Mathilde Bigot

Fabien Chauveau

Camille Amaz

Ralph Sinkus

Olivier Beuf

Simon A Lambert
email: simon.lambert@creatis.insa-lyon.fr

The apparent mechanical effect of isolated Aβ and α-Syn aggregates revealed by multifrequency MRE

Keywords: MR Elastography, fibril aggregates, amyloid, phantom, multi-frequency, power law exponent, dementia, fractal-like

Several biological processes are involved in dementia, and fibrillar aggregation of misshaped endogenous proteins appears to be an early hallmark of neurodegenerative disease. A recently developed means of studying neurodegenerative diseases is Magnetic Resonance Elastography (MRE), an imaging technique investigating the mechanical properties of tissues. Although mechanical changes associated with these diseases have been detected, the specific signal of fibrils has not yet been isolated in clinical or pre-clinical studies. The present study aims to exploit the fractal-like properties of fibrils to separate them from nonaggregated proteins using a multi-frequency MRE power law exponent in a phantom study.

Two types of fibril, α-Syn and Aβ, and a non-aggregated protein, Bovine Serum Albumin, used as control were incorporated in a dedicated non-dispersive agarose phantom. Elastography was performed at multiple frequencies between 400 and 1,200 Hz. After 3D-direct inversion, storage modulus G', phase angle ϕ, wave speed c and the power law exponent y were computed.

No significant changes in G' and ϕ were detected. Both α-Syn and Aβ inclusions showed significantly higher y values than control inclusions (p = 0.005) but did not differ between each other.

The present phantom study highlighted a specific biomechanical effect of α-Syn and Aβ aggregates which was better captured with the power law exponent derived from multi-frequency MRE than with single frequency derived parameters.

Graphical abstract

The apparent mechanical effect of isolated Aβ and α-Syn aggregates revealed by multi-frequency MRE M. Bigot, F. Chauveau, R. Sinkus, O. Beuf, S. A. Lambert Multi-frequency MR Elastography was used on agarose phantoms with small inclusions containing protein aggregates, early hallmarks of dementias. α-Syn and Aβ, the two types of studied aggregates had significantly high power law exponent compared with a non-aggregated protein used as control, while MRE parameters at a single frequency did not discriminate those aggregates from control. Their fractal-like structure based on β-sheets might be the source of their effect on power law exponent.

Introduction

Dementias are neurodegenerative diseases characterized by a decline in mental ability. Progression involves several biological processes such as impaired neurotransmission, inflammation, neurodegeneration and, from the early stages, fibrillar aggregation of misshaped endogenous proteins [START_REF] Seelaar | Clinical, genetic and pathological heterogeneity of frontotemporal dementia: a review[END_REF][START_REF] Braak | Staging of brain pathology related to sporadic Parkinson's disease[END_REF][START_REF] Nelson | Correlation of Alzheimer Disease Neuropathologic Changes With Cognitive Status: A Review of the Literature[END_REF] .

These fibrils appear to be hallmarks of dementia: in the wide span of neurodegenerative diseases, distinct endogenous proteins, such as amyloid-β, α-synuclein or Tau, share a common mechanism of pathological aggregation and spreading across brain areas [START_REF] Jucker | Self-propagation of pathogenic protein aggregates in neurodegenerative diseases[END_REF] . They also share a microstructural feature: all misshape to form β-sheets.

A recent means of studying neurodegenerative diseases is Magnetic Resonance Elastography (MRE), an imaging technique investigating mechanical properties of tissues [START_REF] Muthupillai | Magnetic resonance elastography by direct visualization of propagating acoustic strain waves[END_REF] . The most popular MRE method uses a monochromatic shear wave propagated through tissue and synchronized with oscillating motion encoding gradients (MEG). The wave fields acquired are typically processed using a viscoelastic wave equation, leading to mechanical property maps: i.e., information on elasticity and viscosity. MRE has been used in both preclinical and clinical studies on dementia. Recent clinical studies showed that brain stiffness varied between areas according to pathology [START_REF] Elsheikh | MR Elastography Demonstrates Unique Regional Brain Stiffness Patterns in Dementias[END_REF] . Stiffness was decreased in frontal and temporal lobes in frontotemporal dementia [START_REF] Huston | Magnetic resonance elastography of frontotemporal dementia[END_REF] , lentiform nucleus in Parkinson's disease 8 , and frontal, temporal and parietal lobes in Alzheimer's disease (AD) [START_REF] Murphy | Regional brain stiffness changes across the Alzheimer's disease spectrum[END_REF] . Small-animal studies confirmed decreased stiffness and further suggested an association with neuronal loss [START_REF] Majumdar | In vivo cerebral MR elastography in a mouse model of Alzheimer's disease: preliminary results[END_REF][START_REF] Klein | Enhanced Adult Neurogenesis Increases Brain Stiffness: In Vivo Magnetic Resonance Elastography in a Mouse Model of Dopamine Depletion[END_REF][START_REF] Hain | Dopaminergic Neurodegeneration in the Mouse Is Associated with Decrease of Viscoelasticity of Substantia Nigra Tissue[END_REF][START_REF] Munder | MR elastography detection of early viscoelastic response of the murine hippocampus to amyloid β accumulation and neuronal cell loss due to Alzheimer's disease[END_REF][START_REF] Murphy | Magnetic resonance elastography of the brain in a mouse model of Alzheimer's disease: initial results[END_REF][START_REF] Bigot | Magnetic Resonance Elastography of Rodent Brain[END_REF] . A biomechanical study in a multiple sclerosis mouse model showed that phase angle was sensitive to demyelination and remyelination [START_REF] Schregel | Demyelination reduces brain parenchymal stiffness quantified in vivo by magnetic resonance elastography[END_REF] .

Few studies have specifically focused on the influence of AD amyloid β (Aβ) fibrils on MRE parameters. In humans, a study comparing AD patients testing positive versus negative for presence of Aβ according on PET-CT suggested that amyloid load alone could not explain the changes in stiffness observed [START_REF] Murphy | Decreased Brain Stiffness in Alzheimer's Disease Determined by Magnetic Resonance Elastography[END_REF] . In a mouse model of AD, increased storage modulus (representing elasticity) correlated with intracellular fibril load during adolescence, attributed to change in cell structure. However, in older mice, no link was found between extracellular presence of Aβ and viscoelastic parameters [START_REF] Munder | MR elastography detection of early viscoelastic response of the murine hippocampus to amyloid β accumulation and neuronal cell loss due to Alzheimer's disease[END_REF] . Overall, most studies suggested that protein aggregation might not be the main factor driving brain stiffness changes. However, concurrent pathological phenomena such as inflammation, neurodegeneration or vascular impairment cannot be easily disentangled in living organisms.

In the last decade, various teams reported that multi-frequency MRE could reveal the underlying microstructure of matter at a macroscopic scale. Whatever the mechanical model, the frequency-dependence of the MRE-derived parameters could be fitted to a power law, and the exponent of this power law could be linked to the underlying microarchitecture. The exponent was reported to be reduced in chronic multiple sclerosis while remaining constant in relapsing-remitting multiple sclerosis. Altered cerebral structures in the chronic case were a possible explanation for this difference [START_REF] Wuerfel | MR-elastography reveals degradation of tissue integrity in multiple sclerosis[END_REF][START_REF] Streitberger | Brain viscoelasticity alteration in chronic-progressive multiple sclerosis[END_REF] . To establish the link between the power law exponent and the underlying microarchitecture, some studies used calibrated phantoms, made with a single type of microstructure such as paper strips [START_REF] Guo | Fractal network dimension and viscoelastic powerlaw behavior: II. An experimental study of structure-mimicking phantoms by magnetic resonance elastography[END_REF] , microvasculature [START_REF] Jugé | Microvasculature alters the dispersion properties of shear waves--a multi-frequency MR elastography study[END_REF] or polystyrene microsphere distribution [START_REF] Lambert | Bridging Three Orders of Magnitude: Multiple Scattered Waves Sense Fractal Microscopic Structures via Dispersion[END_REF] . In a polystyrene microsphere phantom, Lambert et al. used wave propagation to show that multiple wave scattering on microscopic rigid scatterers resulted macroscopically in power law frequency-dependence of wave speed. The exponent of the power law was linked to the apparent fractality of the medium. Importantly, for the geometrically induced power law to be established, the scattering objects need to be not too small compared to the shear wavelength; otherwise, apparent attenuation vanishes and only the material's constitutive dispersion matters.

Like microspheres and vessels, fibrils are more rigid than their surrounding glial cells [START_REF] Smith | Characterization of the nanoscale properties of individual amyloid fibrils[END_REF][START_REF] Lu | Viscoelastic properties of individual glial cells and neurons in the CNS[END_REF] and can induce multiple scattering of propagating shear waves. Moreover, their repetitive β-sheet structure over a large scale (from protein diameter to a few mm) confers fractal-like properties. This fractality in the empirical data can be observed only over a limited range [START_REF] Hamburger | Apparent fractality emerging from models of random distributions[END_REF][START_REF] Ciccotti | Pernicious effect of physical cutoffs in fractal analysis[END_REF] as long as it is sufficiently large: i.e., according to Lazzari et al., the upper limit should be "20 times larger than the primary particle" [START_REF] Lazzari | Fractal-like structures in colloid science[END_REF] . We propose here to exploit these properties, using multi-frequency MRE to study the effect of two types of fibril on shear wave scattering in a non-dispersive agarose medium. This phantom study seeks to improve understanding of the specific contribution of protein fibrils to mechanical changes observed in the brain.

Methods

Sample preparation

Dedicated phantoms with well-delineated fibril inclusions were produced to isolate the effect of fibrils using a purely elastic agarose base. In this lossless environment, any frequency dependence of speed mainly originates from multiple scattering of waves due to the presence of obstacles, such as fibrils [START_REF] Lambert | Bridging Three Orders of Magnitude: Multiple Scattered Waves Sense Fractal Microscopic Structures via Dispersion[END_REF][START_REF] Yamakoshi | Ultrasonic imaging of internal vibration of soft tissue under forced vibration[END_REF] .

Fibrils of human recombinant proteins (rPeptide), amyloid-β peptide 1-40 (Aβ) and full-length A53Tmutated α-synuclein (α-Syn) were prepared as previously described [START_REF] Verdurand | Differential effects of amyloid-beta 1-40 and 1-42 fibrils on 5-HT1A serotonin receptors in rat brain[END_REF][START_REF] Verdurand | In Silico, in Vitro, and in Vivo Evaluation of New Candidates for α-Synuclein PET Imaging[END_REF] . Aggregation was controlled with thioflavin-T fluorimetry and transmission electron microscopy after uranyl-acetate staining.

Phantoms were made with 1 % agarose powder (Sigma A9414) diluted in artificial cerebrospinal fluid [START_REF] Alzet® | Preparation of Artificial CSF[END_REF] , to have composition and mechanical properties similar to the brain. In each phantom, a maximum of four 3

x 3 x 3 mm 3 inclusions were made, using a mold (Fig. 1). Each inclusion was filled as follows: 21% of one type of aggregated fibrils at high concentration in PBS solution (initial monomer concentration, 200 µM), 5% thioflavin-T solution (0.01% in PBS) for fluorescence imaging, and 74% agarose. Agarose mass concentration was 1.35% in the 4 inclusions, to compensate the effect of dilution on mechanical properties. One inclusion was used as control, containing Bovine Serum Albumin (BSA, Carl Roth, ref.

3737), a non-aggregated protein, at a mass concentration of 4.3 mg/mL in PBS. Red Congo staining was used to visually locate this control inclusion and hence facilitate phantom positioning on MRI (Fig. 1). Altogether, 7 phantoms were produced, with 11 α-Syn inclusions, 10 Aβ inclusions and 7 BSA inclusions.

Phantoms were directly molded in their holder and covered by paraffin wax (DTM 133460) to prevent dehydration and bulk motion during actuation.

Macrofluorescence

To check that fibrils did not migrate out of the inclusions, fluorescence microscopy (Leica Macrofluo Z16 APA A) was performed on the whole phantom after MRE imaging. Thioflavin-T, which binds to fibrils and emits green fluorescence, was used as fluorochrome (Fig. 2).

MRE set-up

Experimental set-up

The sample holder was slid inside a homemade 3D-printed Helmholtz coil (Fig. 3) designed to maximize uniformity and filling factor (Fig. S1) and to facilitate vertical handling of small samples.

The volume coil was simply taped to a dedicated elastography bench, so that a needle pierced the sample.

The needle was fixed on a Plexiglas flap, linked to an air-cooled piezoelectric driver (APA100M Cedrat Technologies) via a star-shaped plastic rod. Actuator frequency and amplitude were set by a function generator (33220A Agilent Technologies) through an amplifier (LA75 Cedrat Technologies).

Data acquisition

All acquisitions were performed on a pre-clinical 4.7 T MRI scanner (BioSpec Bruker system, Ettlingen, Germany). The MRE sequence was derived from a RARE sequence [START_REF] Van Reeth | Constant gradient elastography with optimal control RF pulses[END_REF] , and was performed sequentially at 600-800-900-1,000-400-1,200-600 Hz. Acquisition at 600 Hz was performed at the beginning and repeated at the end of elastography to check that the sample's mechanical properties did not alter over time (MRE exam duration, 1h45). Sinusoidal Motion Encoding Gradients (MEG) with 180 mT.m -1 amplitude were synchronized with the piezoelectric driver through a trigger sent by the MRI console. To establish steadystate regime before phase encoding, 40 wave periods were performed ahead of MEG. For each frequency, MRE acquisition was repeated 3 times, with change in MEG orientation (slice, phase, read) to acquire the 3 directions of motion. Number of MEG periods and echo time were adjusted according to frequency so as to maximize SNR (SNR varying between 23 at 400 Hz and 17 at 1,000 Hz). To minimize acquisition time, a RARE factor of 2 was used.

An MSME (multi-slice multi-echo) sequence was acquired to locate inclusions on the computed T2-maps and facilitate ROI positioning, as explained below.

Further details on both MRE and MSME sequences are given in Table 1.

Post-processing

Data processing was performed as described by Sinkus et al. [START_REF] Sinkus | Imaging anisotropic and viscous properties of breast tissue by magnetic resonance-elastography[END_REF] . Raw phase images were unwrapped, filtered with a 3D-Gaussian filter (kernel = 2 x 2 x 2, σ = 0.39 x 0.39 x 1 mm 3 ) and Fourier transformed into 3D displacement fields. 3D curl inversion was applied to the fields to calculate the complex shear modulus G* and the complex wave number k at each acquired frequency.

Image analysis

ROIs were manually drawn only on the central slice, excluding inclusion borders, to avoid bias due to reconstruction artifacts or slice positioning error (Fig. 2b). In-plane, inclusions were located on T2-maps and reported on MRE acquisitions. Only central inclusion areas were included, to avoid inhomogeneity at the corners of the inclusions and edge reconstruction effects, leading to ROI sizes ranging between 10 and 38 pixels. One ROI close in size was also drawn in simple agarose for each sample. only valid when ϕ is independent of frequency [START_REF] Holm | A unifying fractional wave equation for compressional and shear waves[END_REF][START_REF] Hirsch | Magnetic Resonance Elastography: Physical Background and Medical Applications[END_REF][START_REF] Hiscox | Magnetic resonance elastography (MRE) of the human brain: technique, findings and clinical applications[END_REF] .

Statistics

All results are reported as mean ± standard deviations. Statistical tests used OriginPro 2016 and R statistical software version 3.3.3. Only the first of the two 600 Hz acquisitions was taken into account, so as to have the same number of data points as for the other frequencies. The significance threshold α was set at 0.05/3 = 0.017 to correct for multiple comparisons between the three groups (Aβ, α-Syn, BSA).

First, differences in the microstructure-sensitive parameter y were assessed on non-parametric analysis of variance (Kruskal-Wallis test), followed by Kolmogorov-Smirnov tests.

Second, G', Phi, c were evaluated by using linear mixed-effects models that included groups (α-Syn, Aβ, BSA) and frequency as fixed-effect and random effect grouped by phantom used. Results are reported as effect size β, confidence interval CI, and p-value. For relevant parameters identified by the linear mixed model, exploratory comparisons were also performed at each frequency to simulate the fastest and most commonly performed single-frequency acquisitions.

Finally, to confirm the validity of the mathematical relation between ϕ and y, an additional Kruskal-Wallis test was performed to evaluate the frequency dependence of ϕ.

Results

Macrofluorescence results

Macrofluorescence revealed no fibril leakage out of the inclusion areas (Fig. 2b). However, the fluorescence distribution was non-homogeneous for some inclusions.

Differences in elastic properties between fibrils and BSA

Mean relative difference in kr measured in inclusions between the first and last 600-Hz acquisitions was 3±2%. A homogeneous 1% agarose phantom without inclusion was also characterized at 600 Hz over a 23hour period (15 measures) to demonstrate the mechanical stability of agarose over time: the standard deviation of kr was only 1% of the mean value.

Linear mixed-effects models showed no group effect on wave speed c (Fig. 5a), or on storage modulus (Fig. 5b), even though G' and c tend to be smaller in the fibril groups at low frequency. Datasets and results of linear mixed-effects models are provided as supplemental Excel file.[MB1]

Difference in structural parameters y and ϕ between fibrils and BSA

Both α-Syn and Aβ inclusions showed significantly higher values of exponent y than BSA inclusions (yα-Syn = 0.22±0.08; yAβ = 0.26±0.15; yBSA = 0.11±0.05 -p = 0.005, Fig. 6a), but did not differ between each other. Of note, no difference was observed between BSA inclusions and simple agarose outside the inclusions (p = 0.963; complete results extracted from agarose ROIs outside the inclusions are reported in Supplementary Table S2).

Parameter ϕ was not significantly different between both fibril groups and BSA. A trend toward higher values in fibril groups is however noticeable (Fig. 6b).

[MB2]Frequency had an impact on ϕ distribution (Kruskal-Wallis test on pooled values from all frequencies, p < 0.0003).

Discussion

Detection of a fibril-specific effect

Fibrils are macromolecular protein assemblies of β-sheet stacks and are considered to be an early hallmark of neurodegenerative diseases [START_REF] Seelaar | Clinical, genetic and pathological heterogeneity of frontotemporal dementia: a review[END_REF][START_REF] Braak | Staging of brain pathology related to sporadic Parkinson's disease[END_REF][START_REF] Nelson | Correlation of Alzheimer Disease Neuropathologic Changes With Cognitive Status: A Review of the Literature[END_REF] . Their specific effect on MRE parameters could not easily be inferred from previous studies of transgenic mice and patients [START_REF] Munder | MR elastography detection of early viscoelastic response of the murine hippocampus to amyloid β accumulation and neuronal cell loss due to Alzheimer's disease[END_REF][START_REF] Murphy | Decreased Brain Stiffness in Alzheimer's Disease Determined by Magnetic Resonance Elastography[END_REF] . In the present, we simplified the fibril environment to isolate a potential effect on various MRE-derived parameters. More specifically, we used their fractal-like distribution [START_REF] Lazzari | Fractal-like structures in colloid science[END_REF] as a source of multiple scattering of shear waves that leaves its footprint in the exponent of the frequency power law. Surprisingly, MRI becomes sensitive to nanometric structures because of the spatial extent induced by apparent fractality. Based upon stiffness itself, fibrils are difficult to distinguish: no difference was found between the two fibril groups versus BSA using single-frequency MRE parameters. Therefore y is really an added value to characterize the underlying microarchitecture: e.g., here for fibril detection.

BSA, a non-aggregated protein, did not modify the dispersion properties of agarose, while the two fibril types were able to significantly increase the power law exponent y, confirming that fibrils are a dispersive obstacle for shear waves. Aβ and α-Syn are made of different proteins, and their aggregation leads to fibrils harboring different widths and lengths, as visible on transmission electron microscopy images (provided as Supplementary Figure S3). No significant difference in y was found between the two types of fibril, suggesting that their common architectural base, β-sheet stacks, behaves as a fractal-like object, leading to the same frequency-dependence of mechanical properties.

Based on our results, predicting a potential overruling of fibril effect by other modifications in the case of neurodegenerative diseases remains delicate, since the sensitivity of detection in this study was difficult to estimate: firstly because the exact concentration of fibrils was unknown (monomer concentration was 200 µM, but the number of monomers per fibril unit could not be easily estimated), and secondly because of fluorescence inhomogeneities, probably deriving from the difficulty of mixing fibrils with agarose. Such a multi-frequency MRE experiment could be used on ex vivo and in vivo brain to screen for an effect of fibrils on mechanical properties. However, demonstrating a specific effect of fibrils among all the other microstructural changes involved in neurodegenerative disease and in in vivo conditions might require full brain characterization at the microscopic scale using quantitative and 3D histological methods (neuronal loss, glial activation, etc.) [START_REF] Coelho | Local volume fraction distributions of axons, astrocytes, and myelin in deep subcortical white matter[END_REF] .

Relevance of multi-frequency study

Multi-frequency MRE allowed an effect of isolated fibrils on mechanical parameters to be shown for the first time. The power law exponent was significantly altered by fibril presence, unlike the storage modulus, wave speed and phase angle.

A tendency for G' to be smaller in the fibril groups at low frequency was noted. We think that this observation cannot be explained by a change in microstructure, and rather arise from an interference with the agarose polymerization process.[MB3] In addition, the changes in wave speed and elasticity appear to be very similar on graphs, possibly because of similar absorption (i.e. imaginary part of k) in both BSA and fibrils [START_REF] Sinkus | Rheological determinants for simultaneous staging of hepatic fibrosis and inflammation in patients with chronic liver disease[END_REF] . The absence of a fibril effect at high frequencies on viscoelasticity parameters corroborates preclinical [START_REF] Munder | MR elastography detection of early viscoelastic response of the murine hippocampus to amyloid β accumulation and neuronal cell loss due to Alzheimer's disease[END_REF] and human 17 findings.

Like the power law exponent, ϕ globally increased, however non significantly, in the presence of fibrils.

Since both parameters were reported to provide information on microstructure [START_REF] Schregel | Demyelination reduces brain parenchymal stiffness quantified in vivo by magnetic resonance elastography[END_REF][START_REF] Jugé | Microvasculature alters the dispersion properties of shear waves--a multi-frequency MR elastography study[END_REF] , this similarity was expected. Some limitations in the use of ϕ have to be pointed out: the dependence of ϕ on G'' value was problematic, since the phantoms were almost elastic: i.e., G'' was small compared to G'. ϕ values might thus have been affected by noise. As underlined by Dittman et al., ϕ estimation is not robust to noise, worsening this effect [START_REF] Dittmann | In vivo wideband multifrequency MR elastography of the human brain and liver[END_REF] . This explains why the difference between fibrils and BSA was not significant when ϕ was considered only at a given frequency.

Another aspect ruling out ϕ as a reliable dispersion parameter is the frequency-dependence demonstrated here, which makes ϕ mathematically unrelated to y [START_REF] Hiscox | Magnetic resonance elastography (MRE) of the human brain: technique, findings and clinical applications[END_REF] . As expected, the data exhibited a frequency power law, expressed by y, which stems from a fit of the wave speed to the six frequencies. If data additionally followed the rheological model of a spring pot, slope y could be calculated from phase angle at each individual frequency. Our data do not support this hypothesis, as shown by the fact that ϕ did not show the same separation for different material constructs as y. A theoretical model matching this observation is when both the temporal and the spatial operator in equation 𝜑 = 𝑡𝑎𝑛 -1 ( 𝐺" 𝐺 ′ ) are of fractional order. This yields one parameter modifying the slope of the frequency behavior and an additional parameter impacting the ratio of G'' to G' [START_REF] Valtorta | Measurement of rheological properties of soft biological tissue with a novel torsional resonator device[END_REF] .

It seems difficult to dispense with multi-frequency MRE and to extract the power law parameter for fibril characterization. However, multi-frequency MRE is long to perform, ex or in vivo. It could benefit from sequences with reduced acquisition time [START_REF] Klatt | Sample interval modulation for the simultaneous acquisition of displacement vector data in magnetic resonance elastography: theory and application[END_REF][START_REF] Kearney | Simultaneous 3D MR elastography of the in vivo mouse brain[END_REF][START_REF] Guenthner | Ristretto MRE: A generalized multi-shot GRE-MRE sequence[END_REF] . We used six frequencies with a rather large range to enhance fitting quality, but the number of acquisitions could be reduced according to Clauset et al., since the power law model is well established [START_REF] Clauset | Power-Law Distributions in Empirical Data[END_REF] .

Perspectives for preclinical follow-up

In this study performed on phantoms, the presence of fibrils was the only factor affecting y. Under in vivo conditions, additional tissue complexity could mask the fibril effect that we measured in a simple phantom, and neurodegenerative diseases moreover involve several additional microscopic changes in brain.

Microstructural changes such as demyelination [START_REF] Schregel | Demyelination reduces brain parenchymal stiffness quantified in vivo by magnetic resonance elastography[END_REF] or age-related decrease in neuron density [START_REF] Sack | The impact of aging and gender on brain viscoelasticity[END_REF] have also been shown to induce mechanical changes. Such variability makes the relationship between MRE measurements and biological processes even more difficult to elucidate. In the case of chronic multiple sclerosis, the specific structure triggering a modification of power law exponent is still unclear. Similarly, it is difficult to link phase angle variations to a specific physiological or pathological change in the brain: ϕ was found unchanged by an increase in cerebral blood flow [START_REF] Chatelin | Cannabinoid receptor activation in the juvenile rat brain results in rapid biomechanical alterations: Neurovascular mechanism as a putative confounding factor[END_REF] , and decrease in neuron number [START_REF] Freimann | MR elastography in a murine stroke model reveals correlation of macroscopic viscoelastic properties of the brain with neuronal density[END_REF] and inflammation [START_REF] Millward | Tissue structure and inflammatory processes shape viscoelastic properties of the mouse brain[END_REF] , while changes in ϕ were reported in rodents for demyelination [START_REF] Schregel | Demyelination reduces brain parenchymal stiffness quantified in vivo by magnetic resonance elastography[END_REF] and in the hippocampal area of AD patients, regardless of amyloid status [START_REF] Gerischer | High resolution MR elastography of the hippocampus as a novel biomarker for Alzheimer's disease?[END_REF] . ϕ also correlated with memory performance in healthy volunteers in the hippocampal area [START_REF] Schwarb | Medial temporal lobe viscoelasticity and relational memory performance[END_REF] . Finally, the heterogeneity of the structural composition in the brain may make the fibril effect on y or ϕ brain-area dependent.

Conclusion

In conclusion, the power law exponent derived from multi-frequency MRE data showed significant changes due to the presence of α-Syn and Aβ aggregates immersed in a homogeneous non-viscous medium. This phantom study isolated a specific effect of fibrils, independently of other changes associated with their accumulation in the brain.

  The mean and standard deviation of the real and imaginary parts of G* (storage modulus G' standing for elasticity and loss modulus G" standing for viscosity) and real part of k (kr) inside the ROIs were computed. Mean phase angle 𝜑 = 𝑡𝑎𝑛 -1 ( 𝐺" 𝐺 ′ ) and wave speed 𝑐 = 2𝜋𝑓 𝑘 𝑟 were derived. The relation between wave speed c and excitation frequency f was fitted on a power law: 𝑐 = 𝑎𝑓 𝑦 . Unwrapped phase and wave speed at 400 and 1,200 Hz are shown as an example Figure 4. ϕ and y are two variables shown to be related to underlying microstructure in previous studies. The relation ϕ = πy can be deduced from literature, but is

FiguresFigure 1 :

 1 Figures

Figure 2 :

 2 Figure 2: a) Bright field image of a phantom with a 1x magnification. A needle (presented in Figure 3) mark is shown by a white arrow. b) Macrofluorescence image of an α-Syn inclusion with a 5X magnification. Green fluorescence, appearing here in white, indicates binding of thioflavin-T to fibrils. In this inclusion, the edges of the inclusion are clearly visible, indicating that no leakage of fibrils has happened. An ROI is drawn, avoiding the edges of the inclusion (white dashed square).

Figure 3 :

 3 Figure 3: Elastography bench dedicated to small samples: 1) a piezoelectric driver for wave generation (APA100M Cedrat Technologies), cooled by 2) an air intake; 3) a star-shaped rod transmitted movement to 4) a cactus needle, in direct contact with the sample; 5) a 36 mm diameter Helmholtz coil and 6) associated coupling loop used for signal acquisition and transmission. 7) The sample holder was slid inside the coil.

Figure 4 :

 4 Figure 4: MRE unwrapped phase and wave speed obtained at 400 and 1,200 Hz (motion encoding direction orthogonal to the slice). Aβ, α-Syn and BSA inclusions and agarose ROIs are represented in green, red, blue and black, respectively.

Figure 5 :

 5 Figure 5: Boxplots of a) wave speed c and b) storage modulus G' computed from ROI included in inclusions containing α-Syn, Aβ fibrils and BSA for frequencies varying between 400 and 1,200 Hz. For each frequency, 11, 10 and 7 values are represented for α-Syn, Aβ40 and BSA, respectively. Each box of the plot contains values from the 25 th to the 75 th percentile. Maximum and minimum values are symbolized by crosses, mean value by a square and median by a line in the box.

Figure 6 :

 6 Figure 6: a) Boxplots of exponent y for α-Syn, Aβ and BSA. 11, 10 and 7 values are represented as dots for α-Syn, Aβ and BSA, respectively. b) Boxplots of phase angle ϕ for α-Syn, Aβ and BSA inclusions (11, 10 and 7 values for each box plot, for α-Syn, Aβ and BSA, respectively). Each box of the plot contains values from the 25 th to the 75 th percentile. Maximum and minimum values are symbolized by crosses, mean value by a square and median by a line in the box.

Table 1 :

 1 MR acquisition parameters. MRE acquisitions were repeated 3 times at each frequency, with MEGs in slice, phase and read directions (NA, not available).

		MSME	MR elastography
	Field of view	25 x 25 mm²	25 x 25 mm²
	Matrix	128 x 128	64 x 64
	Spatial resolution	195 x 195 µm²	391 x 391 µm²
	Slice thickness	1 mm	1 mm
	Number of slices	7	7
	Number of phase offset images	NA	8
	Mechanical excitation frequency (Hz) NA	400 600 800 900 1000 1200
	Number of MEG periods	NA	1	2	8	9	10	8
	Echo time (ms)	12 -24 -36 -48 -60	15	15 28 30 28	21
		-72
	Repetition time (ms)	5033	1200 1000
	Acquisition time	8'3"	5'3" 4'16"

Acknowledgments

This work was supported by the LABEX PRIMES (ANR-11-LABX-0063) of Université de Lyon, (France), within the Programme Investissements d'Avenir (ANR-11-IDEX-0007) and CNRS PEPS Balanced. It benefited from experimental expertise acquired on the PILoT platform, a member of the France Life Imaging network (ANR-11-INBS-0006) and on the LYMIC -PLATIM platform. We thank the RHU MARVELOUS (ANR-16-RHUS-0009) for providing a framework where statistical analysis was performed. We also thank Kevin Tse Ve Koon and Pauline Lefebvre for MRE sequence implementation and fruitful discussions.

Abbreviations used

Supplementary materials

Comparison of a volume coil and a handmade Helmholtz coil

BSA and agarose

Variable Frequency range Agarose BSA G' 400-1200 Hz 5.7 ± 0.8 -6.7 ± 0.6 kPa 5.2 ± 1.4 -6.6 ± 1.0 kPa c 400-1200 Hz 2.3 ± 0.1 -2.6 ± 0.1 m.s -1 2.4 ± 0.3 -2.6 ± 0.2 m.s -1 ϕ 400-1200 Hz 0.09 ± 0.02 -0.20 ± 0.05 rad 0.17 ± 0.06 -0.35 ± 0.10 rad y / 0.12 ± 0.02 0.11 ± 0.05 Table S2: Range of mean ± standard deviation obtained for G', c and ϕ for all investigated frequencies and for y in the agarose ROI, far from all inclusions and in the BSA inclusion, used as control for the evaluation of a fibril effect.

Transmission electronic microscopy of α-Syn and Aβ fibrils


M Égane Bournissou

QUADRATIC BEHAVIORS OF THE 1D LINEAR SCHR ÖDINGER EQUATION WITH BILINEAR CONTROL

Keywords: 1991 Mathematics Subject Classification. Primary: 93B05, 93C20; Secondary: 81Q93 Exact controllability, Schrödinger equation, bilinear control, obstruction, power series expansion

We consider a 1D linear Schrödinger equation, on a bounded interval, with Dirichlet boundary conditions and bilinear control. We study its controllability around the ground state when the linearized system is not controllable and wonder whether the quadratic term can help to recover the directions lost at the first order. More precisely, in this paper, we formulate assumptions under which the quadratic term induces a drift which prevents the small-time local controllability (STLC) of the system in appropriate spaces.

For finite-dimensional systems, quadratic terms induce coercive drifts in the dynamic, quantified by integer negative Sobolev norms, along explicit Lie brackets which prevent STLC.

In the context of the bilinear Schrödinger equation, the first drift, quantified by the H -1 -norm of the control, was already observed in [8] and used to deny STLC with controls small in L ∞ . In this article, we improve this result by denying STLC with controls small in W -1,∞ .

Furthermore, for any positive integer n, we formulate assumptions under which one may observe a quadratic drift quantified by the H -n -norm of the control and we use it to deny STLC in suitable spaces.

1. Introduction 1.1. Description of the control system. Let T > 0. In this paper, we consider the 1D linear Schrödinger equation given by, i∂ t ψ(t, x) = -∂ 2

x ψ(t, x) -u(t)µ(x)ψ(t, x), (t, x) ∈ (0, T ) × (0, 1), ψ(t, 0) = ψ(t, 1) = 0, t ∈ (0, T ).

(

) 1
Such an equation arises in quantum physics to describe a quantum particle stuck in an infinite potential well and subjected to a uniform electric field whose amplitude is given by u(t). The function µ : (0, 1) → R depicts the dipolar moment of the particle. This equation is a bilinear control system where the state is the wave function ψ such that for all time ψ(t) L 2 (0,1) = 1 and u : (0, T ) → R denotes a scalar control.

1.2. Functional setting. Unless otherwise specified, in space, we will work with complex valued functions. The Lebesgue space L 2 (0, 1) is equipped with the hermitian scalar product given by f, g := 1 0 f (x)g(x)dx, f, g ∈ L 2 (0, 1).

Let S be the unit-sphere of L 2 (0, 1). The operator A is defined by Dom(A) := H 2 (0, 1) ∩ H 1 0 (0, 1), Aϕ := -

d 2 ϕ dx 2 .
Its eigenvalues and eigenvectors are respectively given by ∀j ∈ N * , λ j := (jπ) 2 and ϕ j := √ 2 sin(jπ•).

The family of eigenvectors (ϕ j ) j∈N * is an orthonormal basis of L 2 (0, 1). We denote by, ∀j ∈ N * , ψ j (t, x) := ϕ j (x)e -iλ j t , ∀(t, x) ∈ R × [0, 1],

the solutions of the Schrödinger equation [START_REF] Ball | Controllability for distributed bilinear systems[END_REF] with u ≡ 0 and initial data ϕ j at time t = 0. When k = 1, ψ 1 is called the ground state. We also introduce the normed spaces linked to the operator A, given by, for all s ≥ 0, .

H s (
Finally, for T > 0 and u ∈ L 1 (0, T ), the family (u n ) n∈N of the iterated primitives of u is defined by induction as, u 0 := u and ∀n ∈ N, u n+1 (t) := t 0 u n (τ )dτ, t ∈ [0, T ].

1.3. Hypotheses on the dipolar moment. Let us make precise the assumptions on the dipolar moment µ we shall consider in the following. In all this paper, we will consider a dipolar moment µ at least in H 3 ((0, 1), R). Let n ∈ N * . The functional setting under which the n-th quadratic obstruction occurs in the vicinity of the ground state for the Schrödinger equation ( 1) is the following. (H1) K There exists K ∈ N * such that µϕ 1 , ϕ K = 0. (H2) K,n The sequence (c j := µϕ 1 , ϕ j µϕ K , ϕ j ) j∈N * satisfies

+∞ j=1 j 4n |c j | < +∞, (2)
A p K := (-1) p-1 +∞ j=1 λ j -λ 1 + λ K 2 (λ K -λ j ) p-1 (λ j -λ 1 ) p-1 c j = 0, ∀p ∈ {1, . . . , n -1},

A n K := (-1) n-1 +∞ j=1 λ j - λ 1 + λ K 2 (λ K -λ j ) n-1 (λ j -λ 1 ) n-1 c j = 0. (3)
Notice that by [START_REF] Beauchard | Local controllability of a 1-D Schrödinger equation[END_REF], all the series considered in (3) and (4) converge absolutely. Moreover, the existence of a function µ satisfying (H1) K -(H2) K,n is proved in Appendix A.

Remark 1.1. For smooth vector fields X, Y in C ∞ (R p , R p ), the Lie bracket [X, Y ] is defined as the following smooth vector field: [X, Y ](x) := X (x)Y (x) -Y (x)X(x). We also define by induction ad 0 X (Y ) := Y and ∀k ∈ N, ad k+1 X (Y ) := [X, ad k X (Y )]. With f 0 := -iA and f 1 := iµ, (3) and (4) can be written formally in terms of Lie brackets as ∀p = 1, . . . , n-1, [ad p-1 f 0 (f 1 ), ad p f 0 (f 1 )]ϕ 1 , ϕ K = 0 and [ad n-1 f 0 (f 1 ), ad n f 0 (f 1 )]ϕ 1 , ϕ K = 0. These Lie brackets are exactly those along which the quadratic order adds a drift, denying W 2n-3,∞ -STLC (see Definition 1.2) for finite-dimensional systems ẋ = f 0 (x) + uf 1 (x) in [START_REF] Beauchard | Quadratic obstructions to small-time local controllability for scalar-input systems[END_REF]Theorem 3]. The previous formal computations can become rigorous for instance when, for some p ∈ N * , µ is in H 2p (0, 1) with its p -1 first odd derivatives vanishing at x = 0 and 1. Indeed, in this case, we get

A p K = (-1) p-1 2 [ad p-1 A (µ), ad p A (µ)]ϕ 1 , ϕ K ,
where the iterated Lie brackets are well-defined (for all k ∈ N * , to compute ad k A (µ), one needs to check that ad k-1 A (µ) is in Dom A) and denote commutators of operators (see Proposition A.2 for more details).

At a heuristic level, assumptions (H1) K -(H2) K,n correspond to the fact that, in the asymptotic of small controls, the solution ψ of the Schrödinger equation [START_REF] Ball | Controllability for distributed bilinear systems[END_REF] satisfies

ψ(T ), ϕ K e -iλ 1 T ≈ -A n K u n 2
L 2 (0,T ) . 1.4. Main result. First, we define the notion of small-time local controllability (STLC) used in this paper, stressing the regularity imposed on the control as it plays a crucial role in the validity of controllability results. Definition 1.2. Let (E T , • E T ) be a family of normed vector spaces of real functions defined on [0, T ] for T > 0. The system ( 1) is said to be E-STLC around the ground state if there exists s ∈ N such that for every T > 0, for every ε > 0, there exists δ > 0 such that for every

ψ f ∈ S with ψ f -ψ 1 (T ) H s (0) (0,1) < δ, there exists u ∈ L 2 ((0, T ), R) ∩ E T with u E T < ε such that the solution ψ of (1) associated to the initial condition ϕ 1 satisfies ψ(T ) = ψ f .
Often, to prove STLC, one can use the linear test: if the linearized system is controllable, one can hope to prove the STLC of the nonlinear system through a fixed-point theorem. When the linearized system misses some directions, one can wonder whether the following term in the expansion of the solution can help to recover controllability along the lost direction.

For the Schrödinger equation (1), since [START_REF] Beauchard | Local controllability of 1D Schrödinger equations with bilinear control and minimal time[END_REF], it is known that when one of the coefficients µϕ 1 , ϕ j vanishes, one can give explicit impossible motions in small time, due to a drift quantified by the H -1 -norm of the control which denies L ∞ -STLC. However, Beauchard and Marbach highlighted in [START_REF] Beauchard | Quadratic obstructions to small-time local controllability for scalar-input systems[END_REF] that for control-affine systems in finite dimension, to observe the first drift, the optimal norm for the smallness assumption on the control is the W -1,∞ one. Hence, the goal was to improve the result of [START_REF] Beauchard | Local controllability of 1D Schrödinger equations with bilinear control and minimal time[END_REF] and show that the first drift can be used to deny W -1,∞ -STLC instead. Moreover, we give assumptions on µ under which one may observe a quadratic drift quantified by any H -n -norm of the control, allowing to deny H 2n-3 -STLC.

Theorem 1.3. Let (K, n) ∈ N * 2 . Assume that µ satisfies (H1) K -(H2) K,n . If n ≥ 2 (resp. n = 1), then the Schrödinger equation (1) is not H 2n-3 -STLC (resp. W -1,∞ -STLC).
More precisely, there exists C > 0 such that for every A ∈ (0,

|A n K | 4
), there exists T * > 0 such that for every T ∈ (0, T * ), there exists η > 0 such that for every u ∈ H 2n-3 (0, T )

(resp. u ∈ L 2 (0, T )) with u H 2n-3 (0,T ) ≤ η (resp. u 1 L ∞ (0,T ) ≤ η), the solution ψ of (1) with initial data ϕ 1 satisfies -sign(A n K ) ψ(T ), ϕ K e -iλ 1 T ≥ A u n 2 L 2 (0,T ) -C (ψ -ψ 1 )(T ) 2 L 2 (0,1) . (5)
Remark 1.4. Theorem 1.3 entails that, under hypotheses (H1) K -(H2) K,n , there exists R > 0 such that the following targets cannot be reached by the solution of (1),

∀δ ∈ (0, R), ψ f = 1 -δ 2 ϕ 1 + i sign(A n K )δϕ K e -iλ 1 T . Indeed, if there exists a control u such that ψ(T ) = ψ f , then (5) leads to -δ ≥ A u n 2 L 2 -2Cδ 2 ,
which doesn't hold for small δ.

Example 1.5. For µ(x) = x -1 2 , one can compute that µϕ 1 , ϕ 1 = 0 and A 1 1 = 1. Hence, µ satisfies (H1) 1 -(H2) 1,1 and Theorem 1.3 applies. Thus, for the negative STLC result presented by Coron in [START_REF] Coron | On the small-time local controllability of a quantum particle in a moving onedimensional infinite square potential well[END_REF], for a quantum particle in a moving one-dimensional infinite square potential well, the control does not need to be small in L ∞ but rather small in W -1,∞ .

The main ideas of this paper are presented on a toy-model in finite dimension in Section 2. Then, we recall the well-posedness of the Schrödinger equation in Section 3. In Section 4, error estimates on the expansion of the solution are given. The coercivity of the second-order term of the expansion is studied in Section 5. Finally, Theorem 1.3 is proved in Section 6.

State of the art.

Local exact controllability results. First, Turinici in [START_REF] Turinici | On the controllability of bilinear quantum systems[END_REF] deduced a negative control result for the Schrödinger equation ( 1) from the work [START_REF] Ball | Controllability for distributed bilinear systems[END_REF] of Ball, Marsden and Slemrod on the controllability of bilinear control systems. This result was later completed by Boussaid, Caponigro and Chambrion in [START_REF] Boussaïd | Regular propagators of bilinear quantum systems[END_REF].

However, these negative results are due to an 'unfortunate' choice of functional setting in which controllability doesn't hold. It may not be due to a deep non-controllability. Thus, exact local controllability results for 1D models have been proven under a more appropriate functional setting by Beauchard in [START_REF] Beauchard | Local controllability of a 1-D Schrödinger equation[END_REF][START_REF] Beauchard | Controllability of a quantum particle in a 1D variable domain[END_REF], later improved by Beauchard and Laurent in [START_REF] Beauchard | Local controllability of 1D linear and nonlinear Schrödinger equations with bilinear control[END_REF].

Morancey and Nersesyan also proved the controllability of one Schrödinger equation with a polarizability term [START_REF] Morancey | Global exact controllability of 1D Schrödinger equations with a polarizability term[END_REF] and of a finite number of equations with one control [START_REF] Morancey | Simultaneous local exact controllability of 1D bilinear Schrödinger equations[END_REF][START_REF] Morancey | Simultaneous global exact controllability of an arbitrary number of 1D bilinear Schrödinger equations[END_REF]. Puel [START_REF] Puel | Local exact bilinear control of the Schrödinger equation[END_REF] also proved the local exact controllability for a Schrödinger equation, in a bounded regular domain of dimension N ≤ 3, in a neighborhood of an eigenfunction corresponding to a simple eigenvalue, for controls u = u(t, x).

Global approximate results. Chambrion, Mason, Sigalotti and Boscain proved in [START_REF] Chambrion | Controllability of the discretespectrum Schrödinger equation driven by an external field[END_REF] the approximate controllability of Schrödinger in L 2 , using Galerkin approximations, under hypotheses later refined by Boscain, Caponigro, Chambrion and Sigalotti in [START_REF] Boscain | A weak spectral condition for the controllability of the bilinear Schrödinger equation with application to the control of a rotating planar molecule[END_REF]. Similar results were proved for one [START_REF] Boussaïd | Regular propagators of bilinear quantum systems[END_REF] or a finite number of equations [START_REF] Boscain | Multi-input Schrödinger equation: controllability, tracking, and application to the quantum angular momentum[END_REF] in more regular Sobolev spaces. Such results can also stem from exact controllability results in infinite time [START_REF] Nersesyan | Global exact controllability in infinite time of Schrödinger equation: multidimensional case[END_REF] or from variational arguments [START_REF] Vahagn Nersesyan | Growth of Sobolev norms and controllability of the Schrödinger equation[END_REF].

About quadratic obstructions. When the linearized system is not controllable, the strategy of performing a power series expansion of the solution, presented in [START_REF] Coron | Control and nonlinearity[END_REF]Chap. 8] for finite-dimensional control system, can be used to prove positive controllability results as in [START_REF] Beauchard | Controllability of a quantum particle in a 1D variable domain[END_REF][START_REF] Beauchard | Local controllability of 1D Schrödinger equations with bilinear control and minimal time[END_REF][START_REF] Cerpa | Exact controllability of a nonlinear Korteweg-de Vries equation on a critical spatial domain[END_REF][START_REF] Cerpa | Boundary controllability for the nonlinear Korteweg-de Vries equation on any critical domain[END_REF][START_REF] Coron | Exact boundary controllability of a nonlinear KdV equation with critical lengths[END_REF] or stabilization results as in [START_REF] Chowdhury | Open loop stabilization of incompressible Navier-Stokes equations in a 2d channel using power series expansion[END_REF][START_REF] Coron | Quadratic approximation and time-varying feedback laws[END_REF][START_REF] Coron | Local exponential stabilization for a class of Korteweg-de Vries equations by means of time-varying feedback laws[END_REF].

It can also be used to deny STLC. In [START_REF] Beauchard | Quadratic obstructions to small-time local controllability for scalar-input systems[END_REF], the authors proved that, in finite dimension, for scalar-input differential systems, when the linear test fails, STLC cannot be recovered from the quadratic term. Indeed, the second-order term adds a drift quantified by the H -n -norm of the control, along an explicit Lie bracket, denying W 2n-3,∞ -STLC for the nonlinear system. This phenomenon was already observed in infinite dimension, for a Schrödinger equation, by Coron in [START_REF] Coron | On the small-time local controllability of a quantum particle in a moving onedimensional infinite square potential well[END_REF] and by Beauchard and Morancey in [START_REF] Beauchard | Local controllability of 1D Schrödinger equations with bilinear control and minimal time[END_REF]. Indeed, the authors proved a coercivity type inequality, similar to [START_REF] Beauchard | Local controllability of 1D linear and nonlinear Schrödinger equations with bilinear control[END_REF], to explicit impossible motions in small time. In that case, those motions were given, at least formally, by the same Lie brackets as in [START_REF] Beauchard | Quadratic obstructions to small-time local controllability for scalar-input systems[END_REF]. In [START_REF] Marbach | An obstruction to small-time local null controllability for a viscous Burgers' equation[END_REF], for a Burgers equation, those Lie brackets vanish and don't lead to obstruction. However, STLC is still denied proving an inequality analogous to [START_REF] Beauchard | Local controllability of 1D linear and nonlinear Schrödinger equations with bilinear control[END_REF] but involving a fractional Sobolev norm of the control instead. In [START_REF] Beauchard | Unexpected quadratic behaviors for the small-time local null controllability of scalar-input parabolic equations[END_REF], obstructions caused by both quadratic integer and fractional drifts are proven on a scalar-input parabolic equation. Finally, a similar result has also been proved on a KdV system, for boundary controls in [START_REF] Coron | On the small-time local controllability of a kdv system for critical lengths[END_REF] by Coron, Koenig and Nguyen.

A finite-dimensional counterpart

Let p ∈ N * and H 0 , H 1 ∈ M p (R) symmetric matrices. Consider Schrödinger control systems of the form

iX (t) = H 0 X(t) -u(t)H 1 X(t), (6)
where the state is X(t) ∈ C p and the control is u(t) ∈ R. We write (ϕ 1 , . . . , ϕ p ) for an orthonormal basis of eigenvectors of H 0 , (λ 1 , . . . , λ p ) for its eigenvalues and denote by X 1 (t) := ϕ 1 e -iλ 1 t . We take X(0) = ϕ 1 as initial data for [START_REF] Beauchard | Quadratic obstructions to small-time local controllability for scalar-input systems[END_REF]

H 1 ϕ 1 , ϕ K = 0 and a 1 K := [H 1 , [H 0 , H 1 ]] ϕ 1 , ϕ K = 0. ( 7
)
Then the system (6) is not W -1,∞ -STLC: there exists C > 0 such that for all a ∈ (0,

|a 1 K | 8
), there exists T * > 0 such that for all T ∈ (0, T * ), there exists η > 0 such that for all u ∈ L 1 (0, T ) with u 1 L ∞ (0,T ) < η, the solution X of (6) with initial data ϕ 1 satisfies

-sign(a 1 K ) X(T ), ϕ K e -iλ 1 T ≥ a u 1 2 L 2 (0,T ) -C (X -X 1 )(T ) 2 . ( 8
)
Remark 2.2. This result is contained in [START_REF] Beauchard | Quadratic obstructions to small-time local controllability for scalar-input systems[END_REF]Theorem 3] where the authors gave Lie brackets along which the quadratic order induces a drift, denying STLC of the nonlinear system. However, the proof is still done here in the easier setting of Schrödinger ODEs as it provides a guideline to study obstructions for the Schrödinger PDE [START_REF] Ball | Controllability for distributed bilinear systems[END_REF]. Moreover, a new tool, introduced in [START_REF] Beauchard | On expansions for nonlinear systems, error estimates and convergence issues[END_REF], called the Magnus expansion in the interaction picture, could also be used to prove this result. However, this tool seems to not be suitable for an infinite-dimensional framework, thus is not discussed here.

Theorem 2.1 relies on the power series expansion of the solution of (6) around the trajectory (X 1 , u ≡ 0). The first and second-order terms of the expansion are respectively solutions of,

iX L = H 0 X L -u(t)H 1 X 1 , with X L (0) = 0, (9)
and

iX Q = H 0 X Q -u(t)H 1 X L , with X Q (0) = 0. ( 10
)
We prove that the solution of the linearized system misses the K-th direction and that the second-order term is, up to some negligible terms, a quadratic form, with a coercivity quantified by the H -1 -norm of the control. We also prove estimates on the cubic remainder of the expansion to show that the quadratic term allows to deny STLC for the full nonlinear equation.

2.1.

Error estimates on the expansion of the solution. The goal of this section is to give a rigorous meaning to the following expansion

X ≈ X 1 + X L + X Q . (11)
In this section, we consider the following asymptotic.

Definition 2.3. Given two scalar quantities A(T, u) and B(T, u), we write A(T, u) = O (B(T, u)) if there exists C, T * > 0 such that for any T ∈ (0, T * ), there exists η > 0 such that for all u ∈ L 1 (0, T ) with

u 1 L ∞ (0,T ) < η, we have |A(T, u)| ≤ C|B(T, u)|.
Thus, the notation O refers to the convergence u 1 L ∞ (0,T ) → 0 but with controls u in L 1 (0, T ) to ensure the well-posedness of ( 6) and all the equations considered. Moreover, this convergence holds uniformly with respect to the final time on a small time interval [0, T * ].

2.1.1. Error estimates for the auxiliary system. We need sharp error estimates to prove that the cubic remainder of the expansion (11) can be neglected in front of the drift u 1 2 L 2 . Therefore, classical error estimates involving the L 2 -norm of the control u are not enough. One can compute estimates involving rather the L 2 -norm of the time-primitive u 1 of the control by introducing the new state

X(t) := e -iH 1 u 1 (t) X(t). (12)
Such an idea was introduced in [START_REF] Coron | On the small-time local controllability of a quantum particle in a moving onedimensional infinite square potential well[END_REF] and later used in [START_REF] Beauchard | Local controllability of 1D Schrödinger equations with bilinear control and minimal time[END_REF] for the Schrödinger equation. This strategy can also be found in [START_REF] Beauchard | Quadratic obstructions to small-time local controllability for scalar-input systems[END_REF] to study the quadratic behavior of differential systems or in [START_REF] Beauchard | On expansions for nonlinear systems, error estimates and convergence issues[END_REF] to give refined error estimates for various expansions of scalar-input affine control systems. The new state X solves the following ODE, called the auxiliary system,

X (t) = -ie -iH 1 u 1 (t) H 0 e iH 1 u 1 (t) X(t) = -i +∞ k=0 (-iu 1 (t)) k k! ad k H 1 (H 0 ) X(t). ( 13
) Remark 2.4. Taking • a sub-multiplicative norm on M p (R), one gets ∀t ∈ [0, T ], u 1 (t) k k! ad k H 1 (H 0 ) ≤ H 0 (2|u 1 (t)| H 1 ) k k! ,
proving the absolute convergence of the series given in [START_REF] Cerpa | Exact controllability of a nonlinear Korteweg-de Vries equation on a critical spatial domain[END_REF]. The second equality of ( 13) is obtained showing that the matrices on both sides satisfy the following Cauchy problem

M (t) = -iu(t) ad H 1 (M (t)) with M (0) = H 0 ,
giving that e -iH 1 u 1 (t) H 0 e iH 1 u 1 (t) = exp(-iu 1 (t) ad H 1 )H 0 .

By the Duhamel formula, the solution of the auxiliary system satisfies

X(t) = X 1 (t) -i t 0 e -iH 0 (t-τ ) +∞ k=1 (-iu 1 (τ )) k k! ad k H 1 (H 0 ) X(τ )dτ. ( 14
)
The first-order term X L and the second-order term X Q of the expansion of X around the trajectory (X 1 , u ≡ 0) are given by, for all t ∈ [0, T ],

X L (t) = - t 0 e -iH 0 (t-τ ) u 1 (τ ) ad 1 H 1 (H 0 )X 1 (τ )dτ, (15)
X Q (t) = t 0 e -iH 0 (t-τ ) -u 1 (τ ) ad 1 H 1 (H 0 ) X L (τ ) + iu 1 (τ ) 2 2 ad 2 H 1 (H 0 )X 1 (τ ) dτ. ( 16
)
In the following, for a function f :

[0, T ] → C p , we denote by f ∞ := sup t∈[0,T ] f (t) .
Proposition 2.5. The following error estimates on the expansion of the auxiliary system (13) hold

X -X 1 ∞ = O u 1 L 1 (0,T ) , (17)
X -X 1 -X L ∞ = O u 1 2 L 2 (0,T ) , (18)
X -X 1 -X L -X Q ∞ = O u 1 3 L 3 (0,T ) . (19)
Proof. Bound on the solution. Taking the scalar product of ( 6) with X and then the imaginary part of the corresponding equality, one gets d dt X(t) 2 = 0 and thus ∀t ∈ [0, T ],

X(t) = ϕ 1 .
Thus, as H 1 is symmetric and u is real-valued, the definition (12) of the auxiliary system entails that, for all t ∈ [0, T ], X(t) = ϕ 1 .

Proof of [START_REF] Coron | On the small-time local controllability of a quantum particle in a moving onedimensional infinite square potential well[END_REF]. With ( 14) and the boundness of X, one directly has,

( X -X 1 )(t) = t 0 e -iH 0 (t-τ ) O(|u 1 (τ )|) X(τ )dτ = O u 1 L 1 (0,T ) .
Proof of [START_REF] Coron | Control and nonlinearity[END_REF]. Putting together ( 14) and ( 15), one gets

( X -X 1 -X L )(t) = - t 0 e -iH 0 (t-τ ) u 1 (τ ) ad 1 H 1 (H 0 )( X -X 1 )(τ )dτ -i t 0 e -iH 0 (t-τ ) +∞ k=2 (-iu 1 (τ )) k k! ad k H 1 (H 0 ) X(τ )dτ = O u 1 L 1 X -X 1 ∞ + u 1 2 L 2 X ∞ .
Thus, the boundness of X and estimate (17) of X -X 1 lead to [START_REF] Coron | Control and nonlinearity[END_REF]. Proof of [START_REF] Coron | Exact boundary controllability of a nonlinear KdV equation with critical lengths[END_REF]. The proof is similar using equations ( 14)-( 16) and estimates ( 17)-( 18).

2.1.2.

Error estimates for the initial system.

Proposition 2.6. The following error estimates on the expansion of the initial system (6) hold

(X -X 1 ) (T ) = O ( u 1 L 1 + |u 1 (T )|) , (X -X 1 -X L ) (T ) = O u 1 2 L 2 + |u 1 (T )| 2 , (X -X 1 -X L -X Q ) (T ) = O u 1 3 L 3 + |u 1 (T )| 3 .
Proof. As the proofs of the three estimates are similar, we only prove here the second one. Identifying the expansions of the initial and auxiliary systems in [START_REF] Boussaïd | Regular propagators of bilinear quantum systems[END_REF], one gets,

X L (T ) = X L (T ) + iu 1 (T )H 1 X 1 (T ). (20)
Thus, the relations ( 12) and (20) lead to,

(X -X 1 -X L ) (T ) ≤ e iH 1 u 1 (T ) ( X -X 1 -X L )(T ) + (e iH 1 u 1 (T ) -1) X L (T ) + (e iH 1 u 1 (T ) -1 -iH 1 u 1 (T ))X 1 (T ) .
The first term of the right-hand side is estimated by u 1 2 L 2 using estimate [START_REF] Coron | Control and nonlinearity[END_REF]. Doing an expansion of the exponential, the last term is estimated by |u 1 (T )| 2 and the second term is estimated by X L (T ) |u 1 (T )|. Finally, looking at [START_REF] Chambrion | Controllability of the discretespectrum Schrödinger equation driven by an external field[END_REF], the norm of X L (T ) is estimated by u 1 L 1 .

To conclude the estimate of the cubic remainder, it only remains to show that the boundary term u 1 (T ) can be neglected. This can be done by noticing that such a term appears in the dynamic of the linearized system. Proposition 2.7. If H 0 and H 1 satisfy a 1 K = 0, then the solution X of (6) with initial data ϕ 1 satisfies

|u 1 (T )| = O √ T u 1 L 2 (0,T ) + (X -X 1 )(T ) . (21)
Proof. Notice that the assumption a 1 K = 0 on H 0 and H 1 entails the existence of j ∈ {1, . . . , p} such that

H 1 ϕ 1 , ϕ j = 0 (indeed, if H 1 ϕ 1 = 0 then [H 1 , [H 0 , H 1 ]]ϕ 1 = 0). Set such j. Solving (9), straightforward computations give, (X -X 1 ) (T ), ϕ j e -iλ 1 T = i H 1 ϕ 1 , ϕ j T 0 u(t)e i(λ j -λ 1 )(t-T ) dt+O ( (X -X 1 -X L )(T ) ) .
Yet, if j = 1, doing one integration by parts and using Cauchy-Schwarz inequality, one gets

T 0 u(t)e i(λ j -λ 1 )(t-T ) dt = u 1 (T ) + O( √ T u 1 L 2 (0,T ) ).
This also holds if j = 1. Therefore, using Proposition 2.6 to estimate (X -X 1 -X L )(T ), one gets

|u 1 (T )| = O √ T u 1 L 2 + u 1 2 L 2 + |u 1 (T )| 2 + (X -X 1 )(T ) .
By definition of the notation O, those estimates are done under the asymptotic u 1 L ∞ going to zero. Thus, such estimate entails (21).

Corollary 2.8. If H 0 and H 1 satisfy a 1 K = 0, the following error estimate holds on the expansion of the solution of (6),

(X -X 1 -X L -X Q ) (T ) = O u 1 3 L 3 + (X -X 1 )(T ) 3 . (22)
2.2. Study of the quadratic term. The goal of this section is to prove that the secondorder term of the expansion (11) has a drift. First, notice that the solution of the linearized system ( 9) is given by,

X L (t) = i p j=1 H 1 ϕ 1 , ϕ j t 0 u(τ )e i(λ j -λ 1 )τ dτ ϕ j e -iλ j t , t ∈ (0, T ). ( 23
)
Therefore, plugging this expression into the second-order system [START_REF] Boscain | Multi-input Schrödinger equation: controllability, tracking, and application to the quantum angular momentum[END_REF], one has

X Q (T ), ϕ K e -iλ 1 T = - p j=1 H 1 ϕ 1 , ϕ j H 1 ϕ K , ϕ j T 0 u(t) t 0 u(τ )e i(λ j (τ -t)+λ K (t-T )+λ 1 (T -τ )) dτ dt.
The coercivity quantified by the H -1 -norm of the control u is revealed by integrations by parts as follows,

X Q (T ), ϕ K e -iλ 1 T = Q(u 1 ) + u 1 (T ) p j=1 [H 0 , H 1 ]ϕ 1 , ϕ j H 1 ϕ K , ϕ j T 0 u 1 (τ ) cos[(λ j -λ 1 )(τ -T )]dτ,
where Q is a quadratic form on L 2 (0, T ) given by,

Q(u 1 ) := - a 1 K 2 T 0 u 1 (t) 2 cos[(λ K -λ 1 )(t -T )]dt + p j=1 [H 0 , H 1 ]ϕ 1 , ϕ j [H 0 , H 1 ]ϕ j , ϕ K × T 0 u 1 (t) t 0 u 1 (τ ) sin (λ j (τ -t) + λ K (t -T ) + λ 1 (T -τ )) dτ dt. ( 24
)
Using Proposition 2.7 to neglect the boundary term u 1 (T ) and Cauchy-Schwarz inequality, one gets,

X Q (T ), ϕ K e -iλ 1 T = Q(u 1 ) + O T u 1 2 L 2 + (X -X 1 )(T ) 2 . ( 25
)
Moreover, as stated below, the quadratic form Q has a coercivity quantified by the L 2 -norm of u 1 .

Lemma 2.9. There exists T * > 0 such that for every T ∈ (0, T * ) and u 1 ∈ L 2 (0, T ),

-sign(a 1 K )Q(u 1 ) ≥ |a 1 K | 8 T 0 u 1 (t) 2 dt.
Proof. Assume a 1 K > 0 (the case a 1 K < 0 is similar). Let T ∈ (0, T * ) with T * to be determined and u 1 ∈ L 2 (0, T ). Upper negative bound on the first term of

Q. If T * ≤ π 3(λ K -λ 1 ) , as cosine is decreasing on [0, π 3 ], one gets, - a 1 K 2 T 0 u 1 (t) 2 cos[(λ K -λ 1 )(t -T )]dt ≤ - a 1 K 4 T 0 u 1 (t) 2 dt. ( 26
)
Estimate of the second term of Q. If we denote by

C 1 K := p j=1 | [H 0 , H 1 ]ϕ 1 , ϕ j [H 0 , H 1 ]ϕ j , ϕ K | ,
by Cauchy-Schwarz inequality, the absolute value of the second term in ( 24) is bounded by,

C 1 K T T 0 u 1 (t) 2 dt ≤ a 1 K 8 T 0 u 1 (t) 2 dt, (27)
if we choose

T * ≤ a 1 K 8C 1 K .

Conclusion.

Putting together ( 26), [START_REF] Vahagn Nersesyan | Growth of Sobolev norms and controllability of the Schrödinger equation[END_REF] with the definition of Q (24), the proof follows with

T * := |a 1 1 | 8C 1 1 , if K = 1 and T * := min |a 1 K | 8C 1 K ; π 3(λ K -λ 1 )
, if K = 1.

2.3.

Proof of the first quadratic obstruction on Schrödinger ODEs. First, using the explicit form of X L given in [START_REF] Marbach | An obstruction to small-time local null controllability for a viscous Burgers' equation[END_REF], when H 1 ϕ 1 , ϕ K = 0, one notices that the K-th direction is lost at the linear level in the sense that

X L (T ), ϕ K = 0.
Thus, using the work on the quadratic term [START_REF] Morancey | Global exact controllability of 1D Schrödinger equations with a polarizability term[END_REF] and the estimate on the cubic remainder [START_REF] Coron | Local exponential stabilization for a class of Korteweg-de Vries equations by means of time-varying feedback laws[END_REF], one has

X(T ), ϕ K e -iλ 1 T = X Q (T ), ϕ K e -iλ 1 T + O ( (X -X 1 -X L -X Q )(T ) ) = Q(u 1 ) + O T u 1 2 L 2 + u 1 3 L 3 + (X -X 1 )(T ) 2 .
Expanding the notation O (see Definition 2.3), this means that there exists C, T 1 > 0 such that for all T ∈ (0, T 1 ), there exists η 1 > 0 such that for all u ∈ L 1 (0, T ) with u 1 L ∞ (0,T ) < η 1 , one has,

X(T ), ϕ K e -iλ 1 T -Q(u 1 ) ≤ C T u 1 2 L 2 + u 1 3 L 3 + (X -X 1 )(T ) 2 .
Let a ∈ (0,

|a 1 K | 8 ), T * := min(T 1 , |a 1 K | 16C -a 2C
), T ∈ (0, T * ) and η := min(η 1 ,

|a 1 K | 16C -a 2C ).
Then, for all u ∈ L 1 (0, T ) with u 1 L ∞ (0,T ) < η, one has,

X(T ), ϕ K e -iλ 1 T -Q(u 1 ) ≤ |a 1 K | 8 -a u 1 2 L 2 + C (X -X 1 )(T ) 2 .
Together with the coercivity of Q given in Lemma 2.9, this concludes the proof of Theorem 2.1.

Well-posedness of the Schrödinger equation

In this section, we recall the result about the well-posedness of the following Cauchy problem, given in [5, Proposition 2] and later extended in [START_REF] Bournissou | Local controllability of the bilinear 1D Schrodinger equation with simultaneous estimates[END_REF],

   i∂ t ψ(t, x) = -∂ 2 x ψ(t, x) -u(t)µ(x)ψ(t, x) -f (t, x), (t, x) ∈ (0, T ) × (0, 1), ψ(t, 0) = ψ(t, 1) = 0, t ∈ (0, T ), ψ(0, x) = ψ 0 (x),
x ∈ (0, 1).

(

) Proposition 3.1. Let T > 0, µ ∈ H 3 ((0, 1), R), ψ 0 ∈ H 3 (0) (0, 1), f ∈ L 2 ((0, T ), H 3 ∩ H 1 0 ) and u ∈ L 2 ((0, T ), R). 28
There exists a unique weak solution of (28) i.e a function ψ ∈ C 0 ([0, T ], H 3 (0) ) such that the following equality holds in H 3 (0) (0, 1) for every t ∈ [0, T ],

ψ(t) = e -iAt ψ 0 + i t 0 e -iA(t-τ ) u(τ )µψ(τ ) + f (τ ) dτ.
Moreover, for every R > 0, there exists C = C(T, µ, R) > 0 such that if u L 2 (0,T ) < R then this weak solution satisfies

ψ C 0 ([0,T ],H 3 (0) (0,1)) ≤ C ψ 0 H 3 (0) (0,1) + f L 2 ((0,T ),H 3 ∩H 1 0 (0,1)) . If f ≡ 0, then ψ(t) L 2 (0,1) = ψ 0 L 2 (0,1) , ∀t ∈ [0, T ].
Therefore, from now on, we will always work with controls u at least in L 2 (0, T ) and with µ at least in H 3 (0, 1) to ensure the well-posedness of all the equations considered. We will also always take ϕ 1 as the initial condition of (1). Besides, as highlighted in [START_REF] Bournissou | Local controllability of the bilinear 1D Schrodinger equation with simultaneous estimates[END_REF], the solution of ( 28) and more precisely its regularity, relies strongly on the control and the dipolar moment. However, for sake of simplicity, in the following, we will not mention this dependency.

4.

Error estimates on the expansion of the solution 4.1. Formal expansion of the solution. The main tool to prove Theorem 1.3 is the power series expansion of the solution of the Schrödinger equation (1) around the ground state. By Proposition 3.1, for u ∈ L 2 (0, T ) and µ ∈ H 3 (0, 1), one may consider,

• the first-order term Ψ, in C 0 ([0, T ], H 3 (0) ), solution of the linearized equation given by,   

i∂ t Ψ = -∂ 2 x Ψ -u(t)µ(x)ψ 1 , Ψ(t, 0) = Ψ(t, 1) = 0, Ψ(0, x) = 0, (29)
which can be explicitly computed as,

Ψ(t) = i +∞ j=1 µϕ 1 , ϕ j t 0 u(τ )e i(λ j -λ 1 )τ dτ ψ j (t), t ∈ [0, T ], (30)
• and the second-order term,

ξ in C 0 ([0, T ], H 3 (0) ), which is the solution of,    i∂ t ξ = -∂ 2 x ξ -u(t)µ(x)Ψ, ξ(t, 0) = ξ(t, 1) = 0, ξ(0, x) = 0. ( 31
)
The proof of Theorem 1.3 is in two steps.

• First, we understand in which way the following expansion holds rigorously:

ψ ≈ ψ 1 + Ψ + ξ.
• Then, we prove that the quadratic term entails a drift, quantified by the H -nnorm of the control, preventing H 2n-3 -STLC when n ≥ 2 and W -1,∞ -STLC when n = 1, for the full nonlinear system. First, we specify the smallness assumption on the controls that we use in all the following. Definition 4.1. Given two scalar quantities A(T, u) and B(T, u), we write A(T, u) = O (B(T, u)) if there exists C, T * > 0 such that for any T ∈ (0, T * ), there exists η > 0 such that for all u ∈ L 2 (0, T ) with u 1 L ∞ (0,T ) < η, we have |A(T, u)| ≤ C|B(T, u)|.

As in Definition 2.3, this notation refers to the convergence u 1 L ∞ (0,T ) → 0 uniformly with respect to the final time. However, to ensure the well-posedness of the Schrödinger equation ( 1), one asks for the controls u to be at least in L 2 (0, T ) and not just in L 1 (0, T ) as in finite dimension. 4.2. The auxiliary system. As discussed in Section 2.1.1, to prove that the behavior of the nonlinear solution is driven by the quadratic term of the expansion, one needs to compute sharp error estimates, not quantified with respect to the control u but with respect to the time-primitive u 1 of the control. This is done again by the means of an auxiliary system: if ψ is a solution of (1), we consider a new state ψ given by,

ψ(t, x) := ψ(t, x)e -iu 1 (t)µ(x) , (t, x) ∈ [0, T ] × [0, 1], (32)
which is a weak solution of

     i∂ t ψ = -∂ 2 x ψ -iu 1 (t)(2µ (x)∂ x ψ + µ (x) ψ) + u 1 (t) 2 µ (x) 2 ψ, ψ(t, 0) = ψ(t, 1) = 0, ψ(0, x) = ϕ 1 (x). (33)
The well-posedness of this equation is stated below as in [8, Proposition 2]. Proposition 4.2. Let T > 0, µ ∈ H 3 ((0, 1), R), u 1 ∈ H 1 ((0, T ), R) with u 1 (0) = 0. There exists a unique weak solution ψ ∈ C 0 ([0, T ], H 3 ∩ H 1 0 (0, 1)) of (33). Moreover, for every R > 0, there exists C = C(T, µ, R) > 0 such that, if u 1 H 1 (0,T ) < R, then this weak solution satisfies ψ C 0 ([0,T ],H 3 ∩H 1 0 (0,1)) ≤ C. Remark 4.3. Because of the term ∂ x ψ in (33), it does not seem possible to use a fixedpoint theorem to prove directly the well-posedness of (33) when u 1 ∈ L 2 ((0, T ), R). Thus, up to now, the solution of (33) is only understood through its link (32) with the Schrödinger equation [START_REF] Ball | Controllability for distributed bilinear systems[END_REF]. Thus, the regularity and the bound on ψ stated in Proposition 4.2 follow from Proposition 3.1 and therefore hold under assumptions on u 1 and not just on u 1 .

Furthermore, when u 1 is in H 1 ((0, T ), R), ψ is a weak solution of (33) in the sense that the following equality holds in H 1 0 (0, 1) for every t ∈ [0, T ],

ψ(t) = ψ 1 (t) - t 0 e -iA(t-τ ) u 1 (τ ) 2µ ∂ x + µ ψ(τ ) + iu 1 (τ ) 2 µ 2 ψ(τ ) dτ.
Notice that the right-hand side of the equality is indeed in H 1 0 (0, 1) thanks to the smoothing effect stated below in Lemma 4.4, which was highlighted in [START_REF] Beauchard | Local controllability of 1D linear and nonlinear Schrödinger equations with bilinear control[END_REF] and later used in [START_REF] Bournissou | Local controllability of the bilinear 1D Schrodinger equation with simultaneous estimates[END_REF].

Moreover, when u 1 is in C 1 ([0, T ], R), ψ and thus ψ are in C 1 ([0, T ], L 2 (0, 1)) and the first equation of (33) is satisfied in L 2 (0, 1) at every time.

Lemma 4.4. There exists a nondecreasing function C : R + → R * + such that for all T ≥ 0 and f in L 2 ((0, T ), H 1 (0, 1)), the function

G : t → t 0 e -iA(t-τ ) f (τ )dτ is in C 0 ([0, T ], H 1 0 (0, 1)) with G C 0 ([0,T ],H 1 0 (0,1)) ≤ C(T ) f L 2 ((0,T ),H 1 (0,1)) .
Now, we want to study the expansion around the ground state of the solution ψ of the auxiliary system (33).

First-order term. Linearizing (32), the first-order term Ψ of the expansion of ψ is given by

Ψ(t, x) = Ψ(t, x) -iu 1 (t)µ(x)ψ 1 (t, x), ( 34
)
where Ψ is the solution of (29). Thus, Ψ is in C 0 ([0, T ],

H 3 ∩ H 1 0 ) and is a weak solution of    i∂ t Ψ = -∂ 2 x Ψ -iu 1 (t) (2µ ∂ x ψ 1 + µ ψ 1 ) , Ψ(t, 0) = Ψ(t, 1) = 0, Ψ(0, x) = 0. (35)
Second-order term. Doing an expansion of order 2 of (32), the second-order term ξ is given by,

ξ(t, x) = ξ(t, x) -iu 1 (t)µ(x) Ψ(t, x) + u 1 (t) 2 2 µ(x) 2 ψ 1 (t, x), ( 36
)
where ξ is the solution of [START_REF] Turinici | On the controllability of bilinear quantum systems[END_REF]. Notice that ξ is in C 0 ([0, T ],

H 3 ∩ H 1 0 ) and is a weak solution of,      i∂ t ξ = -∂ 2 x ξ -iu 1 (t)(2µ ∂ x Ψ + µ Ψ) + u 1 (t) 2 µ 2 ψ 1 , ξ(t, 0) = ξ(t, 1) = 0, ξ(0, x) = 0.
(37) Proposition 4.5. The first and second-order terms of the expansion of ψ satisfy

Ψ L ∞ ((0,T ),H 1 0 (0,1)) = O ( u 1 L 2 ) and ξ L ∞ ((0,T ),L 2 (0,1)) = O u 1 2 L 2 . ( 38
)
Proof. First, solving (35), the following equality holds in H 1 0 (0, 1),

Ψ(t) = - t 0 e -iA(t-τ ) u 1 (τ ) 2µ ∂ x + µ ψ 1 (τ )dτ, t ∈ [0, T ].
As the function τ → u 1 (τ ) (2µ

∂ x + µ ) ψ 1 (τ ) is in L 2 ((0, T ), H 1 (0, 1
Ψ C 0 ([0,T ],H 1 0 ) ≤ C(T ) u 1 2µ ∂ x + µ ψ 1 L 2 ((0,T ),H 1 ) ≤ C(T ) µ H 3 (0,1) u 1 L 2 (0,T )
, also using the algebra structure of H 3 (0, 1). As the constant is nondecreasing with respect to the final time, one gets the first estimate of (38). Moreover, solving (37), one gets

ξ(t) = - t 0 e -iA(t-τ ) u 1 (τ ) 2µ ∂ x + µ Ψ(τ ) + iu 1 (τ ) 2 µ 2 ψ 1 (τ ) dτ, t ∈ [0, T ].
Therefore, using the triangular inequality together with the fact that for all s ∈ R, e iAs is an isometry from L 2 (0, 1) to L 2 (0, 1), one gets,

ξ(t) L 2 (0,1) ≤ t 0 |u 1 (τ )| 2µ ∂ x + µ Ψ(τ ) L 2 (0,1) + |u 1 (τ )| 2 µ 2 ψ 1 (τ ) L 2 (0,1) dτ ≤ µ H 3 u 1 L 1 (0,T ) Ψ L ∞ ((0,T ),H 1 0 ) + µ 2 L ∞ u 1 2 L 2 (0,T ) ψ 1 L ∞ ((0,T ),L 2 ) .
Thus, the estimate on Ψ allows to conclude the proof of the second estimate of (38).

4.3.

Energy estimates on the auxiliary system. One of our goals is to prove that the first obstruction can occur assuming only that u 1 L ∞ is small. Hence, we seek to prove estimates on ψ not assuming the boundness of u 1 in H 1 (0, T ) as required in Proposition 4.2.

Lemma 4.6. For every T > 0, µ ∈ H 3 ((0, 1), R) and u 1 ∈ H 1 ((0, T ), R), the solution ψ of the auxiliary system (33) satisfies

∀t ∈ [0, T ], ψ(t) L 2 (0,1) = 1. ( 39
)
This first result follows directly from the definition of the auxiliary system (32) and the conservation of the L 2 -norm of the solution of the Schrödinger equation given in Proposition 3.1.

Proposition 4.7. For every µ ∈ H 3 ((0, 1), R), the solution ψ of the auxiliary system (33) satisfies

ψ -ψ 1 L ∞ ((0,T ),L 2 (0,1)) = O u 1 L 2 (0,T ) . ( 40
)
Proof. The estimate is computed first for regular controls so that the equation (33) holds in L 2 (0, 1) at every time (see Remark 4.3) and then deduced by density. Denote by R := ψ -ψ 1 . Looking at (33), R is the solution of

i∂ t R = -∂ 2 x R -iu 1 (t) 2µ ∂ x + µ R -iu 1 (t) 2µ ∂ x + µ ψ 1 + u 1 (t) 2 µ 2 ψ, (41)
with Dirichlet boundary conditions and initial condition R(0, •) = 0. Let t ∈ [0, T ]. The proof consists in taking the L 2 -scalar product of (41) with R, integrating over [0, t] and taking the imaginary part. First, notice that

i t 0 ∂ t R(τ ), R(τ ) dτ = 1 2 t 0 d dt R(τ ) 2 L 2 (0,1) dτ = 1 2 R(t) 2 L 2 (0,1) . ( 42
)
Moreover, as for every τ ∈ [0, T ], R(τ ) is in H 1 0 (0, 1), one integration by parts gives,

- t 0 ∂ 2 x R(τ ), R(τ ) dτ = t 0 ∂ x R(τ ) 2 L 2 (0,1) dτ = 0. ( 43
)
Besides, as the operator 2µ ∂ x + µ is skew-Hermitian on H 1 0 (0, 1) and u 1 is real-valued,

i t 0 u 1 (τ ) 2µ ∂ x + µ R(τ ), R(τ ) dτ = 0. ( 44
)
Moreover, using Young and Cauchy-Schwarz inequalities, for every control such that

u 1 L ∞ ≤ 1, t 0 -iu 1 (τ ) 2µ ∂ x + µ ψ 1 (τ ) + u 1 (τ ) 2 µ 2 ψ(τ ), R(τ ) dτ ≤ 1 2 u 1 2 L 2 + 1 2 (2µ ∂ x + µ )ψ 1 2 L ∞ (L 2 ) + µ 2 ψ 2 L ∞ (L 2 ) t 0 R(τ ) 2 L 2 dτ. (45)
Thus, the equation (41) together with estimates (42)-(45) and estimate (39) on ψ, one gets the existence of C = C(µ) > 0 such that, for all u 1 such that

u 1 L ∞ ≤ 1, R(t) 2 L 2 (0,1) ≤ C u 1 2 L 2 + C t 0 R(τ ) 2 L 2 (0,1) dτ.
Therefore, Gronwall's Lemma leads to (40), as the definition of O means that we work in the asymptotic u 1 L ∞ small (see Definition 4.1).

4.4.

Error estimates for the auxiliary system. Proposition 4.8. For every µ ∈ H 3 ((0, 1), R) and p ∈ N * , the following scalar error estimates hold

e iu 1 (T )µ ( ψ -ψ 1 -Ψ)(T ), ϕ p = O u 1 2 L 2 (0,T ) , (46)
e iu 1 (T )µ ( ψ -ψ 1 -Ψ -ξ)(T ), ϕ p = O u 1 3 L 2 (0,T ) . ( 47
)
Proof. Proof of (46). Solving (33) and ( 35), the following equality holds in H 1 0 (0, 1),

( ψ -ψ 1 -Ψ)(T ) = - T 0 e -iA(T -t) u 1 (t) 2µ ∂ x + µ ( ψ -ψ 1 )(t) + iu 1 (t) 2 µ 2 ψ(t) dt.
(48) Thus, e iu 1 (T )µ ( ψ -ψ 1 -Ψ)(T ), ϕ p = I 1 + I 2 , where

I 1 := - T 0 u 1 (t) e -iA(T -t) (2µ ∂ x + µ )( ψ -ψ 1 )(t), e -iu 1 (T )µ ϕ p dt, I 2 := -i T 0 u 1 (t) 2 e -iA(T -t) µ 2 ψ(t), e -iu 1 (T )µ ϕ p dt.
Using Cauchy-Schwarz's inequality, that e iAs : L 2 → L 2 is an isometry and estimate (39) on ψ, one gets

|I 2 | ≤ T 0 u 1 (t) 2 e -iA(T -t) µ 2 ψ(t) L 2 (0,1) e -iu 1 (T )µ ϕ p L 2 (0,1) dt ≤ µ 2 L ∞ u 1 2 L 2 .
Moreover, using that the operator 2µ ∂ x +µ is skew-Hermitian on H 1 0 and Cauchy-Schwarz inequality, one gets

|I 1 | = T 0 u 1 (t) ( ψ -ψ 1 )(t), (2µ ∂ x + µ )e iA(T -t) e -iu 1 (T )µ ϕ p dt ≤ u 1 L 1 ψ-ψ 1 L ∞ ((0,T ),L 2 ) (2µ ∂ x +µ )e iA(T -t) e -iu 1 (T )µ ϕ p L ∞ ((0,T ),L 2 ) = O √ T u 1 2 L 2 ,
using estimate (40) on ψ -ψ and because, for all time t,

(2µ ∂ x + µ )e iA(T -t) e -iu 1 (T )µ ϕ p L 2 ≤ 3 µ H 3 e -iu 1 (T )µ ϕ p H 1 0 = 3 µ H 3 e -iu 1 (T )µ ϕ p -iu 1 (T )µ ϕ p L 2 = O(1), (49)
using that e iAs : H 1 0 (0, 1) → H 1 0 (0, 1) is an isometry and recalling that we work with u 1 L ∞ small and thus bounded by definition of O.

Proof of (47). Solving (33), ( 35) and (37), one gets

e iu 1 (T )µ ( ψ -ψ 1 -Ψ -ξ)(T ), ϕ p = J 1 + J 2 ,
where

J 1 := - T 0 u 1 (t) e -iA(T -t) 2µ ∂ x + µ ( ψ -ψ 1 -Ψ)(t), e -iu 1 (T )µ ϕ p dt, J 2 := -i T 0 u 1 (t) 2 e -iA(T -t) µ 2 ( ψ -ψ 1 )(t), e -iu 1 (T )µ ϕ p dt.
As before, using (40) to estimate ψ -ψ 1 , one gets,

|J 2 | ≤ µ 2 L ∞ u 1 2 L 2 ψ -ψ 1 L ∞ ((0,T ),L 2 ) = O u 1 3 L 2 . (50)
Moreover, as the operator 2µ ∂ x + µ is skew-Hermitian on H 1 0 , J 1 is given by

J 1 = T 0 u 1 (t) ( ψ -ψ 1 -Ψ)(t), (2µ ∂ x + µ )[e iA(T -t) e -iu 1 (T )µ ϕ p ] dt.
Recalling the computation of ψ -ψ 1 -Ψ given in (48), one can write

J 1 = J 1,1 + J 1,2 with J 1,1 := - T 0 u 1 (t) t 0 u 1 (τ )× e -iA(t-τ ) 2µ ∂ x + µ ( ψ -ψ 1 )(τ ), (2µ ∂ x + µ )[e iA(T -t) e -iu 1 (T )µ ϕ p ] dτ dt, J 1,2 := -i T 0 u 1 (t) t 0 u 1 (τ ) 2 e -iA(t-τ ) µ 2 ψ(τ ), (2µ ∂ x + µ )[e iA(T -t) e -iu 1 (T )µ ϕ p ] dτ dt.
Using Cauchy-Schwarz inequality in L 2 (0, 1), estimate (39) on ψ and estimate (49), one gets

|J 1,2 | = O u 1 L 1 (0,T ) u 1 2 L 2 (0,T ) ψ L ∞ ((0,T ),L 2 (0,1)) = O √ T u 1 3 L 2 . (51)
Besides, for all (t, τ ) ∈ [0, T ], using Cauchy-Schwarz inequality and estimate (49), one gets

u 1 (t)u 1 (τ ) e -iA(t-τ ) 2µ ∂ x + µ ( ψ -ψ 1 )(τ ), (2µ ∂ x + µ )[e iA(T -t) e -iu 1 (T )µ ϕ p ] = O |u 1 (t)||u 1 (τ )| ( ψ -ψ 1 )(τ ) H 1 0 (0,1) ∈ L 1 ((0, T ) × (0, T )), as u 1 is in L 2 (0, T ) and ( ψ -ψ 1 ) is in C 0 ([0, T ], H 3 ∩ H 1 0 (0, 1)
). Thus, one can apply Fubini's theorem to write J 1,1 as

J 1,1 = - T τ =0 u 1 (τ ) 2µ ∂ x + µ ( ψ -ψ 1 )(τ ), F (τ ) dτ with F (τ ) := T t=τ e iA(t-τ ) u 1 (t)(2µ ∂ x + µ )[e iA(T -t) e -iu 1 (T )µ ϕ p ]dt.
As in estimate (49), for all time t, e iA(T -t) e -iu 1 (T )µ ϕ p H 2 (0) = O (1) . Thus, the function

u 1 (•)(2µ ∂ x + µ )[e iA(T -•) e -iu 1 (T )µ ϕ p ] is in L 2 ((0, T ), H 1 ). So, by Lemma 4.4, F is in C 0 ([0, T ], H 1 0 ) with F C 0 ([0,T ],H 1 0 ) ≤ C u 1 (2µ ∂ x + µ )[e iA(T -t) e -iu 1 (T )µ ϕ p ] L 2 ((0,T ),H 1 ) = O ( u 1 L 2 )
. (52) Therefore, using Cauchy-Schwarz inequality and that 2µ ∂ x + µ is skew-Hermitian, (40) and (52) lead to

|J 1,1 | ≤ C u 1 L 1 ψ -ψ 1 L ∞ ((0,T ),L 2 ) F L ∞ ((0,T ),H 1 0 ) = O √ T u 1 3 L 2 . (53)
Estimates (50), ( 51) and (53) conclude the proof of (47).

4.5.

Error estimates for the Schrödinger equation. Now, from estimates on the auxiliary system, one can deduce estimates for the Schrödinger equation.

Proposition 4.9. For every µ ∈ H 3 ((0, 1), R) and p ∈ N * , the following error estimates hold

ψ -ψ 1 L ∞ ((0,T ),L 2 (0,1)) = O ( u 1 L 2 + |u 1 (T )|) , (ψ -ψ 1 -Ψ)(T ), ϕ p = O u 1 2 L 2 + |u 1 (T )| 2 , (ψ -ψ 1 -Ψ -ξ)(T ), ϕ p = O u 1 3 L 2 + |u 1 (T )| 3 .
Proof. As the proof of the three estimates is similar, we give here only the proof of the last estimate. Using the links (32), (34) and (36) between the several systems, we have,

(ψ-ψ 1 -Ψ-ξ)(T ) = (e iu 1 (T )µ -1-iu 1 (T )+ u 1 (T ) 2 2 µ 2 )ψ 1 (T )+(e iu 1 (T )µ -1-iu 1 (T )µ) Ψ(T ) + (e iu 1 (T )µ -1) ξ(T ) + e iu 1 (T )µ ( ψ -ψ 1 -Ψ -ξ)(T ).
We conclude as in the proof of Proposition 2.6: doing an expansion of e iu 1 (T )µ , the p-th coordinate of the first term (resp. second and third term) can be estimated by |u 1 (T )| 3 (resp. by |u 1 (T )| 2 Ψ L 2 and by |u 1 (T )| ξ L 2 ). Then, using (38) to estimate Ψ(T ) and ξ(T ) and (47) to estimate the last term, one concludes the proof.

To conclude the error estimate of the cubic remainder, one needs to be able to neglect the boundary term u 1 (T ). As in Proposition 2.7, it can be done by noticing that such a term arises in the dynamic of the linearized system. The proof is exactly the same with the L 2 (0, 1)-scalar product instead of the C p -one and with µ instead of H 1 and thus is left to the reader. Proposition 4.10. If µ satisfies (4), then the solution ψ of (1) with initial data ϕ 1 satisfies

u 1 (T ) = O √ T u 1 L 2 (0,T ) + (ψ -ψ 1 )(T ) L 2 (0,1) . (54)
Corollary 4.11. For every µ ∈ H 3 ((0, 1), R) satisfying (4) and p ∈ N * , the following error estimates hold

(ψ -ψ 1 -Ψ)(T ), ϕ p = O u 1 2 L 2 (0,T ) + (ψ -ψ 1 )(T ) 2 L 2 (0,1) , ( 55
) (ψ -ψ 1 -Ψ -ξ)(T ), ϕ p = O u 1 3 L 2 (0,T ) + (ψ -ψ 1 )(T ) 3 L 2 (0,1) . (56)

Coercivity of the quadratic term

Most objects defined in this section have a dependency with respect to the final time T , the index K of the lost direction and the index n of the obstruction considered. To lighten the notations, we will only mention the dependency with respect to n. The goal of this section is to prove that, under (H2) K,n , in an appropriate sense, the quadratic term has the following drift

ξ(T ), ϕ K e -iλ 1 T ≈ -A n K u n 2
L 2 (0,T ) . Plugging the explicit form of the first-order term Ψ (30) into the second-order system [START_REF] Turinici | On the controllability of bilinear quantum systems[END_REF], computations lead to

ξ(T ), ϕ K e -iλ 1 T = T 0 u(t) t 0 u(τ )h(t, τ )dτ dt, (57)
where the quadratic kernel h is given by

h(t, τ ) := - +∞ j=1 µϕ K , ϕ j µϕ 1 , ϕ j e i[λ K (t-T )+λ j (τ -t)+λ 1 (T -τ )] , ∀(t, τ ) ∈ [0, T ] 2 . ( 58
)
By the assumption (2) on µ, h is bounded in C 2n (R 2 , C). This regularity is the key to perform integrations by parts to reveal coercive drifts, quantified by any integer negative Sobolev norm.

Proposition 5.1. Let n ∈ N and H ∈ C 2n (R 2 , C). There exists a quadratic form Q n on C 2n such that for all T > 0 and u ∈ L 1 (0, T ),

T 0 u(t) t 0 u(τ )H(t, τ )dτ dt = n p=1 T 0 u p (t) 2 1 2 d dt (∂ p-1 1 ∂ p-1 2 H(t, t)) -∂ p 1 ∂ p-1 2 H(t, t) dt + T 0 u n (t) t 0 u n (τ )∂ n 1 ∂ n 2 H(t, τ )dτ dt + Q n (u 1 (T ), . . . , u n (T ), γ n 0 , . . . , γ n n-1 ),
where, for all p = 0 . . . n -1, we denote by γ n p :=

T 0 u n (τ )∂ p 1 ∂ n 2 H(T, τ )dτ.
Proof. This result is proved by induction on m ∈ {0, . . . , n}. The equality is clear for m = 0 with the convention that the sum is empty and taking Q 0 = 0. Assume it holds for m ∈ {0, . . . , n -1}. Integrations by parts show that

T 0 u m (t) t 0 u m (τ )∂ m 1 ∂ m 2 H(t, τ )dtdτ = -u m+1 (T ) T 0 u m+1 (τ )∂ m 1 ∂ m+1 2 H(T, τ )dτ + u 2 m+1 (T ) 2 ∂ m 1 ∂ m 2 H(T, T ) + T 0 u m+1 (t) 2 1 2 d dt (∂ m 1 ∂ m 2 H(t, t)) -∂ m+1 1 ∂ m 2 H(t, t) dt + T 0 u m+1 (t) t 0 u m+1 (τ )∂ m+1 1 ∂ m+1 2 H(t, τ )dτ dt.
The conclusion of the induction follows after noticing that, doing once again integrations by parts, ∀i = 0, . . . , m -1,

γ m i = u m+1 (T )∂ i 1 ∂ m 2 H(T, T ) -γ m+1 i .
With the expression of h (58) and the definition of the coefficients A p K (4), one computes that, ∀p ∈ {1, . . . , n}, 1 2

d dt (∂ p-1 1 ∂ p-1 2 h(t, t)) -∂ p 1 ∂ p-1 2 h(t, t) = -iA p K e i(λ K -λ 1 )(t-T ) .
Therefore, applying Proposition 5.1 to (57), under hypothesis (H2) K,n on µ, one gets

ξ(T ), ϕ K e -iλ 1 T = Q n (u n ) + O   n p=1 |u p (T )| 2 + T 0 u n (τ )∂ p 1 ∂ n 2 h(T, τ )dτ 2   , (59)
where Q n is the following quadratic form defined by, for s in L 2 (0, T ),

Q n (s) := -A n K T 0 s(t) 2 cos[(λ K -λ 1 )(t -T )]dt + T 0 s(t) t 0 s(τ )k n (t, τ )dτ dt, (60)
and where the real quadratic kernel k n is given by,

k n (t, τ ) := (-1) n+1 +∞ j=1 (λ K -λ j ) n (λ j -λ 1 ) n µϕ K , ϕ j µϕ 1 , ϕ j × sin (λ K (t -T ) + λ j (τ -t) + λ 1 (T -τ )) . (61)
The following lemma states the coercivity of the quadratic form Q n .

Lemma 5.2. Assuming (2) and (4) on µ, there exists T * > 0 such that for every T ∈ (0, T * ) and s ∈ L 2 (0, T ),

-sign(A n K )Q n (s) ≥ |A n K | 4 T 0 s(t) 2 dt.
Proof. The proof is the same as the proof of Lemma 2.9 with

T * := |A n 1 | 4C n 1 , if K = 1 and T * := min |A n K | 4C n K ; π 3(λ K -λ 1 ) , if K = 1,
where the constant C n K bounds the kernel k n and is given by

C n K := +∞ j=1 |(λ K -λ j ) n (λ j -λ 1 ) n c j | ,
where (c j ) j∈N * is defined in (H2) K,n . Notice that C n K is finite by ( 2) and non-vanishing by (4).

As in Proposition 4.10, the boundary terms (u p (T )) p=1,...,n arising in (59) can be neglected. To that end, we prove first that the linearized system can move in at least n directions.

Lemma 5.3. Under (H2) K,n , there exists at least n values of j in N * such that c j = 0, where the sequence (c j ) j∈N * is defined in (H2) K,n .

Proof. Assume by contradiction that there exists at most n -1 values of j in N * such that c j = 0. Let denote by (c j i ) i=1,...n-1 such values. Then, for all p ∈ {1, . . . , n},

A p K = (-1) p-1 n-1 i=1 λ j i - λ 1 + λ K 2 v p-1 j i c j i , where v j := (λ K -λ j )(λ j -λ 1 ). ( 62
)
Our goal is to prove that in that case,

-A n K = n-1 p=1 σ p A n-p K where ∀p = 1, . . . , n -1, σ p := 1≤i 1 <...<ip≤n-1 v j i 1 . . . v j ip , (63)
which leads to an absurdity thanks to (3) and ( 4) and thus concludes the proof.

(62) into the right-hand side of (63), one gets,

n-1 p=1 σ p A n-p K = (-1) n-1 n-1 i=1 λ j i - λ 1 + λ K 2 c j i   n-1 p=1 (-1) p σ p v n-p-1 j i   .
Yet, Vieta's formulas give that, for all x ∈ R,

(x -v j 1 ) . . . (x -v j n-1 ) = x n-1 + n-1 p=1 (-1) p σ p x n-1-p .
Using this formula for x = v j i for every i ∈ {1, . . . , n -1}, one gets,

n-1 p=1 σ p A n-p K = (-1) n-1 n-1 i=1 λ j i - λ 1 + λ K 2 c j i (-v n-1 j i ) = -A n K ,
using once again (62), which concludes the proof of (63) and thus the proof.

Proposition 5.4. Under (H1) K -(H2) K,n , the solution ψ of (1) with initial data ϕ 1 satisfies,

n p=1 |u p (T )| = O √ T u n L 2 (0,T ) + u 1 2 L 2 (0,T ) + (ψ -ψ 1 )(T ) L 2 (0,1) .
Proof. By Lemma 5.3, there exists J ⊂ N * of cardinal n such that for all j ∈ J, c j = 0. Besides, thanks to (H1) K , J ⊂ N * -{1, K}. Using the explicit form of Ψ (30), one gets, for j ∈ J,

(ψ -ψ 1 )(T ), ϕ j e -iλ 1 T = i µϕ 1 , ϕ j T 0 u(t)e i(λ j -λ 1 )(t-T ) dt + O sup j∈J | (ψ -ψ 1 -Ψ)(T ), ϕ j | .
Yet, as j = 1, doing n integrations by parts and using Cauchy-Schwarz inequality, one gets

T 0 u(t)e i(λ j -λ 1 )(t-T ) dt = n p=1 (-i[λ j -λ 1 ]) p-1 u p (T ) + O( √ T u n L 2 ).
Using (55) to estimate the linear remainder ψ -ψ 1 -Ψ (as J is finite), writing U := (u p (T )) p=1...n and V := ((-i[λ j -λ 1 ]) p-1 ) (j,p)∈J ×{1,...,n} , the first equation can be written as

V U = O √ T u n L 2 + u 1 2 L 2 + (ψ -ψ 1 )(T ) L 2 .
The invertibility of the Vandermonde matrix V concludes the proof. Therefore, in (59), neglecting the boundary terms (u p (T )) p=1,...,n thanks to Proposition 5.4 and neglecting the other boundary terms thanks to Cauchy-Schwarz inequality and the boundness of h, one gets,

ξ(T ), ϕ K e -iλ 1 T = Q n (u n ) + O T u n 2 L 2 + u 1 4 L 2 + (ψ -ψ 1 )(T ) 2 L 2 , (64)
where Q n is the quadratic form (60), which is coercive as stated in Lemma 5.2.

Proof of the obstructions caused by quadratic integer drifts

From now on, we assume that (H1) K -(H2) K,n hold and we work in the asymptotic u small in W -1,∞ if n = 1 and small in H 2n-3 if n ≥ 2. The proof of Theorem 1.3 consists in describing what happens at each order of the expansion of the solution. Under (H1) K , recalling the explicit computation of Ψ given in (30), we have

Ψ(T ), ϕ K e -iλ 1 T = 0.
Thus, the study of the quadratic term given by (64) and estimate (56) of the cubic remainder entail

ψ(T ), ϕ K e -iλ 1 T = ξ(T ), ϕ K e -iλ 1 T + O ( (ψ -ψ 1 -Ψ -ξ)(T ), ϕ K ) = Q n (u n ) + O T u n 2 L 2 + u 1 3 L 2 + (ψ -ψ 1 )(T ) 2 L 2 , (65)
thanks to Definition 4.1 of O.

First obstruction. When n = 1, one directly gets that

ψ(T ), ϕ K e -iλ 1 T -Q 1 (u 1 ) = O (T + u 1 L ∞ ) u 1 2 L 2 + (ψ -ψ 1 )(T ) 2 L 2 .
As in Section 2.3, expanding the notation of O, this leads to the existence of C > 0 such that for all A ∈ (0,

|A 1 K | 4
), there exists T * > 0 such that for all T ∈ (0, T * ), there exists η > 0 such that for all u ∈ L 2 (0, T ) with

u 1 L ∞ (0,T ) < η, ψ(T ), ϕ K e -iλ 1 T -Q 1 (u 1 ) ≤ |A 1 K | 4 -A u 1 2 L 2 + C (ψ -ψ 1 )(T ) 2 L 2 .
This inequality, together with the coercivity of Q 1 given in Lemma 5.2, concludes the proof of Theorem 1.3 for n = 1.

Second obstruction and the following. When n ≥ 2, it remains to prove that the cubic term u 1 3 L 2 can be, in a good functional setting, neglected in front of the drift u n 2 L 2 . This is done thanks to Gagliardo-Nirenberg inequalities [29, Theorem p.125]. Lemma 6.1. Let n ∈ N, n ≥ 2. There exists C > 0 such that for all T > 0 and v ∈ H 3(n-1) (0, T ), we have,

v (n-1) 3 L 2 (0,T ) ≤ C v 3(n-1) L 2 (0,T ) v 2 L 2 (0,T ) + CT -3(n-1) v 3 L 2 (0,T )
. Applying this inequality to u n , we have

u 1 3 L 2 (0,T ) ≤ C u (2n-3) L 2 (0,T ) + T -2n+3 u L 2 (0,T ) u n 2 L 2 (0,T ) .
using Leibniz formula. And the heredity holds after a shift of indexes in the second sum by definition (70) of the (α p k ) k=0,...,p . Moreover, one can also prove by induction that for all p ∈ N, As for all (n, m) = (0, 0), 4p-2-n-m 2 ≤ 2p -2, thanks to (73), every term of the two sums linked to some (k, n, m) such that (n, m) = (0, 0) can be bounded by µ 2 H 2p-2 . Therefore, one gets the existence of C > 0 such that ad p-1 A (µ), ad p A (µ) ϕ 1 , ϕ K -

p-1 k=0 α p-1 k α p 0 µ (2p-2-k) µ (2p+k) ϕ 1 , ϕ K + p k=0 α p k α p-1 0 µ (2p-k) µ (2p-2+k) ϕ 1 , ϕ K ≤ C µ 2 H 2p-2 . ( 74
)
Moreover, every remaining scalar product in (74) can be written as ± µ (2p-1) 2 ϕ 1 , ϕ K up to some negligible terms, as stated by the following estimates, ∀k = -p, . . . , p-1, µ (2p-2-k) µ (2p+k) ϕ 1 , ϕ K -(-1) k+1 µ (2p-1) 2 ϕ 1 , ϕ K ≤ C µ 2 H 2p-2 .

These estimates are proved by integrations by parts and using (73). Therefore, one gets the existence of C > 0 such that ad p-1 A (µ), ad p A (µ) ϕ 1 , ϕ K -α p where g is in C ∞ c (0, 1) such that 1 0 g (2n-1) (y) 2 dy = 1 and x(a) is defined by x(a) = x + 1l a>0 + x -1l a<0 , where x ± are in J ± \ ∪ n i=0 I ± i such that ϕ K (x + ) > 0 (and ϕ K (x -) < 0). First, µ a is supported on (x --a, x -) if a < 0 and (x + , x + + a) if a > 0. Thus, for a small enough, the support of µ a is in J ± \ ∪ n i=0 I ± i . Then, for example when a > 0, a change of variables and a Taylor expansion with respect to a give, ( µ (2n-1) a ) 2 ϕ 1 , ϕ K = 1 ϕ 1 (x + )ϕ K (x + )

x + +a x + g (2n-1) x -x + a 2 ϕ 1 (x)ϕ K (x)dx = a ϕ 1 (x + )ϕ K (x + ) 1 0 g (2n-1) (y) 2 ϕ 1 (x + + ay)ϕ K (x + + ay)dy

= a + O(|a|) 3 .
Moreover, for all k = 0, . . . , 2n -2, similarly, one gets (80) Moreover, the previous computations prove the continuity of a → F ( µ a ) on R * . Besides, by (80), F ( µ a ) → 0 when a → 0 so the map can be extended continuously at 0.

µ (k)
Step 3: F is onto by Brouwer. By Step 1 and Step 2, and more precisely with (78), (79) and (80), there exists δ > 0 such that, for all a ∈ R n+2 such that a = 0 and |a| < δ, one gets the existence of C > 0 such that |F (µ a ) -(a 0 , . . . , a n+1 )| ≤ C |a 0 , . . . , a n+1 | 3 with µ a := μa 0 ,...,an + µ a n+1 -A n+1 K (μa) . (81) Let ρ > 0 such that Cρ 2 < 1 2 and x ∈ R n+2 such that |x| < ρ 2 . Then, the map G x : a → a -F (µ a ) + x maps the ball B(0, ρ) to itself and is continuous. Therefore, Brouwer's fixed-point theorem entails the existence of a such that F (µ a ) = x. This holds for every x such that |x| < ρ 2 . Thus, taking x = (0, . . . , 0, ρ 4 ), one gets the existence of µ + such that F (µ + ) = (0, . . . , 0, ρ 4 ). Hence, the function µ + n+1 := 1 √ ρ/4

µ + satisfies F (µ + n+1 ) = (0, . . . , 0, 1). The function µ - n+1 is constructed similarly. And this ends the proof.

  )), by Lemma 4.4, one gets the existence of C = C(T ) > 0 such that

0 -( 1 2 H a+b 2 .

 0122 is true for p = 0 by definition of α 0 0 . Moreover, if (71) is true for some p ∈ N * , then -α p 0 ) + 2(1 -(-1) p α p p ) + (-1) p+1 α p+1 p+1 = 1,using the definition (70) of the coefficients (α p k ) k=0,...,p and the statement (71) for p.Step 2: Semi-explicit formula for the Lie brackets ([ad p-1 A (µ), ad p A (µ)]) p∈N * . Using the explicit formula (69) and Leibniz formula, one can compute thatad p-1 A (µ), ad p A (µ) f = n µ (2p-2-k) µ (2p-n+k-m) f (n+m) 2p-k) µ (2p-2-n+k-m) f (n+m) .(72)Many terms in these sums can be neglected. Indeed, for all a, b ∈ N, for all f ∈ C ∞ ([0, 1]), there exists C > 0 such that µ (a) µ (b) , f ≤ C µ

0 p- 1 k=0(- 1 ) 1 k-α p- 1 0 p k=0 (- 1 )Step 2 : 1 |ϕ 1

 1111k=01211 k+1 α p-k+1 α p k µ (2p-1) 2 ϕ 1 , ϕ K ≤ C µ 2 H 2p-2 . (75) Surjectivity of the last component of F . Let a ∈ R * . Define, µ a (x) := |a| 2n-(x(a))ϕ K (x(a))| g x -x(a) |a| ,

a 2 L 2

 22 (0,1) = |a| 4n-2k-1 |ϕ 1 (x(a))ϕ K (x(a))| 1 0 g (k) (y) 2 dy = O(|a| 3 ).Hence, by Proposition A.3 giving a semi-explicit formula for the quadratic forms (A p K ) p∈N * , one gets F ( µ a ) = (0, . . . , 0, a) + O(|a| 3 ).

  . In this section, the commutator of H 0 and H 1 is denoted by [H 0 , H 1 ] := H 0 H 1 -H 1 H 0 and •, • denotes the classical hermitian scalar product on C p . Assume that there exists K ∈ {1, . . . , p} such that H 0 and H 1 satisfy

	Theorem 2.1.

This work benefits from the support of ANR project TRECOS, grant ANR-20-CE40-0009.

So, this Gagliardo-Nirenberg inequality, together with (65) gives,

As before, expanding the notation of O, it means that there exists C, T 1 > 0 such that for all T ∈ (0, T 1 ), there exists η 1 > 0 such that for all u ∈ H 2n-3 (0, T ) with u 1 L ∞ (0,T ) < η 1 ,

Let A ∈ (0,

4 ), T * := min(T 1 ,

, T ∈ (0, T * ) and η := min(η 1 ,

Then, for all u ∈ H 2n-3 (0, T ) with u H 2n-3 (0,T ) < η, one has

This inequality, together with the coercivity of Q n given in Lemma 5.2 concludes the proof of Theorem 1.3 for n ≥ 2.

Remark 6.2. Notice that for n ≥ 2, the smallness assumption on the control depends on the final time T , which is not the case for n = 1. Such a phenomenon already appears in finite dimension (see [START_REF] Beauchard | Quadratic obstructions to small-time local controllability for scalar-input systems[END_REF]Section 2.4.4] for example).

The goal of this appendix is to prove the following theorem.

Theorem A.1. For all K ∈ N * , K ≥ 2 and n ∈ N * , there exists µ ∈ C ∞ c (0, 1) satisfying (H1) K -(H2) K,n .

Notice that in all this section, the coefficients (A p K ) p∈N * given in (3) and ( 4) are seen as quadratic forms with respect to µ.

A.1. Strategy of the proof. The existence of µ ∈ C ∞ c (0, 1) satisfying (H1) K -(H2) K,n can be brought down to three steps.

• First, instead of working with the series given in (3) and (4), we prove that the coefficients (A p K ) p∈N * can be namely written as a sum of L 2 (0, 1)-scalar products of two derivatives of µ with ϕ 1 ϕ K .

• Then, putting together the terms of the same order, we prove that the quadratic forms A p K can be written as

• Therefore, one can construct oscillating functions µ such that

The negligible terms are dealt with a Brouwer fixed-point theorem.

A.2. Computations on Lie brackets. In this appendix, we work with µ in C ∞ c (0, 1) to ensure the well-posedness of all the Lie brackets considered and also to not worry about boundary terms when performing integrations by parts. However, the following results hold with weaker regularity and fewer boundary conditions on µ.

Proposition A.2. For all p ∈ N * , the quadratic form A p K defined in (3) can be written as

Proof. First, one can prove by induction on q in N that for all a, b ∈ N * ,

Indeed, it holds for q = 0 by definition of ad 0 A (µ). Moreover, as the functions ϕ j are the eigenvectors of A which is a symmetric operator, the heredity follows after noticing that 3) and (67), one deduces that 2(-1) p-1 A p K is given by

which gives (66) using the symmetry/skew-symmetry of the operators ad k A (µ).

Proposition A.3. For all p ∈ N * , there exists a constant C > 0 and Q p a quadratic form such that for all µ ∈ C ∞ c (0, 1),

Proof. Let µ ∈ C ∞ c (0, 1).

Step 1: Computations of the Lie brackets (ad p A (µ)) p∈N . First, one can prove by induction on p ∈ N that

where the coefficients (α p k ) k=0,...,p are defined by induction as α 0 0 := 1 and for all p ∈ N * ,

Indeed, (69) holds for p = 0 by definition of ad 0 A (µ). Moreover, if (69) is true for some p ∈ N * , then

Using (71) and that α p 0 = (-1) p by the first equality of (70), one gets that

which concludes the proof of (68) using (66).

A.3. Proof of the existence of µ satisfying (H1) K -(H2) K,n .

Theorem A.4. Let K ∈ N * , K ≥ 2 and x ∈ (0, 1) such that sin(Kπx) = 0. There exists δ > 0 such that for every n ∈ N * , for every

As ϕ 1 > 0 on (0, 1), by definition of x and by continuity, there exists δ > 0 such that ϕ 1 ϕ K > 0 on (x, x + δ) and ϕ 1 ϕ K < 0 on (x -δ, x) (or conversely, but it works the same). Let us prove the statement by induction on n.

Initialization. Let J + be an open interval of (x, x+δ). One can construct µ + 1 = 0 supported on J + such that µ + 1 ϕ 1 , ϕ K = 0. Moreover, looking at (66), A 1 K (µ + 1 ) = (µ + 1 ) 2 ϕ 1 , ϕ K > 0 and thus after rescaling satisfies A 1 K (µ + 1 ) = 1. One can construct µ - 1 similarly. Heredity. Assume that the statement of the theorem holds for 1, . . . , n and we prove it for n + 1. Let J + (resp. J -) open interval of (x, x + δ) (resp. (x -δ, x)). There exists (I ± i ) i=0,...,n open intervals of J ± , two by two disjoints. By induction, for all i = 1, . . . , n, there exists µ ± i supported on

If needed, we denote by µ 0 i ≡ 0 for all i = 1, . . . , n. Moreover, one can also choose µ 0 supported on

We prove that F : µ → ( µϕ 1 , ϕ K , A 1 K (µ), . . . , A n+1 K (µ)) is onto and thereby prove the heredity.

Step 1: Surjectivity of the first n + 1-components of F . Let a = (a 0 , a 1 , . . . , a n ) ∈ R n+1 . As µ 0 and (µ ± i ) i=1,...,n have all disjoint supports and satisfy respectively (77) and ( 76