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quantum inversion has many applications in nuclear physics . most of the potential descriptions of the few - body nuclear scattering base on the inversion of the scattering data . though nucleon - nucleon interaction is used as input for all nuclear calculations , here we always have some fitted parameters and this fitting is , of course inversion . moreover all modern high - precision nucleon - nucleon potentials are fitted to scattering data and are now perceived as phenomenology . the problem here is `` that quantitative models for the nuclear force have only a poor theoretical background , while theory based models yield only poor results '' @xcite . at the same time the nucleon - nucleon phase shift analysis data are smooth in all investigated energy region up to 3 gev @xcite this fact justifies the potential description without explicit internal degrees of freedom . it is commonly supposed that nucleon - nucleon potential is nonlocal . however local configuration space potentials simplify nuclear calculations greatly . nonlocality effects may be treated as corrections due to the internal degrees of freedom . for these reasons construction of the high - precision local nn potential describing at least nn scattering data is necessary for exact calculations in nuclear theory . recently such results appeared in the literature @xcite . to simplify the following investigations we worked out the presented simple algorithm which allows to invert scattering data above inelasticity limit and to get the corresponding optical potential .    in sect . 2 we describe our inversion algorithm , that allows to get the configuration space potential from phase shift analysis data neglecting inelasticity . in sect . 3 we show how to get an optical complex - valued potential from the real - valued one . this optical potential describes phase shift analysis data and loss of flux due to inelastic processes . in sect . 4 we apply the worked out method to the nucleon - nucleon scattering data . we extract two different optical potentials corresponding to the different asymptotic behavior of the scattering data . the input data of the marchenko inversion are @xmath5    where @xmath6  is the scattering matrix dependance on the momentum @xmath7 , @xmath8 , @xmath9 is the energy of the j - th bound state , so that @xmath10 , @xmath11  is the particle ( reduced ) mass . the @xmath12 matrices give the asymptotic behavior of the corresponding normalized bound states . these are output data of the partial scattering schrdinger equation @xmath13\psi(r , q)=q^2\psi(r , q ) \label{schr}\ ] ]    to illustrate the worked out algorithm first we consider the one channel case . we proceed from the marchenko equation @xmath14    where    @xmath15    @xmath16 are the riccati - hankel functions . solution of the eq . ( [ eq2 ] ) is the function @xmath17 , which gives the reconstructed potential @xmath18 for the eq . ( [ schr ] )    @xmath19    contrary to the the algorithm used in @xcite we use the algebraic approximant not for the phase shifts @xmath20 but for the @xmath0-matrix    @xmath21    or @xmath22    in the approximant ( [ eq5a ] ) @xmath23 and @xmath24 are an odd and even polynomials of @xmath7 , which do not turn to zero at the real axis simultaneously ( for @xmath25 , in the case then there is a bound state with @xmath26 , conversely @xmath23 and @xmath27 are even and odd polynomials ) . in the approximant ( [ eq5b ] ) @xmath28 and @xmath27 must be of different parity , but we can not describe the case with zero energy bound state . the important feature of approximants ( [ eq5a ] ) and ( [ eq5b ] ) is that these are the most common pad approximants for the @xmath0-matrix consistent with its properties . both approximants lead to the finite - dimensional kernel @xmath29 of the equation ( [ eq2 ] ) . the function @xmath30 and potential @xmath31 are expressed through the elementary functions ( @xmath32 , @xmath33 and powers of @xmath34 ) , so we get bargmann potentials . choice ( [ eq5a ] ) or ( [ eq5b ] ) is fixed for needed ( phenomenological ) dependance @xmath6 by numerical experiment . we choose the relation which approximates @xmath6 better with less number of @xmath35 poles ( taking into account their multiplicity ) . the less number of these poles gives the more simple potential . the properties of the pad approximants are well known , and there are broad enough functional classes , for which this approximation converges everywhere besides poles of these functions . in particular , the consequence of the pad - hypothesis @xcite is that if a function is analytical at point @xmath36 and meromorphic at circle @xmath37 , containing this point , then its diagonal pad approximations @xmath38 $ ] ( where @xmath39 is the number of the approximation poles ) converges to this function on compact subsets of @xmath37 , which do not contain the poles of this function . obviously for the major part of nuclear physics problems the @xmath0-matrix does satisfy these conditions . particularly for the short range potentials the @xmath0-matrix is meromorohic at all complex plane . another remark concerns the @xmath0-matrix poles positions . our approximants for @xmath0-matrix ( [ eq5a ] ) , ( [ eq5b ] ) suggest that @xmath0-matrix poles may be positioned everywhere besides the real axis . previously at numerical approaches the following @xmath0-matrix approximant was used @xcite :    @xmath40    with real values @xmath41 , what assumes that @xmath0-matrix poles are positioned only at the imaginary axis . it is obvious , our approximants are more common and allow to reconstruct the potentials of more complex form .    the approximants ( [ eq5a ] ) and ( [ eq5b ] ) lead to the following expressions for the phase shifts @xmath42 @xmath43    or @xmath44    having the scattering data set @xmath45 ( @xmath46 , we get system of @xmath47 linear equations which define @xmath47 unknown coefficients of @xmath48 from ( [ eq6a ] ) and ( [ eq6b ] ) . the number of experimental phase shifts differs generally from the chosen number @xmath47 . therefore the phenomenological dependance @xmath49 must be approximated by appropriate spline . the @xmath47 needed values @xmath50 are defined by some selection from this spline . the increase in number of precise experimental phase shifts would make approximants ( [ eq5a ] ) , ( [ eq5b ] ) more close to the true @xmath0-matrix . the scattering data are defined experimentally and they have errors . the arbitrary selection of @xmath51 ( @xmath46 from the experimentally defined allowable region may lead to close roots of the polynomials @xmath23 and @xmath27 , i.e. to false @xmath0-matrix poles . therefore @xmath23 and @xmath27 must be factored and close roots particularly those situated near the real axis must be eliminated . this elimination corresponds to elimination of the pad approximant defects . furthermore , using the approximant ( [ eq5a ] ) ( in case of ( [ eq5b ] ) the solution is a particular case of the considered below solution for two bound channels ) and calculating the integral in the eq . ( [ eq3 ] ) using the residue theorem , we get the following expression for the kernel @xmath29 of eq . ( [ eq1 ] )    @xmath52    where @xmath53 ( @xmath54 are the @xmath0-matrix poles . summing is taken over all the @xmath0-matrix poles @xmath55 positioned above the real axis and over bound states @xmath56 : @xmath57 . so in this case the kernel of the eq . ( [ eq2 ] ) is finite - dimensional . it is known that solution of such equations reduces to the linear equations solving . we present the specific solution of this equation for our case . we search the solution of eq . ( [ eq2 ] ) in the form    @xmath58    where @xmath59 are unknown coefficients . substituting ( [ eq7 ] ) and ( [ eq8 ] ) into ( [ eq2 ] ) we get the following system of linear equations    @xmath60    where @xmath61    the functional coefficients @xmath59 are defined by ( [ eq9 ] )    @xmath62    then @xmath30 and the potential @xmath63 from ( [ eq8 ] ) and ( [ eq4 ] ) . the worked out technique of the inversion problem was approved on restoration of the square pit potentials . the accuracy of the results is illustrated by initial and restored potentials in fig.1 . the convergence of the restoration algorithm with increasing of power n of the @xmath64 } } } \right . \kern-\nulldelimiterspace } { n]}$ ] pad approximant of the @xmath0-matrix is also shown here .    in case of two bound channels the system of the partial schrdinger equations is    @xmath65    @xmath66    where @xmath67 , @xmath68 are potentials in channels 1 and 2 , @xmath69 is potential bounding them , @xmath70 and @xmath71 are channel wave functions . by analogy with ( [ eq5b ] ) we approximate the @xmath0-matrix by the following expression    @xmath72    here we again choose the most general form of pad approximant in contrast to the form used in @xcite . the positions of @xmath0-matrix poles are determined from the equations analogous to ( [ eq6b ] ) , but with additional equations for the mixing parameter    @xmath73    with scattering data @xmath74 .    in case of bound channels and @xmath75 , in the original marchenko theory it was proposed to use transformation of the initial eqs . ( [ eq10 ] ) with @xmath76 . the same approach was used in @xcite . nevertheless , from the numerical point of view this approach is less effective than the direct solution of the inverse problem with generalized marchenko equation @xcite . formally it has the former view    @xmath77    but functions involved are matrices @xmath78 @xmath79h\left ( { qy } \right)dq } + \sum\limits_{i = 1}^{n_{b } } { h\left ( { \beta _ i x } \right)m_i h\left ( { \beta _ i y } \right ) } , \ ] ]    where    @xmath80    using the chosen @xmath0-matrix approximant and the residue theorem we get the following expression for @xmath29    @xmath81    here @xmath82 are constant matrices ,    @xmath83    @xmath53 are the @xmath0-matrix poles , values @xmath84 correspond to bound states , @xmath85 is the set of @xmath0-matrix poles of the second order positioned above the real axis . we solve eq . ( [ eq14 ] ) using substitution    @xmath86    where summing is taken over all the @xmath0-matrix poles @xmath55 positioned above the real axis and over all values @xmath84 . this construction leads to the system of linear equations for the functional @xmath87 matrix - coefficients @xmath59 , @xmath88    @xmath89    where    @xmath90    @xmath91    @xmath92    @xmath93    @xmath94 is the unit matrix . the integrals in expressions ( [ eq19 ] ) are easily calculated analytically , but are cumbersome so we do not present them .    having solved this linear equation system we get the sought - for potential from ( [ eq18 ] ) and ( [ eq4 ] ) . the multichannel generalization is made analogously . the marchenko inversion does not give the needed optical potential . but found potential may serve as the initial potential for the iteration procedure that converts it into the optical ( complex - valued ) potential that describes inelastic processes . first we consider the one channel problem . the phase equation @xcite for the initial potential @xmath95 obtained by some inversion procedure ( say from marchenko equation ) is    @xmath96    where @xmath97 is riccati - bessel amplitude @xcite . let us consider the complex - valued potential @xmath98 obtained from @xmath95 by transformation    @xmath99    where @xmath100 is some real parameter . evidently the phase equation for this potential is    @xmath101    from eqs . ( [ eq20 ] ) and ( [ eq22 ] ) we get    @xmath102    for smooth enough potentials and @xmath103 not rapidly increasing with increasing of @xmath7 the right side of eq . ( [ eq23 ] ) rapidly decreases comparing with @xmath104 and @xmath105 , because under the integral in ( [ eq23 ] ) there is more rapidly oscillating function than in ( [ eq20 ] ) and ( [ eq22 ] ) . then as a first approximation we may take    @xmath106    for inelastic scattering the @xmath0-matrix is expressed through the real inelastic parameter @xmath107 and the real phase shift @xmath108    @xmath109    so it is easily arrived at    @xmath110    @xmath111    whence it follows that @xmath112 . the formula ( [ eq27 ] ) allows to calculate the parameter @xmath113 from the known values @xmath107 and @xmath114 . this approximation works well enough for many quantum mechanical problems . for example in case of nucleon - nucleon scattering @xmath115 below the inelasticity limit which is high enough and @xmath107 grows slowly enough above this limit . nevertheless in some instances the initial potential @xmath116 must be corrected . in this case the following iteration procedure may be efficient . first from the formula ( [ eq27 ] ) values @xmath117 are calculated from @xmath118 and @xmath20 . then with the potential @xmath119 , defined by formula ( [ eq21 ] ) new phase shifts @xmath120 are calculated from ( [ eq22 ] ) . if they do not satisfy phase shift analysis data @xmath121 then changing the input data to @xmath122 we repeat inversion ( eqs . ( [ initial_for_mar])-([final_mar ] ) ) and determine new @xmath100 from formula ( [ eq27 ] ) . the case of two channels is considered in a like manner , though the final expressions are more complex . by analogy with the one channel case the following generalization for the optical potential is derived ( the mixing parameter @xmath123 is considered small ) :    @xmath124    the first @xmath0-matrix approximation is    @xmath125    as all the parameters and phase shifts are real , then eq . ( [ eq29 ] ) links between parameters @xmath100 and @xmath126 with the mixing parameters @xmath127 and phase shifts @xmath128 . this consideration may be generalized to the multichannel case though calculations become much more complex . we applied the described algorithm of inversion to reconstruction of the nucleon - nucleon potential . as input data for this reconstruction we used modern phase shift analysis data up to 1100 mev for @xmath129 state and up to 3 gev for @xmath130 state of nucleon - nucleon system @xcite . the deuteron properties were taken from @xcite . the relativistic effects were taken into account in frames of relativistic quantum mechanics of systems with a fixed number of particles ( point form dynamics ) . the review of this approach can be found in @xcite . this approach is based on the assumption that at not high energies we may consider the number of particles fixed , but the invariance group is the poincare group . a system of two particles is described by the wave function , which is an eigenfunction of the mass operator or the mass squared operator . in this case we may represent this wave function as a product of the external and internal wave functions @xcite . the internal wave function is also an eigenfunction of the mass operator or the mass squared operator . it is shown that the mass squared method is consistent with conventional fitting of the lorentz invariant cross section as a function of laboratory energy @xcite . we consider system of two particles ( nucleons ) with equal masses . then the internal wave function @xmath131 satisfies the following equation        the eq . ( [ eq30b ] ) formally coincides with the schrdinger equation . in eqs . ( [ eq30 ] ) , ( [ eq30b ] ) @xmath135 is the mass squared operator , @xmath11 is the mass of nucleon , @xmath136 is the nucleon - nucleon potential , @xmath137 is the momentum operator of one of the nucleons in the center of masses system . we use system @xmath138 . the quasicoordinate representation corresponds to the realization @xmath139 , @xmath140 . in @xcite we showed that this formalism can be easily generalized to the case of inelastic channels , particularly it allows to take into account the isobar channels in nn scattering . this formal coincidence allows us to apply our inversion algorithm . the modern phase shift analysis data @xcite allow to construct nucleon - nucleon potentials of two different kinds , depending on the asymptotic behavior of the phase shifts above investigated energy region of 3 gev . we constructed the nucleon - nucleon optical potentials of two kinds for @xmath141 and @xmath142 partial waves . these potentials describe the deuteron properties and the phase shift analysis data up to 3 gev and they have different behavior at short distances . one is a repulsive core potential and another is a moscow attractive potential with forbidden states . the moscow potential was introduced in @xcite . these potentials are not phase equivalent and their @xmath0-matrices differ even bellow 3 gev , but within the experimental errors . the @xmath141 phase shift of moscow potential begins from @xmath143 . @xmath144 phase shifts of moscow potential begin from @xmath145 . the mixing parameter @xmath146 of moscow potential differs from this of repulsive core potential by sign . above 3 gev these potentials give different @xmath0-matrices . the real parts of the constructed partial potentials are presented in fig . 2 and fig . they may be downloaded from www.physics.khstu.ru in numerical form . standard notation for central and tensor parts is used , so for @xmath142      the imaginary parts of potentials are defined by eqs . ( [ eq21 ] , [ eq28 ] ) , where parameters @xmath148 and @xmath149 can be easily calculated from the phase shift analysis data @xcite . the phase shifts and mixing parameter are compared with the phase shift analysis data @xcite in fig . 4 . in table 1 the results of our deuteron properties calculations are compared with the experimental data @xcite . both kinds of the constructed partial potentials describe the data well in the limits of the experimental errors . we conclude that the available data of the nn scattering and the deuteron properties are not discriminative with respect to the kind of the nn potential . to discriminate between the considered two kinds of nucleon - nucleon potential we need careful experimental examination of other inelastic processes such as @xmath150 @xcite or @xmath151 . arndt , i.i . strakovsky and r.l . workman , phys . rev . * c62 * , 034005 ( 2000 ) ; r.a . arndt , j.s . hyslop and l.d . roper , phys . rev . * d35 * , 128 ( 1987 ) ; j bystricky , c. lechanoine - leluc , f. lehar , eur . j. * c4 * , 607 ( 1998 ) ; j. ball , j. bystricky , j .- fontaine et al , nuovo cimento * a111 * , 13 ( 1988 ) . calculation + & & with repulsive & with moscow + & & core potential & potential + energy ( mev ) & 2,22458900(22 ) & 2,2246@xmath152 & 2,2246@xmath152 + @xmath153 ( fm@xmath154 ) & 0,2859(3 ) & 0,2639@xmath155 & 0,277@xmath155 + a@xmath156 ( fm@xmath157 & 0,8802(20 ) & 0,8802 & 0,8802 + r@xmath158 ( fm ) & 1,9627(38 ) & 1,951 & 1,956 + @xmath159 & 0,02714 & 0,02714 & 0,02714 + @xmath160 & 0,857406(1 ) & 0,8497@xmath155 & 0,859@xmath155 +
the numerical algorithm of the inverse quantum scattering is developed . 
 this algorithm is based on the marchenko theory , and includes three steps . 
 the first one is the algebraic pad approximation of the unitary s - matrix , what is realized by solving a system of linear equations . 
 second step is the exact solution of the marchenko equation . 
 the used approximant reduces it to another system of linear equations . at this step 
 we get the real - valued potential . 
 it is shown numerically that the developed algorithm is able not only to generate the given @xmath0-matrix dependence , but converges to the initial potential . at third step 
 we construct the optical complex - valued potential which gives the needed s - matrix . 
 it is shown that the modern phase shift analysis data allow to construct the nucleon - nucleon optical potentials of two kinds . 
 these potentials describe the deuteron properties and the phase shift analysis data up to 3 gev and they have different behavior at short distances . 
 one is a repulsive core potential and another is a moscow attractive potential with forbidden states . 
 * reconstruction of the optical potential in the inverse quantum scattering . 
 application to the relativistic inelastic nn scattering *    n.a . 
 khokhlov@xmath1 , v.a . 
 knyr@xmath2     + @xmath3 _ e - mail : khokhlov@fizika.khstu.ru_ ; @xmath4 _ e - mail : knyr@fizika.khstu.ru_