Shridevi Shenoi Perdoor

Fabien Dubois
email: fabien.dubois@neel.cnrs.fr

Aude Barbara

Christophe Nguyen

Lamiaa M A Ali

Magali Gary-Bobo

Ines Hristovska

Kassandre Combet

Olivier Pascual

Guillaume Micouin

Akos Banyasz

Yann Bretonnière

Chantal Andraud

Alain Ibanez

Xavier Cattoën
email: xavier.cattoen@neel.cnrs.fr

Ultrabright Silica-Coated Organic Nanocrystals for Two-Photon In Vivo Imaging

Keywords: Nanoparticles, bio-imaging, two-photon absorption, dynamic light scattering, colloidal stabilization

come

Introduction

Optical imaging techniques offer several advantages for early diagnosis of tumors and imagingguided therapy compared to alternative approaches, such as high sensitivity, high spatial resolution, high signal to noise ratio in the near infrared spectral region and the use of low concentrations of contrast agent, [START_REF] Choy | Current Advances in Molecular Imaging: Noninvasive in Vivo Bioluminescent and Fluorescent Optical Imaging in Cancer Research[END_REF][START_REF] Hadjipanayis | Current and Future Clinical Applications for Optical Imaging of Cancer: From Intraoperative Surgical Guidance to Cancer Screening[END_REF][START_REF] Pierce | Optical Contrast Agents and Imaging Systems for Detection and Diagnosis of Cancer[END_REF][START_REF] Luker | Optical Imaging: Current Applications and Future Directions[END_REF][START_REF] Sun | J-Aggregate Squaraine Nanoparticles with Bright NIR-II Fluorescence for Imaging Guided Photothermal Therapy[END_REF] . However, optical imaging has been primarily limited to dermal and ocular applications, due to autofluorescence, light scattering and attenuation by tissues and the absorption of light by hemoglobin, lipids and water, leading to low penetration depths. [START_REF] Thomas | Advances and Challenges in Label-Free Nonlinear Optical Imaging Using Two-Photon Excitation Fluorescence and Second Harmonic Generation for Cancer Research[END_REF][START_REF] Hemmer | Exploiting the Biological Windows: Current Perspectives on Fluorescent Bioprobes Emitting above 1000 Nm[END_REF] Optical imaging in the near infrared (NIR) region (650 nm -1700 nm), where light scattering and photon absorption are minimal, enables deeper tissue penetration and improved signal to noise ratios. [START_REF] Thomas | Advances and Challenges in Label-Free Nonlinear Optical Imaging Using Two-Photon Excitation Fluorescence and Second Harmonic Generation for Cancer Research[END_REF][START_REF] Hemmer | Exploiting the Biological Windows: Current Perspectives on Fluorescent Bioprobes Emitting above 1000 Nm[END_REF] Nonlinear optics and the development of multiphoton microscopy have further helped to achieve deeper tissue penetration, thereby expanding the scope of in vivo optical approaches for deep tissue imaging. Two-photon microscopy, which exploits the simultaneous absorption of two photons, each with half the energy (and twice the wavelength) of the corresponding single photon transition, to excite the fluorophore, enables deeper tissue penetration as it typically shifts the excitation wavelength from the visible to the biological transparency window in the NIR range. Additionally, photodamage is greatly minimized due to the use of lower energy photons and confinement of fluorescence emission to a diffraction-limited spot. [START_REF] Svoboda | Principles of Two-Photon Excitation Microscopy and Its Applications to Neuroscience[END_REF] This makes two-photon microscopy ideal for high-resolution (µm-scale), deep-tissue, real-time imaging and tracking of dynamic processes in cell biology. [START_REF] Thomas | Advances and Challenges in Label-Free Nonlinear Optical Imaging Using Two-Photon Excitation Fluorescence and Second Harmonic Generation for Cancer Research[END_REF] In order to realize deep-tissue optical imaging, it is crucial to develop tracers that not only exhibit the required photophysical properties (such as two-photon absorption and a fluorescence emission in the NIR), but also good biocompatibility, high sensitivity and specificity. Many organic dyes which show two-photon absorption are hydrophobic, leading to molecular aggregation in aqueous media and associated fluorescence quenching, which in turn limits their use as fluorescent probes for imaging.

The development of new types of fluorophores relying on aggregation-induced emission (AIE) has

overcome the problem of aggregation-induced quenching. In particular, organic dyes that form J-aggregates have shown great potential for use in bio-imaging. J-aggregation refers to the selfassembly of molecules in a head-to-tail arrangement, resulting in delocalization of the π-electrons over the self-assembled molecular aggregate, which in turn leads to constructive coupling of the excited state transition dipoles. [START_REF] Würthner | From Serendipitous Discovery to Supramolecular Engineering of Functional Dye Materials[END_REF][START_REF] Chen | Shortwave Infrared Imaging with J-Aggregates Stabilized in Hollow Mesoporous Silica Nanoparticles[END_REF] This results in several interesting photophysical characteristics, such as red-shifted absorption and emission spectra, narrow absorption and emission bands, high absorption coefficients and good quantum yields. [START_REF] Würthner | From Serendipitous Discovery to Supramolecular Engineering of Functional Dye Materials[END_REF][START_REF] Chen | Shortwave Infrared Imaging with J-Aggregates Stabilized in Hollow Mesoporous Silica Nanoparticles[END_REF][START_REF] Song | J-Aggregates of Organic Dye Molecules Complexed with Iron Oxide Nanoparticles for Imaging-Guided Photothermal Therapy under 915-Nm Light[END_REF] Several organic dyes designed to form J-aggregates have been reported in the literature, such as perylene bisimide derivatives [START_REF] Kaiser | Supramolecular Construction of Fluorescent J-Aggregates Based on Hydrogen-Bonded Perylene Dyes[END_REF] , pyrrolopyrrole cyanine derivatives [START_REF] Yang | Robust Colloidal Nanoparticles of Pyrrolopyrrole Cyanine J-Aggregates with Bright Near-Infrared Fluorescence in Aqueous Media: From Spectral Tailoring to Bioimaging Applications[END_REF] , heptamethine cyanine dyes [START_REF] Chen | Shortwave Infrared Imaging with J-Aggregates Stabilized in Hollow Mesoporous Silica Nanoparticles[END_REF][START_REF] Song | J-Aggregates of Organic Dye Molecules Complexed with Iron Oxide Nanoparticles for Imaging-Guided Photothermal Therapy under 915-Nm Light[END_REF][START_REF] Song | Nano-Assemblies of J-Aggregates Based on a NIR Dye as a Multifunctional Drug Carrier for Combination Cancer Therapy[END_REF] and porphyrin-based dyes [START_REF] Biswas | Two-Photon Absorption Enhancement of Polymer-Templated Porphyrin-Based J-Aggregates[END_REF][START_REF] Gurunatha | Tuning the Optical Coupling between Molecular Dyes and Metal Nanoparticles by the Templated Silica Mineralization of J-Aggregates[END_REF][START_REF] Shakiba | Stable J-Aggregation Enabled Dual Photoacoustic and Fluorescence Nanoparticles for Intraoperative Cancer Imaging[END_REF] . These chromophores are often complexed with polymers or encapsulated in nanoparticles (NPs) to impart stability and enhance their functionality. J-aggregates based on heptamethine cyanine dyes complexed with amphiphilic polymers to form stabilized polymeric micelles have been reported by Liu and coworkers. [START_REF] Song | J-Aggregates of Organic Dye Molecules Complexed with Iron Oxide Nanoparticles for Imaging-Guided Photothermal Therapy under 915-Nm Light[END_REF][START_REF] Song | Nano-Assemblies of J-Aggregates Based on a NIR Dye as a Multifunctional Drug Carrier for Combination Cancer Therapy[END_REF] These polymer-stabilized J-aggregates have been further complexed with iron oxide nanoparticles [START_REF] Song | J-Aggregates of Organic Dye Molecules Complexed with Iron Oxide Nanoparticles for Imaging-Guided Photothermal Therapy under 915-Nm Light[END_REF] and anticancer drugs [START_REF] Song | Nano-Assemblies of J-Aggregates Based on a NIR Dye as a Multifunctional Drug Carrier for Combination Cancer Therapy[END_REF] for photothermal and combination cancer therapy, respectively. Polymeric-micelle-stabilized J-aggregates of pyrrolopyrrole cyanine [START_REF] Yang | Robust Colloidal Nanoparticles of Pyrrolopyrrole Cyanine J-Aggregates with Bright Near-Infrared Fluorescence in Aqueous Media: From Spectral Tailoring to Bioimaging Applications[END_REF] for in vivo imaging and squaraine [START_REF] Sun | J-Aggregate Squaraine Nanoparticles with Bright NIR-II Fluorescence for Imaging Guided Photothermal Therapy[END_REF] for image guided photothermal therapy have also been reported.

Organic dyes exhibiting AIE have also been encapsulated in protein nanoparticles [START_REF] Qin | Biocompatible Nanoparticles with Aggregation-Induced Emission Characteristics as Far-Red/near-Infrared Fluorescent Bioprobes for in Vitro and in Vivo Imaging Applications[END_REF][START_REF] Geng | Lipid-PEG-Folate Encapsulated Nanoparticles with Aggregation Induced Emission Characteristics: Cellular Uptake Mechanism and Two-Photon Fluorescence Imaging[END_REF] to enhance uptake by cancer cells and enable two-photon in vivo imaging. Silica-based NPs are also interesting options for the encapsulation of AIE based dyes, due to the availability of well-established synthesis protocols for controlling their size, their low intrinsic toxicity and ease of surface modification to increase their colloidal stability. [START_REF] Montalti | Dye-Doped Silica Nanoparticles as Luminescent Organized Systems for Nanomedicine[END_REF][START_REF] Rampazzo | NIR Fluorescent Dye Doped Silica Nanoparticles for in Vivo Imaging, Sensing and Theranostic[END_REF] In addition, the inert silica matrix can shield encapsulated dye molecules from the external environment, providing stability, optical transparency, and biocompatibility. [START_REF] Rampazzo | NIR Fluorescent Dye Doped Silica Nanoparticles for in Vivo Imaging, Sensing and Theranostic[END_REF][START_REF] Bonacchi | Luminescent Silica Nanoparticles: Extending the Frontiers of Brightness[END_REF] Furthermore, the silicate matrix can be readily functionalized through silica surface chemistry, thereby providing an efficient strategy for the encapsulation of hydrophobic nanocrystals to obtain water-soluble nanomaterials. [START_REF] Hanske | Silica-Coated Plasmonic Metal Nanoparticles in Action[END_REF][START_REF] Selvan | Non-Cytotoxic, Silica-Coated CdSe Quantum Dots with Efficient Photoluminescence[END_REF][START_REF] Ang | Silica-Coated Mn-Doped ZnS Nanocrystals for Cancer Theranostics[END_REF] AIE based dyes have also been incorporated into silica NPs for bio-imaging. [START_REF] Wang | Multi-Stable Fluorescent Silica Nanoparticles Obtained from in Situ Doping with Aggregation-Induced Emission Molecules[END_REF][START_REF] Wang | Aggregation-Induced Emission Luminogen-Embedded Silica Nanoparticles Containing DNA Aptamers for Targeted Cell Imaging[END_REF][START_REF] Li | Aggregation-Induced Emission Enhancement and Cell Imaging of a Novel (Carbazol-N-Yl)Triphenylamine-BODIPY[END_REF][START_REF] Wei | Surface Modification and Biological Imaging of Aggregation-Induced Emission (AIE) Dye Doped Silica Nanoparticles[END_REF] J-aggregates of a heptamethine dye have been prepared inside hollow mesoporous silica nanoparticles for NIR imaging by Sletten and coworkers. [START_REF] Chen | Shortwave Infrared Imaging with J-Aggregates Stabilized in Hollow Mesoporous Silica Nanoparticles[END_REF] The silica nanoparticles have been shown to stabilize J-aggregates and facilitate high resolution in vivo imaging in mice. [START_REF] Chen | Shortwave Infrared Imaging with J-Aggregates Stabilized in Hollow Mesoporous Silica Nanoparticles[END_REF] Prasad and coworkers have developed organically-modified silica (ORMOSIL) NPs loaded with a high concentration of AIE-based dyes for two-photon fluorescence imaging [START_REF] Kim | Aggregation-Enhanced Fluorescence in Organically Modified Silica Nanoparticles: A Novel Approach toward High-Signal-Output Nanoprobes for Two-Photon Fluorescence Bioimaging[END_REF] and photodynamic therapy when co-encapsulated with a photosensitizing drug. [START_REF] Kim | Organically Modified Silica Nanoparticles Co-Encapsulating Photosensitizing Drug and Aggregation-Enhanced Two-Photon Absorbing Fluorescent Dye Aggregates for Two-Photon Photodynamic Therapy[END_REF] Encapsulation of organic crystal-state emitters in sol-gel matrices is an efficient strategy to produce bright nanotracers for bio-imaging applications. [START_REF] Philippot | Fluorescent Organic Nanocrystal Confined in Sol-Gel Matrix for Bio-Imaging[END_REF][START_REF] Dubuisson | Fluorescent Molecular Nanocrystals Anchored in Sol-Gel Thin Films: A Label-Free Signalization Function for Biosensing Applications[END_REF][START_REF] Ibanez | Controlled Nanocrystallization of Organic Molecules in Sol-Gel Glasses[END_REF][START_REF] Monnier | Confined Nucleation and Growth of Organic Nanocrystals in Sol-Gel Matrices[END_REF][START_REF] Shenoi-Perdoor | Red-Emitting Fluorescent Organic@silicate Core-Shell Nanoparticles for Bio-Imaging[END_REF] As described in our previous work, [START_REF] Shenoi-Perdoor | Red-Emitting Fluorescent Organic@silicate Core-Shell Nanoparticles for Bio-Imaging[END_REF] fluorescent organic@silicate core-shell nanoparticles (NPs) containing a crystalline organic core surrounded by a silicate shell, with a high loading of dyes (30-40 wt %) are promising for bioimaging applications. These recently-described core-shell NPs, which comprise a nanocrystalline organic-dye core (ca 40-50 nm) surrounded by a silicate shell, are prepared by a novel spray-drying technique which combines sol-gel chemistry and crystallization. [START_REF] Philippot | Fluorescent Organic Nanocrystal Confined in Sol-Gel Matrix for Bio-Imaging[END_REF][START_REF] Shenoi-Perdoor | Red-Emitting Fluorescent Organic@silicate Core-Shell Nanoparticles for Bio-Imaging[END_REF][START_REF] Philippot | New Core-Shell Hybrid Nanoparticles for Biophotonics: Fluorescent Organic Nanocrystals Confined in Organosilicate Spheres[END_REF][START_REF] Philippot | Polymorphism of CMONS Nanocrystals Grown in Silicate Particles through a Spray-Drying Process[END_REF] The organic dyes used in the core-shell NPs are based on the D-π-A dipolar structure and are specifically designed to exhibit two-photon absorption in the crystal state. [START_REF] Redon | Tuning the Solid-State Emission of Small Push-Pull Dipolar Dyes to the Far-Red through Variation of the Electron-Acceptor Group[END_REF] The permanent dipole moment associated with such structures results in strong dipole-dipole interactions, which in turn can induce specific organization and orientation of the molecules in the solid state. [START_REF] Philippot | Polymorphism of CMONS Nanocrystals Grown in Silicate Particles through a Spray-Drying Process[END_REF] These core-shell NPs show fluorescence emission in the red, close to the biological transparency window, making them interesting candidates for deep-tissue imaging using two-photon microscopy.

In this study, we evaluate the potential of our novel organic@silicate core-shell NPs for two-photon in vivo fluorescence imaging. The CuAAC (Copper Catalyzed Azide to Alkyne Cycloaddition) click reaction [START_REF] Shenoi-Perdoor | Click Functionalization of Sol-Gel Materials[END_REF] was used to graft PEG chains onto the core-shell NPs to enhance their colloidal stability and "stealth" in blood by minimizing protein adsorption and opsonization. [START_REF] Ma | Elucidating the Mechanism of Silica Nanoparticle PEGylation Processes Using Fluorescence Correlation Spectroscopies[END_REF][START_REF] Rio-Echevarria | Highly PEGylated Silica Nanoparticles: "Ready to Use" Stealth Functional Nanocarriers[END_REF][START_REF] Cauda | Impact of Different PEGylation Patterns on the Long-Term Bio-Stability of Colloidal Mesoporous Silica Nanoparticles[END_REF][START_REF] Adumeau | Impact of Surface Grafting Density of PEG Macromolecules on Dually Fluorescent Silica Nanoparticles Used for the in Vivo Imaging of Subcutaneous Tumors[END_REF][START_REF] Şen Karaman | Rational Evaluation of the Utilization of PEG-PEI Copolymers for the Facilitation of Silica Nanoparticulate Systems in Biomedical Applications[END_REF][START_REF] Nissinen | Tailored Dual PEGylation of Inorganic Porous Nanocarriers for Extremely Long Blood Circulation in Vivo[END_REF][START_REF] He | The Effect of PEGylation of Mesoporous Silica Nanoparticles on Nonspecific Binding of Serum Proteins and Cellular Responses[END_REF] Classical DLS experiments coupled with small-volume DLS (confocal geometry) was used to characterize the colloidal stability of the NPs in different biological media with increasing complexity, starting with simulated body fluid (SBF) and various plasma proteins in increasing concentrations. Finally, we present, two-photon fluorescence properties in solution to demonstrate the potential of these NPs for two-photon fluorescence imaging both in vitro and in vivo. The sols and the core-shell NPs (NP-I-N3) were prepared according to the published protocol 36 using a 2:1 molar ratio of tetramethoxysilane (TMOS) and 1,2-bis(trimethoxysilyl)ethane (TMSE), with 3% AzPTES with respect to Si and dilute aqueous HCl (1 M HCl) in THF. Briefly, dilute HCl was added to the silicate precursors, previously dispersed in 5 mL of THF, under vigorous stirring, followed by aging for 3 days at 60 °C. Finally, additional THF was added after aging to reach a final volume of 60 mL. The as-prepared sols were stored at 4 °C. The dye was dissolved in the sol just before nebulization and spray-drying and the sol was nebulized under the conditions described in our previous work, yielding organic nanocrystals covered with a silicate layer. [START_REF] Shenoi-Perdoor | Red-Emitting Fluorescent Organic@silicate Core-Shell Nanoparticles for Bio-Imaging[END_REF] The crystalline nature of the organic core was ascertained by electron diffraction and by PXRD (Figure S1). A controlled partial dissolution of the silicate shell of the sprayed-dried NPs was performed by sonicating the NPs in water (60 mL) for 20 minutes in an ultrasonic bath (150 W), before increasing the solution pH to 12 by adding 1 M NaOH. The resulting suspension was stirred for 24 h in a closed bottle. The NP suspension was then subjected to centrifugation for 5 minutes at 5000 rpm and the supernatant was filtered using 5 µm, 1.2 µm and 0.8 µm syringe filters, to remove large aggregates. This step was followed by neutralization of the NP suspension to pH 7.4 by the dropwise addition of 1 M HCl. The resulting NP suspensions were used for fluorescence spectroscopy measurements. The pH of the neutralized core-shell NPs for click functionalization was further stabilized using 0.133 M phosphate buffer by adding a volume corresponding to 10% of the volume of the NP suspension.

Experimental section

Details of materials, reagents and characterization techniques are given in the supporting information

Synthesis of core-shell NPs with clickable functions

CuAAC functionalization of core-shell NPs

The treated core-shell NPs were functionalized using the CuAAC click reaction, following the protocol described by Moitra et al. [START_REF] Moitra | Facile Route to Functionalized Mesoporous Silica Nanoparticles by Click Chemistry[END_REF] Five equivalents of propargyl-PEG 5000-OMe for each equivalent of AzPTES were first weighed into a bottle in which the treated NP suspension was added under constant stirring. One equivalent each of tris-(3-hydroxypropyltriazolylmethyl)amine (THPTA) and of the pre-catalyst, copper sulphate (CuSO4.5H2O), were then added to this solution.

Argon was bubbled into the colloidal suspension for a few minutes to remove any traces of oxygen before the addition of four equivalents of sodium ascorbate. More Ar was bubbled in the suspension and the bottle was then closed and kept under constant stirring at room temperature for 24 hours.

Purification of the functionalized NPs (NP-I-PEG) was performed by dialyzing the NP suspension (10 mL in Pur-A-Lyzer tubes with 3.5 kDa pore size) against 2.5 L of 0.0133 M PB for two days.

The purified NP suspensions were concentrated using Vivaspin 2 (2 mL) concentrators.

Centrifugation was performed at 3000 rpm for 3 minutes on volumes of 2 mL. Successive centrifugations were performed to increase the concentration of particles in suspension.

Steady-state spectroscopic experiments under resonant one-photon absorption conditions

The two-photon excitation cross-section measurements were performed by detecting the twophoton excitation-induced fluorescence (TPEF) of the NPs compared to the signal of a pH 13 aqueous solution of fluorescein (c=10 -5 mol/L) used as a standard. [START_REF] Xu | Measurement of Two-Photon Excitation Cross Sections of Molecular Fluorophores with Data from 690 to 1050 Nm[END_REF][START_REF] De Reguardati | High-Accuracy Reference Standards for Two-Photon Absorption in the 680–1050 Nm Wavelength Range[END_REF] The experimental setup is based on a femtosecond Ti:Sapphire laser (Chameleon Ultra I, Coherent, 80 MHz pulse repetition rate, 140 fs pulse duration) allowing for tunable excitation between 700 and 1040 nm. The laser intensity was adjusted by using a continuously variable neutral density filter to ensure constant excitation power in the tuning range. The laser power was monitored by a NIST traceable thermal detector (S405C, Thorlabs). The laser beam was focused into the 10 × 10 mm quartz (QZS) sample cell by a 100 mm focal length lens. The fluorescence signal was detected at 90 o to the excitation beam. The cell position was adjusted so that the optical path of both the excitation beam and the fluorescence emission was minimized in the cell, thus reducing experimental artefacts such as inner-filter effects, scattering and solvent absorption. The emitted fluorescence was filtered using a low-pass interference filter and delivered to a fiber optic spectrometer (AvaSpec-EVO, Avantes).

In vitro and in vivo assays

For in vitro experiments, MCF-7 human breast cancer cells were used (purchased from ATCC) and cultured in Dulbecco's modified Eagle's medium (DMEM) supplemented with 10% fetal bovine serum and 50 µg mL -1 gentamycin and maintained under a humidified atmosphere at 37°C containing 5% CO2 (incubator). Cytotoxic studies were realized by seeding MCF-7 on 96-well plates (2000 cells/well). The day after seeding, cells were incubated with increasing concentrations of NP-I-PEG (from 0.05 to 50 µg mL -1 ). One day after, the plates intended for "light cytotoxicity" study were exposed during 4 h at the sunlight, then they were replaced in the incubator in the dark for the remaining 48 h. The plates "dark cytotoxicity" stayed in the incubator 72 h without interruption. At the end of the 3 days, a MTT assay was performed to determine the cell viability.

Briefly, cells were incubated for 4 h with 0.5 mg mL -1 of MTT (3-(4,5-dimethylthiazol-2-yl)-2,5-diphenyltetrazolium bromide; Promega) in culture medium. The MTT/medium solution was then removed and the precipitated crystals were dissolved in EtOH/ DMSO (v/v). The solution absorbance was read at 540 nm in a microplate reader. All values are reported in relation to the control values (without any treatment) which are considered as 100% living cells.

For fluorescence imaging on living cells, MCF-7 cells were seeded in glass bottom dishes (World Precision Instrument, Stevenage, UK), one day prior to commencing the experiment, at a density of 10 6 cells.cm -2 . Cells were then washed once and incubated in 1 mL culture medium with or without NP-I-PEG at a concentration of 10 or 25 µg mL -1 for 24 h. Two hours before the end of the incubation, cells were loaded with LysoTracker green (1 µM, Molecular Probes, Life technologies, Oregon, USA); 15 minutes before the end of incubation, cells were loaded with Hoechst 33342 (10 µg mL -1 , Invitrogen, Cergy Pontoise, France) for nuclear staining, or with green CellMask (5 µg mL -1 , Invitrogen, Cergy Pontoise, France) for membrane staining. Before microscope observation, cells were gently washed with cell culture medium and scanned with a LSM 780 LIVE confocal microscope (Carl Zeiss, Le Pecq, France), at 450 nm (continuous) or 900 nm using the Chameleon (pulsed) laser for NP-I-PEG visualization. Samples were excited at 561 nm for membranes, at 760 nm for nuclei and at 488 nm for lysosomes.

All images were performed with a high magnification (63×/1.4 OIL DIC Plan-Apo).

For mice studies, all experimental procedures were carried out in accordance with European regulations for animal use and were authorized by the local Ethics Committees (CEEA-55/CETEA-2015-0038) and the French Ministry of National Education and Research (APAFIS# DR2017-58).

For two-photon in vivo imaging, 8-week-old CX3CR1 eGFP/+ mice, expressing GFP only in microglia in the central nervous system (SNC), were used. For transcranial imaging, a thinnedskull cortical window preparation was performed over the somatosensory cortex, where the skull was thinned to 20-30 µm. Mice were deeply anesthetized with isoflurane (3-4%, Isovet, Piramal Healthcare, UK Ltd.) and mounted in a stereotaxic apparatus (D. Kopf Instruments). Carprofen (5 mg/kg s.c.) was injected at the beginning of the surgery to reduce post-surgical pain and inflammation. After the skull was exposed, a thin custom-made metal implant was glued allowing the area over the somatosensory cortex to be delimited. The skull was then carefully thinned using a high-speed dental drill. To avoid heat-induced damage, drilling was repeatedly interrupted and cold sterile saline was applied. When a 20 to 30 µm skull thickness was reached, a thin layer of cyanoacrylate glue was applied and a cover glass was placed on top of the thinned skull.

For in vivo imaging sessions, mice were anesthetized with a mixture of ketamine (100 mg/kg) and xylazine (10 mg/kg). Their body temperature was maintained at 37 °C using a heating pad. A retroorbital injection of 200 µL of a NP solution at a concentration of 80 µg/mL was performed while the mouse was deeply anesthetized. Two-photon microscopy observations were performed on a Bruker Ultima microscope (Insight 3X laser, Spectra Physics), equipped with a 20× waterimmersion objective (0.95 N.A. Olympus). Green (CX3CR1 eGFP/+ cells) and red (NPs) fluorescence were separated by a 560 nm dichroic mirror coupled to 525/50 nm and 650/40 emission filters.

Images were acquired every 0.9 seconds over an area of 200×200 µm 2 and a resolution of 521x521 pixels, and laser power was maintained below 30

Results and discussion

Due to the presence of organic nanocrystals at the core of the NPs, surface functionalization had to be performed in aqueous solution to minimize dissolution of the hydrophobic core. Our previous protocol [START_REF] Shenoi-Perdoor | Red-Emitting Fluorescent Organic@silicate Core-Shell Nanoparticles for Bio-Imaging[END_REF] was thus adapted to incorporate an azide functionality within the NPs during sol-gel synthesis (figure 1), [START_REF] Moitra | Facile Route to Functionalized Mesoporous Silica Nanoparticles by Click Chemistry[END_REF][START_REF] Bürglová | A General Method for Preparing Bridged Organosilanes with Pendant Functional Groups and Functional Mesoporous Organosilicas[END_REF] for subsequent functionalization with alkyne-modified moieties via the CuAAC reaction in pure water. In addition to TMOS (2 eq.) and TMSE (1 eq.), 3 mol% of (3-azidopropyl)triethoxysilane (AzPTES) was added to the starting sol. Using dye I, the morphology and size distribution of the NPs (NP-I-N3) obtained (FESEM image, Figure 2) were unaltered after the introduction of this new silicate precursor into the sol (Figure S2). Fourier transform infrared spectroscopy (FTIR) confirmed the presence of a high loading of dye (30-40 wt% as previously determined by TGA [START_REF] Shenoi-Perdoor | Red-Emitting Fluorescent Organic@silicate Core-Shell Nanoparticles for Bio-Imaging[END_REF] ) and the azide groups within the NPs (supporting information, Figure S3). Indeed, the bands corresponding to -CN (2210 cm -1 ) and C=C (1573 cm To disperse the dry NP powder obtained by spray-drying, a partial dissolution of the silicate shell with sodium hydroxide at pH 12 was performed for 24 h in order to separate the chemically aggregated NPs and impart a strongly negative ζ potential. After this treatment, the suspension was centrifuged to sediment the largest particles and aggregates, then the supernatant containing the desired NPs was filtered using 5 µm to 0.8 µm cellulose acetate membranes and finally neutralized to pH 7.5.

The suspension of NPs as prepared in pure water was characterized by DLS measurements. The auto-correlation functions of the measured intensities were derived and fitted to determine the average size of the NPs and the width of their size distribution. The latter is shown in Figure S4-a.

in the supporting information, as a function of the number of NPs and as a function of their scattered intensities; the latter favors the largest NPs, since scattering intensity increases with increasing particle size. These suspensions were stable for several hours, although the subsequent addition of sodium chloride up to 0.9 wt %, which corresponds to blood salinity, had a strong negative impact on the colloidal stability. This effect was monitored by DLS measurements, performed every minute over two hours, that showed a progressive increase of the average size of the scattering species as a function of time (SI, Figure S4-b). This reflects the salt-induced aggregation of NPs, which was further confirmed by observing the NP sedimentation on a longer time scale, namely during fluorescence spectroscopy experiments performed over 5 hours (SI, Figure S4-b). This colloidal instability in saline environments makes it necessary to functionalize the surface of the NPs.

Accordingly, we covalently grafted polyethylene glycol (Mn = 5000) onto the organosilica surface of the core-shell NPs using the CuAAC click reaction of NP-I-N3 with propargyl-PEG-OMe (Figure 3). To further characterize the suspensions, DLS and FCS experiments under a confocal microscope were also performed. [START_REF] Barbara | SERS Correlation Spectroscopy of Silver Aggregates in Colloidal Suspension: Quantitative Sizing Down to a Single Nanoparticle[END_REF] The experiments involve measuring the elastic (for DLS This is attributed to the weakness of the fluorescence signal that favours the detection of the largest NPs and to a possible retardation of the NPs due to weak optical trapping within the confocal volume resulting from the much larger incident power used as compared to that in DLS. [START_REF] Hosokawa | Cluster Formation of Nanoparticles in an Optical Trap Studied by Fluorescence Correlation Spectroscopy[END_REF] Blood is a complex medium containing mainly high amounts of salts (ionic strength around 0.15 M), proteins and cells. To predict the colloidal stability in blood, we undertook a series of DLS analyses in environments of increasing chemical complexity, approaching step by step the plasma composition.

The first step was to check the stability of the colloidal solution of PEGylated NPs (NP-I-PEG) in the presence of a salty medium and in the presence an ionic environment simulating that of blood at 37°C. DLS measurements were performed in solutions of the functionalized NP-I-PEG in pure water and at 25 °C and were compared to those obtained at 37 °C in pure water, with 0.9 wt% NaCl, for the salty medium, and with simulated body fluid (SBF) which mimics blood's ionic composition (Table S1). The size distributions of the NPs obtained from these DLS measurements are presented in supporting information (Figure S6). Neither the temperature, the salty medium nor the SBF influences the size distribution of the NPs, which are similar to that of the NP-I-PEG in pure water and at 25°C. Moreover, in all situations, the functionalized NP-I-PEG were stable with a mean size lower than 200 nm, in contrast to the unfunctionalized NP-I-N3.

The colloidal stability of the NPs was then tested in the presence of plasma proteins at 37 °C, at least 15 minutes after addition. The latter constitute a major fraction of blood, with serum albumins constituting 55% of the total amount of blood plasma proteins, globulins constituting 38% and the remaining 7% constituted mainly by fibrinogen. It is now well documented that the interaction of NPs with those proteins lead to the formation of a protein corona [START_REF] Ke | A Decade of the Protein Corona[END_REF] , which may lead to aggregation or recognition by the macrophages. It is therefore essential to check the colloidal stability and possible aggregation of NP-I-PEG in the presence of plasma proteins. Incremental experiments were performed: (i) the interaction of the NPs with bovine serum albumin (BSA) as a model albumin was first probed; (ii) the effect of the second most abundant protein in the blood plasma, namely γ-globulin, was then investigated; and finally, (iii) the colloidal stability of the functionalized NPs was investigated in an environment composed of a combination of the individually tested proteins (SBF, BSA and γ-globulin). As previously, the colloidal stability was investigated by DLS measurement performed at 37°C. During the first step, an increasing amount of BSA in SBF, ranging from 6 to 60 mg/mL, was added to the NP-I-PEG suspensions. The limiting concentration of 60 mg/mL corresponds to the maximum concentration encountered in blood. DLS measurements were also performed on a pure BSA solution as a reference. b,c, both as number-and intensity-weighted distributions. At low concentrations of BSA (6-15 mg/mL), the size distributions are very similar to those obtained for the NPs in pure water, confirming that no aggregation occurred. At higher concentrations, a peak at 8 nm appears in the number-weighted distribution, while that of the NPs disappears. This corresponds to the observation of the diffusion of the albumin molecules whose concentration becomes important.

The situation is different in the case of the intensity-weighted distributions, as the latter favors the larger objects due to their larger contribution to the observed signal. A slight aggregation was observed only at the highest concentration of BSA (60 mg/mL), resulting in a size increase from 220 nm to ~ 260 nm, while only a slight broadening of the distribution is observed for the 30 mg/mL concentration in BSA. However, the observation of aggregation at a BSA concentration of 60 mg/mL is not problematic, as this concentration is higher than the upper limit of the normal albumin concentration in the body. The stability of the functionalized NPs suspension under nominal conditions is therefore confirmed. Finally, the colloidal stability of the functionalized NPs was investigated in an environment composed of a combination of individually tested proteins at their highest concentrations, ie in SBF with 60 mg/mL of BSA and 45 mg/mL of γ-globulin. A small quantity (1 mg/mL) of glucose was also added to better simulate the plasma composition. Figure S8 presents the measured ACF together with the calculated fit. The latter was obtained by least square fitting and considering two independent exponential functions: one for the contribution of the small molecules and one for the contribution of the NPs. The derived characteristic time for the NPs is τ = 1140 µs, which corresponds to a mean hydrodynamic diameter of 250 nm. This is in agreement with the previous observation of a slight aggregation upon addition of 60 mg/mL albumin to the solution. This result is promising as it indicates that no further aggregation appears under these drastic conditions.

The previous measurements highlight a particularly good colloidal stability, with only minor aggregation of the NPs observed upon adding BSA at 60 mg/mL. However, since standard DLS measurements are carried out in a large observation volume of several mm 3 , many NPs are illuminated at the same time and the measured size distributions average the different sizes of the NPs. Therefore, possible NP size heterogeneities that appear in the early stages of the aggregation process could be missed. On the contrary, when performing DLS experiments in a small observation volume, only a few tens of NPs diffuse during the whole acquisition time of a few seconds and the presence of a unique aggregate that scatters much more light than the small NPs is sufficient to modify the entire ACF. The former can therefore be detected even at very low concentrations.

To support the results obtained from the standard DLS measurements, DLS studies under our confocal microscope were therefore undertaken on the NP solutions containing SBF and an increasing concentration of BSA, ranging from 6 to 60 mg/mL. As before, ten ACFs were measured for each BSA concentration and the mean ACFs are shown in Figure 6. The latter were normalized to ease the comparison of the characteristic times. At 6 and 15 mg/mL of BSA the ACFs present a mean characteristic time 𝜏𝜏 𝐷𝐷 = 3.2 ms and 𝜏𝜏 𝐷𝐷 = 3.5 ms, respectively, slightly higher than the mean characteristic time, 𝜏𝜏 𝐷𝐷 = 3 ms, of the NPs with no BSA, but within the range of 2.5-4 ms due to the natural size dispersion of the NPs. At 30 and 60 mg/mL of BSA, the ACFs exhibit a mean characteristic time 𝜏𝜏 𝐷𝐷 = 4.2 ms and 𝜏𝜏 𝐷𝐷 = 5.5 ms, respectively, which are higher than the value for the non-aggregated NPs and increase with increasing concentration of BSA. These first results confirm the previous observation of the absence of aggregation for 6 and 15 mg/mL of BSA, the broadening of the size distribution at 15 mg/mL corresponding to a slightly larger mean characteristic time; and the start of an aggregation process at 60 mg/mL of BSA. The ten different ACFs performed for each concentration were also analyzed individually. For each concentration, the measured ACFs were separated into two categories: those with a characteristic time within the range 2.5-4 ms for the individual NPs and those with higher characteristic time. The mean ACF of each category in displayed in Figure S10, for the four concentrations in BSA and in Figure 6-b for the 60 mg/mL concentration. Up to 30 mg/mL, the majority of the ten measured ACFs exhibit characteristic times consistent with the diffusion of individual NPs with mean diffusion times 𝜏𝜏 𝐷𝐷 = 2.8 -3.2 ms, corresponding to the expected hydrodynamic diameter dH =150-180 nm. For these three concentrations, only one or two ACFs out of ten exhibit a higher characteristic time corresponding to the diffusion of small aggregates typically composed of 2-5 NPs. [START_REF] Barbara | SERS Correlation Spectroscopy of Silver Aggregates in Colloidal Suspension: Quantitative Sizing Down to a Single Nanoparticle[END_REF][START_REF] Adamczyk | Hydrodynamic Radii and Diffusion Coefficients of Particle Aggregates Derived from the Bead Model[END_REF] At a BSA concentration of 60 mg/mL, we observe a larger number of ACFs exhibit characteristic times which are not consistent with the diffusion of individual NPs, as their value range from 𝜏𝜏 𝐷𝐷 = 4.5 to 11 ms.

This indicates that more and larger aggregates are formed at this concentration as compared to the lower concentrations, thereby confirming the beginning of an aggregation process. shown earlier, [START_REF] Redon | Tuning the Solid-State Emission of Small Push-Pull Dipolar Dyes to the Far-Red through Variation of the Electron-Acceptor Group[END_REF] this large absorbance is associated with a charge transfer transition characteristic of donor-acceptor type structures. The absorbance spectrum is further broadened by light scattering due to the relatively large particles giving a non-negligible contribution even above 600 nm. The two-photon absorbance spectrum (Figure 7-a) peaks at 930 nm with a σ2 value of 49±15 GM per chromophore. This spectrum appears to be narrower than its linear counterpart, which we attribute to the stronger influence of scattering on the latter. The spectral overlap between both spectra is due to the excitation of the lowest excited state of the molecule in both cases, in agreement with its symmetry. The one and two-photon excitation induced fluorescence (TPEF) spectra exhibiting maxima at around 630 nm are in good agreement (Figure 7-b). The small discrepancy between the two spectra is probably due to the strong scattering of the exciting laser in the TPEF experiments. The overall NP brightness (φf ×σ2 max ×NNP), which is the main parameter to demonstrate the potential of these objects for imaging applications, was calculated from the fluorescence quantum yield, the twophoton cross section and the estimated number of fluorophores per NP. [START_REF] Pawlicki | Two-Photon Absorption and the Design of Two-Photon Dyes[END_REF] This yields a very high value of (3.0+0.9) x 10 6 GM per NP, due to the very large number of dye molecules present in each NP (ca 2x10 6 fluorophores/NP in a 170 nm NP; see supporting information). [START_REF] Shenoi-Perdoor | Red-Emitting Fluorescent Organic@silicate Core-Shell Nanoparticles for Bio-Imaging[END_REF] This represents a significant advantage of these NP, since they will be imaged as a very bright spot, in contrast to the case of NPs where the dyes are diluted within a matrix.

To demonstrate the imaging potential of these nanoparticles, we first decided to study their ability

Conclusion

We demonstrated here the potential of ultrabright silica-coated organic nanocrystals for bioimaging. These NPs containing crystalline emitters, specifically engineered to exhibit two-photon absorption and emission properties for two-photon bio-imaging. The NPs were functionalized with PEG chains (Mn: 5000), using CuAAC click chemistry to render them "stealthy" within the blood

Figure 1

 1 Figure 1 schematically summarizes the preparation of nanoparticles by spray-drying and the

Figure 1 :

 1 Figure 1 : Preparation of nanoparticles

Figure 2 :

 2 Figure 2 : (a) Structure of the dye (E)-2-(benzo[d]thiazol-2-yl)-3-(4-diphenylamino)phenyl)acrylonitrile . (b) FESEM imaging of as-synthesized NP-I-N3. (C) NP-I-PEG after size selection and PEGylation (right).

- 1 )

 1 from the dye are clearly visible in the spectrum of NP-I-N3, and absent in dye-free NPs. The integrity of the encapsulated dye molecule after synthesis was demonstrated by absorption and fluorescence spectroscopy after dissolving the NP-I-N3 NPs in ethanol.

Figure 3 :

 3 Figure 3 : CuAAC click functionalization of NPsThe comparison of the FTIR spectra before and after functionalisation (SI -FigureS3)

  ) or inelastic scattering (here fluorescence for FCS) of the NPs as their Brownian motion brings them to diffuse through a small observation volume where they are illuminated and scatter light. The measured intensities exhibit temporal fluctuations related to an increase (or decrease) of signal each time a NP enters (or leaves) the confocal volume. Consequently, the amplitudes and duration of these fluctuations are related to the mean number, 〈𝑁𝑁〉, of NPs, and to their residence time, 𝜏𝜏 𝐷𝐷 , respectively, within the observation volume. Both parameters can be obtained by adjusting their values through fitting of the intensity auto-correlation function, 𝐴𝐴𝐴𝐴𝐴𝐴 = 𝐺𝐺(𝑡𝑡) -1, where 𝐺𝐺(𝑡𝑡) = 〈𝐼𝐼(𝜏𝜏)𝐼𝐼(𝑡𝑡+𝜏𝜏)〉 〈𝐼𝐼(𝜏𝜏)〉 2 . The size and the concentration of the NPs are further obtained from the experimental determination of 〈𝑁𝑁〉 and 𝜏𝜏 𝐷𝐷 . The detailed procedure is given in supporting information.A series of ten DLS experiments were performed with this setup, leading to a slightly different ACF for each measurement due to the dispersion of the NPs and to the fact that only a few of them are measured simultaneously, in contrast to the conventional DLS equipment. An averaged ACF is presented in Figure4-a together with the calculated one using 𝐺𝐺(0) = 21.8 and 𝜏𝜏 𝐷𝐷 = 3 ms.Knowing the confocal volume 𝑉𝑉 𝑐𝑐 = 0.7 µm 3 and its radius r0 = 190 ± 30 nm from calibration measurements, we obtain a mean concentration 𝐴𝐴 = 2.1 × 10 10 p/mL and a mean hydrodynamic diameter dH = 160 ± 30 nm. The different ACFs present characteristic residence times ranging from 2.5 to 4 ms, which correspond to hydrodynamic diameters ranging from 80 to 300 nm, and 〈𝑁𝑁〉 value ranging from 0.012 to 0.024, corresponding to a concentration 𝐴𝐴 = 2.5 ± 0.8 × 10 10 p/mL. These results are in full agreement with the NP sizes obtained from the commercial DLS, and with the initial concentration of 80 µg/mL corresponding to a concentration of 2 × 10 10 p/mL when considering NPs of 170 nm in diameter and of density of 1.5.

Figure 4 :

 4 Figure 4 : (a) DLS experimental ACF averaged over ten measurements (dots) and calculated fit (line) with 𝐺𝐺(0) = 21.8 and 𝜏𝜏 𝐷𝐷 = 3 ms and (b) FCS experimental ACF (dots) averaged over ten measurements and calculated one (line) with 𝐺𝐺(0) = 2.4 and 𝜏𝜏 𝐷𝐷 = 5 ms.FCS measurements were analysed using exactly the same experimental procedure. As can

Figure 5

 5 -a presents the normalized ACFs of the BSA and of the NPs in SBF together with their fit calculated for characteristic time decays of 18 µs for BSA and 910 µs that corresponds to hydrodynamic diameters of 8.4 nm and 200 nm, respectively (which are the expected values). As the characteristic residence time of the BSA and that of the NPs are very different, their respective contribution to the ACF are not correlated and therefore simply sum to yield the final ACF. At short times, the time decay due to the BSA becomes more visible as its concentration increases. At longer time, and at low concentrations in BSA (6 and 15 mg/mL), the ACFs are very similar to that of the NPs only in the presence of SBF, indicating the stability of the colloidal solutions. At higher BSA concentrations (30 and 60 mg/mL), a slight shift of the ACFs is observed at longer times, consistent with an increase in the polydispersity of the solutions. To obtain more quantitative results, the ACFs were fitted and the NPs size distributions were obtained. The latter are displayed in Figures5-

Figure 5 :

 5 Figure 5 : (a) Normalized autocorrelation data corresponding to NP-I-PEG in SBF with BSA at a concentration of 0 mg/mL (orange), 6mg/mL (black), 15 mg/mL (red), 30 mg/mL (blue) and 60 mg/mL (pink); of pure BSA in PBS (green). Fits of the green and orange plots (black) give a characteristic time of 18 ms for BSA and 910 ms for the NP-I-PEG. (b) Number and (c) Intensity distributions of NP-I-PEG in SBF at 37 ᵒC with the different concentrations of BSA.

Finally, the concentration

  of the individual NPs within the four different solutions was estimated from the ACFs with no aggregates. The obtained mean and individual values are displayed in Figure 6-c for each concentration in BSA as well as for the NPs solution with no BSA.The blue rectangle indicates the width of the error bar on the estimation of the initial NP concentration. One can observe that the concentration of the individual NPs is stable at 6 to 15 mg/mL in BSA and decreases slightly at 30 mg/mL (although the measured value is still the same within the error bars), excluding the coagulation of the NPs. On the contrary, an actual decrease in the concentration of individual NPs is observed at 60 mg/mL in BSA, which again is consistent with an aggregation process. However, even in this latter case, almost half of the initial individual NPs are still present as isolated NPs in the solution.

Figure 6 :

 6 Figure 6 : (a) Mean ACFs normalized to one measure of the NPs solutions with 0 (black line), 6 (light red), 15 (dark red), 30 (light blue) and 60 mg/mL of BSA (dark blue). (b) at 60 mg/mL of BSA, the average ACF is the sum of two of ACFs: one corresponding to the diffusion of isolated NPs with a mean characteristic time of 2.5 ms (dark blue) and one corresponding to the diffusion of small aggregates with a mean characteristic time of 7.5 ms and values ranging from 4.5 to 11 ms depending on the different ACFs. (c) Mean concentration of the isolated NPs (gray dots) and concentration extracted from the different ACFs (Blues crosses) as a function of the BSA concentration.

Figure 7

 7 Figure 7 shows the resonant one-photon absorbance spectrum of the aqueous solution of the

Figure 7 :

 7 Figure 7 : a) Absorbance spectrum of aqueous solution of NP-I-PEG (black) superposed with their two-photon absorption spectrum per chromophore unit in the NP (red). b) One-(black) and two-photon (red) absorption induced fluorescence spectra of the NP-I-PEG.

  photon excitation (900 nm), the NP-I-PEG demonstrated a much higher brightness, resulting in

Figure 8 :

 8 Figure 8 : Subcellular localization of NP-I-PEG in living breast cancer cells (MCF-7). MCF 7 cells were incubated for 24 h with 25 µg.mL -1 of NP-I-PEG. Cells were washed twice and imaged using confocal microscope Zeiss LSM 780 laser beam at 450 nm or at 900 nm (pulsed laser, Chameleon), magnification ×63. Cell membranes were stained with cell mask (green); nuclei were stained by Hoechst 33342 (blue); lysosomes were stained with LysoTracker green and compounds were visualized in red. Scale bar: 10 µm.

Figure 9 :

 9 Figure 9 : Circulation of NPs in blood vessels; (a) green channel (GFP microglia in the brain) and red channel (blood vessel due to presence of NP; b) red channel only. Image size 200×200 µm² A significant drop in fluorescence intensity (Figure 11-a) was recorded within 15-20 minutes of

  photon excitation induced fluorescence spectra peaking at ~630 nm, with a maximal σ2 value of

  mW. The open source software packages ImageJ

	and	Icy,	created	by	BioImage	Analysis	Lab,	Institut	Pasteur	(France;

http://icy.bioimageanalysis.org/) were used for image treatment, while drift correction was applied using custom-written Matlab software to correct image drift during acquisition.

Acknowledgements A.B., G.M. and Y.B. acknowledge the contribution of SFR Biosciences (UMS3444/CNRS, US8/Inserm, ENS de Lyon, UCBL) facilities PLATIM, for access to the femtosecond laser system and for the technical assistance. M.G.B. and C.N acknowledge the imaging facility MRI, member of the national infrastructure France-BioImaging supported by the French National Research Agency (ANR-10-INBS-04, «Investments for the future»). The authors thanks O. Stephan (Liphy laboratory, UMR5588/CNRS-UGA Grenoble) for the utilization of the commercial Malvern Zetasizer Nano ZS apparatus and B. Van Der Sanden (TIMC-IMAG Lab) for helpful discussions. Prof J. R. Bartlett (Western Sydney University) is warmly acknowledged for carefully reading this manuscript.

Supporting Information Available: DLS and FCS mathematical model used, IR and DLS characterizations. This material is available free of charge via the Internet at http://pubs.acs.org.


Jean-François Bony

Setsuro Fujiié
email: fujiie@fc.ritsumei.ac.jp

Thierry Ramond
email: thierry.ramond@universite-paris-saclay.fr

Maher Zerzeri
email: zerzeri@math.univ-paris13.fr

Jean-Franc ¸ois Bony

Setsuro Fujii É

An example of resonance instability

Keywords: 2010 Mathematics Subject Classification. 35B34, 35B35, 81Q20, 37C25, 35J10, 35P20 Resonances, semiclassical asymptotics, microlocal analysis, spectral instability, Schrödinger operators

published or not. The documents may come

Introduction

In this note, we consider semiclassical Schrödinger operators P on L 2 (R n ), n ≥ 1, (1.1)

P = -h 2 ∆ + V (x),
where V ∈ C ∞ 0 (R n ; R) is a real-valued smooth compactly supported potential. Depending on the situation, one may also work with such operators outside a compact smooth obstacle with Dirichlet boundary condition. Since P is a compactly supported perturbation of -h 2 ∆, the resonances of P near the real axis are well-defined through the analytic distortion method or using the meromorphic extension of its truncated resolvent. We send back the reader to the books of Sjöstrand [START_REF] Sjöstrand | Lectures on resonances[END_REF] or Dyatlov and Zworski [START_REF] Dyatlov | Mathematical theory of scattering resonances[END_REF] for a general presentation of resonance theory, and we denote Res(P ) the set of resonances of P .

The stability of the resonances is a rather touchy question. Indeed, we do not know yet whether the concept of resonance persists under the perturbation by a non-analytic non-exponentially decreasing potential. Therefore, we only consider here perturbations of Schrödinger operators (1.1) by subprincipal real-valued smooth compactly supported potentials of the form h τ W (x) with τ > 0 and W ∈ C ∞ 0 (R n ; R). But even in this setting, the stability of resonances is a subtle problem since stability results and instability results can be obtained for the same operator.

On one hand, the resonances tend to be stable as other spectral objects like the eigenvalues. This is particularly clear when the resonances are defined by complex distortion, since the usual perturbation theory of discrete spectrum can be directly applied to the distorted operator. But, even if the resonances are defined as the poles of the meromorphic extension of some weighted resolvent, Agmon [START_REF] Agmon | A perturbation theory of resonances[END_REF][START_REF] Agmon | Erratum: "A perturbation theory of resonances[END_REF] has proved their stability. On the other hand, the resonances can be unstable since they do not come from a self-adjoint problem. Thus, some typical non self-adjoint effects may occur concerning the resonances even if P is self-adjoint. For instance, the distorted operator may have a Jordan block or the truncated resolvent may have a pole of algebraic order greater than 1 (see e.g. Sjöstrand [START_REF] Sjöstrand | Semiclassical resonances generated by nondegenerate critical points[END_REF]Section 4]).

Our instability result is the following. Theorem 1.1 (Resonance instability). In dimension n = 2, one can construct an operator P and a potential W as above satisfying the following property for all δ > 0 small enough. There exist a set H ⊂]0, 1] with 0 ∈ H and constants D 0 , E 0 , α > 0 such that, for all C > 0 and -C ≤ A < B ≤ C, i) On one hand, P has no resonance z with Re z ∈ E 0 + [-Ch, Ch] and

(1.2) Im z ≥ -D 0 h -αh, for h ∈ H small enough.
ii) On the other hand, the resonances z of P + h 1+δ W with Re z ∈ E 0 + [Ah, Bh] closest to the real axis satisfy

(1.3) Im z ∼ -D 0 h -δh, for h ∈ H small enough.
The result is illustrated in Figure 1. Theorem 1.1 ii) provides at least one resonance z of P + h 1+δ W satisfying Re z ∈ E 0 + [Ah, Bh] and Im z ∼ -D 0 h -δh. But its proof shows that the number of such resonances is at least of order | ln h|. In particular, the essential quantum trapping in E 0 + [Ah, Bh] defined by (1.4) ess-qt(Q) = lim n→+∞ lim sup

h→0 h∈H inf z 1 ,...,zn∈Res(Q) Re z•∈E 0 +[Ah,Bh] sup z∈Res(Q)\{z 1 ,...,zn} Re z∈E 0 +[Ah,Bh] h | Im z| ,
increases by at least (α -δ)(D 0 + α) -1 (D 0 + δ) -1 when we add the perturbation h 1+δ W to the operator P . Thus, the resonance instability described here is not an anomaly due to an exceptional resonance or a Jordan block but a phenomenon mixing geometry and analysis.

In the statement of the previous result, we do not specify the subset of semiclassical parameters H . In fact, depending on the geometric situation, the resonance instability may occur on the whole interval H =]0, 1] or only near a sequence H like {j -1 ; j ∈ N * }. Operators corresponding to these different situations are given at the end of Section 2.

For 0 < κ ≪ 1 fixed, one can show that the resonances z of P + κhW with Re z ∈ E 0 + [Ah, Bh] closest to the real axis satisfy Im z ∼ -D 0 h for h ∈ H small enough. The proof of this point is similar to that of Theorem 1.1. On the contrary, for larger values of κ, some cancellations may appear and P + κhW may have a resonance free region of size D 0 h + αh below the real axis as for P . The constructions in Theorem 1.1 can be realized in any dimension n ≥ 2, but our method of proof does not work in dimension n = 1. Indeed, the Hamiltonian vector field must have an anisotropic hyperbolic fixed point. Nevertheless, we do not know yet if the resonance instability phenomenon described here occurs in dimension one.

Let P θ denote the operator P in the proof of Theorem 1.1 after a complex distortion of angle θ = h| ln h|. Its resolvent satisfies a polynomial estimate in

Ω = E 0 + [-Ch, Ch] + i[-D 0 h -αh, h].
This means that, for some M > 0, we have

(1.5) (P θ -z) -1 h -M ,
uniformly for z ∈ Ω. By the usual perturbation argument, it implies that P + Q has no resonance in Ω for any distortable perturbation Q of size o(h M ). The stability of resonances under small enough perturbations has already been observed (see e.g. Agmon [START_REF] Agmon | A perturbation theory of resonances[END_REF][START_REF] Agmon | Erratum: "A perturbation theory of resonances[END_REF]). Summing up, the resonances of P are stable for perturbations of size o(h M ) and unstable for some perturbations of size h 1+δ (showing that M ≥ 1 + δ).

The present result is obtained for a Schrödinger operator whose trapped set at energy E 0 consists of a hyperbolic fixed point and homoclinic trajectories, following our recent paper [START_REF] Bony | Resonances for homoclinic trapped sets[END_REF]. In fact, the instability phenomenon obtained here does not hold in the geometric settings previously studied (see Figure 2). In the "well in the island" situation, the resonances are known to be exponentially close to the real axis (see Helffer and Sjöstrand [START_REF] Helffer | Résonances en limite semi-classique[END_REF] for globally analytic potentials and Lahmar-Benbernou, Martinez and the second author [START_REF] Fujiié | Width of shape resonances for non globally analytic potentials[END_REF] for potentials analytic at infinity). Adding a subprincipal real potential hW (x) does not change this properties. When the trapped set at energy E 0 consists of a non-degenerate critical point (say at (x 0 , 0) ∈ T * R n ), Sjöstrand [START_REF] Sjöstrand | Semiclassical resonances generated by nondegenerate critical points[END_REF] has proved that the resonances form, modulo o(h), a quarter of a rectangular lattice which is translated by hW (x 0 ) when a subprincipal potential hW (x) is added. Finally, the asymptotic of the resonances generated by a hyperbolic closed trajectory has been obtained by Gérard and Sjöstrand [START_REF] Gérard | Semiclassical resonances generated by a closed trajectory of hyperbolic type[END_REF] (see also Ikawa [11] and Gérard [START_REF] Gérard | Asymptotique des pôles de la matrice de scattering pour deux obstacles strictement convexes[END_REF] for obstacles). Modulo o(h), they form half of a rectangular lattice which is translated by a real quantity after perturbation by a real potential hW (x). Summing up, the imaginary part of the resonances is very stable in the three previous examples: it moves only by o(h) when a perturbation by a real potential of size h is applied. In other words, if the quantum trapping (or maximum of the quantum lifetime) in E 0 + [-Ch, Ch] of an operator Q is defined by

(1.6) qt(Q) = lim sup h→0 sup z∈Res(Q) Re z∈E 0 +[-Ch,Ch] h | Im z| ,
with the conventions that qt(Q) = +∞ if the limit diverges and qt(Q) = 0 if Q has no resonance, we have qt(P ) = qt(P + hW ) in these examples. The situation is completely opposite in Theorem 1.1 since a self-adjoint perturbation of size o(h) induces a change of size 1 of the quantum trapping. By definition, we always have qt(Q) ∈ [0, +∞] and qt(Q) ≥ ess-qt(Q). Moreover, if the resonance expansion of the quantum propagator holds, we have χe -itQ/h ϕ(Q)χ ≈ e t/ qt(Q) for t ≫ 1 and h in an appropriate sequence, justifying the name of quantum trapping. Other results in scattering theory provide resonance free regions, that is upper bounds on the quantum trapping, under geometric assumptions. In general, the bounds obtained do not depend on the subprincipal symbol, assumed to be self-adjoint in an appropriate class (see for instance Section 3.2 of Nonnenmacher and Zworski [START_REF] Nonnenmacher | Quantum decay rates in chaotic scattering[END_REF]). In the present setting, Section 3.1 of [START_REF] Bony | Resonances for homoclinic trapped sets[END_REF] implies qt(P ) ≤ D -1 0 , but Theorem 1.1 i) shows that this inequality is not sharp. Theorem 1.1 may seem natural since the distorted resolvent is generally large in the unphysical sheet and small perturbations may produce eigenvalues. More precisely, the norm of the distorted resolvent is known to be larger than h -1 , that is

(P θ -z) -1 ≫ h -1 ,
with Im z < 0, in many cases (see e.g. Burq and two of the authors [START_REF] Bony | Minoration de la résolvante dans le cas captif[END_REF] or Dyatlov and Waters [START_REF] Dyatlov | Lower resolvent bounds and Lyapunov exponents[END_REF]). By the pseudospectral theory (see e.g. Section I.4 of Trefethen and Embree [START_REF] Trefethen | The behavior of nonnormal matrices and operators[END_REF]), there exists a bounded operator Q θ of size o(h) such that z is precisely an eigenvalue of

P θ + Q θ . Nevertheless, it is not clear that Q θ is the distortion of some operator Q, that Q is a potential and that Q is self-adjoint.
In fact, as explained in the previous paragraph, this is not always the case. This instability phenomenon is due to the non self-adjoint nature of the resonances (even for self-adjoint operators). Such a property never holds for the usual spectrum in the selfadjoint framework. Indeed, for any self-adjoint operator P and any bounded perturbation W , the spectrum of P + W satisfies

σ(P + W ) ⊂ σ(P ) + B(0, W ).
Thus, a perturbation of size h 1+δ of a self-adjoint operator can not lead to a perturbation of size h of its spectrum.

The operator P and the potential W are constructed in Section 2. The instability phenomenon stated in Theorem 1.1 is proved in Section 3.

Construction of the operators

To construct a Schrödinger operator P = -h 2 ∆+V (x) as in (1.1) with unstable resonances, we follow Example 4.23 and Example 4.24 (B) of [START_REF] Bony | Resonances for homoclinic trapped sets[END_REF]. We send back the reader to this paper for a slightly different presentation, some close geometric situations and general results about resonances generated by homoclinic trajectories. As usual, p(x, ξ) = ξ 2 + V (x) denotes the symbol of P , its associated Hamiltonian vector field is

H p = ∂ ξ p • ∂ x -∂ x p • ∂ ξ = 2ξ • ∂ x -∇V (x) • ∂ ξ ,
and the trapped set at energy E for P is

K(E) = (x, ξ) ∈ p -1 (E); t → exp(tH p )(x, ξ) is bounded .
Recall that K(E) is compact and stable by the Hamiltonian flow for E > 0.

0 supp Vtop πx(γ2) πx(γ3) supp V ref supp W {V ref ≥ E0} πx(γ1) (a, 0) Figure 3. The potentials V = V top + V ref and W .
In dimension n = 2, we consider the potential (2.1)

V (x) = V top (x) + V ref (x),
as in Figure 3 and described below. On one hand, the potential

V top is of the form V top (x) = V 1 (x 1 )V 2 (x 2 )
where the functions V • ∈ C ∞ 0 (R) are single barriers (see Figure 4) with

V 1 (x 1 ) = E 0 - λ 2 1 4 x 2 1 + O(x 3 1 ) and V 2 (x 2 ) = 1 - λ 2 2 4E 0 x 2 2 + O(x 3
2 ), near 0 and 0 < λ 1 < λ 2 . In particular, V top is an anisotropic bump,

V top (x) = E 0 - λ 2 1 4 x 2 1 - λ 2 2 4 x 2 2 + O(x 3 ),
near 0 and (0, 0) is a hyperbolic fixed point for H p . The stable/unstable manifold theorem ensures the existence of the incoming/outgoing Lagrangian manifolds Λ ± characterized by Λ ± = (x, ξ) ∈ T * R 2 ; exp(tH p )(x, ξ) → (0, 0) as t → ∓∞ .

They are stable by the Hamiltonian flow and included in p -1 (E 0 ). Eventually, there exist two smooth functions ϕ ± , defined in a vicinity of 0, satisfying

(2.2) ϕ ± (x) = ± 2 j=1 λ j 4 x 2 j + O(x 3 ),
and such that Λ ± = {(x, ξ); ξ = ∇ϕ ± (x)} near (0, 0).

0 0 V1(x1) x1 ∈ R x2 ∈ R 1 V2(x2) E0 Figure 4. The potentials V 1 and V 2 . supp V add 0 O supp Vtop πx(γ2) πx(γ3) πx(γ1) supp W supp V abs (a, 0)
Figure 5. A realization with a potential V top and an obstacle O.

On the other hand, the reflecting potential V ref is non-trapping and localized near (a, 0) ∈ R 2 with a large enough. A dynamical result (Lemma B.1 of [START_REF] Bony | Resonances for homoclinic trapped sets[END_REF]) ensures that no Hamiltonian trajectory of energy E 0 can start from the support of V ref , touch the support of V top and then come back to the support of V ref . Thus, a trapped trajectory of energy E 0 is either {(0, 0)} or a Hamiltonian trajectory starting asymptotically from the origin, touching the support of V ref and coming back to the origin; these latter trajectories are called homoclinic. In other words, K(E 0 ) satisfies

K(E 0 ) = Λ -∩ Λ + ,
and H = Λ -∩ Λ + \ {(0, 0)} denotes the set of homoclinic trajectories.

Giving to V ref the form of a "croissant" barrier, we can make sure that H consists of a finite number of trajectories {γ 1 , . . . , γ K } on which Λ -and Λ + intersect transversally. In the sequel, we will need at least two homoclinic trajectories, that is K ≥ 2. Such a geometric configuration can also be realized replacing the potential barrier V ref by an obstacle O having essentially the form of {V ref (x) > E 0 }, the operator being P = -h 2 ∆ R 2 \O + V top (x) with Dirichlet boundary condition (see Figure 5). In that case, one can easily realize a situation where K = 3 whereas it seems complicated to have K = 2 (see Example 4.14 of [START_REF] Bony | Resonances for homoclinic trapped sets[END_REF]).

As explained in the next section (see also Example 4.24 of [START_REF] Bony | Resonances for homoclinic trapped sets[END_REF]), the resonance instability is governed by the function

(2.3) µ(τ, h) = Γ λ 1 + λ 2 2λ 1 -i τ λ 1 e -πτ 2λ 1 K k=1 e iA k /h B k e iT k τ ,
for τ ∈ C, where A k , B k , T k are dynamical quantities related to the curve γ k = (x k , ξ k ). We recall quickly how these quantities are defined and send back the reader to [START_REF] Bony | Resonances for homoclinic trapped sets[END_REF] for the proof of convergence of the various objects. First,

A k = γ k ξ • dx,
is the action along γ k . Let ν k denote the Maslov index of Λ + along γ k . The function x k (t) has the following asymptotics

x k (t) = g k ± e ±λ 1 t + o e ±λ 1 t ,
as t → ∓∞ for some vector g k ± ∈ R 2 . As a matter of fact, g k ± is collinear to the first vector of the canonical basis (1, 0) and do not vanish. Eventually, if γ k (t, y) = (x k (t, y), ξ k (t, y)) : R×R -→ T * R 2 is a smooth parametrization of Λ + by Hamiltonian curves such that γ k (t, 0) = γ k (t), the limits

M + k = lim s→-∞ det ∂x k (t, y) ∂(t, y) | t=s, y=0 e -s λ 1 +λ 2 2 , M - k = lim s→+∞ det ∂x k (t, y) ∂(t, y) | t=s, y=0 e -s λ 2 -λ 1 2 ,
exist and belong to ]0, +∞[. With these notations,

(2.4)

B k = λ 1 2π M + k M - k e -π 2 (ν k + 1 2 )i g k -iλ 1 |g k + ||g k -| - λ 1 +λ 2 2λ 1 , T k = ln(λ 1 |g k + ||g k -|) λ 1 . Note that B k ∈ C \ {0} and T k ∈ R.
The idea is to find a geometric situation and a set H ⊂]0, 1] with 0 ∈ H such that (2.5) µ(τ, h) = 0, for all τ ∈ C and h ∈ H . For simplicity, we take in the sequel K = 3 as in Figure 3 or 5 and assume that the trajectories γ 1 and γ 3 are symmetric. In particular, A 1 = A 3 , B 1 = B 3 and T 1 = T 3 . We consider two situations:

Case (I): A 1 = A 2 (say A 2 > A 1 ), 2B 1 = B 2 e iν , ν ∈ R, and T 1 = T 2 . Using (2.
3) and the symmetry of γ 1 and γ 3 , these relations imply that (2.5) holds true with

H = A 2 -A 1 (2j + 1)π + ν ; j ∈ N .
The required relations can be realized since T 2 is only given by the potential V on the line

R × {0} if ∂ x 2 V (x 1 , 0) = 0 for all x 1 ∈ R, whereas B 2 is given by ∂ 2 x 2 V on R × {0}. If V ref is replaced
by an obstacle O, one may need an additional potential V add in order to satisfy these relations (see Figure 5).

Case (II):

A 1 = A 2 , 2B 1 = -B 2 and T 1 = T 2 .
In this setting, (2.5) holds true with H =]0, 1]. These relations can been obtained as before. More precisely, one can adjust V ref on R × {0} with ∂ x 2 V = 0 on R × {0} in order to have A 1 = A 2 and T 1 = T 2 . Then, playing on the Maslov index and on ∂ 2

x 2 V ref on R × {0}, one can obtain 2B 1 = -B 2 .
Adding an absorbing potential -ih| ln h|V abs , with V abs ≥ 0, it is possible to artificially remove a homoclinic trajectory and thus to work with only K = 2 trajectories (see Remark 2.1 ii) and Example 4.14 of [START_REF] Bony | Resonances for homoclinic trapped sets[END_REF]). The resulting operator will be non self-adjoint (dissipative) but the conclusions of Theorem 1.1 will still hold.

For the perturbation W , we take any non-negative C ∞ 0 (R 2 ; R) function supported away from the support of V and non-zero on the base space projection of only one homoclinic trajectory. In the sequel, we will assume that this trajectory is γ 1 as in Figures 3 and5. We assume that W = 0 near the support of V only to simplify the discussion. The same way, W ≥ 0 and W non-zero on π x (γ 1 ) can be weakened to R W (x 1 (t)) dt = 0. Finally,

W = c 1 W 1 + c 2 W 2 + c 3 W 3 ,
with W j non-zero only on γ j , may be suitable for Theorem 1.1 generically with respect to c j ∈ R.

Proof of the spectral instability

We consider the operators constructed in the previous section. In particular, we work in dimension n = 2, the trapped set of energy E 0 > 0 has K = 3 homoclinic trajectories and (2.5) holds true for h ∈ H . Following Chapter 4 of [START_REF] Bony | Resonances for homoclinic trapped sets[END_REF], the resonances of P closest to the real axis are given by the 3 × 3 matrix Q whose entries are

(3.1) Q k,ℓ (z, h) = e iA k /h Γ S(z, h)/λ 1 λ 1 2π M + k M - k e -π 2 (ν k + 1 2 )i g ℓ -iλ 1 |g k + ||g ℓ -| -S(z,h)/λ 1 ,
with rescaled spectral parameter

(3.2) S(z, h) = λ 1 + λ 2 2 -i z -E 0 h .
The same way, the entries of the corresponding matrix for P = P + h 1+δ W are

(3.3) Q k,ℓ (z, h) = e -iwh δ Q k,ℓ (z, h) if k = 1, Q k,ℓ (z, h) if k = 1,
with the notation w = R W (x 1 (t)) dt = 0.

Lemma 3.1. The matrices Q and Q are of rank one with non-zero entries. Moreover, Q 2 (z, h) = 0 for all z ∈ C and h ∈ H .

Proof. Since M ± • = 0 and g • ± = 0, the entries of Q and Q are always non-zero. From (3.1), the entries of Q can be written Q k,ℓ = α k β ℓ for some α k , β k ∈ C \ {0}. In particular, Q = α(β, •) with α, β ∈ C 3 \ {0} and Q is of rank one (the same thing for Q). Thus, 0 is an eigenvalue of Q of multiplicity at least 2 and the last eigenvalue is given by its trace, that is

tr(Q) = Γ S(z, h)/λ 1 3 k=1 e iA k /h λ 1 2π M + k M - k e -π 2 (ν k + 1 2 )i g k -iλ 1 |g k + ||g k -| - λ 1 +λ 2 2λ 1 +i z-E 0 λ 1 h = µ z -E 0 h , h .
For h ∈ H , all the eigenvalues of Q are zero from (2.5) and

Q is nilpotent. Since Q j = α(β, α) j-1 (β, •), Q is nilpotent iff (β, α) = 0 iff Q 2 = 0.
Let W be the 3 × 3 diagonal matrix W = diag(-iw, 0, 0). The eigenvalues of WQ(z, h) are

(3.4) -iwQ 1,1 (z, h), 0, 0.
In the present setting, the quantization rule for P takes the following form: we say that z is a pseudo-resonance of P when

(3.5) 1 ∈ sp h S(z,h)/λ 1 -1/2+δ WQ(z, h) .
The set of pseudo-resonances is denoted by Res 0 ( P ). Since (3.

E 0 + [-Ch, Ch] + i - λ 2 2 + δλ 1 h -C h | ln h| , h , with Re z ∈ E 0 + τ h + hε(h)[-1, 1] satisfy z = z q (τ ) + o(h| ln h| -1 ) with (3.7) z q (τ ) = E 0 - A 1 λ 1 | ln h| + 2qπλ 1 h | ln h| -ih λ 2 2 + δλ 1 + i ln( µ(τ ))λ 1 h | ln h| , and
µ(τ ) = wΓ 1 2 -δ -i τ λ 1 λ 1 2π M + 1 M - 1 e -π 2 (ν 1 + 3 2 )i g 1 -iλ 1 |g 1 + ||g 1 -| -1 2 +δ+i τ λ 1 ,
for some q ∈ Z. On the other hand, for each τ ∈ [-C, C] and q ∈ Z such that z q (τ ) belongs to (3.6) with a real part lying in

E 0 + τ h + hε(h)[-1, 1]
, there exists a pseudo-resonance z satisfying z = z q (τ ) + o(h| ln h| -1 ) uniformly with respect to q, τ . Moreover, there exists M > 0 such that, for all z ∈ (3.6), we have

(3.8) dist z, Res 0 ( P ) > β h | ln h| =⇒ 1 -h S/λ 1 -1/2+δ WQ -1 ≤ M.
In the lemma, we have used that the eigenvalues of WQ are explicitly given by (3.4) and that two of them are zero. On the contrary, note that µ(τ ) is a smooth function which does not vanish and that there are a lot of pseudo-resonances in (3.6). The assumption 0 < δ < 1/2 allows to avoid the poles of the Γ function. This result implies the following resolvent estimates at the classical level. Lemma 3.3. For all 0 < δ < 1/2, ν = λ 1 /4 + λ 2 /2 and C, β > 0, the following properties are satisfied for h ∈ H small enough.

i) For all z ∈ E 0 + [-Ch, Ch] + i[-νh, h], we have

(3.9) 1 -h S/λ 1 -1/2 Q -1 max 1, h λ 2 2λ 1 + Im z λ 1 h .
ii) For all z ∈ (3.6) with dist(z, Res 0 ( P )) > βh| ln h| -1 , we have

(3.10) 1 -h S/λ 1 -1/2 Q -1 h -δ .
The particular value of ν in Lemma 3.3 has no particular meaning. We only need ν > D 0 = λ 2 /2 for Theorem 1.1 and ν < λ 1 /2 + λ 2 /2 to avoid the poles of Γ.

Proof. Since Q 2 = 0 by Lemma 3.1, we get

(3.11) 1 -h S/λ 1 -1/2 Q -1 = 1 + h S/λ 1 -1/2 Q. Using that |h S/λ 1 -1/2 | = h λ 2 2λ 1 + Im z λ 1 h and that Q(z, h) is uniformly bounded for z ∈ E 0 + [-Ch, Ch] + i[-νh, h],
this identity yields (3.9).

On the other hand, (3.3), e -iwh δ = 1 -iwh δ + O(h 2δ ) and [START_REF] Dyatlov | Mathematical theory of scattering resonances[END_REF]. Combining these estimates with (3.8), (3.9) and (3.11), (3.12) implies

Q 2 = 0 give 1 -h S/λ 1 -1/2 Q = 1 -h S/λ 1 -1/2 Q -h S/λ 1 -1/2+δ WQ + O(h λ 2 2λ 1 + Im z λ 1 h +2δ )Q = 1 -h S/λ 1 -1/2+δ WQ + O(h λ 2 2λ 1 + Im z λ 1 h +2δ )Q 1 -h S/λ 1 -1/2 Q = 1 + O(h λ 2 2λ 1 + Im z λ 1 h +2δ ) 1 -h S/λ 1 -1/2+δ WQ -1 1 -h S/λ 1 -1/2+δ WQ 1 -h S/λ 1 -1/2 Q . (3.12) We have |h S/λ 1 -1/2 | = h λ 2 2λ 1 + Im z λ 1 h ≤ h -δ for z ∈ (3.
1 -h S/λ 1 -1/2 Q -1 = 1 -h S/λ 1 -1/2 Q -1 1 -h S/λ 1 -1/2+δ WQ -1 1 + O(h δ ) -1 = 1 + h S/λ 1 -1/2 Q 1 -h S/λ 1 -1/2+δ WQ -1 + O(1), (3.13)
if dist(z, Res 0 ( P )) > βh| ln h| -1 . Then (3.10) follows.

The next result provides a resonance free region for P and the asymptotic of the resonances closest to the real axis for P = P + h 1+δ W . Combined with Lemma 3.2, it implies directly Theorem 1.1 with D 0 = λ 2 /2 if we choose λ 1 = 1. Lemma 3.4. There exists α > 0 such that, for all δ > 0 small enough and C > 0, the following properties hold for h ∈ H small enough.

i) P has no resonance in The proof of Lemma 3.4 gives a polynomial estimate of the resolvents in the corresponding domains (at distance larger than h| ln h| -1 from the pseudo-resonances of P ).

E 0 + [-Ch, Ch] + i - λ 2 2 + α h, h .
Proof. The first point of the lemma has already been obtained in Lemma 12.1 of [START_REF] Bony | Resonances for homoclinic trapped sets[END_REF]. In order to show the second point, we follow the strategy of Chapters 11 and 12 of [START_REF] Bony | Resonances for homoclinic trapped sets[END_REF] and summarized in the introduction of [START_REF] Bony | Resonances for homoclinic trapped sets[END_REF]. Then, we first prove that P has no resonance and we show a polynomial estimate of its resolvent away from the pseudo-resonances.

Lemma 3.5. For δ > 0 small enough, C, β > 0 and h ∈ H small enough, P has no resonance in the domain

(3.14) E 0 + [-Ch, Ch] + i - λ 2 2 + δλ 1 h -C h | ln h| , h \ Res 0 ( P ) + B 0, β h | ln h| , ρ k + 0 γ k supp W ρ k - Figure 6
. The geometric setting in the proof of Lemma 3.5.

and there exists M > 0 such that the distorted operator P θ of angle θ = h| ln h| satisfies

( P θ -z) -1 h -M ,
uniformly for h ∈ H small enough and z ∈ (3.14).

Proof of Lemma 3.5. This result is just an adaptation of Proposition 11.4 of [START_REF] Bony | Resonances for homoclinic trapped sets[END_REF]. We only give the changes which have to be made in the present setting, sending back the reader to Section 11.2 of [START_REF] Bony | Resonances for homoclinic trapped sets[END_REF] for the technical details. From the general arguments of Chapter 8 of [START_REF] Bony | Resonances for homoclinic trapped sets[END_REF], it is enough to show that any u = u(h) ∈ L 2 (R 2 ) and z = z(h) ∈ (3.14) with

(3.15) ( P θ -z)u = O(h ∞ ), u L 2 (R 2 ) = 1,
vanishes microlocally near each point of K(E 0 ). For k = 1, 2, 3, let u k ± be microlocal restrictions of u near ρ k ± , where ρ k -(resp. ρ k + ) is a point on γ k just "before" (resp. "after") (0, 0) (see Figure 6). As in [4, (11.23)], they are Lagrangian distributions u k -∈ I(Λ 1,k + , h -N ) and u k + ∈ I(Λ 0 + , h -N ), associated to the Lagrangian manifold Λ + just after (0, 0) (denoted Λ 0 + ) and after a turn along γ k (denoted Λ 1,k + ) for some N ∈ R. After an appropriate renormalization (see [4, (11.25

)]), the symbols a k -(x, h) ∈ S(h -N ) of u k -satisfy the relation (3.16) a k -(x, h) = h S(z,h)/λ 1 -1/2 3 ℓ=1 P k,ℓ (x, h)a ℓ -(x ℓ -, h) + S(h -N +ζ-δ ), near x k -= π x (ρ k -).
In this expression, the symbols P k,ℓ ∈ S(1) (resp. the constant ζ > 0) are independent of u (resp. δ, u) and

P k,ℓ (x k -, h) = Q k,ℓ (z, h). Compared with [4, (11.27)], Q is replaced by Q in P k,ℓ (x k -, h).
Indeed, no change has to be made for the propagation through the fixed point (0, 0) since W is supported away from V top (see [4, (11.29)]), but the usual transport equation near γ

k 2∇ϕ + • ∇a k -+ (∆ϕ + -iσ)a k -= O(h -N +1 ), with σ = (z -E 0 )/h is replaced by 2∇ϕ + • ∇a k -+ (∆ϕ + -iσ + ih δ W )a k -= O(h -N +1 ), giving on the curve γ k ∂ t a k -(x k (t)) + (∆ϕ + -iσ + ih δ W )a k -(x k (t)) = O(h -N +1 ),
and leading to the additional factor e -ih δ W (x k (t)) dt in the quantization matrix Q (see [4, (11.31)]). On the other hand, the remainder term O(h -N +ζ-δ ) in (3.16) comes from the fact that |h S/λ

= x k -, we get 1 -h S(z,h)/λ 1 -1/2 Q(z, h) a -(x -, h) = O(h -N +ζ-δ ),
where a -(x -, h) is a shortcut for the 3-vector with coefficients a

k -(x k -, h). From (3.10), it yields |a -(x -, h)| h -N +ζ-2δ ,
uniformly for z ∈ (3.14). Using again (3.16), we deduce a k -∈ S(h -N +ζ-3δ ) ⊂ S(h -N +ζ/2 ) for δ > 0 small enough. Thus, starting from a k -∈ S(h -N ), we have proved a k -∈ S(h -N +ζ/2 ). By a bootstrap argument (see the end of Chapter 9 of [START_REF] Bony | Resonances for homoclinic trapped sets[END_REF]), we obtain u = O(h ∞ ) microlocally near K(E 0 ) and the lemma follows.

To finish the proof of Lemma 3.4, it remains to show that P has a resonance near each pseudo-resonance. That is Lemma 3.6. For δ > 0 small enough, C, β > 0 and h ∈ H small enough, the operator P has at least one resonance in B(z, βh| ln h| -1 ) for any pseudo-resonance z ∈ (3.6).

Proof of Lemma 3.6. This result is equivalent to Proposition 11.6 of [START_REF] Bony | Resonances for homoclinic trapped sets[END_REF] in the present setting, and we only explain how to adapt its proof. If Lemma 3.6 did not hold, there would exist a sequence z = z(h) ∈ (3.6) of pseudo-resonances where h ∈ H goes to 0 such that (3.17) P has no resonance in D = B z, β h | ln h| .

We now construct a "test function". Let v be a WKB solution near x 1 -of

( P -z) v = 0 near x 1 -, v(x) = e iϕ 1,1 + (x)/h on |x| = |x 1 -| near x 1 -,
for z ∈ ∂D, where ϕ 1,1 + is a generating function of Λ 1,1 + . Note that P = P near x 1 -and that v can be chosen holomorphic with respect to z near D. After multiplication by a renormalization factor as in [4, (11.44)], this function is denoted v. Consider cut-off functions χ, ψ ∈ C ∞ 0 (T * R 2 ) such that χ = 1 near ρ 1 -and ψ = 1 near the part of the curve supp(∇χ)∩γ 1 before ρ 1 -. Then, we take as "test function" v = Op(ψ) P , Op(χ) v.

Let u ∈ L 2 (R 2 ) be the solution of (3.18) ( P θ -z)u = v, for z ∈ ∂D. From Lemmas 3.2 and 3.5, u is well-defined and polynomially bounded. Let u k be a microlocal restriction of u near ρ k -as before. Working as in Lemma 11.10 of [START_REF] Bony | Resonances for homoclinic trapped sets[END_REF], one can show that u k -∈ I(Λ , where a k denotes the symbol of v near x k -. In particular, a k (x, h) = 0 for k = 1 and a 1 (x 1 -, h) = 1. This relation is obtained using the proofs of (3.16) and Lemma 11.8 of [START_REF] Bony | Resonances for homoclinic trapped sets[END_REF]. To obtain a contradiction with (3.17 Applying (3.19) with x = x k -leads to

1 -h S(z,h)/λ 1 -1/2 Q(z, h) a -(x -, h) = a(x -, h) + O(h ζ-3δ ),
where c(x -, h) is a generic shortcut for the 3-vector with coefficients c k (x k -, h). It implies

a -(x -, h) = 1 -h S/λ 1 -1/2 Q -1 a(x -, h) + O(h ζ-4δ ) = 1 + h S/λ 1 -1/2 Q 1 -h S/λ 1 -1/2+δ WQ -1 a(x -, h) + O(1) + O(h ζ-4δ ).
from (3.10) and (3.13). We deduce

Wa -(x -, h) = W 1 -h S/λ 1 -1/2+δ WQ -1 a(x -, h) -h -δ a(x -, h) + h -δ 1 -h S/λ 1 -1/2+δ WQ -1 a(x -, h) + O(1) + O(h ζ-4δ ).

Inserting this expression in (3.21) and using (3.8) yield

Wb(x -, h) = h -δ 2iπ ∂D 1 -h S/λ 1 -1/2+δ WQ -1 a(x -, h) d z + O h | ln h| + O h 1+ζ-4δ | ln h| .
Note that a(x -, h) = t (1, 0, 0) is an explicit eigenvector of WQ associated to its non-zero eigenvalue -iwQ 1,1 (z, h) (see (3.4)). Thus, computing the integral as in [4, (11.67)], we get

Wb(x -, h) = iλ 1 h 1-δ | ln h| a(x -, h) + o h 1-δ | ln h| + O h | ln h| + O h 1+ζ-4δ
| ln h| .

Taking δ > 0 small enough and using that Wb ∈ S(h 1-2δ | ln h| -1 ), the previous asymptotic shows that Wb = 0 so that I = 0. On the other hand, since P has no resonance in D (see (3.17)), the function u defined by (3.18) is holomorphic in D and (3.20) gives I = 0. Eventually, we get a contradiction and the lemma follows.

The second point of Lemma 3.4 is a direct consequence of Lemmas 3.5 and 3.6.

Figure 1 .

 1 Figure 1. The spectral setting of Theorem 1.1.

Figure 2 .

 2 Figure 2. The resonances generated by (A) a well in the island, (B) a nondegenerate critical point and (C) a hyperbolic closed trajectory.

5 )Lemma 3 . 2 .

 532 is similar to Definition 4.2 of[START_REF] Bony | Resonances for homoclinic trapped sets[END_REF], we can adapt Proposition 4.3 and Lemma 11.3 of[START_REF] Bony | Resonances for homoclinic trapped sets[END_REF] in our case and obtain the following asymptotic of the pseudo-resonances. Let 0 < δ < 1/2, C, β > 0 and ε(h) be a function which goes to 0 as h → 0. Then, uniformly for τ ∈ [-C, C], the pseudo-resonances z of P in(3.6)

ii)

  In the domain (3.6), we have dist Res( P ), Res 0 ( P ) = o h | ln h| . As in Definition 4.4 of [4], the notation dist(A, B) ≤ ε in C means that ∀a ∈ A ∩ C, ∃b ∈ B, |a -b| ≤ ε, and ∀b ∈ B ∩ C, ∃a ∈ A, |a -b| ≤ ε.

  From the properties of u k -and |∂D| = 2πβh| ln h| -1 , we haveI ∈ I(Λ + , h 1-2δ | ln h| -1 ) microlocally near ρ k -, where its renormalized symbol b k (x, h) satisfies (3.21) b k (x, h) = 1 2iπ ∂D a k -(x, h) d z.

  1,k + , h -2δ ) with renormalized symbol a k -. Moreover, as in (3.16), we get

		3
	(3.19)	a k -(x, h) = h S(z,h)/λ 1 -1/2	P k,ℓ (x, h)a ℓ -(x ℓ -, h) + a k (x, h) + S(h ζ-3δ ),
		ℓ=1
	near x k -