 ( ΩR X(x) ) equals λ 1 (Ω), in adequation with the theoretical result (3.1).

7 ABX 9 at x = a 2 , 6 (resp. λ ABX 8 )ABX 7 (

 792687 for x = a 1 , with a 1 ∈ ] 42 100 , 43 100 [ and to a crossing between λ with a 2 ∈ ] 28 100 , 29 100 [. The nodal sets of the corresponding eigenfunctions are given in Figures19 and 20. The first line of the figure gives the eigenvalues λ ABX and the nodal set of the associated eigenfunction and the second line λ resp. λ ABX 9

Figure 19 :

 19 Figure 19: Nodal set for the eigenfunctions associated with λ ABX(x) k

Figure 20 :

 20 Figure 20: Nodal set for the eigenfunctions associated with λ ABX(x) k

2 , λ 2

 22 (Ω)] and the eigenfunctions are antisymmetric. The equality between λ 3 ( ΩR X(x) ) and λ AB X(x) 2observed numerically was also rigourously proved in (3.1).

9 at x = c 3 ,

 93 with c 3 ∈ ] 23 100 , 24 100 [. The nodal set of the corresponding eigenfunctions are given in Figures 23, 24 and 25. We can verify that there exists an exchange of symmetry permitting to predict the existence of the points C 1 , C 2 and C 3 .

Figure 25 :

 25 Figure 25: Nodal set for the eigenfunctions associated with λ ABX k , k = 8, 9 for poles X = (i, i)/100, 21 ≤ i ≤ 26.

5 and λ ABX 6 leads 5 =

 65 us to guess numerically a double eigenvalue at the center. As for the pairs λ ABX 3 and λ ABX4which lead us to the family of candidates for the minimal 3-partitions of the square (see Subsection 4.4), we are led to produce a candidate for a minimal 5-partition for the square, with the property that it is minimal inside the class of 5-partitions which can be lifted to ΩR C . Although λ ABC λ 11 ( ΩR C ), which is not Courant-sharp, we observe that it is Courant-sharp for the AB-Hamiltonian.

Figure 28 :

 28 Figure 28: Three candidates for the 5-partition of the square.

Figure 29 :

 29 Figure 29: Two candidates for the 5-partition of the disk.

Table 1 :

 1 Eigenvalues of the Dirichlet Laplacian on ΩR C for two meshes.

	# elements	3110	4410	6194	3102	4362	6254
	1	19.739	19.739	19.739	19.739	19.739	19.739
	2	33.518	33.509	33.510	33.517	33.511	33.510
	3	33.521	33.512	33.512	33.519	33.514	33.510
	4	49.348	49.348	49.348	49.348	49.348	49.348
	5	49.348	49.348	49.348	49.348	49.348	49.348
	6	66.581	66.581	66.581	66.581	66.581	66.581
	7	66.581	66.581	66.581	66.581	66.581	66.581
	8	78.957	78.957	78.957	78.957	78.957	78.957
	9	98.696	98.696	98.696	98.696	98.696	98.696
	10	98.696	98.696	98.696	98.696	98.696	98.696
	11 111.910 111.910 111.910 111.910 111.910 111.910
	12 111.910 111.910 111.910 111.910 111.910 111.910
	13 122.231 122.209 122.210 122.229 122.212 122.209
	14 122.240 122.215 122.217 122.235 122.220 122.211

Table 2 :

 2 First thirteen eigenvalues of the Dirichlet Laplacian on Ω and nodal set for the associated basis φ mn .

	2 ≃ 167.783	(4, 1) , (1, 4)
	λ 11 (Ω) = 18π 2 ≃ 177.653	(3, 3)
	λ 12 (Ω) = λ 13 (Ω) = 20π 2 ≃ 197.392	(4, 2) , (2, 4)

  which gives the first 12 eigenvalues of the Dirichlet Laplacian on ΩR X for three points X: one near the boundary denoted by A, one at the center denoted by C and one other denoted by B.

				C )
	1	19.739	19.739	19.739
	2	19.739	20.269	33.528
	3	49.348	49.325	33.534
	4	49.348	49.348	49.348
	5 6	49.348 49.348	49.348 51.480	49.348 66.581	C
	7 8	78.957 78.957	78.957 79.536	66.581 78.957	B
	9 10 98.696 98.696	98.658 98.696	98.696 98.696	A
	11 98.696	98.696 111.910
	12 98.696 102.647 111.910

n λ n ( ΩR A ) λ n ( ΩR B ) λ n ( ΩR

Table 3 :

 3 First 12 eigenvalues of the Dirichlet Laplacian on ΩR

	A , ΩR B and ΩR C , with

  2390 and 151.40 ≤ λ 16 ( ΩR C 3 ) ≤ 151.42 , 0.2829 ≤ a 2 ≤ 0.2830 and 150.03 ≤ λ 16 ( ΩR A 2 ) ≤ 150.05 . Consequently the extrema seem to be global at C 3 and local at A 2 and A ′ 2 . Furthermore, λ 16 ( ΩR

X ) tends to λ 9 (Ω) as X tends to the boundary or to the center C.

  suggests that there are 3 points, C 1 , C 2 and C 3 on the diagonal such that λ 8 ( ΩR C 1 ), λ 12 ( ΩR C 2 ) and λ 16 ( ΩR C 3 ) are eigenvalues of multiplicity 2. This corresponds to a crossing between λ

	AB X(x) 4 100 [. Similarly, there is a crossing between λ 100 , 29 c 1 ∈ ] 28 AB X(x) and λ 6 and λ AB X(x) 5 AB X(x) 7 c 2 ∈ ] 36 100 , 37 AB X(x) 100 [, and also between λ 8	at x = c 1 , with at x = c 2 , with

  that the triple point is very unstable and only appears for t ≃ 96 Nodal sets of the linear combination of u 11 and u 12 tu 11 + (1 -t)u 12 with t = 1 200 (0, 25, 55, 96, 125, 138, 175, 194, 200).

	and t ≃ 194 200 when we consider 0 ≤ t ≤ 1.	200
	Figure 27:

We sometimes speak more shortly of spectrum of the square.

] and the eigenfunctions are symmetric. This observation shows that the theoretical upper-bound λ 11 ( ΩR X ) ≤ λ 7 (Ω) deduced from (3.5) can not be improved.

Look at the horizontal nodal line joining the pole to the boundary!

Acknowledgements.

The authors would like to thank Thomas Hoffmann-Ostenhof, Luc Hillairet and Susanna Terracini for useful discussions around this topics. We have been in particular stimulated by successive versions of [18]. Discussions about numerics with François Alouges and Grégory Vial were very fruitful. G. Vial helps us also by realizing numerous meshes for the computations and also for the detection of the nodal lines. This paper has been partially written during the stay of the authors at the Erwin Schrödinger Institute from May to July 2009 and the authors are grateful for the very good working conditions and also for the fruitful discussions with other participants of the workshop on Topics in Spectral Theory.

is strictly increasing from [0 , 1 2 ] onto [λ 1 (Ω) , λ ABC 1 ] and the eigenfunctions are symmetric. The equality between λ 2 ( ΩR X(x) ) and λ ABX(x) 1 was proved in (3.1).

is strictly decreasing from [0, 1 2 ] onto [λ ABC

2

, λ 2 (Ω)] and the eigenfunctions are antisymmetric. The equality between λ 3 ( ΩR X(x) ) and λ ABX(x) 2

was proved in (3.1).

(d) λ 4 ( ΩR X(x) ) = λ 5 ( ΩR X(x) ) = λ 2 (Ω). This numerical observation is more accurate than the theoretical result deduced from (3.2): λ 4 ( ΩR

] and the eigenfunctions are symmetric.

is strictly decreasing from [0, 1 2 ] onto [λ ABC

4

, λ 4 (Ω)] and the eigenfunctions are antisymmetric.

(g) λ 8 ( ΩR X(x) ) = λ 4 (Ω). We have proved in (3.8) that λ 8 ( ΩR X(x) ) ≤ λ 4 (Ω) and we observe that this upper-bound is actually an equality. We notice that there is a gap between λ 4 (Ω) and λ 5 (Ω) where there is no eigenvalue λ k ( ΩR X ) for X on the perpendicular bisector. This observation is no more true for poles on the diagonal (see Figure 22).

is strictly increasing from [0 , a 1 ] onto [λ 6 (Ω) , λ ABA ] and the eigenfunctions are antisymmetric. This illustrates theoretical result deduced from (3.9): λ 12 ( ΩR X(x) ) ≤ λ 7 (Ω) and shows that this result is optimal.

] and the eigenfunctions are antisymmetric. It is strictly increasing from

] and the eigenfunctions are symmetric. We observe then that λ 13 ( ΩR X ) can be bounded from above by λ 7 (Ω) whereas we have proved in (3.6) the upper-bound by λ 9 (Ω).

] with symmetric eigenfunctions and strictly decreas-

] with antisymmetric eigenfunctions.

Figure 22: Moving the poles on the diagonal.

. Theoretically, we have proved, see (3.2), that λ 4 ( ΩR X(x) ) ≤ λ 2 (Ω). This relation seems to be an equality, see Subsection 4.3 .

(e) x → λ 6 ( ΩR

is strictly increasing from [0 , 1 2 ] onto [λ 3 (Ω) , λ ABC

3

] and the eigenfunctions are symmetric.

where it is strictly decreasing onto [λ ABC 4 , λ 4 (Ω)] and the eigenfunctions are antisymmetric. This numerical computations show that the theoretical estimate λ 7 ( ΩR (i) λ 10 ( ΩR X(x) ) = λ 5 (Ω) .

(j) x → λ 11 ( ΩR X(x) ) equals λ 5 (Ω) on [0 , d 2 ] and λ

] and the eigenfunctions are symmetric. It is strictly decreasing from

] and the eigenfunctions are antisymmetric.

and the eigenfunctions are antisymmetric. It is strictly increasing from

] and the eigenfunctions are symmetric. We then observe λ 13 ( ΩR X(x) ) ≤ λ 7 (Ω) whereas we have proved the weaker upper-bound by λ 9 (Ω) in (3.6). on [0 , 1 2 ]. It is strictly decreasing from [0 , c 3 ] onto [λ ABC 3 9 , λ 9 (Ω)] with antisymmetric eigenfunctions and strictly increasing from [c 3 , 1 2 ] onto [λ ABC 3

Appendix. Additional numerics for the first Aharonov-Bohm eigenvalues

Figures A1-A4 give the nodal set for the eigenfunctions associated with λ AB k ( ΩX ) with k = 3, 4, 5, 6 and X ∈ P 0 = {(i, j)/20, 1 ≤ i, j ≤ 10}. Line i and column j correspond to the pole X = ( i 20 , j 20 ). As mentioned in (4.1), the fourth and fifth eigenvalues, λ 4 ( ΩR X ) and λ 5 ( ΩR X ), are equal to the second eigenvalue of the square, hence the associated eigenfunctions of the Dirichlet-Laplacian on the square have two nodal domains and those of the Dirichlet-Laplacian on the double covering ΩR X have three or four nodal domains according to the location of X. We do not represent them here. We have observed that λ ABX 4 equals λ 7 ( ΩR X ) when X is close to the center C and λ 8 ( ΩR X ) for X close to the corner X = (0, 0). The transition between these two eigenvalues is represented in Figure A2 We verify the property proved in [START_REF] Helffer | Nodal sets for groundstates of Schrödinger operators with zero magnetic field in nonsimply connected domains[END_REF] that from the pole X only an odd number of lines can start for the nodal set corresponding to an ABX-eigenfunction.