Dr Daniela ; Munteanu
email: munteanu@up.univ-mrs.fr

Jean-Luc L Autran

Xavier Loussier

Samuel Harrison

Robin Cerutti

Luc

Compact Modeling of Symmetrical Double-Gate MOSFETs Including Carrier Confinement and Short-Channel Effects

Keywords: compact modeling, Double-Gate MOSFET, quantum effects, shortchannel effects compact modeling, Double-Gate MOSFET, quantum effects, short-channel effects

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and an impressing number of studies have confirmed its enormous potentiality to push back the integration limits to which conventional devices are subjected [START_REF] Hisamoto | Short course IEDM Tech Dig[END_REF][START_REF] Frank | Monte Carlo simulation of a 30nm dual-gate MOSFET: How short can Si go?[END_REF][START_REF] Taur | Analytic Solutions of Charge and Capacitance in Symmetric and Asymmetric Double-Gate MOSFETs[END_REF][START_REF] Chan | Quasi-2D Compact Modeling for Double-Gate MOSFET[END_REF]. The main advantage of this architecture is to offer a reinforced electrostatic coupling between the conduction channel and the gate electrode. In other terms, a double-gate structure can efficiently sandwich (and thus very well control, electrostatically speaking) the semiconductor element playing the role of the transistor channel, which can be a Silicon thin layer or nanowire, a Carbon nanotube, a molecule or an atomic linear chain. The MOSFET operation of such ultimate DG devices with a single quantum conduction channel has been theoretically demonstrated in recent works [START_REF] Bescond | Atomic-scale Modeling of Source-to-Drain Tunneling in Ultimate Schottky Barrier Double-Gate MOSFET's[END_REF][START_REF] Bescond | Atomic-scale modeling of Double-Gate MOSFETs using a tightbinding Green's function formalism[END_REF].

Although the operation of DG transistor is similar to the conventional MOSFET, the physics of DG MOSFET is more complicated. Moreover, physical phenomena such as 2D electrostatics or carrier quantization have to be considered, since DG structure will be precisely used to design very integrated devices (with short channel and extremely thin films). Therefore, new compact models, dedicated the circuit simulation, have to be developed for DG MOSFET [START_REF] Chan | Quasi-2D Compact Modeling for Double-Gate MOSFET[END_REF]. Several interesting models have been proposed for the classical (i.e. without quantum effects) drain current in long channels DG [START_REF] Taur | Analytic Solutions of Charge and Capacitance in Symmetric and Asymmetric Double-Gate MOSFETs[END_REF][START_REF] Chan | Quasi-2D Compact Modeling for Double-Gate MOSFET[END_REF][START_REF] Taur | A Continuous, Analytic Drain-Current Model for DG MOSFETs[END_REF][START_REF] Jiménez | Continuous Analytic I-V Model for Surrounding-Gate MOSFETs[END_REF][START_REF] Ortiz-Conde | Rigorous analytic solution for the drain current of undoped symmetric dual-gate MOSFETs[END_REF] or for short channel DG operating in the subthreshold regime [START_REF] Liang | A 2-D Analytical Solution for SCEs in DG MOSFETs[END_REF]. Carrier quantization effects have been considered for the first time in [START_REF] Baccarani | A Compact Double-Gate MOSFET Model Comprising Quantum-Mechanical and Nonstatic Effects[END_REF]. In this work, we propose a compact model which combines short-channel with quantum-mechanical effects and applies to all operation regimes. In addition the model is continuous over all gate and drain bias range, which makes it very suitable for implementation in circuit simulators.

The development is based on the calculation of the 2D potential distribution in the device taking into account the quantum-evaluated inversion charge. A full 2-D quantum mechanical numerical simulation code [START_REF] Munteanu | Two-dimensional Modeling of Quantum Ballistic Transport in Ultimate Double-Gate SOI Devices[END_REF] is used for completely validating the model. The drain current as predicted by the model is compared with experimental data measured on scaled DG devices fabricated using the SON (Silicon-on-Nothing) process [START_REF] Harrison | Electrical characterization and modeling of high-performance SON DG MOSFETs[END_REF][START_REF] Cerutti | New Design Adapted Planar Double Gate Process for Performant Low Standby Power Application[END_REF]. Finally, the drain current model is supplemented by a node charge model and further, the entire DG model is successfully implemented in Eldo IC analog simulator.

Drain current modeling

The schematic of a symmetric DG structure and its parameters are shown in Figure 1a.

Figure 1b illustrates the band diagrams in an horizontal cross-section together with the first energy subbands. The drain current modelling starts with the calculation of the 2D potential distribution in the DG transistor. For this purpose several methods have been proposed, the most complete being the evanescent-mode analysis, where the potential is [START_REF] Oh | Analytic Description of Short-Channel Effects in Fully-Depleted Double-Gate and Cylindrical, Surrounding-Gate MOSFETs[END_REF]. The first term represents the long channel solution and the second term takes into account short-channel behaviour. This last term is then approximated by retaining only the lowest-order mode from a Fourier expansion of modes. The method can be very powerful for taking into account short-channel effects in the evaluation of the threshold voltage [START_REF] Oh | Analytic Description of Short-Channel Effects in Fully-Depleted Double-Gate and Cylindrical, Surrounding-Gate MOSFETs[END_REF], but the mathematical development is complicated. For simplifying the calculation, in this work we assume the following dependence for the potential:

divided into two different parts ) y , x ( * ) y ( ) y , x ( L Ψ + Ψ = Ψ
) y , x ( VE ) x ( ) y , x ( s × Ψ = Ψ (1)
where Ψ S is the surface potential and VE(x,y) is the vertical distribution envelope function. The 2D potential distribution is thus obtained by modulating the surface potential by an envelope function containing the potential dependence in the vertical direction. VE(x,y) is then given by:

) 0 y , x ( P ) y , x ( P ) y , x ( VE = = (2)
where P(x,y) is calculated as in [START_REF] Ortiz-Conde | Analytic Solution of the Channel Potential in Undoped Symmetric Dual-Gate MOSFETs[END_REF]:

F o r P e e r R e v i e w O n l y ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ - ε β - ψ = - ψ β ) 2 t y ( e kT 2 n q cos ln 2 ) y , x ( P Si 2 )) x ( QFL 0 ( Si i 2 0 (3)
where β=q/kT, QFL(x) is the quasi-Fermi level and ψ 0 are calculated as shown in [START_REF] Ortiz-Conde | Analytic Solution of the Channel Potential in Undoped Symmetric Dual-Gate MOSFETs[END_REF].

For calculating the vertical distribution envelope function VE(x,y), the expression of QFL(x) is needed. An analytical expression of QFL(x) has been proposed in [START_REF] Van Overstraeten | Theory of the MOS transistor in weak inversionnew method to determine the number of surface states[END_REF] for bulk MOSFET, depending on the x position in the channel, on the channel length and on the drain voltage. However, our detailed investigation by numerical simulation showed that the quasi-Fermi level in DG MOSFET also depends on the gate voltage and on the film thickness. Therefore, we adopted here a quasi-empirical expression (equation 4) inspired from that proposed in [START_REF] Van Overstraeten | Theory of the MOS transistor in weak inversionnew method to determine the number of surface states[END_REF] and extensively verified by numerical simulation:

( ) c 3 D V Si 1 FB V G V c D at 1 L x 1 q / kT m / V exp ln m q kT 2 ) x ( QFL × ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - = - - (4)
where ( )

FB G V V b 2 m - + = , 1 nm 2 . 0 a - = , 1 V 5 . 7 b - = and V 1 c = .
The last term to be calculated for obtaining the 2D potential distribution is the surface potential Ψ S (x). As presented in [START_REF] Loussier | Compact model of drain-current in Double-Gate MOSFETs including carrier quantization and short-channel effects[END_REF], for obtaining the expression of Ψ S (x) the Gauss's law is applied to the particular closed surface shown in Figure 1a:

Si i Si Si A S Si Si 2 dx ) x ( Q 2 dx t qN dx ) x ( E 2 t ) dx x ( E 2 t ) x ( E ε - ε - = - + + - (5)
where E(x) is the electric field, E S (x) is the surface electric field at the Si/SiO 2 interface and N A is the channel doping. Q i (x) is the inversion charge density in the x point of the channel, calculated by the integration of the electron charge over the Si film thickness.

In the right hand side of equation ( 5), the first term corresponds to the depletion charge and the second term corresponds to the mobile inversion charge. In has been shown in reference [START_REF] Harrison | Electrical characterization and modeling of high-performance SON DG MOSFETs[END_REF] that for very thin films (<15nm), the electric field E(x) in equation ( 5) can be approximated as:

dx ) x ( d ) x ( E s Ψ - ≈ (6)
The following equation can also be written for the electric field:

dx ) x ( dE dx ) dx x ( E ) x ( E = + + - (7)
In equation ( 10), the surface electric field at the interface Si/SiO 2 , E S (x), is obtained from the boundary conditions at the interface:

F s s ox ox Si FB G E t V V φ + Ψ + ε ε = - (8)
where V FB is the flat-band voltage and φ F is the Fermi potential. Replacing ( 6) in ( 7) and then in (5) and using (8), we obtain the following differential equation for the surface potential Ψ S :

[ ] i F FB G ox Si A Si Si s Si Si ox 2 s 2 Q ) V V ( C 2 t qN t 1 t C 2 dx d + φ - - - ε = Ψ ε - Ψ (9)
An approximative analytical solution of equation ( 9) is given by:

2 1 1 2 1 1 s m ) x ( R ) x m exp( C ) x m exp( C ) x ( - - + = Ψ (10)
with C 1 , C 2 , m 1 and R(x) calculated for filling the boundary conditions Ψ S (x=0)=φ S and

Ψ S (x=L)=φ S +V D : ) L m sinh( 2 m ) L m exp( 1 ) 0 ( R V )] L m exp( 1 [ C 1 2 1 1 D 1 S 2 , 1 m m - + + - φ ± = ( 11
) Si Si i F FB G ox Si A t ) x ( Q ) V V ( C 2 t qN ) x ( R ε + φ - - - = ( 12
) ) t /( C 2 m Si Si ox 1 ε = (13)
( ) The evaluation of R(x) requires to know the value of the inversion charge density Q i (x), which can be calculated in two different cases: (a) the "classical" case, i. e. without quantum confinement effects and (b) the quantum case. In the classical case, the inversion charge is given by the following equation, assuming a Boltzmann distribution for the carriers in the channel:

2 i SD A S n / N N ln ) q / kT ( = φ (14)
( ) [
]

dy e ni q ) x ( Q Si t 0 ) x ( QFL ) y , x ( kT q i ∫ = - Ψ (15)
In the quantum case, the inversion charge Q i (x) is given by:

⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + Ψ - + ξ β - + × ∑ ∑ π = ) x ( QFL ) x ( 2 E exp 1 ln g m qkT ) x ( Q S g i t , l l , t t , l i l , t D 2 2 i h ( 16
)
where m t * =0.19×m 0 , m l * =0.98×m 0 , g l =2, g t =4, β=q/kT,

* t l D 2 m m = , * t * l t D 2 m m m =
. In equation ( 16) ξ i l,t are the energy levels calculated using a standard method for first-order perturbation applied to the energy levels of an infinite rectangular well (as shown in [START_REF] Harrison | Electrical characterization and modeling of high-performance SON DG MOSFETs[END_REF]):

i i t , l i t , l ) r ( ξ ∆ + ξ = ξ (17)
where (ξr) i l,t are the energy levels of an infinite rectangular well. (ξr) i l,t is given by the well-known equation:

2 Si * t , l 2 2 2 i t , l t qm 2 i ) r ( π = ξ h ( 18
)
and i i i H ϕ ϕ = ξ ∆ ( 19
)
where H is the Hamiltonian of the perturbation and φ i are the electron wave functions associated to energy levels ξ i l,t . In equation [START_REF] Baccarani | A Compact Double-Gate MOSFET Model Comprising Quantum-Mechanical and Nonstatic Effects[END_REF], R(0) is calculated considering Q i (0)

given by equation ( 15) or ( 16) with Ψ S (0)=φ S . Since Ψ S (x) given by equation ( 10) depends on Q i (x), replacing [START_REF] Liang | A 2-D Analytical Solution for SCEs in DG MOSFETs[END_REF] and ( 17) in [START_REF] Ortiz-Conde | Analytic Solution of the Channel Potential in Undoped Symmetric Dual-Gate MOSFETs[END_REF] leads to an implicit equation on Q i (x), which is solved numerically for obtaining Q i (x).

Finally, for calculating the drain current in DG MOSFET we express the current density (including both the drift and the diffusion components) as:

dx ) x ( dQFL ) y , x ( qµn J - = (20)
which is then integrated in y and z directions:

dx ) x ( dQFL ) x ( µWQ ) x ( I i d = ( 21
)
Current continuity requires the drain current be independent of x and therefore, integrating equation ( 21) in x direction from x=0 to x=L gives the final expression of I D :

∫ = D V 0 i D ) x ( dQFL ) x ( Q L W µ I (22)

Model validation by numerical simulation: short-channel, quantum effects and volume inversion

The model was validated by an extensive comparison with quantum numerical simulation using a full 2-D Poisson-Schrödinger code [START_REF] Munteanu | Two-dimensional Modeling of Quantum Ballistic Transport in Ultimate Double-Gate SOI Devices[END_REF]. In a first step, the potential distribution as given by equation (1) has been extensively validated for various structure parameters and biases. An example is shown in Figures 2a, where the surface potential as given by the model for L=50nm and L=10nm is compared with numerical simulation.

In Figure 2b, the potential distribution in a vertical cut-line perpendicular to the Si film (in the middle of the channel, y direction) is illustrated. A good agreement is obtained In a second step, the drain current expression has been completely validated by numerical simulation, for channel lengths varying between 30nm and 200nm and film thicknesses from t Si =15nm down to t Si =2nm. Figure 4 shows an example of this validation step DG MOSFET with different channel lenghts (a constant mobility is considered in equation ( 22)). Short channel behaviour of the quantum drain current is also checked in Figure 4: the model reproduces very well the simulation (even for L=30nm), in both weak and strong inversion regimes. The extensive investigation of additional I D (V D ) curves has shown that the model is completely valid in both linear and saturation regimes.

The validation procedure was continued by an in-depth investigation of the model capability to take into account carrier quantization effects. For this purpose the inversion charge density Q i (x) (in both classical and quantum case) in long and short channels has been compared to numerical results and very good agreement has been found (Figure 5). Further the classical and quantum drain current were calculated as a function of the channel thickness t Si . Figure 6 shows that the quantum model perfectly reproduces two essential phenomena:

(1) the impact of quantum effects quantum effects, increasingly significant when t Si is scaled down. The shift between classical and quantum I D (V G ) curves increases for thinner Si channels. In the same way, the shift between the classical and the quantum threshold voltage is clearly higher for t Si =2nm than that for t Si =10nm.

(2) the drain current dependence on the channel thickness in the subthreshold region, as a manifestation of the volume inversion, which is a key phenomenon in symmetrical

Compact model versus experimental data

Finally, the model was used to fit drain current measured [START_REF] Harrison | Electrical characterization and modeling of high-performance SON DG MOSFETs[END_REF][START_REF] Cerutti | New Design Adapted Planar Double Gate Process for Performant Low Standby Power Application[END_REF] on DG devices (Figure 7). The match between experiment and model is very good, especially in the subthreshold regime. Above threshold the model slightly overestimates the current due to the use of a constant mobility and no series resistances. For improving the model accuracy the next step will be to consider a realistic mobility model [START_REF] Alessandrini | Development of an analytical mobility model for the simulation of ultra-thin single-and double-gate SOI MOSFETs[END_REF] and to include the effect of series resistances.

The proposed compact model can easily be used to obtain all main performance indicators of DG MOSFET, such as the threshold voltage V T , the subthreshold swing S, the DIBL (Drain-Induced-Barrier-Lowering) effect on the threshold voltage, the threshold voltage roll-off, I on and I off currents and the CV/I metric. In addition, the model can be directly implemented in circuit simulation software and used for the simulation of DG MOSFET based-circuits, as will be shown in following paragraphe.

Model implementation in Eldo TM IC analog simulator

The drain current model presented previously has been implemented in a circuit simulator in order to evaluate the performances of simple DG MOSFET-based circuits.

For this purpose, the model was firstly supplemented by a charge model including the expressions of charges on the device terminals. The schematic description of the entire model is given in Figure 8a as well as the symbol of a DG transitor with n-channel. In this figure Q G is the total charge on the two gates, Q D is the charge on the drain terminal and Q S is the charge on the source terminal. The starting point for calculating the gate charge is the neutrality condition which requires that the total charge in the device be always zero:

0 Q Q I G = + (24)
In (24) Q I is the total inversion charge obtained by the integration of relation ( 16) from 0 to L:

∫ = L 0 i I dx ) x ( Q Q (25)
The gate charge Q G is then obtained from equation ( 24). Under normal bias conditions, the inversion charge is not uniformly distributed along the channel except for V D =V S .

Because of this bias dependence, Q i (x) contributes differently to the source and drain charges. Various approaches have been proposed for sharing the inversion charge between the source and drain nodes [START_REF] Robinson | A general four-terminal charging-current model for the insulated-gate field-effect transistor-I[END_REF][21][22]. In our development we have adopted the approach given in [22] and also presented in [START_REF] Loussier | Compact model of drain-current in Double-Gate MOSFETs including carrier quantization and short-channel effects[END_REF]. At low drain voltage, the inversion charge is equally shared between the source and drain. When the drain voltage increases, the drain charge is strongly reduced and the source charge becomes close to the inversion charge Q I .

It is important to note that our compact model is completely continuous over all operation regimes and the drain current and node charges equations are derivable and their derivatives are also continuous over all bias regimes. We have also verified that the source and the drain electrodes can be permuted. rectangular input voltage. This results demonstrates that the model can be perfectly used to the simulation of small circuit based on DG MOSFETs.

Conclusion

In this paper we developped a compact model for the drain current and node charges in Other parameters are: t Si =5nm, V G =1V, V D =0.4V, intrinsic channel.

(N SD ) t Si L V D V G t ox V G Gate (φ M ) Channel (N A ) Ε S Ε(x+dx) Ε(x) Ε=0 t Si /2 x y 0 L x PE E 1 t E 1 l E FS = 0 E FD A-A qV D E FN = qV F (x) EC(x) A A qφ S qΨ(x) E Fi (x) t ox Source (N SD ) Drain (N SD ) t Si L V D V G t ox V G Gate (φ M ) Channel (N A ) Ε S Ε(x+dx) Ε(x) Ε=0 t Si /2 x y 0 L x PE E 1 t E 1 l E FS = 0 E FD A-A qV D E FN = qV F (x) EC(x) A A qφ S qΨ(x) E Fi (x) t ox

  (DG) structure has been in the last years the object of intensive research

  In the classical case and considering the Boltzmann distribution for the carriers, equation (21) becomes[START_REF] Liang | A 2-D Analytical Solution for SCEs in DG MOSFETs[END_REF]:

  and the numerical simulation. The variation of the quasi-Fermi level (equation 4) was also validated as presented in Figure3. Equation (3) has been derived under classical assumptions, but we verified by quantum numerical simulation that this equation still applies in the quantum case.

  DG transistors. Above threshold the drain current does not depend much on the Si channel thickness[START_REF] Taur | A Continuous, Analytic Drain-Current Model for DG MOSFETs[END_REF].

  The compact model described previously for the n-channel DG transistor (NMOS) has been implemented in Eldo IC simulator. A similar model has been considered for the DG MOSFET with p-channel (PMOS). The model has been used further to simulate DC and transient response of a three-stage inverter chain containing DG MOSFETs (the schematic of this circuit is shown in Figure8b). The Figure8cshows the time response of the two outputs voltages (the output voltage of the second and of the third stages) to a

symmetricalFigure 1 .

 1 Figure captions

Figure 2 .

 2 Figure 2. (a) Surface potential variation along the channel from source to drain for

Figure 3 .

 3 Figure 3. Variation with x of the normalized quasi-Fermi level QFL/V D for L=200nm

Figure 4 .

 4 Figure 4. Drain current in long and short channel DG transistors as calculated by model

Figure 5 .

 5 Figure 5. Variation of the inversion charge density Q i (x) along the channel from source

Figure 6 .

 6 Figure 6. Impact of film thickness on the subthreshold operation of L=50nm DG

Figure 7 .

 7 Figure 7. Compact model versus experimental data measured on DG transistors

Figure 8 .

 8 Figure 8. (a) Schematic description of the DG model implemented in Eldo IC analog

Figure 1 .Figure 2 .

 12 Figure 1. Munteanu et al.

Figure 3 .Figure 4 .

 34 Figure 3. Munteanu et al.

Figure 5 .

 5 Figure 5. Munteanu et al.

Figure 6 .Figure 7 .Figure 8 .

 678 Figure 6. Munteanu et al.

http://mc.manuscriptcentral.com/tandf/jenmol

Acknowledgements

This work was supported by the Network of Excellence SINANO and by the French National Research Agency (ANR) under the PNANO-MODERN project.


Yann Rébillé

A Radon-Nikodym approach to measure information


Véronique Flambard
email: veronique.flambard@icl-lille.fr

Nicolas Vaillant
email: nicolas.vaillant@icl-lille.fr

François-Charles Wolff

DATING AS LEISURE

Introduction

Becker's analysis of marriage (1973,1974) is built on the hypothesis that the social process by which men and women meet is a market phenomenon, implying both material and psychological gains for (potential) partners. This market, frequently referred to the "marriage market", can be analyzed by defining social relationships as a special case of consumer behavior. [START_REF] Levy | Marketing society and conflict[END_REF] have termed exchanges on this market "intimate marketing". One of the most cited papers on this issue is that of [START_REF] Hirschman | People as products: analysis of a complex marketing exchange[END_REF], who analyzed the behavior of advertisers using "lonely hearts" columns. Hirschman's analysis of dating advert users explores the role of consumption in human courtship and the idea of "marital exchange" through the developments of the theory of resource exchange [START_REF] Foa | Resource theory of social exchanges, Contemporary topics in social psychology[END_REF].

Hirschman deals with the demand for partners in terms of the Lancaster's economic theory of consumer behavior (1966). Each potential partner is treated as a bundle of characteristics, including age, level of education, physical appearance and intelligence among others.

Consumption of these characteristics during the dating process will, to some extent, be a form of leisure.

Becker's seminal analysis of marriage (see [START_REF] Grossbard-Shechtman | Marriage market models[END_REF][START_REF] Grossbard-Shechtman | A consumer theory with competitive markets for work in marriage[END_REF] has been followed by a number of works looking at the economics of the partner selection process more generally. There is empirical work on personal advertisements by [START_REF] Cameron | Estimates of a hedonic ageing equation for partner search[END_REF][START_REF] Cameron | Looks unimportant? A demand function for male attractiveness by female personal advertisers[END_REF], 2000a, 2000b) and on marriage bureaux-Le Guirriec and [START_REF] Le Guirriec | From libertinism to marital commitment: an economic analysis of the marital research's differentiated objectives[END_REF], [START_REF] Cameron | A Goods Characteristics Model of the Hedonic Ageing Equation: Evidence From a French Marriage Bureau[END_REF]. Using data from a speed-dating experiment, where random matching of subjects were generated, Fisman et al. (2006) show that women put greater weight on the intelligence and race of partners, while men respond more to physical attractiveness. Moreover, men do not value women's intelligence or ambition when it exceeds their own and women exhibit a preference for men who grew up in affluent neighborhoods.

Finally, male selectivity is invariant to group size, contrary to female selectivity. In another contribution, Fisman et al. (2008) extend their findings by examining racial preferences in dating. Gender, but also subjects' backgrounds, age and physical appearance seem to affect racial preferences.

Partner selection occurs naturally and informally, but it may also be organized through commercial intermediaries facilitating mating and dating [START_REF] Quah | Optimal search, matchmaking and the factors affecting age at first marriage[END_REF], Vaillant, 2004a).

According to [START_REF] Ahuvia | Formal intermediaries in the marriage market: A typology and review[END_REF], people who use commercial intermediaries do so in exchange for three services: i) searching, i.e. defining who are potential mates; ii) matching, i.e. obtaining information about the potential right partners; and iii) interaction, i.e. interacting in order to form or reject a relationship. Matchmaking agencies, personal advertisements and online dating services allow individuals to reduce the extensive costs of mate search, by locating easily potential partners [START_REF] Vaillant | Discrimination in matchmaking: Evidence from a French marriage bureau[END_REF]. Therefore, weighting costs and benefits, individuals with a relatively high income level experience substantial opportunity costs by searching for the right mate [START_REF] Gronau | Leisure, Home Production and Work --The Theory of the Allocation of Time Revisited[END_REF]. They are more likely to use commercial intermediaries to reduce them, except if their high opportunity cost is more than compensated by a rich social life. Indeed, networks increase their pool of potential partners, especially when they seek partners with similar professional characteristics [START_REF] Batabyal | On the likelihood of finding the right partner in an arranged marriage[END_REF].

Furthermore, people who are characterized by features of low market value, or people who are picky about their prospective partners are more likely to buy matchmaking services of one sort or another. Speed-dating drastically reduces searching and matching costs, by eliminating a large amount of leisure interaction per date. Moreover, it greatly increases the number of dates per unit of effort. It might then be aimed at increasing the efficiency of dating leisure by attempting to increase matching.

Recent institutional, technical and social changes have altered the pool of individuals looking for dates, at all ages and at different moments in their lives [START_REF] Stevenson | Marriage and divorce: changes and their driving forces[END_REF]. In particular, the rise in household appliances over the last century have made leisure and consumption complementarities more important drivers of couple matching than the production capabilities emphasized by [START_REF] Becker | A treatise on the family[END_REF]. [START_REF] Gould | Waiting for Mr Right: rising inequality and declining marriage rates[END_REF] argue that rising wage inequalities have increased the return of a partner search and decreased the value of remaining unmarried. [START_REF] Caucutt | Why do women wait? Matching, Wage, Inequalities, and the Incentives for Fertility Delay[END_REF] claim that rising returns to labor market experience raise the incentive for highly educated women to develop stronger prospects in both the labor and marriage markets. Clearly, people using online services may be distinguished according to at least two categories: candidates for marriage who are involved in mating rather than dating on the one hand, i.e. people who are trying to "invest" in marriage; individuals who have short-run intentions on the other hand, i.e. people trying to "consume" human relationships. Thus, the continued existence of online dating services simply reaffirms that it meets the needs of individuals of both categories.

From an economic viewpoint, time devoted to dating can be analyzed either as a cost for individuals who are searching to marry, or reciprocally as a consumption good for dating consumers, insofar as this activity provides in itself satisfaction. In the first case, the cost can be both explicit, subscriptions to an online site or restaurant expenses for example, and implicit (since time spent on dating cannot be devoted to paid work (it is leisure time). In the second case, dating is consumption good, a form of leisure. Of course, other activities can serve as substitute or complement to dating activities (such as social networks, hotline, sex toys, porn movies…), but their impact on dating is not discussed here.

In this chapter, we specifically consider individuals who express a demand for dates.

The remainder of our text is organized as follows. In the next Section, we present a theoretical model of dating behavior explaining why improved economic conditions may have ambiguous effects on dating. We turn to an empirical analysis in Section 2. Using a vector error correction model and original French time series data, we investigate the relationship between online dating, the economic sentiment (a proxy for economic wealth and people confidence in the future) and decrease in dating opportunities (measured by the lagged fertility rate). Concluding comments are in Section 3.

The economics of dating

To explain optimal investment in dating activities by individuals, we turn to the expected utility theory and consider a simple microeconomic model of time allocation à la [START_REF] Gronau | Leisure, Home Production and Work --The Theory of the Allocation of Time Revisited[END_REF]. Dating is viewed as an additional possible time use, which extends the usual tradeoff between leisure and working activities. The model could be modified to consider not only the implicit cost of dating (time), but also the associated expenses. Here, instead the latter are pulled with private consumption without allowing any distinction on marginal utility for different expenses. For example, one euro spent on a movie or visit to the theatre may (or may not) bring the same marginal utility than one euro spent in a bar, but here the marginal utility is treated as identical.

The model

We assume that there are only two states of the world in our model: either the representative agent is successful in meeting singles (state s ), or he/she is unsuccessful (state u ). Let q be the probability of the « bad » outcome, so that q 1 is the probability associated to s . Whatever the outcome, the individual level of utility U is expected to depend on his/her level of private consumption C , on leisure l and on the number of potential partners met ( u D or s D ) . We denote respectively ) , , (

u u s s D l C U q D l C U q EU (1)
We assume that the numbers of meetings in successful and unsuccessful situations (respectively s D and u D ) are affected by time spent in dating activity. We rely on the following parametric specification:

D s n 0 N(d) (2) D u n 0 (3)
where n 0 is an exogenous variable reflecting the number of random meetings, independently on the effort of the agent.

With more time spent in dating (denoted by d ), the individual is expected to meet more potential partners. We rely in the sequel on decreasing returns for these dating activities, i.e. the dating technology is concave ( 0 ' N and 0 ' ' N

).

Let us now turn to the individual resources constraints. First, the total amount of time T is devoted to hours of work h , leisure l and dating activities d , so that d l h T .

Second, there is the standard budget constraint such that the individual devotes his/her income to private consumption C (prices being normalized to one). There are two sources of earnings: labor income h w. , with w the hourly wage rate, and an exogenous non-labor income Y (including bequests or public transfers for instance)2 . From the corresponding constraint y h w C

. and using the time constraint, we get the following full-budget constraint:

y T w d w l w C . . . (4)
The maximum attainable level of income is given by Y y T w.

. From (4), it is clear that the opportunity cost of one hour spent in dating activities and the opportunity cost of leisure are similar, given by the wage rate.

The problem for the consumer is hence to maximize his/her expected utility level by choosing the amount of time to spend in dating and leisure. Formally, the optimization

problem is: ) , , . . ( . )] ( , , . . [ . ) 1 ( max , n l d w l w Y U q d N n l d w l w Y U q EU u s d l (5)
In what follows, we denote by ). We further assume some complementary between private consumption and the time inputs ( 0

1 S i U , 0 1 u i U , 3 , 2 i
), but some substituability between the time inputs l and d ( 0

23 s U , 0 23 u U ).
The corresponding first-order conditions 0 / l EU and 0 / d EU are:

0 ) . ( . ) . ( . ) 1 ( 2 1 2 1 u u s s U U w q U U w q (6) 0 ) . ( . ) ' . ( . ) 1 ( 1 3 1 u s s U w q U N U w q (7)
The interpretation of these conditions is as follows. From ( 6), the marginal expected cost of one hour of leisure in terms of foregone income (1 q) w U 1

s q w U 1 u is equal to its marginal expected benefit u s U q U q 2 2 . . ) 1 (
since utility is increasing with leisure. From (7), the marginal expected cost of one hour spent in dating activities (1 q) w U 1

s q w U 1 u is equal to its marginal expected benefit s U N q 3 ' . ) 1 (
. Combining ( 6) and ( 7) leads to:

s u s U N q U q U q 3 2 2 ' . ) 1 ( . ) ( . ) 1 ( (8)
Equation ( 8) means that the marginal expected benefit from one hour of leisure

u s U q U q 2 2 . ) ( . ) 1 (
should be equal to the marginal expected benefit from one hour spent in dating activities

s U N q 3 ' . ) 1 (
at the equilibrium. When this equality does not hold, a better outcome can be reached by reallocating time between leisure and dating.

Economic conditions and dating

We are interested in the comparative static properties of the model in order to know the impact of a change in the exogenous income Y and the wage rate w on both the optimal amounts of leisure and dating. From the second-order derivatives, it can be shown that the derivatives Y d / and w d / can be either positive or negative. As in standard models of time allocation, the impact of a change in the individual level of income on dating activities will be given by the sum of a substitution effect and an income effect.

Consider an agent with little economic resources and consider an increase in his/her wage rate. The substitution effect will first dominate and the individual will increase his/her number of hours worked. This mechanically reduces time devoted to dating, especially if dating and other leisure are complementary goods. Now, with a substantial rise in income, the income effect is expected to be larger than the substitution effect. The additional utility gained from an extra hour of dating become greater than the utility derived from the additional income earned, and more effort will be devoted to dating activities. Note that this marginal benefit also depends on the effectiveness of the dating activity, measured through the shape of the dating technology (.) N .

The relationship of online dating with the business cycle is not documented either.

During a recession, people tend to rethink their priorities. It is true in business, but also in the personal live sphere. The need to get through it together, to feel part of a community, to create social and affective links, can be stronger in difficult periods. Moreover, for those looking for a mate, the perspective of income and risk pooling may also be an incentive during economic slowdown. In an article published in the New York Time, [START_REF] Ellin | The Recession. Isn't it romantic?[END_REF] At the same time, it is also possible that during economic recession, people feel less confident and that, as a result, socialization and in particular dating are less successful. Dating (online or otherwise) may hence be correlated in a complex way with the income level as we just discussed, but also with the level of uncertainty, the attitude of people and their confidence. It follows that the sign of the correlation between the business cycle and online dating remains ambiguous even if anecdotal evidence suggests a negative correlation between online dating and the wealth of the economy.

The empirical model

Our empirical analysis is based on a time series analysis. We use a French monthly dataset covering the period from January 2004 to August 2008 (56 observations). Specifically, we focus on the relationships between demand for dating (MEETIC), economic sentiment (ESI) and a proxy of the decrease in dating opportunities (measured by the lagged fertility rate denoted invOP) in France.

Data

In order to test the relationship between online dating and income, one could use micro data with information on time (and more generally financial resources) allocation between work, consumption and dating as well as information about wealth and labour income. Such data have never been collected as far as we know. Alternatively, we may rely on macro data, which will be relevant to assess the relationship between confidence and the business cycle. By adding other variables, we could extend the analysis to other covariates.

For example, it would be interesting to study the link between dating interest, confidence and sexual health (STIs, HIV, AIDS, pregnancies). Using data from the San Francisco Stop Aids Project, [START_REF] Geoffard | Comportements sexuels risqués et incitations : L'impact des nouveaux traitements sur la prévention du VIH Revue économique[END_REF] show that improvements in treatments against HIV (in 1996) provoked an increase in the risk level, by diminishing the cost associated with risk for those individuals who have opted for testing. Therefore, although they studied the relationship between sexual health and one aspect of confidence (confidence in physicians competence which may also maybe reflect a general form of optimism), they did not study the relationship with dating interest. Such a work will be interested to carry on.

Existing surveys provide some descriptive statistics, but do not offer any analysis and explanations. For instance, the Durex Sexual Well being Global Survey presents findings from 26 countries every year, investigating people's attitudes to sex. Since 2005, Durex

Company also publishes yearly reports called "Face of global sex", with a different question each year. In 2008, directly relevant to a study that would investigate confidence, dating interest and sexual health, the focus was on confidence. A sexual confidence scale assesses confidence aspects of sexual health (avoiding STIs/HIV/AIDS and pregnancy) as well as other dimensions of confidence, i.e. sexual fulfillment and where to find guidance on various sexual issues.

The endogenous covariates

The variable MEETIC was built using http://www.google.fr/trends, which counts the number of times the word "MEETIC" has been searched on google over one month period.

"Meetic" is the dating site by far the most viewed in Europe and France3 . Interestingly, the use of online web search queries, submitted daily by millions of users around the world, is employed by public health specialists to monitor health-seeking behavior. Indeed, the relative frequency of certain queries (for example, cough) is highly correlated with the percentage of physician visits for that condition (that is to say, for cough in the example). [START_REF] Carroll | Does consumer sentiment forecast household spending? If so, why?[END_REF][START_REF] Ludvigson | Consumer confidence and consumer spending[END_REF].

Finally, dating also depends on the opportunities of success (correlated with the number of people looking for a date). As a proxy for the decrease opportunities of dating (invOP) in France, we use the t-9 month rate of fecundity. We can reasonably think that a woman who is pregnant is less likely to look for a date. Therefore, when the share of pregnant woman increases, the share of single women looking for a date is decreasing. This proxy variable does not take into account the proportion of male that are available, but since two persons are needed for a date, it is related to the availability of possible matches. It should be kept in mind that this remains only an approximation. According to data from Forrester

Research, over one-third of those using online dating are currently married. Hence, we cannot rule out that some of them may be married with women that are expecting a baby.

All of the variables were expressed as logarithms and seasonally adjusted. Note that we add +38 to each observation of the economic indicator, insofar as it presents negative values (min(ESI)=-37). Figure 1 depicts the evolution of the various variables.

[Insert Figure 1 about here]

When studying times series, we need to know first of all if the variables are integrated or not. If they are integrated of the same order, a stable long-run relationship may exist which can be taken into acount using error correction models. We therefore apply tests of unit roots to establish the order of integration in each of the four variables. Specifically, we turn to the unit root test developed by [START_REF] Phillips | Testing for a Unit Root in Time Series Regressions[END_REF]. Basically, the Phillips-Perron test is a modification of the standard test proposed by [START_REF] Dickey | Distribution of the estimators for autoregressive time series with a unit root[END_REF]. The Phillips-Perron test estimates the Dickey-Fuller test and modifies the t-ratio of the unit root coefficient so that serial correlation does not affect the test statistic. Table 1 presents the results for the unit root test for a model with an intercept.

[Insert Table 1 about here]

The variables log(MEETIC) and log(ESI) appear to be non-stationary in levels, and stationary when differenced. In other words, we can conclude that these series are integrated of order one (I( 1)). The Phillips-Perron on the variable log(invOP) rejects the existence of a unit root. We thus conduct a Dickey-Fuller GLS test, which is a modified Dickey-Fuller t-test for a unit root in which the series have been transformed by a generalized least-squares regression. The DF-GLS statistic for the series does not exceed the critical values in absolute terms at the 1% level of significance (-2.36>-2.62). When we take the first difference of each variable, it does, which means that log(invOP) is integrated of order one (or I( 1)).

A VECM approach

The results described above indicate that there may be one or several stable long-run relationships between the variables and that we should test for cointegration before specifying the multivariate model. The concept of cointegration was formalized by Engle and Granger (1987). We say that variables are cointegrated when we can form a linear combination of them that is stationary, this linear combination can be considered as a long-run relationship between the variables. If one or several cointegrating relationships are identified in a multivariate system, the long-run relationship can be included as error correction terms in the vector autoregressive setup. Engle and Granger (1987) and Johansen [START_REF] Johansen | Statistical analysis of cointegrating vectors[END_REF][START_REF] Johansen | Maximum Likelihood Estimation and Inference on Cointegration-With Applications to the Demand for Money[END_REF] tests can be used to determine the number of relations of cointegration. The latter is often preferred due to its one-step approach and its ability to handle more than one cointegrating relationship. Another drawback with the original Engle and Granger test is the importance of the choice of dependent variables. In the Johansen test, it is assumed that all variables are endogenous and the test is therefore not sensitive to structure choice. where the vector of coefficients represents the cointegrating vector defining the linear combination of variables that are I(0). There can be described as long-run equilibrium.

First, the dating demand is negatively related both to the economic sentiment and the fertility rate, which indicates that it is positively related with the number of potential partners.

The estimated coefficient associated to log(ESI) indicates that a 1% increase in ESI will decrease the demand for dating by 3% in the long run. When people are more confident in the economy and have more income, they tend to spend less time on meetic. On the contrary, during economic downturns, they appear to increase their demand for online dating.

Our results are consistent with anecdotal evidence suggesting that some people have more time to invest in online dating (those that are not employed full time) and that the marginal utility of dating increases during economic recession ("people want to go through it together"). Besides, the dating demand increases with the number of potential partners : the latter raises the marginal utility of dating, which can explain why it becomes more attractive to invest in dating4 .

The next stage consists in studying the nature of the short-run relationship using a VECM. In this where ect t-1 is the lagged error correction term and the vector e represents the weight or adjustment speed. In other words, it measures how fast the stationary variable ect t-1 feedbacks onto the system. The t are serially independent errors with mean zero and finite covariance matrix.

[Insert Table 4 about here]

Table 4 presents estimates of the short-run equations ( 10), ( 11) and ( 12). We can observe that the demand for meetic and the fertility rate appear to be the variables that adjust to any disequilibrium from the long-run relation (through the error correction term) at the previous period. Standard Granger-type causality tests can be performed in an ECM.

Following [START_REF] Granger | Investigating causal relations by econometric models and cross-spectral methods[END_REF], ESI causes InvOP if the prediction of InvOP is improved when including past values of ESI than when not including them. In the short-run, the t-test applied to the lag of each explanatory variable in each of the three equations indicates the statistical significance of the short-run causal effects (and the strict Granger exogeneity or endogeneity of the dependent variables).

Here, the variable ESI is deemed to cause InvOP if one or more of the lagged coefficient of ESI is statistically different from zero. Our data suggests that this is indeed the case. In other words, when economic agents are optimist about the economy, the fertility rate tends to increase and the opportunities of dating falls. The other variables appear to be strictly exogenous to each other. Again, this result is robust to other specifications, for instance the one including a model with one lag and one cointegration relation.

Discussion and concluding comments

With some exceptions, little attention has been paid to the question of time devoted to a partner choice. However, recent institutional, technical and social changes have altered the pool of individuals looking for dates, at all ages and at different moments in their lives. It should renew the interest for research on partner choices and the time devoted to it.

Search models, risk-pooling models and expected utility models of arbitrage between working time and leisure focus on different dimensions of choice of the individual. In particular, the first two can explain the different mechanisms that tend to extend the search period and that modify the costs and benefits of marriage (or cohabitation). Rising returns to labour market experience raises the incentive for highly educated women of search to develop stronger prospects in both the labor and the marriage market. If individuals view marriage as an insurance against income risks, those who are more risk tolerant will accept to delay marriage. But clearly, not all people using online services are looking for a life partner. The motivations are therefore very different from the ones just traditionally analyzed. To take into account both forms of dating, we use a model where the time has to be allocated between leisure and work. We have chosen to explore this last dimension, which was yet to be explored, by focusing on one growing form of partner search: online dating.

In this work, we seek to understand the relationship, if any, between dating and the business cycle, and more exactly between online dating and the business cycle. Online search offers a larger set of potential partners and anonymity than offline search. Moreover, it is likely to become more attractive in an economic recession as the former appears as a cheaper substitute to the latter.

We rely on a simple microeconomic model of time allocation. Dating is viewed as an additional possible time use, which extends the usual tradeoff between leisure and working activities. As in standard models of time allocation, the impact of a change in the individual level of income on dating activities is given by the sum of a substitution effect and an income effect. Hence, in general, we cannot predict the direction of the effect of a change in the economic sentiment on the demand for dating. In simple microeconomic models where workers are supposed to choose their hours, the relationship between time devoted to dating and income should be U-shaped. What should happen then in aggregate under real world conditions? From our own viewpoint, additional issues should lead to a much more complex, undetermined relationship. As a matter of fact, the marginal utility of dating for an individual also depends on the effectiveness of the dating activities, on its risk aversion, on its own risk of income variation and on its desire to pool risk with a mate among other factors.

The economic theory is therefore not able to predict the effect of a recession for the industry of online dating, as we could for a traditional normal good or inferior good. It is therefore interesting to explore empirically this question and to study whether, as reported in newspapers, economic recession tend to stimulate the online dating industry on the aggregate.

For that purpose, we have used monthly observations covering the period from January 2004 to August 2008 on the demand for online dating in France through the MEETIC website, on the economic sentiment (ESI) and on a proxy of the rate of missed opportunities of dating (invOP) in France through the lagged fertility rate. The results are as follows. The three series are integrated of order one. One relation of cointegration is identified, which can be described as a long-run equilibrium between the three variables. The dating demand is negatively related to both the economic sentiment and the fertility rate. So, during economic downturns, people appear to increase their demand for online dating, which is consistent with anedoctal evidences suggesting that some people have more time to invest in online dating and that the marginal utility of dating increases during economic recession. In conclusion, our empirical tests indicate that for the last recession the demand for online dating increased with the deterioration of the economic sentiment in the long-run trend. But nothing can be said about short-run fluctuations around the equilibrium, regarding the relationship between the economic sentiment and the demand for dating. To conclude, our empirical analysis shows that during the last recession, the demand for online dating has increased with the deterioration of the economic sentiment in the long-run trend. However, nothing can be said about short-run fluctuations around the equilibrium, regarding the relationship between the economic sentiment and the demand for dating. Our study offers a first investigation of this interesting and puzzling question of the effect of the income on the demand for dating. More work needed on this topic to better understand how income may have an impact on the demand for dating, both theoretically and empirically. It would be worthwhile to collect both macro data in order to predict long-term trends for the dating industry, but also micro data on both users and non-users of these online dating services. This would allow to shed light on the selection process into online dating and to better understand individual strategies to find a partner through these services. All these issues are of interest for future work.

  With the increasing value of the search of a partner, online dating becomes more and more attractive. David Evans, an online dating consultant, estimates at 30 million the number of persons that will log on in 2009 to one of the 1500 estimated online dating services in the United States. European Internet users are far more engaged in online dating than those in North America. According to ComScore World Metrix, 18 percent of European Internet users (38.2 millions) visit online personals sites each month (in 2006). France, Germany and England represent the largest market shares. According to JupiterResearch, online dating services are the third largest attractor in Internet users for paid content, after music and games, earning ten percent of the online audience in 2007. Online search offers a larger set of potential partners and anonymity than offline search. Those facing difficulties meeting potential mates are more likely to use Internet. Online search also allows for the selection of partners with similar leisure preferences (for example, with the use of conditions "must love travel").

  in both states 1 . The individual's expected level of utility EU may be expressed as:

  reports that dating interest is up, both with online and offline matchmakers, during the fourth quarter of 2008 (in a recession). Schwartz, a professor of sociology and the relationship expert at perfectmatch.com, reports a 51 percent increase in new members in the fourth quarter of 2008 compared to 2007. During the previous economic downturn in 2001, a similar increased interest for dating was observed. With the average cost of offline dating being higher than the monthly cost to online dating, both the substitution and income effects tend to encourage online dating compared to dating during hardship. Unemployed and underemployed people, with more time on their hands, can devote more time surfing the web.

  Economic conditions are captured by the economic sentiment indicator of consumers from the "Joint Harmonised EU Programme of Business and Consumer Surveys" (see http://ec.europa.eu/economy_finance/db_indicators/surveys/documents/userguide_en.pdf). About 40 000 consumers are currently surveyed across the EU. Answers obtained from these questions are aggregated in the form of "balances" which are constructed as the difference between the percentages of respondents giving positive and negative replies. The Commission calculates confidence indicators as arithmetic means of answers (seasonally adjusted balances) to a selection of questions related to the confidence indicator they are supposed to monitor. Sentiment indices are widely believed to have predictive power for the performance of the macroeconomy. Some empirical support for this belief is provided by research findings, which indicate that the sentiment index contains information about future changes in the economy beyond what is contained in past values of other available indicators

FIGURESFigure 1 .

 1 FIGURES

  representation, short-run fluctuations are represented with the lagged first differences. Any changes in the dependent variables are a function of the level of disequilibrium in the cointegration relationships (measured with the error correction term ect):

		4			4			4
	Meetic t a Meetic	b i, Meetic	Meetic t i	c i, Meetic	ESI t i	d i, Meetic	invOP t i
		i 1			i 1			i 1
	e Meetic ect t 1 1t					(10)
		4				4		4
	invOPP t a Meetic		b i,invOPP	Meetic t i	c i,invOPP	ESI t i	d i,invOPP	invOP t i
		i 1			i 1		i 1
	e invOPP ect t 1 2t				(11)
	4			4			4
	ESI t a ESI	b i, ESI	Meetic t i	c i, ESI	ESI t i		d i, ESI	invOP t i
	i 1			i 1			i 1
	e ESI ect t 1 3t					(12)

These utility functions are supposed to be continuous and twice differentiable.

Note that we do not account for the potential income of the spouse in case of a successful meeting.

See http://www.businessweek.com/bwdaily/dnflash/feb2005/nf2005028_4806_db089.htm or for a more recent source: http://www.onlinepersonalswatch.com/files/meetic_res-q309_vfengfinale.pdf .

When estimating a model with one lag and one cointegration relation, we reach very similar conclusions.

Before performing the Johansen procedure, it is necessary to specify the lag length of the vector auto-regressive model. The results of diagnostic tests proposed by [START_REF] Sims | Macroeconomics and Reality[END_REF] are shown in Table 2.

[Insert Table 2 about here]

The final prediction error (FPE), the Schwarz's Bayesian information criterion (SBIC), the Hannan and Quinn information criterion (HQIC) and the Akaike's information criterion (AIC) indicate vector autoregressions of order 1, whereas the likelihood-ratio test statistics (LR) concludes to a four lag-order. Both lag 1 and lag 4 will be considered in the Johansen procedure, but we only report the results for four lags. The Johansen procedure produces Trace and Maximum Eigen-value tests presented in Table 3 from which the number of cointegrating vectors can be identified. We proceed sequentially by first testing for H 0 : r 0, with r the number of cointegrating vectors. If H 0 is rejected, we then test for r 1 and so on until the null hypothesis could not be rejected.

[Insert Table 3 about here]

The Johansen procedure clearly rejects the null hypothesis of zero cointegrating vector against the alternative, at both 1% and 5% levels when considering only one lag in the underlying VAR model (results are not presented). Table 3 indicates that the null hypothesis of no cointegration relation cannot be rejected at 1%, but is rejected at 5% in favor of at most one statistically significant cointegrating vector. Considering the results of the Johansen procedure that rejects the null hypothesis of no cointegration relation at 5% (with one or four lags), we will consider a model with four lags and one relation of cointegration.

TABLES


Erwan Gautier

Hervé Le Bihan

published or not. The documents may come


A V Kazeykina

R G Novikov

A large time asymptotics for transparent potentials for the Novikov-Veselov equation at positive energy

In the present paper we begin studies on the large time asymptotic behavior for solutions of the Cauchy problem for the Novikov-Veselov equation (an analog of KdV in 2 + 1 dimensions) at positive energy. In addition, we are focused on a family of reflectionless (transparent) potentials parameterized by a function of two variables. In particular, we show that there are no isolated soliton type waves in the large time asymptotics for these solutions in contrast with well-known large time asymptotics for solutions of the KdV equation with reflectionless initial data.

Introduction

We consider the scattering problem for the two-dimensional Schrödinger equation

-∆ψ + v(x)ψ = Eψ, x ∈ R 2 , E = E f ixed > 0 (1.1)
at a fixed positive energy, where

v(x) = v(x), v ∈ L ∞ (R 2 ), |v(x)| < q(1 + |x|) -2-ε , ε > 0, q > 0. (1.2)
It is known that for any k ∈ R2 , such that k 2 = E, there exists a unique bounded solution ψ + (x, k) of equation (1.1) with the following asymptotics

ψ + (x, k) = e ikx -iπ √ 2π e -iπ 4 f k, |k| x |x| e i|k||x| |k||x| + + o 1 |x| , |x| → +∞. (1.3)
This solution describes scattering of incident plane wave e ikx on the potential v(x). The function

f = f (k, l), k ∈ R 2 , l ∈ R 2 , k 2 = l 2 = E, arising in (1.
3), is the scattering amplitude for v(x) in the framework of equation (1.1).

In the present paper we are focused on transparent (or invisible) potentials v for equation (1.1). We say that v is transparent if its scattering amplitude f is identically zero at fixed energy E. We consider transparent potentials for equation (1.1) as analogs of reflectionless potentials for the one-dimensional Schrödinger equation at all positive energies; see, for example, [START_REF] Segur | The Korteweg-de Vries equation and waves. I. Solutions of the equation[END_REF], [START_REF] Zakharov | Asymptotic behaviour of non-linear wave systems integrated by the inverse scattering method[END_REF] as regards reflectionless potentials in dimension one.

In [START_REF] Grinevich | Transparent potentials at fixed energy in dimension two. Fixed energy dispersion relations for the fast decaying potentials[END_REF] it was shown that 1. There are no nonzero transparent potentials for equation (1.1), where

v(x) = v(x), v ∈ L ∞ (R 2 ), |v(x)| < αe β|x| , α > 0, β > 0. (1.4) 2.
There is a family of nonzero transparent potentials for equation (1.1), where v(x) = v(x), v ∈ S(R 2 ), (1.5) and S denotes the Schwartz class.

In the present paper, in addition to the scattering problem for (1.1), we consider its isospectral deformation generated by the following (2 + 1)dimensional analog of the KdV equation:

∂ t v = 4Re(4∂ 3 z v + ∂ z (vw) -E∂ z w), ∂ z w = -3∂ z v, v = v,
(1.6) v = v(x, t), w = w(x, t), x = (x 1 , x 2 ) ∈ R 2 , t ∈ R, where

∂ t = ∂ ∂t , ∂ z = 1 2 ∂ ∂x 1 -i ∂ ∂x 2 , ∂ z = 1 2 ∂ ∂x 1 + i ∂ ∂x 2 .
Equation (1.6) is contained implicitly in the paper of S.V. Manakov [START_REF] Manakov | The inverse scattering method and two-dimensional evolution equations[END_REF] as an equation possessing the following representation:

∂(L -E) ∂t = [L -E, A] + B(L -E), (1.7)
(Manakov L-A-B triple), where L = -∆+v(x, t) or, in other words, L (at fixed t) is the Schrödinger operator of (1.1), A and B are suitable differential operators of the third and the zero order respectively. Equation (1.6) was written in an explicit form by S.P. Novikov and A.P. Veselov in [START_REF] Veselov | Finite-zone, two-dimensional, potential Schrödinger operators. Explicit formula and evolutions equations[END_REF], [START_REF] Veselov | Finite-zone, two-dimensional Schrödinger operators. Potential operators[END_REF], where higher analogs of (1.6) were also constructed. Note that both Kadomtsev-Petviashvili equations can be obtained from (1.6) by considering an appropriate limit E → ±∞ (V.E. Zakharov).

In terms of scattering data the nonlinear equation (1.6), where E = E f ixed > 0, v is sufficiently regular and has sufficient decay as |x| → ∞, w is decaying as |x| → ∞, (1.8) takes the form (2.1)-(2.2) (see Section 2) and, in particular,

∂f (k, l, t) ∂t = 2i k 3 1 -3k 1 k 2 2 -l 3 1 + 3l 1 l 2 2 f (k, l, t), (1.9
)

k = (k 1 , k 2 ) ∈ R 2 , l = (l 1 , l 2 ) ∈ R 2 , k 2 = l 2 = E, where f (•, •, t)
is the scattering amplitude for v(•, t). Equation (1.9) implies that the nonlinear evolution equation (1.6) under assumptions (1.8) preserves the transparency (or invisibility) property of v(x, 0) in the framework of the scattering problem for (1.1).

In the present paper we begin studies on the large time asymptotic behavior for solutions of the Cauchy problem for (1.6) under assumptions (1.8). We give a large time estimate for the family of solutions of (1.6) with E = E f ixed > 0 given by (2.11)-(2.13) and parameterized by a function of two variables. All potentials of this family are transparent at fixed t and E. In addition, this family contains all solutions of (1.6) with E = E f ixed > 0 such that:

• v(•, 0) ∈ S(R 2 )
, where S denotes the Schwartz class, (1.10)

• v(•, 0) is transparent for (1.1), (1.11) • v(•, 0) satisfies (1.
2), where (1.12)

q < Q(E, ε) "small norm" condition, (1.13)
and Q is a special real function with the properties

Q(E, ε) > 0 as E > 0, ε > 0, Q(E, ε) → +∞ for fixed ε > 0 as E → +∞, • v, w ∈ C ∞ (R 2 × R), ∂ j x v(x, t) = O (|x| -3 ), w(x, t) = o(1), |x| → ∞, for t ∈ R, j ∈ (N 0) 2 .
The aforementioned family of solutions of (1.6) was considered for the first time in [START_REF] Grinevich | Transparent potentials at fixed energy in dimension two. Fixed energy dispersion relations for the fast decaying potentials[END_REF]. In the present work we prove the following estimate

|v(x, t)| const(v) ln(3 + |t|) 1 + |t| , x ∈ R 2 , t ∈ R (1.14)
for each v of this family.

Estimate (1.14) implies that there are no isolated soliton type waves in the large time asymptotics for v(x, t), in contrast with large time asymptotics for solutions of the KdV equation with reflectionless initial data. Apparently, it is not difficult to obtain the estimate (1.14), where the right-hand side is replaced by const(v) 1+|t| and even to give precise expression for the leading term of the asymptotics of v(x, t) as t → ∞. However, already (1.14) in its present form implies the aforementioned absence of isolated soliton-type waves in the large-time asymptotics for v(x, t).

Studies on the large time asymptotics for solutions of the Cauchy problem for the Kadomtsev-Petviashvili equations were fulfilled in [START_REF] Manakov | An asymptotic behavior of the solutions of the Kadomtsev-Petviashvili equations[END_REF], [START_REF] Hayashi | Asymptotics for large time of global solutions to the generalized Kadomtsev-Petviashvili equation[END_REF], [START_REF] Kiselev | Asymptotics of a solution of the Kadomtsev-Petviashvili-2 equation[END_REF].

Estimate (1.14) is proved in sections 3-4, using the stationary phase method, techniques developed in [START_REF] Grinevich | Transparent potentials at fixed energy in dimension two. Fixed energy dispersion relations for the fast decaying potentials[END_REF] and [START_REF] Kiselev | Asymptotics of a solution of the Kadomtsev-Petviashvili-2 equation[END_REF] and an analysis of some cubic algebraic equation depending on a complex parameter.

Transparent Potentials and Inverse Scattering Transform

In order to study the large time behavior of the family of transparent potentials described in the previous section, we will use the inverse scattering transform for the two-dimensional Schrödinger equation (1.1) described in [START_REF] Grinevich | Transparent potentials at fixed energy in dimension two. Fixed energy dispersion relations for the fast decaying potentials[END_REF]. First, we give the complete definition of scattering data for (1.1). Let k ∈ C 2 , k 2 = E, Imk = 0 and v(x) satisfy conditions (1.2) and (1.13). Then there exists a unique solution of (1.1) such that

ψ(x, k) = e ikx (1 + o(1)), |x| → ∞.
It can be shown ( [START_REF] Novikov | The inverse scattering problem on a fixed energy level for the two-dimensional Schrödinger operator[END_REF]) that for E ∈ R, Imk = 0 the function ψ(x, k) can be expanded as

ψ(k, x) = e ikx -πsgn(Imk 2 , k1 )e ikx a(k) -k 2 x 1 + k 1 x 2 + e -2iRekx b(k) -k2 x 1 + k1 x 2 + o 1 |x| .
For E > 0 the function b(k) is considered to be scattering data for (1.1) in addition to the scattering amplitude f (k, l) arising in (1.3). It was shown in [START_REF] Grinevich | The inverse scattering problem for the two-dimensional Schrödinger operator, the ∂-method and non-linear equations[END_REF][START_REF] Novikov | Reconstruction of a two-dimensional Schrödinger operator from the scattering amplitude at fixed energy[END_REF][START_REF] Novikov | The inverse scattering problem on a fixed energy level for the two-dimensional Schrödinger operator[END_REF] that at fixed positive energy f (k, l) and b(k) uniquely determine the potential v(x) satisfying (1.2), (1.13) (while f (k, l) alone is insufficient for this purpose).

If potential v(x, t) satisfies the Novikov-Veselov equation (1.6) under assumptions (1.8), then dynamics of the scattering data is described by the following formulas -3 ).

∂b(k, t) ∂t = 2i k 3 1 + k3 1 -3k 1 k 2 2 + 3 k1 k2 2 b(k, t), k ∈ C 2 , Imk = 0, k 2 = E, (2.1) ∂f (k, l, t) ∂t = 2i k 3 1 -3k 1 k 2 2 -l 3 1 + 3l 1 l 2 2 f (k, l, t), k, l ∈ R 2 , k 2 = l 2 = E. ( 2
In the present paper we are concerned with transparent potentials, i.e. in the further considerations we assume that f (k, l, t) ≡ 0. We will also put

E = 1 (2.3)
without loss of generality (the case of an arbitrary fixed positive energy may be reduced to (2.3) by scaling transformation). Along with the function ψ(x, k) we will consider the function µ(x, k) related to ψ(x, k) by the following expression ψ(x, k) = e ikx µ(x, k).

(2.4)

It is also convenient in the two-dimensional scattering theory to introduce new notations

z = x 1 + ix 2 , z = x 1 -ix 2 , λ = k 1 + ik 2 .
Then

k 1 = 1 2 λ + 1 λ , k 2 = i 2 1 λ -λ
and we will consider that

ψ = ψ(z, λ, t), µ = µ(z, λ, t), b = b(λ, t).
In new notations the Schrödinger equation takes the form

Lψ = Eψ, L = -4∂ z ∂ z + v(z, t), z ∈ C, equation (2.1) is written as ∂b(λ, t) ∂t = i λ 3 + 1 λ 3 + λ3 + 1 λ3 b(λ, t),
and (2.4) takes the form ψ(z, λ, t) = e i 2 (λz+z/λ) µ(z, λ, t).

We also note that in the present paper notation f (z) does not imply that f is holomorphic on z, i.e. we omit the dependency on z in the notations. Let a transparent potential v(x, t) satisfy at t = 0 conditions (1.2), (1.13). Then the function µ(z, λ, t) has the following properties (see [START_REF] Grinevich | Transparent potentials at fixed energy in dimension two. Fixed energy dispersion relations for the fast decaying potentials[END_REF]):

1. µ(z, λ, t) is continuous on λ ∈ C; 2. ∀λ ∈ C, |λ| = 1, the function µ(z, λ, t) satisfies the equation ∂µ(z, λ, t) ∂ λ = r(λ, z, t)µ(z, λ, t), (2.5)
where

r(λ, z, t) = exp(iS(λ, z, t))r(λ), r(λ) = πsgn(λ λ -1) λ b(λ, 0), (2.6)
S(λ, z, t) = - 1 2 λz + λz + z λ + z λ + t λ 3 + λ3 + 1 λ 3 + 1 λ3 ;
(2.7)

3. µ(z, λ, t) → 1 as λ → 0, ∞.

Properties 1-3 uniquely determine µ(z, λ, t) for all λ ∈ C.

Under the same assumptions on the potential and if v(x, 0) ∈ S(R 2 ), the function b(λ, t) has the following properties: for every t ∈ R b(•, t) ∈ S(C);

(2.8)

b(1/ λ, t) = b(λ, t), b(-1/ λ, t) = b(λ, t); (2.9) ∂ m λ ∂ n λ b(λ, t) |λ|=1 = 0 for all m, n ≥ 0. (2.10)
The reconstruction of the transparent potential v(z, t) from these scattering data is based on the following scheme.

1. Function µ(z, λ, t) is constructed as the solution of the following integral equation

µ(z, λ, t) = 1 - 1 π C r(ζ, z, t)µ(z, ζ, t) dReζdImζ ζ -λ . ( 2
f (λ) = - 1 π D (∂ζf (ζ)) dReζdImζ ζ -λ + 1 2πi ∂D f (ζ) dζ ζ -λ . 2. Expanding µ(z, λ, t) as λ → ∞, µ(z, λ, t) = 1 + µ -1 (z, t) λ + o 1 |λ| , (2.12)
we define v(z, t) as v(z, t) = 2i∂ z µ -1 (z, t).

(2.13)

3. It can be shown ( [START_REF] Grinevich | The inverse scattering problem for the two-dimensional Schrödinger operator, the ∂-method and non-linear equations[END_REF]) that

Lψ = ψ where ψ(z, λ, t) = e i 2 (λz+z/λ) µ(z, λ, t), L = -4∂ z ∂ z + v(z, t), v(z, t) = v(z, t), v(z, t) is transparent.
3 Estimate for the linearized case

Consider I(t, z) = C f (ζ) exp(iS(ζ, z, t))dReζdImζ, J(t, z) = -3 C ζ ζ f (ζ) exp(iS(ζ, z, t))dReζdImζ, (3.1)
where

f (ζ) ∈ L 1 (C), S is defined by (2.7). If v(z, t) = I(t, z), w(z, t) = J(t, z), where (|ζ| 3 + |ζ| -3 )f (ζ) ∈ L 1 (C)
as a function of ζ, and, in addition,

f (ζ) = f (-ζ) and/or f (ζ) = -|ζ| -4 f - 1 ζ , then v, w satisfy the linearized Novikov-Veselov equation (1.6) with E = 1. In addition, v(p, t) ≡ 0 for |p| < 2, t ∈ R, where v(•, t) is the Fourier transform of v(•, t), that is v(•, t) is transparent in the Born approximation at energy E = 1 for each t ∈ R.
The goal of this section is to give, in particular, a uniform estimate of the large-time behavior of the integral I(t, z) of (3.1) under the assumptions that

f ∈ C ∞ (C), ∂ m λ ∂ n λ f (λ) = O (|λ| -∞ ) as |λ| → ∞, O (|λ| ∞ ) as |λ| → 0, ∂ m λ ∂ n λ f (λ)| |λ|=1 = 0 (3.2)
for all m, n 0. Applying the classical stationary phase method to (3.1), (3.2) (see, for example, [START_REF] Fedoryuk | Method of steepest descent[END_REF]) yields

|I(t, z)| = O 1 |t| , t → ∞, (3.3)
uniformly on z ∈ K, where K is any compact set of the complex plane. This is not sufficient to guarantee the absence of soliton-type waves in the large time asymptotics of the potential v(z, t) = I(t, z). So our further reasoning will be devoted to obtaining an estimate like (3.3) uniformly on z ∈ C.

For this purpose we introduce parameter u = z t and write the integral I in the following form

I(t, u) = C f (ζ) exp(itS(u, ζ))dReζdImζ, (3.4)
where

S(u, ζ) = - 1 2 ζu + ζ ū + u ζ + ū ζ + ζ 3 + ζ3 + 1 ζ 3 + 1 ζ3 . (3.5)
We will start by studying the properties of the stationary points of the function S(u, ζ). These points satisfy the equation

S ′ ζ = - ū 2 + u 2ζ 2 + 3ζ 2 - 3 ζ 4 = 0. (3.6)
The degenerate stationary points obey additionally the equation

S ′′ ζζ = - u ζ 3 + 6ζ + 12 ζ 5 = 0. (3.7)
We denote ξ = ζ 2 and

Q(u, ξ) = - ū 2 + u 2ξ + 3ξ - 3 ξ 2 .
For each ξ, a root of the function Q(u, ξ), there are two corresponding stationary points of

S(u, ζ), ζ = ± √ ξ. The function S ′ ζ (u, ζ
) can be represented in the following form

S ′ ζ (u, ζ) = 3 ζ 4 (ζ 2 -ζ 2 0 (u))(ζ 2 -ζ 2 1 (u))(ζ 2 -ζ 2 2 (u)). (3.8)
We will also use hereafter the following notations:

U = {u = 6(2e -iϕ + e 2iϕ ), ϕ ∈ [0, 2π)} and U = {u = re iϕ : r ≤ |6(2e -iϕ + e 2iϕ )|, ϕ ∈ [0, 2π)},
the domain limited by the curve U (see also Figure 1).

Lemma 3.1.

1. If u = 18e 2πik 3 , k = 0, 1, 2, then ζ 0 (u) = ζ 1 (u) = ζ 2 (u) = e -πik 3
and S(u, ζ) has two degenerate stationary points, corresponding to a third-order root of the function Q(u, ξ), ξ 1 = e -2πik 3 .

2. If u ∈ U ( i.e. u = 6(2e -iϕ + e 2iϕ ) ) and u = 18e

2πik 3 , k = 0, 1, 2, then ζ 0 (u) = ζ 1 (u) = e iϕ/2 , ζ 2 (u) = e -iϕ .
Thus S(u, ζ) has two degenerate stationary points, corresponding to a second-order root of the function Q(u, ξ), ξ 1 = e iϕ , and two nondegenerate stationary points corresponding to a first-order root,

ξ 2 = e -2iϕ . 3. If u ∈ intU, then ζ i (u) = e iϕ i , and ζ i (u) = ζ j (u) for i = j.
In this case the stationary points of S(u, ζ) are non-degenerate and correspond to the roots of the function Q(u, ξ) with absolute values equal to 1.

4. If u ∈ C\U, then ζ 0 (u) = (1 + ω)e iϕ/2 , ζ 1 (u) = e -iϕ , ζ 2 (u) = (1 + ω) -1 e iϕ/2
for certain ϕ and ω > 0.

In this case the stationary points of the function S(u, ζ) are non-degenerate, and correspond to the roots of the function Q(u, ξ) that can be expressed as

ξ 0 = (1 + τ )e iϕ , ξ 1 = e -2iϕ , ξ 2 = (1 + τ ) -1 e iϕ , (1 + τ ) = (1 + ω) 2 .
Lemma 3.1 is proved in section 5. Formula (3.8) and Lemma 3.1 give a complete description of the stationary points of the function S(u, ζ).

In order to estimate the large-time behavior of the integral having the form

I(t, u, λ) = C f (ζ, λ) exp(itS(u, ζ))dReζdImζ (3.9)
uniformly on u, λ ∈ C, in the present and the following sections we will use the following general scheme.

1. Consider D ε , the union of disks with a radius of ε and centers in singular points of function f (ζ, λ) and stationary points of S(u, ζ).

2. Represent I(t, u, λ) as the sum of integrals over D ε and C\D ε :

I(t, u, λ) = I int + I ext ,
where

I int = Dε f (ζ, λ) exp(itS(u, ζ))dReζdImζ, I ext = C\Dε f (ζ, λ) exp(itS(u, ζ))dReζdImζ.
(3.10)

3. Find an estimate of the form

|I int | = O (ε α ) , as ε → 0 (α ≥ 1)
uniformly on u, λ, t.

4. Integrate I ext by parts using Stokes formula

I ext = 1 2t ∂Dε f (ζ, λ) exp(itS(u, ζ)) S ′ ζ (u, ζ) d ζ- 1 it C\Dε f ′ ζ (ζ, λ) exp(itS(u, ζ)) S ′ ζ (u, ζ) dReζdImζ- - 1 it C\Dε f (ζ, λ) exp(itS(u, ζ))S ′′ ζζ (u, ζ) (S ′ ζ (u, ζ)) 2 dReζdImζ = = - 1 t (I 1 -I 2 -I 3 ). (3.11) 5.
For each I i find an estimate of the form

(a) |I i | = O ln 1 ε or (b) |I i | = O 1 ε β , as ε → 0. 6. In case (a) set ε = 1 |t| which yields the overall estimate |I(t, u, λ)| = O ln(|t|) |t| , as t → ∞.
In case (b) set ε = 1 |t| k , where k(α + β) = 1, which yields the overall estimate

|I(t, u, λ)| = O 1 |t| α α+β , as t → ∞.
Using this scheme we obtain, in particular, the following result Lemma 3.2. Under assumptions (3.2), (3.4),

|I(t, u)| = O ln(3 + |t|) 1 + |t| for t ∈ R uniformly on u ∈ C.
A detailed proof of Lemma 3.2 is given in section 5.

Estimate for the non-linearized case

In this section we prove estimate (1.14) for the solution v(x, t) of the Cauchy problem for the Novikov-Veselov equation at positive energy with the initial data v(x, 0) satisfying properties (1.10)-(1.13) or, more generally, for v(x, t) constructed by means of (2.6)-(2.13).

We proceed from the formulas (2.12), (2.13) for the potential v(z, t) and the integral equation (2.11) for µ(z, λ, t).

We write (2.11) as

µ(z, λ, t) = 1 + (A z,t µ)(z, λ, t), (4.1)
where

(A z,t f )(λ) = ∂ -1 λ (r(λ) exp(itS(u, λ))f (λ)) = - 1 π C r(ζ) exp(itS(u, ζ))f (ζ) ζ -λ dReζdImζ and S(u, ζ) is defined by (3.5), u = z t .
Equation (4.1) can be also written in the form

µ(z, λ, t) = 1 + A z,t • 1 + (A 2 z,t µ)(z, λ, t). (4.2)
According to the theory of the generalized analytic functions (see [START_REF] Vekua | Generalized analytic functions[END_REF]), equations (4.1), (4.2) have a unique solution for all z, t. This solution can be written as µ(z, λ, t) = (I -A 2 z,t ) -1 (1 + A z,t From estimate (4.10) given below it follows that (4.4) uniformly converges for sufficiently large t. We will also write formula (4.4) in the form

µ(z, λ, t) = 1 + A z,t • 1 + R, (4.5)
where

R = ∞ k=1 A 2k z,t (1 + A z,t • 1
). In addition to A z,t we introduce another integral operator B z,t defined as

B z,t • f = C r(ζ) exp(itS(u, ζ))f (ζ)dReζdImζ. ( 4
uniformly on u ∈ C; (b) |B z,t • A z,t • 1| = O ln(|t|) |t| , as t → ∞ (4.8)
uniformly on u ∈ C;

(c)

|(A 2 z,t • 1)(λ)| β |t| 1/14 , |t| 1, (4.9)
where β is a constant independent of u and λ;

(d)

|(A n z,t • 1)(λ)| β n-1 |t| ⌊n/2⌋/14 , |t| 1; (4.10) (e) |B z,t • A n-1 z,t • 1| β n-1 ln(3 + |t|) |t| 1+⌊(n-2)/2⌋/14 , |t| 1. (4.11)
Proof of Lemma 4.1. We proceed according to the scheme described in the previous section.

(a) This point follows from Lemma 3.2.

(b) As (A z,t • 1)(λ) ∈ C(C), we take D ε to be the union of disks of a radius ε centered in the stationary points of S(u, ζ). We note that (A z,t • 1)(λ) = O(1) uniformly on u (or, equivalently, on z) and λ. Thus the integral I int (as in (3.10)) can be estimated as

I int = Dε r(ζ, ζ) exp(itS(u, ζ, ζ))(A z,t • 1)(ζ)dReζdImζ = O(ε 2 ).
Now let us estimate the integral I ext (as in (3.10)). For this purpose we apply the Stokes formula (as in (3.11)) taking into consideration that

∂λ(A z,t • f )(λ) = r(λ) exp(itS(u, λ))f (λ): C\Dε r(ζ) exp(itS(u, ζ))(A z,t • 1)(ζ)dReζdImζ = = ∂Dε r(ζ) exp(itS(u, ζ))(A z,t • 1)(ζ) 2tS ′ ζ (u, ζ) dζ- - C\Dε r ′ ζ (ζ) exp(itS(u, ζ))(A z,t • 1)(ζ) itS ′ ζ (u, ζ) dReζdImζ+ + C\Dε r(ζ) exp(itS(u, ζ))(A z,t • 1)(ζ)S ′′ ζ ζ (u, ζ) it(S ′ ζ (u, ζ)) 2 dReζdImζ- - C\Dε r 2 (ζ)(exp(itS(u, ζ))) 2 itS ′ ζ (u, ζ) dReζdImζ. (4.12)
Now, proceeding as in the proof of Lemma 3.2, we obtain (4.8). For the integral I int we have

|I int | = Dε r(ζ) exp(itS(u, ζ)) ζ -λ (A n-1 z,t • 1)(ζ)dReζdImζ β n-2 |t| ⌊(n-1)/2⌋/14 Dε r(ζ) exp(itS(u, ζ)) ζ -λ dReζdImζ β n-1 ε |t| ⌊(n-2)/2⌋/14 . (4.13)
To estimate I ext we use the following representation

C\Dε r(ζ) exp(itS(u, ζ))(A n-1 z,t • 1)(ζ) ζ -λ dReζdImζ = = ∂Dε r(ζ) exp(itS(u, ζ))(A n-1 z,t • 1)(ζ) 2t(ζ -λ)S ′ ζ (u, ζ) dζ- - C\Dε r ′ ζ (ζ) exp(itS(u, ζ))(A n-1 z,t • 1)(ζ) it(ζ -λ)S ′ ζ (u, ζ) dReζdImζ+ + C\Dε r(ζ) exp(itS(u, ζ))(A n-1 z,t • 1)(ζ)S ′′ ζ ζ (u, ζ) it(ζ -λ)(S ′ ζ (u, ζ)) 2 dReζdImζ- - C\Dε r 2 (ζ)(exp(itS(u, ζ))) 2 (A n-2 z,t • 1)(ζ) it(ζ -λ)S ′ ζ (u, ζ) dReζdImζ = = J 1 + J 2 + J 3 + J 4 . (4.14)
The integrals J i can be estimated in the following way

|J 1 | β n-2 |t| • |t| ⌊(n-1)/2⌋/14 1 ε 7 ∂Dε |r(ζ)|dζ β n-1 |t| • |t| ⌊(n-1)/2⌋/14 1 ε 7 .
Similarly,

|J 2 | β n-1 |t| • |t| ⌊(n-1)/2⌋/14 1 ε 7 , |J 3 | β n-1 |t| • |t| ⌊(n-1)/2⌋/14 1 ε 13 , |J 4 | β n-1 |t| • |t| ⌊(n-2)/2⌋/14 1 ε 7 .
Thus,

|I ext | β n-1 |t| • |t| ⌊(n-2)/2⌋/14 • ε 13 .
Now we set ε = |t| -1/14 and obtain the overall estimate

|(A n z,t • 1)(λ)| β n-1 |t| 1/14 |t| ⌊(n-2)/2⌋/14 = β n-1 t ⌊n/2⌋/14 .
(e) Proceeding from (d) this point is proved similarly to (b). Lemma 4.2. Under the assumptions of Lemma 4.1, we have that:

(a) A z,t • 1 = a 1 (z,t) λ + O 1 |λ| 2 for λ → ∞, where |∂ z a 1 (z, t)| = O ln(|t|) |t| , as t → ∞ (4.15) uniformly on z ∈ C. (b) A 2 z,t • 1 = a 2 (z,t) λ + O 1 |λ| 2 for λ → ∞,
uniformly on z ∈ C. (c) R = q(z,t) λ + O 1 |λ| 2
as λ → ∞, where

|∂ z q(z, t)| = O ln(|t|) |t| , as t → ∞ (4.17) uniformly on z ∈ C.
Proof of Lemma 4.2. The asymptotics for A z,t • 1, A 2 z,t • 1 and R follow from the definitions of A z,t and R, formula (2.12) and properties (2.8), (2.9). The rest of the proof consists in the following.

(a) Estimate (4.15) follows from (4.7) and the formula

a 1 (z, t) = 1 π C r(ζ) exp(itS(u, ζ))dReζdImζ = B z,t • 1.
(b) Estimate (4.16) follows from (4.8) and the formula

a 2 (z, t) = 1 π C r(ζ, ζ) exp(itS(u, ζ, ζ))(A z,t •1)(ζ)dReζdImζ = B z,t •A z,t •1. (c) We note that q(z, t) = ∞ k=2 a k (z, t) where a k (z, t) is defined (A k z,t • 1)(λ) = a k (z, t) λ + O 1 |λ| 2 .
Next,

a k (z, t) = 1 π C r(ζ) exp(itS(u, ζ))(A k-1 z,t • 1)(ζ)dReζdImζ.
Thus, proceeding as in point (e) of lemma 4.

1 + |t| , x ∈ R 2 , t ∈ R.
5 Proofs of Lemmas 3.1 and 3.2

Proof of Lemma 3.1. Under the additional assumption that ξ = 0 the system of equations (3.6)-(3.7) is equivalent to the following system

ξ 3 -ū 6 ξ 2 + u 6 ξ -1 = 0, ξ 3 -u 6 ξ + 2 = 0.
(5.1)

We claim that ξ = ξ 1 corresponds to a degenerate stationary point of (3.5), iff the polynomial

P (ξ) = ξ 3 - ū 6 ξ 2 + u 6 ξ -1 can be represented in the form P (ξ) = (ξ -ξ 1 ) 2 (ξ -ξ 2 ) .
(5.2)

Indeed, if ξ = ξ 1 = 0 is a zero of the function Q(u, ξ) = -ū 2 + u 2ξ +3ξ -3 ξ 2 = 3P (ξ)/ξ 2 , then Q(u, ξ) is holomorphic in a certain neighborhood of ξ = ξ 1
and can be expanded into the following Taylor series

Q(u, ξ) = c 1 (ξ -ξ 1 ) + c 2 (ξ -ξ 1 ) 2 + c 3 (ξ -ξ 1 ) 3 + . . .

Thus S ′

ζ can be represented as

S ′ ζ (u, ζ) = c 1 (ζ 2 -ξ 1 ) + c 2 (ζ 2 -ξ 1 ) 2 + c 3 (ζ 2 -ξ 1 ) 3 + . . .
After differentiating with respect to ζ we obtain

S ′′ ζζ (u, ζ) = 2c 1 ζ + 4c 2 (ζ 2 -ξ 1 )ζ + 6c 3 (ζ 2 -ξ 1 ) 2 ζ + . . .
The stationary point corresponding to ξ = ξ 1 can be degenerate if and only if c 1 = 0. So for the polynomial P (ξ) we get the following representation in the neighborhood of ξ = ξ 1

P (ξ) = c 2 (ξ -ξ 1 ) 2 ξ 2 + O(|ξ -ξ 1 | 3 ).
As P (ξ) is a third-order polynomial, it follows that P (ξ) = (ξξ 1 ) 2 (ξξ 2 ).

Expanding the expression for P (ξ) and equating coefficients for the corresponding powers of ξ results in the following system

     2ξ 1 + ξ 2 = ū 6 , 2ξ 1 ξ 2 + ξ 2 1 = u 6 , ξ 2 1 ξ 2 = 1.
(5.3)

Excluding u from the system yields

2ξ 1 ξ 2 + ξ 2 1 = 2 ξ1 + ξ2 , ξ 2 1 ξ 2 = 1.
(5.4)

We represent ξ 1 in the form ξ 1 = re iϕ . Then from the second equation in (5.4) we get that ξ 2 = r -2 e -2iϕ . Substituting this into the first equation of (5.4) yields (r 2r -2 )e 2iϕ = (rr -1 )e -iϕ .

(5.5)

For r = 1 equation (5.5) holds for all ϕ. In addition, if r = 1, r > 0, then equation (5.5) can be rewritten as (r + r -1 )e 3iϕ = 2 and has no solutions for real ϕ.

Using now the second equation in (5.3) one can see that the set of u values, for which (5.4), (5.3) are solvable and (5.2) holds, is a curve on the complex plane described in the parametric form by u = 6(2e -iϕ + e 2iϕ ) (see Figure 1). This curve has three singular points corresponding to ϕ = 2πk 3 , k = 0, 1, 2. For these values of ϕ we have that e iϕ = ξ 1 = ξ 2 = e -2iϕ , and P (ξ) can be represented in the form P (ξ) = (ξ-ξ 1 ) 3 . For ϕ = 2πk 3 , k = 0, 1, 2, we have that P (ξ) = (ξξ 1 ) 2 (ξξ 2 ) where e iϕ = ξ 1 = ξ 2 = e -2iϕ . The first two statements of the Lemma 3.1 are proved.

u = 18 u = 18e 2πi 3 u = 18e 4πi 3 U 0 U 2 U 1
Figure 1: The curve U on the complex plane.

Let us now fix ξ = e iϕ and find the set of u for which this ξ is a root of the polynomial P (ξ).

One can see that ξ is the root of P (ξ) iff

ξ u 6 -ξ 2 ū 6 = 1 -ξ 3 . (5.6)
We now solve the homogeneous equation

e iϕ u 6 -e 2iϕ ū 6 = 0 (5.7)
with respect to u to find the plausible perturbations of u for which ξ = e iϕ remains a root of P (ξ). From (5.7) we get u ū = e iϕ , and thus u = se iϕ/2 for s ∈ R. So for all u that belong to the line u(ϕ, s) = 6(2e -iϕ + e 2iϕ ) + se iϕ/2 , s ∈ R (5.8) one of the roots of P (ξ) is equal to e iϕ . Now we note that the tangent vector to U

d dϕ (2e -iϕ + e 2iϕ ) = 2i(e 2iϕ -e -iϕ )
is collinear to the perturbation vector se iϕ/2 for all ϕ = 2πik 3 , k = 0, 1, 2. Thus u(ϕ, s) given by (5.8) is the tangent line to U passing through the point 6(2e -iϕ + e 2iϕ ).

We note that for each u ∈ intU there exist two different tangents to the curve U passing through this u. Indeed, note that the tangent lines to the curve U passing through the points u = 18e 2πik 3 , k = 0, 1, 2, divide the domain intU into three parts

U 0 = {u = re iϕ , 0 r < 6(2e -iϕ + e 2iϕ ), -π 3 ≤ ϕ ≤ π 3 }, U 1 = {u = re iϕ , 0 < r < 6(2e -iϕ + e 2iϕ ), π 3 < ϕ ≤ π}, U 2 = {u = re iϕ , 0 < r < 6(2e -iϕ + e 2iϕ ), π < ϕ < 5π 3 }
(see Figure 1). We first study U 0 . Let us consider K 01 and K 02 which are the sets of points of the following pencils of tangent lines:

K 01 = u ∈ C : u = 6(2e -iϕ + e 2iϕ ) + se iϕ/2 , s ∈ R, - 2π 3 ϕ < 0 , K 02 = u ∈ C : u = 6(2e -iϕ + e 2iϕ ) + se iϕ/2 , s ∈ R, 0 < ϕ 2π 3 .
It is easily seen that U 0 ⊂ K 01 , U 0 ⊂ K 02 , i.e. each point from U 0 is covered by a certain tangent line from both pencils K 01 and K 02 . It can be shown similarly that each point of U j , j = 1, 2, is covered twice by the corresponding tangent lines. Thus every point from intU is covered twice which means that for each u in the domain limited by the curve U there exist two different roots of P (ξ) equal to 1 in absolute value. As the product of the roots of the polynomial P (ξ) is equal to 1, the third root of the polynomial is also equal to 1 in absolute value. We do not consider the values u from the boundary of U which means that the above-mentioned roots correspond to non-degenerate stationary points. Thus the third statement of Lemma 3.1 is proved. Now suppose u ∈ C\U. We note that every such point u belongs to one and only one tangent line to the curve U. That means that for every u ∈ C\U one of the roots of the polynomial P (ξ), ξ 1 , is such that

|ξ 1 | = 1. As u ∈ U, ξ 0 = ξ 1 , ξ 2 = ξ 1 .
Besides |ξ 0 | = 1 because otherwise there would be two different tangent lines passing through the corresponding point u. That means that for every u ∈ C\U there exists a root of the polynomial P (ξ), namely ξ = ξ 0 , such that |ξ 0 | = 1.

Considering equation (5.6) and its conjugate yields the following system of linear equations for u and ū

au + bū = c, bū + āū = c, where a = ξ 6 , b = -ξ 2 6 , c = 1 -ξ 3 for each ξ ∈ C, |ξ| = 1, ξ = 0. Thus u = cā -cb aā -b b = 6 ξ -ξξ 3 + ξ 2 -ξ 2 ξ3 ξ ξ(1 -ξ ξ) = 6 1 ξ + ξ + ξ ξ
for each ξ ∈ C, |ξ| = 1, ξ = 0. Now let us consider ξ = ξ 0 = (1 + τ )e iϕ , 0 < τ < +∞. Then the corresponding value of the parameter u is

u = 6 2e -iϕ + e 2iϕ + τ 2 1 + τ e -iϕ .
In addition to ξ = ξ 0 , for this value of the parameter u polynomial P (ξ) has also a root ξ = ξ 1 = e -2iϕ . Indeed, if u = 6(2e -iϕ + e 2iϕ ), then ξ 1 = e -2iϕ is a root of the polynomial P (ξ), and the plausible perturbation of this u for which ξ 1 remains a root of P (ξ) is equal to se -iϕ . In addition, as the product of the roots of P is equal to 1, the third root is ξ 2 = (1 + τ ) -1 e iϕ . The fourth statement of Lemma 3.1 is proved.

Proof of lemma 3.2. In this case D ε is the union of disks with a radius of ε centered in the stationary points of S(u, ζ). The integral I int (as in (3.10)) is estimated as

|I int | = Dε r(ζ) exp(itS(u, ζ))dReζdImζ const Dε dReζdImζ = O(ε 2 ).
The estimate for I ext (as in (3.10)) is proved separately for u ∈ U and u ∈ C\U.

I. u ∈ U:

In this case all ζ ∈ D ε lie in the ε-neighborhood of the unit circle. Consequently, from (2.6), (2.8) and (2.10) it follows that for any N ∈ N and any ε 0 ∈ (0, 1/2] there exists C = C(ε 0 , N) such that

|r(ζ)| Cρ N , |r ′ ζ (ζ)| Cρ N for all ζ ∈ D ρ , ρ ε 0 .
The function S ′ ζ (u, ζ) can be estimated as

|S ′ ζ (u, ζ)| 3 ε 6 0 |ζ| 4 for ζ ∈ C\D ε 0 , and
|S ′ ζ (u, ζ)| 3 ρ 6 |ζ| 4 for ζ ∈ ∂D ρ , ε ρ ε 0 .
Taking N = 5 results in the following estimate for I 1

|I 1 | 1 2 ∂Dε |r(ζ)| |S ′ ζ (u, ζ)| d ζ const ε N ε 6 ∂Dε |ζ| 4 d ζ const ε N ε ε 6 (1 + ε) 4 = O(1), as ε → 0.
When estimating I 2 and I 3 we integrate separately over D ε 0 \D ε and C\D ε :

|I 2 | Dε 0 \Dε r ′ ζ (ζ) exp(itS(u, ζ)) S ′ ζ (u, ζ) dReζdImζ+ C\Dε 0 r ′ ζ (ζ) exp(itS(u, ζ)) S ′ ζ (u, ζ) dReζdImζ const ε 0 ε ρ N ρ ρ 6 dρ+const C\Dε 0 |r(ζ)||ζ 4 |dReζdImζ = O(1), as ε → 0, |I 3 | Dε 0 \Dε r(ζ) exp(itS(u, ζ))S ′′ ζζ (u, ζ) (S ′ ζ (u, ζ)) 2 dReζdImζ+ + C\Dε 0 r(ζ) exp(itS(u, ζ))S ′′ ζζ (u, ζ) (S ′ ζ (u, ζ)) 2 dReζdImζ const ε 0 ε ρ N ρ ρ 12 dρ+const C\Dε 0 |r(ζ)||ζ 3 |dReζdImζ N =11
= O(1), as ε → 0.

Setting finally ε = 1 |t| yields

I(t, u) = O 1 |t| , as t → ∞
uniformly on u ∈ U.

II. u ∈ C\U:

Let us divide the complex plane into six sets, each containing one and only one stationary point of S(u, ζ):

C = 2 k=0 Z + k Z - k .
We define the set Z ± k as the set of points of the complex plane to which the stationary point ±ζ k is the closest:

Z + k = {ζ ∈ C : |ζ -ζ i | |ζ -ζ k |, |ζ + ζ j | |ζ -ζ k |, i, j ∈ {0, 1, 2}}, Z - k = {ζ ∈ C : |ζ -ζ i | |ζ + ζ k |, |ζ + ζ j | |ζ + ζ k |, i, j ∈ {0, 1, 2}}
where k ∈ {0, 1, 2}. We will estimate the integral over each Z + k separately. The integrals over Z - k are treated similarly. Let us first take ζ ∈ Z + 1 . Using the definition of Z + 1 and the property that all ζ ∈ D ε Z + 1 lie in the ε-neighborhood of the unit circle one can see that the scheme of reasoning for the case I is applicable.

Now let us consider

ζ ∈ Z + 0 Z + 2 .
(A) First, we will study the set of values of parameter ω for which .11) In order to get rid of this member in the denominator, let us represent r(ζ) by the Taylor formula in the neighborhood of (1 + ω)e iϕ/2 : r(ζ) = r(e iϕ/2 +ωe iϕ/2 )+r ′ (e iϕ/2 +ωe iϕ/2 +se iθ )ρ, s = λρ for some λ ∈ [0, 1],

|ζ+ζ 1 | 2-3ε 0 , |ζ+ζ 0 | > 2-ε 0 , |ζ+ζ 2 | 1+ε 0 , ξ ∈ Z + 0 D ε 0 . ( 5
-ζ 2 | 1 2 |ζ 0 -ζ 2 | = ω(2 + ω) 2(1 + ω) , ζ ∈ Z + 0 . ( 5
where ′ denotes the derivative with respect to s and where λ depends, in particular, on ρ.

For an arbitrary value of ω the following estimates hold:

|r(e iϕ/2 + ωe iϕ/2 )| const|ω| N , (5.12) For an arbitrary value of ω > 0 (satisfying

|r ′ (e iϕ/2 + ωe iϕ/2 )| const|ω| N . ( 5

( c )

 c In this case we build D ε as the union of disks with a radius of ε and centers in λ and stationary points of S(u, ζ). The integral I int over D ε behaves asymptotically as O(ε). When estimating the integral I ext over C\D ε we use (3.8), (3.11) and the following inequalities |ζ -λ| ε, |ζζ i | ε (ζ i are stationary points of S(u, ζ)) which hold for all ζ ∈ C\D ε . Thus we obtain that the asymptotical behavior of I ext is at most O 1 |t|ε 13 . Then, as proposed by the scheme, we choose ε = |t| -1/14 and obtain the required estimate. (d) This point is proved by induction. As in point (c) D ε is the union of disks with a radius of ε and centers in λ and stationary points of S(u, ζ).

ζ 2 (

 2 u), -ζ 2 (u) lie outside the 2ε 0 -neighborhood of zero, i.e. 1 1+ω > 2ε 0 . We will consider ζ ∈ Z + 0 (the case ζ ∈ Z + 2 is treated similarly). If ζ ∈ Z + 0 D ε 0 (for a certain ε 0 ), then it can be represented as ζ = (1 + ω)e iϕ/2 + ρe iθ , ρ ε 0 . Let us estimate the ratio r(ζ) S ′ ζ (ζ, u) in D ε 0 . If ζ 0 belongs to the 2ε 0 -neighborhood of ζ 1 , then all ζ ∈ Z + 0 D ε 0 belong to the 3ε 0neighborhood of ζ 1 . The following estimates hold:

. 9 )

 9 Further, we note that for any N ∈ N there exists a function r(ζ) that can be represented in the form (2.6) with a certain b(ζ) satisfying properties (2.8)-(2.10), such that |r(ζ)| |ζζ 1 | N |r(ζ)| for ζ belonging to the 3ε 0 -neighborhood of ζ 1 . This and (5.9) imply that r(ζ) S ′ ζ (ζ, u) const |r(ζ)||ζ| 4 |ζζ 2 ||ζζ 0 | . (5.10) A similar reasoning holds for the case when ζ 0 belongs to the 2ε 0 -neighborhood of -ζ 1 . Now if ζ 0 does not belong to the 2ε 0neighborhood of ζ 1 and -ζ 1 , then |ζζ 1 | ε 0 and |ζ + ζ 1 | ε 0 for all ζ ∈ Z + 0 D ε 0 . Two last estimates of (5.9) hold and thus (5.10) holds with r(ζ) = r(ζ). The difference |ζζ 2 | can be estimated |ζ

. 13 )D ε 0 .For I 2 D ε 0 ,( 2 |ζ -ζ 0 | 2 ++ 2 |ζ -ζ 0 | 2 +D(D ε 0 . 2 De iϕ/ 2 +

 130202222022 This finally yields|r(ζ)||ζ| 4 |ζζ 2 ||ζζ 0 | const |r(e iϕ/2 + ωe iϕ/2 )| ω(2+ω) 2(1+ω) |ζζ 0 | + |r ′ (e iϕ/2 + ωe iϕ/2 + λρe iθ )|ρ ρ|ζζ 0 | const |ζζ 0 | , ζ ∈ Z + 0Now we are ready to estimate I 1 : we note that the estimater ′ ζ (ζ) S ′ ζ (u, ζ) const |ζζ 0 | , ζ ∈ Z + 0can be obtained using the same reasoning as for the ratior(ζ) S ′ ζ (u, ζ) Dε 0 \Dε) dReζdImζ |ζζ 0 | = O(1) as ε → 0.In order to estimater(ζ) (S ′ ζ (u, ζ))2 in D ε 0 we take the members in the Taylor formula for r(ζ) up to the second order: r(ζ) = r(e iϕ/2 +ωe iϕ/2 )+r ′ (e iϕ/2 +ωe iϕ/2 )ρ+ 1 2 r′′ (e iϕ/2 +ωe iϕ/2 +λρe iθ )ρ 2 .(5.14) From formulas (5.10)-(5.14) it follows thatr(ζ) (S ′ ζ (u, ζ)) 2 |r(ζ)||ζ| 8 |ζζ 2 | 2 |ζζ 0 | 2 const |r(e iϕ/2 + ωe iϕ/2 )| |r ′ (e iϕ/2 + ωe iϕ/2 )| ρ |r ′′ (e iϕ/2 + ωe iϕ/2 + λρe iθ )|ρ 2 2ρ 2 |ζζ 0 | 2 const |ζζ 0 | 2 , ζ ∈ Z + 0 Dε 0 \Dε) r(ζ) exp(itS(u, ζ))S ′′ ζζ (u, ζ) (S ′ ζ (u, ζ)) Setting ε = 1 |t| yields I 3 = O(ln(|t|)), as t → ∞. (B) Now let 1 1 + ω 2ε 0 . If ζ ∈ Z + 0 D ε 0 , then the following estimates hold |ζ ± ζ 1 | 1 2ε 0 -1ε 0 , |ζ ± ζ 2 | 1 2ε 0 -3ε 0 , |ζ + ζ 0 | > 2ε 0 .Consequently,|S ′ ζ (u, ζ)| const |ζ| 4 |ζζ 0 |, ζ ∈ Z + 0and the part of the integral I ext over Z + 0 for this case can be estimated, proceeding as in the previous section, asO ln(|t|) |t| , t → ∞. If ζ ∈ Z + ε 0 , then the following estimates hold |ζ ± ζ 1 | 1 -3ε 0 , |ζ ± ζ 0 | 1 -3ε 0 ,and thus |S ′ ζ (u, ζ)| const |ζ| 4 |ζζ 2 ||ζ + ζ 2 |. We can estimate |ζ + ζ 2 | |ζ 2 | = 1 1 + ω . Now let us expand r(ζ) into Taylor formula in the neighborhood of 1 1+ω e iϕ/2 : λρe iθ ρ, λ ∈ [0, 1].

  .6) To study (4.4) we will need some estimates on the values of operators A z,t and B z,t .

	(a)
	|B z,t • 1| = O	ln(|t|) |t|	, as t → ∞	(4.7)

Lemma 4.1. Under assumptions (2.8)-(2.10), the following estimates hold:

  |r(ζ)||ζ| 8 |ζ + ζ 2 | 2 |ζζ 2 | 2 const |r( 1 1+ω e iϕ/2 )| |ζζ 2 | 2 + |r ′′ ( 1 1+ω e iϕ/2 + λρe iθ )|ρ 2 2ρ 2 |ζζ 2 | 2 const |ζζ 2 | 2 .Following further the reasoning from the case (A) we obtain that I

	This yields
	r(ζ) S ′ ζ (u, ζ)	const	|r( 1 1+ω e iϕ/2 )| 1 1+ω |ζ -ζ 2 |	+	|r ′ ( 1 1+ω e iϕ/2 + λρe iθ )|ρ ρ|ζ -ζ 2 |	const |ζ -ζ 2 |	.
	In the same manner
	r(ζ) ζ (u, ζ)) 2 (S ′ + 1+ω e iϕ/2 )| ρ |r ′ ( 1 1 2				1 1+ω	2 |ζ -ζ 2 | 2	+
	1+ω
	ing estimate holds:					1 1+ω	2ε 0 ) the follow-
		r	1 1 + ω	e iϕ/2	const	1 1 + ω

N . ext = O ln(|t|) |t| , as t → ∞

uniformly on u ∈ C.

Centre des Mathématiques Appliquées, Ecole Polytechnique, Palaiseau, 91128, France email: kazeykina@cmap.polytechnique.fr

Centre des Mathématiques Appliquées, Ecole Polytechnique, Palaiseau, 91128, France email: novikov@cmap.polytechnique.fr