Ashish Arora
email: ashish.arora@lncmi.cnrs.fr

Maciej Koperski

Karol Nogajewski

Jacques Marcus

Clément Faugeras

Marek Potemski
email: marek.potemski@lncmi.cnrs.fr

Excitonic resonances in thin films of WSe 2 : From monolayer to bulk material

We present optical spectroscopy (photoluminescence and reflectance) studies of thin layers of the transition metal dichalcogenide WSe 2 , with thickness ranging from mono-to tetra-layer and in the bulk limit. The investigated spectra show the evolution of excitonic resonances as a function of layer thickness, due to changes in the band structure and, importantly, due to modifications of the strength of Coulomb interaction as well. The observed temperature-activated energy shift and broadening of the fundamental direct exciton are well accounted for by standard formalisms used for conventional semiconductors. A large increase of the photoluminescence yield with temperature is observed in WSe 2 monolayer, indicating the existence of competing radiative channels. The observation of absorption-type resonances due to both neutral and charged excitons in WSe 2 monolayer is reported and the effect of the transfer of oscillator strength from charged to neutral exciton upon increase of temperature is demonstrated.

Introduction

Atomically thin layers of semiconducting transition metal dichalcogenides (SC-TMDCs) of the form MX 2 (such as MoS 2 , MoSe 2 , WS 2 and WSe 2 ) attract a considerable research interest stimulated by the scientific curiosity to study a new class of two-dimensional (2D) semiconductors and by their potential applications in optoelectronics, photonics and the development of valleytronics. [START_REF] Mak | Atomically Thin MoS 2 : A New Direct-Gap Semiconductor[END_REF][START_REF] Wang | Electronics and Optoelectronics of Two-Dimensional Transition Metal Dichalcogenides[END_REF][START_REF] Eda | Two-Dimensional Crystals: Managing Light for Optoelectronics[END_REF][START_REF] Butler | Challenges, and Opportunities in Two-Dimensional Materials Beyond Graphene[END_REF][START_REF] Xiao | Coupled Spin and Valley Physics in Monolayers of MoS 2 and Other Group-VI Dichalcogenides[END_REF][START_REF] Jones | Optical Generation of Excitonic Valley Coherence in Monolayer WSe 2[END_REF][START_REF] Tonndorf | Photoluminescence Emission and Raman Response of Monolayer MoS 2 , MoSe 2 , and WSe 2[END_REF] Bulk MX 2 crystals, largely investigated in the past, [START_REF] Wilson | The Transition Metal Dichalcogenides Discussion and Interpretation of the Observed Optical, Electrical and Structural Properties[END_REF][START_REF] Beal | Transmission Spectra of Some Transition Metal Dichalcogenides: II. Group VIA: Trigonal Prismatic Coordination[END_REF][START_REF] Beal | Krönig analysis of the reflectivity spectra of 3R-WSe 2[END_REF] consist of weakly bonded X-M-X monolayer units (three atomic planes). Monolayer (1 ML) and few-, N-monolayer (N ML) flakes of SC-TMDCs can be conveniently extracted from bulk crystals and deposited on substrates using the methods of mechanical exfoliation. [START_REF] Novoselov | Two-Dimensional Atomic Crystals[END_REF] Since very first works on MoS 2 layers, [START_REF] Mak | Atomically Thin MoS 2 : A New Direct-Gap Semiconductor[END_REF] it is now well established that the band structure of N MLs of SC-TMDCs critically depends on N. Monolayers of SC-TMDCs are believed to be direct band gap (or nearly direct band gap) semiconductors and undergo a crossover to the indirect band gap structure of bulk SC-TMDCs with increasing N. [START_REF] Yun | Thickness and Strain Effects on Electronic Structures of Transition Metal Dichalcogenides: 2H-MX 2 Semiconductors (M = Mo, W; X = S, Se, Te)[END_REF][START_REF] Zhao | Evolution of Electronic Structure of Atomically Thin Sheets of WS2 and WSe2[END_REF][START_REF] Ramasubramaniam | Large Excitonic Effects in Monolayers of Molybdenum and Tungsten Dichalcogenides[END_REF] Strong excitonic (Coulomb) effects, whose strength may also vary with N, is another relevant ingredient which determines the optical properties of thin films of SC-TMDCs. [START_REF] Ramasubramaniam | Large Excitonic Effects in Monolayers of Molybdenum and Tungsten Dichalcogenides[END_REF] A thorough understanding of basic optical properties of these films is essential for advancing their further explorations, towards efficient future applications.

In this paper, we report on optical investigations of thin films of WSe 2 with thicknesses ranging from 1 ML to 4 ML and bulk flakes, using micro-photoluminescence (µPL), microphotoluminescence excitation (µPLE) and microreflectivity contrast (µRC) spectroscopy techniques, in 5 K-300 K temperature range. Our Nlayer WSe 2 structures are first characterized with low temperature, band-edge µPL and absorptiontype experiments which depict the characteristic indirect to (nearly) direct band gap crossover, when approaching from thicker to the 1 ML film. We then focus on the evolution of absorption resonances as a function of N in a broader energy range, as they are derived from µRC measurements. The ground state A and spin-orbit (SO) split B excitons associated with the K-point of the Brillouin zone are each accompanied by the excited state resonances A * and B * , respectively. Energy separation between A and A * , and between B and B * varies with N, which reflects the changes in the strength of excitonic binding as well as in the character of excitonic states, from non-Rydberg states in 1 ML to rather Rydberg like states in bulk. Another clear absorption resonance C, observed in the upper spectral energy range and most sensitive to N, is tentatively assigned to the transition at the M-point of the Brillouin zone. Energy shift and broadening of the ground state A exciton resonance with increasing temperature are both well accounted for by conventional formulae for the band gap shrinkage and optical phonon mediated broadening of free exciton emission, which have been widely applied to other semiconductors. Strikingly, the integrated PL intensity in the 1 ML of WSe 2 is found to increase considerably with temperature, whereas opposite is observed in conventional direct band gap semiconductors. Although neutral excitons (which dominate the PL spectra at high temperature) have higher emission efficiency than the bound/charged excitons (which dominate the PL at low temperatures), [START_REF] Wang | Valley Dynamics Probed Through Charged and Neutral Exciton Emission in Monolayer WSe 2[END_REF][START_REF] Aleiner | Threeparticle Complexes in Two-Dimensional Seminconductors[END_REF] this observation of an enhancement of integrated intensity is a thinkable sign of the presence of a radiatively inefficient channel at low temperature, which upon thermalization, transfers the carriers to radiatively efficient channel at higher temperatures. It opens up scope for further investigations and discussions on this material. Finally, both neutral (A) and charged (trion, T) [START_REF] Kheng | Observation of Negatively Charged Excitons Xin Semiconductor Quantum Wells[END_REF][START_REF] Buhmann | Electron-concentrationdependent Quantum-well Luminescence: Evidence for a Negatively Charged Exciton[END_REF][START_REF] Mak | Tighly Bound Trions in Monolayer MoS 2[END_REF][START_REF] Finkelstein | Optical Spectroscopy of a Two-Dimensional Electron Gas Near the Metal-Insulator Transition[END_REF] excitonic resonances are observed in 1 ML flake not only in emission, but in absorption-type µRC spectra as well. This allows us to demonstrate the effect of transfer of oscillator strength from trion to exciton, upon increasing the temperature. [START_REF] Ciulin | Radiative Behavior of Negatively Charged Excitons in CdTe-Based Quantum Wells: A Spectral and Temporal Analysis[END_REF][START_REF] Kossacki | Optical Studies of Charged Excitons in II-VI Semiconductor Quantum Wells[END_REF][START_REF] Suris | Excitons and Trions Modified by Interaction with a Two-Dimensional Electron Gas[END_REF][START_REF] Esser | Theory of Trion Spectra in Semiconductor Nanostructures[END_REF] Further studies of the effects of the presence of free carriers on the optical response of the WSe 2 monolayers are highly desirable.

Samples and characterization

For preliminary characterization of the flakes, we performed atomic force microscopy (AFM) and µPL measurements on the flakes. Insets to the Fig. 1 (c-e) show the optical images of the flakes. The thicknesses of the 1 ML, 2 ML, 3 ML, 4 ML as well as the bulk flake were found to be ∼0.7 nm, ∼1.4 nm, ∼2.1 nm, ∼2.8 nm and ∼20 nm respectively using AFM (data not shown). The PL spectra of the flakes have been shown as solid curves in Fig. 1(a-e). The 1 ML thick flake is found to emit light most efficiently, and the emission intensity reduces considerably as the thickness of the flakes is increased. A reduction in intensity by 4 orders of magnitude is observed when the flake thickness is increased from 1 ML to 20 nm. It is because the nature of the band gap changes from direct to indirect for thickness greater than 1 ML. [START_REF] Zhao | Evolution of Electronic Structure of Atomically Thin Sheets of WS2 and WSe2[END_REF] The emission also shifts to lower energy with increasing flake thickness. This indicates the reduction of the fundamental bandgap with an increased flake thickness, in analogy to the effect of quantum confinement in semiconductor quantum wells.

To gain more insight into the near band-edge properties of our layers, apart from µPL, we performed µPLE and µRC spectroscopy studies on the same flakes. The well pronounced, lowest energy absorption resonance seen in these spectra (dashed blue and dashed red traces in Fig. 1) is identified with the so-called A exciton associated to the direct band alignment of WSe 2 at Kpoint of the Brillouin zone. Its double peak structure in the monolayer (trion T resonance on the low energy side) and in the bulk sample (excited exciton state A * on the higher energy side) will be discussed later on. A small shift in the absorption resonances seen in the PLE spectra and those extracted from µRC measurements is likely due to effectively higher sample temperature in the course of PLE measurements which imply the use of laser excitation and unavoidable local heating effects, and/or due to energy sensitive carrier relaxation process which may also alter the shape of absorption as reflected in PLE spectra. One of the prominent and clear features of the data shown in Fig. 1 is the red shift of the PL spectra with respect to the fundamental absorption resonances. This shift progressively increases with the number of layers. It is pretty large (∼150 meV) for the WSe 2 bilayer and up to ∼500 meV in the bulk sample. The logic of indirect and low energy transitions giving rise to the PL spectra but direct and higher energy transitions determining the absorption resonances is consistent with the indirect fundamental band gap in all WSe 2 layers with N≥2. Instead, however, the monolayer WSe 2 is believed to be a direct band gap semiconductor. Its strong emission intensity and partially overlapping emission and absorption spectra (practically common emission and absorption peak due to A exciton) are in favor of this scenario. The main part of the low temperature emission spectrum of WSe 2 monolayer appears still below the fundamental absorption resonance, what is, however, commonly attributed to the recombination process due to localized/bound and/or charged excitonic complexes.

Absorption resonances versus layer thickness

In Fig. 2(a), we present the results of RC experiments performed at T = 5 K for the same flakes in a wide range of energy (spectra are shifted vertically for clarity). Three dominant resonances, labelled A, B and C are observed for all the different flakes. A and B are attributed to the two ground state (n = 1) excitonic absorptions at the K-point of the Brillouin zone while the origin of the excitonic absorption C (labeled as A in some works following Ref. 8) is still a matter of debate. [START_REF] Wilson | The Transition Metal Dichalcogenides Discussion and Interpretation of the Observed Optical, Electrical and Structural Properties[END_REF][START_REF] Zhao | Evolution of Electronic Structure of Atomically Thin Sheets of WS2 and WSe2[END_REF][START_REF] Zhao | Lattice Dynamics in Monoand Few-Layer Sheets of WS 2 and WSe 2[END_REF][START_REF] Del Corro | Excited Excitonic States in 1L, 2L, 3L and Bulk WSe 2 Observed by Resonant Raman Spectroscopy[END_REF][START_REF] Kormányos | Theory for Two-Dimensional Transition Metal Dichalcogenide Semiconductors[END_REF] Recent k•p calculations predict van-Hove singularities at both K and M points, with a transition energy ∼3 eV for 1 ML. [START_REF] Kormányos | Theory for Two-Dimensional Transition Metal Dichalcogenide Semiconductors[END_REF] Considering an excitonic binding energy equal to that of Aexciton (0.37 eV) [START_REF] He | Tightly Bound Excitons in Monolayer WSe 2[END_REF] as a first approximation, then the exciton transition energy of ∼2.6 eV agrees with our observation of C-exciton. Another report predicted theoretically that the possible transition at M-point should red shift by ∼100 meV when the flake thickness is increased from 2 ML to bulk. [START_REF] Zhao | Origin of Indirect Optical Transitions in Few-Layer MoS 2 , WS 2 , and WSe 2[END_REF] This is in qualitative agreement with our observed red shift of ∼140 meV in C-exciton indicating that this feature possibly originates from the M-point of the Brillouin zone. In addition, the behaviour of C-exciton is in contrast with that shown by the A-exciton transition where the ob- shows the absorption coefficient of the flakes derived from the lineshape fittings. In both the plots, the spectra for the layers with thickness ≥2 ML have been shifted along the y-axis for clarity.

served shift is only a ∼20 meV, and the former is therefore unlikely to have the much debated similar origins as A-exciton i.e. at the K-point. In all the flakes, A and B excitons are accompanied by a broad shoulder at high energy side of the excitons.

We associate these features with the contributions due to the excited state (n = 2 and onwards) A * and B * exciton transitions. This is in accordance with a recent study where the excited states of A exciton in WSe 2 MLs were observed at an energy range lying ∼160 meV to ∼370 meV above the A resonance. [START_REF] He | Tightly Bound Excitons in Monolayer WSe 2[END_REF] A collective contribution of all these states gives rise to a rather broad feature in RC spectra. Furthermore, in the case of 1 ML thick flake, a feature appears towards lower energy side of the A-exciton. As discussed later, we identify this shoulder as the spectral contribution due to the formation of the trion T (cherged exciton). Additional high energy features appear in the RC spectra of the flakes thicker than 2 ML. An assignment of these features requires a more detailed theoretical and experimental work.

For the lineshape analysis of the RC spectra, we followed a method similar to that described in Ref. 31. In this method, we considered the excitonic contribution to the dielectric response function to be given by a Lorentz oscillator like model as

ε(E) = (n b + ik b ) 2 + ∑ p A p (E 2 p -E 2 -iγ p E) . (1)
where n b + ik b represents the background complex refractive index of WSe 2 in the absence of excitons and was assumed to be equal to that of the bulk material 10 in the simulations. The index p stands for the type of exciton characterized by a resonance energy of E p , with amplitude A p and a phenomenological broadening parameter γ p [equal to full width at half maximum (FWHM) of the lorentzian function]. The RC was then calculated using the transfer matrix formalism. To fully reproduce our experimental data, we introduced three additional oscillators in the dielectric function of the WSe 2 layers to account for the two observed exciton excited states and for the trion feature. The refractive indices of Si and the SiO 2 layers were obtained from Ref. 32. The results of this calculation for our different flakes are presented in Fig. 2(a) as dashed gray lines, together with the corresponding experimental data. We notice that the overall background shape of the reflectance contrast spectra has been reproduced quite well using transfer matrix method. Also, the Lorentz oscillator model has been able to reproduce the exciton lineshapes nicely. It is then tempting to express these results in the form of the absorption coefficient α(λ ), defined as

α(λ ) = 4πk λ . ( 2
)
where k is the imaginary part of the refractive index. The calculated wavelength dependence of the absorption coefficient for the flakes are presented in Fig. 2(b). The transition energies derived from the modelling of our data are presented in Fig. 3 as a function of the flake thickness. Both A and C-exciton energies decrease with increasing the number of layers, the effect being most pronounced for C. The energy of the B exciton appears to be only weakly dependent on the number of layers. Interestingly, the energy of both excited states A * and B * increases from 1 ML to 2 ML and then decreases for larger thicknesses. This peculiar evo- lution observed only for the exciton excited states is due to two competing effects. The first effect is the reduction of the exciton binding energy, possibly due to the change of the dielectric environment and of the dimensionality of the material from 2D-towards a more 3D system. The second effect is characteristic of TMDCs monolayers and arises from the nonlocal character of screening of the Coulomb interactions in such systems. [START_REF] He | Tightly Bound Excitons in Monolayer WSe 2[END_REF][START_REF] Chernikov | Exciton Binding Energy and Nonhydrogenic Rydberg Series in Monolayer WS 2[END_REF][START_REF] Qiu | Optical Spectrum of MoS 2 : Many-Body Effects and Diversity of Exciton States[END_REF] Indeed, exciton excited states in 1 ML WSe 2 strongly deviate from the hydrogenic Rydberg model, [START_REF] He | Tightly Bound Excitons in Monolayer WSe 2[END_REF] verified for conventional 2D semiconductors. They were experimentally observed much closer in energy to the ground state than expected. This effect dominates in the case of the 1 ML flake but ceases to contribute when the number of layers is increased. When going from the monolayer to the thicker flakes, the excited state energy increases in a first step because the exciton structure recovers its Rydberg character but simultaneously, the exciton binding energy decreases when going towards the 3D limit. The latter effect dominates for stacks thicker than 3 ML and the excited state energy then decreases. This scenario has to be confirmed on theoretical grounds.

Assuming that the dominant contribution to the A * structure for the bulk comes from the n = 2 state of the exciton, we find an A exciton binding energy of ∼60 meV following the ideal Rydberg model for 3D case. This is in agreement with a previous report. [START_REF] Beal | Transmission Spectra of Some Transition Metal Dichalcogenides: II. Group VIA: Trigonal Prismatic Coordination[END_REF] We notice that for 2 ML flake, the B * resonance merges with the C exciton peak and is omitted from the fitting procedure. For the bulk flake, the B * structure is too weak to be included in the analysis. From the RC fitting of the 1 ML thick flake, the trion (T) binding energy is estimated to be ∼30 meV which agrees quite well with a recently reported value. [START_REF] Wang | Valley Dynamics Probed Through Charged and Neutral Exciton Emission in Monolayer WSe 2[END_REF]

Ground state exciton: Temperature activated energy shift and broadening

To explore the properties of A exciton and T trions further, we performed µRC spectroscopy of the 1 ML thick flake as a function of temperature. In addition, we performed temperature dependent µRC measurements of A exciton in a 2 ML and a 20 nm thick bulk flake. Figures 4(a-c) show the evolution of the RC spectra around the A exciton peak as a function of temperature (5 K to 300 K) for 1 ML, 2 ML and 20 nm thick flake respectively, along with the calculated spectra. In the fitting procedure, we have also considered T for 1 ML and A * for the bulk flake. Commonly for the three studied films, the exciton resonance energies undergo a red shift with increasing temperature as shown in Fig. 4(d). This evolution, very typical for semiconductors, can be reproduced by the bandgap versus temperature relations, equally well when using the Varshni [START_REF] Varshni | Temperature Dependence of the Energy Gap in Semiconductors[END_REF] or O'Donnell et al. [START_REF] O'donnell | Temperature Dependence of Semiconductor Band Gaps[END_REF] formulae. Varshni's relation is given by:

E g (T ) = E 0 -(αT 2 )/(T + β ). ( 3
)
where E 0 is the band gap at absolute zero, α and β are the fitting parameters related to the temperature-dependent dilatation of the lattice and Debye temperature respectively. Although, this relationship describes the evolution of band gap, it reproduces the exciton resonance energies equally well, which suggests the independence of exciton binding energy on temperature. The parameters which fit the data are given in Table 1. We no- tice that the parameter α shows a reduction as the thickness of the flake is increased from 1 ML to 2 ML and bulk. Notably, there is a fundamental change in the crystal lattice when the thickness is increased from 1 ML thickness onwards. However, the β parameter is ∼170 K for all the three kinds of flakes, and it was fixed to this value during the fitting. The effective Debye temperature seems therefore to be independent of the film thickness. Our estimation of β parameter of WSe 2 is in agreement with a previous report of Hu et al. where a value of β = 160 K was found from the photoconductivity measurements on bulk crystals. [START_REF] Hu | Urbach Tail in the Absorption Spectra of 2H-WSe 2 Layered Crystals[END_REF] The relation provided by O'Donnell et al. describes the band gap dependence on temperature in terms of average phonon energy hω , as the following

E g (T ) = E 0 -S hω [coth( hω /2kT ) -1]. ( 4
)
where S is a coupling constant. The parameters obtained for the best fit are given in Table 1. Similar to β parameter, hω stays nearly the same ∼15 meV, and its value was fixed during fitting. The S parameter shows a gradual decrease from the monolayer to the bulk. Both relations produce seemingly indistinguishable fits to the data, and the curves corresponding to Eq. 4 are shown in Fig. 4(d). Figure 4: (a-c) show the reflectance contrast spectra of the 1 ML, 2 ML and bulk flakes as a function of temperature, in an energy region around the lowest energy exciton peak. The dashed lines represent the corresponding lineshape fittings to the spectra. The spectra were measured at 5 K (not available for 1 ML in this set of measurements), 20 K and then upto 300 K in steps of 20 K. In all the cases, the spectra have been shifted along the y-axis for clarity. (d) Filled squares, circles and triangles show the variation of the exciton band gap of 1 ML, 2 ML and the bulk flake respectively, as a function of temperature. The grey lines are the fits to the data using Eq. 4.

from the fits in Figs. 4(a-c). In semiconductors, the temperature dependence of the width of the n = 1 ground state exciton has the form 37

γ(T ) = γ 0 + σ T + γ 1 e hω/kT -1 . ( 5
)
where γ 0 is the broadening at 0 K, the term linear in T depicts the interaction of excitons with acoustic phonons and the last term arises from interaction with LO (longitudinal optical) phonons. The former is neglected being negligibly small compared to the latter, which is proportional to the Bose function for LO-phonon occupation. hω is the LO-phonon energy and is taken to be equal to 31.25 meV (250 cm -1 ) for WSe 2 . [START_REF] Tonndorf | Photoluminescence Emission and Raman Response of Monolayer MoS 2 , MoSe 2 , and WSe 2[END_REF] The fits to the data are shown as solid lines in Fig. 4(e) and the fitting parameters are shown in Table 1. We notice that γ 0 , which depends on the film quality, is different for the three films studied. However, γ is similar for 1 ML and 2 ML flakes, whereas it is larger for the bulk film. It can be understood as follows. For 1 ML and 2 ML films, the binding energy of the exciton is much larger than the LO phonon energy, therefore the probability of scattering of the ground state exciton to its excited bound and continuum states through the LOphonons is negligible. [START_REF] Rudin | Temperature-Dependent Exciton Linewidths in Semiconductors[END_REF] In such a case, the scattering of the exciton assisted by an annihilation of LO-phonons is mostly permissible within the n = 1 state, to higher in energy excitonic states with larger center-of mass momentum. However, for the bulk case, the exciton binding energy is comparable to that of LO-phonon energy. Therefore, the ground state exciton has a significant probability to scatter to the energetically accessible excited exciton states through annihilation of an LOphonon. Such additional contributions give rise to an increase in broadening in the bulk flake as compared to the 1 and 2 ML flakes at elevated temperatures, in turn increasing the slope of the γ(T) curve.

Excitons and charged excitons in 1 ML flake

The monolayers of SC-TMDCs are commonly accepted to be the direct band gap semiconductors and their optical response in the vicinity of the band edge is understood in terms of the associated excitonic resonances (free-, charged-and localized/bound excitons). At low temperatures and in the presence of impurities and/or free carries, the localized/bound and/or charged exciton states (trions) usually dominate the PL response of direct gap semiconductors whereas the free exciton emission is rather weakly pronounced. This picture can also be applied to account for low temperature spectra of our WSe 2 monolayer, as shown in the bottom traces in Fig. 5(a). These spectra consist of up to four emission peaks which, in line with the previous studies, can be attributed to the neutral exciton (A), charged exciton (T, trion) and two localized/bound excitons (L1 and L2). [START_REF] Wang | Valley Dynamics Probed Through Charged and Neutral Exciton Emission in Monolayer WSe 2[END_REF] The observed evolution of the µPL spec-tra with temperature is, however, intriguing. To begin with, when the temperature is raised, the L1 and L2 peaks disappear from the spectra (at ∼40 K) initially whereas the higher energy trion emission survives up to considerable higher temperatures (up to ∼200 K). The opposite would be expected when naively assuming that lower in energy excitonic complexes should have higher dissociation/activation energies. This assumption may not, however, apply to TMDCs' monolayers for which the T complexes have been speculated to exhibit higher dissociation energies than the impurity-bound excitons, [START_REF] Aleiner | Threeparticle Complexes in Two-Dimensional Seminconductors[END_REF] in overall agreement with our observation. Regardless of the apparent dissociation energies, the PL dynamics is another relevant parameter which has to be taken account when discussing the PL changes with temperature. Time resolved PL studies of WSe 2 monolayers show that A-exciton emission decays rather fast (few picoseconds decay time) whereas the decay time of the subsequent lower in energy PL peaks progressively increases up to ∼100 ps for the L2 peak. [START_REF] Wang | Valley Dynamics Probed Through Charged and Neutral Exciton Emission in Monolayer WSe 2[END_REF] The temperature-activated population of higher energy states, which presumably give rise to the emission more effectively could also qualitatively account for the observed temperatureactivated redistribution of the PL intensity among different emission peaks. This redistribution itself can be qualitatively understood but the observed rise of the integrated PL intensity with temperature (inset to Fig. 5(a)) is a veritable puzzle. Typically in semiconductors, a quenching of the PL intensity is observed with increasing temperature. The observed rise of the PL yield (integrated intensity) is particularly striking because it appears at relatively higher temperatures where all the emission appears to arise due to recombination of free excitons. We do not have a solid explanation for this effect but cannot restrain from speculating on its possible origin. We put forward an hypothesis where we speculate an existence of certain dark, optically inactive states in the WSe 2 monolayer, below the ground state A exciton. Those dark states could be associated with the excitons indirect in k-space, which is however in contradiction with a general belief that the WSe 2 monolayer is a direct gap semiconductor, [START_REF] Zhao | Evolution of Electronic Structure of Atomically Thin Sheets of WS2 and WSe2[END_REF] (though the consensus in this matter may not have been reached so far [START_REF] Ramasubramaniam | Large Excitonic Effects in Monolayers of Molybdenum and Tungsten Dichalcogenides[END_REF] ). Optionally one could spec- ulate about other dark excitonic states, similar to the exchange-split dark excitons in conventional semiconductors, [START_REF] Chen | Exchange Effects on Excitons in Quantum Wells[END_REF] in case of the specific alignment of the signs of the spin-orbit splitting in the conduction and valence bands of the WSe 2 monolayer. [START_REF] Kośmider | Large Spin Splitting in the Conduction Band of Transition Metal Dichalcogenide Monolayers[END_REF] The existence of such states could also qualitatively account for a double exponential decay of the A-emission reported recently from time resolved experiments. [START_REF] Wang | Valley Dynamics Probed Through Charged and Neutral Exciton Emission in Monolayer WSe 2[END_REF] Nonetheless, firm conclusions about the fine structure of the fundamental exciton in WSe 2 monolayer require further experiments and theoretical studies.

The appearance of charged excitons in SC-TMDCs has been largely discussed in literature, [START_REF] Jones | Optical Generation of Excitonic Valley Coherence in Monolayer WSe 2[END_REF][START_REF] Mak | Tighly Bound Trions in Monolayer MoS 2[END_REF] though their identification has been so far based on the PL spectroscopy only. Most convincing with this respect are the PL experiments carried out as a function of the gate voltage (carrier concentration), [START_REF] Jones | Optical Generation of Excitonic Valley Coherence in Monolayer WSe 2[END_REF] nevertheless the distinction between the bound/localized and (free) trion states as observed in low temperature PL is rather a subtle issue. Vast studies of conventional two dimensional structures (e.g. CdTe [START_REF] Kheng | Observation of Negatively Charged Excitons Xin Semiconductor Quantum Wells[END_REF] and GaAs [START_REF] Finkelstein | Optical Spectroscopy of a Two-Dimensional Electron Gas Near the Metal-Insulator Transition[END_REF] show that charged excitons are characterized by the significant oscillator strength and should be also visible in absorption type spectra, whereas bound/localized excitons may be predominant in the PL spectra. With the results shown in Fig. 5, obtained for one of our 1 ML layer flakes, we identified the trion peak (T) on the low energy side of the A exciton in the low temperature PL [Fig. 5(a)] and in the absorption type spectra [Fig. 5(b)] as well. Observation of trions in absorption spectra is further confirmed by spectral evolution with temperature. A characteristic transfer of the oscillator strength from the trion to neutral exciton is observed upon increasing the temperature. This effect is analogous to what has been largely discussed in the context of optical studies of (weakly doped) CdTe and GaAs quantum wells. [START_REF] Finkelstein | Optical Spectroscopy of a Two-Dimensional Electron Gas Near the Metal-Insulator Transition[END_REF][START_REF] Ciulin | Radiative Behavior of Negatively Charged Excitons in CdTe-Based Quantum Wells: A Spectral and Temporal Analysis[END_REF][START_REF] Kossacki | Optical Studies of Charged Excitons in II-VI Semiconductor Quantum Wells[END_REF][START_REF] Suris | Excitons and Trions Modified by Interaction with a Two-Dimensional Electron Gas[END_REF][START_REF] Esser | Theory of Trion Spectra in Semiconductor Nanostructures[END_REF] Generally speaking, the loss and gain of the oscillator strength of charged and neutral exciton with temperature, respectively, is due to the thermal redistribution of free carriers from Fermi-Dirac statistics at low temperature towards Boltzmann distribution at higher temperatures. The increase of temperature opens the k-space around the band edge for the efficient formation of the neutral exciton which leads to its increased oscillator strength. On the other hand, the weakening of the oscillator strength of charged excitons associated with carriers with higher k-wave vectors explains the progressive disappearance of the trion absorption with temperature. Quantitative analysis of the transfer of the oscillator strength from trion to neutral exciton remains to be done in future, on samples with the controlled carrier concentration. Interestingly, the evolution of the optical response with carrier concentration may scale very differently in SC-TMDCs monolayers than in conventional semiconductors, due specific form of Coulomb interaction and non-Rydberg character of excitons in the former systems. [START_REF] He | Tightly Bound Excitons in Monolayer WSe 2[END_REF][START_REF] Chernikov | Exciton Binding Energy and Nonhydrogenic Rydberg Series in Monolayer WS 2[END_REF]

Conclusions

In conclusion, we have studied the excitonic band structure of layered WSe 2 semiconductor as a function of layer thickness and temperature. We presented a comprehensive lineshape analysis of the RC spectra, and deduced the exciton transition energies. The exciton transitions taking place at the K-point and M-point of the Brillouin zone were observed in the spectral energy range of ∼1.6 -2.7 eV. The parameters dictating the temperature activated energy shift and broadening of the fundamental absorption resonance have been determined for 1 ML, 2 ML and bulk WSe 2 samples. The formation of a charged exciton (trion) in WSe 2 monolayer has been traced with absorption type experiments and the transfer of the oscillator strength from the trion to exciton has been observed upon increasing the temperature. The observation of a striking rise of the PL yield with temperature in a monolayer WSe 2 opens up a scope for further theoretical and experimental investigations of this system.

Methods

The monolayer (ML) and a few layer (FL) WSe 2 flakes were obtained on Si/(100 nm)SiO 2 substrate using mechanical exfoliation techniques. [START_REF] Novoselov | Two-Dimensional Atomic Crystals[END_REF] The flakes of interest were first identified by visual inspection under a microscope. A transfer matrix based calculation was performed to predict the color of the flake as a function of its thickness. [START_REF] Hecht | [END_REF] The observed color of the flake was then compared with the theoretically predicted one to roughly estimate the number of layers in the flakes. For performing µPL measurements, the 632.8 nm radiation from He-Ne laser was focused on the flake using a 50x long working distance objective. The sample was mounted on the cold finger of a continuous flow liquid helium cryostat, at a temperature of ∼5 K. The spot diameter was ∼2 µm and the light power focused on the sample was 5 µW for 1 ML, 50 µW for 2 ML and 250 µW for the thicker layers. The PL emission from the sample was dispersed using a 0.5 m focal length monochromator and detected using a liquid nitrogen cooled Si charge coupled device camera. The setup for performing µPLE was similar to the one used for µPL, except that a tunable Ti-sapphire laser with wavelength ranging from 700 nm to 800 nm was used for excitation. The intensity of the emission at a certain detection wavelength for each flake was monitored as a function of the excitation wavelength to record the PLE spectrum. For performing the µRC measurements, the light from a 100 W tungsten halogen lamp was focused on a pinhole of 150 µm diameter. The light was then collimated and focused (spot size ∼4 µm) on the sample. The reflected light was detected using a setup similar to the one used for performing µPL. If R(λ ) and R 0 (λ ) are the reflectance spectra of the WSe 2 flake and the Si/SiO 2 substrate respectively, as a function of the wavelength (λ ), then the percentage reflectance contrast spectrum is defined as follows:

C (λ ) = R(λ ) -R 0 (λ ) R(λ ) + R 0 (λ ) . (6)
The measurements were performed at temperatures ranging from 5 K to 300 K. It must be mentioned that the RC spectrum shown for 1 ML flake in Fig. 2 was measured after approximately 1 month than the spectra shown in Fig. 4(a). We notice that the overall spectral intensity of the RC exciton features was reduced to about half when the sample was around 1 month old due to the effects of sample aging. In addition, the oscillator strength of the trion peak showed an increase after 1 month, which is possibly due the spontaneous doping of the sample in atmosphere. However, the energy positions of the spectral signatures did not show any noticeable shift.

Figure 1 :

 1 Figure 1: Solid lines in (a-e) show the PL spectra for a 1 ML, 2 ML, 3 ML, 4 ML and 20 nm thick flakes respectively. Dashed red lines in (a-c) are the PLE spectra of the corresponding flakes, obtained by monitoring the PL emission intensity at an energy shown by the arrows. The dotted curves are the absorption spectra simulated from the modeling of the reflectance contrast spectra of the corresponding flakes. The insets to (c) and (d) show the optical image of a 3 ML and 4 ML thick flakes whereas the inset to (e) shows a flake which contains 1 ML, 2 ML and 20 nm thick bulk flakes used for the study.

Figure 2 :

 2 Figure 2: (a) represents the reflectance contrast spectra of the flakes shown in Fig. 1 with sample kept at 5 K temperature. The dashed lines are the lineshape fits to the corresponding spectra. (b)shows the absorption coefficient of the flakes derived from the lineshape fittings. In both the plots, the spectra for the layers with thickness ≥2 ML have been shifted along the y-axis for clarity.

Figure 3 :

 3 Figure 3: Transition energies of the features obtained after modeling the reflectance contrast spectra, for the excitons A, B and C, the excited state contributions of the excitons A * and B * and the trion T as a function of number of layers.

Figure 4 (

 4 e) shows the temperature dependence of linewidth broadenings (γ p in Eq. 1

Figure 5 :

 5 Figure 5: (a) and (b) show the PL and absorption spectra (derived from reflectance) for 1 ML thick flake as a function of temperature, in an energy region around the lowest energy exciton peak. The arrows in (a) represent the energy positions of the indicated spectral features at 20K and the inset shows the integrated PL intensity as a function of temperature. (c) Squares and circles show the temperature dependence of the oscillator strength parameter of the exciton, A and the trion, T. Triangles show the total oscillator strength parameter.

Table 1 :

 1 Fitting parameters as obtained from fitting of temperature dependence of the ground state exciton resonance energies and lineshape broadenings for 1 ML, 2 ML and bulk WSe 2 flakes

	Parameter	1 ML 2 ML Bulk
	Using Equation 3
	E 0 (eV)	1.744 1.717 1.692
	α(×10 -4 eV K -1 ) 4.24 3.58 3.44
	β (K)	170	170	170
	Using Equation 4
	E 0 (eV)	1.742 1.716 1.691
	hω (meV)	15	15	15
	S	2.06 1.75 1.67
	Using Equation 5
	γ 0 (meV)	15	34	26
	γ (meV)	78	75	105

Acknowledgement We thank Piotr Kossacki for useful discussions, Ivan Breslavetz for technical support and Sandip Ghosh for valuable suggestions during building of the µRC setup. We acknowledge the support from the EC Graphene Flagship project (No. 604391) and the European Research Council (MOMB project No. 320590).


Zhenzhen You
email: zhenzhen.you@cea.fr

Michel Vandenberghe

Yael Balbastre

Nicolas Souedet

Anne-Sophie Herard

Thierry Delzescaux
email: thierry.delzescaux@cea.fr

Cellular morphometric analysis: from microscopic scale to whole mouse brains

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

Neurodegenerative diseases occur when neurons in the brain and spinal cord begin to deteriorate.

In certain cases such as Alzheimer's disease, cell morphology and function are disturbed (Fig. 1) and pathological aggregates form in the brain. Characterizing the relationship between these anomalies is important to understand the mechanisms involved in this pathology.

Quantifying the morphological changes is crucial and Whole-slide imaging (WSI) offers the unique opportunity to analyze whole brain sections at the cellular level using various histological markers. However, this technique generates terabytes of data which can be difficult to analyze. • This original approach enables:

-to extract and summarize pertinent information from high-resolution qualitative images, -to dramatically reduce (ratio = 65536) the amount of information to be processed.

• Analysis has been extended to other staining of interest (Fig. 4 cells: Nissl staining, amyloid plaques: 6E10 staining) and from brain sections to the entire reconstructed brains in 3D using our in-house software BrainVISA (http://brainvisa.info). • 3D Voxel-wise statistical studies will be realized to investigate cellular structural alterations without a priori between groups as already performed on autoradiography data [START_REF] Dubois | [END_REF] . • The possibility to correlate 3D whole-brain parametric maps with in vivo imaging modalities (MRI, fMRI, PET, in vivo microscopy, etc.) will improve the understanding of the relationship between brain structure and function in disease. • Each section (Fig. 3a) was segmented to detect microglial cells by a machine learning classifier [START_REF] Vandenberghe | EMBC[END_REF] (Fig. 3b). • The microglial cells were labelled (Fig. 3c) and corresponding Voronoi partitions were calculated to estimate their spatial influence i.e. each point was attributed to the closest cell (Fig. 3d). • Based on the original RGB image, labelled image and Voronoi image, parameters of interest (Fig. 2) were calculated for each microglial cell and stored in the form of a table (Fig. 3e). • According to this parametric information, we spatially summarized information (Fig. 3f-g) by generating quantitative heat maps (Fig. 3h-n) at a lower resolution (112 µm) for each parameter of interest. antibody to mark microglial cells and scanned using an AxioScan.Z1 (Zeiss) at a resolution of 0.44 µm (x10) (Fig. 3a).

  g) : Extract information from several pixels in a high resolution image and summarize it in a low resolution image in both horizontal and vertical direction partitions corresponding to each microglial cell to estimate their spatial influence

Figure 1 :

 1 a) microglial cells (brown) in a normal mouse b) activated microglial cells in a mouse model of Alzheimer's disease Summarize information (f-g) based on c) and e)

  a) Supplementary staining tested, b) 3D density maps obtained in whole mouse brains Cellular morphometric analysis: from microscopic scale to whole mouse brains Zhenzhen YOU 1 , Michel VANDENBERGHE 1 , Yael BALBASTRE 1 , Nicolas SOUEDET 1 , Anne-Sophie HERARD 1 , Thierry DELZESCAUX 1 1 CEA/DSV/I²BM/MIRCen, Fontenay-aux-Roses, France Le 23ème Colloque Médecine et Recherche en Neurosciences de la Fondation IPSEN

Figure 2 :Figure 3 :

 23 Figure 2: Parameters of interest. o: mass center, o ͞ p: mean radius, Orientation (θ): angle between the direction of maximal Feret diameter and horizontal axis


N Arrizabalaga

L Le Treust
email: loic.letreust@univ-rennes1.fr

N Raymond

ON THE MIT BAG MODEL IN THE NON-RELATIVISTIC LIMIT

Keywords: 2010 Mathematics Subject Classification. 35J60, 35Q75, 49J45, 49S05, 81Q10, 81V05, 35P15, 58C40 Dirac operator, Hadron bag model, Relativistic particle in a box, MIT bag model, Robin Laplacian, Spectral theory

This paper is devoted to the spectral investigation of the MIT bag model, that is, the Dirac operator on a smooth and bounded domain of R 3 with certain boundary conditions. When the mass m goes to ˘8, we provide spectral asymptotic results.

1. Introduction 1.1. The physical context. In elementary particle physics [START_REF] Griffiths | Introduction to elementary particles[END_REF], the strong force is one of the four known fundamental interaction forces along with the electromagnetism, the weak interaction and the gravitation. It is responsible for the confinement of the quarks inside composite particles called hadrons, such as protons, neutrons or mesons. Its force-carrying (gauge bosons) particles are called the gluons (the force-carrying particles of the electromagnetism are the photons) and they carry together with the quarks, a type of charges called the color charges. Their interactions are detailed in the theory of quantum chromodynamics (the theory of electromagnetism is called quantum electrodynamics).

1.1.1. The standard model. Following the work of Gell-Mann and Zweig and the deep inelastic scattering experiments held at the Stanford Linear Accelerator Center in the 601 s, some physicists introduced in the mid-70 1 s the standard model [START_REF] Hosaka | Quarks, baryons and chiral symmetry[END_REF] in an attempt to give an unified framework for the elementary particle physics. It turned out that this model has been very fruitful since it allows to predict the existence of many particles. Despite of its success, the confinement of the quarks remains badly understood because of the complexity of the associated equations.

1.1.2. An attempt to understand better the confinement of quarks. In parallel to the introduction of the standard model and relying on the work of Bogoliubov [5, Section IV], Chodos, Jaffe, Johnson, Thorn, and Weisskopf [START_REF] Chodos | New extended model of hadrons[END_REF][START_REF] Chodos | Baryon structure in the bag theory[END_REF][START_REF] Chodos | Field-theoretic Lagrangian with baglike solutions[END_REF][START_REF] Johnson | The MIT bag model[END_REF][START_REF] Hosaka | Quarks, baryons and chiral symmetry[END_REF], physicists at the MIT, developped a simplified phenomenological model to get a better understanding of the phenomenons involved in the quark-gluon confinement. Following the results of the experiments held at that time, they chose to include several qualitative properties of the quarks: -the perfect confinement of the quarks inside the hadrons 1 , -the relativistic nature of the quarks2 . The region of space Ω where the quarks live is called the bag and the model is called the MIT bag model. Let us remark that the MIT bag model can also be viewed as a model for a relativistic particle confined in a box. In the non-relativistic setting, the Dirac operator is replaced by the Dirichlet Laplacian and the associated model appears in many courses of introduction to quantum physics [START_REF] Messiah | Quantum mechanics[END_REF]. This model has been successfully used to predict many properties of hadrons (see for instance [START_REF] Degrand | Masses and other parameters of the light hadrons[END_REF]). Let us also mention that the equivalent of the MIT bag model in dimenssion two appears in the study of the graphene and it is referred to it as the infinite mass boundary condition (see [START_REF] Benguria | Spectral gaps of Dirac operators with boundary conditions relevant for graphene[END_REF][START_REF] Stockmeyer | Infinite mass boundary conditions for Dirac operators[END_REF] and the references therein).

1.2. The MIT bag Dirac operator. In the whole paper, Ω denotes a fixed bounded domain of R 3 with regular boundary and m is a real number. The Planck's constant and the velocity of light are assumed to be equal to 1. Let us recall the definition of the Dirac operator associated with the energy of a relativistic particle of mass m and spin 1 2 (see [START_REF] Thaller | The Dirac equation[END_REF]). The Dirac operator is a first order differential operator, acting on L 2 pΩ, C 4 q in the sense of distributions, defined by (1.1) H " α ¨D `mβ , D " ´i∇ , where α " pα 1 , α 2 , α 3 q, β and γ 5 are the 4 ˆ4 Hermitian and unitary matrices given by

β " ˆ12 0 0 ´12 ˙, γ 5 " ˆ0 1 2 1 2 0 ˙, α k " ˆ0 σ k σ k 0 ˙for k " 1, 2, 3 .
Here, the Pauli matrices σ 1 , σ 2 and σ 3 are defined by

σ 1 " ˆ0 1 1 0 ˙, σ 2 " ˆ0 ´i i 0 ˙, σ 3 " ˆ1 0 0 ´1 ˙,
and α ¨X denotes ř 3 j"1 α j X j for any X " pX 1 , X 2 , X 3 q. Let us now impose the boundary conditions under consideration in this paper and define the associated unbounded operator.

Notation 1.1. In the following, Γ :" BΩ and for all x P Γ, npxq is the outwardpointing unit normal vector to the boundary. Definition 1.2. The MIT bag Dirac operator pH Ω m , DpH Ω m qq is defined on the domain DompH Ω m q " tψ P H 1 pΩ, C 4 q : Bψ " ψ on Γu , with B " ´iβpα ¨nq , by H Ω m ψ " Hψ for all ψ P DompH Ω m q. Observe that the trace is well-defined by a classical trace theorem. Notation 1.3. We will denote H " H Ω m when there is no risk of confusion. We denote x¨, ¨y the C 4 scalar product (antilinear w.r.t. the left argument) and x¨, ¨yU the L 2 scalar product on the set U.

Remark 1.4. The operator B defined for all x P Γ is an Hermitian matrix which satisfies B 2 " 1 4 , so that, its spectrum is t˘1u. Both eigenvalues have multiplicity two. Thus, the MIT bag boundary condition imposes to the wavefunction ψ to be an eigenvector of B associated with the eigenvalue `1. This boundary condition is chosen by the physicists, [START_REF] Johnson | The MIT bag model[END_REF], in order to get a vanishing normal flow at the bag surface, ´in ¨j " 0 at the boundary Γ, where the current density j is defined by j " xψ, αψy .

The opposite boundary condition

ψ P kerp1 4 `Bq is discussed in Section 1.3.2.
The following theorem gathers some fundamental properties of the MIT bag Dirac operator that are related to its self-adjointness. iii. Each eigenvalue of H has even multiplicity. iv. For each ψ P DompHq, we have

(1.2) }Hψ} 2 L 2 pΩq " }α ¨∇ψ} 2 L 2 pΩq `m}ψ} 2 L 2 pBΩq `m2 }ψ} 2 L 2 pΩq
and

(1.3) }α ¨∇ψ} 2 L 2 pΩq " }∇ψ} 2 L 2 pΩq `1 2 ż BΩ κ|ψ| 2 ds
where κ is the trace of the Weingarten map:

dn s : T s BΩ ÝÑ R 3 v Þ ÝÑ B v npsq .
Remark 1.6. This theorem is a special case of results in spin geometry (see for instance the survey [START_REF] Bär | Boundary value problems for elliptic differential operators of first order[END_REF]). Let us comment the different points.

i. The proof of the self-adjointness can be found in [START_REF] Bär | Guide to Elliptic Boundary Value Problems for Dirac-Type Operators[END_REF]Theorem 4.11]. Note that self-adjointness results have also been obtained in the case of C 8 -boundaries in [START_REF] Booß-Bavnbek | The Calderón projection: new definition and applications[END_REF] through Calderón projections (see [START_REF] Ourmières-Bonafos | A strategy for self-adjointness of Dirac operators: applications to the MIT bag model and shell interactions[END_REF] where these projections are used in relation with the MIT bag) and sophisticated pseudo-differential techniques, and in two dimensions, with C 2 -boundaries by using Cauchy kernels and the Riemann mapping theorem, [START_REF] Benguria | Spectral gaps of Dirac operators with boundary conditions relevant for graphene[END_REF] (see also [START_REF] Stockmeyer | Infinite mass boundary conditions for Dirac operators[END_REF]). Let us also mention that more general local boundary conditions are considered in [START_REF] Benguria | Spectral gaps of Dirac operators with boundary conditions relevant for graphene[END_REF][START_REF] Booß-Bavnbek | The Calderón projection: new definition and applications[END_REF]. The compactness of the resolvent follows from a classical embedding theorem (see also [START_REF] Bär | Guide to Elliptic Boundary Value Problems for Dirac-Type Operators[END_REF]Corollary 5.6]). ii. The results about the symmetry of the spectrum and the multiplicity follows from the symmetries of the operator. iii. The formula for the square of H is a particular case of a more general formula in [16, p.379]. Basic properties about the Weingarten map may be found in [START_REF] Spivak | A comprehensive introduction to differential geometry[END_REF].

We refer to Appendix A where, for the convenience of the reader, we recall the proofs of some points. From the spectral point of view, we have the following asymptotic result.

Theorem 1.7. Let ´∆Dir be the Laplacian with domain H 2 pΩ, C 4 q X H 1 0 pΩ, C 4 q, and let pµ Dir n q ně1 be the non-decreasing sequence of its eigenvalues. For all n ě 1, we have

µ n pmq ´ˆm `1 2m µ Dir n ˙" mÑ`8 o ˆ1 m ˙.
It is actually possible to describe the next term in the expansion of the first positive eigenvalue.

Theorem 1.8. Let u 1 P H 1 0 pΩ, Cq be a L 2 -normalized eigenfunction of the Dirichlet Laplacian associated with its lowest eigenvalue µ Dir 1 . We have

µ 1 pmq ´ˆm `1 2m µ Dir 1 ´1 2m 2 ż Γ |B n u 1 | 2 dΓ ˙" mÑ`8 o ˆ1 m 2 ˙.
Remark 1.9. This asymptotic expansion of µ 1 pmq coincides with the one of the first eigenvalue of the operator a m 2 ´∆Rob 2m where ´∆Rob 2m is the Robin Laplacian of mass 2m, i.e., the operator of L 2 pΩ, Cq whose quadratic form is defined for u P H 1 pΩ, Cq by

u Þ ÝÑ ż Ω |∇u| 2 dx `2m ż Γ |u| 2 dΓ. 1.3.2.
The MIT bag model with negative mass. Let us now describe our result related to the MIT bag model with "negative mass". This "negative mass" may be understood in two equivalent ways.

i. When we investigate the case Ω " R 3 , the Dirac operators α ¨D `mβ and α ¨D ´mβ are unitarily equivalent. Thus, in the case of a general Ω, one may be tempted to consider α ¨D ´mβ with the MIT bag condition B. ii. Since we have γ 5 pα ¨D `mβq γ 5 " α ¨D ´mβ , γ 5 Bγ 5 " ´B , we notice that α ¨D ´mβ with boundary condition B is unitarily equivalent to α ¨D `mβ with boundary condition ´B. In this case, the flux ´in ¨j also vanishes at the boundary and the justification given by the physicists [START_REF] Johnson | The MIT bag model[END_REF] of the MIT bag boundary condition can also be applied here (see Remark 1.4). Of course, these changes of signs have no effect on the self-adjointness.

Remark 1.10. From the physical point of view, the fact that H Ω 0 does not commutes with the chirality matrix γ 5 (see [START_REF] Thaller | The Dirac equation[END_REF]) is called the chiral symmetry violation [START_REF] Thomas | Advances in nuclear physics: volume 13, chapter Chiral symmetry and the bag model: a new starting point for nuclear physics[END_REF][START_REF] Hosaka | Quarks, baryons and chiral symmetry[END_REF]. The chiral symmetry is supposed to be a property approximately satisfied by light quarks and exactly satisfied for quarks of mass 0.

In this paper, we will show that the limit m Ñ ´8 for the operator H Ω m turns out to be a semiclassical limit and not of perturbative nature as when m Ñ `8. It will be shown that the boundary is attractive for the eigenfunctions with eigenvalues lying essentially in the Dirac gap r´|m|, |m|s and that their distribution is governed by the operator

(1.4) L Γ ´κ2 4 `K ,
where κ and K are the trace and the determinant of the Weingarten map, respectively, and where L Γ is defined as follows.

Definition 1.11. The operator pL Γ , DpL Γ qq is the self-adjoint operator associated with the quadratic form

Q Γ pψq " ż Γ |∇ s ψ| 2 dΓ , @ψ P H 1 pΓ, Cq 4 X kerpB ´14 q .
As a consequence of our investigation, we will get the following lower bound of the quadratic form Q Γ . Proposition 1.12. We have

@ψ P H 1 pΓ, Cq 4 X kerpB ´14 q , Q Γ pψq ě ż Γ ˆκ2 4 ´K˙| ψ| 2 dΓ .
Taking advantage of semiclassical technics, we will establish the following uniform eigenvalues estimate. Theorem 1.13. Let ε 0 P p0, 1q and N ε 0 ,m :" tn P N ˚: µ n p´mq ď m ? 1 ´ε0 u .

There exist positive constants C ´, C `, m 0 such that, for all m ě m 0 and n P N ε 0 ,m , µ ń pmq ď µ n p´mq ď µ ǹ pmq , with µ n pmqbeing the n-th eigenvalue of the operators L Γ,m of L 2 pΓ, Cq 4 defined by

L Γ,ḿ " ˆr1 ´C´m ´1 2 sL Γ ´κ2 4 `K ´C´m ´1˙1 2 `, L Γ,m " ˆr1 `C`m ´1 2 sL Γ ´κ2 4 `K `C`m ´1˙1 2 .
Remark 1.14. By Proposition 1.12,

r1 `C`m ´1 2 sL Γ ´κ2 4 `K `C`m ´1 ě 0 ,
so that the square root is well-defined. In the expression of L Γ,ḿ , we are obliged to take the non-negative part.

Rewriting the previous theorem in term of asymptotic expansions of the eigenvalues, we get the following result (see for instance [START_REF] Kachmar | Weyl formulae for the Robin Laplacian in the semiclassical limit[END_REF]Corollary 3.2]).

Corollary 1.15. For all n P N ˚, we have that

µ n p´mq " mÑ`8 r µ 1 2 n `Opm ´1 2 q,
where pr µ n q nPN ˚is the non-decreasing sequence of the eigenvalues of the following non-negative operator on L 2 pΓ, Cq 4 X kerp1 4 ´Bq:

L Γ ´κ2 4 `K .
Let us describe the spectrum of the effective operator on the boundary in the case when Ω is a ball (see [START_REF] Thaller | The Dirac equation[END_REF]Section 4.6]).The proof of the following proposition just follows from straightforward computations. Proposition 1.16. Assume that Ω " Bp0, Rq with R ą 0. Let A " βp1 `2S ¨Lq be the "spin-orbit" operator where S " 1 2 γ 5 pα 1 , α 2 , α 3 q and L " x ˆD. We have AB " BA ,

L Γ ´κ2 4 `K " R ´2A 2 ,
and its spectrum is tn 2 {R 2 , n P N ˚u.

1.4. Remarks. Let us conclude this introduction with some comments related to Robin Laplacians, δ-interactions and shell-interactions.

1.4.1. Comparison to Robin Laplacians. Theorem 1.13 shares common features with the known results about the Robin Laplacian in the strong coupling limit (see [START_REF] Pankrashkin | An effective Hamiltonian for the eigenvalue asymptotics of the Robin Laplacian with a large parameter[END_REF] for the asymptotic of individual eigenvalues and [START_REF] Kachmar | Weyl formulae for the Robin Laplacian in the semiclassical limit[END_REF] in relation with the spectral uniformity and the semiclassical point of view). But two major differences have to be emphasized. Firstly, the effective operator is not semiclassical in our case (it looks like the effective operator in the case of a Schrödinger operator with a strong attractive δ-interaction on Γ, see [START_REF] Dittrich | On eigenvalue asymptotics for strong δ-interactions supported by surfaces with boundaries[END_REF]). Secondly, the effective operator in our case is a quadratic function of the principal curvatures (and not a linear one as in the Robin case). These differences are crucially related to the vectorial nature of the Dirac operator with the MIT conditions: they lead to a kind of semiclassical degeneracy. It is also rather surprising that the order of this degeneracy is still less than the order of the famous Born-Oppenheimer correction. Here, by the Born-Oppenheimer method, we mean a semiclassical method of reduction to the boundary explained in Sections 3 and 4 (see also [START_REF] Lampart | The adiabatic limit of the Laplacian on thin fibre bundles[END_REF], [START_REF] Jecko | On the mathematical treatment of the Born-Oppenheimer approximation[END_REF], [START_REF] Raymond | Bound States of the Magnetic Schrödinger Operator[END_REF]Chapter 13], and the references therein).

Shell interactions.

There is a close relation between the MIT bag model that we study in this work and the shell interactions for Dirac operators studied in [START_REF] Arrizabalaga | Shell interactions for Dirac operators: on the point spectrum and the confinement[END_REF]. In [1, Theorem 5.5], the authors prove that H `Ves generates confinement with respect to Γ for λ 2 e ´λ2 s " ´4, where

V es ψ " 1 2
pλ e `λs βqpψ ``ψ ´q dΓ , λ e , λ s P R, ψ ˘are the non-tangential boundary values of ψ on Γ and dΓ is the surface measure on Γ. By using [1, Proposition 3.1], it is possible to see that the existence of eigenvalues for H `Ves is equivalent to a spectral property of some bounded operators on Γ. More precisely,

(1.5) kerpH `Ves ´µq ‰ 0 ðñ kerpλ s β ´λe `4C σ,µ q ‰ 0 , where C σ,µ is a Cauchy-type operator defined on Γ in the principal value sense.

In the regime λ 2 e ´λ2 s " ´4, the right hand side of p1.5q is also equivalent to the existence of a solution ψ P H 1 pΩ, C 4 q of the boundary value problem pH ´µqψ " 0 in Ω and ψ " i 2 pλ e ´λs βqpα ¨nqψ on Γ. Observe that when λ e " 0 and λ s " 2 we recover the MIT bag model given in Definition 1.2. It is worth pointing out that the right hand side of p1.5q does not hold for λ s ą 0 if µ P r´m, ms. So the eigenvalues must belong to Rzr´m, ms for λ s ą 0, as we already know from [23, Section 5] in the case λ e " 0 and λ s " 2.

1.5. Organization of the paper. Section 2 is devoted to the proofs of Theorems 1.7 and 1.8. The remaining sections are concerned with the case of the large negative mass. In Section 3, we explain the main steps towards the proof of Theorem 1.13. In Section 4, we prove the propositions and theorems stated in Section 3.

Large positive mass

This section is devoted to the proofs of Theorems 1.7 and 1.8. For that purpose, one will work with the square of the Dirac operator H 2 appearing in Theorem 1.5 and determine the asymptotic expansion of its lowest eigenvalues.

For m ą 0 and ψ P D " tψ P H 1 pΩ, C 4 q, ψ P ker pB ´14 q on Γu, we let

Q m pψq " }∇ψ} 2 `żΓ ´m `κ 2 ¯|ψ| 2 dΓ .
In addition, we also define, for ψ P H 1 0 pΩ, C 4 q, Q 8 pψq " }∇ψ} 2 .

Let us denote by pλ j pQ m qq jě1 and pλ j pQ 8 qq jě1 , the ordered sequence of eigenvalues related to the operators associated with the quadratic forms Q m and Q 8 . Their respective L 2 -normalized eigenfunctions are denoted by ψ j,m and ψ j,8 .

2.1. First non-trivial term in the asymptotic expansion. Theorem 1.7 is a consequence of the following proposition and of Theorem 1.5.

Proposition 2.1. For all j ě 1, we have lim mÑ`8 λ j pQ m q " λ j pQ 8 q .

Proof. Since H 1 0 pΩ, C 4 q Ă D, we have, for all n ě 1, λ n pQ m q ď λ n pQ 8 q .

Let us fix N ě 1 and consider an orthonormal family pψ j,m q 1ďjďN such that ψ j,m is an eigenfunction of the operator related to Q m and associated with its j-th eigenvalue.

We set E N pmq " span pψ j,m q 1ďjďN . We easily get that, for all ψ P E N pmq,

Q m pψq ď λ N pQ m q}ψ} 2 ď λ N pQ 8 q}ψ} 2 .
Let us first prove that λ 1 pQ m q converges towards λ 1 pQ 8 q. For that purpose, let us establish that the only accumulation point of pλ 1 pQ m qq mě0 is λ 1 pQ 8 q. Since pψ 1,m q is bounded in H 1 pΩq, we may assume, up to a subsequence extraction, that ψ 1,m converges weakly to ψ 1,8 P H 1 pΩq. But, we have ż

Γ |ψ 1,m | 2 dΓ " Opm ´1q ,
and by the Fatou lemma, ψ 1,8 " 0 on Γ so that ψ 1,8 P H 1 0 pΩq. Then, we get λ 1 pQ 8 q ě lim mÑ`8

λ 1 pQ m q ě lim inf mÑ`8 }∇ψ 1,m } 2 ě }∇ψ 1,8 } 2 ě λ 1 pQ 8 q .
We deduce that ψ 1,8 is an eigenfunction of the Dirichlet Laplacian associated with λ 1 pQ 8 q. Therefore, we obtain the convergence result for the first eigenvalue. We also get that pψ 1,m q converges strongly in H 1 pΩq to ψ 1,8 .

Let us now proceed by induction. Let N ě 1. Assume that, for all j P t1, . . . , Nu, pλ j pQ m qq converges to λ j pQ 8 q. Assume also that, up to a subsequence extraction, pψ j,m q converges to an eigenfunction associated with λ j pQ 8 q, ψ j,8 . As above, we may assume that pψ N `1,m q converges weakly to some ψ N `1,8 P H 1 pΩq and that its trace on Γ is zero. We also get, by convergence in L 2 pΩq, that ψ N `1,8 P ˆspan 1ďjďN ψ j,8 ˙K .

By the min-max principle, it follows that

λ N `1pQ 8 q ě lim mÑ`8 λ N `1pQ m q ě lim inf mÑ`8 }∇ψ N `1,m } 2 ě }∇ψ N `1,8 } 2 ě λ N `1pQ 8 q .
From these last inequalities, we infer that ψ N `1,8 is an eigenfunction of the Dirichlet Laplacian associated with λ N `1pQ 8 q, that pλ N `1pQ m qq converges to λ N `1pQ 8 q and pψ N `1,m q converges strongly in H 1 pΩq to ψ N `1,8 .

Asymptotic expansion of the first eigenvalue.

The following lemma will be used in the proof of Theorem 1.5. Proof. We have ∇u " pB n uq n so that by integration by parts, we get ż

Γ |B n u| 2 n dΓ " ż Γ |∇u| 2 n dΓ " ż Ω ∇|∇u| 2 dx " ˆżΩ 2∇u ¨∇B k u dx ˙k"1,2,3 " 2 ˆżΩ p´∆uqB k u dx `żΓ B n uB k u dΓ ˙k"1,2,3 " 2 ˆλj pQ 8 q{2 ż Ω B k |u| 2 dx `żΓ B n uB k u dΓ ˙k"1,2,3 " 2 ż Γ B n u∇u dΓ " 2 ż Γ |B n u| 2 n dΓ ,
and the conclusion follows.

Theorem 1.8 is a consequence of the following proposition and of Theorem 1.5.

Proposition 2.3. Let u 1 P H 1 0 pΩq be an L 2 -normalized eigenfunction of the Dirichlet Laplacian associated with its lowest eigenvalue λ 1 pQ 8 q. We have that

λ 1 pQ m q " λ 1 pQ 8 q ´1 2m ż Γ |B n u 1 | 2 dΓ `Opm ´2q .
Remark 2.4. In the case of the Robin Laplacian, we obtain

λ Rob 1 pQ m q " λ 1 pQ 8 q ´1 m ż Γ |B n u 1 | 2 dΓ `Opm ´2q,
and we recover asymptotically the fact that λ Rob 1 pQ m q ď λ 1 pQ m q. Proof. The proof of this result is divided into three steps: (a) we perform a formal study of the asymptotic expansion of λ 1 pQ m q, (b) we build rigorously a test function based on Step (a) to get the upper bound, (c) we study the lower bound.

Step (a). We look for quasi-eigenvalues and quasi-eigenfunctions in the form

λ app 1 pQ m q " λ 1 pQ 8 q `λ m `Opm ´2q , ψ app 1,m " ψ 1,8 `m´1 ϕ `Opm ´2q ,
where λ and ϕ are unknown.

We recall that ψ 1,m and ψ 1,8 satisfy ´∆ψ 1,m " λ 1 pQ m qψ 1,m , on Ω , ψ 1,m P kerpB ´14 q, on Γ , pB n `κ{2 `mqψ 1,m P kerpB `14 q , on Γ . and ´∆ψ 1,8 " λ 1 pQ 8 qψ 1,8 , on Ω,

ψ 1,8 " 0 , on Γ .
Then, we want that p´∆ ´λ1 pQ 8 qq ϕ " λψ 1,8 , on Ω , ϕ P kerpB ´14 q, on Γ , B n ψ 1,8 `ϕ P kerpB `14 q, on Γ .

(2.1) Denoting for all s P Γ, P `psq " 1´Bpsq 2 , the orthogonal projection on kerpB ´14 q, we get that 0 " P `pB n ψ 1,8 `ϕq " P `Bn ψ 1,8 `ϕ .

Taking the scalar product of equation ( 2 Let us now consider λ. Note that for the eigenfunction ψ 1,8 of the Dirichlet Laplacian in L 2 pΩ, C 4 q associated with its lowest eigenvalue λ 1 pQ 8 q, there exists a P C 4 such that |a| " 1 and ψ 1,8 " au 1 . Then, we have

λ " ´1 2 ż Γ |B n u 1 | 2 p1 `xa, Bayq dΓ " ´1 2
ż Γ |B n u 1 | 2 dΓ ´1 2 xa, ´iβα ¨ˆż Γ |B n u 1 | 2 n dΓ ˙ay .
Finally, using Lemma 2.2, we obtain that

λ " ´1 2
ż Γ |B n u 1 | 2 dΓ.
Step (b). Let ψ 1,8 " au 1 be an eigenfunction of the Dirichlet Laplacian associated with λ 1 pQ 8 q and w P H 2 pΩq be such that w " ´P`Bn ψ

Q m pψ 1,8 `m´1 ϕ 1 q " λ 1 pQ 8 q `m´1 ˆ2Re x∇ψ 1,8 , ∇ϕ 1 y Ω `żΓ |ϕ 1 | 2 dΓ ˙`Opm ´2q " λ 1 pQ 8 q}ψ 1,m } 2 L 2 ´m´1 ż Γ |P `pB n ψ 1,8 q | 2 dΓ `Opm ´2q, so that (2.4) λ 1 pQ m q ď λ 1 pQ 8 q ´m´1 ż Γ |P `pB n ψ 1,8 q | 2 dΓ `Opm ´2q .
Step (c). Let us now study the lower bound. The sequence pψ 1,m q is uniformly bounded in H 1 pΩq. We extract a subsequence pm k q kPN such that lim inf

mÑ`8 m pλ 1,m ´λ1,8 q " lim kÑ`8 m k pλ 1,m k ´λ1,8 q
and pψ 1,m k q kPN converges strongly in H 1 pΩq to ψ 1,8 P H 1 0 pΩq and pB n ψ 1,m k q converges to pB n ψ 1,8 q in H ´1{2 pΓq. Integrating by parts yields

pλ 1,m k ´λ1,8 q xψ 1,m k , ψ 1,8 y Ω " ´mk ´1 xpκ{2m k `1q ´1B n ψ m k ,8 , P `Bn ψ 1,8 y Γ , (2.
m pλ 1,m ´λ1,8 q " ´}P `Bn ψ 1,8 } 2 L 2 pΓq ě ´1 2 ż Γ |B n u 1 | 2 dΓ
and the result follows.

3. Large negative mass: main steps in the proof of Theorem 1.13

In this section, we study the non-relativistic limit m Ñ `8 of the nonnegative eigenvalues of the MIT bag Dirac operator H Ω ´m. For the sake of readability, we present the main ingredients used in the proof of Theorem 1.13. Part of the ideas are related to recent results about the semiclassical Robin Laplacian (see [START_REF] Helffer | Tunneling for the Robin Laplacian in smooth planar domains[END_REF]Section 7], [START_REF] Helffer | Eigenvalues for the Robin Laplacian in domains with variable curvature[END_REF] and [START_REF] Kachmar | Weyl formulae for the Robin Laplacian in the semiclassical limit[END_REF]). The detailed proofs will be given in Section 4.

3.1. Semiclassical reformulation and boundary localization. The main objective of this section is to get boundary localization results of Agmon type. For that purpose, we will rather consider `HΩ ´m˘2 and introduce the semiclassical parameter h " m ´2 Ñ 0 .

3.1.1. The semiclassical operator. In order to lighten the presentation, it will also be more convenient to work with the following operator

(3.1) L h " h 2 ppH Ω ´mq 2 ´m2 1 4 q ,
whose domain is given by DompL h q " DomppH Ω ´mq 2 q " ! ψ P H 2 pΩq : ψ P kerpB ´14 q, ´Bn `κ 2 ´h´1 2 ¯ψ P kerpB `14 q, on Γ

) .

The associated quadratic Q h form is defined by

(3.2) @ψ P DompQ h q , Q h pψq " h 2 }∇ψ} 2 L 2 pΩq `żΓ ´κ 2 h 2 ´h3 2 ¯|ψ| 2 dΓ ,
where DompQ h q " DompH Ω ´mq " ψ P H 1 pΩq : ψ P kerpB ´14 q on Γ ( .

In other words, the operator L h is the semiclassical Laplacian with combined MIT bag and Robin conditions on the boundary.

Relations between the eigenvalues of L h and H Ω

´m. Let us describe the relations between the spectra of our operators. Let us recall that the spectrum of H Ω ´m is discrete, symmetric with respect to 0 and with pair multiplicity. The spectrum of H Ω ´m lying in r´m, ms is given by ! ˘ah ´2λ n phq `h´1 : n P Nzt0u , ´h ď λ n phq ď 0

) , where λ n phq denotes the n-th eigenvalue of L h . Therefore, we shall focus on the study of the negative eigenvalues of L h .

Agmon type localization estimates.

The estimates given in Proposition 3.1 are a consequence of the fact that the Laplacian is a non-negative operator.

Proposition 3.1. Let ǫ 0 P p0, 1q and γ P p0, ? ε 0 q. There exists C ą 0 such that for any h P p0, 1s, any eigenvalue λ ď ´ε0 h of L h and any eigenfunction ψ h of L h associated with λ, we have

› › › › ψ h exp ˆγdp¨, Γq h 1{2 ˙› › › › 2 L 2 pΩq `h´1 ˇˇˇQ h ˆψh exp ˆγdp¨, Γq h 1{2 ˙˙ˇˇˇˇď C}ψ h } 2 L 2 pΩq .
3.2. The operator near the boundary. Relying on Proposition 3.1, we introduce the operator near the boundary. Given δ P p0, δ 0 q (with δ 0 ą 0 small enough), we introduce the δ-neighborhood of the boundary

(3.3) V δ " tx P Ω : distpx, Γq ă δu ,
and the quadratic form, defined on the variational space V δ " ! u P H 1 pV δ q : upxq " 0 for all x P Ω such that distpx, Γq " δ and Bu " u on Γ

) , by the formula

@u P V δ , Q tδu h puq " ż V δ |h∇u| 2 dx `żΓ ´κ 2 h 2 ´h3 2 ¯|u| 2 dΓ .
We denote by L tδu h the corresponding operator.

The operator near the boundary in tubular coordinates.

Let ι be the canonical embedding of Γ in R 3 and g the induced metric on Γ. pΓ, gq is a C 3 Riemannian manifold, which we orientate according to the ambient space. Let us introduce the map Φ : Γ ˆp0, δq Ñ V δ defined by the formula Φps, tq " ιpsq ´tnpsq .

The transformation Φ is a C 3 diffeomorphism for any δ P p0, δ 0 q provided that δ 0 is sufficiently small. The induced metric on Γ ˆp0, δq is given by

G " g ˝pId ´tLpsqq 2 `dt 2 ,
where Lpsq " dn s is the second fundamental form of the boundary at s. Let us now describe how our MIT bag -Robin Laplacian is transformed under the change of coordinates. For all u P L 2 pV δ q, we define the pull-back function (3.4) r ups, tq :" upΦps, tqq.

For all u P H 1 pV δ q, we have " ´h2 ã´1 ∇ s pãg ´1∇ s q ´h2 ã´1 B t pãB t q .

(3.5) ż V δ |u| 2 dx " ż Γˆp0,
In these coordinates, the Robin condition becomes

h 2 B t u " ´κ 2 h 2 ´h3
2 ¯u on t " 0 .

We introduce, for δ P p0, δ 0 q, (3.8) r V δ " tps, tq : s P Γ and 0 ă t ă δu , r V δ " tu P H 1 p r V δ q, up¨, 0q P ker pB ´14 q , up¨, δq " 0u , . Using smooth cut-off functions, the min-max principle and the Agmon estimates of Proposition 3.1, it is standard to deduce the following proposition (see [START_REF] Helffer | Semi-classical analysis for the Schrödinger operator and applications[END_REF]). Proposition 3.2. Let ǫ 0 P p0, 1q and γ P p0, ? ǫ 0 q. There exist constants C ą 0, h 0 P p0, 1q such that, for all h P p0, h 0 q, δ P p0, δ 0 q, n ě 1 such that λ n phq ď ´ǫ0 h, (3.9)

r D δ " tu P H 2 p r V δ q X r V δ : B t up¨
λ n phq ď λ tδu n phq ď λ n phq `C exp ´´γδh ´1 2 ¯.

In the following, it is sufficient to choose (3.10) δ " h

3.3.

The rescaled MIT bag operator in boundary coordinates. Looking at the rate of convergence obtained in Proposition 3.1, we perform a change of scale in the normal direction that allows us to see something at the limit. We introduce the rescaling ps, τ q " ps, h ´1 2 tq , the new semiclassical parameter " h 1 4 and the new weights (3.11) p a ps, τ q " ãps, h 1 2 τ q , p g ps, τ q " gps, h 1 2 τ q .

We also introduce the parameter (3.12) T " δh ´1 2 " h ´1 4 " ´1 (see (3.10)). We consider rather the operator

(3.13) x L " h ´1 Ă L h ,
acting on L 2 p p V , p a dΓ dτ q and expressed in the rescaled coordinates ps, τ q.

As in (3.8), we let (3.14) p V " tps, τ q : s P Γ and 0 ă τ ă ´1u , p V " tu P H 1 p p V ; p a dΓ dτ q, up¨, 0q P ker pB ´14 q , up¨, ´1q " 0u , p D " tu P H 2 p p V ; p a dΓ dτ q X p V : B τ up¨, 0q ´´κ 2 2 ´1¯u p¨, 0q P kerpB `14 qu ,

p Q puq " ż p V ´ 4 x∇ s u, p g ´1 ∇ s uy `|B τ u| 2 ¯p a dΓ dτ `żΓ ´κ 2 2 ´1¯| ups, 0q| 2 dΓ , x L " ´ 4 p a ´1 ∇ s pp a p g ´1 ∇ s q ´p a ´1 B τ p a B τ .
3.4. Contribution of the normal variable. Notice that the first order terms in (3.14) are related to the normal variable. Hence, we are naturally led to introduce the following quadratic form gathering all the terms acting in the normal direction:

(3.15)

p V 1 " tu P L 2 p p V ; p a dΓ dτ q, B τ u P L 2 p p V ; p a dΓ dτ q,
up¨, 0q P ker pB ´14 q , up¨, ´1q " 0u ,

x Q 1 puq " ż Γ ´ż ´1 0 |B τ u| 2 p a dτ `´κ 2 2 ´1¯| ups, 0q| 2 ¯dΓ .
The goal of this section is to study the lowest part of the spectrum of the operator x L 1 associated with the quadratic form x Q 1 .

Diagonalization of the boundary condition.

Without the gradient term in the s-direction appearing in p Q puq, the MIT bag boundary condition can be diagonalized for every s P Γ. Let us introduce for all s P Γ, the unitary 4 ˆ4 matrix

P n :" 1 ? 2 ˆ12 iσ ¨n iσ ¨n 1 2 ˙.
We have

P ´1 n BP n " β , thus, for all ψ P p V 1 ,
ϕ " P n ψ P tu P L 2 p p V ; p a dΓ dτ q, B τ u P L 2 p p V ; p a dΓ dτ q, xup¨, 0q, e 3 y " xup¨, 0q, e 4 y " 0, up¨, ´1q " 0u , where xu, e k y is the k-th component of the vector u P C 4 . Since P n is unitary and does not depend on the variable τ , we get that

p Q 1 puq " p Q 1 pP n uq .
Up to this change of variable, the first two components satisfy the following Robin boundary condition ˆBτ `1 ´κ 2 2 ˙up¨, 0q " 0 , whereas the last two satisfy the Dirichlet boundary condition.

The Robin Laplacian on the half-line.

Let C 0 ą 0, κ, K P p´C 0 , C 0 q and 0 ą 0 such that for all P p0, 0 q, a ,κ,K pτ q " 1 ´ 2 κτ ` 4 Kτ 2 P p´1{2, 1{2q .

We introduce the following operator in one dimension (and valued in C), defined on the Hilbert space L 2 pp0, ´1q; a ,κ,K dτ q by (3.16) H Rob ,κ,K " ´a´1 ,κ,K pτ qB τ pa ,κ,K pτ qB τ q " ´B2 τ ` 2 κ ´2 4 Kτ a ,κ,K pτ q B τ , with domain DompH Rob ,κ,K q " tψ P H 2 pp0, ´1q, Cq :

ψp ´1q " ˆBτ `1 ´ 2 κ 2 ˙ψp0q " 0u .
For the associated quadratic form Q Rob ,κ,K , we have, DompQ Rob ,κ,K q " tψ P H 1 pp0, ´1q, Cq , ψp ´1q " 0u ,

Q Rob ,κ,K pψq " ż ´1 0 |B τ ψ| 2 a ,κ,K dτ `ˆ´1 ` 2 κ 2 ˙|ψp0q| 2 .
Let us notice that our Robin Laplacian H Rob ,κ,K on a weighted space looks like the one introduced by Helffer and Kachmar in [START_REF] Helffer | Eigenvalues for the Robin Laplacian in domains with variable curvature[END_REF]. But, here, we have an additional term κ 2 2 in the boundary condition which will have an important impact on the spectrum in the limit Ñ 0. We can also observe that `HRob ,κ,K ˘κ,K is an analytic family of type (B) in the sense of Kato (see [START_REF] Kato | Perturbation theory for linear operators[END_REF]). Let us now describe the bottom of the spectrum of H Rob ,κ,K when goes to 0. Proposition 3.4. The lowest eigenpair pλ 1 `HRob ,κ,K ˘, u ,κ,K q of H Rob ,κ,K satisfies the following. Let ε 0 P p0, 1q. There exist 0 , C ą 0 such that for all P p0, q, there holds ˇˇˇλ 1 `HRob

,κ,K ˘´ˆ´1 ` 4 ˆK ´κ2 4 ˙˙ˇˇˇˇď C 6 , λ 2 `HRob ,κ,K ˘ě ´ε0 {2 ,
and }u ,κ,K ´ψ0 } H 1 pp0, ´1q;a ,κ,K dτ q ď C 2 , where ψ 0 pτ q " ? 2e ´τ . The constants 0 , C ą 0 do not depend on κ, K but depend on C 0 . Notation 3.5. In the following, we use κ " κpsq and K " Kpsq and we let u ,κpsq,Kpsq pτ q " v ps, τ q , λ j `HRob ,κpsq,Kpsq ˘" λ R j ps, q . The asymptotic expansion of the eigenfunction in Proposition 3.4 leads to the following remark (v ps, τ q does not depend very much on s in the semiclassical limit). Remark 3.6. We introduce the "Born-Oppenheimer correction": R psq " }∇ s v } 2 L 2 pp0, ´1q;p a dτ q . It can be shown that (3.17) }R } L 8 pΓq " Op 4 q , by using straightforward adaptations of [START_REF] Helffer | Tunneling for the Robin Laplacian in smooth planar domains[END_REF]Lemma 7.3]. By using (3.17) and an induction procedure, it is also possible to show the same estimate for the second order derivatives: sup sPΓ }∇ 2 s v } L 2 pp0, ´1q;p a dτ q " Op 2 q .

3.4.3. Spectrum of x L 1 . Since the spectrum of the Dirichlet Laplacian is non-negative, Proposition 3.4 gives us immediately the following result. Proposition 3.7. Let ε 0 P p0, 1q. There exist C, 0 ą 0 such that for any P p0, 0 q, we have

spp x L 1 q Ă p´1 ´C 4 , ´1 `C 4 q Y r´ε 0 , `8q.
The L 2 p p V ; p a dΓ dτ q 4 -spectral projection Π :" χ p´1´C 4 ,´1`C 4 q p x L 1 q satisfies Ran Π " tps, τ q P p V Þ Ñ f psqv ps, τ q, f P L 2 pΓ; dΓq 4 X kerp1 4 ´Bqu .

Remark 3.8. Since, s Þ Ñ v ps, ¨q is regular, we also have

Π ψ P H 1 p p V ; p a dΓ dτ q 4
for any ψ P p V . Actually, we can give an explicit expression of Π by using the diagonalization of the MIT condition of Section 3.4.1:

(3.18) Π ψ " v P n ˆ12 0 0 0 ˙P n xψ, v y L 2 pp0, ´1q,p a dτ q ,
where xψ, v y L 2 pp0, ´1q,p a dτ q " pxψ j , v y L 2 pp0, ´1q,p a dτ q q jPt1,...,4u . By taking the derivative of (3.18) with respect to s, by using the Leibniz formula and (3.17), we get the commutator estimate, for ψ P p V , }r∇ s , Π sψ} L 2 p p V ,p a dτ dΓq ď C}ψ} L 2 p p V ,p a dτ dΓq . 3.5. Effective operator on Ran Π . In this section, we compare the lower part of the spectrum of the operator x L with the one of the operator x L eff acting on Ran Π , whose quadratic form gathers all the terms of orders lower or equal than 4 and which is defined by (3.19) p V eff " tu P H 1 p p V ; p a dΓ dτ q 4 , up¨, 0q P ker pB ´14 q , up¨, ´1q " 0u X Ran Π ,

y Q eff puq " p Q 1 puq ` 4 ż p V
|∇ s u| 2 p a dΓ dτ .

We get the following result.

Theorem 3.9. For ε 0 P p0, 1q, ą 0, we let p N ǫ 0 , " tn P N ˚: p λ n p q ď ´ε0 u .

There exist positive constants 0 , C, such that, for all P p0, 0 q and n P p

N ε 0 , , (3.20)
p λ ń p q ď p λ n p q ď p λ ǹ p q , where p λ n p q is the n-th eigenvalue of x L eff,˘ whose quadratic form is defined for all

u P p V eff,˘ " p V eff by p Q eff,˘ puq " p Q 1 puq ` 4 ż p V p1 ˘C q|∇ s u| 2 p a dΓ dτ ˘C 6 ż p V
|u| 2 p a dΓ dτ .

3.6.

Effective operator on the boundary. The aim of this section is to exhibit an effective operator on the boundary Γ. To do so, we will have to study the Born-Oppenheimer correction terms. The effective operator up to the order 4 on the boundary has the following quadratic form:

(3.21) p V Γ,eff " H 1 pΓq X kerp1 4 ´Bq , p Q Γ,eff pf q " ´}f } 2 L 2 pΓq ` 4 ż Γ ´|∇ s f | 2 `´´κ psq 2 4 `Kpsq ¯|f | 2 ¯dΓ .
More precisely, we obtain the following result.

Theorem 3.10. For ε 0 P p0, 1q, ą 0, we let p N ǫ 0 , " tn P N ˚: p λ n p q ď ´ε0 u .

There exist positive constants 0 , C such that, for all P p0, 0 q and n P p

N ε 0 , , (3.22)
p λ Γ,ń p q ď p λ n p q ď p λ Γ,ǹ p q , where p λ Γ,n p q is the n-th eigenvalue of x L Γ,eff,˘ whose quadratic form is defined by: Let us now give the proof of Proposition 3.1.

p V Γ,eff,˘" p V Γ,eff , p Q Γ,eff,˘ pf q " ´}f } 2 L 2 pΓq ` 4 ż Γ ´p1 ˘C q|∇ s f | 2 `´´κ psq 2 4 `Kpsq ˘C ¯|f | 2 ¯dΓ .
Proof. First, we notice that by (3.2), (4.1)

L h ě ´h.

Let us denote by a h p¨, ¨q the sesquilinear form associated with Q h defined in (3.2). Let us define the following Lipschitzian functions x P Ω Þ Ñ Φpxq " γdistpx, BΩq P R and

x P Ω Þ Ñ χ h pxq " e Φpxqh ´1{2 P R . Since χ h is real-valued and Lipschitzian, we get that χ 2 h ψ h belongs to DpQ h q. We have that

a h pψ h , χ 2 h ψ h q " Re xL h ψ h , χ 2 h ψ h y Ω " Re " h 2 x∇ψ h , ∇pχ 2 h ψ h qy Ω `żΓ ´κ 2 h 2 ´h3 2 ¯|χ h ψ h | 2 dΓ * .
By Lemma 4.1, we get that a h pψ h , χ 2 h ψ h q " Q h pχ h ψ h q ´h2 }ψ h ∇χ h } 2 L 2 pΩq . Recall that ψ h is an eigenfunction of L h associated with the eigenvalue λ, so that (4.2)

Q h pχ h ψ h q ´h2 }ψ h ∇χ h } 2 L 2 pΩq " λ}χ h ψ h } 2 L 2 pΩq . Let R ě 1 and c ą 1. We introduce a quadratic partition of unity of Ω,

χ 2 1,h,R

`χ2

2,h,R " 1 , in order to study the asymptotic behavior of ψ h in the interior and near the boundary Γ separately. We assume that χ 1,h,R satisfies

χ 1,h,R pxq " # 1 if distpx, Γq ě h 1{2 R 0 if distpx, Γq ď h 1{2 R{2,
and that, maxp|∇χ 1,h,R pxq|, |∇χ 2,h,R pxq|q ď 2ch ´1{2 {R , for all x P Ω. Using again Lemma 4.1, we get

Q h pχ h ψ h q " ÿ k"1,2 Q h pχ k,h,R χ h ψ h q ´h2 }χ h ψ h ∇χ k,h,R } 2 L 2 pΩq .
We have Q h pχ 1,h,R χ h ψ h q ě 0 because of a support consideration. Let us also remark that

h 2 }χ h ψ h ∇χ k,h,R } 2 L 2 pΩq ď h4c 2 {R 2 }χ h ψ h } 2 L 2 pΩq and h 2 }ψ h ∇χ h } 2 L 2 pΩq ď hγ 2 }χ h ψ h } 2 L 2 pΩq . We deduce from (4.2) that λ}χ h ψ h } 2 L 2 pΩq ě Q h pχ 2,h,R χ h ψ h q ´h}ψ h χ h } 2 L 2 pΩq `γ2 `8r c 2 R ´2˘, thus, (4.3) hpε 0 ´γ2 ´8r c 2 R ´2q}χ h ψ h } 2 L 2 pΩq ď ´Qh pχ 2,h,R χ h ψ h q . By Lemma 4.1, we get that Q h pχ 2,h,R χ h ψ h q " a h pψ h , pχ 2,h,R χ h q 2 ψ h q `h2 }ψ h ∇pχ 2,h,R χ h q} 2 L 2 pΩq " λ}χ 2,h,R χ h ψ h } 2 L 2 pΩq `h2 }ψ h χ h ∇χ 2,h,R } 2 L 2 pΩq `h}ψ h χ 2,h,R χ h ∇Φ} 2 L 2 pΩq `h3{2 2Re xψ h e 2Φh ´1{2 ∇χ 2,h,R , ψ h χ 2,h,R ∇Φy L 2 pΩq ě λ}χ 2,h,R χ h ψ h } 2 L 2 pΩq ´h4c{Rγ}χ h ψ h } 2 L 2 pΩq
. Hence, we obtain by (4.3) and (4.1) that `ε0

´γ2 ´8r c 2 R ´2 ´4cR ´1γ ˘}χ h ψ h } 2 L 2 pΩq ď }χ 2,h,R χ h ψ h } 2 L 2 pΩq ď }ψ h } 2 L 2 pΩq e 2Rγ .
Let us fix R ą 0 so that `ε0 ´γ2 ´8r c 2 R ´2 ´4cR ´1γ ˘ě pε 0 ´γ2 q{2 ą 0 .

We get that }χ h ψ h } L 2 pΩq ď C}ψ h } L 2 pΩq , and the conclusion follows by (4.2).

Proof of Proposition 3.4.

Proof. The proof follows from the method used in [START_REF] Helffer | Eigenvalues for the Robin Laplacian in domains with variable curvature[END_REF] by Helffer and Kachmar. Let us recall the strategy. The operator is

H Rob ,κ,K " ´a´1 ,κ,K pτ qB τ pa ,κ,K pτ qB τ q " ´B2 τ ` 2 κ ´2 4 Kτ a ,κ,K pτ q B τ , ˆBτ `1 ´κ 2 2
˙up¨, 0q " 0 .

We look for quasi-eigenvalues and quasi-eigenfunctions expressed as formal series:

λ " λ 0 ` 2 λ 1 ` 4 λ 2 , ψ " ψ 0 ` 2 ψ 1 ` 4 ψ 2 .
By writing the formal eigenvalue equation, expanding the operator and the boundary condition in powers of 2 , we get the following equations. In the following, the integration interval is p0, `8q. The first equation is

´B2 τ ψ 0 " λ 0 ψ 0 , pB τ `1qψ 0 p0q " 0 .
We get that the solution is λ 0 " ´1 and ψ 0 pτ q " ? 2e ´τ . Then, the second one is `´B 2 τ `1˘ψ

1 " pλ 1 ´κB τ q ψ 0 , pB τ `1qψ 1 p0q ´κ 2 ψ 0 p0q " 0 .

By taking the scalar product with ψ 0 , we find (by the Fredholm alternative) that there is a solution if and only if

x `´B 2 τ `1˘ψ 1 , ψ 0 y L 2 p0,`8q " xpλ 1 ´κB τ q ψ 0 , ψ 0 y L 2 p0,`8q

holds. Note that xB τ ψ 0 , ψ 0 y L 2 p0,`8q " ´1 and that, by integration by parts,

x `´B 2 τ `1˘ψ 1 , ψ 0 y L 2 p0,`8q " xpB τ `1q ψ 1 p0q, ψ 0 p0qy `xψ 1 , `´B 2 τ `1˘ψ 0 y L 2 p0,`8q " κ 2 |ψ 0 p0q| 2 " κ ,
so that λ 1 " 0. We may actually give an explicit expression for a function ψ 1 satisfying `´B 2 τ `1˘ψ

1 " κψ 0 , pB τ `1qψ 1 p0q ´κ 2 ψ 0 p0q " 0 .

The functions κ ´τ ?

2

`c¯e ´τ are a solution for all c P R. We choose c " 0, so that ψ 1 pτ q " κτ ? 2 e ´τ . We can now consider the crucial step. We write `´B 2 τ `1˘ψ

2 " λ 2 ψ 0 ´κB τ ψ 1 ´τ p´2K `κ2 qB τ ψ 0 , pB τ `1qψ 2 p0q ´κ 2 ψ 1 p0q " 0 .

As in the previous case, it is sufficient to find λ 2 such that

x `´B 2 τ `1˘ψ 2 , ψ 0 y L 2 p0,`8q " xλ 2 ψ 0 ´κB τ ψ 1 ´τ p´2K `κ2 qB τ ψ 0 , ψ 0 y L 2 p0,`8q . holds. We have x `´B

˘.

Therefore, the only eigenvalue in the spectrum of H Rob ,κ,K that is close to ´1 ` 4 ´K ´κ2 4 ¯is the first one. The approximation of u ,κ,K follows from elementary arguments and the Agmon estimates (to deal with the cutoff functions). 4.3. Proof of Theorem 3.9. Let us denote Π K " Id ´Π .

Main Lemma.

The proof of the theorem relies on the following lemma (see also [START_REF] Kachmar | Weyl formulae for the Robin Laplacian in the semiclassical limit[END_REF]). Lemma 4.2. There exist C, 0 ą 0 such that the following holds for all P p0, 0 q and all u P p V ,

x Q pΠ uq ď p Q 1 pΠ uq ` 4 p1 `C q ż p V |∇ s Π u| 2 p a dΓ dτ and x Q puq ě p Q 1 pΠ uq ` 4 p1 ´C q ż p V |∇ s Π u| 2 p a dΓ dτ ´C 6 }Π u} 2 L 2 p p V ;p a dΓ dτ q `p Q 1 pΠ K uq ` 4 p1 ´C q ż p V |∇ s Π K u| 2 p a dΓ dτ ´C 2 }Π K u} 2 L 2 p p V ;p a dΓ dτ q ,
Proof. Let us remark first that there exist C, 0 ą 0 such that for all P p0, 0 q, ˇˇż p V ´|∇ s u| 2 p a ´x∇ s u, p a p g ´1 ∇ s uy ¯dΓ dτ ˇˇď C ż p V |∇ s u| 2 p a dΓ dτ , since 0 ă τ ă ´1. Then, the upper bound follows. Let us now focus on the lower bound. Since Π h is a spectral projection of x L 1 , we get that for all u P p V ,

x Q 1 puq " x Q 1 pΠ uq `x Q 1 pΠ K uq .
We also have ż

p V |∇ s u| 2 p a dΓ dτ " ż p V |∇ s `Π u `ΠK u ˘|2 p a dΓ dτ " ż p V |∇ s pΠ uq| 2 p a dΓ dτ `żp V |∇ s pΠ K uq| 2 p a dΓ dτ `2Re ż p V
x∇ s pΠ uq, ∇ s pΠ K uqy p a dΓ dτ .

Let us analyze the double product. We have ż p V x∇ s pΠ uq, ∇ s pΠ K uqy p a dΓ dτ "

ż p V x∇ s ppΠ q 2 uq, ∇ s p `ΠK ˘2 uqy p a dΓ dτ " ż p V xΠ ∇ s pΠ uq, Π K ∇ s pΠ K uqy p a dΓ dτ `żp V xΠ ∇ s pΠ uq, " ∇ s , Π K ‰ Π K uy p a dΓ dτ `żp V xr∇ s , Π s Π u, Π K ∇ s pΠ K uqy p a dΓ dτ `żp V xr∇ s , Π s Π u, " ∇ s , Π K ‰ Π K uy p a dΓ dτ .
Since Π is an orthogonal projection of L 2 p p V , p a dΓ dτ q, we get that Re

ż p V xΠ ∇ s pΠ uq, Π K ∇ s pΠ K uqy p a dΓ dτ " 0 .
Moreover, by commuting Π K and ∇ s , by using an integration by parts and Remark 3.6 (see also Remark 3.8), we have ˇˇż

p V xr∇ s , Π s Π u, Π K ∇ s pΠ K uqy p a dΓ dτ ˇď C ´}Π u} 2 L 2 p p
V ;p a dΓ dτ q `}∇ s Π u} 2 L 2 p p V ;p a dΓ dτ q ¯1{2 }Π K u} L 2 p p V ;p a dΓ dτ q . Using the inequality |2ab| ď 2 a 2 ` ´2b 2 , we obtain that ż

p V |∇ s u| 2 p a dΓ dτ ě ż p V |∇ s pΠ uq| 2 p a dΓ dτ `żp V |∇ s pΠ K uq| 2 p a dΓ dτ ´C ´}Π u} 2 L 2 p p V ;p a dΓ dτ q `}∇ s Π u} 2 L 2 p p V ;p a dΓ dτ q ¯1{2 }Π K u} L 2 p p V ;p a dΓ dτ q ě p1 ´C 2 q ż p V |∇ s pΠ uq| 2 p a dΓ dτ ´C 2 }Π u} 2 L 2 p p V ;p a dΓ dτ q `żp V |∇ s pΠ K uq| 2 p a dΓ dτ ´C ´2}Π K u} 2 L 2 p p V ;p a dΓ dτ q
and the result follows.

4.3.2. Proof of Theorem 3.9. The upper bound of Theorem 3.9 follows immediately from the min-max principle. Let us focus on the lower bound. By Proposition 3.4, we have that there exist 0 , C ą 0, such that, for all P p0, 0 q and all u P p V ,

p Q 1 pΠ K uq ` 4 p1 ´C q ż p V |∇ s Π K u| 2 p a dΓ dτ ´C 2 }Π K u} 2 L 2 p p V ;p a dΓ dτ q ě ´3 4 ε 0 }Π K u} 2 L 2 p p V ;p a dΓ dτ q Hence, Lemma 4.2 ensures that x Q puq ě p Q eff,´ pΠ uq ´3 4 ε 0 }Π K u} 2 L 2 p p V ;p a dΓ dτ q .
Since Π is an orthogonal projection of L 2 p p V ; p a dΓ dτ q, we get that the spectrum of x L lying below ´ε0 is discrete and coincides with the one of x L eff,´ .

4.4. Proof of Theorem 3.10.

Proof. We first notice that, by definition of v (see Propositions 3.4 and 3.7), The operators C and T , resp. CT are anti-unitary, resp. unitary transformations that leave DompHq invariant and satisfy C 2 " ´T 2 " 1 4 , CT " T C, HC " ´CH, HT " T H and H pCT q " ´pCT q H . Moreover, for any ψ P C 4 , we have that xψ, T ψy " 0.

p Q 1 pf v q " ż Γ λ R
We can relate the mean curvature to the commutator between the boundary condition and a Dirac derivative parallel to the boundary. Let n 1 and n 2 be two eigenvectors of the Weingarten map dn s , whose respective eigenvalues are denoted by pλ 1 , λ 2 q and, such that, pn, n 1 , n 2 q is an orthonormal basis of R 3 . We have α ¨pn ˆ∇q " α ¨n2 B n 1 ´α ¨n1 B n 2 .

Then, by the Leibniz formula and Lemma A.1, it follows that pα ¨n ˆ∇qpα ¨n ψq " ´α ¨n pα ¨n2 B n 1 ´α ¨n1 B n 2 q ψ `ppα ¨n2 qpα ¨Bn 1 nq ´pα ¨n1 qpα ¨Bn 2 nqq ψ , and thus, again by Lemma A.1, pα ¨n ˆ∇qpα ¨n ψq " ´α ¨n pα ¨n2 B n 1 ´α ¨n1 B n 2 q ψ ´ipλ 1 `λ2 qγ 5 α ¨n .

We deduce that α ¨pn ˆDqBψ " Bpα ¨n ˆDqψ ´ipλ 1 `λ2 qβγ 5 α ¨n , and the conclusion follows.

A.2. Formula for the square. Let us finally consider Point iv in Theorem 1.5.

In the following lines, we assume that ψ P DompHq. First, we expand the square to get Since the α-matrices are Hermitian and thanks to the Green-Riemann formula we have:

(A.1) @ϕ, ψ P H 1 pΩ, C 4 q , xα ¨Dϕ, ψy Ω " xϕ, α ¨Dψy Ω `xp´iα ¨nqϕ, ψy BΩ .

Then, we use (A.1) with ϕ " βψ, and by using that α anticommutes with β, we find

Lemma 2 . 2 .

 22 Let u P H 1 0 pΩ, Cq be an L 2 -normalized eigenfunction of the Dirichlet Laplacian on Ω. Then ż Γ |B n u| 2 n dΓ " 0 .

  5) thus, by Step (a), lim inf mÑ`8

Notation 3 . 3 .

 33 The function u ,κ,K denotes the first positive eigenfunction of H Rob ,κ,K normalized in L 2 pp0, ´1q, a ,κ,K dτ q.

Theorem 1 . 4 .4. 1 .

 141 13 is a consequence of the semiclassical reformulation in Section 3.1, Proposition 3.2, the rescaling of Section 3.3 and Theorem 3.10. Proof of the results stated in Section 3 Proof of the Agmon estimates of Proposition 3.1. Before stating the proof, let us recall the following lemma.

Lemma 4 . 1 .

 41 Let χ and ψ be Lipschitzian functions on Ω, we have Re x∇ψ, ∇pχ 2 ψqy " }∇pχψq} 2 ´}ψ∇χ} 2 .

  |∇ s v | 2 |f | 2 p a dΓ dτ " ż Γ R |f | 2 dΓ , and ˇˇ2Re ż p V xv ∇ s f, f ∇ s v y p a dΓ dτ ˇˇď 2 ˆżΓ R |f | 2 dΓ ˙1{2 ˆżΓ |∇ s f | 2 dΓ ˙1{2 ď ´2 ż Γ R |f | 2 dΓ ` 2 ż Γ |∇ s f | 2 dΓ ,where R is defined in Remark 3.6. The result follows.Lemma A.2 (Discrete symmetries). Let us introduce three operators defined for ψ P C 4 byCψ " iβα 2 ψ, Charge conjugation operator, T ψ " ´iγ 5 α 2 ψ, Time reversal-symmetry operator, CT ψ " βγ 5 ψ, CT-symmetry operator.

Lemma A. 3 (

 3 Mean curvature as commutator). We have rα ¨pn ˆDq, Bs " ´κγ 5 B .Proof. Let s P BΩ. By anticommutation between α and β, we have α ¨pn ˆDqBψ " β α ¨pn ˆ∇qpα ¨n ψq .

}Hψ} 2 L 2

 22 pΩq " xα ¨Dψ, α ¨Dψy Ω `m2 xβψ, βψy Ω `2m Re xβψ, α ¨Dψy Ω .

  Theorem 1.5. Let Ω be a nonempty, bounded and regular open set in R 3 and m P R. The following properties hold true.i. The operator pH, DompHqq is a self-adjoint operator with compact resolvent. ii. Let us denote by pµ n pmqq ně1 Ă R ˚the non-decreasing sequence of eigenval-

ues of |H| counted with multiplicity. The spectrum of H, denoted by sppHq, is symmetric with respect to 0 (with multiplicity) and sppHq " t˘µ n pmq, n ě 1u .

  1,8 . Let us study the existence of a solution ϕ 1 of equation (2.2). We denote by p´∆q ´1 the inverse of the Dirichlet Laplacian and v " ϕ 1 ´w so that `Id ´λ1 pQ 8 qp´∆q Let ψ P ker pId ´λ1 pQ 8 qp´∆q ´1q K . We have by integrations by parts that xψ, p´∆q ´1 pλψ 1,8 ´p´∆ ´λ1 pQ 8 qq wqy Ω " λ 1 pQ 8 q ´1 `xψ, λ 1 ψ 1,8 y Ω ´xp´∆ ´λ1 pQ 8 qq ψ, wy Ω ´xB n ψ, wy Γ " λ 1 pQ 8 q ´1 `xψ, λ 1 ψ 1,8 y Ω `xB n ψ, P `Bn ψ 1,8 y Γ ˘.

	Hence, we get
		0 " xψ 1,8 , p´∆q ´1 pλψ 1,8 ´p´∆ ´λ1,8 q wqy Ω
	provided that
	(2.3)	λ "	´żΓ	|P `pB n ψ 1,8 q | 2 dΓ .

´1˘v

" p´∆q ´1λψ 1,8 ´p´∆q ´1 p´∆ ´λ1 pQ 8 qq w . By the Fredholm alternative, there exists such a function v if and only if p´∆q ´1 pλψ 1,8 ´p´∆ ´λ1 pQ 8 qq wq P ker `Id ´λ1 pQ 8 qp´∆q ´1˘K . Let a, b P C 4 be such that xa, by " 0, |a| " |b| " 1, ψ 1,8 " au 1 and ψ " bu 1 . We have 0 " xψ, p´∆q ´1 pλ 1 ψ 1,8 ´p´∆ ´λ1 pQ 8 qq wqy Ω since 0 " xB n ψ, P `Bn ψ 1,8 y Γ " 1 2 xb, ´iβα ¨ˆż Γ |B n u 0 | 2 n dΓ ˙ay . Hence, assuming that (2.3) is true, we get that system (2.2) has a solution ϕ 1 . ψ 1,8 `m´1 ϕ 1 can be used as a test function and we have

  and x´κB τ ψ 1 , ψ 0 y L 2 p0,`8q " κ xψ 1 , B τ ψ 0 y L 2 p0,`8q `κ xψ 1 p0q, ψ 0 p0qy " ´κ xψ 1 , ψ 0 y L 2 p0,`8q ,

	" ´κ2 x´τ p´2K `κ2 qB τ ψ 0 , ψ 0 y L 2 p0,`8q " 2p´2K `κ2 q ż `8 0 τ e ´2τ dτ " ´κ2 , 4	ż `8 0	τ e ´2τ dτ " ´K	`κ2 2	.
	Hence, it follows that
		λ 2 " K	´κ2 4	.
	By using convenient cutoff functions (to satisfy the Dirichlet condition near ´1)
	and the spectral theorem, we easily get that dist ˆ´1 ` 4 ˆK ´κ2 4 ˙, sp `HRob ,κ,K ˘˙ď C 6 .
	Then, by using straightforward adaptations of the results in [20, Appendix] (we deal
	with the additional term in the boundary condition as a perturbation), we get the `HRob lower bound for λ 2 ,κ,K
	2 τ	`1˘ψ

2 , ψ 0 y L 2 p0,`8q " xpB τ `1q ψ 2 p0q, ψ 0 p0qy " κ 2 xψ 1 p0q, ψ 0 p0qy " 0 ,

  1 ps, q|f pσq| 2 dΓ .|∇ s pf v q| 2 p a dΓ dτ " |∇ s f | 2 |v | 2 p a dΓ dτ `żp V |∇ s v | 2 |f | 2 p a dΓ dτBy Proposition 3.4, we get that }v ps, ¨q} 2 L 2 pp0, ´1q,p a psq dτ q " 1, for all s P Γ. Hence,ż p V |∇ s f | 2 |v | 2 p a dΓ dτ "

	Then, we have ż	ż
	p V	ż	p V
	`2Re	p V
			ż

xv ∇ s f, f ∇ s v y p a dΓ dτ . Γ |∇ s f | 2 dΓ ,

2

  Re xβψ, α ¨Dψy Ω " xiα ¨nβψ, ψ,y BΩ " x´iβα ¨nψ, ψy BΩ " }ψ} 2 L 2 pBΩq . It remains to use that β is unitary to deduce (A.2) }Hψ} 2 L 2 pΩq " }α ¨Dψ} 2 L 2 pΩq `m2 }ψ} 2 L 2 pΩq `m}ψ} 2 L 2 pBΩq . Moreover, assume that ψ P H 2 pΩq. Then, we use again the Green-Riemann formula (A.1) and we get xα ¨Dψ, α ¨Dψy Ω " xψ, pα ¨Dq 2 ψy Ω `xp´iα ¨nqψ, α ¨Dψy BΩ . By noticing that pα ¨Dq 2 " 1 4 D 2 and by using integration by parts again, we find xα ¨Dψ, α ¨Dψy Ω " xDψ, Dψy Ω `i xψ, ppα ¨nqpα ¨Dq ´pn ¨Dqq ψy BΩ . Since H 2 pΩq is dense in H 1 pΩq, we get that this formula holds for any u P DompHq. We shall now investigate the boundary term by using the first algebraic relation in Lemma A.1: i xψ, ppα ¨nqpα ¨Dq ´pn ¨Dqq ψy BΩ " ´xψ, γ 5 α ¨pn ˆDqψy BΩ " ´xγ 5 ψ, α ¨pn ˆDqψy BΩ . It remains to study the term xγ 5 ψ, α ¨pn ˆDqψy BΩ . Since ψ belongs to DompHq, we have xγ 5 ψ, α ¨pn ˆDqψy BΩ " xγ 5 ψ, rα ¨pn ˆDq, Bsψy BΩ `xγ 5 ψ, Bα ¨pn ˆDqψy BΩ , and, since B is a symmetric operator, we get xγ 5 ψ, Bα ¨pn ˆDqψy BΩ " xBγ 5 ψ, α ¨pn ˆDqψy BΩ " ´xγ 5 Bψ, α ¨pn ˆDqψy BΩ . Hence, we deduce that xγ 5 ψ, α ¨pn ˆDqψy BΩ " 1 2 xγ 5 ψ, rα ¨pn ˆDq, Bsψy BΩ . Finally, by using Lemma A.3, we get i xψ, ppα ¨nqpα ¨Dq ´pn ¨Dqq ψy BΩ " ´1 2 xγ 5 ψ, ´κγ 5 ψy BΩ , and the conclusion follows.

No isolated quark has been observed yet.

For light quarks, the non-relativistic approximation E " mc 2 is not valid and the Schrödinger operator ´∆ has to be replaced by the Dirac one to describe the kinetic energy.

Acknowledgments References

Acknowledgments

This work was partially supported by the Henri Lebesgue Center (programme "Investissements d'avenir" -n o ANR-11-LABX-0020-01). L. L.T. was supported by the ANR project Moonrise ANR-14-CE23-0007-01. N. A. was partially supported by ERCEA Advanced Grant 669689-HADE, MTM2014-53145-P (MICINN, Gobierno de España) and IT641-13 (DEUI, Gobierno Vasco). N.A. wishes to thank the IRMAR (Université de Rennes 1) where the paper was written, for the invitation and hospitality. The authors would also like to thank Albert Mas for the considerations in Section 1.4.2 and for many stimulating discussions. Finally, the authors are deeply grateful to the anonymous referees who helped improving the paper by pointing out fundamental references.

A.2. Formula for the square Appendix A. About Theorem 1.5

In this appendix we discuss some aspects mentioned in Theorem 1.5. Since the formula for the square of the MIT bag operator plays an important role in this paper, we recall its proof.

A.1. Elementary properties. Let us recall the following elementary algebraic properties.

Lemma A.1. For all x, y P R 3 , we have pα ¨xqpα ¨yq " px ¨yq1 4 `iγ 5 α ¨px ˆyq , βpα ¨xq " ´pα ¨xqβ , βγ 5 " ´γ5 β , γ 5 pα ¨xq " pα ¨xqγ 5 .

Proof. We refer to [31, Appendix 1.B].

Points ii and iii of Theorem 1.5 are immediate consequences of the following lemma (see [START_REF] Thaller | The Dirac equation[END_REF]Section 1.4.6] and [START_REF] Reiher | Relativistic quantum chemistry: the fundamental theory of molecular science[END_REF]Section 10.4.5]).