O Lancry

J.-L Farvacque

E Pichonat

C Gaquière

Indirect interband transition in hexagonal GaN

In this paper, we report on optical investigations with Raman experiment to underline a new Ultra Violet (UV) luminescence band in hexagonal Gallium Nitride (GaN) at 4.56 eV. GaN is a direct band gap semiconductor, the photoluminescence peak corresponding to the energy gap at 3.43 eV dominates the spectrum. Nevertheless, other electronic interband transitions can appear on the spectrum: the electronic indirect interband transitions. We attribute one of them to the observed new photoluminescence band at 4.56 eV. This interpretation is supported by photoluminescence spectra obtained on three different samples at room temperature and at -50°C with UV excitation source: mbd-266 nm solid laser (4.66 eV) and by the study of three criteria: the partly opposite parities of initial and final wave function, the implication of acoustic phonons and temperature control.

Introduction

Gallium Nitride and its alloys such as Al x Ga 1-x N (Aluminium Gallium Nitride) which remain direct gap for x up to 50% have been used for the fabrication of optoelectronic devices [START_REF] Lim | High responsitivity intrinsic photoconductors based on Al x Ga 1-x[END_REF][START_REF] Nakamura | Room-temperature continuous-wave operation of InGaN multi-quantum-well structure laser diodes with a lifetime of 27 hours[END_REF]. The interest of such materials results from the considerable improvements of growth techniques and advances in device applications. Consequently, there are many investigations of their basic physical properties. One of the most attractive properties is the direct character of such wide band gap materials (3.43 eV for GaN up to 4.53 eV for Al x Ga 1-x N with x = 0.5) [START_REF] Vurgaftman | Band parameters for nitrogen-containing semiconductors[END_REF][START_REF] Brunner | Optical constants of epitaxial AlGaN films and their temperature dependence[END_REF] allowing the fabrication of UV detectors as well as blue lasers and electronic devices [START_REF] Duccatteau | Output power density of 5.1/mm at 18 GHz with an AlGaN/GaN HEMT on Si substrate[END_REF] with high mechanical and chemical stability. A useful technique to study the wide band gap is photoluminescence [START_REF] Shigekawa | Device temperature measurement of highly biased AlGaN/GaN High-Electron-Mobility Transistors[END_REF] where light emission energy gives information about crystalline quality, defects, donors, acceptors, temperature, stress, alloy composition …In view to study the photoluminescence spectra of GaN and its alloys such as Al x Ga 1- x N up to x = 0.5, we used a very high energetic laser (4.66 eV) as excitation source. This, fortunately, enabled us to observe a new emission line at 4.56 eV in the GaN spectrum which we attribute to an indirect transition as will be shown afterward.

Then, we studied two different AlGaN samples grown on a GaN substrate with approximately the same x composition (13 and 15 %) but with different thicknesses (140 nm and 1 µm), in view to clarify the identification of the various luminescence peaks concerning such composite materials.

Experimental setup

The micro-photoluminescence measurements were performed at room temperature and at -50°C using a Horiba Jobin Yvon LabRam HR UV Micro-Raman system. Photoluminescence was excited by the mbd-266 nm solid laser (4.66 eV). The emission of the samples was then dispersed by a 800 mm focal length monochromator and detected with a backthinned illuminated CCD. A block diagram of the measurement system is drawn in figure 1. Spectral resolution after post mathematical treatment is 0.01 eV. The three different samples investigated in this paper are grown on sapphire substrates. The first one (sample A) is formed of only a non intentional doped (nid) 3 µm-thick GaN layer, the second one (sample B) is composed of an undoped 140 nm-thick Al 0.13 Ga 0.87 N layer (13% aluminium content) grown on a nid GaN layer and the third one (sample C) of an undoped 1 µm-thick Al 0.15 Ga 0.85 N layer (15% aluminium content) grown on a nid GaN layer. The substrate do not have any effect on the characteristics of the UV luminescence band in hexagonal GaN at 4.56 eV.

Results and discussion

Figure 2 shows luminescence spectrum of sample A composed essentially of Gallium Nitride (3 µmthick) on which we can identify the luminescence band gap at 3.43 eV [START_REF] Vurgaftman | Band parameters for nitrogen-containing semiconductors[END_REF] and another UV luminescence band at 4.56 eV. The latter is more energetic than the band gap transition, intensity and full width at half maximum (FWHM) of this band are lower than those of band gap luminescence, we attribute this band to a possible indirect interband transition. The very energetic laser, 4.66 eV, used to measure Al x Ga 1-x N photoluminescence for x up to 50% allowed us to discover this indirect transition. The phenomenon can be explained as follows. The incident laser beam excites the electrons of the GaN valence band which after thermalization can populate the various minima present in the conduction band as long as the thermal energy (here the room temperature energy 0.025 eV) remains weaker than the energy barrier E surrounding these secondary minima (cf. figure 3). Thus, electrons recombine emitting mainly photons whose energy is equal to the energy gap but also a weaker proportion of photons whose energy is equal to a possible indirect interband transition energy. Otherwise, if the condition E > kT is not satisfied, excited electrons in the conduction band have enough thermal energy to thermalize at the conduction band minimum prior any direct recombination. In the GaN band diagram, the only possible indirect interband transition at 4.56 eV is the one whose valence band minimum is located on the A point of the Brillouin zone (cf. figure 4).

Note that an indirect transition is possible as long as three criteria are satisfied:

1) The initial and final wave functions must have partly opposite parities. The parity of the conduction wave function at Point A and of the valence band at the point should be checked by ab initio calculations.

2) An indirect transition involves acoustic phonons in order to conserve crystal momentum. Thus, its intensity is generally very weak but should increase proportionally with kT. Such a temperature dependence should be checked experimentally.

3) However, the temperature must not be too important to avoid thermalization from the secondary minima towards the point minimum, which renders point 2 difficult to check experimentally. Table 1 displays temperature dependence of the 4.56 eV peak intensity. Measurements were performed at room temperature and at -50°C, photoluminescence peak areas were compared. The measurement system does not allow to compare absolute intensities of spectra peaks, it's necessary to compare relative intensities. When GaN sample is cooled to -50°C, ratio between 3,43 eV-luminescence band gap area and 4,56 eV-luminescence band area decreases compared with the same ratio obtained at room temperature. These results confirm the criteria of temperature which must be not too high. This criteria is essential for electrons to remain in the secondary minima and recombine emitting photons whose energy is equal to 4.56 eV. This finding enabled us to identify clearly the various luminescence peaks concerning AlGaN materials. Figure 5 and figure 6 show photoluminescence spectra of respectively sample B and sample C. The two samples are composed of Al x Ga 1-x N layer grown on GaN with approximately the same x composition (respectively 13 and 15 %) but with different thicknesses (respectively 140 nm and 1 µm). As the penetration depth of the UV laser (266 nm) in Al x Ga 1-x N with x 15 % is around 60 nm [START_REF] Perkowitz | Optical characterization of semiconductors[END_REF][START_REF] Muth | Absorption coefficient, energy gap, exciton binding energy, and recombination lifetime of GaN obtained from transmission measurements[END_REF][10][START_REF] Demiryont | Optical properties of aluminium oxynitrides deposited by laser-assisted CVD[END_REF], only Al x Ga 1-x N layer is probed. When we supposed to obtain the same photoluminescence spectra for the two samples, we have in fact two different spectra. For sample C, Al 0.15 Ga 0.85 N layer luminescence band appears at 3.7 eV (cf. figure 6). This value matches well with the calculated value of the Al 0.13 Ga 0.87 N direct band gap denotated E g (x) obtained with the formula [START_REF] Brunner | Optical constants of epitaxial AlGaN films and their temperature dependence[END_REF]: a non-linearity parameter. For sample B, like in the previous spectrum, luminescence band gap of Al 0.15 Ga 0.85 N is obtained at 3.7 eV. Furthermore, two other peaks are also identified: the first one, which is also the more intense, located at 3.4 eV, corresponds to the GaN direct band gap, the other one, clearly less intense and located at 4.56 eV, corresponds to the indirect interband transition described previously. Indeed, since the penetration depth is around 60 nm in AlGaN, the UV beam intensity which reaches the GaN layer, after the 140 nm-thick AlGaN layer crossing, is given by Lambert's law in an absorbent material:

) x 1 ( bx x E ) x 1 ( E ) x ( E AlN g GaN g g - - + - = (
with I 0 the incident beam intensity. This estimated value of the UV beam intensity which reaches the GaN layer is sufficient to obtain its photoluminescence spectrum. The study of these three samples clarifies the identification of various luminescence peaks obtained with a very energetic laser.

Conclusion

Optical investigations by photoluminescence enabled us to observe a new luminescence UV emission at 4.56 eV in hexagonal GaN. This band is attributed to electronic indirect interband transition. This transition was discovered thanks to very energetic laser as excitation source used to characterize by photoluminescence the Al x Ga 1-x N material with x up to 50 %. This interpretation is supported by photoluminescence spectra obtained on three different samples. Photoluminescence spectra of such composite materials are now easily treated, the identification of various luminescence peaks is now clearly demonstrated. This study is, to our knowledge, the only one which displays this indirect interband transition at 4.56 eV in GaN material by photoluminescence experiments. Table 1. Temperature dependence of the 4.56 eV peak intensity observed in photoluminescence spectra of the three samples obtained at -50°C and at room temperature.

Figure 1 .

 1 Figure 1. Block diagram of the measurement system.

Figure 2 .Figure 3 .

 23 Figure 2. Sample A photoluminescence spectrum. To obtain a better spectral resolution between 4.4 eV and 4.7 eV, illumination time of the sample under UV light is 600 times longer than illumination time used to obtain the whole photoluminescence spectrum.

Figure 4 .

 4 Figure 4. GaN band diagram [7].

Figure 6 .Figure 5 .

 65 Figure 6. Sample C photoluminescence spectrum. To obtain a better spectral resolution between 4.4 eV and 4.7 eV, illumination time of the sample under UV light is 600 times longer than illumination time used to obtain the whole photoluminescence spectrum.


Alain Audouard
email: alain.audouard@lncmi.cnrs.fr

Fabienne Duc

David Vignolles

Rustem B Lyubovskii

Laure Vendier

Gena V Shilov

Elena I Zhilyaeva

Rimma N Lyubovskaya

Enric Canadell

R N Lyubovskaia

O A Dyachenko

V V Gritsenko

S G Mkoyan

L O Atovmyan

Temperature-and pressure-dependent metallic states in (BEDT-TTF) 8 [Hg 4 Br 12 (C 6 H 5 Br) 2 ]

Temperature-driven metal-insulator and pressure-driven insulator-metal transitions observed in (BEDT-TTF) 8 [Hg 4 X 12 (C 6 H 5 Y) 2 ] with X = Y = Br are studied through band structure calculations based on X-ray crystal structure determination and Shubnikov-de Haas (SdH) oscillations spectra, respectively. In connection with chemical pressure effect, the transition, which is not observed for X = Cl, is due to gap opening linked to structural changes as the temperature decreases.

Even though many body interactions can be inferred from the pressure dependence of the SdH oscillations spectra, all the data can be described within a Fermi liquid picture. M X Y S a a a) (Cl, Br) 293 K e) (Cl, Br) 293 K b) (Br, Br) 200 K c) (Br, Br) 150 K d) (Br, Br) 120 K f) (Br, Br) 200 K g) (Br, Br) 150 K -7.7 -8.2 -8.7 Energy (eV) -7.6 -8.2 -8.8 Energy (eV) Energy (eV) -7.

I. INTRODUCTION

According to room temperature X-ray diffraction data 1-3 , the four charge transfer salts (BEDT-TTF) 8 [Hg 4 X 12 (C 6 H 5 Y) 2 ] (where X, Y = Cl, Br and BEDT-TTF stands for bisethylenedithio-tetrathiafulvalene) are isostructural compounds. In the following, they are referred to as (X, Y). Even though a metallic ground-state is observed for both (Cl, Cl) and (Cl, Br), i.e. X = Cl, a metal-insulator transition is reported as the temperature is lowered for X = Br. 4 Band structure calculations yield Fermi surface (FS) composed of one electron and one hole compensated quasi-two-dimensional orbit for both (Cl, Cl) 5 and (Cl, Br) 6 , referred to as the a orbits (see Fig. 1). Quantum oscillations spectra of these compounds yield frequencies F a = 241.5 ± 2.0 T for (Cl, Cl) 7 and 235.5 ± 1.0 T for (Cl, Br) 6,8,9 which corresponds to orbits area of 11% and 16% of the first Brillouin zone (FBZ) area, respectively, in satisfactory agreement with band structure calculations. In large enough magnetic field, the orbits are connected by magnetic breakdown (MB) junctions. As a result, these FS can be regarded as textbook cases for the study of quantum oscillations in networks of compensated orbits. 10 In line with this statement, numerous Fourier components that are linear combinations of F a and of the frequencies linked to the δ and ∆ pieces (see Fig. 1) are observed in the quantum oscillations spectra of these two compounds.

Oppositely, both (Br, Cl) and (Br, Br) have an insulating ground state. 4 Nevertheless, a metallic ground state is recovered for (Br, Br) under an applied pressure of 0.4 GPa and Shubnikov-de Haas (SdH) oscillations have been observed at 0.8 GPa in fields below 15 T. 11 Their frequency (F = 215 T) is close to that linked to the closed orbits a of the compounds with X = Cl, i.e. with metallic ground state at ambient pressure. This could suggest that the pressure-induced metallic state in (Br, Br) is similar to the ambient pressure ground state of (Cl, Cl) and (Cl, Br) and offer the possibility to study the applied pressure-induced evolution of the ground state. In that respect, metal-insulator transition in strongly correlated fermions systems have received much attention. As an external parameter such as doping or applied pressure is varied, numerous phases such as Mott insulator, superconducting and density wave states, eventually including quantum critical points, are observed.

As examples, quantum critical points have been inferred from doping dependence of the effective mass in both superconducting cuprates 12 and iron pnictides. [START_REF] Shishido | 14 CrysAlisPro[END_REF] In this work, we first focus on the temperature dependence of atomic and electronic structures of (Br, Br) in order to get information about the metal-insulator transition. It is concluded that the insulating state at ambient pressure is related to progressive band separation linked to chemical pressure as the temperature decreases, followed by a subtle conformational change of two of the donors in the vicinity of the transition, leading to gap opening. In a second step, the pressure-induced metallic state is considered. The FS topology deduced from SdH oscillations spectra is different from the FS in the metallic state at room temperature but similar to that of the compounds with X = Cl, indicating that applied pressure and temperature increase have different effects. Finally, even though an increase of many body effects is observed as approaching the insulating state as the applied pressure decreases, the data are in line with a Fermi liquid picture throughout the pressure range explored. The crystals studied were synthesized by the electrocrystallization technique reported in Ref. 4. Their mean dimensions were 0.1 × 0.1 × 0.02 mm 3 for X-ray diffraction experiments and 0.5 × 0.5 × 0.02 mm 3 for magnetoresistance measurements.

Single crystal X-ray diffraction data were collected at selected temperatures (100, 120, 150, 200, 293 K) with an Oxford Diffraction Xcalibur CCD diffractometer equipped with a Cryojet cooler device from Oxford Instruments and using a graphite-monochromated Mo-Kα radiation source (λ = 0.71073 Å). The unit cell determination and data reduction were carried out using the CrysAlisPro package. 14 Absorption correction was performed using the multiscan procedure. All the structures were solved by direct methods using SIR92 15 and refined with the JANA2006 program. [START_REF] Petricek | The crystallographic computing system[END_REF] All atoms except carbon were refined anisotropically.

The hydrogen atoms were not placed in either structure. Experimental details on the crystallographic data and structure refinements are given in Table I. Besides the complete data collections described above, the temperature dependence of cell parameters was obtained by refining the position of 250 to 400 reflections between 100 and 300 K with temperature steps in the range 10 to 25 K.

Tight-binding band structure calculations were based upon the effective one-electron Hamiltonian of the extended Hückel method. [START_REF] Whangbo | [END_REF] A modified Wolfsberg-Helmholtz formula was used to calculate the non-diagonal H ij values. 18 All valence electrons were taken into account in the calculations and the basis set consisted of Slater-type orbitals of double-ζ quality for C 2s and 2p, S 3s and 3p and of single-ζ quality for H. The ionization potentials, contraction coefficients and exponents were taken from previous works. 5,19 Alternating current (1 µA, 77 Hz) and (5 to 10 µA, 20 kHz) was injected parallel to the normal to the conducting plane for measurements of the interlayer zero-field resistance and magnetoresistance, respectively. Electrical contacts to the crystal were made using annealed platinum wires of 20 µm in diameter glued with graphite paste. Hydrostatic pressure was applied up to 1.0 GPa in an anvil cell designed for isothermal measurements in pulsed magnetic fields. 20 In the following, the pressure applied at room temperature is considered.

Magnetoresistance experiments were performed up to 55 T in pulsed magnetic field with pulse decay duration of 0.36 s, in the temperature range from 1.5 K to 4.2 K. Magnetic field was applied normal to the conducting plane. A lock-in amplifier with a time constant of 30 µs was used to detect the signal across the potential contacts. Analysis of the oscillatory magnetoresistance is based on discrete Fourier transforms of the data, calculated with a Blackman window.

III. ZERO-FIELD PROPERTIES

At ambient pressure, a strong increase of the zero-field resistance is observed in Fig. 2 To gain some insight on the origin of this change, the evolution of the crystal structure can be considered as the temperature is lowered from room temperature. However, in view of the large number of structural parameters, associated with the big unit cell (V cell = 3720 Å3 at room temperature), that should account for the relatively small change evidenced in the data of Fig. 3, it seems difficult to extract information from a structural study alone. Under such circumstances it is usually more informative to consider first the electronic structure.

Because of the directionality of the S and C p orbitals leading to the HOMO of BEDT-TTF, the HOMO...HOMO intermolecular interactions are quite sensitive to small structural variations and their analysis is extremely useful for understanding the relationship between features of the crystal and electronic structures (see e. the temperature dependence of the HOMO energy of B and D is strongly non-monotonic as well. Therefore, it is clear that some abrupt change affecting the electronic parameters occurs in the region highlighted in Fig. 4 around the four donors of type B and D.

Taking advantage of these results, we can examine the evolution of the crystal structure as the temperature is varied. At variance with the other structural parameters, the temperature According to band structure calculations reported in Fig. 1, a metallic state is obtained in the high temperature range. At room temperature and 200 K, the FS is composed of one elongated electron orbit with an area A e =13 % of the FBZ area and two hole orbits with area A h =A e /2. Although this topology is different from that of (Cl, Br) displayed in Fig. 1(e) it can be regarded as closely related, the only qualitative difference being a splitting of the hole orbit. In particular, we are still dealing with a compensated metal.

As the temperature is lowered, a sizeable shrinkage of the orbits area is observed since, at 150 K, the electron orbits are also split in two parts with an area which reduces to only 1 % of the FBZ area each. Further temperature lowering finally leads to gap opening (E g = 38 meV at 120 K, see Fig. 1(d)) and to the orbits suppression, accounting for the metal- as defined in Fig. 2. Lines are guides to the eye.

insulator transition observed in the ambient pressure data of Fig. 2. The difference between the behavior of (Cl, Br) and (Br, Br) can be accounted for by chemical pressure. Indeed, since Br atoms entering the HgX 3 anions are larger than Cl atoms, the distance between BEDT-TTF molecules is larger for X = Br than for X = Cl. As a result, band overlapping is less pronounced in (Br, Br), leading to a less conducting compound and leaving enough room for molecular changes or sliding as the temperature decreases. Coming back to the origin of the metal-to-insulator transition, we conclude that thermal contraction prepares the system by considerably reducing the bands overlap. At T c , this overlap is already quite small and a sudden conformational change of half of the donor molecules lead to the band gap opening, hence to the abrupt change in conductivity regime reported in Fig. 2.

As the applied pressure increases, T c decreases, the transition broadens and the ground state becomes clearly not insulating at pressures of a few tenth of a GPa, as displayed in Fig. 2(b). As a result, the temperature at which the resistance start to increase on cooling (T onset ) is significantly higher than T c in this pressure range. Nevertheless, both T c and T onset go to zero at P c = 0.34 ± 0.02 GPa as observed in Fig. 6. The temperature dependence of the resistivity in the pressure-induced metallic state is very similar to the data obtained for (Cl, Br). 6 More precisely, a bump is observed followed by a metallic behavior (see Fig. 7(a)). This behavior, observed in many correlated organic conductors, is attributed to a combined influence of disorder and electron correlations. 6,[24][25][26][27] Below about 25 K, the resistivity follows a T 2 variation (ρ = ρ 0 + AT 2 , see Fig. 7(b)) which is generally regarded as a signature of a correlated Fermi liquid. As already observed for the (Cl, Br) compound, the A coefficient strongly decreases as the applied pressure increases, suggesting a reduction of many body strength under pressure. 6 IV.

FIELD-DEPENDENT PROPERTIES UNDER PRESSURE

As displayed in Fig. 8, SdH oscillations are observed in the pressure-induced metallic state. Fourier analysis reveal a mean peak that ranges from 205 ± 2 T at 0.4 GPa to 225.5

± 1.0 T at 1.0 GPa (see Fig. 9). These values correspond to 9.5 % and 10.5 % of the ambient pressure FBZ area at 200K, respectively, which is close to the a orbit area of (Cl, Cl) and (Cl, Br) and to the ambient pressure electron orbit area of (Br, Br) deduced from band structure calculations at 200 K. The pressure sensitivity of this frequency is much larger than for (Cl, Br), namely, d[ln(F a )]/dP = 0.20 GPa -1 and 0.04 GPa -1 for (Br, Br) and (Cl, Br), respectively. In the case of (Cl, Br), the frequency linked to the quantum interferometer b, the area of which is just equal to the FBZ area (F b = 2F a +F δ +F ∆ ), is detected in the spectra. Since the same pressure dependence is observed for F b and F a , it can be concluded that the pressure dependence of F a is the same as that of the FBZ area.

In the case where this statement holds for (Br, Br) it indicates that the FBZ area, hence the unit cell parameters of (Br, Br), are more strongly pressure sensitive. Effect of chemical pressure can also be invoked to account for this difference in pressure sensitivity.

Two other points must be noticed. First, band structure calculations relevant to the metallic state at ambient pressure in the temperature range above T c predict split hole orbits (see Fig. 1), hence a frequency F a /2. Secondly, oscillatory spectra of Fig. 8 contain much less Fourier components than in the case of (Cl, Cl) and (Cl, Br). Indeed, apart the frequency F a and its second harmonics, only a frequency at 770 ± 20 T is observed at 1 GPa. Regarding the first point, it can be remarked that, due to the rather small field range in which oscillations are observed, Fourier peaks are rather broad which makes necessary to check if they actually correspond to a single frequency (see e.g. Ref. 28).

More insight about these two points can be obtained through field and temperature dependence of the a oscillations amplitude. In the following, this is considered in the framework oscillatory magnetoresistance and the corresponding Fourier analysis. This result demonstrates that no Fourier component with frequency F a /2 enters the oscillatory spectrum. The data are therefore more in line with a Fermi surface such as that of (Cl, Cl) and (Cl, Br) i.e. composed of one electron and one hole compensated orbit. As reported in Fig. 9(b), m * (a) strongly increases as the applied pressure decreases, as it is observed for (Cl, Br). In this latter case, the A coefficient of the zero-field resistivity is proportional to [m * (a)] 2 which has been interpreted on the basis of a pressure-induced decrease of electron correlations. 6,25 Although only a very rough agreement with this proportionality is observed for (Br, Br), A strongly decreases as the applied pressure decreases, as observed in Fig. 7(b). This feature suggests that applied pressure induces a decrease of the electron correlations, as well.

In the pressure range from 0.4 GPa to 0.7 GPa, Dingle plots of Fig. 11 exhibit a linear behavior which allow to deduce Dingle temperatures in the range 6 to 7 K. In contrast, a clear curvature is observed at 1.0 GPa. This curvature which is concomitant with the appearance of the frequency at 770 ± 20 T in the spectra, can be accounted for by the decrease of the MB field, hence of the field-dependent MB damping factor which includes four Bragg reflections for the a oscillations (R M B = [1-exp(-B 0 /B)] 2 ). 8 Within this assumption, the frequency at 770 T might correspond to the MB orbit 2a+δ, in which case the frequency linked to δ amounts to F δ = 319 ± 22 T which is very large compared to the data of (Cl, Br). [START_REF]Recall that this latter frequency corresponds to an 'orbit' which is forbidden[END_REF] Nonetheless, the effective mass linked to this frequency is 1.3 ± 0.2 which is close to 2m * (a), as expected for m * (2a + δ). 8 Unfortunately, reliable value of the MB field cannot be derived from the data due to the large Dingle temperature. 30 Nevertheless, it can be inferred that the MB field decreases as the applied pressure increases, as already observed for (Cl, Br). However, the MB field is much larger in (Br, Br) than in (Cl, Br) at a given applied pressure (see 6 and open symbols in Fig. 11) which justifies the absence of frequency mixing in (Br, Br).

Finally, as it is the case for (Cl, Br), the effective mass strongly decreases as the applied pressure increases. Such a variation, which is much too large to be accounted for by band mass decrease under pressure 6,25 , can be attributed to a pressure-induced decrease of electron correlations since the A coefficient of the T 2 law of the zero-field resistivity also strongly decreases under pressure. Phase diagram of strongly correlated systems, in which insulating, magnetic, superconducting and non-Fermi liquids states are observed, also exhibits large overlinear increase or even divergence of the effective mass as approaching the metal-insulator transition. As pointed out in Ref. 12, this behavior can be more clearly evidenced considering an effective Fermi energy E * F = ehF a /m * (a) which includes both the effective mass, eventually renormalized by interactions, and the electronic density through the oscillation frequency. As examples, this parameter goes to zero at specific doping values in underdoped superconducting cuprates for which quantum critical points are suggested 12 and at the superconducting-SDW transition in iron pnictides BaFe 2 (As 1-x P x ) 2 . [START_REF] Shishido | 14 CrysAlisPro[END_REF] At variance with the above examples and with numerous strongly correlated fermion systems, the studied compound contains no magnetic atom and no superconducting ground state is observed in the (temperature-pressure) phase diagram, at least down to 1.5 K. As evidenced in Fig. 9, despite the large pressure sensitivity of the effective mass, E * F remains finite at P c which is in line with a Fermi liquid picture.

V. SUMMARY AND CONCLUSION

According to band structure calculations relevant to the ambient pressure state in the high temperature range, the Fermi surface of (Br, Br) is composed of one electron orbit with area A e = 13 % of the FBZ area and two hole orbits with area A e /2. This topology is closely related to that of (Cl, Cl) and (Cl, Br), although the hole orbit is not split in the latter cases.

As the temperature decreases, the orbits area strongly decreases leading to gap opening below T c = 122 K, in line with the temperature dependence of the resistance. This behavior, which is not observed in (Cl, Cl) and (Cl, Br) can be attributed to a chemical pressure effect in which Br atoms of the [HgBr 3 ] -anions, larger than Cl atoms, reduce the compression of donors molecules. As a result, band overlapping is reduced which accounts for the different FS topology at room temperature. Besides, as the temperature is lowered, a change of two of the BEDT-TTF molecules conformation and a non-monotonous temperature dependence of the lattice parameters around T c are observed.

A pressure-induced metallic ground state is observed above P c = 0.34 GPa. SdH oscillations spectra observed in the pressure range between 0.4 GPa and 0.7 GPa exhibit only one Fourier component and its second harmonic. Frequencies are in agreement with both band structure calculations in the high temperature range for the electron orbits of (Br, Br) and data for the compensated orbits of (Cl, Cl) and (Cl, Br). No Fourier component with frequency F a /2 is observed which suggests that the FS under pressure is similar to that of (Cl, Cl) and (Cl, Br). The large pressure sensitivity of F a , and likely of the FBZ area, can be ascribed to the reduced chemical pressure as well.

Effective mass and coefficient of the T 2 law of the zero-field resistivity strongly decrease as the applied pressure increases. As in the case of (Cl, Br) this can be attributed to a pressure-induced decrease of many body effects. Nevertheless, at variance with more strongly correlated systems for which a superconducting state is generally observed in the vicinity of the insulating state in phase diagram, the effective Fermi energy remain finite at P c which is in line with a Fermi liquid picture throughout the phase diagram of the studied charge transfer salt family.

to a maximum at a field B max which should be lower than the maximum field reached in the experiment in order to derive reliable B 0 value. For the a oscillations that involves four Bragg reflections, B max = B 0 /ln(1 + B 0 eh/2π 2 k B m e m * a T D ) 31 which leads to B max as high as 56 T for the data at 1.0 GPa.

  FIG. 1. (color online) Band structure of (a) (Cl, Br) at 293 K, (b) to (d) (Br, Br) at 200 K, 150 K and 120 K, respectively. Dashed lines mark the Fermi level in (a), (b) and (c) and the top of the HOMO band in (d). Corresponding Fermi surfaces of (e) (Cl, Br) at 293 K, (f) and (g) (Br, Br) at 200 K and 150 K, respectively. Dark green [dark grey] and tan [light grey] areas stand for electron and hole orbits, respectively. Dashed lines in (e) mark the magnetic breakdown gaps; labels a depict the closed electron and hole orbits (arrows indicate the quasi particles path) which have the same area; δ and ∆ are 'forbidden orbits' liable to enter magnetic breakdown orbits. Data for (Cl, Br) are from Ref. 6.

  (a) as the temperature is lowered. No hysteresis is detected upon heating. Arrhenius plot of this data exhibits a clear inflection point as reported in Fig.2(b). The metal-insulator transition temperature (T c = 122 ± 5 K) can therefore be obtained through the sharp maximum of the parameter d[ln(R)]/d(1/T). More insight on this behavior can be obtained through Xray diffraction data. (BEDT-TTF) 8 [Hg 4 Br 12 (C 6 H 5 Br) 2 ] crystallizes in the triclinic crystal system, space group P1. Upon lowering the temperature down to 100 K, no symmetry change is observed. However, the temperature dependence of the cell parameters and volume reported in Fig.3exhibit a kink at T c which indicates that the transition is linked to a structural change.

FIG. 2 .

 2 FIG. 2. (color online) (a) Temperature dependence of the zero-field resistance at few applied pressures. Arrows indicate the onset of the resistance rise on cooling (T onset ). (b) Arrhenius plot of the resistance at ambient pressure and 0.33 GPa. The maximum of the logarithmic derivative displayed in the inset yields the temperature T c .

  g. Refs. 21 and 22). The so-called β HOM O-HOM O intermolecular interaction energies 23 for the different donor...donor interactions as well as the HOMO energies of the different BEDT-TTF donors as a function of temperature are considered in the following. As shown in Fig. 4, the donor lattice of the

FIG. 4 .

 4 FIG. 4. (color online) Donor lattice of (BEDT-TTF) 8 [Hg 4 Br 12 (C 6 H 5 Br) 2 ] where capital letters and roman numerals label the different types of donors and intermolecular interactions, respectively. The ellipse in dotted line highlights the part of the lattice where main structural changes should occur according to the electronic structure data.

FIG. 5 .Fig. 5 .

 55 FIG. 5. (color online) Evolution of the BEDT-TTF molecules conformation as a function of temperature. Labels are the same as in Fig. 4. Ellipses in dotted lines mark the C-C bonds that experience the most significant changes as the temperature varies (see text).

FIG. 6 .

 6 FIG. 6. (color online) Applied pressure dependence of T c (solid symbols) and T onset (open symbols)

FIG. 7 .

 7 FIG. 7. (color online) (a) Temperature dependence of the resistivity in the pressure-induced metallic state above P c and (b) low temperature part of the data plotted as a function of T 2 .

FIG. 8 .

 8 FIG. 8. (color online) Fourier analysis of the magnetoresistance data at 1.5 K displayed in the insert. Labels refer to the Fermi surface displayed in Fig. 1.

FIG. 9 .

 9 FIG. 9. (color online) Applied pressure dependence of (a) the oscillations frequency F a and (b) the effective mass m * (a). The Fermi energy calculated as E * F = ehF a /m * (a) is displayed in the insert. Solid and open symbols stand for (Br, Br) and (Cl, Br), respectively. Arrows marks the critical pressure P c . Data for (Cl, Br) are from Ref. 6.

Fig. 10 FIG. 10

 1010 Fig.10displays oscillatory magnetoresistance data at 0.4 GPa, which is the studied applied pressure closest to P c , and best fits to the data assuming only one Fourier component is involved. A good agreement between fits and experimental data is observed for both the

FIG. 11 .

 11 FIG. 11. (color online) Dingle plots of the a oscillations. Thermal damping factors (R T ) are calculated with effective masses reported in Fig. 9. Solid and empty symbols are experimental data of (Br, Br) and (Cl, Br), respectively, for various temperatures in the range 1.5 to 4.2 K.Solid and dashed lines are best fits of the Lifshits-Kosevich formalism to the data, including and excluding the contribution of magnetic breakdown, respectively. Magnetic breakdown field (B 0 ) is infinitely large for (Br, Br) below 1 GPa (i.e. R M B =1), while B 0 = 50 T and 21 T for (Br, Br) at 1.0 GPa and (Cl, Br) at 1.1 GPa 6 , respectively.

  31 D. Vignolles, A. Audouard, V. N. Laukhin, J. Béard, E. Canadell, N. G. Spitsina, and E. B. Yagubskii, Eur. Phys. J. B 55, 383 (2007).

TABLE I .

 I Experimental data and agreement factors for the structure refinements of (BEDT-TTF) 8 [Hg 4 Br 12 (C 6 H 5 Br) 2 ] at 100, 120, 150, 200 and 293 K. Reflections with I > 3σ(I) were considered as observed (σ(I) corresponds to the estimated standard deviations of measured individual intensities I).

	Temperature (K)	100	120	150	200	293
	Maximum θ (deg)	32	29	32	29	32
	Redundancy	1.65	2.81	1.67	2.83	1.70
	Number of measured reflections 18078/37105 24105/48089 18582/38435 19996/48673 13408/40197
	(observed/all)
	Unique reflections (observed/all) 10560/22447 9025/17107 10796/22940 7345/17186 7151/23540
	Rint (observed/all)	6.82/8.79	3.98/4.89	3.96/4.97	4.15/5.70	6.09/9.00
	Rwobs/Rwall	8.75/9.14	2.93/3.10	5.91/6.13	3.76/4.02	8.81/8.89
	g.o.f.obs/ g.o.f.all	3.57/2.53	1.16/0.88	2.40/1.69	1.47/1.01	3.71/2.11
	Number of refined parameters	554	554	554	554	554
	II. EXPERIMENTAL

TABLE II .

 II Calculated values of | β HOM O-HOM O | for the fourteen different donor...donor interactions and HOMO energies (eV) of the four different donors of (BEDT-TTF) 8 [Hg 4 Br 12 (C 6 H 5 Br) 2 ]. Labels are the same as in Fig. 4.

		| β HOM O-HOM O | (eV)
	Interaction 200 K	150 K	120 K
	I	0.1312	0.1385	0.1383
	II	0.1093	0.1113	0.1000
	III	0.1434	0.1560	0.1525
	IV	0.1328	0.1416	0.1418
	V	0.1197	0.1290	0.1222
	VI	0.1132	0.1319	0.1345
	VII	0.1299	0.1061	0.1284
	VIII	0.0719	0.0863	0.0847
	IX	0.1135	0.1158	0.1193
	X	0.2915	0.2938	0.2861
	XI	0.1649	0.1565	0.1312
	XII	0.2257	0.2385	0.2364
	XIII	0.2897	0.2928	0.2820
	XIV	0.3066	0.3273	0.3242
		HOMO energy (eV)
	Donor	200 K	150 K	120 K
	A	-8.4917	-8.5048	-8.5767
	B	-8.4808	-8.3775	-8.4266
	C	-8.4687	-8.4443	-8.4492
	D	-8.4738	-8.4561	-8.6004

In the case where Bragg reflections are involved in a MB orbit, the parameter A a /R T goes

ACKNOWLEDGMENTS

This work has been supported by FP7 I3 EuroMagNET II, DGI-Spain (Grants No. FIS2009-12721-C04-03 and CSD2007-00041) and RFBR grant 11-03-01039a.


Liviu I Ignat

AND Ademir F Pazoto
email: ademir@im.ufrj.br

Lionel Rosier
email: rosier@iecn.u-nancy.fr

INVERSE PROBLEM FOR THE HEAT EQUATION AND THE SCHR ÖDINGER EQUATION ON A TREE

Keywords: Carleman estimate, tree, network, heat equation, Schrödinger equation, inverse problem

In this paper we establish global Carleman estimates for the heat and Schrödinger equations on a network. The heat equation is considered on a general tree and the Schrödinger equation on a star-shaped tree. The Carleman inequalities are used to prove the Lipschitz stability for an inverse problem consisting in retrieving a stationary potential in the heat (resp. Schrödinger) equation from boundary measurements.

Introduction

In this paper we consider two inverse problems on a network formed by the edges of a tree. The problems we address here enter in the framework of quantum graphs. The name quantum graph is used for a graph considered as a one-dimensional singular variety and equipped with a differential operator. Those quantum graphs are metric spaces which can be written as the union of finitely many intervals, which are compact or [0, ∞) and any two of these intervals are either disjoint or intersect only at one of their endpoints.

Quantum graphs arise as simplified models in mathematics, physics, chemistry, and engineering (e.g., nanotechnology and microelectronics), when one considers propagation of waves through a quasi-one-dimensional system that looks like a thin neighborhood of a graph. We can mention in particular the quantum wires and thin waveguides. Differential operators on metric graphs arise in a variety of applications, to quote a few: carbon nano-structures [START_REF] Kuchment | On the spectra of carbon nano-structures[END_REF], photonic crystals [START_REF] Figotin | Spectral properties of thin-film photonic crystals[END_REF], high-temperature granular superconductors [START_REF] Alexander | Superconductivity of networks. A percolation approach to the effects of disorder[END_REF], quantum waveguides [START_REF] Carini | Bound states in waveguides and bent quantum wires. I. Applications to waveguide systems[END_REF], freeelectron theory of conjugated molecules in chemistry, quantum chaos, etc. For more details we refer the reader to the review papers [START_REF] Kuchment | Graph models for waves in thin structures[END_REF], [START_REF] Kuchment | Quantum graphs. I. Some basic structures[END_REF], [START_REF] Kuchment | Quantum graphs: an introduction and a brief survey[END_REF], [START_REF] Exner | Vertex coupling in quantum graphs: approximations by scaled Schrödinger operators[END_REF] and the references therein for more informations on this topic.

To be more precise we consider the heat equation on a 1-D network Γ given by the edges of a general tree and the Schrödinger equation on a star-shaped tree.

The first system we consider is the following one

         u t -∆ Γ u + pu = 0, in Γ × (0, T ), u = h, on ∂Γ × (0, T ), u(•, 0) = u 0 , in Γ, (1.1)
where ∆ Γ is the Laplace operator on the network Γ. The system is closed with the coupling conditions at the internal nodes of the tree, namely the continuity and the Kirchhoff's law on the flux at all internal vertices of Γ. Here, u is a collection of functions u α each of them satisfying a heat equation on some edge of the network.

Simultaneously with problem (1.1) we consider the following problem

         iu t + ∆ Γ u + pu = 0, in Γ × (0, T ), u = h, on ∂Γ × (0, T ), u(•, 0) = u 0 , in Γ, (1.2)
under similar coupling conditions as in the previous model.

In both cases we are interested in determining the potential p, a collection of functions defined on the edges of Γ, from boundary measurements. In the case of the first system, we are able to prove that we can recover p using only N -1 measurements, where N is the total number of exterior nodes of the network Γ. However, in the case of the second system, besides of the fact that we need to deal with a star-shaped network, we only can recover the potential p from measurements performed at all the exterior nodes of Γ.

The use of Carleman estimates to achieve uniqueness and stability results in inverse problems is well known. Some authors use local Carleman inequalities and deduce uniqueness and Hölder estimates. Others make use of global Carleman inequalities and deduce Lipschitz stability results and hence uniqueness results. We shall follow that second approach.

Inverse problems with a finite number of measurements have been widely studied by Bukhgeim and Klibanov (see [START_REF] Bukhgeȋm | Uniqueness in the large of a class of multidimensional inverse problems[END_REF], [START_REF] Klibanov | Inverse problems in the "large" and Carleman estimates[END_REF], and [START_REF] Klibanov | Inverse problems and Carleman estimates[END_REF]) by means of Carleman estimates (see also the book [START_REF] Isakov | Inverse Problems for Partial Differential Equations[END_REF] and the references therein). For a wide class of partial differential equations, their method provides the stability in the inverse problem, whenever a suitable Carleman estimate is available. Since [START_REF] Bukhgeȋm | Uniqueness in the large of a class of multidimensional inverse problems[END_REF], there have been many works based upon their methodology.

The theory of global Carleman estimates for parabolic operator has been largely developed since the work by Fursikov-Imanuvilov [START_REF] Fursikov | Controllability of evolution equations[END_REF] and it has been applied to many situations (e.g. to prove the controllability along the trajectories or the stability in inverse problems). Since a complete list of references is too long we refer the reader to [START_REF] Yamamoto | Carleman estimates for parabolic equations and applications[END_REF] for a quite complete review of the state of art.

Concerning the Schrödinger equation we refer to [START_REF] Baudouin | An inverse problem for Schrödinger equations with discontinuous main coefficient[END_REF][START_REF] Baudouin | Uniqueness and stability in an inverse problem for the Schrödinger equation[END_REF][START_REF] Cardoulis | Inverse problem for the Schrödinger operator in an unbounded strip[END_REF][START_REF] Cardoulis | Simultaneous identification of the diffusion coefficient and the potential for the Schrödinger operator with only one observation[END_REF][START_REF] Mercado | Inverse problems for the Schrödinger equation via Carleman inequalities with degenerate weights[END_REF] where Carleman estimates are proved and used to establish the stability for some inverse problems (see also [START_REF] Isakov | Carleman type estimates in an anisotropic case and applications[END_REF][START_REF] Lasiecka | Global uniqueness, observability and stabilization of nonconservative Schrödinger equations via pointwise Carleman estimates: Part I. H1-estimates[END_REF] for some other Carleman estimates for Schrödinger equation).

The same approach has given many results for the wave equation. Since a complete list is too long we quote only some of them, related to the same inverse problem consisting in retrieving a stationary potential in wave equation: [START_REF] Puel | Generic well-posedness in a multidimensional hyperbolic inverse problem[END_REF] and [START_REF] Yamamoto | Uniqueness and stability in multidimensional hyperbolic inverse problems[END_REF] for Dirichlet boundary data and a Neumann measurement and [START_REF] Yu | Global uniqueness and stability in determining coefficients of wave equations[END_REF] for Neumann boundary data and a Dirichlet measurement. These references are based on the use of local or global Carleman estimates. In the framework of Carleman inequalities on networks we mention the recent paper [START_REF] Baudouin | Global Carleman estimate on a network for the wave equation and application to an inverse problem[END_REF] where the authors establish a global Carleman estimate for the wave equation on a star-shaped tree and used it to derive the Lipschitz stability in an inverse problem. The Carleman estimate in [START_REF] Baudouin | Global Carleman estimate on a network for the wave equation and application to an inverse problem[END_REF] involves some positive definite matrix introduced in [START_REF] Benabdallah | Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications to controllability and an inverse problem[END_REF] to derive a Carleman estimate for the one-dimensional heat equation with discontinuous coefficients.

As far as we know, the determination of a time-independent potential for the heat or Schrödinger equation in a network-like structure has not been addressed in the literature yet. This type of problems has been studied for example for membranes or elastic strings (see for instance [START_REF] Avdonin | On an inverse problem for tree-like networks of elastic strings[END_REF] and the references therein).

Let us now state the main results of the paper. For a given initial data u 0 and a given boundary data h, we denote by u(p) the solution of the above systems associated with the potential p ∈ L ∞ (Γ, R). We introduce the space

H 2,1 (Γ × (0, T )) := L 2 (0, T ; H 2 (Γ)) ∩ H 1 (0, T ; L 2 (Γ)).
(See below Section 2 for the definition of H 2 (Γ).) We also introduce the ball B m (0

) := {q ∈ L ∞ (Γ, R); ||q|| L ∞ (Γ) ≤ m}.
Then the following stability results hold.

Theorem 1.1. Assume that p ∈ L ∞ (Γ), u 0 = u 0 (x), h = h(x, t) and r > 0 are such that the solution u(p) of (1.1) fulfills u(p) ∈ H 2,1 (Γ × (0, T )), ∂ t u(p) ∈ H 2,1 (Γ × (0, T ))
, and such that for some t 0 ∈ (0, T ) it holds |u(p)(•, t 0 )| ≥ r a.e. on Γ.

Then, for any m > 0 there exists a constant

C = C(m, ||∂ t u(p)|| L ∞ (Γ×(0,T )) , r) such that for any q ∈ B m (0) satisfying ∂ x [u(p) -u(q)](v, .) ∈ H 1 (0, T ) for all exterior nodes v, we have p-q L 2 (Γ) ≤ C [u(p) -u(q)](•, t 0 ) H 2 (Γ) + v∈E ∂ x [u(p) -u(q)](v, •) H 1 (0,T ) ,
where E denotes the set of all the exterior vertices of Γ except one.

For the second system, under the assumption that the network is a star-shaped tree, we can prove a similar stability result. Theorem 1.2. Assume that p ∈ L ∞ (Γ; R), u 0 = u 0 (x), h = h(x, t) and r > 0 are such that the solution of (1.2) satisfies

• u 0 (x) ∈ R or iu 0 (x) ∈ R a.e. in Γ, • |u 0 (x)| ≥ r > 0 a.e. in Γ, and • ∂ t u(p) ∈ H 2,1 (Γ × (0, T )).
Then, for any m ≥ 0, there exists a constant

C = C(m, ||∂ t u(p)|| H 2,1 (Γ×(0,T )) , r) > 0 such that for any q ∈ B m (0) satisfying ∂ t u(q) ∈ H 2,1 (Γ × (0, T )), we have ||p -q|| L 2 (Γ) ≤ C v∈∂Γ ||∂ x [u(p) -u(q)](v, .)|| H 1 (0,T ) •
The above theorems extend to networks classical results on inverse problems. To prove those results, we need to establish (new) global Carleman estimates for the heat (resp. the Schrödinger) equation on trees. Note that if we impose Kirchhoff-type conditions to the weight function at the internal vertices, the Carleman estimate cannot be derived. In our Carleman estimates, the weight function has to fulfill some nonlinear flux condition at each internal vertex. On the other hand, for the Schrödinger equation posed on a star-shaped tree with N external vertices, we consider a combination of N weight functions in order to cancel some "bad" terms at the internal vertex involving time derivatives. That strategy was used in [START_REF] Baudouin | A global Carleman estimate in a transmission wave equation and application to a one-measurement inverse problem[END_REF], with two different weight functions, in order to improve the observation region for the wave equation.

The article is organized as follows. In Section 2 we introduce the notations and some classical facts about the heat and Schrödinger equations on trees. Section 3 presents the analysis in the case of the heat equation. The Schrödinger equation is considered in Section 4. Finally we discuss some open problems in Section 5.

Notations and Preliminaries

Let Γ = (V, E) be a graph where V is the set of vertices and E the set of edges. The edges are assumed to be of finite length and their ends are the vertices of V . For each v ∈ V we denote

E v = {e ∈ E : v ∈ e}.
The multiplicity of a vertex of Γ is equal to the number of edges that branch out from it. If the multiplicity is equal to one, the vertex is said to be exterior, otherwise it is said to be interior. We assume that Γ does not contain vertices with multiplicity two, since they are irrelevant for our models.

From now on, we assume that Γ is a tree, that is, Γ is a planar finite connected graph without circuit (closed path). We fix an orientation of Γ and for each oriented edge e, we denote by I(e) its initial vertex and by T (e) its terminal one.

We identify every edge e of Γ with an interval I e , where I e = [0, l e ], l e being the length of e. This identification introduces a coordinate x e along the edge e.

Let v be a vertex of V and e be an edge in E v . We set

i(v, e) = 0 if v = I(e), l e if v = T (e).
We identify any function u on Γ with a collection {u e } e∈E of functions u e defined on the edges e of Γ. Each u e can be considered as a function on the interval I e . In fact, we use the same notation u e for both the function on the edge e and the function on the interval I e identified with e. For a function u : Γ → C, u = {u e } e∈E , we denote by f (u) : Γ → C the family {f (u e )} e∈E , where f (u e ) : e → C.

A function u = {u e } e∈E is continuous if and only if u e is continuous on I e for every e ∈ E, and u is continuous at the vertices of Γ:

u e (i(v, e)) = u e ′ (i(v, e ′ )), ∀ e, e ′ ∈ E v .
The space L p (Γ), 1 ≤ p < ∞ consists of all the functions u = {u e } e∈E on Γ that belong to L p (I e ) for each edge e ∈ E. That space is endowed with the norm

u p L p (Γ) = e∈E u e p L p (Ie) < ∞.
Similarly, the space L ∞ (Γ) consists of all the functions u = {u e } e∈E that belong to L ∞ (I e ) for each edge e ∈ E. The corresponding norm is

u L ∞ (Γ) = sup e∈E u e L ∞ (Ie) < ∞.
The Sobolev space H m (Γ), with m ∈ N * , consists of all the continuous functions on Γ that belong to H m (I e ) for each e ∈ E. It is endowed with the norm

u 2 H m (Γ) = e∈E u e 2 H m (e) < ∞.
The spaces L 2 (Γ) and H m (Γ) are Hilbert spaces when endowed with the inner products

(u, v) L 2 (Γ) = e∈E (u e , v e ) L 2 (Ie) = e∈E Ie u e (x)v e (x)dx and (u, v) H m (Γ) = e∈E (u e , v e ) H m (Ie) = e∈E m k=0 Ie d k u e dx k d k v e dx k dx.
H 1 0 (Γ) denotes the set of functions in H 1 (Γ) that vanish at the exterior vertices. We now introduce the Laplace operator ∆ Γ on the tree Γ. Even if it is a standard procedure, we prefer to recall it following [START_REF] Cattaneo | The spectrum of the continuous Laplacian on a graph[END_REF], for the sake of completeness. Consider the sesquilinear continuous form ϕ on H 1 0 (Γ) defined by

ϕ(u, v) = (u x , v x ) L 2 (Γ) = e∈E Ie u e x (x)v e x (x)dx.
We denote by D(∆ Γ ) the set of all the functions u ∈ H 1 0 (Γ) such that the linear map v ∈

H 1 0 (Γ) → ϕ u (v) := ϕ(u, v) satisfies |ϕ u (v)| ≤ C v L 2 (Γ)
for all v ∈ H 1 0 (Γ). For u ∈ D(∆ Γ ), we can extend ϕ u to a linear continuous mapping on L 2 (Γ). There is a unique element in L 2 (Γ), denoted by ∆ Γ u, such that

ϕ(u, v) = -(∆ Γ u, v) L 2 (Γ) for all v ∈ H 1 0 (Γ)
. We now define the normal exterior derivative of a function u = {u e } e∈E at the endpoints of the edges. For each e ∈ E and v an endpoint of e we consider the normal derivative of the restriction of u to the edge e of E v evaluated at i(v, e) to be defined by:

∂u e ∂n e (i(v, e)) = -u e x (0+) if i(v, e) = 0, u e x (l e -) if i(v, e) = l e .
With this notation it is easy to characterize D(∆ Γ ) (see [START_REF] Cattaneo | The spectrum of the continuous Laplacian on a graph[END_REF]): In other words D(∆ Γ ) is the space of all the continuous functions u = {u e } e∈E on Γ, such that for each edge e ∈ E, u e ∈ H 2 (I e ), and which vanish at each exterior node and fulfill the following Kirchhoff-type condition e∈E; T (e)=v u e x (l e -) -e∈E; I(e)=v u e x (0+) = 0 at each interior node v. It is easy to verify that (∆ Γ , D(∆ Γ )) is a linear, unbounded, self-adjoint, dissipative operator on L 2 (Γ), i.e. ℜ(∆ Γ u, u) L 2 (Γ) ≤ 0 for all u ∈ D(∆ Γ ).

D(∆ Γ ) = u = {u e } e∈E ∈ H 2 (Γ) ∩ H 1 0 (Γ);
3. The heat equation 3.1. Preliminaries and notations. In this section we introduce the notations for the elements of the considered tree. We mainly follow the notations of [START_REF] Dáger | Wave propagation, observation and control in 1-d flexible multi-structures[END_REF].

We first describe the procedure to index the edges and vertices of the tree. We first choose an exterior vertex, called the root of the tree and denoted by R. The remaining edges and vertices will be denoted by e α and O α , respectively, where α = (α 1 , . . . , α k ) is a multi-index (taking value in {1} ∪ k≥2 N k ). The multi-indices are defined by induction in the following way. For the edge containing the root R we choose the index 1. That edge is denoted by e 1 and its second end is denoted by O 1 . Assume now that the interior vertex O α , which is the end of the edge e α , has multiplicity equal to m α + 1. The m α edges, different from e α , that branch out from O α are denoted by e αβ with β ∈ {1, . . . , m α }. (See Figure 1.) Let now I be the set of the interior vertices of Γ and E be the set of the exterior vertices of Γ, R being excepted. We denote by

I I = {α, O α ∈ I}, I E = {α, O α ∈ E}.
the sets of the indices for the interior and exterior vertices (except the root R). With these notations I = I I ∪ I E is the set of the indices of all the vertices except the root R.

The length of the edge e α will be denoted by l α . Each e α is parameterized by the interval [0, l α ], so that the end O α of e α corresponds to x = l α while the origin of e α corresponds to x = 0.

3.2.

Carleman estimate for the heat equation. In this section we derive a Carleman estimate for the heat equation on a tree. The following properties for a function u = {u α } α∈I : Γ → R will be relevant for our work. (C1) Continuity condition at the internal vertices:

u α (l α ) = u αβ (0) for all α ∈ I I and β ∈ [[1, m α ]].
(C2) Flux condition at the internal vertices:

u α x (l α ) = m α β=1 u αβ x (0) for all α ∈ I I .
(C3) Vanishing condition at the root R and at the external vertices: u(v) = 0 for all v ∈ {R}∪E.

We introduce the set

Z = {u = {u α } α∈I : Γ × [0, T ] → R; u α ∈ C 2,1 ([0, l α ] × [0, T ]), u(•, t) satisfies (C1)-(C3)}. Note that u(•, t) ∈ D(∆ Γ ) for u ∈ Z and t ∈ [0, T ].
θ(x, t) = e λψ(x) t(T -t) , ϕ(x, t) = e λC ψ -e λψ(x) t(T -t) , x ∈ Γ, t ∈ (0, T ),
we have the following Carleman estimate.

Proposition 3.1. There exist a continuous function ψ : Γ → (0, +∞) and some positive constants λ 0 , s 0 , C such that for all λ ≥ λ 0 , s ≥ s 0 and q ∈ Z, it holds

T 0 Γ (sθ) -1 (|q t | 2 + |∆ Γ q| 2 ) + λ 2 (sθ)|q x | 2 + λ 4 (sθ) 3 |q| 2 e -2sϕ dxdt + T 0 λ(sθ)(|q x | 2 e -2sϕ )(R, t)dt ≤ C T 0 Γ |q t + ∆ Γ q| 2 e -2sϕ dxdt + v∈E T 0 λ(sθ)(|q x | 2 e -2sϕ )(v, t)dt . (3.3)
In the above proposition we have used the following notations

|q| 2 = {|q α | 2 } α∈I , |q t | 2 = {|q α t | 2 } α∈I , |q x | 2 = {|q α x | 2 } α∈I , etc. and Γ u dx = α∈I I α u α dx. Remark 3.2.
(1) The same inequality holds for the operator

∂ t -∆ Γ instead of ∂ t + ∆ Γ just by changing t into T -t.
(2) In the definition of Z, we can replace C 2,1 by H 2,1 , as well.

(3) We note that the result is false with only N -2 measurements for a star-shaped tree with N edges of length 1. Indeed, a non trivial solution of the heat equation that vanishes on N -2 edges, at the internal vertex and at all the external vertices does exist.

Proof. Let us consider the operator P = ∂ t + ∆ Γ . Set u = e -sϕ q and w = e -sϕ P (e sϕ u).

Following [START_REF] Rosier | Null controllability of the complex Ginzburg-Landau equation[END_REF] we obtain

w = M u = u t + sϕ t u + (∆ Γ u + 2sϕ x u x + s(∆ Γ ϕ)u + s 2 |ϕ x | 2 u) = M 1 u + M 2 u,
where

M 1 u = ∆ Γ u + sϕ t u + s 2 |ϕ x | 2 u (3.4)
and

M 2 u = u t + 2sϕ x u x + s(∆ Γ ϕ)u (3.5)
are the self-adjoint and skew-adjoint parts of M , respectively. Then

w 2 = M 1 u + M 2 u 2 = M 1 u 2 + M 2 u 2 + 2(M 1 u, M 2 u),
where • and (•, •) denote the norm and the inner product of L 2 (Γ × (0, T )), respectively.

Step 1. Exact computation of (M 1 u, M 2 u). Recall that

(M 1 u, M 2 u) = α∈I T 0 l α 0 (M 1 u) α (M 2 u) α dxdt.
We compute the integral term in the r.h.s. of the above identity only for one (arbitrary) edge e α , that we denote by e for simplicity. We assume that e is parameterized by x ∈ [0, l]. Also, where there is no confusion, we use the symbols and to denote T 0 l 0 and T 0 , respectively. We write

T 0 l 0 (M 1 u)(M 2 u) dxdt = I 1 + I 2 + I 3 + I 4 with I 1 = T 0 l 0 u xx u t , I 2 = T 0 l 0 u xx (2sϕ x u x + sϕ xx u), I 3 = T 0 l 0 (sϕ t u + s 2 ϕ 2 x u)(u t + 2sϕ x u x ), I 4 = T 0 l 0 (sϕ t u + s 2 ϕ 2 x u)(sϕ xx u).
For I 1 we have that

I 1 = - u x u xt + u x u t l 0 = u x u t l 0 .
We write the second term as

I 2 = I 1 2 + I 2 2
where

I 1 2 = 2s u xx ϕ x u x and I 2 2 = s u xx ϕ xx u.
Thus

I 1 2 = -s ϕ xx |u x | 2 + s ϕ x |u x | 2 l 0 and I 2 2 = -s u x (ϕ 3x u + ϕ xx u x ) + s u x ϕ xx u l 0 = s 2 ϕ 4x u 2 -s ϕ 3x u 2 2 l 0 -s ϕ xx |u x | 2 + s ϕ xx uu x l 0 . I 3 is decomposed as I 3 = I 1 3 + I 2 3
where

I 1 3 = (sϕ t u + s 2 |ϕ x | 2 u)u t I 2 3 = (sϕ t u + s 2 |ϕ x | 2 u)(2sϕ x u x ).
Then

I 1 3 = - (sϕ tt + 2s 2 ϕ x ϕ xt ) |u| 2 2 , I 2 3 = sϕ x (sϕ t + s 2 |ϕ x | 2 )∂ x (|u| 2 ) = -s 2 (ϕ x ϕ t ) x + s(ϕ 3 x ) x |u| 2 + sϕ x (sϕ t + s 2 |ϕ x | 2 )|u| 2 l 0 .
Finally,

I 4 = (s 2 ϕ t + s 3 |ϕ x | 2 )ϕ xx |u| 2 .
We conclude that for the edge e,

T 0 l 0 M 1 uM 2 u dxdt = -2s T 0 l 0 ϕ xx |u x | 2 + T 0 l 0 |u| 2 s 2 (ϕ 4x -ϕ tt ) -s 2 (|ϕ x | 2 ) t -s 3 ϕ x (|ϕ x | 2 ) x + T 0 u x u t + sϕ xx uu x + s|u x | 2 ϕ x + |u| 2 - s 2 ϕ 3x + s 2 ϕ x ϕ t + s 3 (ϕ x ) 3 l 0 . (3.6)
Summing now the above identity over all the edges {e α } α∈I we obtain the exact expression of the scalar product (M 1 u, M 2 u):

(M 1 u, M 2 u) = -2s T 0 Γ (∆ Γ ϕ)|u x | 2 + T 0 Γ |u| 2 s 2 (ϕ 4x -ϕ tt ) -s 2 (|ϕ x | 2 ) t -s 3 ϕ x (|ϕ x | 2 ) x + α∈I T 0 u α x u α t + sϕ α xx u α u α x + s|u α x | 2 ϕ α x + |u α | 2 - s 2 ϕ α 3x + s 2 ϕ α x ϕ α t + s 3 (ϕ α x ) 3 l α 0 . (3.7)
Step 2. Terms in the inner product related to the internal nodes. Let us pick an internal node O α . Using our previous notations, its parent edge is e α and its children edges are denoted by e αβ with β ∈ [[1, m α ]]. Let us denote by X α the sum of the boundary terms involving this internal node O α in the right hand side of (3.7). Thus

X α = T 0 u α x u α t + sϕ α xx u α u α x + s|u α x | 2 ϕ α x + |u α | 2 - s 2 ϕ α 3x + s 2 ϕ α x ϕ α t + s 3 (ϕ α x ) 3 (l α , t)dt - T 0 β∈[[1,m α ]] u αβ x u αβ t + sϕ αβ xx u αβ u αβ x + s|u αβ x | 2 ϕ αβ x + |u αβ | 2 - s 2 ϕ αβ 3x + s 2 ϕ αβ x ϕ αβ t + s 3 (ϕ αβ x ) 3 (0, t)dt.
Moreover, in (3.7) we also have contributions from the exterior nodes in E and from the root R. These contributions are given by

Y = -s T 0 |u 1 x | 2 ϕ 1 x (0, t)dt + s α∈I E T 0 |u α x | 2 ϕ α x (l α , t)dt. (3.8)
Let us now define the weight function ψ = {ψ α } α∈I on the tree as follows. The components

ψ α : [0, l α ] → R are chosen in such a way that ψ α ∈ C ∞ ([0, l α ]) and (B1) |ψ α x (x)| 2 + ψ α xx (x) ≥ 0 on [0, l α ], (B2) ψ α x > 0 on [0, l α ], (B3) 3 4 C ψ ≥ ψ α > 2 3 C ψ on [0, l α ], for some positive constant C ψ , (B4) |ψ α xx | ≤ Kψ α x on [0, l α ] for some positive constant K, (B5) ψ α (l α ) = ψ αβ (0) for all α ∈ I I , β ∈ [[1, m α ]], (B6) ψ αβ x (0) -(m α + 1)ψ α x (l α ) > 0 for all α ∈ I I , β ∈ [[1, m α ]], (B7) β∈[[1,m α ]] (ψ αβ x (0)) 3 -(ψ α x (l α )) 3 -(m α +1)ψ α x (l α ) ψ α x (l α )- β∈[[1,m α ]] ψ αβ x (0) 2 > 0 for all α ∈ I I .
Finding a set of functions as above is easy. We can even take ψ α to be affine, ψ α (x) = a α x+b α . The coefficients a α and b α are positive numbers that satisfy (P1) 3 4

C ψ ≥ a α l α + b α > b α > 2 3 C ψ for all α ∈ I, (P2) a α l α + b α = b αβ for all α ∈ I I and β ∈ [[1, m α ]], (P3) a αβ -(m α + 1)a α > 0 for all α ∈ I I and β ∈ [[1, m α ]], (P4) β∈[[1,m α ]] (a αβ ) 3 -(a α ) 3 -(m α + 1)a α a α - β∈[[1,m α ]] a αβ 2 > 0 for all α ∈ I I .
Let us first deal with the conditions (P2)-(P4). We define the constants corresponding to the edge e 1 by a 1 = 2 and b 1 = 1. Assuming that we have already constructed a α and b α for some multi-index α, then b αβ is given by (P2). Next, we have to find a αβ large enough to satisfy (P3)-(P4). Let us choose a αβ = r α a α . Obviously, for large enough r α , depending on m α , conditions (P3) and (P4) are satisfied. Finally, assume that all the coefficients a α and b α have been defined to satisfy (P2)-(P4). Adding 2 3 C ψ to all the b αβ , we see that (P1) is fulfilled for C ψ large enough, while (P2)-(P4) still hold true.

Let us split X α into X α = X α 1 + X α 2 + X α 3 + X α 4
, where

X α 1 := T 0 [u α x u α t ](l α , t) - β∈[[1,m α ]] [u αβ x u αβ t ](0, t) dt, X α 2 := T 0 [sϕ α xx u α u α x ](l α , t) - β∈[[1,m α ]] [sϕ αβ xx u αβ u αβ x ](0, t) dt, X α 3 := T 0 [s|u α x | 2 ϕ α x ](l α , t) - β∈[[1,m α ]] [s|u αβ x | 2 ϕ αβ x ](0, t) dt, X α 4 := T 0 |u α | 2 - s 2 ϕ α 3x + s 2 ϕ α x ϕ α t + s 3 (ϕ α x ) 3 (l α , t) - β∈[[1,m α ]] |u αβ | 2 - s 2 ϕ αβ 3x + s 2 ϕ αβ x ϕ αβ t + s 3 (ϕ αβ x ) 3 (0, t) dt.
We now estimate each term X α i , i = 1, ..., 4. Using the definition of the function u we have for any index α ∈ I the following identities u α

x = e -sϕ α (-sϕ α x q α + q α x ), u α t = e -sϕ α (-sϕ α t q α + q α t ).

Let us set

u(O α , t) = u α (l α , t) = u αβ (0, t) and ϕ(O α , t) = ϕ α (l α , t) = ϕ αβ (0, t) for any α ∈ I I and β ∈ [[1, m α ]]
. With these notations we have

X α 1 = T 0 u α x u α t (l α , t) - β∈[[1,m α ]] u αβ x u αβ t (0, t) dt = T 0 u t (O α , t)e -sϕ(O α ,t) -sϕ α x q α (l α ) + s β∈[[1,m α ]] ϕ αβ x q αβ (0) + q α x (l α ) - β∈[[1,m α ]] q αβ x (0) dt = T 0 u t (O α , t)e -sϕ(O α ,t) -sϕ α x q α (l α ) + s β∈[[1,m α ]] ϕ αβ x q αβ (0) dt = T 0 u t (O α , t)u(O α , t) -sϕ α x (l α ) + s β∈[[1,m α ]] ϕ αβ x (0) dt = - T 0 -sϕ α xt (l α , t) + s β∈[[1,m α ]] ϕ αβ xt (0, t) |u(O α , t)| 2 2 dt. (3.9)
Let us estimate X α 2 . Using property (B4) we infer that

|ϕ α xx | = λe λψ α t(T -t) |λ(ψ α x ) 2 + ψ α xx | ≤ T 2 4 |ϕ α x | 2 + K|ϕ α x |.
This gives that

|X α 2 | = T 0 sϕ α xx u α u α x (l α , t) -s β∈[[1,m α ]] ϕ αβ xx u αβ u αβ x (0, t) dt ≤ s 2 2 T 0 |u α | 2 ( T 2 4 |ϕ α x | + K) 2 |ϕ α x |(l α , t)dt + 1 2 T 0 |ϕ α x ||u α x | 2 (l α , t)dt + β∈[[1,m α ]] s 2 2 T 0 |u αβ | 2 ( T 2 4 |ϕ αβ x | + K) 2 |ϕ αβ x |(0, t)dt + 1 2 T 0 |ϕ αβ x ||u αβ x | 2 (0, t)dt = s 2 2 T 0 |u(O α , t)| 2 ( T 2 4 |ϕ α x | + K) 2 |ϕ α x |(l α , t) + β∈[[1,m α ]] ( T 2 4 |ϕ αβ x | + K) 2 |ϕ αβ x |(0, t) dt + 1 2 T 0 |ϕ α x ||u α x | 2 (l α , t)dt + β∈[[1,m α ]] |ϕ αβ x ||u αβ x | 2 (0, t) dt. (3.10)
To estimate the term

T 0 |ϕ α x ||u α x | 2 (l α , t
)dt which occurs in (3.10) and in X α 3 , we notice that

u α x (l α , t) - β∈[[1,m α ]] u αβ x (0, t) = e -sϕ(O α ,t) (-s) ϕ α x (l α , t) - β∈[[1,m α ]] ϕ αβ x (0, t) q(O α , t), hence |u α x (l α , t)| 2 ≤ (m α + 1) β∈[[1,m α ]] |u αβ x (0, t)| 2 + s 2 |u(O α , t)| 2 ϕ α x (l α , t) - β∈[[1,m α ]] ϕ αβ x (0, t) 2 .
(3.11) We infer that

T 0 |ϕ α x ||u α x | 2 (l α , t)dt ≤ (m α + 1) T 0 |ϕ α x (l α , t)| β∈[[1,m α ]] |u αβ x (0, t)| 2 + s 2 |u(O α , t)| 2 ϕ α x (l α , t) - β∈[[1,m α ]] ϕ αβ x (0, t) 2 (3.12)
Combined to (3.10) and to the fact that ϕ αβ x (0, t) ≤ 0 by (B2), this yields

X α 2 + X α 3 ≥ - s 2 2 T 0 |u(O α , t)| 2 ( T 2 4 |ϕ α x | + K) 2 |ϕ α x |(l α , t) + β∈[[1,m α ]] ( T 2 4 |ϕ αβ x | + K) 2 |ϕ αβ x |(0, t) dt + (s - 1 2 )
T

0 β∈[[1,m α ]] |ϕ αβ x ||u αβ x | 2 (0, t)dt -(s + 1 2 )(m α + 1) T 0 |ϕ α x (l α , t)| β∈[[1,m α ]] |u αβ x (0, t)| 2 + s 2 |u(O α , t)| 2 ϕ α x (l α , t) - β∈[[1,m α ]]
ϕ αβ x (0, t) 2 .

(3.13)

Using the definition of X α and estimates (3.9), (3.13) we obtain that

X α ≥ Z α 1 + Z α 2
where

Z α 1 = - T 0 -sϕ α xt (l α , t) + s β∈[[1,m α ]] ϕ αβ xt (0, t) |u(O α , t)| 2 2 dt - s 2 2 T 0 |u(O α , t)| 2 ( T 2 4 |ϕ α x | + K) 2 |ϕ α x |(l α , t) + β∈[[1,m α ]] ( T 2 4 |ϕ αβ x | + K) 2 |ϕ αβ x |(0, t) dt -(s 3 + 1 2 s 2 )(m α + 1) T 0 |u(O α , t)| 2 |ϕ α x (l α , t)| ϕ α x (l α , t) - β∈[[1,m α ]] ϕ αβ x (0, t) 2 dt + T 0 |u(O α , t)| 2 | - s 2 ϕ α 3x + s 2 ϕ α x ϕ α t + s 3 (ϕ α x ) 3 (l α , t)dt - T 0 |u(O α , t)| 2 β∈[[1,m α ]] - s 2 ϕ αβ 3x + s 2 ϕ αβ x ϕ αβ t + s 3 (ϕ αβ x ) 3 (0, t)dt and Z α 2 = (s- 1 2
)

β∈[[1,m α ]] T 0 |ϕ αβ x ||u αβ x | 2 (0, t)dt-(s+ 1
2 )(m α +1) T 0 |ϕ α x (l α , t)| β∈[[1,m α ]] |u αβ x (0, t)| 2 dt.
We notice that |ϕ α x | 3 ≥ c(λθ) 3 while, with (B3),

|ϕ α x | + |ϕ α x | 2 + |ϕ α xt | + |ϕ α x ϕ α t | + |ϕ α 3x | ≤ c(λθ) 3 . It follows that Z α 1 = T 0 |u(O α , t)| 2 × × s 3 (ϕ α x ) 3 (l α , t) - β∈[[1,m α ]] (ϕ αβ x ) 3 (0, t) -(m α + 1)|ϕ α x (l α , t)| ϕ α x (l α , t) - β∈[[1,m α ]] ϕ αβ x (0, t) 2 + . . . = T 0 |u(O α , t)| 2 × × (sλθ) 3 -(ψ α x ) 3 (l α ) + β∈[[1,m α ]] (ψ αβ x ) 3 (0) -(m α + 1)|ψ α x (l α )| ψ α x (l α ) - β∈[[1,m α ]] ψ αβ x (0) 2 + O(s 2 λ 3 θ 3 )
and

Z α 2 = T 0 β∈[[1,m α ]] |u αβ x (0, t)| 2 sλθ ψ αβ x (0) -(m α + 1)|ψ α x (l α )| + O(λθ) dt.
Looking at the coefficient of s 3 in Z α 1 and of s in Z α 2 and using (B6), (B7), and (3.12), we obtain that for s ≥ s 0 and λ ≥ λ 0 (with s 0 , λ 0 large enough)

X α ≥ Z α 1 + Z α 2 ≥ C T 0 s 3 λ 3 θ 3 |u(O α , t)| 2 dt + C T 0 sλθ |u α x (l α , t)| 2 + β∈[[1,m α ]] |u αβ x (0, t)| 2 dt. (3.14)
In particular, X α > 0.

Step 3. Estimation of the integrals along the edges. We need the following lemma.

Lemma 3.3. [32, Claim 1]

There exist λ 1 ≥ λ 0 , s 1 ≥ s 0 and A > 0 such that for all λ ≥ λ 1 , s ≥ s 1 , it holds

T 0 Γ |u| 2 s 2 (ϕ 4x -ϕ tt ) -s 2 (|ϕ x | 2 ) t -s 3 ϕ x (|ϕ x | 2 ) x dxdt ≥ Aλs 3 T 0 Γ |u| 2 |ϕ x | 3 dxdt. (3.15)
As the proof of [32, Claim 1] does not involve any integration by parts in x, it is still valid in our context.

The following lemma is inspired by [START_REF] Rosier | Null controllability of the complex Ginzburg-Landau equation[END_REF]Claim 2]. Lemma 3.4. There exist s 2 ≥ s 1 , λ 2 ≥ λ 1 , and a positive constant C such that for all λ ≥ λ 2 and s ≥ s

2 λs T 0 Γ |ϕ x ||u x | 2 + λs -1 T 0 Γ |ϕ x | -1 |∆ Γ u| 2 ≤ C s -1 M 1 u 2 + λs 3 T 0 Γ |ϕ x | 3 |u| 2 + λs T 0 α∈I |ϕ α x u α u α x |(0) + |ϕ α x u α u α x |(l α ) + λs T 0 α∈I |(|ϕ α x |) x ||u α | 2 (0) + |(|ϕ α x |) x ||u α | 2 (l α ) . (3.16)
Proof. Let us pick any edge e α . To simplify the writing, we remove the index α in our computations. Using the definition of M 1 u (see (3.4)) we have that

s -1 T 0 l 0 |ϕ x | -1 |u xx | 2 = s -1 T 0 l 0 |ϕ x | -1 |M 1 u -sϕ t u -s 2 ϕ 2 x u| 2 ≤ Cs -1 T 0 l 0 |ϕ x | -1 |M 1 u| 2 + s 2 |ϕ t | 2 u 2 + s 4 |ϕ x | 4 u 2 .
Using property (B3) we get that |ϕ t | ≤ C|ϕ x | 2 . Therefore, we have for some constant A > 0

s -1 T 0 l 0 |ϕ x | -1 |u xx | 2 ≤ A M 1 u 2 λs + s 3 T 0 l 0 |ϕ x | 3 u 2 .
(3.17)

The first term in the left hand side of (3.16) satisfies

λs |ϕ x ||u x | 2 = λs |ϕ x |(-u xx )u - (|ϕ x |) x u x u + T 0 |ϕ x |uu x l 0 ≤ λ 2s |ϕ x | -1 |u xx | 2 + λs 3 2 |ϕ x | 3 |u| 2 + λs 2 (|ϕ x |) xx |u| 2 + λs T 0 |ϕ x |uu x -(|ϕ x |) x |u| 2 2 l 0 ≤ λ 2 A M 1 u 2 λs + s 3 |ϕ x | 3 u 2 + λs 3 2 |ϕ x | 3 |u| 2 + λs 2 (|ϕ x |) xx |u| 2 + λs T 0 |ϕ x |uu x -(|ϕ x |) x |u| 2 2 l 0 ≤ A 2 s -1 M 1 u 2 + s 3 λ |ϕ x | 3 u 2 + λs 3 2 |ϕ x | 3 |u| 2 + λs 2 (|ϕ x |) xx |u| 2 + λs T 0 |ϕ x ||u||u x | + |(|ϕ x |) x | |u| 2 2 (0) + λs T 0 |ϕ x ||u||u x | + |(|ϕ x |) x | |u| 2 2 (l).
(3.18)

The claim follows by summing (3.17) and (3.18) over all the edges.

Step 4. Conclusion. By (3.7), (3.15) and (B1), we get for λ ≥ 1

w 2 = M 1 u + M 2 u 2 = M 1 u 2 + M 2 u 2 + 2(M 1 u, M 2 u) = M 1 u 2 + M 2 u 2 + 2 α∈I I X α + Y -2s T 0 Γ ϕ xx |u x | 2 + T 0 Γ |u| 2 s 2 (ϕ 4x -ϕ tt ) -s 2 (|ϕ 2 x |) t -s 3 ϕ x (|ϕ x | 2 ) x ≥ M 1 u 2 + M 2 u 2 + 2 α∈I I X α + Y + 2s T 0 Γ (λ 2 ψ 2 x + λψ xx )θ|u x | 2 + Aλs 3 T 0 Γ |u| 2 |ϕ x | 3 ≥ M 1 u 2 + M 2 u 2 + α∈I I X α + Y + Aλs 3 T 0 Γ |u| 2 |ϕ x | 2 . ( 3
M 2 u 2 + M 1 u 2 (1 - A 2s ) + Aλs 3 2 |u| 2 |ϕ x | 3 + Aλs 2C |ϕ x ||u x | 2 + Aλ 2sC |ϕ x | -1 |∆ Γ u| 2 + α∈I I X α + Y ≤ w 2 + A 2 B, (3.20)
where

B = B 1 + B 2 = λs T 0 α∈I |ϕ α x u α u α x |(0) + |ϕ α x u α u α x |(l α ) + λs T 0 α∈I |(|ϕ α x |) x ||u α | 2 (0) + |(|ϕ α x |) x ||u α | 2 (l α ) .
We now prove that for s large enough, the term B is small compared to α∈I I X α , so that B can be absorbed by the left hand side of (3.20). Using (3.14) and the fact that u vanishes at the vertices of E ∪ R = ∂Γ, we see that

B 2 ≤ Cλ 3 s T 0 α∈I θ|u(O α , t)| 2 dt = Cλ 3 s T 0 α∈I I θ|u(O α , t)| 2 dt ≤ C s 2 α∈I I X α . (3.21)
Using again the fact that u vanishes at the vertices of ∂Γ, we obtain with (3.14) that

B 1 ≤ Cλs T 0 α∈I I |ϕ α x u α |(O α , t) |u α x (l α , t)| + β∈[[1,m α ]] |u αβ x (0, t)| ≤ C T 0 α∈I I (sλ) 2 |ϕ α x ||u α | 2 (O α , t) + |ϕ α x (O α , t)| |u α x (l α , t)| 2 + β∈[[1,m α ]] |u αβ x (0, t)| 2 dt ≤ C s α∈I I X α . (3.22)
Gathering together (3.20), (3.21) and (3.22), we obtain

M 2 u 2 + M 1 u 2 (1 - A 2s ) + Aλs 3 2 T 0 Γ |u| 2 |ϕ x | 3 + Aλs 2C |ϕ x ||u x | 2 + Aλ 2sC T 0 Γ |ϕ x | -1 |∆ Γ u| 2 + (1 - C s ) α∈I I X α + Y ≤ w 2 .
Writing explicitly the term Y and tacking into account the sign of the functions ψ α x occurring in Y , we get for s and λ large enough

M 1 u 2 + M 2 u 2 + λs 3 T 0 Γ |u| 2 |ϕ x | 3 + λs T 0 Γ |ϕ x ||u x | 2 + λs -1 T 0 Γ |ϕ x | -1 |∆ Γ u| 2 + α∈I I X α + T 0 λsθ 1 |u x | 2 (R, t)dt ≤ C w 2 + T 0 α∈I E λsθ α |u α x | 2 (l α , t)dt . (3.23)
Finally, using the definition of M 2 we get

λs -1 T 0 Γ |ϕ x | -1 |u t | 2 dxdt ≤ Cλs -1 T 0 Γ |ϕ x | -1 |M 2 u| 2 + s 2 |ϕ x | 2 |u x | 2 + s 2 |ϕ xx | 2 |u| 2 ≤ C T 0 Γ s -1 |M 2 u| 2 + λs|ϕ x ||u x | 2 + λs|ϕ x | -1 |ϕ xx | 2 |u| 2 . (3.24)
From (3.23) and (3.24), we infer that for s ≥ s 3 and λ ≥ λ 3 (with s 3 , λ 3 large enough) we have that

M 1 u 2 + M 2 u 2 + λs 3 T 0 Γ |u| 2 |ϕ x | 3 + λs T 0 Γ |ϕ x ||u x | 2 + λs -1 T 0 Γ |ϕ x | -1 (|∆ Γ u| 2 + |u t | 2 ) + α∈I I X α + T 0 λsθ 1 |u x | 2 (R, t)dt ≤ C w 2 + T 0 α∈I E λsθ α |u α x | 2 (l α , t)dt .
Replacing u by e -sϕ q in the last inequality, we readily obtain (3.3).

Inverse problem.

Before proving the stability result in Theorem 1.1 we need to analyze the following system:

                             u α t (x, t) = u α xx (x, t) + b α (x)u α (x, t) + R α (x, t)f α (x), (x, t) ∈ (0, l α ) × (0, T ), α ∈ I, u α (l α , t) = 0, t ∈ (0, T ), α ∈ I E , u 1 (0, t) = 0, t ∈ (0, T ), u α (l α , t) = u αβ (0, t), t ∈ (0, T ), α ∈ I I , β ∈ [[1, m α ]], u α x (l α , t) = m α β=1 u αβ x (0, t), t ∈ (0, T ), α ∈ I I , (3.25) where b = {b α } α∈I ∈ L ∞ (Γ). Proposition 3.5. Assume that u = {u α } α∈I is a solution of (3.25) which satisfies u t ∈ H 2,1 (Γ× (0, T )). If R = {R α (x, t)} α∈I is such that R t ∈ L ∞ (Γ × (0, T )) and that |R(x, t 0 )| ≥ r > 0,
for a.e. x ∈ Γ and some t 0 ∈ (0, T ), (3.26) then there exists a positive constant

C = C(||R t || L ∞ (Γ×(0,T )) , ||b|| L ∞ (Γ) , r) such that f L 2 (Γ) ≤ C u(•, t 0 ) H 2 (Γ) + v∈E ∂ xt u(v, •) L 2 (0,T ) (3.27)
for any f ∈ L 2 (Γ).

Proof. We proceed as in [START_REF] Yamamoto | Carleman estimates for parabolic equations and applications[END_REF].

Set z = ∂ t u. Then z = {z α } α∈I satisfies                              z α t (x, t) = z α xx (x, t) + b α (x)z α (x, t) + R α t (x, t)f α (x), (x, t) ∈ (0, l α ) × (0, T ), α ∈ I, z α (l α , t) = 0, t ∈ (0, T ), α ∈ I E , z 1 (0, t) = 0, t ∈ (0, T ), z α (l α , t) = z αβ (0, t), t ∈ (0, T ), α ∈ I I , β ∈ [[1, m α ]], z α x (l α , t) = m α β=1 z αβ x (0, t), t ∈ (0, T ), α ∈ I I .
(3.28) On the other hand,

R(x, t)f (x) = ∂ t u(x, t) -(∆ Γ u)(x, t) -bu(x, t). (3.29)
Using a change of variables it is sufficient to prove (3.27) when t 0 = T /2.

We now apply the Carleman estimate (3.3) with q = z = ∂ t u (and some fixed λ > 0)

T 0 Γ (sθ) -1 |∂ tt u| 2 + sθ|∂ xt u| 2 + (sθ) 3 |∂ t u| 2 e -2sϕ + T 0 (sθ)(|∂ xt u| 2 e -2sϕ )(R, t)dt ≤ C T 0 Γ |∂ tt u -∆ Γ ∂ t u| 2 e -2sϕ + v∈E T 0 sθ(|∂ xt u| 2 e -2sϕ )(v, t)dt ≤ C T 0 Γ |(∂ t R)f | 2 + |∂ t u| 2 e -2sϕ + v∈E T 0 sθ(|∂ xt u| 2 e -2sϕ )(v, t)dt ,
which gives, for s large enough,

T 0 Γ (sθ) -1 |∂ tt u| 2 + sθ|∂ xt u| 2 + (sθ) 3 |∂ t u| 2 e -2sϕ + T 0 (sθ)(|∂ xt u| 2 e -2sϕ )(R, t)dt ≤ C T 0 Γ |(∂ t R)f | 2 e -2sϕ + v∈E T 0 sθ(|∂ xt u| 2 e -2sϕ )(v, t)dt .
Since lim t→0 e -2sϕ(x,t) = 0 for x ∈ Γ and |ϕ t (x, t)| ≤ Cθ 2 (x, t) for all x ∈ Γ and t > 0, we get

Γ |∂ t u(x, T 2 )| 2 e -2sϕ(x, T 2 ) dx = T /2 0 ∂ ∂t Γ |∂ t u(x, t)| 2 e -2sϕ(x,t) dx dt = T /2 0 Γ 2∂ t u∂ tt u -2s∂ t ϕ|∂ t u| 2 e -2sϕ(x,t) dxdt ≤ T /2 0 Γ 2|∂ t u||∂ tt u| + Csθ 2 |∂ t u| 2 e -2sϕ(x,t) dxdt ≤ C T /2 0 Γ (s 2 θ) -1 |∂ tt u| 2 + (sθ) 2 |∂ t u| 2 e -2sϕ(x,t) dxdt ≤ C s T 0 Γ |(∂ t R)f | 2 e -2sϕ
Γ |f (x)| 2 e -2sϕ(x,T /2) dx ≤ C Γ |R(x, T /2)f (x)| 2 e -2sϕ(x,T /2) dx ≤ C Γ |∂ t u(x, T 2 )| 2 + |∆ Γ u(x, T 2 )| 2 + |u(x, T 2 )| 2 e -2sϕ(x,T /2) dx ≤ C s T 0 Γ |(∂ t R)f | 2 e -2sϕ dxdt + Ce -Cs v∈E T 0 |∂ xt u| 2 (v, t)dt + C u(•, T 2 ) 2 H 2 (Γ) . Since ∂ t R ∈ L ∞ (Γ × (0, T )), it holds Γ |f (x)| 2 e -2sϕ(x,T /2) dx ≤ C s T 0 Γ |f | 2 e -2sϕ dxdt + Ce -Cs v∈E T 0 |∂ xt u| 2 (v, t)dt + C u(•, T 2 ) 2 H 2 (Γ) .
It follows from the definition of ϕ that

ϕ(x, T 2 ) ≤ ϕ(x, t) for all (x, t) ∈ Γ × (0, T ), so that T 0 Γ |f (x)| 2 e -2sϕ(x,t) dxdt ≤ T Γ |f (x)| 2 e -2sϕ(x,T /2) dt. Therefore (1 - CT s ) Γ |f (x)| 2 e -2sϕ(x,T /2) dt ≤ C v∈E T 0 |∂ xt u| 2 (v, t)dt + C u(•, T 2 ) 2 H 2 (Γ) .
The desired inequality follows for s large enough.

We are now able to prove the stability result for system (1.1).

Proof of Theorem 1.1. Let us denote

w = u(p) -u(q).
It satisfies the following system

w t = ∆ Γ w -qw + Rf in Γ × (0, T ), w(x, t) = 0, on ∂Γ × (0, T ), where f = q -p, R = u(p). Note that R ∈ C([0, T ]; H 1 (Γ)) ⊂ C(Γ × [0, T ]), for u ∈ L 2 (0, T ; H 2 (Γ)
) and u t ∈ L 2 (0, T ; L 2 (Γ)). Using our hypothesis, we see that |R(•, t 0 )| ≥ r > 0 on Γ, hence we can apply Proposition 3.5 to obtain

p -q L 2 (Γ) ≤ C [u(p) -u(q)](•, t 0 ) H 2 (Γ)) + v∈E ∂ x [u(p) -u(q)](v, •) H 1 (0,T ) ,
where

C = C( ∂ t u(p) L ∞ (Γ×(0,T )) , q L ∞ (Γ) , r
). The proof is now completed.

Schrödinger equation on a star-shaped tree

In this section, we consider a network Γ which is a star-shaped tree constituted by N edges e j (with N ≥ 3) connected at the internal node O. Here, the parameterization of the edge e j is chosen so that the origin O of e j corresponds to x = 0, while the endpoint O j of e j corresponds to x = l j , for all j ∈ [[1, N ]].

iy j,t + y j,xx + p j (x)y j = f j (x, t), x ∈ (0, l j ), j ∈ [[1, N ]], t ∈ (0, T ), (4.31)
y j (0, t) = y l (0, t), t ∈ (0, T ), j, l ∈ [[1, N ]], (4.32)
1≤j≤N y j,x (0, t) = 0, t ∈ (0, T ), (

y(l j , t) = 0, j ∈ [[1, N ]], t ∈ (0, T ), (4.34) y(x, 0) = y 0 (x), x ∈ Γ, 4.33)
where p = {p j } j=1,N ∈ L ∞ (Γ) is some given potential function. Our main aim is to prove the stability for the inverse problem consisting in retrieving the potential p from the measurement of y x (l j , t) for j ∈ [ [1, N ]]. This is done thanks to some Carleman estimate in following the classical Bukhgeim-Klibanov method.

The first step will be the proof of a Carleman inequality on Γ. The key point is that choosing only one weight function ψ = {ψ j } j=1,N : Γ → R as in the case of the heat equation is not convenient since we fail to control some boundary terms. Instead, we consider a family of weights {ψ k } k=1,N allowing us to get rid of some bad boundary terms. 4.1. Carleman estimate. Assume given a family (ψ k j ) 1≤j,k≤N of functions fulfilling the following properties

ψ k j : [0, l j ] → R is of class C 2 , ∀j, k ∈ [[1, N ]], (4.36)
ψ k 1 j 1 (0) = ψ k 2 j 2 (0), ∀j 1 , k 1 , j 2 , k 2 ∈ [[1, N ]], (4.37)
|(ψ k j ) ′ (x)| 2 + (ψ k j ) ′′ (x) ≥ 0, ∀x ∈ [0, l j ], ∀j, k ∈ [[1, N ]], (4.38)
(ψ k j ) ′ (x) = 0, ∀x ∈ [0, l j ], ∀j, k ∈ [[1, N ]], (4.39) C 2 ≥ ψ k j (x) > C 3 , ∀x ∈ [0, l j ], ∀j, k ∈ [[1, N ]], (4.40)
where C > 0 is some positive constant. We also assume that the following flux conditions at x = 0 are satisfied:

1≤j≤N (ψ k j ) ′ (0) = 0, ∀k ∈ [[1, N ]], (4.41) 1≤k≤N (ψ k
j ) ′ (0) = 0, ∀j ∈ [[1, N ]], (4.42) 1≤k≤N |
(ψ k j ) ′ (0)| 2 = C 1 , ∀j ∈ [[1, N ]], (4.43) 1≤k≤N
(ψ k j ) ′′ (0) = C 2 , ∀j ∈ [[1, N ]], (4.44)
1≤k≤N

[(ψ k j ) ′ (0)] 3 > 0, ∀j ∈ [[1, N ]], (4.45)
for some constants C 1 > 0 and C 2 ∈ R. Such a family of weights functions (ψ k j ) 1≤j,k≤N exists. It is sufficient to pick (affine) functions of the form ψ k j (x) = a k j x + 5 12 C with C >> 1 and

a k j := N -1 if j = k, -1 if j = k.
Let us introduce the families of weights

θ k j (x, t) = e λψ k j (x) t(T -t) , ϕ k j (x, t) = e λC -e λψ k j (x) t(T -t) ,
and the class of functions

Z = {q = (q j ) j=1,N ∈ C(Γ×[0, T ]); q j ∈ C 2,1 ([0, l j ]×[0, T ]) ∀j ∈ [[1, N ]]
, and (4.32)-(4.34) hold}.

Proposition 4.1. Assume that the family of weights (ψ j k ) fulfills (4.36)-(4.45). Then there exist some constants λ 0 ≥ 1, s 0 ≥ 1 and C 0 > 0 such that for all λ ≥ λ 0 , all s ≥ s 0 , and all q ∈ Z, it Let us fix any pair (j, k) of indices in [ [1, N ]] and let us compute

T 0 l j 0 (M k 1 u k ) j (M k 2 u k ) j dxdt.
For the sake of simplicity, we shall drop the indices j and k during the computations, and we shall write u for T 0 l j 0 u(x, t)dxdt, and h for

T 0 h(t)dt. Then 2Re T 0 l j 0 (M k 1 u k ) j (M k 2 u k ) j dxdt = 2Re [i(2sϕ x u x + sϕ xx u) + sϕ t u][u t -i(u xx + s 2 |ϕ x | 2 u)] = I 1 + I 2 + I 3
where

I 1 = 2Re i(2sϕ x u x + sϕ xx u)(u t -i(u xx + s 2 |ϕ x | 2 u)), I 2 = 2Re sϕ t u(u t -iu xx ), I 3 = 2Re sϕ t u(-is 2 |ϕ x | 2 u).
Obviously, I 3 = 0. Let us begin with the computation of I 1 .

I 1 = 2Re (2sϕ x u x + sϕ xx u)(u xx + s 2 |ϕ x | 2 u) +2Re i(2sϕ x u x + sϕ xx u)u t = I 1 1 + I 2 1 .
To calculate I 1 1 , we need to evaluate the real part of the integral term J := u xx ϕ x u x . Integrating by part yields

J = - u x (ϕ xx u x + ϕ x u xx ) + ϕ x |u x | 2 l 0 ,
where l stands for l j . On the other hand 2Re

u x ϕ x u xx = ϕ x (u x u xx + u x u xx ) = ϕ x ∂ x |u x | 2 = - ϕ xx |u x | 2 + ϕ x |u x | 2 l 0 . Therefore 2Re J = -2 ϕ xx |u x | 2 + ϕ xx |u x | 2 -ϕ x |u x | 2 l 0 + 2 ϕ x |u x | 2 l 0 = - ϕ xx |u x | 2 + ϕ x |u x | 2 l 0 .
It follows that

I 1 1 = 2 Re 2sJ + s ϕ xx uu xx + 2s 3 (ϕ x ) 3 u x u + s 3 ϕ xx |ϕ x | 2 |u| 2 = 4s Re J -2s Re (ϕ 3x u + ϕ xx u x )u x + 2s Re ϕ xx uu x l 0 + 2s 3 (ϕ x ) 3 ∂ x |u| 2 +2s 3 ϕ xx |ϕ x | 2 |u| 2 = 2s{- ϕ xx |u x | 2 + ϕ x |u x | 2 l 0 } + s ϕ 4x |u| 2 -[ϕ 3x |u| 2 ] l 0 -2 ϕ xx |u x | 2 +2s Re ϕ xx uu x l 0 -6s 3 (ϕ x ) 2 ϕ xx |u| 2 + 2s 3 (ϕ x ) 3 |u| 2 l 0 + 2s 3 ϕ xx |ϕ x | 2 |u| 2 = -4s ϕ xx |u x | 2 + s ϕ 4x |u| 2 -4s 3 (ϕ x ) 2 ϕ xx |u| 2 + [2sϕ x |u x | 2 + (-sϕ 3x + 2s 3 (ϕ x ) 3 )|u| 2 + 2sϕ xx Re (uu x )] l 0 .
On the other hand,

I 2 1 = i (2sϕ x u x + sϕ xx u)u t -i (2sϕ x u x + sϕ xx u)u t = -i (2sϕ xt u x + 2sϕ x u xt + sϕ xxt u + sϕ xx u t )u +i 2s(ϕ xx u t + ϕ x u xt )u -i 2sϕ x uu t l 0 -i sϕ xx uu t = -i (sϕ xxt |u| 2 + 2sϕ xt u x u) -i 2sϕ x uu t l 0 = i sϕ xt (uu x + u x u) -i sϕ xt |u| 2 l 0 -2is ϕ xt u x u -i 2sϕ x uu t l 0 = i sϕ xt (uu x -u x u) + i [sϕ x (uu t -u t u)] l 0 .
It remains to estimate I 2 .

I 2 = sϕ t (uu t + uu t ) + sϕ t (-iuu xx + iuu xx ) =: I 1 2 + I 2 2 .
We find that

I 1 2 = - sϕ tt |u| 2 , I 2 2 = is (ϕ xt u + ϕ t u x )u x -is ϕ t uu x l 0 -is (ϕ tx u + ϕ t u x )u x + is ϕ t uu x l 0 = 2 Re (is)ϕ xt uu x + 2 Re (-is)ϕ t uu x l 0 .
Thus

I 1 + I 2 + I 3 = -4s ϕ xx |u x | 2 + s ϕ 4x |u| 2 -4s 3 |ϕ x | 2 ϕ xx |u| 2 +i sϕ xt (uu x -u x u) -s ϕ tt |u| 2 + 2 Re (is)ϕ tx uu x + [2sϕ x |u x | 2 + (-sϕ 3x + 2s 3 (ϕ x ) 3 )|u| 2 + 2sϕ xx Re (uu x ) +isϕ x (uu t -u t u) + 2 Re {(-is)ϕ t uu x }] l 0 .
We conclude that (with the indices written again)

1≤k≤N ||w k || 2 = 1≤k≤N ||M k 1 u k || 2 + ||M k 2 u k || 2 + 1≤j,k≤N -4s T 0 l j 0 ϕ k j,xx |u k j,x | 2 -4s Im T 0 l j 0 ϕ k j,xt u k j u k j,x + T 0 l j 0 |u k j | 2 [s(ϕ k j,4x -ϕ k j,tt ) -4s 3 (ϕ k j,x ) 2 ϕ k j,xx ] + T 0 [2sϕ k j,x |u k j,x | 2 + (-sϕ k j,3x + 2s 3 (ϕ k j,x ) 3 )|u k j | 2 + 2sϕ k j,xx Re (u k j u k j,x ) +2sϕ k j,t Re (-iu k j u k j,x ) + isϕ k j,x (u k j u k j,t -u k j,t u k j )] l 0 . (4.51)
Step 2. Estimation of the boundary terms at the internal node O.

We estimate each term in

j,k (-2s) T 0 ϕ k j,x (0)|u k j,x (0)| 2 + j,k T 0 (sϕ k j,3x (0) -2s 3 (ϕ k j,x (0)) 3 )|u(0)| 2 + j,k (-2s) T 0 ϕ k j,xx (0) Re (u(0)u k j,x (0)) + j,k (-2s) T 0 ϕ k j,t (0) Re (-iu(0)u k j,x (0)) + j,k T 0 (-is)ϕ k j,x (0)(u(0)u t (0) -u t (0)u(0)) =: J 1 + J 2 + J 3 + J 4 + J 5 .
In the above equation and in the following ones, we write merely u(0) := u k j (0, t), ϕ(0) = ϕ k j (0, t), etc. Using (4.42), (4.43), and (4.33) (for the q j 's) we see that

J 1 = (-2s) j T 0 k ϕ k j,x (0)|e -sϕ(0) (-sϕ k j,x q j + q j,x )| 2 = -2s 3 T 0 j,k (ϕ k j,x (0)) 3 |u(0)| 2 -2s j T 0 ( k ϕ k j,x (0))e -2sϕ(0) |q j,x (0)| 2 +4s 2 Re j T 0 ( k [ϕ k j,x (0)] 2 )u(0)q j,x (0) = -2s 3 T 0 j,k (ϕ k j,x (0)) 3 |u(0)| 2 .
Therefore

J 1 + J 2 = j,k T 0 (sϕ k j,3x (0) -4s 3 (ϕ k j,x (0)) 3 )|u(0)| 2 . (4.52)
On the other hand, using (4.43), (4.44) and (4.33), we have that

J 3 = -2s j,k T 0 ϕ k j,xx (0) Re [u(0)e -sϕ(0) (-sϕ k j,x (0)q j (0) + q j,x (0))] = 2s 2 Re T 0   j,k ϕ k j,xx (0)ϕ k j,x (0)   |u(0)| 2 +2sλ Re T 0 u(0)e -sϕ(0) e λψ(0) t(T -t) j k (ψ k j ) ′′ (0) + λ k [(ψ k j ) ′ (0)] 2 q j,x (0) = 2s 2 Re T 0   j,k ϕ k j,xx (0)ϕ k j,x (0)   |u(0)| 2 . ( 4
J 1 + J 2 + J 3 ≥ cs 3 λ 3 j,k T 0 e λψ k j (0) t(T -t) 3 |u(0)| 2 . (4.54)
Finally, we claim that J 4 = J 5 = 0. Indeed, using (4.33), we obtain that

J 4 = -2s Im T 0 ϕ t (0)u(0) j,k u k j,x (0)dt = -2s Im T 0 ϕ t (0)u(0) j,k
(-sϕ k j,x (0)q(0) + q j,x (0))e -sϕ(0) dt = 0, while J 5 = 0 by (4.41). Thus we conclude that

J 1 + J 2 + J 3 + J 4 + J 5 ≥ cs 3 λ 3 T 0 e λψ(0) t(T -t) 3 |u(0)| 2 . (4.55) for s ≥ s 1 , λ ≥ λ 1 .
Step 3. Estimation of the integrals along the edges. Direct estimations as in [30, Proposition 2.1] (without any integration by parts) yield that for some constant A > 0

j,k {(-4s) T 0 l j 0 ϕ k j,xx |u k j,x | 2 -4s Im T 0 l j 0 ϕ k j,xt u k j u k j,x + T 0 l j 0 |u k j | 2 [s(ϕ k j,4x -ϕ k j,tt ) -4s 3 (ϕ k j,x ) 2 ϕ k j,xx ]} ≥ A j,k {λ 2 s T 0 l j 0 e λψ k j t(T -t) |(ψ k j ) ′ u k j,x | 2 + λs 3 T 0 l j 0 |ϕ k j,x | 3 |u k j | 2 } (4.56)
provided that s ≥ s 2 , λ ≥ λ 2 . Combining (4.51), (4.55) and (4.56), we infer that

j,k T 0 l j 0 [|(M k 1 u k ) j | 2 + |(M k 2 u k ) j | 2 ] + λ 2 s T 0 l j 0 e λψ k j t(T -t) |(ψ k j ) ′ u k j,x | 2 + λs 3 T 0 l j 0 |ϕ k j,x | 3 |u k j | 2 +cs 3 λ 3 T 0 e λψ(0) t(T -t) 3 |u(0)| 2    ≤ c j,k T 0 l j 0 |w k j | 2 + s T 0 |ϕ k j,x (l j )| |u k j,x (l j )| 2 dt . ( 4
q j ∈ C 2,1 ([0, l j ] × [0, T ]) ∀j ∈ [[1, N ]] by q ∈ H 2,1 (Γ × (0, T )).
4.2. The boundary problem. We consider the following boundary initial-value problem

           iu j,t + u j,xx + p j (x)u j = 0, x ∈ (0, l j ), j ∈ [[1, N ]], t ∈ (0, T ), u j (0, t) = u l (0, t), j, k ∈ [[1, N ]], t ∈ (0, T ), 1≤j≤N u j,x (0, t) = 0, t ∈ (0, T ), u j (l j , t) = h j (t), j ∈ [[1, N ]], t ∈ (0, T ), u(x, 0) = u 0 (x), x ∈ Γ. (4.58)
In what follows we fix the initial data u 0 and the boundary data h = {h j } j=1,N , and we denote by u(p) the solution of the system (4.58) associated with the potential p ∈ L ∞ (Γ).

Theorem 4.3. Assume that p ∈ L ∞ (Γ; R), u 0 ∈ L ∞ (Γ) and r > 0 are such that • u 0 (x) ∈ R or iu 0 (x) ∈ R a.e. in Γ, • |u 0 (x)| ≥ r > 0 a.e. in Γ, and • ∂ t u(p) ∈ H 2,1 (Γ × (0, T )). Then, for any m ≥ 0, there exists a constant C = C(m, ||∂ t u(p)|| H 2,1 (Γ×(0,T )) , r) > 0 such that for any q ∈ B m (0) ⊂ L ∞ (Γ; R) satisfying ∂ t u(q) ∈ H 2,1 (Γ × (0, T )), we have that ||p -q|| L 2 (Γ) ≤ C 1≤j≤N ||∂ x [u(p) -u(q)] j (l j , .)|| H 1 (0,T ) •
Proof. Pick any p, q as in the statement of the theorem, and introduce the difference y := u(p)u(q) of the corresponding solutions of (4.58). Then y fulfills the system

           iy j,t + y j,xx + q j (x)y j = f j (x)R j (x, t), x ∈ (0, l j ), j ∈ [[1, N ]], t ∈ (0, T ), y j (0, t) = y l (0, t), j, k ∈ [[1, N ]], t ∈ (0, T ), 1≤j≤N y j,x (0, t) = 0, t ∈ (0, T ), y j (l j , t) = 0, j ∈ [[1, N ]], t ∈ (0, T ), y(x, 0) = 0, x ∈ Γ, (4.59)
with f j = q j -p j (real valued) and R j := (u(p)) j . To complete the proof of Theorem 4.3, we need the following result.

Proposition 4.4. Suppose that R = {R j } j=1,N satisfies • R(x, 0) ∈ R or iR(x, 0) ∈ R a.e. in Γ, • |R(x, 0)| ≥ r > 0 a.e. in Γ, • R ∈ H 1 (0, T ; L ∞ (Γ)), and
• ∂ t y ∈ H 2,1 (Γ × (0, T )). Then for any m ≥ 0 there exists a constant C = C(m, ||R t || L 2 (0,T ;L ∞ (Γ)) , r) such that for any q ∈ L ∞ (Γ, R) with ||q|| L ∞ (Γ) ≤ m and for all f ∈ L 2 (Γ, R), the solution y of (4.59) satisfies ||f || L 2 (Γ) ≤ C 1≤j≤N ||y j,x (l j , .)|| H 1 (0,T ) • (4.60)
Proof of Proposition 4.4. Let f ∈ L 2 (Γ; R) and R ∈ H 1 (0, T ; L ∞ (Γ)) be such that R(x, 0) ∈ R a.e. in Γ, and let y be the solution of (4.59). We take the even-conjugate extensions of y and R to the interval (-T, T ); i.e., we set y(x, t) = y(x, -t) for t ∈ (-T, 0) and similarly for R. Since R(x, 0) ∈ R a.e. in Γ, we have that R ∈ H 1 (-T, T ; L ∞ (Γ)), and y satisfies the system (4.59) in Γ × (-T, T ). In the case when R(x, 0) ∈ iR a.e. in Γ, the proof is still valid by taking odd-conjugate extensions. Let z(x, t) = y t (x, -t). Then z satisfies the following system:            z j,t + iz j,xx + iq j (x)z j = if j (x)R j,t (x, t), x ∈ (0, l j ), j ∈ [[1, N ]], t ∈ (-T, T ), z j (0, t) = z l (0, t), j, k ∈ [[1, N ]], t ∈ (-T, T ), 1≤j≤N z j,x (0, t) = 0, t ∈ (-T, T ), z j (l j , t) = 0, j ∈ [[1, N ]], t ∈ (-T, T ), z(x, 0) = -i f (x)R(x, 0),

x ∈ Γ.

(4.61)

We apply Proposition 4.1, but on the time interval (-T, T ) instead of (0, T ). Therefore, here we consider where we used the fact that θ k j ≥ T -2 . From (4.63), the Carleman estimate (4.46) (applied on the interval (-T, T ) instead of (0, T )), and the fact that ϕ k j (x, 0) ≤ ϕ k j (x, t) for all (x, t) ∈ (0, l j ) × (-T, T ), that θ k j e -2sϕ k j is bounded from above in (0, l j ) × (-T, T ), that q ∈ L ∞ (Γ), and that R t ∈ L 2 (-T, T ; L ∞ (Γ)), we infer that for s and λ large enough This completes the proof of Proposition 4.4 and of Theorem 4.3.

θ k j (x, t) =
|J| ≤ cλ -2 s -

Open problems

We now mention a few open problems related to our work. One of them is whether it is possible to reduce the number of measurements at the boundaries. It could be interesting to combine the ideas of the paper with those appearing in [START_REF] Dáger | Observation and control of vibrations in tree-shaped networks of strings[END_REF], [START_REF] Dáger | Wave propagation, observation and control in 1-d flexible multi-structures[END_REF] where less measurements on the boundary are needed but some rationality assumptions on the lengths of the edges have to be made. For the Schrödinger equation, the question whether a Carleman estimate on a tree with N exterior vertices can be written with only one weight function and N -1 boundary observations seems to be challenging.

The extension of the present work to more general graphs with other kind of coupling is also an open problem. We recall here the works of Kostrykin and Schrader [START_REF] Kostrykin | Kirchhoff's rule for quantum wires[END_REF][START_REF] Kostrykin | Laplacians on metric graphs: eigenvalues, resolvents and semigroups[END_REF] where self-adjoint Laplace operators with general coupling conditions are introduced.

  v, e)) = 0 for any interior vertex v and (∆ Γ u) e = (u e ) xx for all e ∈ E, u ∈ D(∆ Γ ).

Figure 1 .

 1 Figure 1. A tree with 10 edges.

  set

2 Figure 2 .

 22 Figure 2. A star-shaped tree with 4 edges

2 2 j,k l j 0 e) 3 e

 203 t)(T -t) , ϕ k j (x, t) = e λC -e λψ k j (x) (T + t)(T -t) , ∀(x, t) ∈ Γ × (-T, T ).As in the proof of Proposition 4.1, we introducew k j = e -sϕ k j z j , ( M k 2 z) j = e sϕ k j (M k 2 w k ) j and (M k 2 w k ) j = w k j,t + i(w k j,xx + s 2 |ϕ k j,x | 2 w k j ). Next, we set k j,x | 2 + s 2 |ϕ k j,x | 2 |w k j | 2 )dxdt + iNote that, by (4.61) and (4.41),w k j,x (0)w k j (0) = j (z j,x (0) -sϕ k j,x (0)z(0))e -2sϕ(0) z(0) k j (x,0) |f (x)| 2 |R(x, 0)| 2 dx.Using the hypothesis on R(x, 0), we infer thatRe (J) ≥ r -2sϕ k j (x,0) |f (x)| 2 dx. (4.62)On the other hand, we have that -2sϕ k j |z j | 2 dxdt , (4.63)

,k l j 0 e

 0 k j |f R t | 2 dxdt + λs T -T θ k j e -2sϕ k j |z j,x (l j , t)| 2 dt ≤ cλ -2 s -k j (x,0) |f | 2 dx + cλ -1 s -j,x (l j , t)| 2 dt. (4.64)It follows from (4.62), (4.64), and the fact that z(x, t) = -z(x, -t) for (x, t) ∈ Γ × (-T, 0), that for s and λ large enoughj-2sϕ k j (x,0) |f (x)| 2 dx ≤ c j 0 -T |z j,x (l j , t)| 2 dt.(4.65)Then (4.60) follows from (4.65) since e -2sϕ k j (x,0) ≥ e -2sT -2 (e λC -1) .

  .53) Combining (4.52), (4.53) and (4.45), we obtain that for s ≥ s 1 and λ ≥ λ 1 ,

Acknowledgements

The authors wish to thank Institut Henri Poincaré (Paris, France) for providing a very stimulating environment during the "Control of Partial Differential Equations and Applications" program in the Fall 2010. LI is supported by the grants PCE-2011-3-0075, "Analysis, control and numerical Approximations of PDE" and TE-4/2010 of CNCSIS Romania. LR is partially supported by the Agence Nationale de la Recherche, Project CISIFS, grant ANR-09-BLAN-0213-02. AP is partially supported by CNPq, Math-Amsud project CIP-PDE, and French-Brasil agreement.

)

Proof. In what follows, the letter c will denote a constant (independent of s, λ, q, j, k) which may vary from line to line. Let q ∈ Z be given, and for

, the operator M k acting simply on the components of u k along the different edges. Let M k 1 and M k 2 denote respectively the (formal) adjoint and skew-adjoint parts of the operator M k . We readily obtain that

Letting ( M k 1 q) j := e sϕ k j (M k 1 u k ) j and ( M k 2 q) j := e sϕ k j (M k 2 u k ) j , we easily check that (4.47) and (4.48) hold. On the other hand,