Leila Lefebvre

Jérome Chevalier

Laurent Gremillard
email: laurent.gremillard@insa-lyon1.fr

Rachid Zenati

Gilbert Thollet

Didier Bernache-Assollant

Alexandre Govin

Jérôme Chevalier

Laurent Grémillard

RACHID Zenati

Gilles Thollet

Bernache-Assollant

Structural transformations of bioactive glass 45S5next term with thermal treatments

Keywords: Silicates, Bioactive glasses, Crystallization, Differential scanning calorimetry, X-ray diffraction

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

I. Introduction

Bioactive glasses are biocompatible and exhibit a strong interfacial bonding with bone. Their bioactivity is attributed to the formation on their surface of a hydroxycarbonated apatite (HCA) layer similar to a large extent to the mineral part of bone. The rate of tissue bonding appears to depend on the rate of HCA formation, which follows a sequence of reactions between the implanted material and the surrounding tissues and physiologic fluids [1]. Hench et al. [2] proposed a three-step mechanism of HCA formation when the bioactive glass comes into contact with physiologic fluids: ion exchange, dissolution and precipitation. Ion exchange occurs at the bioactive glass surface: cations such as Na + and Ca 2+ from the glass exchange with H + from the surrounding solution. Network dissolution occurs from the breakage of Si-O-Si bonds by the action of hydroxyl ions (OH -). The hydrated silica (SiOH) formed on the glass surface undergoes rearrangement by polycondensation of neighbouring silanols, resulting in a silica-rich gel layer. Precipitation of the calcium and phosphate ions released from the glass together with those from the solution form a calcium-phosphate-rich layer (CaP) on the surface. The reactions involved in this process were observed by several authors in vitro as well as in vivo [3], [4] and [5]. The study of different compositions in the ternary Na 2 O, SiO 2 , CaO system with 6 wt.% P 2 O 5 showed that the 45S5 Bioglass ® (45 wt.% SiO 2 , 24.5 wt.% Na 2 O, 24.5 wt.% CaO, 6 wt.% P 2 O 5 ) is the most bioactive glass. It is nowadays used successfully as middle ear and dental implants [6].

The poor mechanical strength of bioactive glasses is a major problem that limits their application as load-bearing implants. Approaches to achieve enhanced mechanical and biochemical properties include transformation of bioactive glasses into glass-ceramic. In this technique, the glasses are subjected to thermal treatments which may affect the materials microstructures and hence their mechanical properties, but also their biological activity.

Most of the existing literature on the transformation of 45S5 Bioglass ® shows that thermal treatments above 600 °C result in the formation of Na 2 Ca 2 Si 3 O 9 as the main crystalline phase [7], [8], [9], [10], [11] and [12]. A secondary minor phase, Na 2 CaSi 3 O 8 , is often suggested to match the stoichiometry of the initial glass [13]. For some authors [14], raising the temperature to 800 °C for a long period leads to the development of calcium phosphate crystals with a structure similar to hydroxyapatite. However, the real composition of the crystals appearing during thermal treatments of Bioglass ® still remains unclear. As an example, Lin et al. [15] proposed Na 2 CaSi 2 O 6 rather than Na 2 Ca 2 Si 3 O 9 to be the major crystalline phase. Moreover, to our knowledge, there is no detailed investigation on the lattice parameters of the proposed crystalline phases and their evolution versus temperature. In addition, the location of phosphorous ions in the structure and its influence on the transformation processes is still not understood.

45S5 Bioglass ® has the potential to be used in many more bioactive applications than hydroxyapatite. Our global strategy is therefore to extend the application fields of 45S5 Bioglass ® by the processing, from powders, of porous blocks for bone substitution applications and tissue engineering [START_REF] Zenati | Porous Bioglass and preparation thereof[END_REF]. This necessitates the optimization of the mechanical and biological properties of porous bioactive glasses via a careful understanding of the process-microstructure-biomechanical properties relations. In particular, it is crucial to control bioactivity and mechanical properties by a full knowledge of microstructural evolutions during thermal treatments. The aim of the present paper is therefore to provide a precise understanding of the major transformation processes occurring during thermal treatment of Bioglass ® and to control the major phase crystallization kinetics via temperature-time-transformation (TTT) curves.

II. Materials and methods

I II I. .1 1. . B Bi io og gl la as ss s ® ® p pr re ep pa ar ra at ti io on n High purity SiO 2 , Na 2 CO 3 , CaCO 3 and P 2 O 5 powders were weighted and mixed to obtain 45S5 Bioglass ® (45 SiO 2 , 24.5 CaO, 24.5 Na 2 O, 6 P 2 O 5 in wt.%). The powders were melted in a Pt crucible for 4 h at 1400 °C with a decarbonatation step (5 h at 950 °C). The melt was then quenched in water and ground in ethanol to a fine powder (1 μm). Quenching was fast enough to retain a completely amorphous material, as verified by X-ray diffraction (XRD). The composition was also checked by chemical analysis, to ascertain that no impurity was present after the different preparation steps. I II I. .2 2. . T Tr ra an ns sf fo or rm ma at ti io on ns s p pr ro oc ce es ss se es s II. 2

.1 In situ follow-up

The transformations of the glass powder were followed with two methods:  Simultaneous thermal gravimetric analysis-differential thermal analysis (TGA-DTA; SETARAM TG-DTA 92, Caluire, France) on powders, for which 20 mg of powder was heated in a 20% N 2 -80% O 2 atmosphere at 5 °C min -1 up to 1300 °C.

 Environmental scanning electron microscopy (ESEM; FEI, XL30, Eindhoven, Netherlands), with an in situ heating stage. In order to get an easier observation, larger grains (on the order of 10 μm) were introduced into a Pt crucible. They were heated up to 950 °C (heating rate 5 °C min -1 ) in the microscope. Measurements were conducted with a 1.9 Torr (250 Pa) water pressure.

II.2.2 Characterization of thermally treated powders

Several thermal treatments for 5 min (heating rate 5 °C min -1 ) in the range of 550-950 °C were carried out on the glass powder. Two types of investigations were then performed:  XRD (RIGAKU vertical diffractometer, Kent, UK), to identify the crystalline phases and their evolution versus temperature. Samples were obtained by pressing a mixture of 45S5 Bioglass ® powder with an amorphous binder (PEG 1500) into 1 cm diameter pieces. These pieces were then analysed with Cu Kα radiations from 2θ = 10° to 2θ = 90° at 2° min -1 for the crystalline phases determination and 0.2° min -1 for Rietveltd analysis and crystallite size calculation. The crystallite size of the main phase, Na 2 CaSi 2 O 6 , at the different temperatures was estimated via the Scherrer equation on the peaks of higher intensity (2θ  34°) . .cos

k     (1)



where β is the full width at half maximum of the peak, ς is the crystallite size in meters, λ the wavelength of the Cu Kα line (1.5406 × 10 -10 m) and k is the Scherrer constant equal to 0.89. Rietveltd analysis was performed with the software WINPLOTR ® to evaluate the lattice parameters of the main phase. The diffractograms conducted at different temperature were compared with theoretical diffractograms of Na 2 CaSi 2 O 6 established from the structure determined by Oshato and Maki [START_REF] Oshato | [END_REF].

 Fourier transformed infrared analysis (FTIR; Nicolet Magna-IR 550 spectrometer, Madison, Wisconsin) was performed to identify the nature of the chemical bonds between atoms. The samples were small pellets, of 0.5 cm diameter, obtained by pressing the glass powder with KBr.

3 I II I. .3 3. . C Cr ry ys st ta al ll li iz za at ti io on n k ki in ne et ti ic cs s: : T TT TT T c cu ur rv ve es s The crystallization kinetics of the major phase were investigated by differential scanning calorimetry (DSC). Several isothermal measurements were conducted between 620 and 680 °C on 10 mg of powder. The samples were subjected to a first heating until 460 °C at 15 °C min-1, followed by a plateau of 15 min in order to stabilize the temperature and a second heating at 50 °C min-1 to the desired temperature. The samples were then held for 90 min at the isothermal temperature. The total area of exothermic crystallization peaks was measured for each temperature. The area of the peak at each time for a given temperature normalized by the total area was calculated. These results allow the determination of a degree of transformation for a given time and a given temperature, i.e. TTT curves. The degree of transformation at a given time for a given temperature, α T (t), was calculated from the following expression:

0 0 ( ). ( ) ( ). f t f T t f T H t dt t H t dt               (2)
where H f is the heat flow, t the time and t f the duration of the dwell at a given temperature (i.e. 90 min).

III. Results

I II II I. .1 1. . I

In n s si it tu u f fo ol ll lo ow w--u up p o of f t tr ra an ns sf fo or rm ma at ti io on n p pr ro oc ce es ss se es s Figure 1 shows the TGA-DTA curve. TGA shows a total 2.8% weight reduction, which can be divided in two main weight losses, at 100 and 400 °C. They are due respectively to the departure of free water and -OH groups. The small apparent weight loss at 610 °C may be attributed to the onset of crystallization. DTA shows an endothermic effect at T g1 = 550 °C caused by the glass transition, followed by an exothermic peak beginning at T c1 = 610 °C. These two events were already well identified in the literature [7], [9] and [14]. A second small endothermic effect is observed at T g2 = 850 °C. This event had already been observed [9] and [14] but not discussed in the existing literature. Finally, melting takes place in the 1070-1278 °C range. Two endothermic peaks (maximum signal respectively at T m1 = 1192 °C and T m2 = 1235 °C) may be attributed to the melting of two different crystalline phases. Figure 2 shows ESEM micrographs taken at low magnification on one Bioglass ® grain during in situ heating from room temperature to 950 °C. Three micrographs have been selected to show the evolution of grain morphology with temperature: 110 °C (similar to the morphology at room temperature) (Figure 2a), around T g1 = 550 °C (Figure 2b) and around T g2 = 850 °C (Figure 2c). These micrographs show two steps of particle rounding at T g1 and T g2 . Between T g1 and T g2 the rounding of the particles seems to slow down. These two temperatures must therefore correspond to two glassy transitions. Figure 3 shows ESEM micrographs taken in situ at higher magnification at the surface of the same grain. Figure 3a, taken below the glass transition (T g1 ), exhibits the features of a faceted ground powder. Figure 3b, taken at T g1 = 550 °C, shows a significant modification of its morphology. The grain is more rounded, confirming the glassy transition. An additional event is observed at T s = 570 °C (Figure 3c), prior to crystallization, which was not observed on the DTA thermograms: the surface, which was completely smooth after the glassy transition, presents a specific morphology, with the creation of domains. This may be attributed to the glass-in-glass phase separation expected when two high valence ions such as Si 4+ and P 5+ are present simultaneously in a glass [START_REF] Doremus | Glass science[END_REF] and [START_REF] Rawson | Inorganic glass forming systems[END_REF]. Each ion type tends to concentrate in a separate phase. Figure 3d, taken at 700 °C, shows the result of crystallization, with the formation of 'craters' at the surface. At 800 °C, Figure 3e shows the migration of a second phase with a different morphology and structure. The present micrograph was obtained via charge contrast imaging, which relates to electrical conductivity variations in the sample. The last figure, Figure 3f, shows the resulting surface after thermal treatment at 950 °C (the micrograph was taken at ambient temperature, after cooling, to ensure thermal stability and better contrast and resolution). On this micrograph, the separation between the two types of domain is still more pronounced, with a spinodal like morphology. Large crystals (about 500 nm diameter) are observed. They may be attributed to a heterogeneous grain growth at high temperature (Ostwald ripening). I II II I. .2 2. . C Ch ha ar ra ac ct te er ri iz za at ti io on n o of f t th he er rm ma al ll ly y t tr re ea at te ed d p po ow wd de er rs s Figure 4 shows the X-ray diffractograms of the powder before and after thermal treatments between 550 and 950 °C. The results obtained for the powder before any treatment confirm the amorphous nature of Bioglass ® after quenching and grinding. Crystallization is observed at 600 °C, which is in agreement with the results obtained by TGA-DTA analysis. The crystalline phase is identified as Na 2 CaSi 2 O 6 rather than Na 2 Ca 2 Si 3 O 9 , as shown previously by Lin et al. [15]. An apparent broadening of the main peak (at 2θ = 33.75°) occurs at 750 °C; this is in fact the beginning of the separation into two peaks, still both attributed to Na 2 CaSi 2 O 6 . The separation is clearly observed at 800 °C and intensifies at higher temperatures. Concurrently, in the 800-950 °C range, a new small peak is observed (arrow), suggesting that a secondary crystalline phosphate phase appears. This peak was attributed to silicorhenanite (Na 2 Ca 4 (PO 4 ) 2 SiO 4 ). In order to understand the separation of the two main peaks, more precise XRD analyses (at lower scan speed and better angular resolution) were performed on powders treated between 750 and 950 °C. The diffractogram between 33° and 34.5° was fitted by two pseudo-Voigt functions, and the distance between the two peaks calculated for each temperature. Figure 5 shows the separation of these two peaks (Δ2θ) as a function of temperature. A linear dependency is observed. Thus it can be inferred that the peak separation is due to a variation of lattice parameters rather than to a phase change. A Rietveltd refinement was applied to the different diffractograms to measure the lattice parameters (a and c). The evolution is represented in Figure 6. The figure shows that a decreases gradually with temperature, while a steep increase of c is noticed around 800 °C, when the secondary phase appears. Table 1 summarizes the Na 2 CaSi 2 O 6 crystal size evaluated from XRD, calculated via Equation (1). It varies from 15 to 41 nm between 650 and 950 °C. FTIR spectra taken on the powder before and after thermal treatments at 800 °C are shown in Figure 7. The main absorption bands for the amorphous Bioglass ® are observed at 1024, 926 and 480 cm -1 . They are commonly attributed to Si-O-Si and Si-O stretching modes and Si-O-Si bending mode, respectively. These bands are generally observed in amorphous silica glasses [START_REF] Filgueiras | [END_REF] and [21]. The band at 600 cm -1 is related to amorphous phosphate. The FTIR spectra of the powder treated at 800 °C shows that the broad band at 1024 cm -1 is split into two bands due to the combination of isolated tetrahedral Si. At the same time, the band at 926 cm -1 , which is attributed to the Si-O bond with non-bonding oxygen, increases in intensity. This increase is due to the crystallization of the major phase Na 2 CaSi 2 O 6 [12], in agreement with XRD data. The presence of a crystalline calcium phosphate (apatite like) phase is suggested by the new bands at 620, 580 and 530 cm -1 since the dual peaks at about 620 and 580 cm -1 are attributed to P-O bending vibration [14]. This is also in agreement with XRD and the identification of the silicorhenanite phase. Cr ry ys st ta al ll li iz za at ti io on n k ki in ne et ti ic cs s: : T TT TT T c cu ur rv ve es s Few papers report on the crystallization kinetics of bioactive glasses [22], [23] and [24]. In particular, TTT curves on bioactive glass 45S5 have never been established, although this type of curve can serve as a basis for the elaboration and thermal treatments of porous blocks. Crystallization isotherms were conducted between 620 °C (onset of Na 2 CaSi 2 O 6 crystallization) and 680 °C (to avoid the apparition of the secondary phase). DSC measurements made on the isothermal plateau at temperature in the 620-680 °C range (Figure 8) show exothermic peaks of crystallization. As expected when a single transformation mechanism is involved in a given range of temperature, the peak intensity increases with temperature, while the area of the peak remains constant. The enthalpy of crystallization is given by the area of the peaks, and is found to be constant (84 ± 10 J g -1 ), independent of temperature. The plot of α T (t) vs. time obtained from the DSC measurements and from Equation ( 2) is given in Figure 9. The knowledge of the degree of transformation (α T (t)) calculated at given times and temperatures allows the plotting of TTT curves in Figure 10.

The kinetics of Bioglass ® crystallization have been previously analysed using methods based on the Avrami equation [8] ( ) 1 exp( ( ) )

n T t     b t ( 3
)
where b is given by the Arrhenius relation:

0 .exp Q b b RT         (4)
where b 0 is a constant, Q the activation energy and R the gas constant. The Avrami exponent is generally obtained by plotting

1 ln ln 1             
 as a function of ln(t) and the apparent activation energy given by the plot of ln(b) vs.

T

. Figure 11 summarizes the values obtained for n and ln(b) for each temperature. In agreement with the study of Clupper and Hench [8], conducted with non-isothermal kinetics, the mean value of n (about 1) is consistent with a slow nucleation on the surface and infinitely rapid growth of the nuclei, and the apparent activation energy is close to 280 kJ mol -1 .

IV. Discussion and summary of the structural transformations occurring in Bioglass ® 45S5

The different observations conducted on Bioglass ® with various techniques allow us to propose a possible scenario of the structural transformations occurring during thermal treatments from room temperature to 950 °C. The first step is the glass transition around T g1 = 550 °C, which has already been reported extensively in the literature. It is followed by a glass-in-glass phase separation at T s = 570 °C. This means that the glass is no longer homogeneous but rather consists of two immiscible phases. The viscosity of a phaseseparated silicate glass is usually higher than that of a homogeneous glass of the same composition because of the high viscosity of the silica-rich phase that dominates flow behaviour [START_REF] Rawson | Inorganic glass forming systems[END_REF]. This is consistent with our ESEM observations, which shows that the rounding of the particles starting at T g1 = 550 °C seems to slow down around T s = 570 °C, indicating a decrease of mobility in the material. The formation of phosphate-rich domains during the glass-in-glass phase separation should have a catalytic effect on the nucleation of the major crystalline phase at T c1 = 610 °C as they decrease the necessary energy for nucleation of the surrounding silicate-rich nuclei (the phosphate-rich domains may act as heterogeneous nucleation sites). The Avrami exponent of about 1 is consistent with a predominant slow nucleation on the surface and infinitely rapid growth of the nuclei, in accordance with Clupper and Hench [8]. The major crystalline phase, appearing at First, the composition of Bioglass ® is closer to Na 2 CaSi 2 O 6 . Second, the Rietveldt analysis shows a better agreement of the experimental diffraction pattern with that calculated for Na 2 CaSi 2 O 6 . However, a complete determination of the crystalline structure, including positions of the atoms and taking into account the possible solid solution, is still lacking. The Na 2 CaSi 6 O 16 phase mentioned in other studies was not observed in our material. The crystallization mechanism of the major phase in Bioglass ® is in agreement with the results of Koga et al. [26], who studied the mechanism of crystallization in the Na 2 O-CaO-2SiO 2 system leading to Na 2 CaSi 2 O 6 and found that a surface nucleation takes place for small particles (i.e. smaller than 100 μm). El Ghannam et al. [9] found that the change in bioactive glass microstructure after crystallization alters the corrosion behaviour by accelerating the dissolution kinetics of phosphorus in physiological fluid and slowing the Na and Ca dissolution. This also led to a significant increase in the zeta potential of the material. In conjunction with this increase, they observed a significant decrease in the amount of serum protein adsorbed onto the material surface. On the other hand, crystallization is beneficial for the mechanical properties and helps tailor the dissolution kinetics to the bone growth rate.

T c1 =
In the 800-950 °C range, a secondary phosphate phase crystallizes. It is identified in X-ray diffractograms as silicorhenanite (Na 2 Ca 4 (PO 4 ) 2 SiO 4 ). This phase is isostructural to apatite and was never identified in Bioglass ® after short thermal treatments at 800 °C. Lin et al. [15] showed by Raman analysis that in the 45S5 glass, phosphorus has the same behaviour than in pure silica. It tends to depolymerize the silicate network by entering the network as a fourfold-coordinated cation. However, McMillan [START_REF] Mcmillan | Glass ceramics[END_REF] thought that the double oxygen/phosphorus (P O) bond is favourable to phosphate phase formation in a silica network and thus increases the tendency towards crystallization. Elgayar et al. [START_REF] Elgayar | [END_REF] showed by nuclear magnetic resonance of phosphorus analysis applied to the 45S5 glass that 75% of the phosphorous ions are in an orthophosphate (PO )

 environment, while 25% are under the form of a diphosphate specie  . Using the relative XRD intensity peaks, we found around 4% of phosphate crystalline phase. This indicates that 75% of phosphorus ions in the material are in the crystalline phosphate phase. This is consistent with the fact that in silicorhenanite, phosphorus is in the orthophosphate form. Thus we concluded that the orthophosphate ions in the glass lead to the formation of the crystalline phase whereas the diphosphate species remain in the amorphous phase. significant increase of the cell parameter c around 800 °C is attributed to a departure of orthophosphate in insertion in the silica network and to the replacement of these groups by SiO 4 groups. The orthophosphate species would then form the crystalline phosphate phase. After nucleation of the principal crystalline phase, differential grain growth leads to an Oswalt ripening, where only big crystallites can grow at the expense of smaller crystallites. Then, a migration of the phosphate vitreous phase around the bigger crystallites of Na 2 CaSi 2 O 6 takes place. We can imagine that the new interface formed has a catalytic effect on the crystallization of the phosphate phase (i.e. the Na 2 CaSi 2 O 6 surface acts as nucleation site for silicorhenanite). Höland et al. [29] studied the influence of rhenanite crystallization in bioactive glasses (containing, in wt.%, 6 P 2 O 5 , 58 SiO 2 , 22.8 Na 2 O, 12.9 CaO and 0.3 F) on the crystallization of carbonated hydroxyapatite on their surface. They concluded that this phase acts as heterogeneous nuclei for HCA crystallization and thus should improve the bioactivity. On the other hand, the state of phosphorus ions affects the bioactivity: glass ceramics containing a crystalline phosphate phase are less reactive than materials containing phosphorus in solid solution [11], and dissolution behaviour and bioactivity are decreased. It has still to be ascertained whether the behaviour of silicorhenanite (vs. HCA crystallization) is similar to the behaviour of rhenanite. Finally, melting takes place in the 1070-1278 °C range. The two endothermic peaks on DTA thermograms are related to the melting of the two crystalline phases. A summary of the transformations sequence is shown in Figure 12.

V. Conclusion

The bioactive glass 45S5 crystallizes principally in the Na 2 CaSi 2 O 6 phase rather than Na 2 Ca 2 Si 3 O 9 in the 600-700 °C range. We suspect the existence of a new crystallized phosphate phase, Na 2 Ca 4 (PO 4 ) 2 SiO 4 . After the glassy transition and before the crystallization process, a spinodal transformation of the glassy phase into two immiscible phases occurs, one phase rich in silicon and the other rich in phosphorus. We believe that the first one leads to the major crystalline phase and the second one to the crystalline phosphate phase. The Rietveltd analyses showing the evolution of cell parameters in the major crystalline phase suggest that groups enter the silica network during glass synthesis. At 800 °C, these

5 4
PO  groups are substituted by silica and lead to the crystalline phosphate phase. Bioglass ® crystallization kinetics show a rapid tendency of the material to crystallize. Furthermore, time-temperature-transformation curves were established which can be used to control the degree of crystallinity of porous blocks. We noticed, however, that these results cannot be extended to dense blocks.

( (c c) ) A An n a ad dd di it ti io on na al l e ev ve en nt t i is s o ob bs se er rv ve ed d a at t T T s s = = 5 57 70 0 °°C C, , p pr ri io or r t to o c cr ry ys st ta al ll li iz za at ti io on n, , w wh hi ic ch h w wa as s n no ot t o ob bs se er rv ve ed d o on n t th he e D DT TA A t th he er rm mo og gr ra am ms s. . T Th he e s su ur rf fa ac ce e, , w wh hi ic ch h w wa as s c co om mp pl le et te el ly y s sm mo oo ot th h a af ft te er r t th he e g gl la as ss sy y t tr ra an ns si it ti io on n, , p pr re es se en nt ts s a a s sp pe ec ci if fi ic c m mo or rp ph ho ol lo og gy y, , w wi it th

5 I

 5 II II I. .3 3. . C

Figures 9 F

 9 Figures

F

  Fi ig gu ur re e 1 10 0: : T TT TT T c cu ur rv ve es s o of f 4 45 5S S5 5 B Bi io og gl la as ss s ® ® . . F Fi ig gu ur re e 1 11 1: : V Va al lu ue es s o ob bt ta ai in ne ed d f fo or r n n a an nd d l ln n( (b b) ) f fo or r e ea ac ch h t te em mp pe er ra at tu ur re e. . F Fi ig gu ur re e 1 12 2: : S Su um mm ma ar ry y o of f t th he e s st tr ru uc ct tu ur ra al l t tr ra an ns sf fo or rm ma at ti io on ns s o of f 4 45 5S S5 5 w wi it th h t te em mp pe er ra at tu ur re e. .

  610 °C, is identified by XRD as Na 2 CaSi 2 O 6 rather than Na 2 Ca 2 Si 3 O 9 , as shown previously by Lin et al. [15]. Na 2 CaSi 2 O 6 is isostructural to the high temperature form of Na 2 Ca 2 Si 3 O 9 (2 Na replace 1 Ca ion), which could explain why 45S5 was thought to crystallize in the Na 2 Ca 2 Si 3 O 9 phase [25]. Two arguments are in favour of Na 2 CaSi 2 O 6 rather than Na 2 Ca 2 Si 3 O 9 .

  Ri ie et tv ve el lt td d r re ef fi in ne em me en nt t a ap pp pl li ie ed d t to o t th he e d di if ff fe er re en nt t d di if ff fr ra ac ct to og gr ra am ms s f Is so ot th he er rm ma al l c cr ry ys st ta al ll li iz za at ti io on n p pe ea ak ks s o ob bt ta ai in ne ed d b by y D DS SC C a at t d di if ff fe er re en nt t t te em mp pe er ra at tu ur re es s. .

	F Fi ig gu ur re e 4 4: : D Di if ff fr ra ac ct to og gr ra am ms s o of f r ra aw w a an nd d t th he er rm ma al ll ly y t tr re ea at te ed d b bi io oa ac ct ti iv ve e g gl la as ss s 4 45 5S S5 5. . U Un nd de er r 6 60 00 0 °°C C t th he e B Bi io og gl la as ss s ® ta ak ke es s p pl la ac ce e b by y t th he e f fo or rm ma at ti io on n o of f N Na a 2 2C Ca aS Si i 2 2O O 6 6 ( (• •: : P PD DF F 7 77 7--2 21 18 89 9) ). . A At t 8 80 00 0 °°C C, , t th he e t tw wo o m o of f t th hi is s p ph ha as se e b be eg gi in n t to o b be e d ( : 3 32 2--1 10 05 53 3, , i : P PD DF F ma aj jo or r p pe ea ak ks s fr ro om m 7 75 50 0 t to o 9 95 50 0 °°C C t to o m F Fi ig gu ur re e 8 8: : I
		at t
	8 80 00 0 °°C C. .
	f fu un nc ct ti io on n o of f t	of f t th he e t tw wo o p pe ea ak ks s o of f m ma aj jo or r i in nt te en ns si it ty y o of f N Na a 2 2C Ca aS Si i 2 2O O 6 6 a as s a a
	h t th he e c cr re ea at ti io on n o of f d do om ma ai in ns s. . ( (d d) ) T Th he e r re es su ul lt t o of f c cr ry ys st ta al ll li iz za at ti io on n, , t ta ak ke en n a at t th he e f wi it th h t 70 00 0 °°C C, , w 7 fo or rm ma at ti io on n o of f " "c cr ra at te er rs s" " a at t t th he e s su ur rf fa ac ce e. . ( (e e) ) A At t 8 80 00 0 °°C C, , m mi ig gr ra at ti io on n o a of f a s se ec co on nd d p ph ha as se e w wi it th h a a d di if ff fe er re en nt t m mo or rp ph ho ol lo og gy y a an nd d s st tr ru uc ct tu ur re e. . ( (f f) ) S Su ur rf fa ac ce e a af ft te er r t th he er rm ma al l t tr re ea at tm me en nt t a at t 9 95 50 0 °°C C ( (t th he e m mi ic cr ro og gr ra ap ph h w wa as s t ta ak ke en n a at t a am mb bi ie en nt t t te em mp pe er ra at tu ur re e, , a co oo ol li in ng g, , af ft te er r c t to o e en ns su ur re e t th he er rm ma al l s st ta ab bi il li it ty y a an nd d b be et tt te er r c co on nt tr ra as st t a an nd d r re es so ol lu ut ti io on n) ). .

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Acknowledgements

The authors thank the Region Rhône-Alpes for its financial support.


R G Novikov
email: novikov@cmap.polytechnique.fr

New global stability estimates for the Gel'fand-Calderon inverse problem

New global stability estimates for the Gel'fand-Calderon inverse problem

Roman Novikov

Introduction

We consider the equation

-∆ψ + v(x)ψ = 0, x ∈ D, (1.1)
where

D is an open bounded domain in R d , d ≥ 2, ∂D ∈ C 2 , v ∈ L ∞ (D). (1.2)
Equation (1.1) arises, in particular, in quantum mechanics, acoustics, electrodynamics. Formally, (1.1) looks as the Schrödinger equation with potential v at zero energy. We consider the map Φ such that ∂ψ ∂ν ∂D = Φ(ψ ∂D ) (1.3) for all sufficiently regular solutions ψ of (1.1) in D = D ∪ ∂D, where ν is the outward normal to ∂D. Here we assume also that 0 is not a Dirichlet eigenvalue for the operator -∆ + v in D.

(1.4)

The map Φ is called the Dirichlet-to-Neumann map for equation (1.1) and is considered as boundary measurements for (physical model described by) (1.1). We consider the following inverse boundary value problem for equation (1.1): Problem 1.1. Given Φ, find v. This problem can be considered as the Gel'fand inverse boundary value problem for the Schrödinger equation at zero energy (see [START_REF] Gelfand | Some problems of functional analysis and algebra[END_REF], [START_REF] Novikov | Multidimensional inverse spectral problem for the equation -∆ψ + (v(x) -Eu(x))ψ = 0, Funkt. Anal. i Pril[END_REF]). This problem can be also considered as a generalization of the Calderon problem of the electrical impedance tomography (see [START_REF] Calderón | On an inverse boundary value problem[END_REF], [START_REF] Sylvester | A global uniqueness theorem for an inverse boundary value problem[END_REF], [START_REF] Novikov | Multidimensional inverse spectral problem for the equation -∆ψ + (v(x) -Eu(x))ψ = 0, Funkt. Anal. i Pril[END_REF]).

We recall that the simplest interpretation of D, v and Φ in the framework of the electrical impedance tomography consists in the following (see, for example, [START_REF] Novikov | Multidimensional inverse spectral problem for the equation -∆ψ + (v(x) -Eu(x))ψ = 0, Funkt. Anal. i Pril[END_REF], [START_REF] Nachman | Reconstructions from boundary measurements[END_REF], [START_REF] Mandache | Exponential instability in an inverse problem for the Schrödinger equation[END_REF]): D is a body with isotropic conductivity σ(x) (where σ ≥ σ min > 0),

v(x) = (σ(x)) -1/2 ∆(σ(x)) 1/2 , x ∈ D, (1.5)
Φ = σ -1/2 Λσ -1/2 + ∂σ 1/2 ∂ν , (1.6)
where ∆ is the Laplacian, Λ is the voltage-to-current map on ∂D, and σ -1/2 , ∂σ 1/2 /∂ν in (1.6) denote the multiplication operators by the functions σ -1/2 ∂D , (∂σ 1/2 /∂ν) ∂D , respectively. In addition, (1.4) is always fulfilled if v is given by (1.5) (where σ ≥ σ min > 0 and σ is twice differential in L ∞ (D)).

Problem 1.1 includes, in particular, the following questions: (a) uniqueness, (b) reconstruction, (c) stability.

Global uniqueness theorems for Problem 1.1 (in its Calderon or Gel'fand form) in dimension d ≥ 3 were obtained for the first time in [START_REF] Sylvester | A global uniqueness theorem for an inverse boundary value problem[END_REF] and [START_REF] Novikov | Multidimensional inverse spectral problem for the equation -∆ψ + (v(x) -Eu(x))ψ = 0, Funkt. Anal. i Pril[END_REF]. In particular, according to [START_REF] Novikov | Multidimensional inverse spectral problem for the equation -∆ψ + (v(x) -Eu(x))ψ = 0, Funkt. Anal. i Pril[END_REF], under assumptions (1.2), (1.4), the map Φ uniquely determines v.

A global reconstruction method for Problem 1.1 in dimension d ≥ 3 was proposed for the first time in [START_REF] Novikov | Multidimensional inverse spectral problem for the equation -∆ψ + (v(x) -Eu(x))ψ = 0, Funkt. Anal. i Pril[END_REF]. In its simplest form, this method of [START_REF] Novikov | Multidimensional inverse spectral problem for the equation -∆ψ + (v(x) -Eu(x))ψ = 0, Funkt. Anal. i Pril[END_REF] consists in reducing Problem 1.1 to Problem 4.1 via formulas and equations (5.1), (5.2) for v 2 = v, ψ 2 = ψ, h 1 = h, v 1 ≡ 0, ψ 1 (x, k) = e ikx , R 1 (x, y, k) = G(xy, k), h 1 ≡ 0 and in solving Problem 4.1 via formula (4.10), see Sections 3,4 and 5 of the present paper.

Global stability estimates for Problem 1.1 in dimension d ≥ 3 were given for the first time in [START_REF] Alessandrini | Stable determination of conductivity by boundary measurements[END_REF]. A variation of this result is presented as Theorem 2.1 of Section 2 of the present paper.

We recall that in global results one does not assume that σ or v is small in some sense or that σ or v is peicewise constant or piecewise real-analytic. For peicewise constant or piecewise real-analytic σ the first uniqueness results for the Calderon version of Problem 1.1 in dimension d ≥ 2 were given in [START_REF] Druskin | The unique solution of the inverse problem in electrical surveying and electrical well logging for piecewise-constant conductivity[END_REF], [START_REF] Kohn | Determining conductivity by boundary measurements II, Interior results[END_REF].

As regards global results given in the literature on Problem 1.1 in dimension d = 2, the reader is refered to Corollary 2 of [START_REF] Novikov | Multidimensional inverse spectral problem for the equation -∆ψ + (v(x) -Eu(x))ψ = 0, Funkt. Anal. i Pril[END_REF] and to [START_REF] Nachman | Global uniqueness for a two-dimensional inverse boundary value problem[END_REF], [START_REF] Brown | Uniqueness in the inverse conductivity problem for nonsmooth conductivities in two dimensions[END_REF], [START_REF] Mandache | Exponential instability in an inverse problem for the Schrödinger equation[END_REF], [START_REF] Lassas | Mapping properties of the nonlinear Fourier transform in dimension two[END_REF], [START_REF] Bukhgeim | Recovering a potential from Cauchy data in the two-dimensional case[END_REF] and to references given in [START_REF] Bukhgeim | Recovering a potential from Cauchy data in the two-dimensional case[END_REF].

In the present work we continue studies of [START_REF] Novikov | Multidimensional inverse spectral problem for the equation -∆ψ + (v(x) -Eu(x))ψ = 0, Funkt. Anal. i Pril[END_REF], [START_REF] Novikov | Formulae and equations for finding scattering data from the Dirichletto-Neumann map with nonzero background potential[END_REF], [START_REF] Novikov | On non-overdetermined inverse scattering at zero energy in three dimensions[END_REF], [START_REF] Novikov | On stable determination of potential by boundary measurements[END_REF], [START_REF] Novikov | An effectivization of the global reconstruction in the Gel'fand-Calderon inverse problem in three dimensions[END_REF]. One of the main purposes of these studies was developing an efficient reconstruction algorithm for Problem 1.1 in dimension d = 3. Note that the aforementioned global reconstruction method of [START_REF] Novikov | Multidimensional inverse spectral problem for the equation -∆ψ + (v(x) -Eu(x))ψ = 0, Funkt. Anal. i Pril[END_REF] is fine in the sense that it consists in solving Fredholm linear integral equations of the second type and using explicit formulas but this reconstruction is not optimal with respect to its stability properties. An effectivization of this reconstruction of [START_REF] Novikov | Multidimensional inverse spectral problem for the equation -∆ψ + (v(x) -Eu(x))ψ = 0, Funkt. Anal. i Pril[END_REF] with respect to its stability properties was developed in [START_REF] Novikov | Formulae and equations for finding scattering data from the Dirichletto-Neumann map with nonzero background potential[END_REF], [START_REF] Novikov | On non-overdetermined inverse scattering at zero energy in three dimensions[END_REF], [START_REF] Novikov | An effectivization of the global reconstruction in the Gel'fand-Calderon inverse problem in three dimensions[END_REF]. In [START_REF] Novikov | On stable determination of potential by boundary measurements[END_REF] and in the present work we illustrate our progress in stability by proving new stability estimates for Problem 1.1 in 3D and, in particular, by proving Theorem 2.2 of Section 2 in 3D. For sufficiently regular potentials this theorem is a principle improvement of the aforementioned Alessandrini stability result of [START_REF] Alessandrini | Stable determination of conductivity by boundary measurements[END_REF].

Note that algorithms developed in [START_REF] Novikov | Formulae and equations for finding scattering data from the Dirichletto-Neumann map with nonzero background potential[END_REF], [START_REF] Novikov | On non-overdetermined inverse scattering at zero energy in three dimensions[END_REF], [START_REF] Novikov | An effectivization of the global reconstruction in the Gel'fand-Calderon inverse problem in three dimensions[END_REF] for finding v from Φ in 3D can be used even if v has discontinuities. In this case these algorithms can be considered as methods for finding the smooth part of v from Φ. In particular, the global algorithms of [START_REF] Novikov | Formulae and equations for finding scattering data from the Dirichletto-Neumann map with nonzero background potential[END_REF], [START_REF] Novikov | An effectivization of the global reconstruction in the Gel'fand-Calderon inverse problem in three dimensions[END_REF] can be considered as methods for approximate finding the Fourier transform v(p), |p| ≤ 2ρ, from Φ for sufficiently great ρ, where v(p) is defined by (4.12), d = 3. However, for the case when v is not smooth enough we did not manage yet to formalize our principle progress of [START_REF] Novikov | An effectivization of the global reconstruction in the Gel'fand-Calderon inverse problem in three dimensions[END_REF] in global stability as a rigorous mathematical theorem.

Note that in [START_REF] Novikov | On stable determination of potential by boundary measurements[END_REF] our new stability estimates of Theorem 2.2 of the next section were proved in the Born approximation (that is in the linear approximation near zero potential) only. Besides, a scheme of proof of these estimates was also mentioned in [START_REF] Novikov | On stable determination of potential by boundary measurements[END_REF] for potentials with sufficiently small norm in dimension d = 3. In the present work we give a complete proof of Theorem 2.2 in the general (or by other words global) case in dimension d = 3.

Our new stabilty estimates for Problem 1.1 (that is the estimates of Theorem 2.2) are presented and discussed in Section 2. In addition, relations between Theorems 2.1, 2.2 and global reconstructions of [START_REF] Novikov | Multidimensional inverse spectral problem for the equation -∆ψ + (v(x) -Eu(x))ψ = 0, Funkt. Anal. i Pril[END_REF], [START_REF] Novikov | Formulae and equations for finding scattering data from the Dirichletto-Neumann map with nonzero background potential[END_REF], [START_REF] Novikov | An effectivization of the global reconstruction in the Gel'fand-Calderon inverse problem in three dimensions[END_REF] are explained in Section 3. The proof of Theorem 2.2 for the global case in 3D is given in Section 8.

Stability estimates

As in [START_REF] Novikov | On stable determination of potential by boundary measurements[END_REF] we assume for simplicity that

D is an open bounded domain in R d , ∂D ∈ C 2 , v ∈ W m,1 (R d ) for some m > d, supp v ⊂ D, d ≥ 2, ( 2.1)
where

W m,1 (R d ) = {v : ∂ J v ∈ L 1 (R d ), |J| ≤ m}, m ∈ N ∪ 0, (2.2)
where

J ∈ (N ∪ 0) d , |J| = d i=1 J i , ∂ J v(x) = ∂ |J| v(x) ∂x J 1 1 . . . ∂x J d d . Let v m,1 = max |J|≤m ∂ J v L 1 (R d ) . (2.3)

Let

A denote the norm of an operator

A : L ∞ (∂D) → L ∞ (∂D). (2.4)
We recall that if v 1 , v 2 are potentials satisfying (1.2), (1.3), where D is fixed, then

Φ 1 -Φ 2 is a compact operator in L ∞ (∂D), (2.5)
where Φ 1 , Φ 2 are the DtN maps for v 1 , v 2 respectively, see [START_REF] Novikov | Multidimensional inverse spectral problem for the equation -∆ψ + (v(x) -Eu(x))ψ = 0, Funkt. Anal. i Pril[END_REF], [START_REF] Novikov | Formulae and equations for finding scattering data from the Dirichletto-Neumann map with nonzero background potential[END_REF]. Note also that (2.1) ⇒ (1.2).

Theorem 2.1 (variation of the result of [START_REF] Alessandrini | Stable determination of conductivity by boundary measurements[END_REF]). Let conditions (1.4), (2.1) hold for potentials v 1 and v 2 , where D is fixed,

d ≥ 3. Let v j m,1 ≤ N , j = 1, 2, for some N > 0. Let Φ 1 , Φ 2 denote the DtN maps for v 1 , v 2 , respectively. Then v 1 -v 2 L ∞ (D) ≤ c 1 (ln(3 + Φ 1 -Φ 2 -1 )) -α 1 , (2.6)
where

c 1 = c 1 (N, D, m), α 1 = (m -d)/m, Φ 1 -Φ 2 is
v 1 -v 2 L ∞ (D) ≤ c 2 (ln(3 + Φ 1 -Φ 2 -1 )) -α 2 , (2.8)
where

c 2 = c 2 (N, D, m), α 2 = m -d, Φ 1 -Φ 2 is
α 2 → +∞ as m → +∞, (2.9)
in contrast with (2.7).

In the Born approximation, that is in the linear approximation near zero potential, Theorem 2.2 was proved in [START_REF] Novikov | On stable determination of potential by boundary measurements[END_REF].

For sufficiently small N in dimension d = 3, a scheme of proof of Theorem 2.2 was also mentioned in [START_REF] Novikov | On stable determination of potential by boundary measurements[END_REF]. This scheme involves, in particular, results of [START_REF] Novikov | Formulae and equations for finding scattering data from the Dirichletto-Neumann map with nonzero background potential[END_REF], [START_REF] Novikov | On non-overdetermined inverse scattering at zero energy in three dimensions[END_REF].

In the general (or by other words global) case Theorem 2.2 in dimension d = 3 is proved in Section 8. This proof involves, in particular, results of [START_REF] Novikov | Formulae and equations for finding scattering data from the Dirichletto-Neumann map with nonzero background potential[END_REF], [START_REF] Novikov | An effectivization of the global reconstruction in the Gel'fand-Calderon inverse problem in three dimensions[END_REF]. We see no principal difficulties (except restrictions in time) to give a similar proof in dimension d > 3.

We would like to mention that, under the assumptions of Theorems 2.1 and 2.2, according to the Mandache results of [START_REF] Mandache | Exponential instability in an inverse problem for the Schrödinger equation[END_REF], the estimate (2.8) can not hold with α 2 > m(2d -1)/d. However, we think that this instability result of [START_REF] Mandache | Exponential instability in an inverse problem for the Schrödinger equation[END_REF] can be sharpened considerably and that in this sense our Theorem 2.2 is almost optimal already.

For additional information concerning stability and instability results given in the literature for Problem 1.1 (in its Calderon or Gel'fand form) the reader is refered to [START_REF] Alessandrini | Stable determination of conductivity by boundary measurements[END_REF], [START_REF] Mandache | Exponential instability in an inverse problem for the Schrödinger equation[END_REF], [START_REF] Novikov | On stable determination of potential by boundary measurements[END_REF], [START_REF] Palamodov | Gabor analysis of the continuum model for impedance tomography[END_REF], [START_REF] Novikov | A global stability estimate for the Gel'fand-Calderon inverse problem in two dimensions[END_REF] and references therein. In particular, in [START_REF] Novikov | A global stability estimate for the Gel'fand-Calderon inverse problem in two dimensions[END_REF] estimate (2.6) is proved for d = 2 and any α 1 ∈]0, 1[, under the assumptions that v j ∈ C 2 ( D), v j C 2 ( D) ≤ N (and, at least, under the additional assumption that supp v j ⊆ D0 ⊂ D for some closed D0 ), j = 1, 2, where c 1 depends on D, N , α 1 (and on D0 ).

3. Theorems 2.1 and 2.2 and global reconstructions of [START_REF] Novikov | Multidimensional inverse spectral problem for the equation -∆ψ + (v(x) -Eu(x))ψ = 0, Funkt. Anal. i Pril[END_REF], [START_REF] Novikov | Formulae and equations for finding scattering data from the Dirichletto-Neumann map with nonzero background potential[END_REF], [START_REF] Novikov | An effectivization of the global reconstruction in the Gel'fand-Calderon inverse problem in three dimensions[END_REF] In this section we explain relations between Theorems 2.1, 2.2 and global reconstructions of [START_REF] Novikov | Multidimensional inverse spectral problem for the equation -∆ψ + (v(x) -Eu(x))ψ = 0, Funkt. Anal. i Pril[END_REF], [START_REF] Novikov | Formulae and equations for finding scattering data from the Dirichletto-Neumann map with nonzero background potential[END_REF], [START_REF] Novikov | An effectivization of the global reconstruction in the Gel'fand-Calderon inverse problem in three dimensions[END_REF] for Problem 1.1, where we assume that dimension d = 3. (The case d > 3 is similar to the case d = 3.) The scheme of the aforementioned global reconstructions consists in the following:

Φ → h Ξ → vρ,τ → v, (3.1)
where h is the Faddeev generalized scattering amplitude (at zero energy) defined in the complex domain Θ (see formulas (4.4), (4.5), (4.1)-(4.3) of Section 4 for d = 3), Ξ is an appropriate subset of Θ, for example Ξ is the subset of Θ with the imaginary part length equal to ρ, vρ,τ is an approximation to v on

B 2τ ρ , vρ,τ → v on B 2τ ρ as ρ → +∞, (3.2)
where v is the Fourier transform of v (see formula (4.12), d = 3), B r is the ball of the radius r, τ is a fixed parameter, 0 < τ ≤ 1.

3.1. Theorem 2.1 and global reconstructions of [START_REF] Novikov | Multidimensional inverse spectral problem for the equation -∆ψ + (v(x) -Eu(x))ψ = 0, Funkt. Anal. i Pril[END_REF], [START_REF] Novikov | Formulae and equations for finding scattering data from the Dirichletto-Neumann map with nonzero background potential[END_REF]. For the case when Ξ = bΘ ρ (see formulas (6.2)), (3.3a)

h bΘ ρ is determined from Φ via (5.1), (5.2) considered for v 2 = v, ψ 2 = ψ, h 2 = h, v 1 ≡ 0, ψ 1 (x, k) = e ikx , R 1 (x, y, k) = G(x -y, k), h 1 ≡ 0, (3.3b) vρ,τ (p) = h(k(p), l(p)), p ∈ B 2τ ρ , τ = 1, for some vector -functions k(p), l(p) such that k(p), l(p) ∈ bΘ ρ , k(p) -l(p) = p, (3.3c)
reconstruction (3.1) was, actually, given in [START_REF] Novikov | Multidimensional inverse spectral problem for the equation -∆ψ + (v(x) -Eu(x))ψ = 0, Funkt. Anal. i Pril[END_REF]. In this case (3.2) holds according to (4.10), (4.11). (In the framework of [START_REF] Novikov | Multidimensional inverse spectral problem for the equation -∆ψ + (v(x) -Eu(x))ψ = 0, Funkt. Anal. i Pril[END_REF] less precise version of (4.11) was known.) General formulas and equations (5.1)-( 5.3) for finding h bΘ ρ from Φ with known background potential v 1 ≡ 0 were given for the first time in [START_REF] Novikov | Formulae and equations for finding scattering data from the Dirichletto-Neumann map with nonzero background potential[END_REF]. If unknown potential v is sufficiently close to some known background potential v 1 ≡ 0, then formulas and equations (5.1)-( 5.3) considered for v 2 = v, ψ 2 = ψ, h 2 = h give more stable determination of h bΘ ρ from Φ than these formulas and equations for v 1 ≡ 0 (see [START_REF] Novikov | Formulae and equations for finding scattering data from the Dirichletto-Neumann map with nonzero background potential[END_REF] for more detailed discussion of this issue).

Note that Theorem 2.1 follows from formulas (4.10), (4.11), (5.1) mentioned above in connection with the versions of (3.1), (3.2) of [START_REF] Novikov | Multidimensional inverse spectral problem for the equation -∆ψ + (v(x) -Eu(x))ψ = 0, Funkt. Anal. i Pril[END_REF], [START_REF] Novikov | Formulae and equations for finding scattering data from the Dirichletto-Neumann map with nonzero background potential[END_REF]. [START_REF] Novikov | Formulae and equations for finding scattering data from the Dirichletto-Neumann map with nonzero background potential[END_REF], [START_REF] Novikov | An effectivization of the global reconstruction in the Gel'fand-Calderon inverse problem in three dimensions[END_REF]. The main disadvantage of the global reconstruction (3.1), (3.2) in the framework of [START_REF] Novikov | Multidimensional inverse spectral problem for the equation -∆ψ + (v(x) -Eu(x))ψ = 0, Funkt. Anal. i Pril[END_REF], [START_REF] Novikov | Formulae and equations for finding scattering data from the Dirichletto-Neumann map with nonzero background potential[END_REF] is related with the following two facts:

Theorem 2.2 and global reconstructions of

(1) The determination of h Ξ for Ξ = bΘ ρ or, more generally, for Ξ ⊆ Θρ , Ξ ∩ bΘ ρ = ∅ (see formulas (6.2)) from Φ via formulas and equations (5.1)-(5.3) for v 2 = v, ψ 2 = ψ, h 2 = h with some known background potential v 1 is stable for relatively small ρ, but is very unstable for ρ → +∞ in the points of Ξ with sufficiently great imaginary part (see [START_REF] Novikov | Formulae and equations for finding scattering data from the Dirichletto-Neumann map with nonzero background potential[END_REF], [START_REF] Novikov | On non-overdetermined inverse scattering at zero energy in three dimensions[END_REF], [START_REF] Novikov | An effectivization of the global reconstruction in the Gel'fand-Calderon inverse problem in three dimensions[END_REF] for more detailed discussion of this issue).

(2) For vρ,τ defined as in (3.3c) the decay of the error vvρ,τ on B 2ρτ is very slow for ρ → +∞ (not faster than O(ρ -1 ) even for infinitely smooth compactly supported v) (see [START_REF] Novikov | An effectivization of the global reconstruction in the Gel'fand-Calderon inverse problem in three dimensions[END_REF] for more detailed discussion of this issue).

As a corollary, the global reconstruction (3.1), (3.2) in the framework of [START_REF] Novikov | Multidimensional inverse spectral problem for the equation -∆ψ + (v(x) -Eu(x))ψ = 0, Funkt. Anal. i Pril[END_REF] and even in the framework of [START_REF] Novikov | Formulae and equations for finding scattering data from the Dirichletto-Neumann map with nonzero background potential[END_REF] is not optimal with respect to its stability properties. A principle effectivization of the global reconstruction (3.1), (3.2) with respect to its stability properties was given in [START_REF] Novikov | An effectivization of the global reconstruction in the Gel'fand-Calderon inverse problem in three dimensions[END_REF].

The main new result of [START_REF] Novikov | An effectivization of the global reconstruction in the Gel'fand-Calderon inverse problem in three dimensions[END_REF] consists in stable construction of some vρ,τ on B 2τ ρ from h on bΘ ρ,τ ⊆ bΘ ρ (see definitions (6.2)) in such a way that

|v(p) -vρ,τ (p)| ≤ C(m, µ 0 , τ )N 2 (1 + |p|) -µ 0 ρ -(m-µ 0 ) for p ∈ B 2τ ρ , ρ ≥ ρ 1 and fixed τ ∈]0, τ 1 [, (3.4)

R.G. Novikov

under the assumptions that

v ∈ W m,1 (R 2 ), v m,1 < N (see definitions (2.2), (2.3)), m > 2, 2 ≤ µ 0 < m, τ 1 = τ 1 (m, µ 0 , N ), ρ 1 = ρ 1 (m, µ 0 , N ), (3.5)
where C(m, µ 0 , τ ), τ 1 (m, µ 0 , N ), ρ 1 (m, µ 0 , N ) are special constants and, in particular, 0 < τ 1 < 1, see Theorem 2.1 of [START_REF] Novikov | An effectivization of the global reconstruction in the Gel'fand-Calderon inverse problem in three dimensions[END_REF]. The scheme of this construction of [START_REF] Novikov | An effectivization of the global reconstruction in the Gel'fand-Calderon inverse problem in three dimensions[END_REF] consists in the following:

h bΘ ρ,τ → H bΩ ρ,τ → H bΛ ρ,τ,ν → H 0 Λ ρ,τ,ν → Hρ,τ on Λ ρ,τ,ν → vρ,τ on B 2τ ρ . ( 3
(p) = lim λ→0 Hρ,τ (λ, p), v- ρ,τ (p) = lim λ→∞ Hρ,τ (λ, p), (3.7)
where definitions (3.7) are similar to formulas (7.16). For more detailed discussion of (3.6) see Section 5 of [START_REF] Novikov | An effectivization of the global reconstruction in the Gel'fand-Calderon inverse problem in three dimensions[END_REF]. Note that in [START_REF] Novikov | An effectivization of the global reconstruction in the Gel'fand-Calderon inverse problem in three dimensions[END_REF] we deal with approximate finding v from h bΘ ρ in a similar way with approximate finding v from the scattering amplitude f at fixed energy E of [START_REF] Novikov | The ∂approach to approximate inverse scattering at fixed energy in three dimensions[END_REF]. The parameter ρ of [START_REF] Novikov | An effectivization of the global reconstruction in the Gel'fand-Calderon inverse problem in three dimensions[END_REF] plays the role of the parameter √ E of [START_REF] Novikov | The ∂approach to approximate inverse scattering at fixed energy in three dimensions[END_REF]. One can see that for sufficiently regular potentials v that is for v ∈ W m,1 (R 3 ), where m is sufficiently great, and for vρ,τ constructed from h bΘ ρ,τ via (3.6) as in [START_REF] Novikov | An effectivization of the global reconstruction in the Gel'fand-Calderon inverse problem in three dimensions[END_REF] the decay rate of the error vvρ,τ on B 2τ ρ is O(ρ -(m-2) ) as ρ → +∞ (see (3.4) for µ 0 = 2) and is much faster than for vρ,τ given by (3.3c) as in [START_REF] Novikov | Multidimensional inverse spectral problem for the equation -∆ψ + (v(x) -Eu(x))ψ = 0, Funkt. Anal. i Pril[END_REF], [START_REF] Novikov | Formulae and equations for finding scattering data from the Dirichletto-Neumann map with nonzero background potential[END_REF]. Therefore, for Problem 1.1 in dimension d = 3 the global reconstruction of [START_REF] Novikov | An effectivization of the global reconstruction in the Gel'fand-Calderon inverse problem in three dimensions[END_REF] (using also (5.1)-(5.3) for finding h bΘ ρ,τ from Φ as in [START_REF] Novikov | Formulae and equations for finding scattering data from the Dirichletto-Neumann map with nonzero background potential[END_REF]) has much more optimal stability properties than the global reconstructions of [START_REF] Novikov | Multidimensional inverse spectral problem for the equation -∆ψ + (v(x) -Eu(x))ψ = 0, Funkt. Anal. i Pril[END_REF], [START_REF] Novikov | Formulae and equations for finding scattering data from the Dirichletto-Neumann map with nonzero background potential[END_REF].

Therefore, Theorem 1.2 illustrating the progress in global stability of [START_REF] Novikov | Formulae and equations for finding scattering data from the Dirichletto-Neumann map with nonzero background potential[END_REF], [START_REF] Novikov | An effectivization of the global reconstruction in the Gel'fand-Calderon inverse problem in three dimensions[END_REF] is much more optimal than Theorem 1.1 related with global reconstruction results of [START_REF] Novikov | Multidimensional inverse spectral problem for the equation -∆ψ + (v(x) -Eu(x))ψ = 0, Funkt. Anal. i Pril[END_REF] and [START_REF] Novikov | Formulae and equations for finding scattering data from the Dirichletto-Neumann map with nonzero background potential[END_REF] only.

Faddeev functions

We consider the Faddeev functions G, ψ and h (see [START_REF] Faddeev | Growing solutions of the Schrödinger equation[END_REF], [START_REF] Faddeev | Inverse problem of quantum scattering theory II[END_REF], [START_REF] Henkin | The ∂equation in the multidimensional inverse scattering problem[END_REF], [START_REF] Novikov | Multidimensional inverse spectral problem for the equation -∆ψ + (v(x) -Eu(x))ψ = 0, Funkt. Anal. i Pril[END_REF]):

ψ(x, k) = e ikx + R d G(x -y, k)v(y)ψ(y, k)dy, (4.1)
G(x, k) = e ikx g(x, k), g(x, k) = -(2π) -d R d e iξx dξ ξ 2 + 2kξ , (4.2)
where

x ∈ R d , k ∈ Σ, Σ = {k ∈ C d : k 2 = k 2 1 + . . . + k 2 d = 0}; (4.3) h(k, l) = (2π) -d R d e -ilx v(x)ψ(x, k)dx, (4.4)
where

(k, l) ∈ Θ, Θ = {k ∈ Σ, l ∈ Σ : Im k = Im l}. (4.5)
One can consider (4.1), (4.4) assuming that v is a sufficiently regular function on R d with sufficient decay at infinity. For example, one can consider (4.1), (4.4) assuming that (1.2) holds.

We recall that:

∆G(x, k) = δ(x), x ∈ R d , k ∈ Σ; (4.6)
formula (4.1) at fixed k is considered as an equation for

ψ = e ikx µ(x, k), (4.7)
where µ is sought in L ∞ (R d ); as a corollary of (4.1),(4.2), (4.6), ψ satisfies (1.1); h of (4.4) is a generalized "scattering" amplitude in the complex domain at zero energy. Note that, actually, G, ψ, h of (4.1)-(4.5) are zero energy restrictions of functions introduced by Faddeev as extentions to the complex domain of some functions of the classical scattering theory for the Schrödinger equation at positive energies. In addition, G, ψ, h in their zero energy restriction were considered for the first time in [START_REF] Beals | Multidimensional inverse scattering and nonlinear partial differential equations[END_REF]. The Faddeev functions G, ψ, h were, actually, rediscovered in [START_REF] Beals | Multidimensional inverse scattering and nonlinear partial differential equations[END_REF].

We recall also that, under the assumptions of Theorem 2.1,

µ(x, k) → 1 as |Im k| → ∞ (uniformly in x) (4.8)
and, for any σ > 1,

|µ(x, k)| < σ for |Im k| ≥ r 1 (N, D, m, σ), (4.9)
where

x ∈ R d , k ∈ Σ; v(p) = lim (k,l)∈Θ, k-l=p |Im k|=|Im l|→∞ h(k, l) for any p ∈ R d , (4.10
)

|v(p) -h(k, l)| ≤ c 3 (D, m)N 2 ρ for (k, l) ∈ Θ, p = k -l, |Im k| = |Im l| = ρ ≥ r 2 (N, D, m), (4.11)
where

v(p) = 1 2π d R d e ipx v(x)dx, p ∈ R d . (4.12)
Results of the type (4.8), (4.9) go back to [START_REF] Beals | Multidimensional inverse scattering and nonlinear partial differential equations[END_REF]. Results of the type (4.10), (4.11) (with less precise right-hand side in (4.11)) go back to [START_REF] Henkin | The ∂equation in the multidimensional inverse scattering problem[END_REF]. Estimates (4.8), (4.11) are related also with some important L 2 -estimate going back to [START_REF] Sylvester | A global uniqueness theorem for an inverse boundary value problem[END_REF] on the Green function g of (4.1).

Note also that in some considerations it is convenient to consider h on Θ as H on Ω, where

h(k, l) = H(k, k -l), (k, l) ∈ Θ, H(k, p) = h(k, k -p), (k, p) ∈ Ω, (4.13
)

Ω = {k ∈ C d , p ∈ R d : k 2 = 0, p 2 = 2kp}. (4.14)
For more information on properties of the Faddeev functions G, ψ, h, see [START_REF] Henkin | The ∂equation in the multidimensional inverse scattering problem[END_REF], [START_REF] Novikov | Formulae and equations for finding scattering data from the Dirichletto-Neumann map with nonzero background potential[END_REF], [START_REF] Novikov | An effectivization of the global reconstruction in the Gel'fand-Calderon inverse problem in three dimensions[END_REF] and references therein.

In the next section we recall that Problem 1.1 (of Introduction) admits a reduction to the following inverse "scattering" problem: Problem 4.1. Given h on Θ, find v on R d . [START_REF] Novikov | Multidimensional inverse spectral problem for the equation -∆ψ + (v(x) -Eu(x))ψ = 0, Funkt. Anal. i Pril[END_REF], [START_REF] Novikov | Formulae and equations for finding scattering data from the Dirichletto-Neumann map with nonzero background potential[END_REF] of Problem 1.1 to Problem 4.1 Let conditions (1.2), (1.4) hold for potentials v 1 and v 2 , where D is fixed. Let Φ i , ψ i , h i denote the DtN map Φ and the Faddeev functions ψ, h for v = v i , i = 1, 2. Let also Φ i (x, y) denote the Schwartz kernel Φ(x, y) of the integral operator Φ for v = v i , i = 1, 2. Then (see [START_REF] Novikov | Formulae and equations for finding scattering data from the Dirichletto-Neumann map with nonzero background potential[END_REF] for details):

Reduction of

h 2 (k, l) -h 1 (k, l) = 1 2π d ∂D ∂D ψ 1 (x, -l)(Φ 2 -Φ 1 )(x, y)ψ 2 (y, k)dydx, (5.1)
where (k, l) ∈ Θ;

ψ 2 (x, k) = ψ 1 (x, k) + ∂D A(x, y, k)ψ 2 (y, k)dy, x ∈ ∂D, (5.2a)
A(x, y, k) = ∂D R 1 (x, z, k)(Φ 2 -Φ 1 )(z, y)dz, x, y ∈ ∂D, (5.2b
)

R 1 (x, y, k) = G(x -y, k) + R d G(x -z, k)v 1 (z)R 1 (z, y, k)dz, x, y ∈ R d , (5.3)
where k ∈ Σ. Note that: (5.1) is an explicit formula, (5.2a) is considered as an equation for finding ψ 2 on ∂D from ψ 1 on ∂D and A on ∂D × ∂D for each fixed k, (5.2b) is an explicit formula, (5.3) is an equation for finding R 1 from G and v 1 , where G is the function of (4.2). Note that formulas and equations (5.1)-(5.3) for v 1 ≡ 0 were given in [START_REF] Novikov | Multidimensional inverse spectral problem for the equation -∆ψ + (v(x) -Eu(x))ψ = 0, Funkt. Anal. i Pril[END_REF] (see also [START_REF] Henkin | The ∂equation in the multidimensional inverse scattering problem[END_REF] (Note added in proof), [START_REF] Nachman | Reconstructions from boundary measurements[END_REF], [START_REF] Nachman | Global uniqueness for a two-dimensional inverse boundary value problem[END_REF]). In this case h 1 ≡ 0, ψ 1 = e ikx , R 1 = G(xy, k). Formulas and equations (5.1)-(5.3) for the general case were given in [START_REF] Novikov | Formulae and equations for finding scattering data from the Dirichletto-Neumann map with nonzero background potential[END_REF].

Formulas and equations (5.1)-(5.

3) with fixed background potential v 1 reduce Problem 1.1 (of Introduction) to Problem 4.1 (of Section 4). In this connection we consider (5.1)-(5.3) for v 2 = v, ψ 2 = ψ, h 2 = h and this reduction consists of the following steps: (1) v 1 → Φ 1 , ψ 1 , R 1 , h 1 via formulas and equations of the diret problem for v 1 ;

(2) Φ, Φ 1 , R 1 ∂D×∂D×Σ → A via (5.2b);

(3) A, ψ 1 ∂D×Σ → ψ ∂D×Σ via (5.2a); (4) h 1 , ψ 1 ∂D×Σ , ψ ∂D×Σ , Φ, Φ 1 → h via (5.1).

Here, on the step (4) we find h on Θ for the unknown potential v of Problem 1.1.

6. Some considerations related with Θ and Ω 6.1 Some subsets of Θ and Ω. Let

B r = {p ∈ R d : |p| < r}, ∂B r = {p ∈ R d : |p| = r}, Br = B r ∪ ∂B r , where r > 0. (6.1)
In addition to Θ of (4.5), we consider, in particular, the following its subsets:

Θ ρ = {(k, l) ∈ Θ : |Im k| = |Im l| < ρ}, bΘ ρ = {(k, l) ∈ Θ : |Im k| = |Im l| = ρ}, Θρ = Θ ρ ∪ bΘ ρ , Θ ∞ ρ,τ = {(k, l) ∈ Θ\ Θρ : k -l ∈ B 2ρτ }, bΘ ρ,τ = {(k, l) ∈ bΘ ρ : k -l ∈ B 2ρτ }, (6.2)
where ρ > 0, 0 < τ < 1, and B r is defined in (6.1).

In addition to Ω of (4.14), we consider, in particular, the following its subsets:

Ω ρ = {(k, p) ∈ Ω : |Im k| < ρ}, bΩ ρ = {(k, p) ∈ Ω : |Im k| = ρ}, Ωρ = Ω ρ ∪ bΩ ρ , Ω ∞ ρ,τ = {(k, p) ∈ Ω\ Ωρ : p ∈ B 2ρτ }, bΩ ρ,τ = {(k, p) ∈ bΩ ρ : p ∈ B 2ρτ }, (6.3)
where ρ > 0, 0 < τ < 1, and B r is defined in (6.1).

Note that

Ω ≈ Θ, Ω ρ ≈ Θ ρ , bΩ ρ ≈ bΘ ρ , Ω ∞ ρ,τ ≈ Θ ∞ ρ,τ , bΩ ρ,τ ≈ bΘ ρ,τ , (6.4)
or, more precisely,

(k, p) ∈ Ω ⇒ (k, k -p) ∈ Θ, (k, l) ∈ Θ ⇒ (k, k -l)
∈ Ω and the same for Ω ρ , bΩ ρ , Ω ∞ ρ,τ , bΩ ρ,τ and Θ ρ , bΘ ρ , Θ ∞ ρ,τ , bΘ ρ,τ , respectively, in place of Ω and Θ.

(6.5)

We consider also, in particular,

Ω ν = {(k, p) ∈ Ω : p / ∈ L ν }, Ω ∞ ρ,τ,ν = Ω ∞ ρ,τ ∩ Ω ν , bΩ ρ,τ,ν = bΩ ρ,τ ∩ Ω ν , (6.6)
where

L ν = {p ∈ R d : p = tν, t ∈ R}, (6.7)
ν ∈ S d-1 , ρ > 0, 0 < τ < 1.

6.2. Coordinates on Ω for d = 3. In this subsection we assume that d = 3 in formulas (4.5), (4.14), (6.1)-(6.7).

For p ∈ R 3 \L ν we consider θ(p) and ω(p) such that θ(p), ω(p) smoothly depend on p ∈ R 3 \L ν , take values in S 2 , and

θ(p)p = 0, ω(p)p = 0, θ(p)ω(p) = 0, (6.8)
where L ν is defined by (6.7) (for d = 3). Assumptions (6.8) imply that

ω(p) = p × θ(p) |p| for p ∈ R 3 \L ν (6.9a) or ω(p) = - p × θ(p) |p| for p ∈ R 3 \L ν , (6.9b)
where × denotes vector product.

To satisfy (6.8), (6.9a) we can take

θ(p) = ν × p |ν × p| , ω(p) = p × θ(p) |p| , p ∈ R 3 \L ν . (6.10)
Let θ, ω satisfy (6.8). Then (according to [START_REF] Novikov | On non-overdetermined inverse scattering at zero energy in three dimensions[END_REF]) the following formulas give a diffeomorphism between Ω ν and (C\0)

× (R 3 \L ν ): (k, p) → (λ, p), where λ = λ(k, p) = 2k(θ(p) + iω(p)) i|p| , (6.11a)
(λ, p) → (k, p), where k = k(λ, p) = κ 1 (λ, p)θ(p) + κ 2 (λ, p)ω(p) + p 2 , κ 1 (λ, p) = i|p| 4 (λ + 1 λ ), κ 2 (λ, p) = |p| 4 (λ - 1 λ ), (6.11b)
where (k, p) ∈ Ω ν , (λ, p) ∈ (C\0)×(R 3 \L ν ). In addition, formulas (6.11a), (6.11b) for λ(k) and k(λ) at fixed p ∈ R 3 \L ν give a diffeomorphism between Z p = {k ∈ C 3 : (k, p) ∈ Ω} for fixed p and C\0.

In addition, for k and λ of (6.11) we have that

|Im k| = |p| 4 |λ| + 1 |λ| , |Re k| = |p| 4 |λ| + 1 |λ| , (6.12)
where (k, p)

∈ Ω ν , (λ, p) ∈ (C\0) × (R 3 \L ν ). Let Λ ρ,ν = {(λ, p) : λ ∈ D ρ/|p| , p ∈ R 3 \L ν }, (6.13) Λ ρ,τ,ν = {(λ, p) : λ ∈ D ρ/|p| , p ∈ R 3 \L ν , |p| < 2τ ρ}, bΛ ρ,τ,ν = {(λ, p) : λ ∈ T ρ/|p| , p ∈ R 3 \L ν , |p| < 2τ ρ}, where ρ > 0, 0 < τ < 1, ν ∈ S 2 , D r = {λ ∈ C\0 : 1 4 (|λ| + |λ| -1
) > r}, r > 0, (6.14)

T r = {λ ∈ C : 1 4 (|λ| + |λ| -1 ) = r}, r ≥ 1/2, (6.15)
L ν is defined by (6.7) (for

d = 3). Note that Λ ρ,τ,ν = Λ + ρ,τ,ν ∪ Λ - ρ,τ,ν , Λ + ρ,τ,ν ∩ Λ - ρ,τ,ν = ∅, bΛ ρ,τ,ν = bΛ + ρ,τ,ν ∪ bΛ - ρ,τ,ν , (6.16)
where

Λ ± ρ,τ,ν = {(λ, p) : λ ∈ D ± ρ/|p| , p ∈ B 2τ ρ \L ν }, bΛ ± ρ,τ,ν = {(λ, p) : λ ∈ T ± ρ/|p| , p ∈ B 2τ ρ \L ν }, (6.17)
D ± r = {λ ∈ C\0 : 1 4 (|λ| + |λ| -1 ) > r, |λ| ±1 < 1}, T ± r = {λ ∈ C : 1 4 (|λ| + |λ| -1 ) = r, |λ| ±1 ≤ 1}, r > 1/2, (6.18)
where ρ > 0, τ ∈]0, 1[, ν ∈ S 2 . Using (6.12) one can see that formulas (6.11) give also the following diffeomorphisms

Ω ν \ Ωρ ≈ Λ ρ,ν , Ω ∞ ρ,τ,ν ≈ Λ ρ,τ,ν , bΩ ρ,τ,ν ≈ bΛ ρ,τ,ν , Z ∞ p,ρ = {k ∈ C 3 : (k, p) ∈ Ω ν \ Ωρ } ≈ D ρ/|p| for fixed p, (6.19)
where ρ > 0, 0 < τ < 1, ν ∈ S 2 (and where we use the definitions (6.3), (6.6), (6.13)).

In [START_REF] Novikov | On non-overdetermined inverse scattering at zero energy in three dimensions[END_REF] λ, p of (6.11) were used as coordinates on Ω. In the present work we use them also as coordinates on Ω\Ω ρ (or more precisely on Ω ν \Ω ρ ).

7. An integral equation of [START_REF] Novikov | An effectivization of the global reconstruction in the Gel'fand-Calderon inverse problem in three dimensions[END_REF] and some related formulas In the main considerations of [START_REF] Novikov | An effectivization of the global reconstruction in the Gel'fand-Calderon inverse problem in three dimensions[END_REF] it is assumed that d = 3 and the basic assumption on v consists in the following condition on its Fourier transform:

v ∈ L ∞ µ (R 3 ) ∩ C(R 3 ) for some real µ ≥ 2, (7.1)
where v is defined by (4.12) (for d = 3),

L ∞ µ (R d ) = {u ∈ L ∞ (R d ) : u µ < +∞}, u µ = ess sup p∈R d (1 + |p|) µ |u(p)|, µ > 0, (7.2)
and C denotes the space of continuous functions.

Note that

v ∈ W m,1 (R d ) =⇒ v ∈ L ∞ µ (R d ) ∩ C(R d ), v µ ≤ c 4 (m, d) v m,1 for µ = m, (7.3)
where W m,1 , L ∞ µ are the spaces of (2.2), (7.2). Let H(λ, p) = H(k(λ, p), p), (λ, p) ∈ (C\0) × (R 3 \L µ ), (7.4) where H is the function of (4.13), λ, p are the coordinates of Subsection 6.2 under assumption (6.9a). Let

L ∞ µ (Λ ρ,τ,ν ) = {U ∈ L ∞ (Λ ρ,τ,ν ) : |||U ||| ρ,τ,µ < ∞}, |||U ||| ρ,τ,µ = ess sup (λ,p)∈Λ ρ,τ,ν (1 + |p|) µ |U (λ, p)|, µ > 0, (7.5)
where Λ ρ,τ,ν is defined in (6.13), ρ > 0,

τ ∈]0, 1[, ν ∈ S 2 , µ > 0. Let v satisfy (7.1) and v µ ≤ C. Let η(C, ρ, µ) def = a(µ)C(ln ρ) 2 ρ -1 < 1, ln ρ ≥ 2, (7.6)
where a(µ) is the constant c 2 (µ) of [START_REF] Novikov | An effectivization of the global reconstruction in the Gel'fand-Calderon inverse problem in three dimensions[END_REF]. Let H(λ, p) be defined by (7.4) and be considered as a function on Λ ρ,τ,ν of (6.13). Then (see Section 5 of [START_REF] Novikov | An effectivization of the global reconstruction in the Gel'fand-Calderon inverse problem in three dimensions[END_REF]):

H = H 0 + M ρ,τ (H) + Q ρ,τ , τ ∈]0, 1[, (7.7)
where

H 0 (λ, p) = 1 2πi T + ρ/|p| H(ζ, p) dζ ζ -λ , (λ, p) ∈ Λ + ρ,τ,ν , (7.8a)
H 0 (λ, p) = - 1 2πi T - ρ/|p| H(ζ, p) λdζ ζ(ζ -λ) , (λ, p) ∈ Λ - ρ,τ,ν , (7.8b)
where Λ ± ρ,τ,ν , T ± r are defined in (6.17), (6.18) (and where the integrals along T ± r are taken in the counter-clock wise direction);

M ρ,τ (U )(λ, p) = M + ρ,τ (U )(λ, p) = - 1 π D + ρ/|p| (U, U ) ρ,τ (ζ, p) dRe ζdIm ζ ζ -λ , (λ, p) ∈ Λ + ρ,τ,ν , (7.9a)
M ρ,τ (U )(λ, p) = M - ρ,τ (U )(λ, p) = - 1 π D - ρ/|p| (U, U ) ρ,τ (ζ, p) λdRe ζdIm ζ ζ(ζ -λ) , (λ, p) ∈ Λ - ρ,τ,ν , (7.9b)
(U 1 , U 2 ) ρ,τ (ζ, p) = {χ 2τ ρ U ′ 1 , χ 2τ ρ U ′ 2 }(ζ, p), (ζ, p) ∈ Λ ρ,τ,ν , χ 2τ ρ U ′ j (k, p) = U j (λ(k, p), p), (k, p) ∈ Ω ∞ ρ,τ,ν , χ 2τ ρ U ′ j (k, p) = 0, |p| ≥ 2τ ρ, j = 1, 2, (7.10)
where U, U 1 , U 2 are test functions on Λ ρ,τ,ν , Ω ∞ ρ,τ,ν is defined in (6.6), λ(k, p) is defined in (6.11a), {•, •} is defined by the formula

{F 1 , F 2 }(λ, p) = - π 4 π -π |p| 2 |λ| 2 -1 λ|λ| (cos ϕ -1) - |p| λ sin ϕ × F 1 (k(λ, p), -ξ(λ, p, ϕ))F 2 (k(λ, p) + ξ(λ, p, ϕ), p + ξ(λ, p, ϕ))dϕ, (7.11)
for (λ, p) ∈ Λ ρ,ν , where F 1 , F 2 are test functions on Ω\ Ωρ , k(λ, p) is defined in (6.11b), Λ ρ,ν is defined in (

= Re k(λ, p)(cos ϕ -1) + k ⊥ (λ, p) sin ϕ, 6.13), ξ(λ, p, ϕ)
k ⊥ (λ, p) = Im k(λ, p) × Re k(λ, p) |Im k(λ, p)| , (7.12)
where × in (7.13) denotes vector product;

H, H 0 , Q ρ,τ ∈ L ∞ µ (Λ ρ,τ,ν ), (7.14)
|||H||| ρ,τ,µ 0 ≤ C 1 -η(C, ρ, µ) , (7.15a)
|||H 0 ||| ρ,τ,µ 0 ≤ C 1 -η(C, ρ, µ) 1 + c 5 (µ 0 )C 1 -η(C, ρ, µ) , (7.15b)
|||Q ρ,τ ||| ρ,τ,µ 0 ≤ 3c 5 (µ 0 )C 2 (1 -η(C, ρ, µ)) 2 (1 + 2τ ρ) µ-µ 0 , (7.15c)
where 2 ≤ µ 0 ≤ µ, c 5 is the constant b 4 of [START_REF] Novikov | An effectivization of the global reconstruction in the Gel'fand-Calderon inverse problem in three dimensions[END_REF], η(C, ρ, µ) is defined by (7.6).

Following [START_REF] Novikov | An effectivization of the global reconstruction in the Gel'fand-Calderon inverse problem in three dimensions[END_REF] we consider (7.7) as an approxiate integral equation for finding H on Λ ρ,τ,ν from H 0 on Λ ρ,τ,ν with unknown remainder Q ρ,τ .

Note also that if v satisfies (7.1), then (see [START_REF] Novikov | On non-overdetermined inverse scattering at zero energy in three dimensions[END_REF], [START_REF] Novikov | An effectivization of the global reconstruction in the Gel'fand-Calderon inverse problem in three dimensions[END_REF])

H(λ, p) → v(p) as λ → 0, H(λ, p) → v(p) as λ → ∞, (7.16)
where p ∈ B 2τ ρ \L ν , H is defined by (7.4) and is considered as a function on Λ ρ,τ,ν , ρ > 0, 0 < τ < 1, ν ∈ S 2 .

Proof of Theorem 2.2 for d = 3

Our proof of Theorem 2.2 for d = 3 is based on formulas (7.16) of Section 7 and on Lemma 8.4 given below in the present section. In turn, Lemma 8.4 follows from formula (7.3) of Section 7 and from Lemmas 8.1, 8.2, 8.3 given below in the present section. In turn:

(1) Lemma 8.1 follows from some estimates and Lemmas of [START_REF] Novikov | An effectivization of the global reconstruction in the Gel'fand-Calderon inverse problem in three dimensions[END_REF] related with studies of the non-linear integral equation given as equation (7.7) of the present paper; (2) Lemma 8.2 does not follow immediately from results of preceding works and its proof is given in Section 9; (3) Lemma 8.3 follows from formulas (4.9), (5.1) and from definitions, see Section 9 for proof of this lemma.

Lemma 8.1. Let vi satisfy (7.1) and vi µ < C, where i = 1, 2. Let

0 < τ ≤ τ 1 (µ, µ 0 , C, δ), ρ ≥ ρ 1 (µ, µ 0 , C, δ), (8.1)
where τ 1 , ρ 1 are the constants of Section 4 of [START_REF] Novikov | An effectivization of the global reconstruction in the Gel'fand-Calderon inverse problem in three dimensions[END_REF] and where δ = 1/2, 2 ≤ µ 0 < µ. Then

|||H 2 -H 1 ||| ρ,τ,µ 0 ≤ 2(|||H 0 2 -H 0 1 ||| ρ,τ,µ 0 + |||Q 2 ρ,τ -Q 1 ρ,τ ||| ρ,τ,µ 0 ), (8.2)
where H i , H 0 i , Q i ρ,τ are the functions of (7.4), (7.7), (7.8), (7.14), (7.15)

for v = v i , i = 1, 2, ||| • ||| ρ,τ,
µ 0 is defined as in (7.5). In addition,

|||Q 2 ρ,τ -Q 1 ρ,τ ||| ρ,τ,µ 0 ≤ 24c 5 (µ 0 )C 2 (1 + 2τ ρ) µ-µ 0 . ( 8.3)
In connection with (8.1) we remind that τ 1 ∈]0, 1[ is sufficiently small and ρ 1 is sufficiently great, see [START_REF] Novikov | An effectivization of the global reconstruction in the Gel'fand-Calderon inverse problem in three dimensions[END_REF]. Lemma 8.1 follows from estimates mentioned as estimates (7.14), (7.15) of the present paper (see estimates (4.3), (5.20), (5.22) of [START_REF] Novikov | An effectivization of the global reconstruction in the Gel'fand-Calderon inverse problem in three dimensions[END_REF]) and from Lemmas A.1, A.2, A.3 and estimate (A.5) (see Appendix to the present paper). Lemma 8.2. Let vi satisfy (7.1) and vi µ < C, where i = 1, 2. Let

η(C, ρ 0 , µ) ≤ 1/2, ln ρ 0 ≥ 2, ( 8.4)
where η is defined in (7.6). Let

0 < τ 0 < 1, 2 ≤ µ 0 < µ, ρ = 2ρ 0 , τ = τ 0 /2. Then |||H 0 2 -H 0 1 ||| ρ,τ,µ 0 ≤ (c 6 + 4c 7 (µ 0 , τ 0 , ρ 0 )C)∆ ρ 0 ,ρ,µ 0 , (8.5)
∆ ρ 0 ,ρ,µ 0 def = |||χ ρ 0 ,ρ (H 2 -H 1 )||| ρ 0 ,µ 0 , (8.6)
where H i , H 0 i are the functions of (4.13), (7.8) for v = v i , i = 1, 2, χ ρ 0 ,ρ is the characteristic function of Ωρ \ Ωρ 0 , ||| • ||| ρ,τ,µ 0 in (8.5) is defined as in (7.5), ||| • ||| ρ 0 ,µ 0 in (8.6) is defined as in (A.12) of Appendix, c 6 is defined by (9.9), c 7 is the constant c 8 of [START_REF] Novikov | An effectivization of the global reconstruction in the Gel'fand-Calderon inverse problem in three dimensions[END_REF] (that is c 7 (µ, τ, ρ) = 3b 1 (µ)τ 2 + 4b 2 (µ)ρ -1 + 4b 3 (µ)τ , where b 1 , b 2 , b 3 are the constants of [START_REF] Novikov | An effectivization of the global reconstruction in the Gel'fand-Calderon inverse problem in three dimensions[END_REF]). Lemma 8.2 is proved in Section 9.

Lemma 8.3. Let the assumptions of Theorem 2.1 hold (for d = 3). Let

ρ 0 ≥ r 1 (N, D, m, σ) for some σ > 1, (8.7)
where r 1 is the number of (4.9). Let 0 < τ 0 < 1, 0 < µ 0 , ρ = 2ρ 0 , τ = τ 0 /2. Then

∆ ρ 0 ,ρ,µ 0 ≤ c 8 σ 2 e 2ρL Φ 2 -Φ 1 (1 + 2ρ) µ 0 , (8.8)
where ∆ ρ 0 ,ρ,µ 0 is defined by (8.6),

c 8 = (2π) -d ∂D dx, L = max x∈∂D |x|, (8.9)
Φ 2 -Φ 1 is defined according to (2.4). Lemma 8.3 is proved in Section 9.

Lemma 8.4. Let the assumptions of Theorem 2.1 hold (for d = 3). Let

0 < τ ≤ τ 2 (m, µ 0 , N ), ρ ≥ ρ 2 (m, µ 0 , N, D, σ), (8.10)
where 2 ≤ µ 0 < m, σ > 1, and τ 2 , ρ 2 are constants such that (8.10) implies that τ ≤ τ 1 (m, µ 0 , c 4 (m, 3)N, 1/2), ρ ≥ ρ 1 (m, µ 0 , c 4 (m, 3)N, 1/2), (8.11a)

τ < 1/2, η(c 4 (m, 3)N, ρ/2, m) ≤ 1/2, ln(ρ/2) ≥ 2, (8.11b) ρ/2 ≥ r 1 (N, D, m, σ), (8.11c) where τ 1 , ρ 1 , η, r 1 are the same that in (8.1), (8.4), (8.7), c 4 is the constant of (7.3). Then

|||H 2 -H 1 ||| ρ,τ,µ 0 ≤ c 9 (N, D, m, µ 0 , σ, τ )e 2Lρ ρ µ 0 Φ 2 -Φ 1 + c 10 (N, m, µ 0 , τ )ρ -(m-µ 0 ) , (8.12
) where c 9 , c 10 are some constants which can be given explicitly. we have that

v 1 -v 2 L ∞ (D) ≤ sup x∈ D R 3 e -ipx (v 2 (p) -v1 (p))dp ≤ I 1 (r) + I 2 (
v 1 -v 2 L ∞ (D) ≤ I 1 (2τ ρ) + I 2 (2τ ρ) ≤ |||H 2 -H 1 ||| ρ,τ,2 |p|≤2τ ρ dp (1 + |p|) 2 + 2c 4 (m, 3)N |p|≥2τ ρ dp (1 + |p|) m ≤ 8πτ ρ|||H 2 -H 1 ||| ρ,τ,2 + 8πc 4 (m, 3)N (m -3)(2τ ) m-3 1 ρ m-3 , (8.18)
at least, under the assumptions of Lemma 8.4 for µ 0 = 2.

Under the assumptions of Lemma 8.4 for µ 0 = 2, using (8.18) and (8.12) we obtain that

v 1 -v 2 L ∞ (D) ≤ c 11 (N, D, m, σ, τ )e 2Lρ ρ 3 Φ 2 -Φ 1 + c 12 (N, m, τ )ρ -(m-3) , (8.19)
where c 11 , c 12 are related in a simple way with c 9 , c 10 for µ 0 = 2. Let now

α ∈]0, 1[, β = 1 -α 2L , δ = Φ 1 -Φ 2 , ρ = β ln(3 + δ -1 ), (8.20)
where δ is so small that ρ ≥ ρ 2 (m, 2, N, D, σ), where ρ 2 is the constant of (8.10). Then, due to (8.19),

v 1 -v 2 L ∞ (D) ≤ c 11 (N, D, m, σ, τ )(3 + δ -1 ) 2Lβ (β ln(3 + δ -1 )) 3 δ+ c 12 (N, D, m, τ )(β ln(3 + δ -1 )) -(m-3) = c 11 (N, D, m, σ, τ )β 3 (1 + 3δ) 1-α δ α (ln(3 + δ -1 )) 3 + c 12 (N, D, m, τ )β -(m-3) (ln(3 + δ -1 )) -(m-3) , (8.21)
where σ, τ are the same that in (8.10) for µ 0 = 2 and where α, β and δ are the same that in (8.20). Using (8.21) we obtain that

v 1 -v 2 L ∞ (D) ≤ c 13 (N, D, m)(ln(3 + Φ 1 -Φ 2 -1 )) -(m-3) (8.22) for δ = Φ 1 -Φ 2 ≤ δ 0 (N, D, m),
(1 + |p|) µ 0 |H 0 2 (λ(1 ∓ 0), p) -H 0 1 (λ(1 ∓ 0), p)| ≤ (c 6 + 4c 7 (µ 0 , τ 0 , ρ 0 )C)∆ ρ 0 ,ρ,µ 0 , (9.1)
where bΛ + ρ,τ,ν , bΛ - ρ,τ,ν are defined in (6.17) (and where

H 0 i (λ(1 -0), p), i = 1, 2, are considered for (λ, p) ∈ bΛ + ρ,τ,ν , H 0 i (λ(1 + 0), p), i = 1, 2, are considered for (λ, p) ∈ bΛ - ρ,τ,ν
). Using (7.8) and the Sohotsky-Plemelj formula, we have that

H 0 (λ(1 -0), p) = 1 2πi T + ρ/|p| H(ζ, p) dζ ζ -λ(1 + 0) + H(λ, p), (λ, p) ∈ bΛ + ρ,τ,ν , (9.2a)
H 0 (λ(1 + 0), p) = - 1 2πi T - ρ/|p| H(ζ, p) λdζ ζ(ζ -λ(1 -0)) + H(λ, p), (λ, p) ∈ bΛ - ρ,τ,ν . (9.2b)
In addition, using the Cauchy-Green formula we have that

H(λ, p) = - 1 2πi T + ρ/|p| H(ζ, p) dζ ζ -λ(1 + 0) + 1 2πi T + ρ/|p| H(ζ, p) dζ ζ -λ - (9.3a) 1 π D + ρ 0 /|p| \D + ρ/|p| ∂H(ζ, p) ∂ ζ dReζdImζ ζ -λ , (λ, p) ∈ bΛ + ρ,τ,ν , H(λ, p) = 1 2πi T - ρ/|p| H(ζ, p) λdζ ζ(ζ -λ(1 -0)) - 1 2πi T - ρ/|p| H(ζ, p) λdζ ζ(ζ -λ) -(9.3b) 1 π D - ρ 0 /|p| \D - ρ/|p| ∂H(ζ, p) ∂ ζ λdReζdImζ ζ(ζ -λ) , (
where H(ζ, p) in the left-hand side of (9.4) is defined according to (7.4), H in the right-hand side of (9.4) is the function of (4.13), {•, •} is defined by (7.11). Using (9.2), (9.3) we obtain that where (λ, p) ∈ bΛ - ρ,τ,ν . Estimates (9.1) follow from formulas (9.6) and from the estimates |A ± (λ, p)| ≤ c 6 (1 + |p|) -µ 0 ∆ ρ 0 ,ρ,µ 0 , (λ, p) ∈ bΛ ± ρ,τ,ν , (9.7)

|B ± (λ, p)| ≤ 4c 7 (µ 0 , τ 0 , ρ 0 )C(1 + |p|) -µ 0 ∆ ρ 0 ,ρ,µ 0 , (λ, p) ∈ bΛ ± ρ,τ,ν , (

where c 6 = sup r∈]1/2,+∞[ q(r) q(r)q(2r) , q(r) = 2r 1 -1 -1 4r 2 1/2 .

(9.9) Note that 0 < c 6 ≤ (2

√ 3 -3) -1 , (9.10)
where c 6 is defined by (9.9). Estimate (9.10) follows from the formulas the properties that H i ∈ C(Λ ρ 0 ,τ 0 ,ν ∪ bΛ ρ 0 ,τ 0 ,ν ), i = 1, 2, |(H 2 -H 1 )(λ, p)| ≤ (1 + |p|) -µ 0 ∆ ρ 0 ,ρ,µ 0 , (λ, p) ∈ bΛ ρ 0 ,τ 0 ,ν

c 6 = 1 1 -2σ , σ = sup τ ∈]0,1[ 1 -(1 -(1/4)τ ) 1/2 1 -(1 -τ ) 1/2 , ( 9
(see formulas (A.8), (A.9), (A.12), (6.19), (7.4)) and from the formulas

1 2π T - r |λ||dζ| |ζ||ζ -λ| = 1 2π T + r |dζ ′ | |ζ ′ -λ ′ | ≤ q(r) q(r) -q(2r) , (9.15)
λ ∈ T - 2r , λ ′ ∈ T + 2r , r > 1/2. In turn, formulas (9.15) follow from the property that z -1 ∈ T + r if z ∈ T - r and from the formula that q(r) is the radius of T + r , where r > 1/2. Estimates (9.8) follow from formulas (A.8), (A.9) (for ρ = ρ 0 ), formula (8.4), Lemma A.5 (for ρ = ρ 0 , µ = µ 0 ), Lemma A.6 (for ρ = ρ 0 ) and the property λ ∈ T ± ρ/|p| ⊂ D ± ρ 0 /|p| if (λ, p) ∈ bΛ ± ρ,τ,ν , ρ = 2ρ 0 > 0, τ = τ 0 /2, 0 < τ 0 < 1. (9.16) Lemma 8.2 is proved.

Lemma 8 . 3 e

 83 4 follows from formula (7.3) and Lemmas 8.1, 8.2, 8.3. The final part of the proof of Theorem 2.2 for d = 3 consists of the following. Using the inverse Fourier transform formula v(x) = R -ipx v(p)dp, x ∈ R 3 , (8.13)

  λ, p) ∈ bΛ - ρ,τ,ν (where the integrals along T ± r are taken in the counter-clockwise direction). In addition (see equation (A.11) of Appendix), ∂H(ζ, p) ∂ ζ = {H, H}(ζ, p), (ζ, p) ∈ Λ ρ 0 ,τ 0 ,ν ,

(H 2 -

 2 |p| \D + ρ/|p| ∂H(ζ, p) ∂ ζ dReζdImζ ζλ , (λ, p) ∈ bΛ + ρ,τ,ν , H 0 (λ(1 + 0), p) = -λ) , (λ, p) ∈ bΛ - ρ,τ,ν .Using (9.5), (9.4) for H 0 = H 0 i , H = H i , i = 1, 2, we obtain that:(H 0 2 -H 0 1 )(λ(1 -0), p) = A + (λ, p) + B + (λ, p),(9.6a)A + (λ, p) = 1 2πi T + ρ 0 /|p| (H 2 -H 1 )(ζ, p) dζ ζλ , B + (λ, p) = -1 π D + ρ 0 /|p| \D + ρ/|p| {H 2 -H 1 , H 2 } + {H 1 , H 2 -H 1 } (ζ, p) dReζdImζ ζλ , where (λ, p) ∈ bΛ + ρ,τ,ν ; (H 0 2 -H 0 1 )(λ(1 + 0), p) = A -(λ, p) + B -(λ, p), (9.6b) A -(λ, p) = -H 1 )(ζ, p) λdζ ζ(ζλ) , B -(λ, p) = -1 π D - ρ 0 /|p| \D - ρ/|p| {H 2 -H 1 , H 2 } + {H 1 , H 2 -H 1 } (ζ, p) λdReζdImζ ζ(ζλ) ,

. 11 )( 1 -τ ) 1 / 2 ≤ 1 - 7 )

 1111217 (1/2)τ, 1 -(1/4)τ ≥ a(1 -(1/4)τ ) + 1a, follow from the property that (ζ, p) ∈ bΛ ρ 0 ,τ 0 ,ν if ζ ∈ T ± ρ 0 /|p| , |p| < 2τ ρ, ρ = 2ρ 0 > 0, τ = τ 0 /2, 0 < τ 0 < 1,(9.13)

  .6) Here h bΘ ρ,τ and H bΩ ρ,τ are related by (4.13) (see also the definitions of bΘ ρ,τ and bΩ ρ,τ

given in (6.2), (6.3)), H bΛ ρ,τ,ν is defined in terms of H bΩ ρ,τ by means of (7.4) (where bΛ ρ,τ,ν is defined in (6.13)),

H 0 Λ ρ,τ,ν

is defined in terms of H bΛ ρ,τ,ν by (7.8) (see also formulas (6.13), (6.16)), Hρ,τ is defined as the solution of (7.7) for H on Λ ρ,τ,ν , where Q ρ,τ is replaced by zero in (7.7), finally, vρ,τ on B 2τ ρ is defined as vρ,τ = v+ ρ,τ or as vρ,τ = vρ,τ , where v+ ρ,τ

  where δ 0 is sufficiently small positive constant. Estimate(8.22) in the general case (with modified c 13 ) follows from(8.22) for δ = Φ 1 -Φ 2 ≤ δ 0 (N, D, m) and the property that v j L ∞ (D) ≤ c 14 (m)N (for d = 3).Thus, Theorem 2.2 for d = 3 is proved.

	9. Proofs of Lemmas 8.2 and 8.3
	Proof of Lemma 8.2. Using the maximum principle for holomorphic functions it is
	sufficient to prove that
	sup
	(λ,p)∈bΛ ± ρ,τ,ν

Proof of Lemma 8.3. Using (A.12), the formulas Ω ≈ Θ, Ω ρ ≈ Θ ρ , Ωρ ≈ Θρ (see (6.4)), and formulas (6.2), (4.13), we have that ∆ ρ 0 ,ρ,µ 0 ≤ sup (k,l)∈ Θρ \Θ ρ 0

where ∆ ρ 0 ,ρ,µ 0 is defined by (8.6), h 1 , h 2 are the functions of Section 5. Estimate (8.8) follows from formulas (9.17), (5.1), (4.7), (4.9), the property that |k -l| ≤ 2ρ for (k, l) ∈ Θρ and the property that

Lemma 8.3 is proved.

Appendix

In this appendix we recall some results of [START_REF] Novikov | An effectivization of the global reconstruction in the Gel'fand-Calderon inverse problem in three dimensions[END_REF] used in Sections 8 and 9 of the present work.

Consider the equation

for unknown U , where U 0 , U are functions on Λ ρ,τ,ν defined in (6.13), M ρ,τ is the map defined by (7.9). In particular, (7.7) can be written as (A.1) with U = H, U 0 = H 0 + Q ρ,τ .

Lemma A.1. Let ρ > 0, ν ∈ S 2 , τ ∈]0, 1[, µ ≥ 2 and 0 < r < (2c 7 (µ, τ, ρ)) -1 , where c 7 is the constant c 8 of [START_REF] Novikov | An effectivization of the global reconstruction in the Gel'fand-Calderon inverse problem in three dimensions[END_REF] (that is c 7 (µ, τ, ρ) = 3b 1 (µ)τ 2 + 4b 2 (µ)ρ -1 + 4b 3 (µ)τ , where b 1 , b 2 , b 3 are the constants of [START_REF] Novikov | An effectivization of the global reconstruction in the Gel'fand-Calderon inverse problem in three dimensions[END_REF]). Let U 0 ∈ L ∞ (Λ ρ,τ,ν ) and |||U 0 ||| ρ,τ,µ ≤ r/2 (see definition (7.5)). Then equation (A.1) is uniquely solvable for U ∈ L ∞ (Λ ρ,τ,ν ), |||U ||| ρ,τ,µ ≤ r, and U can be found by the method of successive approximations, in addition,

where M ρ,τ,U 0 denotes the map

Lemma A.2. Let the assumptions of Lemma A.1 be fulfilled. Let also

Lemma A.2 corresponds to Lemma 4.5 of [START_REF] Novikov | An effectivization of the global reconstruction in the Gel'fand-Calderon inverse problem in three dimensions[END_REF].

Let

where

, where b 1 , b 2 , b 3 are the constants of [START_REF] Novikov | An effectivization of the global reconstruction in the Gel'fand-Calderon inverse problem in three dimensions[END_REF] and n 1 , n 2 , n 3 are the constants of Lemma 11.1 of [START_REF] Novikov | On non-overdetermined inverse scattering at zero energy in three dimensions[END_REF]). In addition, from estimates (7.14), (7.15) of the present paper (see estimates (3.3), (4.20), (4.22) of [START_REF] Novikov | An effectivization of the global reconstruction in the Gel'fand-Calderon inverse problem in three dimensions[END_REF]) it follows that

where τ 1 , ρ 1 are the constants of (4.36) of [START_REF] Novikov | An effectivization of the global reconstruction in the Gel'fand-Calderon inverse problem in three dimensions[END_REF] (in particular, τ 1 ∈]0, 1[ is sufficiently small and ρ 1 is sufficiently great). Then

Lemma A.3 corresponds to estimates (4.36) of [START_REF] Novikov | An effectivization of the global reconstruction in the Gel'fand-Calderon inverse problem in three dimensions[END_REF].

Lemma A.4. Let v satisfy (7.1), v µ ≤ C, ρ satisfy (7.6). Let H be the function of (4.13). Then:

where C denotes the space of continuous functions, Ω, Ω ρ are defined in (4.14), (6.3), η is defined as in (7.6)

where k(λ, p) is defined in (6.11b), Λ ρ,ν is defined in (6.13), {•, •} is defined by (7.11). Lemma A.4 corresponds to formulas (3.2)-(3.4), (3.22) of [START_REF] Novikov | An effectivization of the global reconstruction in the Gel'fand-Calderon inverse problem in three dimensions[END_REF]. Let

where Ω, Ωρ are defined in (4.14), (6.3).

, where ρ > 0, µ ≥ 2. Let {•, •} be defined by (7.11). Then

where Λ ρ,ν is defined in (6.13), b 1 , b 2 , b 3 are the constants of [START_REF] Novikov | An effectivization of the global reconstruction in the Gel'fand-Calderon inverse problem in three dimensions[END_REF]. Lemma A.5 corresponds to estimates (3.26), (3.27) of [START_REF] Novikov | An effectivization of the global reconstruction in the Gel'fand-Calderon inverse problem in three dimensions[END_REF]. Consider , where D ± r are defined in (6.18). Lemma A.6 corresponds to formulas (7.2), (7.10) of [START_REF] Novikov | An effectivization of the global reconstruction in the Gel'fand-Calderon inverse problem in three dimensions[END_REF].


Tina ; Kubitzki

Daniel Minör

Ursula Mackfeld

Marco Oldiges

Thomas Noll

Application of immobilized bovine enterokinase in repetitive fusion protein cleavage for the production of MUC1

Keywords: enterokinase, immobilization, fusion protein cleavage, protein purification, MUC1

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Repeated fusion protein cleavage by immobilized enterokinase in the absence and presence of magnesium. The reaction solution was removed after 24 h and new substrate solution was added to the enzyme-support preparation starting a new cleavage reaction. The set reaction conditions involved 0.625 mg*mL-1 MUC1-IgG2a Fc in 50 mM Tris-HCl, pH 8.0 incubated with 10 U*mL 1*gcarrier-1 (determined with GD4K-2NA) immobilized enterokinase in a reaction volume of 1 mL at 25°C under shaking conditions. The reaction buffer was supplemented with 1 mM magnesium ions.

A) Repetitive batches in the absence of magnesium; B) Repetitive batches in the presence of magnesium. 125x168mm (600 x 600 DPI)

  Application of immobilized bovine enterokinase in repetitive fusion protein cleavage for the production of MUC1

  Application of immobilized enterokinase in repetitive fusion protein cleavage. The reaction solution was removed after 24 h (increased process time of 27 h (#) and 33 h (#')) and new substrate solution was added to the enzyme-support preparation starting a new cleavage reaction. The set reaction conditions involved 1.16 mg*mL-1 MUC1-IgG2a Fc in 50 mM Tris-HCl, pH 8.0 incubated with 10 U*gcarrier-1 (determined with GD4K-2NA) immobilized enterokinase in a reaction volume of 1 mL at 25°C under shaking conditions. 87x63mm (600 x 600 DPI)

Page 1 of 3 Wiley-VCH Biotechnology Journal

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	18 19 20 21 22 23 24 25 26	Journal: Biotechnology Journal Manuscript ID: biot.200900049.R2 F o r Wiley -Manuscript type: Research Article Date Submitted by the Author: 02-Jul-2009 P chromatography. 169x83mm (600 x 600 DPI)
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Scheme of the batch binding chamber used for enterokinase purification with metal chelate affinity