Gilbert El-Hajje

Daniel Ory

Jean-François Guillemoles

Laurent Lombez

Micrometric mapping of absolute trapping defects density using quantitative luminescence imaging

Keywords: CIGS solar cells, time-resolved luminescence, trapping defects quantification, PV performance, carrier dynamics

In the present study, we develop a contactless optical characterization tool that quantifies and maps the trapping defects density within a thin film photovoltaic device. This is achieved by probing time-resolved photoluminescence and numerically reconstructing the experimental decays under several excitation conditions. The values of defects density in different Cu(In,Ga)Se2 solar cells were extracted and linked to photovoltaic performances such as the open-circuit voltage. In the second part of the work, the authors established a micrometric map of the trapping defects density. This revealed areas within the thin film CIGS solar cell with low photovoltaic performance and high trapping defects density. The final part of the work was dedicated to finding the origin of the spatial fluctuations of the thin film transport properties. To do so, we started by establishing a micrometric map of the absolute quasi-Fermi levels splitting within the same CIGS solar cell, using the hyperspectral imager. A correlation is obtained between the map of quasi-Fermi levels splitting of and the map of the trapping defects density. The latter is found to be the origin of the frequently observed spatial fluctuations of thin film materials properties.

I. INTRODUCTION

One of the signature traits of CIGS solar cells is the abnormal variations in its photovoltaic performance following the activation of its metastabilities that are present at the absorber surface and its interface with the buffer layer. By using conventional characterization methods, it has been shown [START_REF] Müller | Effect Of Light Soaking On The Electro-And Photoluminescence Of Cu(In,Ga)Se2 Solar Cells[END_REF][2] that metastabilities exist in different categories. In this work, we aim at developing a contactless optical characterization tool that can quantify and map at the micrometric scale, the defects density that is behind the metastabilities in CIGS devices. In addition, in order to validate our technique we also aim at linking our defects density values to standard PV performance indicators. We will finally attempt to use these defects density maps in order to reveal the origin behind the frequently observed spatial inhomogeneities of thin film solar cells properties. .

II. DESCRIPTION OF APPROACHES & TECHNIQUES

To fulfill the main purpose of this study, the potential of high resolution confocal microscopy and that of hyperspectral imaging of photoluminescence emission were combined and the derived results were quantitatively correlated.

On the one hand, the authors develop a system of rate equations that reconstructs an experimental time-resolved photoluminescence (TRPL) decay obtained under pulsed laser excitation. The corresponding trapping defects density is then derived. We show that our technique is robust as the obtained defects density values are constant regardless of the experimental conditions. In addition, the global reconstruction error does not exceed 7%. Our technique is then applied to a set of five CIGS devices on which J-V measurements were also made [START_REF] El-Hajje | Quantitative luminescence-based measurement of trapping defects density in thin film solar cells[END_REF]. The trapping states are then correlated to global cell properties.

On the other hand, using spatially and time-resolved photoluminescence, we add the spatial dimension to our method. This resulted in a sub-micron map of the material trapping defects density [START_REF] El-Hajje | On the origin of the spatial inhomogeneity of photoluminescence in thin-film CIGS solar devices[END_REF]. This map is compared to a map of absolute values of its quasi-Fermi levels splitting (Δμ) obtained using a hyperspectral imager. The trapping states are then correlated to local cell properties.

III. EXPERIMENTAL RESULTS & DISCUSSIONS

A. Photo-Excitation Power And Voltage-Bias-Dependent TRPL Decays

In this part of the study, the authors acquired TRPL decays under two different excitation sources. We either vary the incident photo-excitation power, or keep the photo-excitation power constant and inject increasing value of current into the sample. Interestingly, in both cases, the initial double exponential decay becomes increasingly a mono exponential decay of photoluminescence. This can be seen in In our photo-excitation power-dependent TRPL experiments, the value of excitation power is increased. As we will see later, the excess photogenerated carrier density becomes progressively dominant over the accessible trap states density. This increasing ratio between the former and the latter densities is at the origin of the observed convergence from a double to a mono-exponential decay [START_REF] El-Hajje | Contactless Characterization Of Metastable Defects In Cu(In,Ga)Se2 Solar Cells Using Time-Resolved Photoluminescence[END_REF]. In the voltage-bias-dependent TRPL experiments, the value of the injected current is increased. It tends to electrically occupy the inactive shallow trap states density present at the absorber surface and its interface with the CdS buffer layer. As the latter density decreases, so does the delay in the luminescence emission induced via carrier trapping. This explains why the TRPL decays in Figure 1.b are becoming shorter under an increasing value of injected current.

Finally, whether it is the current injection, or the excitation power that is being increased, the final state regarding the minority carrier dynamics is the same. The latter is characterized by a total dominance of direct recombination events over the trapping events.

B. Quantification Of Trapping Defects Density

The authors start by a establishing a system of differential rate equations that describes the dynamics of the different populations that are behind the experimental time-resolved luminescence decays. All of the parameters used to solve this system are actual experimental data. Then, an optimization procedure is used in order to reconstruct the experimental time-resolved photoluminescence (TRPL) decay and extract the system trap states density.

The first parameter taken into consideration is the experimental charge carriers' photogeneration. Experimentally, a laser pulse is used in order to induce the photogeneration. Therefore, in order to replicate the experimental excitation conditions of 1.4x10 13 Photons/Pulse/cm 2 delivered within a 6 ps pulse width, the following function is used:

(1) P(t) is the laser pulse temporal profile , P 0 is the actual peak power which is equal to 23.5 mW, τp is the laser pulse width which is equal to 6 ps. Three different populations of states are defined here: N1, N2 and N3. N1 is the population occupying the valence band, N3 is the population occupying the conduction band and N2 is the population occupying the accessible shallow trap states density. The densities of accessible states with the CB and the VB are considered as infinite reservoirs compared to that of the shallow traps which we denote here as Ψ D and aim to determine its value. In order to describe the dynamics behind the experimental decays, the system considers three different types of events: Firstly, direct carrier recombination events, such as SRH and band-to-band recombination. The corresponding lifetime τ Rec is extracted from the initial fast part of the experimental timeresolved photoluminescence decay. The second and third type of events will be related to the existence of the shallow trap states. In particular, traps capture charge carriers and then emit them by means of thermal activation. Consequently, a capture lifetime τ Cap and an emission lifetime τ em are defined, and both are considered equal. These lifetimes will be the characteristic lifetimes of the second slower part of the time-resolved photoluminescence decay. Now that all of the parameters have been defined, we were able to establish the system of differential rate equations that replicate the experimentally probed carrier dynamics:

(2)

The generation which takes place during τ p is written as:

(3) R (λ=700 nm) is the reflection coefficient that is found to be close to 20%. In order for the shallow trap states density to be the only unknown parameter, the capture and the emission lifetimes need to be calculated. The values of these lifetimes are dependent on the properties of the material shallow traps. In other terms, one needs to determine the nature of the shallow trap (donor or acceptor), its energy level E t and its capture cross section σ e . The emission lifetime can be calculated as follows:

(4) υ th is the thermal velocity, k is Boltzmann's constant and T is the temperature. The total system of rate equations becomes now a function of only one variable, the trap states density Ψ D . The numerical procedure consists of varying Ψ D over a large array of different values. For each value, the system of rate equations in (2) results in a decay TRL Num that is a reconstruction of the experimental one TRL Exp . The optimization procedure then selects the value of Ψ D for which the reconstruction error is minimized. The latter error is calculated over the whole set of decay points N p and is expressed as: [START_REF] El-Hajje | Contactless Characterization Of Metastable Defects In Cu(In,Ga)Se2 Solar Cells Using Time-Resolved Photoluminescence[END_REF] We obtain values of ξ varying between 2% and 7%. These operations are repeated for all of the experimental decays. In Fig. 3.a below, we illustrate how the solution of the system of rate equations results in a reconstruction of the experimental TRPL decay. Fig. 3.b illustrates how the reconstruction is made using an array of trapping defects density values and how the optimization algorithm chooses the value that leads to a minimum of reconstruction error. In the present example, the reconstruction error ξ is equal to 5% and the corresponding ΨD is equal to 4.7x10 14 traps/cm 3 . For both sets of experiments, we apply our technique and prove that it is self-consistent with respect to a variation of the incident excitation power and the injected current, when the latter are taken into account. This is proved by the constant value of trapping defects density of 1.055x10 15 traps/cm 3 that it provides, with fluctuations not exceeding 6% with respect to the mean value.

For the excitation power-dependent TRPL experiments, this means that only one contactless measurement of a TRPL decay, at any given photo-excitation flux, is needed for the calculation of the trapping defects density.

We were so far able to prove that our technique can indeed provide precise reconstructions of the experimental timeresolved luminescence decays and quantify the material trapping defects density.

C. effect Of Trapping Defects On Global PV Performances

At this stage we will test if our defect quantification tool can be used as a reliable indicator of thin film PV solar cells performance. In other terms we will try to evidence the existence of a link between a trap states density derived from TRPL experiments using our method, and the standard PV performances.

To do so, standard J(V) characterization are performed on five CIGS solar cell devices. All five devices have the same structure and type of layers.

For each device, all of the performance parameters will be extracted. Then, we perform a single acquisition of a TRPL decay on each CIGS device. Using our previously developed method, we finally extract the value of the trap states density within each device. We deduce that the thin film CIGS device that suffers from the highest trap states density is expected to have the lowest open-circuit voltage. On the one hand these results is in good agreement with the study done in Ref [START_REF] Cao | Defects In Chalcopyrite Semiconductors: Defects In Cu(In,Ga)Se2 Chalcopyrite Semiconductors: A Comparative Study Of Material Properties, Defect States, And Photovoltaic Performance[END_REF] where the highest density of ((In or Ga) Cu) defects ( (InCu++ in our case) induced the lowest Voc.

On the other hand, this validates the type and characteristics of the trapping defects that we are studying here.

For our five CIGS devices, we may not be certain of the exact origins behind each of these different defect densities but we however evidenced their nature, validated their values and their effect on the PV devices performance using a unique approach based on TRPL decays analysis.

D. The Effect Of Trapping Defects On Local PV Performances

We start by performing spatially-resolved measurements of the TRPL by scanning the entire CIGS solar cell surface. Then, by applying our technique to each of the TRPL decays, we obtained a complete micrometric map of the trapping defects density. The map exhibited important fluctuations ranging between 3.5x10 14 cm -3 and 1.05x10 15 cm -3 .

Fig. 5. Micrometric map of the solar cell absolute trapping defects density distriubtion

Finally, based on the hyperspectral imager and the generalized Planck's law for luminescence emission, we establish a map of the absolute value of the quasi-Fermi levels splitting Δμ(eV) within the same solar cell [START_REF] Delamarre | Quantitative Luminescence Mapping Of Cu(In, Ga)Se 2 Thin-Film Solar Cells[END_REF]. A comparison revealed a dependence of the spatial fluctuations of quasi-Fermi levels splitting in a CIGS solar cell on its trapping defects density. This comes in good agreement with the above results where we evidenced a link between the V oc of a solar cell and its trapping defects density. We deduce that the spatial fluctuations of Δμ are dictated by those of the trapping defects density.

This identified link has multiple important implications. First of all, the authors identified at least one of the origins behind the usually observed spatial inhomogeneities of the maximum achievable open circuit voltage of a CIGS solar cell. Second of all, a correlation between the performed experiments is consequently achieved at the global and the micro scale.

  Figures 1.a & 1.b:

Fig. 1 .

 1 Fig. 1.a. Evolution of the TRPL decays behavior as a function of photo-excitation power.

Fig. 1 .

 1 Fig. 1.b. Evolution of the TRPL decays behavior as a function of current injection.

Fig. 2 .

 2 Fig. 2. Schematic illustration of minority carrier dynamics

Fig. 3 .

 3 Fig. 3.a. Illustration of a TRPL decay reconstruction

Fig. 4 .

 4 Fig. 4. Evolution of the open-circuit voltage as a function of the shallow trap states density

Fig. 6 .

 6 Fig. 6. Micrometric map of the solar cell absolute quasi-Fermi levels splitting

978-1-5090-2724-8/16/$31.00 ©2016 IEEE


Dac-Trung Nguyen

Laurent Lombez
email: laurent.lombez@cnrs.fr

François Gibelli

Soline Boyer-Richard

Olivier Durand

Jean-François Guillemoles

Quantitative experimental assessment of hot carrier enhanced solar cells at room temperature

In common photovoltaic devices, the part of the incident energy above the absorption threshold quickly ends up as heat, which limits their maximum achievable efficiency far below the thermodynamic limit for solar energy conversion. Conversely, if the excess kinetic energy of the photogenerated carriers could be converted into additional free energy, it would be possible to approach the thermodynamic limit. This is the principle of hot carrier devices. Unfortunately, such a device operation in conditions relevant for utilisation has never been evidenced. Here we show that the quantitative thermodynamic study of the hot carrier population, with luminance measurements, allows us to discuss the hot carrier contribution to the solar cell performance. We demonstrate that voltage and current can be enhanced in a semiconductor heterostructure due to the presence of the hot carrier population in a single InGaAsP quantum well at room temperature. These experimental results substantiate the potential of increasing photovoltaic performances in the hot carrier regime.

Introduction

Despite their current record power conversion efficiencies (PCE) [START_REF] Green | Solar cell efficiency tables (version 50)[END_REF], solar cells are still far from their performance limits [START_REF] Green | Third generation photovoltaics[END_REF]. In a silicon solar cell, the mainstream technology, about 40% of the absorbed light energy is lost by thermalisation. This one of the main reasons explaining why the maximum achievable PCE of classical photovoltaic device is governed by Shockley-Queisser (SQ) limit of about 31% [START_REF] Shockley | Detailed Balance Limit of Efficiency of p-n Junction Solar Cells[END_REF]. Most of the efforts towards reaching the highest efficiencies possible are based on multijunction solar cells where each individual junction converts a fraction of the solar spectrum: in this approach the average energy converted per junction is modest while full device exhibits a substantial sensitivity to variations of the incident light spectrum. In theory, according to Ross and Nozik [START_REF] Ross | Efficiency of hot-carrier solar energy converters[END_REF], a single diode device can be as efficient as a multijunction device if dissipation of the carrier energy via heat can be avoided (i.e. if kinetic energy of the photogenerated hot carriers could be converted into additional free energy). Indeed thermoelectric devices convert heat gradients into electrical energy and photovoltaic devices convert light flux 1 into electrical energy. Hight lattice thermal conductivity of usual semiconductors would generally prevent a direct use of thermoelectric effect [START_REF] Kettemann | Thermoelectric field effects in low-dimensional structure solar cells[END_REF]. To fight lattice conductivity one option is to keep the lattice isothermal while only the photogenerated carriers can become hot [START_REF] Ross | Efficiency of hot-carrier solar energy converters[END_REF][START_REF] Würfel | Solar energy conversion with hot electrons from impact ionisation[END_REF]. Such decoupling of the lattice and carrier temperature has been observed in nanostructures, such as Quantum Well (QW) structures that have shown slower carrier cooling rates as compared to bulk materials [START_REF] Rosenwaks | Hot-carrier cooling in GaAs: Quantum wells versus bulk[END_REF][START_REF] Xu | Picosecond relaxation of hot carriers in highly photoexcited bulk GaAs and GaAs-AlGaAs multiple quantum wells[END_REF][START_REF] Ryan | Time-Resolved Photoluminescence of Two-Dimensional Hot Carriers in GaAs-AlGaAs Heterostructures[END_REF][START_REF] Shah | Energy-Loss Rates for Hot Electrons and Holes in GaAs Quantum Wells[END_REF][START_REF] Leo | Reduced dimensionality of hot-carrier relaxation in GaAs quantum wells[END_REF][START_REF] Lugli | Nonequilibrium longitudinal-optical phonon effects in GaAs-AlGaAs quantum wells[END_REF].

Today we find a plethora of promising research on the development and characterisation of hot carrier absorber [START_REF] Conibeer | Modelling of hot carrier solar cell absorbers[END_REF][START_REF] Bris | Hot carrier solar cells: Achievable efficiency accounting for heat losses in the absorber and through contacts[END_REF][START_REF] Bris | Thermalisation rate study of GaSb-based heterostructures by continuous wave photoluminescence and their potential as hot carrier solar cell absorbers[END_REF][START_REF] Bris | Hot Carrier Solar Cells: Controlling Thermalization in Ultrathin Devices[END_REF][START_REF] Hirst | Experimental demonstration of hot-carrier photo-current in an InGaAs quantum well solar cell[END_REF][START_REF] Rodière | Experimental evidence of hot carriers solar cell operation in multi-quantum wells heterostructures[END_REF][START_REF] Esmaielpour | Suppression of phonon-mediated hot carrier relaxation in type-II InAs/AlAsxSb1 -x quantum wells: a practical route to hot carrier solar cells[END_REF][START_REF] Zhang | Extended hot carrier lifetimes observed in bulk In0.265±0.02ga0.735n under high-density photoexcitation[END_REF]. However, the optical methods used to obtain carrier temperature can be inaccurate when dealing with nanostructures and high excitation fluxes [START_REF] Tedeschi | Long-Lived Hot Carriers in III-V Nanowires[END_REF][START_REF] Gibelli | Accurate radiation temperature and chemical potential from quantitative photoluminescence analysis of hot carrier populations[END_REF][START_REF] Esmaielpour | Effect of occupation of the excited states and phonon broadening on the determination of the hot carrier temperature from continuous wave photoluminescence in InGaAsP quantum well absorbers[END_REF]. Meanwhile, several studies have been performed towards completed devices; especially electrical investigation to validate energy selective contacts [START_REF] Dimmock | Demonstration of a hot-carrier photovoltaic cell[END_REF][START_REF] Dimmock | Optoelectronic characterization of carrier extraction in a hot carrier photovoltaic cell structure[END_REF][START_REF] Julian | Insights on energy selective contacts for thermal energy harvesting using double resonant tunneling contacts and numerical modeling[END_REF][START_REF] Hanna | Hot carrier solar cells[END_REF].

In this work we study a hot carrier device operating at room temperature under continuous wave illumination with direct measurement of the hot carrier population and assessment of its contribution to power conversion. For this purpose we fabricated a quantum-well-based p/i/n hot carrier solar cell prototype and investigated its optical and electrical properties. To investigate hot carriers effects, we have analysed photoand electroluminescence (PL and EL) spectra from which one may extract thermodynamic quantities such as the temperature and the chemical potential of radiation (using the generalized Planck radiation law [START_REF] Rau | Superposition and Reciprocity in the Electroluminescence and Photoluminescence of Solar Cells[END_REF][START_REF] Lasher | Spontaneous and Stimulated Recombination Radiation in Semiconductors[END_REF]); both are related to the carrier thermodynamic properties. We use a model of the absorption including excitons, free carriers and band filling, which allows us to fit the whole PL spectra over a large range of excitation power [START_REF] Gibelli | Accurate radiation temperature and chemical potential from quantitative photoluminescence analysis of hot carrier populations[END_REF][START_REF] Gibelli | Two carrier temperatures nonequilibrium generalized Planck law for semiconductors[END_REF]. Quantitative optical measurements indicate the presence of a hot carrier population within the quantum well. This gives an increase of the electrochemical potential in the barrier. Such an effect is analogous to a Seebeck conversion process [START_REF] Rodière | Experimental evidence of hot carriers solar cell operation in multi-quantum wells heterostructures[END_REF]. The quantitative optical measurements are then compared to the electrical characteristics. The real impact of the hot carrier population within the structure is then discussed and quantified.

Quantum well hot carrier solar cell

The solar cell prototype is depicted in Figure 1. The multilayer wafer contains a region with an intrinsic InGaAsP-based quantum well/barriers, which acts as the absorber generating hot carriers, while the barriers and claddings play the role of energy semiselective contacts [START_REF] Bris | Hot carrier solar cells: Achievable efficiency accounting for heat losses in the absorber and through contacts[END_REF][START_REF] Feng | Non-ideal energy selective contacts and their effect on the performance of a hot carrier solar cell with an indium nitride absorber[END_REF] (see Table 1). This absorber-contacts layer is sandwiched between two gradually n-and p-doped InP layers for carrier separation and collection.

Current-voltage characteristics are first recorded for cells with diameters ranging from 5 to 200 µm. The results for the 11.0 ± 0.5 µm diameter cell shows the highest efficiency under cw laser excitation 980 nm. The PCE is 11.6 ± 0.5% under focused laser illumination of 1650 W/cm 2 (equivalent to 55 600 Suns and close to the maximum achievable solar concentration, see Supplementary Notes 4 to 6 for more details). Although we measure efficiencies of potential hot carrier solar cell (HCSC), the purpose of the design is to allow a proper optoelectrical measurements. The excellent electrical characteristics stem from an extremely low series resistance, which yields a short-circuit current that is proportional to the laser power within the entire experimental range (see Figure 2). The open circuit voltage exceeds the lowest absorption threshold at laser fluences higher than ∼6000 W/cm 2 , which would indicate a power conversion efficiency above an equivalent thermal equilibrium device. In order to determine whether band filling contributes solely to this high conversion efficiency, further analysis are required as laid out in the following. V o l t a g e ( V ) Indeed, the carrier temperature T as well as the quasi-Fermi level splitting µ = E Fn -E Fp (E Fn and E Fp being the electrons and holes Fermi levels) are determined according to the generalized Planck's law [START_REF] Würfel | The chemical potential of radiation[END_REF][START_REF] Gibelli | Accurate radiation temperature and chemical potential from quantitative photoluminescence analysis of hot carrier populations[END_REF][START_REF] Lasher | Spontaneous and Stimulated Recombination Radiation in Semiconductors[END_REF]:

J s c = -2 . 7 1 A / c m ² V o c = 0 . 7 8 7 V η = 1 1 . 1 3 % ( c ) ( b ) ( a )
Φ(E) = 2A(E) • E 2 h 3 c 2 • 1 exp E -µ k B T -1 (1)
Here Φ is the luminescence emission, E the photon energy, A(E) the absorptivity, h the Planck's constant, c the speed of light in vacuum and k B the Boltzmann constant.

Determination of the absorptivity A(E) is crucial for minimizing the uncertainty in the evaluation of the thermodynamic properties T and µ in the equation 1 (see section Methods and the Supplementary Notes 1 and 2). Indeed A(E) might affect the whole spectral variation of the PL signal including the high energy range. In order to validate the model of the absorption in the sample structure, we compare the absorptivity calculated from luminescence spectra with the external quantum efficiency EQE (see Figure 3). Indeed, if the collection of photo-generated carriers was lossless, the electric current density would correspond to the number of absorbed photons [START_REF] Rau | Superposition and Reciprocity in the Electroluminescence and Photoluminescence of Solar Cells[END_REF]. In the case of relatively shallow wells, as in our samples (∆E c 0.2 eV), the collection efficiency is generally found close to unity [START_REF] Tsui | Determination of the quasi-Fermi-level separation in single-quantum-well p-i-n diodes[END_REF][START_REF] Kluftinger | Temperaturedependent study of the radiative losses in double-quantum well solar cells[END_REF][START_REF] Nelson | Steadystate carrier escape from single quantum wells[END_REF]. Hence photocurrent can be used as an indirect measure of the absorption. We found a good agreement between the EQE and A(E) obtained from our model.

Therefore, luminescence spectra at different laser excitation powers and different applied voltages have been recorded and analysed. In all cases the full-spectrum fit is in good agreement with the experimental data as can be seen in Figures 4 and5.

We optically probe the carrier thermodynamic properties in both the barrier and the QW spectral regions (see Supplementary Note 7 for detailed results). Figure 4c shows the carrier temperature as a function of the excitation power. We observe a temperature increase of more than 1300 K above the room temperature for carrier in the QW (i.e 1300 K above the carrier temperature in the barrier) -a range of temperature already reported in the litterature [START_REF] Tedeschi | Long-Lived Hot Carriers in III-V Nanowires[END_REF][START_REF] Esmaielpour | Effect of occupation of the excited states and phonon broadening on the determination of the hot carrier temperature from continuous wave photoluminescence in InGaAsP quantum well absorbers[END_REF]. We have previously demonstrated that this surplus amount of heat can be converted into a gain in voltage via a Richardson emission of hot carriers (also seen as a pseudo-Seebeck effect) [START_REF] Rodière | Experimental evidence of hot carriers solar cell operation in multi-quantum wells heterostructures[END_REF]. This is confirmed when looking at the electrochemical potentials of the carriers in both spectral region (see Figure 4d): The difference between the two values increases with the carrier temperature in the quantum well. Notably, the value of the electrochemical potential µ b in the barrier region exceeds the lowest absorption threshold of the QW (0.82 eV), which is consistent with the electrical response we observe.

The determined absorptivities for three laser excitation powers are shown in Figure 4b. One can infer that saturation of absorption in the quantum well region is only partial, which indicates that the quantum well would be still optically active for absorption. This observation goes with the hypothesis that the increase of the electrochemical potential in the quantum well µ b is linked to a hot carrier effect rather than a band-filling phenomenon. Therefore the fact that µ b is higher than the fundamental transition E 1 -0 . 7 5 0 . 8 0 0 . 8 5 0 . 9 0 0 . 9 5 1 . 0 0 ). See Supplementary Note 1 for more details on the quantum/well absorption coefficients α w0 and α b0 . The experimental data reflects both qualitatively and quantitatively the absorptivity obtained from PL measurements at low excitation power. The exciton peak is centred at 0.82 eV and the absorption plateaus are observed, corresponding to the fundamental and the first excited levels of the quantum well. The detailed result is presented in the Supplementary Note 3. HH 1 would indeed indicate the barriers are allowing for isentropic cooling of the carriers which corroborates hot carrier solar cell functionality.

1 . 0 5 1 . 1 0 1 0 -4 1 0 -3 1 0 -2 1 0 -1 E Q E F i t t e d a b s o r p t i v i t y E Q E o
We then carried out luminescence experiments under electric bias to quantify the contribution of this hot carrier effect in the obtained device characteristics. Similarly to Figure 4 we display in Figure 5 fits of biased-PL spectra, absorptivity and evolution of the thermodynamic quantities with the applied voltage (see Supplementary Note 8 for detailed results). The incident laser excitation power is set to 1890 W/cm 2 . All the measurements are carried out in a collection regime, at applied bias lower than the V oc = 0.795 V. In such optoelectrical experiment, one might expect values of µ b close to the applied bias as the carriers in this spectral region are supposed to be cold (i.e. fast thermalisation). The green line in Figure 5d shows the condition where V = µ/q. The electrochemical potential µ b in our case is found to be slightly higher than the applied voltage (see Figure 5d blue triangles) and two different regimes can be distinguished. Below a bias voltage V = 0.6 V we see a rather constant value of mu b around 0.7 eV which then increase for higher bias voltages. This can be explained by a weak carrier collection efficiency at high excitation intensities. The effect is confirmed by the observation of a PL signal at all biases, although all the carriers would have been electrically collected (see Figure 5d blue lines). The luminescence signal under bias is the sum of the electroluminescence and PL signal at short circuit current [START_REF] Rau | Superposition and Reciprocity in the Electroluminescence and Photoluminescence of Solar Cells[END_REF], the latter being small but non negligible in our case.

Fixing the laser intensity to a high value and starting at the open-circuit condition, we would, expect T and µ to decrease in the quantum well region with decreasing bias voltage (i.e. we reduce the carrier density by extracting carriers as expected) as illustrated by the results presented in Figure 5 as the bias voltage is varied from 0.8 V to 0.6 V. Thermodynamic values below 0.6 V are once again affected by a reduced carrier collection efficiency.

Contribution of hot carriers to voltage and current gains

A key point in the latter experiment is to probe the hot carrier effect at the maximum power point which is found at 0.651 V (grey zone highlighted in Figure 5d). With the laser excitation condition used here (1890 W/cm 2 on a 11 µm-diameter cell) we obtain a PCE of 11.6%. Here, the electrochemical potential increase from µ w to µ b by 110 mV, and 50 mV for the maximum power point showing an efficient conversion of the temperature difference into electrochemical potential difference [START_REF] Rodière | Experimental evidence of hot carriers solar cell operation in multi-quantum wells heterostructures[END_REF]. Still, we need to estimate the hot carrier contribution to the total voltage and current.

In a first step approach we propose to compare the open circuit voltage in a hot and cold carrier regime. We assess the contribution of the well to the power conversion as a function of hot carriers' temperature, using the fact that the hot carriers are essentially in the well. Two situations are considered: (i) the experimental scenario for which hot carriers are produced in the well, and where electrochemical potential is known in the barriers, and (ii) a virtual situation in which the carriers in the well would be cold and in equilibrium with the barrier. This situation can be simulated using our fitting parameters (see Supplementary Note 9). Figure 6(a) shows the evolution of µ b (i.e. maximum achievable voltage for the case of hot carriers) and the calculated one µ cold for the case of cold carriers. We assumed here the same radiative efficiency in both cases and find that the quasi Fermi level splitting in the barriers to be larger by about 25 to 30 meV with hot carriers as compared to the case of lattice thermalized carriers (see Figure 6(a)). The same simulation in the experiment under bias (and lower laser excitation power ) shows an increase of the QFLS by about 10 meV for V> 0.65V (i.e. where hot carriers are present). This is a strong indication of a voltage gain from the conversion of the hot carriers' thermal energy.

In the next step we probe whether there is any possible gain in current. Although the experiments presented here give a clear picture of the physical phenomena at work, they do not exactly correspond to the operation of a solar cell. To understand the impact of hot carriers in a heterostructure on the conversion of solar energy, it is necessary to extrapolate the previous describes case i.e.

the laser only excites transitions in the barriers, to the case where the solar spectrum excites all optical transitions. In the former case, transitions from the well only contribute to the recombination current while in the latter case they also contribute to the photocurrent. The key point is whether the balance between the additional generation and recombination due to the presence the well yields a positive contribution to solar energy conversion or not.

In order to assess the contribution of the well to current generation, we propose as a metric to consider the marginal gain (additional generation) versus the marginal loss (additional recombination) induced by the well at a given operating point. Subsequently, we evaluate this metric in the presence as well in the absence of hot carriers. All the procedure is detailed in the Supplementary Note 9. On the one hand, the marginal generation is computed for a solar spectrum at a concentration yielding the same photocurrent as measured at the considered working point. We use the measured absorptivity in the spectral range where only well transitions contribute (e.g. E<1.1 eV). As shown in Figure 6(b) the photogenenated generation current increases in the presence of hot carriers and is always higher than the generation current in the equilibrium case (i.e. cold carriers). In that case, the current vanishes because of the net absorption reduction where stimulated emission could occur (i.e. strong band filling effect) [START_REF] Würfel | The chemical potential of radiation[END_REF].

On the other hand, the marginal recombination was computed from radiative recombination assuming a radiative yield (see Figure 6(c)). The recombination current increases in both the hot and the cold regime with the cold regime yielding higher values. Finally to probe what would be the contribution of current due to the quantum well in case of a solar spectrum illumination, we need to subtract the generation current from the recombination current. This gain is much more pronounced in the hot regime. Therefore, the presence of hot carriers is essential for the QW region to act as a current generator.

This finding is very promising since the present structure suffers from weak absorption in the barrier and more importantly, in the single well (which collects only 1% of the incident light, see Figures 4b and5b, a value comparable to monolayers of 2D materials such as graphene). Here, light management in the cell would be beneficial for reducing the effective concentration for the same amount of generated current and thus increase the impact of the QW to solar energy conversion.

Conclusion

We investigated p/i/n single quantum well device by means of complementary optical and electrical measurements. Our device exhibit its maximum efficiency at incident powers equivalent to 50 000 suns, i.e. one order of magnitude higher than current achievement in concentrator photovoltaics. Electrical measurements show Voc above the minimum absorption threshold that could indicate a working condition of this device above the limit for classical device operation where thermal equilibrium is present. To clarify this observation, the variation of the carrier thermodynamic properties as a function of optical excitation power or electrical bias is determined by rigorous fit of the complete luminescence spectra. The optical measurements confirm the electrical behavior and allow to quantify the impact of hot carriers on device functionality. A striking observation is that voltage and current enhancement are observed due to the presence of hot carriers. These findings exemplify that current state-of-the-art semiconductor heterostructures can already be used to boost solar energy conversion by hot carrier mechanisms. The present results will be instrumental to designing a new generation of devices for enhancing the hot carrier conversion contribution, both optically and electrically.

Methods

Sample preparation and hyperspectral imaging

The investigated sample is based on a quaternary In x Ga 1-x As y P 1-y compound grown on lattice-matched InP substrate. Layer composition and thicknesses are given in Figure 1.

The wafer structure has been optimized from the results of our previous work [START_REF] Kettemann | Thermoelectric field effects in low-dimensional structure solar cells[END_REF] which optically showed the evidence of hot carrier effects in a quantum well/barriers absorber. The multilayer wafer contains an intrinsic InGaAsP-based quantum well/barriers region, playing the role of absorber which generates hot charge carriers. The barriers play the role of semi-selective contacts where the carriers are mainly cold. This well/barriers region is sandwiched between two gradually n-and p-doped InP layers for carriers separation and collection. The quantum well has been designed to minimize the number of energy levels in the QW and to spectrally separate absorption in the quantum well and in the barriers, to simplify the spectral analysis. Band structures calculating from k.p formalism is shown in Supplementary Figure 1 indicating that a single optical transition is dominant in the QW region. In our experimental setup, a hyperspectral imaging system from Photon Etc. records quantitative luminescence spectra (luminance), which we subsequently calibrate to obtain absolute values. This allows us to evaluate the thermodynamic properties of the emission that is related to the carrier thermodynamic properties. The excitation laser wavelength is nm. All measurements are carried out at room temperature. More details on the experimental setup are seen in Supplementary Figure 2.

Modelling the absorption

The absorptivity is calculated from absorption coefficients and layer thicknesses. For our well/barriers structure we write:

A(E) = [1 -R(E)] • {1 -exp [-(α w d w + α b d b )]}
Where α w and α b are effective absorption coefficients of well and barriers, d w and d b their thicknesses. We used an absorption model taking into account excitons and free carriers in the well as well as free carriers in the barriers [START_REF] Bris | Thermalisation rate study of GaSb-based heterostructures by continuous wave photoluminescence and their potential as hot carrier solar cell absorbers[END_REF][START_REF] Hirst | Experimental demonstration of hot-carrier photo-current in an InGaAs quantum well solar cell[END_REF][START_REF] Rodière | Experimental evidence of hot carriers solar cell operation in multi-quantum wells heterostructures[END_REF]. This effect of band filling [START_REF] Zitter | Saturated optical absorption through band filling in semiconductors[END_REF] is expressed by introducing occupation functions into the absorption coefficient α s = α 0 • [f e v (E h ) -f e c (E e )]. Detailed expressions of absorption are provided in the see Supplementary Notes 1 and 2.

Evaluation of uncertainties in optical measurements

Temperature T and electrochemical potential µ are determined from fit of luminescence spectra. Uncertainties, in the measurement of T for example, are evaluated by manually assigning it to a fixed value T around its fitted value T 0 and launch the fit again, until the fitted spectrum becomes unsatisfactory. Mathematically, that means ∼ 10% or more of the points in the fitted luminescence spectrum present a relative error |Φ fit (T ) -Φ exp |/Φ exp of at least ∼ 10% compared to experimental data.

Figure 1 :

 1 Figure 1: Description of the hot carrier heterojunction device. (a)Energy band diagram of the heterostructure with 980 nm excitation wavelength, 1890 W/cm 2 and at open circuit condition. E c (E v ) stands for conduction (valence) energy band level. (b) Top view of the micro-solar cell taken with an optical microscope(golden green) and schematic of the opto-electrical characterisation (light red -grey -light blue). Charge carriers are generated by cw laser excitation at 980 nm (blue arrow) and thermodynamic properties of charge carriers are probed by the luminescence induced by radiative recombination (red arrow).

6 E f f e c t i v e s o l a r c o n c e n t r a t i o n f a c t o r 1

 61 r e n t d e n s i t y ( A / c m 2 )

Figure 2 :

 2 Figure 2: Electrical characteristics of the heterojunction device. Variation of short-circuit current density (J sc , blue squares) and open-circuit voltage (V oc , red circles) with laser power. The red horizontal line indicates the E 1 -HH 1 transition energy. The indicative solar illumination concentration incident on the device is obtained by multiplying the laser fluence (in W/cm 2 ) by a factor of 33.7, as J sc = 4.5 mA/cm 2 was measured under AM1.5 solar spectrum excitation. Details on this conversion are given in the Supplementary Note 6. (b) Equivalent conversion efficiency as a function of the effective concentration with 980 nm. (c) current-voltage curve taken at the maximum efficiency.

  r a b s o r p t i v i t y P h o t o n e n e r g y ( e V )

Figure 3 :

 3 Figure 3: External quantum efficiency (EQE) and absorption model. Data are shown from 0.78 eV (1550 nm) to 1.10 eV (1127 nm) where absorption by the quantum well/barriers occurs. The EQE (red circles) is fitted using the expression of unsaturated absorptivity (blue line) A 0(E) = (1 -R) {1 -exp [-(α w0 d w + α b0 d b )]}.The reflectivity R is measured to be about 0.3 over the spectral range [0.75, 1.10] eV. The absorption model includes quantum well interband excitonic and free carrier absorption (in α w ), as well as barrier interband absorption (in α b ). See Supplementary Note 1 for more details on the quantum/well absorption coefficients α w0 and α b0 . The experimental data reflects both qualitatively and quantitatively the absorptivity obtained from PL measurements at low excitation power. The exciton peak is centred at 0.82 eV and the absorption plateaus are observed, corresponding to the fundamental and the first excited levels of the quantum well. The detailed result is presented in the Supplementary Note 3.

F u n d a m e n t a l t r a n s i t i o n E 1 -Figure 4 :

 14 Figure 4: Variation of carriers thermodynamic properties with laser intensity. (a) Three PL spectra (grey circles) and fit at low (black line), medium (magenta line) and strong (red line) laser excitation, in open circuit regime. (b) : Corresponding absorptivity deduced from fit. (c) and (d): Variation of quantum well carrier temperature T w and electrochemical potentials in quantum well µ w (red circles) and barriers µ b (blue triangles) with laser power. Evaluated uncertainties in the optical measurement are displayed by errors bars. See Methods section for details on the evaluation of uncertainties.

Figure 5 :

 5 Figure 5: Variation of carriers thermodynamic properties with bias voltage. The laser intensity is at 1890 W/cm 2 . (a): Three PL spectra (grey circles) with respective fit (continuous lines) at three bias voltages -V = 0.250 V (black line), V = 0.661 V (magenta line) and V = 0.787 V (red line). (b): Corresponding absorptivity deduced from fit at the same bias voltages. (c): Variation of carrier temperature in quantum well (red circles) with bias voltage. The blue line represents ambient temperature around 297 K. (d): Variation of electrochemical potential in quantum well (red circles) and barriers (blue triangles), and of luminescence intensity (black continuous line) with bias voltage. The dashed blue line serves as a visual guide for the variation of the barriers' electrochemical potential and the green continuous line depicts the µ = V dependence. The grey box highlights the maximum output power working point. The evaluation of uncertainties is detailed in Methods section.

Figure 6 :

 6 Figure 6: Assessment of the hot carriers' contribution to power generation . Evaluation of the hot carrier contribution on the electrical properties of the cell with the excitation intensity. (a) Evolution of the electrochemical potential (b) generation current and (c) recombination current in the hot carrier regime (red dots) and cold carrier regime (blue square).

Table 1 : Table detailing the composition layers. The

 1

	Composition	Doping Thickness	Role
	InGaAs	p++	170 nm	Electric contact
	In 0.8 Ga 0.2 As 0.435 P 0.565	p+	50 nm
	InP	p+	50 nm	Cladding
	In 0.8 Ga 0.2 As 0.435 P 0.565	i	120 nm	Barrier
	In 0.78 Ga 0.222 As 0.81 P 0.19	i	7.4 nm	Quantum Well
	In 0.8 Ga 0.2 As 0.435 P 0.565	i	130 nm	Barrier
	InP	n+	401 µm	Cladding
				intrinsic (i) InGaAsP-based
	absorber consists of a 7.4 nm quantum well (E g = 0.78 eV) between 120 and 130 nm
	barriers (E g = 1.05 eV). The absorber is sandwiched between p-and n-doped InP contact
	layers.

Determination of carriers thermodynamic propertiesLuminescence spectra, calibrated in absolute units, are recorded at room temperature and atmospheric pressure in order to determine the carrier thermodyamic properties.

Acknowledgements

This work was carried out in the framework of a project of IPVF (Institut Photovoltaïque d'Île-de-France). This project has been supported by the French Government in the framework of the program of investment for the future (Programme d'Investissement d'Avenir) ANR-IEED-002-0. The authors acknowledge J. Even for the energy band structure simulation, and P. Schultz and J. Connolly for a careful reading of the manuscript.

Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

Authors contribution

JFG and LL did the planning of the study; DTN did the data acquisition; DTN, LL, FG contributed to data treatment; SBR and OD did the sample fabrication; DTN, LL and JFG, contributed to data analysis, modeling and wrote the paper.


Hanen Louati
email: louatihanen42@yahoo.fr

Michel Rouleux
email: rouleux@univ-tln.fr

SEMI-CLASSICAL QUANTUM MAPS OF SEMI-HYPERBOLIC TYPE

Let M = R n or possibly a Riemannian, non compact manifold. We consider semiexcited resonances for a h-differential operator H(x, hD x ; h) on L 2 (M ) induced by a nondegenerate periodic orbit γ 0 of semi-hyperbolic type, which is contained in the non critical energy surface {H 0 = 0}. By semi-hyperbolic, we mean that the linearized Poincaré map dP 0 associated with γ 0 has at least one eigenvalue of modulus greater (or less) than 1, and one eigenvalue of modulus equal to 1, and by non-degenerate that 1 is not an eigenvalue, which implies a family γ(E) with the same properties. It is known that an infinite number of periodic orbits generally cluster near γ 0 , with periods approximately multiples of its primitive period. We construct the monodromy and Grushin operator, adapting some arguments by [NoSjZw], [SjZw], and compare with those obtained in [LouRo], which ignore the additional orbits near γ 0 , but still give the right quantization rule for the family γ(E).

Introduction

Let M be a smooth manifold (for simplicity here M = R n , but our results hold in more general cases, see Examples 1 and 2 below), and H(y, hD y ; h) be a semi-classical Partial Differential Operator of second order, we assume to be self-adjoint on L 2 (M ), and satisfy usual hypotheses required in the framework of resonances. In particular, its Weyl symbol H(y, η; h), in the sense of h-ΨDO, belongs to the class

S N ( η 2 ) = {H ∈ C ∞ (T * R n ) : ∀α ∈ N 2n , ∃C α > 0, |∂ α (y,η) H(y, η; h)| ≤ C α h N η 2 }
i.e. is of growth at most quadratic in momentum at infinity (here η 2 = 1 + |η| 2 ).

a) Main hypotheses

• Hypothesis 1 (Ellipticity, regularity of coefficients and behavior at infinity).

H is elliptic (i.e. |H(y, η; h) + i| ≥ const. η 2 ) and extends analytically in a "conic" neighborhood of the real domain where it has the semi-classical expansion H(y, η; h) ∼ H 0 (y, η) + hH 1 (y, η)

+ • • • , h → 0
To fix the ideas, we assume H(y, η; h)-η 2 → 0 in Γ 0 when | Re y| → ∞, | Im y| ≤ const. Re y .

This assumption can be relaxed, see [HeSj].

Then inf σ ess (H(y, hD h ; h)) = 0 (actually H has only continuous spectrum above 0) and we define the resonances of H near E 0 > 0 by the method of analytic distorsions, as the discrete spectrum of some non self-adjoint extension of H.

Namely, let Γ ⊂ C n be a real, totally real submanifod of dimension n, and H(y, hD y ; h) = |α|≤2 a α (z; h)(hD z ) α a differential operator with C ∞ coefficients in some suitable complex neighborhood Γ C of Γ (here D z denote the holomorphic derivative with respect to coordinates in Γ C ). Then we can define a differential operator H Γ : C ∞ (Γ) → C ∞ (Γ), such that, if u is holomorphic, then (Hu)| Γ = H Γ (u| Γ ). Now assume that H(y, η; h) is defined in Γ 0 as in (1.1).

For 0 ≤ θ ≤ θ 0 , we let Γ = Γ θ be parametrized by

f θ ∈ C ∞ (R n ; C n ) such that f θ (y) = y
for y in a compact set and f θ (y) = e iθ y for large y. The corresponding family of operators H θ = H Γ θ on L 2 is known to be an analytic family of type (A) and σ ess (H θ ) = e -2iθ R + .

Moreover, when θ > 0, H θ is Fredholm and may also have discrete eigenvalues in the lowerhalf plane near E 0 , called (outgoing) resonances. The resonant (or extended) states are the associated eigenfunctions. See [Co], [Va], [ReSi], [BrCoDu], [HeSj], [GéSi] for related approaches, which turn out to be essentially equivalent ( [HeMa]). We follow here mainly [NoSjZw].

Since we shall mostly consider H(y, hD y ; h) as a h-ΨDO, we shall rather denote it by H w (y, hD y ; h).

Locating precisely resonances near E 0 (like Bohr-Sommerfeld quantization conditions) hinges on properties of the Hamiltonian flow on the energy surfaces nearby E 0 . As recalled briefly in Appendix, we need to choose distorsion f θ accurately, as well as other phase-space distorsions, or Lagrangian deformations.

• Hypothesis 2 (Regularity of energy surface)

To save notations we change H 0 to H when considering classical quantities. We fix a regular energy surface {H(y, η) = 0}, and assume there is an energy interval I around E 0 , so that the Hamilton vector field X H has no fixed point on {H(y, η) = E}, for E ∈ I.

Let Φ t = exp(tX H ) : T * R n → T * R n be the Hamiltonian flow and (1.2) K(E) = {ρ ∈ T * R n , H 0 (ρ) = E, Φ t (ρ) doesn't grow to infinity as |t| → ∞}
the trapped set at energy E. Simplest situation holds when K(E) is a fixed point [BrCoDu].

• Hypothesis 3 (Trapped set at energy 0)

We assume here that K 0 = K(0) contains a periodic orbit of primitive period T 0 . The differential of Poincaré map (or first return map) is a symplectic automorphism P 0 of the normal space Σ 0 = Σ(0) of γ 0 in H -1 (0), is a manifold of dimension 2d = 2(n -1), which is called Poincaré section.

Let λ ∈ C be the eigenvalues of A = dP 0 (or Floquet multipliers). The periodic orbit γ 0 is said non degenerate if 1 is not a Floquet multiplier. By Poincaré Continuation Theorem, there is a one parameter family of periodic orbits γ(E) ⊂ H -1 0 (E) containing γ 0 , and γ(E) is non-degenerate for E small enough. By abuse of notations, we shall still call Poincaré section the smooth foliation Σ = E∈I Σ(E) transverse to γ = E∈I γ(E).

An eigenvalue λ ∈ C of A is called elliptic (ee) if |λ| = 1 (λ = ±1) and hyperbolic (he) if |λ| = 1. The corresponding eigenspace will be denoted by F λ .

• Hypothesis 4 (genericity properties of linearized Poincaré map)

F ±1 = {0}, F λ = {0}, ∀λ ≤ 0 In particular, we can define B = log A. Eigenvalues µ = µ(λ) of B (Floquet exponents) verify µ(λ) = µ(λ), µ = log λ. Exponent µ is said ee if Re µ = 0, real-hyperbolic (hr) if Im µ = 0, loxodromic or complex-hyperbolic (hc) if Re µ = 0, Im µ = 0.
Eigenvalues of B have the form µ j , -µ j , µ j , -µ j = 0, Re µ j ≥ 0, with same multiplicity.

For simplicity, assume eigenvalues µ j are distinct.

• Hypothesis 5 (Hyperbolicity)

We are interested in the case where γ 0 is unstable: A is hyperbolic, i.e. has at last one eigenvalue |λ| = 1. We say we have pure (or complete) hyperbolicity iff Re µ j > 0 for all j. In case of complete hyperbolicity, γ 0 is isolated, and we will assume (excluding e.g. symmetries, which would involve tunneling, as in Example 2 below) that the trapped set reduces to K 0 = γ 0 . We say we have partial, or semi-hyperbolicity, iff there exists both j with Re µ j > 0 and k with Re µ k = 0. This is generically the case for hyperbolic systems [Ar].

Recall the Center/Stable/Unstable manifolds Theorem (see e.g. [GéSj2] for a review): Let γ 0 be a non-degenerate periodic orbit, and γ(E) as above. Then there exists a closed symplectic submanifold C (center manifold) containing γ(E) and such that X H 0 (ρ) is tangent to C at every point ρ ∈ C (i.e. C invariant under the Hamiltonian flow). There exist also two vector bundles N ± , such that for all ρ ∈ Σ,

T ρ C σ = N + ρ ⊕ N - ρ (here T ρ C σ is the orthogonal symplectic of T ρ C). Moreover, N ± | γ(E)
are invariant under the Hamiltonian flow, which is contracting on N -| γ E and expansive on N + | γ E . Note that hr and hc components which belong to outgoing/incoming manifolds, differ only by technical aspects, while ee components which belong to the center manifold, play a distinct role.

Here we say that γ 0 (and hence the family γ(E)) is unstable (e.g. in Lyapunov sense) if N ± = 0, i.e. when A is hyperbolic.

• Hypothesis 6 (Strong non-resonance condition and twist condition)

∀k ∈ Z d : d j=1 k j µ j ∈ 2iπZ =⇒ k j = 0, ∀j
Let 0 < < d be the number of elliptic elements, i.e. µ j ∈ iR, Im µ j > 0. Assume moreover that P 0 | C (C the center manifold) is of twist type, i.e. the non linear Birkhoff invariants, are non degenerate. In particular γ(E) is N -fold non-degenerate for all N . By Lewis-Birkhoff Fixed Point Theorem, see [START_REF] Klingenberg | Lectures on closed geodesics[END_REF]Thm.3.3.3], in every neighborhood of (0, 0) ∈ R 2 , there exists infinitely many periodic points (i.e. belonging to periodic orbits). The number of orbits of bounded period is finite.

Applying this Theorem to the normally hyperbolic symplectic invariant manifold C for Poincaré map, we find a sequence of periodic orbits γ k with (primitive) periods T k → ∞ clustering on γ 0 , with T k ≈ kT 0 in the limit k → ∞, and supp(γ k ) → supp(γ 0 ) ("infra-red limit") So we assume the trapped set is of the form (excluding, as we already pointed out,

other components of K 0 in {H = 0}) (1.5) K 0 = k∈N γ k
In particular K 0 is topologically 1-D, and only one Poincaré section (or 2 equivalent Poincaré sections) is needed to describe the dynamics near K 0 , which simplifies the situation presented in [NoSjZw]. Moreover, there is structural stability of K(E) [START_REF] Katok | Introduction to the modern theorey of dynamical systems[END_REF]Thm.18.2.3]: namely the flows Φ t | K(E) and Φ t | K 0 are conjugated, up to time reparametrization, by a homeomorphism close to identity. For instance it could happen that K(E) = k∈N γ k (E), but this is not actually needed, for orbits with large period are unstable. It follows that we can choose Poincaré section Σ transverse to K = E∈I K(E). We shall assume, as in [NoSjZw], that ∂Σ does not intersect K. Our situation is very similar to [NoSjZw], and the more simple structure of the flow allows for some simplifications of the proof.

b) Examples

1) Poincaré example of a pure hyperbolic orbit: H(y, hD y ) is the geodesic flow on onesheeted hyperboloid in R 3 ("diabolo"): the throat circle γ 0 is an unstable hr periodic orbit (geodesic).

2) The geodesic flow H(y, hD y ) on a surface of revolution M embedded in R 4 with axis Oz, projecting on x, z-variables as the "double diabolo", a surface homeomorphic to the one-sheeted hyperboloid in R 3 , but with two throat circles, separated by a crest circle. The effective Hamiltonian has principal symbol

H 0 = η 2 + ζ 2 + (2z 4 -z 2 + 1) cosh y -2 -1, with
(x, ξ) as cyclic variables. For some energy E 1 there are two periodic geodesics of hyperbolic type (the throat circles) situated symmetrically on the hyperplanes z = ± 1 4 (with two hr pairs); our constructions apply modulo tunneling corrections. For some energy E 2 < E 1 there is one periodic geodesic of semi-hyperbolic type (the crest circle), with one ee pair and one hr pair. This is the generic situation, unlike the purely hyperbolic case as in previous Example. See [Chr2,App.C].

3) H(y, hD y ) = -h 2 ∆ y + |y| -1 + ay 1 on L 2 (R n ) (repulsive Coulomb potential perturbed by Stark effect) near an energy level E > 2/ √ a, or more generally, Schrödinger operators with potentials with two or more bumps. Their periodic orbits are generally hr (also called librations). See [GéSj], [Sj3].

4) Non-autonomous case [Tip]: Atom in a periodically polarized electric field

H 1 (t) = -h 2 ∆ + V (|x|) + E(t) • x on L 2 (R 3 ), E(t) = cos ωt x 1 + sin ωt x 2 .
After some transformation, Floquet operator takes the form

U (s + T, s) = e -iω(s+T )L x 3 /h e iT P (x,hD x )/h e iω(s+T )L x 3 /h where T = 2π ω , p(x, ξ) = (ξ -a) 2 + V (|x|) -ω(x 1 ξ 2 -x 2 ξ 1 )
, and a = (1/ω, 0, 0). The operator e iT P (x,hD x )/h is now independent of time, and plays the role of the monodromy operator constructed below.

c) Main result on resonances in the semi-hyperbolic case

Our main result for which we sketch a proof in Sect.2, is a straighforward generalization of [NoSjZw], when allowing for elliptic Floquet exponents, and of [GéSj1] in the hyperbolic case. We summarize it as follows.

Theorem 1.1: Under the Hypotheses 1-6 above, consider the spectral window

W h = [E 0 - ε 0 , E 0 + ε 0 ] -i]0, Ch log(1/h)]. Then if ε 0 , C > 0 are small enough, there is h 0 > 0 small
enough and a family of matrices N (z, h), such that the zeroes of det(Id -N (z, h)) give all resonances of H w (x, hD x ; h) in W h with correct multiplicities. The matrices N (z; h) of order

k h ∼ h -n+1 are of the form N (z; h) = Π h M(z; h)Π h + O(h N ) where Π h : L 2 → L 2 (

d) Bohr-Sommerfeld (BS) quantization rules

BS for an hyperbolic orbit γ(E) are known for a long time, see [GéSj1], [Vo]; in [LouRo1,2] we use the method presented in Sect.3 below, ignoring the orbits accumulating on γ(E). Our proof holds stricto sensu only in the complete hyperbolic case, but the result turns out to be correct otherwise, provided we consider only resonances associated with the family γ(E). A peculiarity of BS rules for resonant spectrum is that they cannot be simply derived from the construction of quasi-modes as in the self-adjoint case (see e.g. [BLaz]). We have: Theorem 1.2 [LouRo]: Under the hypotheses above, let us define the semi-classical action

along γ(E), by S(E; h) = S 0 (E) + hS 1 (E) + O(h 2 ) with S 0 (E) = γ E ξ dx (1.6) S 1 (E) = - T (E) 0 H 1 (y(t), η(t)) dt + 1 2i d j=1 µ j (E) + π g 2 (1.7) Here µ j (E) = µ j + O(E) is Floquet exponent at energy E, g ∈ Z Gelfand-
(E) for E in W h = [E 0 -ε 0 , E 0 + ε 0 ] -i]0, h log(1/h)] are
given by the generalized BS quantization condition

(1.8) 1 2πh S(E; h) + 1 2iπ d j=1 hk j µ j (E) + O(h 2 |k| 2 ) = mh, m ∈ Z, k ∈ N d provided |m|h ≤ ε 0 , |k| ≤ const. log(1/h).
This remains true, at the price of technical difficulties, when replacing the the width

h log(1/h) of W h by h 1-δ , 0 < δ < 1.
We stress that this theorem says in general nothing about other resonances described in Theorem 1.1, unless γ 0 is purely hyperbolic, in which case the periodic orbits γ(E) are isolated, and thus we can assume K(E) = γ(E).

e) Remarks on the trace formulas

In the self-adjoint case (e.g. the geodesic flow on a compact manifold with negative curvature) trace formulas have been considered for hyperbolic or semi-hyperbolic flows. They are expressed in the time variable t (trace of the propagator or wave group, see [Zel]), or in the energy variable E (trace of the semi-classical Green function, see [Vo] and references therein).

In case H is the geodesic flow on a compact Riemannian manifold (M, g), Zelditch [Zel2] computed the singular part of the trace of the wave group U (t). It is obtained as a term e 0 (t) (involving the fixed points of the flow), plus the sum over all periodic geodesics γ on M , of "wave trace invariants" e γ (t), using non commutative residues, that can be computed as an asymptotic series (asymptotics with respect to smoothness). This formula does not involve other periodic orbits (clustering on each γ when γ is of semi-hyperbolic type), but their contribution would appear when investigating "convergence" (in the sense of resurgence) of the series defining e γ (t). Note that the semi-classical parameter is obtained in scaling the variables microlocally near a periodic geodesic to bring the Hamiltonian in BNF. The same situation is likely to appear for resonances in the non compact case.

The trace of the semi-classical Green function instead, near some fixed E can be expressed formally by a sum of terms R γ (E) labelled by the classical periodic orbits γ having energy E.

The poles of R γ (E) are precisely localized by an implicit equation such as Bohr-Sommerfeld quantization condition of Theorem 1.2: since it would give complex energies, it is called by Voros the "generalized quantization condition", to stress that periodic orbits are not necessarily associated with bound states. This paradox could be settled in the framework or resurgence theory.

In the context of resonances the paradox of complex poles disappears. Thus it would be tempting to look for a trace formula as in [SjZw]. First we recall Helffer-Sjöstrand formula

[DiSj]. Let H be a self-adjoint operator, χ ∈ C 2 0 (R) and χ ∈ C 1 0 (C) an almost analytic extension of χ satisfying ∂ χ(z) = O(| Im z|). Then (1.9) χ(H) = - 1 π ∂ χ(z)(z -H) -1 L(dz)
Following the remark after the proof of Theorem 8.1 in [DiSj], this could be generalized to classes of non selfadjoint operators, and applied to H θ . Then we may use (1.9) as a definition.

So let f N ∈ C ∞ (R) be such that supp f N ⊂]0, T 0 N [, φ N ∈ C ∞ 0 (C)
whose support in Im z has to be chosen suitably near 0 (of width O(h)), A N (y, hD y ) be a h-PDO cutoff equal to 1 in a small neighborhod of γ(z), M (z) be the monodromy operator computed either in Sect.2 or Sect.3, and k(N ) → ∞. In the case of resonances, it is plausible to expect a "trace formula" modelled after this of [SjZw], namely

Tr f N (H θ /h)φ N (H θ ; h)A N (y, hD y ) ≈ 1 2iπ N k=1 Tr R f N (z/h)φ N (z; h)M (z) k-1 dM dz (z) dz + O(h k(N ) )
just keeping positive values of N to account for the time reversal symmetry breaking. This however, seems again far from reach, especially because resonances proliferate near the real axis in W h as h → 0.

At last we note that in the framework of resonance scattering outside convex obstacles, trace formulas (or the related zeta function) in the energy representation are given by Ikawa [Ik], and the situation is better understood. It is similar to our case, when Poincaré map has no elliptic element.

2. A hint on the proof of Theorem 1.1.

a) The (absolute) monodromy operator

The energy parameter will be denoted by z, and for the moment we work at a formal level, i.e. H(y, hD y ; h) denotes the self-adjoint operator. We shall follow mainly [SjZw], making use of 2 equivalent Poincaré sections, but taking care eventually of the (semi-) hyperbolic structure of the flow.

Before entering the actual constructions, we recall how to define the monodromy operator and solve Grushin problem in the simple situation (see [SjZw], [IfaLouRo]), where H = hD t acts on L 2 (S 1 ) with periodic boundary condition u(t) = u(t+2π). Solving for (H -z)u(t) = 0, we get two solutions with the same expression but defined on different charts

(2.13) u a (t) = e izt/h , -π < t < π, u a (t) = e izt/h , 0 < t < 2π
indexed by angles a = 0 and a = π on S 1 . All angles will be computed mod 2π. In the following we take advantage of the fact that these functions differ but when z belongs to the spectrum of H.

Let also χ a ∈ C ∞ 0 (S 1 ) be equal to 1 near a, χ a = 1 -χ a . To fix the ideas, we may assume that χ a drops down to 0 near t = -π 2 and t = π 2 . (which belongs to both charts). We set F a ± = i h [P, χ a ] ± u a , where ± denotes the part of the commutator supported in the half circles 0 < t < π and -π < t < 0 mod 2π. Similarly F a ± = i h [P, χ a ] ± u a , and we may assume that χ a drops to 0 near t = π 2 and t = 3π 2 mod 2π. Modulo O(h ∞ ) (as all constructions in this work, so we shall not dwell on this anymore) distributions F a ± , F a ± do not depend on the choice of χ a above since we may expand the commutator when applying to distributions defined on a single chart (2.13) and use that H is self-adjoint.

Remark: It is convenient to view F a + -F a -and F a + -F a -as belonging to co-kernel of H -z in the sense they are not annihilated by H -z. If we form Gram matrix

G (a,a ) (z) = (u a |F a + -F a -) (u a |F a + -F a -) (u a |F a + -F a -) (u a |F a + -F a -)
an elementary computation shows that det G (a,a ) (z) = -4 sin 2 (πz/h), so the condition that u a coincides with u a is precisely that z = kh, with k ∈ Z (see [IfaRLouRo] for details, where a convention slightly different from [SjZw] has been made).

Starting from the point a = 0 we associate with u a the multiplication operator v + → Now we turn to the general case. The situation of [NoSjZw] simplifies since we need only 2 equivalent Poincaré sections (modulo moving around γ 0 ). Let m 0 ∈ γ 0 , and m 1 = exp T 0 2 X H (m 0 ) be two "distinguished" points on γ 0 , which play respectively the role of t = π/2, t = -π/2 mod 2π in the example above. Assume for simplicity that H(y, hD y ; h) verifies time-reversal symmetry, so that m 0 is again the point along γ 0 reached from m 1 within time T 0 /2, where T 0 is the (primitive) period of γ 0 . The corresponding cut-off near the orbit will still be denoted by χ a and χ a .

I a (z)v + = u a (t)v + on C, i.
Since the energy shell E = 0 is non critical, near every m ∈ γ 0 , H(y, hD y ; h) can be reduced microlocally to hD t , i.e. there exists a local canonical transformation κ : T * M → U ⊂ T * R n defined near ((0, 0), m), and a h-FIO T , associated with the graph of κ, elliptic near ((0, 0), m), such that T H(y, hD y ; h) = hD t T near ((0, 0), m).

Fix m, and construct a corresponding T ; if we define

Ker m (H) = T -1 Ker(hD t )
we can identify Ker m (H) with semi-classical distributions on R d (i.e. on a Poincare section) microlocally near (m, (0, 0)); we denote this identification by K :

D (R d ) → Ker m (H). Now we solve (hD t -z)u(x) = 0, u| t=0 = v(x ) in D (R n ) by u(x) = e izt/h v(x ), x = (t, x ). As
in the Example, we obtain this way two Poisson operators v + → I a (z)v + = e izt/h v + when -π < t < π (forward) and v + → I a (z)v + = e izt/h v + when 0 < t < 2π (backward), defined on "Cauchy data" v + ∈ D (R d ). Working locally, we can ignore their domain, and call them both K(z), but moving along the flow in either direction, we introduce new canonical charts (κ, U) and construct new FIO's T accordingly. By compactness, we can cover γ 0 with a finite set of such (κ, U). Assume the intersection of Poincaré sections with the domain of definitions of T contains (strictly) the trapped set.

Instead of specifying γ 0 , it is more convenient to select the orbit γ(z), which is periodic with respect to the Hamilton flow of H at energy z, with period T (z). Accordingly, we change m 0 , m 1 to m 0 (z), m 1 (z). All γ(z) are mapped diffeomorphically to S 1 by the Hamilton flow, so moving once around γ(z) means moving once around S 1 in the Example.

Varying m on the orbit γ(z), we obtain the forward/backward extensions (standing for u a , u a in (2.13)), independent of m(z) ∈ γ(z)

(2.18) I ± (z) : Ker ± m(z) (H -z) → Ker γ(z) (H -z)
where Ker ± m(z) (H -z) denotes the space of forward/backward solutions near m(z). Operators I ± (z) are (microlocally) injective. Thus we obtain the exact sequence (with obvious notations)

(2.19) 0 -→ Ker + m(z) (H -z) ⊕ Ker - m(z) (H -z) -→ Ker γ(z) (H -z) -→ 0
where the 2nd arrow is

I + (z) ⊕ I -(z)
+ m(z) (H -z) ⊕ Ker - m(z) (H -z), e.g. to Ker + m 0 (z) (H -z) ⊕ Ker - m 1 (z) (H -z). Let K f (z) = I + (z)K(z), K b (z) = I + (z)K(z). Since K(z) identifies microlocally Ker ± m(z) (H -z) with D m(z) (R d ) (semi-classical distributions microlocalized on Poincaré section at m(z)), we have also (2.20) 0 -→ D m(z) (R d ) ⊕ D m(z) (R d ) -→ D γ(z) (R n ) -→ 0
where the 2nd arrow is

K f (z) ⊕ K b (z). Let W + , W -be nghbhds of m 0 , m 1 in T * M , such that (2.21) {m : χ a (m) = 1} ∩ {m : χ a (m) = 0} = {m : χ a (m) = 1} ∩ {m : χ a (m) = 0} ⊂ W -∪ W +
Following [SjZw], we define microlocally near W + the (absolute) quantum monodromy operator

M(z) : Ker + m 0 (z) (H -z) → Ker - m 0 (z) (H -z) by (2.23) I + (z)f = I -(z)M(z)f, f ∈ Ker m 0 (z) (H -z), microlocally near W -
Clearly, we can interchange the roles of m 0 (z) and m 1 (z) in this definition: M(z) is independent of the section. We define the quantum monodromy operator M (z) :

D (R d ) → D (R d ) as follows. Let B(z) = K(z) * i h [H, χ a ] W + K(z) : D m 0 (R d ) → D m 0 (R d )
Following [SjZw], we check that B(z) is a h-PDO, defined microlocally near (0, 0) ∈ T * R d , positive and formally self-adjoint. So L(z) = K(z) B(z) -1/2 verifies the "flux norm" identity

(2.25) L(z) * i h [H, χ a ] W + L(z) = Id microlocally near (0, 0) ∈ T * R d Using (2.20), we see that K(z) : D m 0 (z) (R d ) → Ker m 0 (z) (H -z) is invertible, so is L(z) and its inverse is (2.26) L(z) -1 = R + (z) = L(z) * i h [H, χ a ] W +
We call the quantum monodromy operator

(2.28) M (z) = L(z) -1 M(z)L(z)
This is a h-FIO, whose canonical relation is precisely the graph of Poincare map P 0 (z), i.e.

M (z) ∈ I 0 R d × R d ; C ), C = {(x, ξ, x , -ξ ) : (x, ξ) = P 0 (z)(x , ξ )}
We make more precise [START_REF] Sjöstrand | Quantum monodromy and semi-classical trace formulae[END_REF]Prop.4.5] (still before any analytic dilation), taking also into account the hyperbolicity of the flow. Recall the flow is expanding in some direction of Poincaré section, and contracting in the orthogonal one (for the symplectic structure). Denote by D(W + ) ⊂ W + (D like departure) a neighborhood of the outgoing manifold in W + and A(W + ) ⊂ W + (A like arrival) a neighborhood of the incoming manifold in W + (see [NoSjZw],

[NoZw]). By the same letter we denote the space of distributions microlocalized near that set.

Proposition 2.1: For real z, the monodromy operator M (z) is microlocally "unitary" D(W + ) → A(W + ), and similarly for complex z, in the sense that the adjoint of M (z) is

equal to M (z) -1 . Proof: Let v ∈ D m(z) (R d ) microlocally supported near (0,0), and u = L(z)v ∈ D γ(z) (R n ), we
compute (dropping the variable z from the notations) (M v|M v) = (L -1 Mu|L -1 Mu). By inserting (2.26) on the left of the scalar product we get

(M v|M v) = i h [H, χ a ] W + Mu|Mu L 2 (R n ) = (I -) -1 * i h [H, χ a ] W + Mu|I -M(z)u
where we have also introduced the backward extension operator I -= I -(z) as in (2.18). Next we have, for δ > 0 sufficiently small, and 0

< t < T (0)/2 + δ, i h [H, χ] W + = I * - i h [H, χ t ] W t + I -, where χ t = χ • exp -tX H ≡ χ a (• -t)
, and W t + = exp(-tX H )W + , which corresponds to moving supp χ in the direction opposite to the flow of X H , and W + simultaneously so that (2.21) holds. Hence

(2.31) (M v|M v) = i h [H, χ t ] W t + I -Mu|I -Mu L 2 (R n )
Similarly, inserting (2.25) on the left of the scalar product (v|v) = (L -1 u|L -1 u) we get

(2.32) (v|v) = i h [H, χ -t ] W -t + I + u|I + u
For t ∼ T (0)/2, and WF h v sufficiently close to (0,0) so that WF , and comparing (2.31) with (2.32) gives (M v|M v) = (v|v). The Proposition follows easily from the definition of D(W + ), A(W + ). ♣ To fully restore "unitarity" of M , so that Grushin problem be well-posed, we need to introduce the weighted Sobolev spaces, or/and the complex Lagrangian deformations.

h i h [H, χ -t ] W -t + I + u ⊂ W -, we get (v|v) = i h [H, χ t ] W t + I -Mu|I -Mu
Let us conclude by writing M (z) in a form similar to [NoSjZw,(4.33)]. This is done in 2 steps: let K 0 (z) = K(z), K 1 (z) be Poisson operators at m 0 (z), m 1 (z), and L 0 (z), L 1 (z) be the normalized ones. The monodromy operator from m 0 (s

) to m 1 (z) is M 0,1 (z) = L 1 (z) * i h [H, χ a ] W + L 0 (z), and this from m 1 back to m 0 (z), M 1,0 (z) = L 0 (z) * i h [H, χ a ] W -L 1 (z). Then (2.33) M (z) = M 1,0 (z)M 0,1 (z)
This will simplify (for the simplified problem) in action-angle coordinates as we shall see later.

b) Intertwining M(z) with M(w).

Structural stability for hyperbolic flows ( [START_REF] Katok | Introduction to the modern theorey of dynamical systems[END_REF]Thm.18.2.3]) recalled in Sect.1 carries to the monodromy operator. Namely, following [SjZw], let p(z) = H 0 -z. We call a classical time function a solution q(z) (which can be chosen independent of z) of X p(z) q(z) = {p(z), q(z)} = 1 (Lie differentiation). Thus (q(z), p(z)) are just the restriction to γ(z) (in the energy shell p(z) = 0) of (symplectic) Darboux coordinates (t, z) along γ(z), adapted to the Stable/Unstable/Center manifold. Since q(z) is a multi-valued function, we call first return classical time function, and denote by q ∂ (z) its continuation to the second sheet. Thus we have, with a slight abuse of notations

q ∂ (z) m(z) = q • exp T (z)X p(z) m(z) and (2.38) q ∂ (z) -q(z) | γ(z) = T (z)
where T (z) = dJ dz , J(z) = γ z η dy being the classical action along γ(z). We call a quantum time (resp. first return quantum time a solution Q(z), in the h-PDO's sense, of (2.39)

Id = i h [P (z), Q(z)], Id = i h [P (z), Q ∂ (z)]
with principal symbols q(z), q ∂ (z) respectively. Here P (z) = H -z. In the case P (z) is self-adjoint, we can assume Q(z) and Q ∂ (z) are self-adjoint (here again we work formally, but we shall need to take hyperbolicity into account as before). We have

Q ∂ (z) -Q(z) : Ker m(z) (H -z) → Ker m(z) (H -z)
Next we construct h-FIO's that will intertwine Poisson operators at different energies, and consider the following system of equations

(2.41) hD z -Q(z) L U (z, w) = hD z U (z, w) -Q(z)U (z, w) = 0 hD w + Q(w) R U (z, w) = hD z U (z, w) + U (z, w)Q(w) = 0
with initial condition U (0, 0) = Id. We can write (2.41) as L L/R (z, w)U (z, w) = 0, with operator by e it(z-w)/h . We notice that (2.41

L L = hD z -Q(z) L , L R = hD w + Q(w) R ,
) implies U (z, z) = Id, U (z, w)U (w, v) = U (z, v),
and U (w, z) * = U (z, w) when H(y, hD y ; h) is self-adjoint We have K(z) = U (z, w)K(w), and differentiating gives hD z K(z) = Q(z)K(z). Further, varying m, we extend U (z, w) in the forward and backward regions, to U ± (z, w). We have 2.41), we can solve for U ∂ (z, w) with same properties as U (z, w).

(2.42) (H -z)U ± (z, w) = U ± (z, w)(H -w), U (z, w)I ± (z) = I ± (w) Changing Q(z) to Q ∂ (z) in (
There follows the Proposition 2.2: We have the intertwining property

M(z)U (z, w) = U ∂ (z, w)M(w)
and the quantum monodromy operator satisfies the equation

hD z M (z) = K(z) -1 (Q ∂ (z) -Q(z))K(z)M (z) c) Grushin problem
Consider again the model case, with the notations of Sect.2. Introduce the "trace operator" R + (z)u = u(0), if u(t) = e izt/h v with v + = u(0), we check that

K(z) * i h [P, χ a ]u = π -π e -izt/h (χ a ) (t)u(t) dt = v +
Consider also the multiplication operators

E + (z) = χ a I a (z) + (1 -χ a )I a (z), R -(z) = i h [P, χ a ] -I a (z), E -+ (z) = 1 -e 2iπz/h
We claim that

(2.51) i h (P -z)E + (z) + R -(z)E -+ (z) = 0
Namely, evaluating on 0 < t < π, we have I a (z) = e itz/h , I a (z) = e itz/h , while evaluating on -π < t < 0, I a (z) = e itz/h , I a (z) = e i(t+2π)z/h . Now i h (P -z)E + (z) = [P, χ a ] I a (z) -e iπz/h I a (z) vanishes on 0 < t < π, while is equal to R -(z)E -+ (z) on -π < t < 0. So (2.51) follows. Hence Grushin problem (2.52)

P(z; h) u u - = i h (P -z) R -(z) R + (z) 0 u u - = v v + with v = 0 has a solution u = E + (z)v + , u -= E -+ (z)v + ,
P(z) -1 = E(z) E + (z) E -(z) E -+ (z) with (2.55) (P -z) -1 = E(z) -E + (z)E -+ (z) -1 E -(z)
In our case however, because of hyperbolicity, we need to introduce the weighted spaces (or Lagrangian deformations) so that (2.52) be well-posed. Still we start to proceed within the

formalism of Sect.2. Recall R + (z) from (2.26). So if v ∈ D (R d ), u = L(z)v solves near any m(z) (2.57) (H -z)u = 0, R + (z)u = v
To obtain a Cauchy problem globally near γ(z), we need to introduce R -(z). Recall K f /b (z) = I ± (z)K(z), which we normalize to L f /b (z) = I ± (z)L(z) as in (2.26). By (2.23) and (2.28),

we have

(2.58) L f (z) = L b (z)M (z), microlocally near W -× (0, 0)
and solve (2.57) in Ω \ W -(Ω neighborhood of γ(z)) as in the argument after (2.51) by

E + (z)v + = χ a L f (z)v + + (1 -χ a )L b (z)v + so that in particular E + (z)v + = L(z)v + in W + (since L f (z) = L b (z) in W + ), and
R + (z)E + (z)v + = L(z) * i h [H, χ a ] W + L(z)v + = v +
by (2.25). Applying H -z, using (2.58) and (H -z)E + (z)v + = 0 in W + , we find that, with

R -(z) = i h [H, χ a ] W -L b (z), and E -+ (z) = Id -M (z), u = E + (z)v + , u -= E -+ (z)v + solve (formally) the problem P(z) u u -= 0 v + near γ(z)
. This implies that the microlocal inverse of P(z) should be of the form

E(z) = E(z) E + (z) E -(z) E -+ (z)
, and we still have to find E(z), E -(z).

So we try to solve the inhomogeneous problem P(z) u u -= v v + near γ(z), and introduce the forward/backward fundamental solutions of

H -z, namely E f (z) = ∞ 0 e -it(H-z)/h dt, E b (z) = 0
-∞ e -it(H-z)/h dt, which of course assume a simple form after taking H microlocally to hD t . The construction of E(z) is more involved (see [SjZw], [NoSjZw]), but an argument like in Proposition 2.1 leads to

E -(z) = -M (z)K f (z)χ + K b (z) * (1 -χ)
Next we need to specify the right spaces where Grushin problem is well posed. This is done by introducing microlocal weights as in the Appendix, encoding the trapped set. We eventually get Theorem 1.1 as in [NoSjZw]; details will be given elsewhere.

3. An "approximate" theory.

Here we "neglect" the occurrence of infinitely many periodic orbits near γ 0 . It is plausible that this theory would still provide a good description of the resonant spectrum close to the real axis, since orbits with large period are quite unstable and contribute to the spectrum only far away from the real axis. Moreover, it becomes exact in the particular case where there are no elliptic elements, because such periodic orbits are isolated. At last, it provides BS quantization rules for the family γ(E), which are known to hold also in the semi-hyperbolic case.

Using complex coordinates, we may also reduce the center manifold C to γ by moving the elliptic subspaces into N ± .

a) Birkhoff normal form

Our approach relies on the classical BNF for the principal symbol H 0 of H. The first step takes H 0 to the form H 0 (y, η) = -τ + B 0 x, ξ +g(τ )+O(|τ, |x, ξ| 2 | 2 ) (the natural orientation of γ 0 has been reversed). Here (t, τ ) parametrize T * γ, (x, ξ) are transverse variables on Poincare section, g(τ ) = τ + f (-τ ) = O(τ 2 ), and f parametrizes energy according to f ( [GuPa]: Assume that Floquet exponents satisfy the strong non-resonance condition (H.6). Then in a nghbhd of γ 0 , there exists symplectic coordinates (t, τ, x, ξ), t ∈ [0, 2π], such that for all N ≥ 1, we can find a canonical transformation κ N with (3.1)

-τ ) = E; it is related to the period T (E) of γ(E) by f (-τ ) = 2π T •f (-τ ) , with f (0) = 1. Proposition 3.1 [Br],
H 0 • κ N = -τ + d j=1 µ j Q j (x, ξ) + H (N ) 0 (τ ; Q 1 , • • • , Q n ) + O |τ, |x, ξ| 2 | N +1
where H

(N ) 0

(τ ; Q 1 , • • • , Q d ) = O(|τ, Q| 2
) is a polynomial of degree N , and the remainder term

O |τ, |x, ξ| 2 | N +1 is 2π-periodic in t. Here µ j Q j (x, ξ) is a polynomial of the form µ j 2 (ξ 2 j -x 2 j )
very small neighborhood of γ(E), whose size will eventually depend on h, corresponding to θ = -π/4, and that we call the "phase of inflation", T 0 H w (y, hD y ; h)T -1 0 = H(hD t , x, hD x ; E, h) assumes BNF and is approximated at leading order by the Model Hamiltonien. In a somewhat larger neighborhood of γ(E), which we call the "linear phase", we choose θ small enough, and get a new pl.s.h. weight Φ θ (t, x). Farther away from γ(E) (in the "geometric phase") the weight is implied by the escape function. All these weights are patched together in overlapping regions, so to define a globally pl.s.h. function in complex (t, x) (or y) space.

It determines the contour integral for writing realizations of h-FIO's in the complex domain [Sj] in H Φ spaces, conjugating H w (y, hD y ; h) to a h-PDO everywhere elliptic but on γ. In particular near γ

(3.3) |(H -E)| Λ Φ ∼ |x| 2 + | Im t -τ | c) Poisson operator,

its normalisation and the monodromy operator

Let R n t be the section {t} × R d of R n (in BNF coordinates). We look for K(t, E) :

L 2 (R d ) → L 2 (R n t ) (formally), microlocalized near Γ + (E)
, of the form K(t, E)v(x; h) = e i(S(t,x,η)-yη)/h a(t, x, η; E, h)v(y) dy ∧ dη, and such that

H(hD t , x, hD x ; h)K(t, E) = 0, K(0, E) = Id
Considering realizations in the complex domain adapted to the weight Φ θ , we compute most easily K(t, E) in the "phase of inflation". Here, solving eikonal and transport equations, we find that the leading term of S and a with respect to BNF is given by those of the Model Hamiltonian, and K(t, E) is also in BNF. Let χ ∈ C ∞ (R), be equal to 0 near 0, 1 near

[2π, ∞[. There is a h-PDO B(E) = B w (x, hD x ; E) such that L(t, E) = K(t, E)B(E) satisfies as in (2.25) (3.5) i h [H, χ(t)]L(t, E)v|L(t, E)v = v|v
Outside the "phase of inflation" the analysis is somewhat simpler, since H -E is already elliptic (3.3). as an operator on L 2 (R d ), which is a concrete version of (2.28) and (2.33). As a function de χ, M * (E) follows a "0-1 law": it is 0 if supp χ ⊂]0, 2π[, and unitary if χ equals 0 near 0, and 1 near 2π. For the model case one has M * (E)v(x) = e -2iπE/h e πµ v(xe 2πµ ) since χ (t) dt = 1. Unitarity of M * (E) may not be clear in (3.6), but follows from uniqueness of the monodromy operator and Proposition 2.1 (when hypotheses match). Moreover M * (E) is in BNF, so that eigenfunctions of M * (E) are homogeneous polynomials, which leads to Bohr-Sommerfeld quantization rules (see [Lou], [LouRo1,2], and a detailed version [LouRo3] in progress). See also [IfaLouRo] for higher order expansions in the 1-D case. In fact, one can show that M * (E) = e iR w (x,hD x ;E,h)/h , where R is h-PDO in BNF, self-adjoint for real E. This gives another proof for unitarity.

Appendix. A short review on complex scaling

Carrying the arguments of [SjZw] to the framework of resonances, the proof of Theorem 1.1 in Sect.2 requires only some "mild" deformations outside of a neighborhood of γ 0 . Sharper deformations are needed in Sect.4 for Theorem 1.2.

For large x, the "dilated" operator" takes the form H θ (y, hD y ; h) = U * θ H(y, hD y ; h)U θ . Here θ ∈ C is a small parameter (Im θ ≥ 0 for outgoing resonances) that we eventually set to iθ for simplicity).

We say that U θ is an analytic dilation if this is a linear change of variables of the form U θ u(x) = e nθ/2 u(θx), and an analytic distorsion if the change of variables is non linear, but in both cases it is useful to consider the scalar product on L 2 (R n ) as a duality product between L 2 (Γ θ ) and L 2 (Γ θ ) by means of the formula (A.1) u, v θ = e θ R n u(y)v(y) dy

For small θ ∈ C, Γ θ = e θ R n is a totally real manifold, whose cotangent space T * Γ θ , is a IR-manifold (Lagrangian for Im dη ∧ dy, symplectic for Re dη ∧ dy.

It makes no difficulty to extend the notion of "unitary operator" of "self-adjoint" operators in that sense: for instance if U θ , for real θ, is unitary on L 2 (R n ), its adjoint for this duality is the analytic extension (with respect to small θ ∈ C) of U -1 θ , and Q θ is "self-adjoint" means Q θ is the analytic continuation of the self-adjoint operator Q θ for real θ.

Near γ 0 , H θ is defined through microlocally weighted L 2 (or Sobolev) spaces. The microlocal weights G(y, η) are chosen among escape functions, i.e. a smooth functions which is increasing along the flow of X H , and strictly increasing away from the trapped set; they do not depend, locally, on the energy parameter. A general result [GeSj] states that there always exists such a function.

Examples: (1) Let H(y, η) = η 2 , then for any E > 0, K(E) = ∅, and G(y, η) = yη is an escape function since X H G ≥ E when |η 2 -E| ≤ E/2. (2) Let H(y, η) = η 2 + V (y),

  {(y, η) ∈ T * C n : | Im(y, η)| ≤ const. Re(y, η) }

  the weighted Hilbert space) are projectors of rank k h and M(z; h) is the monodromy operator quantizing Poincaré map and computed in Sect.2 below.

  Lidskiy or Cohnley-Zehnder index of γ(E) (depending only on elliptic elements). Then the resonances of H associated with the family γ

  e. Poisson operator with "Cauchy data" u(0) = v + ∈ C. Similarly multiplication by u a defines Poisson operator I a (z)v + = u a (t)v + , which another "Cauchy data" v + at a = π.

  and the solvability condition is ensured by the commutation relation [L L , L R ] = 0. It turns out that U (z, w) can be constructed in the class of h-FIO's on R n , microlocally near m ∈ γ. For the model, U (z, w) is just the multiplication

  We set K 0 (t, E) = K(t, E) where K(t, E) is Poisson operator with Cauchy data at t = 0, andL 0 (t, E) = K 0 (t, E)B(E); we set similarly L 2π (t, E) = K 0 (t -2π, E)B(E)with Cauchy data at t = 2π. The monodromy operator (or semi-classical Poincaré map) is defined by

  and the 3rd arrow is H -z. (2.19) remains true if we change Ker

Acknowlegments: We are grateful to Alain Chenciner, Sergey Bolotin and André Voros for useful information, and to a referee for useful remarks. The second author was partially supported by Grant PRC No. 1556 CNRS-RFBR 2017-2019 "Multi-dimensional semi-classical problems of Condensed Matter Physics and Quantum Mechanics".

(or µ j x j ξ j ) (hr element), -iµ j 2 (ξ 2 j + x 2 j ) (ee element), c j (x 2j-1 ξ 2j-1 + x 2j ξ 2j ) -d j (x 2j-1 ξ 2jx 2j ξ 2j-1 ), µ j = c j + id j (hc or loxodromic elements) which also take the form µ j x j ξ j in complex coordinates. This BNF carries to the semi-classical setting (see also [Zel] for high energy expansions): Proposition 3.2 [GuPa]: Under hypotheses above, conjugating with a h-FIO microlocally unitary near γ 0 , H w (y, hD y ; h) can be taken formally to (3.2)

as a polynomial depending on the n "variables" (hD t , Q w j ), with for instance when µ j is real,

where the . . . stand for terms O(h 2 ), as well as operators with coefficients O(h ∞ ) and periodic in time. N j denotes the order of expansion as a Birkhoff series of Hamiltonian H j , and

any sequence of integers. Moreover, allowing for complex coordinates, one can formally assume that Q w j = 1 2 (x j hD x j + hD x j x j ) for all types of elements (ee or he). Keeping the leading part in (3.2) the Model Hamiltonian,

with periodic boundary conditions on S 1 × R d serves as a guide-line as hD t did in Sect.2.

b) Microlocalisation in the complex domain

Taking into account that there exists an escape function outside the trapped set γ(E), the most relevant region of phase-space for such deformations is a neighborhood of γ(E).

Here we make a complex scaling of the form (x, ξ) → (e iθ x, e -iθ ξ) (independent of E), followed also by a deformation in the (t, τ ) variables. Rather then using weighted spaces as in Sect.2, our main tool is the method of Lagrangian deformations. Namely we perform a FBI transformation (metaplectic FIO with complex phase) which takes the form, in coordinates (s, y; t, x) ∈ T * R n × T * C n adapted to Γ ± as in BNF T 0 u(x, h) = e iϕ 0 (t,s;x,y)/h u(s, y) ds dy, u ∈ L 2 (R n )

where ϕ 0 (t, s; x, y) = ϕ 1 (t, s)+ϕ 2 (x, y), ϕ 1 (t, s) = i 2 (t-s) 2 , ϕ 2 (x, y) = i 2 (x-y) 2 -1 2 x 2 . The corresponding pluri-subharmonic (pl.s.h.) weight is Φ 0 = Φ 1 + Φ 2 = (Im t) 2 /2 + |x| 2 /4. In a