document: the ideas of zero range potential ( zrp ) approach were recently developed to widen limits of the traditional treatment by demkov and ostrovsky @xmath4 and albeverio et al . @xmath5 . 
   the advantage of the theory is the possibility of obtaining an exact solution of scattering problem . 
   the zrp is conventionally represented as the boundary condition on the matrix wavefunction at some point . 
   alternatively , the zrp can be represented as pseudopotential ( breit @xmath6 ) . 
   on the other hand , darboux transformation ( dt ) allows to construct in natural way exactly solvable potentials . 
   general starting point of the theory goes up to matveev theorem ( see @xcite ) . 
   the transformation can be also defined on the base of covariance property of the schrdinger equation with respect to a transformation of wavefunction and potential ( matveev and salle @xmath7 ) . 
   darboux formulas in multi - dimensional space could be applied in the sense of andrianov , borisov and ioffe ideas @xcite . in the circumstances , 
   dt technique can be used so as to correct zrp model . 
   we attempt to dress the zrp in order to improve the possibilities of the zrp model . 
   we use notations and some results from @xcite . 
   dt modifies the generalized zrp ( gzrp ) boundary condition ( section @xmath8 ) and creates a potential with arbitrarily disposed discrete spectrum levels for any angular momentum @xmath9 . in the section @xmath10 we consider @xmath11-representation for a non - spherical potential so as to dress a multi - centered potential , which includes @xmath0 zrps . as an important example , we consider electron scattering by the @xmath1 and @xmath2 structures within the framework of the zrp model ( section @xmath12 ) . in section @xmath13 we present the our calculations for the electron-@xmath3 scattering and discuss them . 
   let us start from the simplest case of a central field . 
   then angular momentum operator commutates with hamiltonian and therefore wavefunction @xmath14 can be expanded in the spherical waves @xmath15 where @xmath16 , @xmath17 is initial particle direction , @xmath18 are partial waves , and @xmath19 are phase shifts . 
   consider the radial schrdinger equation for partial wave with angular momentum @xmath9 . 
   the atomic units are used throughout the present paper , i.e. @xmath20 and born radius @xmath21 . 
   @xmath22 @xmath23 @xmath24 denotes differential operator , and @xmath25 are hamiltonian operators of the partial waves . 
   this equations describe scattering of a particle with energy @xmath26 . 
   the wavefunctions @xmath18 at infinity have the form @xmath27 let us consider gzrp in coordinate origin . 
   this potential is conventionally represented as boundary condition on the wavefunction ( see @xmath28 ) @xmath29 where @xmath30 are inverse scattering lengths . 
   the potential @xmath31 and therefore wavefunctions @xmath18 can be expressed in terms of the spherical functions @xmath32 where spherical functions @xmath33 are related to usual bessel functions as @xmath34 , @xmath35 . in the vicinity of zero they have the asymptotic behavior @xmath36 , and @xmath37 . to substituting the equation @xmath38 into the boundary condition 
   we obtain the elements of @xmath11-matrix @xmath39 the bound states correspond to the poles of the @xmath11-matrix ( i.e the zeros of the denominator @xmath40 ) , which lie on the imaginary positive semi - axis of the complex @xmath41-plane . 
   it is obvious that bound state , with orbital momentum @xmath9 , exists only if @xmath42 ( elsewise an antibound state exists ) and has the energy @xmath43 . 
   thus , spectral problem for gzrp is solved for any value @xmath41 . on the other hand , the equations ( [ e ] ) 
   are covariant with respect to dt that yields the following transformations of the potentials ( coefficients of the operator @xmath25 ) @xmath44 and the wavefunctions @xmath18 @xmath45 where @xmath46 are some solutions of the equations @xmath47 at @xmath48 , and @xmath49 are real parameters , which can be both positive or negative . 
   the dt @xmath50 combines the solutions @xmath18 and a solution @xmath46 that corresponds to another eigen value @xmath51 . repeating the procedure we obtain a chain of the integrable potentials @xmath52 . in general , dressed potential @xmath53 is real for real function @xmath46 . 
   the next step in the dressing procedure of the zero - range potential ( @xmath31 ) is a definition of the free parameters of the solutions @xmath46 . 
   suppose the prop functions @xmath46 satisfy the boundary conditions @xmath54 with @xmath55 . 
   in the simplest case of @xmath56 we have @xmath57 and @xmath58 the dt @xmath50 gives rise to the following requirement on dressed wavefunction @xmath59 the dressed potential @xmath60 is given by @xmath61 it is regular on semiaxis only if @xmath62 . in the limiting case at @xmath63 
   we obtain long - range interaction @xmath64 , which can be regular on semiaxis only if @xmath65 . 
   assuming @xmath66 we get @xmath67 ( trivial transformation ) , and boundary condition can be obtained by the substitution : @xmath68 to dress free wave @xmath69 we obtain zrp at the coordinate origin . 
   thus , zrp can be also introduced in terms of dt . 
   to consider transformation with parameter @xmath70 we obtain regular solution @xmath71 and tangent of phase shift is @xmath72    in the other cases asymptotic of the functions @xmath73 at zero is given by @xmath74 it is clear that the each dt introduces short - range core of centrifugal type ( which depends on angular momentum @xmath9 ) in the potential . in this situation 
   the boundary conditions on the dressed wavefunctions @xmath75 $ ] require modification . 
   thus , in the case @xmath76 the boundary conditions become @xmath77 and in the case @xmath78 we obtain @xmath79    in the generalized case , zrp with angular momentum @xmath9 generates also @xmath80 complex poles of the @xmath11-matrix , which correspond the quasi - stationary states ( resonances ) . 
   the dts @xmath50 with the parameters @xmath49 results in the @xmath11-matrix elements for dressed gzrp @xmath81 we can use darboux transformation in order to add ( or remove ) poles of the @xmath11-matrix . 
   the principal observation allows to built a zero - range potential eigen function in the multi - center problem . 
   let us consider @xmath0 zrps at the points @xmath82 and interaction @xmath83 . 
   the wavefunction @xmath14 can be expressed in terms of the ( outgoing - wave ) green function , defined by the equation @xmath84 the second ( ingoing - wave ) green function is defined by @xmath85 . 
   the partial waves @xmath86 , defined by @xmath87 for multi - centered target which includes @xmath0 zrps and interaction @xmath83 can be expressed as a superposition of the green functions @xmath88 in which @xmath89 are phase shifts , @xmath90 denote @xmath11-matrix orthonormal eigenfunctions , @xmath91 are real numbers . 
   they naturally generalize the spherical partial waves @xmath92 for a non - spherical potential @xmath93 . expanding the partial waves @xmath86 at the infinity we obtain the expressions for @xmath90 @xmath94 where @xmath95 is wavefunction for potential @xmath83 , and @xmath96 is scattering amplitude . 
   scattering amplitude for potential @xmath83 and @xmath0 zrp is given by @xmath97 by the imposition of the boundary conditions @xmath54 the calculation of the partial waves is reduced to the solution of the following system @xmath98 in which we use @xmath99 where @xmath100 , @xmath101 . 
   the tangents @xmath102 can be found from compatibility condition of this system . in simplest case @xmath103 
   @xmath104 the system @xmath105 is reduced to the usual equations of zrp theory @xmath93 . in order to construct functions 
   @xmath106 and @xmath107 for dressed potential , i.e. for @xmath83 , we need to write down green function as single - center expansion over spherical harmonics . in the simplest case , 
   when the prop function is @xmath108 , the green function is given by @xmath109 where @xmath110 the function @xmath111 has the following asymptotic at infinity @xmath112 the integral cross section can be readily derived using the optical theorem : @xmath113 thus , averaged integral cross section is given by @xmath114 
   for purpose of illustration we consider scattering problem for a dressed multi - center potential . 
   the multi - center scattering within the framework of the zrp model was investigated by demkov and rudakov @xmath93 ( 8 centers , cube ) , drukarev and yurova @xmath116 ( 3 centers in line ) , szmytkowski @xmath117 ( 4 centers , regular tetrahedron ) . 
   let structure @xmath1 contains @xmath0 identical scatterers , which involve only @xmath119 waves . 
   denote position vector of the scatterer @xmath120 by the @xmath121 . 
   suppose , for the sake of simplicity , the distance between any two scatterers @xmath122 and @xmath120 is @xmath123 . 
   there are three such structures in three - dimensional space - dome @xmath124 , regular trihedron @xmath125 , regular tetrahedron @xmath126 . 
   the partial waves @xmath127 and phases @xmath89 can be classified with respect to symmetry group representation , degeneracy being defined by the dimension of the representation @xmath93 . 
   the structures @xmath124 , @xmath125 , @xmath126 belong to the @xmath128 point groups respectively . 
   the equation @xmath105 leads to algebraical problem @xmath129 the phases can be readily found from compatibility condition of this system . to factorize the determinant we derive the expressions for the phases @xmath130 @xmath131 thus , assuming @xmath132 we obtain the phases of a regular tetrahedron @xmath117 . 
   the partial and integral cross sections can be expressed as @xmath133      the structures @xmath2 can be used , for instance , to study a slow electron scattering by the polyatomic molecules like @xmath135 , @xmath136 , @xmath137 , etc . 
   let @xmath138 denote the position vectors of the scatterers @xmath139 and @xmath140 denotes the position vector of the scatterer @xmath141 . 
   the scatterers @xmath139 are situated in vertices of a regular structure @xmath1 , i.e. @xmath123 . suppose , for the sake of simplicity , the distance between the scatterer @xmath141 and any scatterer @xmath139 is @xmath142 . 
   therefore the position of the scatterer @xmath141 perfectly fixed only if @xmath132 ( geometric center of the tetrahedron , @xmath143 ) . 
   the partial waves are given by the equation @xmath127 , where the summation should be performed over the @xmath144 . 
   the constants @xmath91 and phases can be derived analytically . 
   thus , we obtain the phases @xmath145 where @xmath146 is inverse scattering length of the scatterers @xmath139 . 
   the @xmath147 obey the quadratic equation @xmath148 where @xmath149 indicates inverse scattering length of the scatterer @xmath141 . in the limiting case when the distance between @xmath139 and @xmath141 scatterers is very large , 
   i.e. @xmath150 , the expression for @xmath151 becomes @xmath152 and @xmath153 . 
   this situation corresponds to independent scattering on the structure @xmath1 and scatterer @xmath141 . 
   the tangent of @xmath154 also reduces to @xmath155 for structure @xmath1 in the limit @xmath156 . 
   the integral cross sections for @xmath3 ( closed - shell ground - state @xmath157 ) are plotted in fig . 
   @xmath158 for a number of values of @xmath159 , which are regarded as constant in the range of interest . 
   the present calculations were carried out with @xmath160 . 
   our calculations were made within the framework of the zrp model and hence they are not expected to be correct for low impact energies ( i.e for energies @xmath161 ev ) where polarization effects ( @xmath3 spherical polarizability @xmath162 ) are known to be important . because induced polarization potentials are always attractive , including polarization algebraically increases the computed phases ( @xmath163 at @xmath164 ) and therefore decreases the integral cross section . 
   thus , fig . 
   @xmath158 clearly shows that polarization effects reproduce the deep minimum ( ramsauer - townsend minimum near @xmath165 ev ) seen in the experimental data @xmath166 and numerical calculations @xmath167 , which are incomparably smaller in the our calculations . 
   the ics can be corrected by the zrps dressing . 
   fig . @xmath168 
   show ics for dressed @xmath169 structure . in our calculation , the parameters @xmath170 , @xmath171 were used . in the higher collision energies our cross section ( fig . 
   @xmath158 ) differs from the experimental results in size but coincides in shape . 
   it is known that role of the higher partial waves is important at the higher collision energies . thus , taking into consideration the @xmath172 waves for the scatterers @xmath139 and @xmath141 we can add in partial waves and improve the agreement both in size at the higher energies and in the position of shape resonance . 
   we demonstrate the posibilities of dt in multi - center scattering problem . 
   thus , these transformations allow to correct the zrp model at low energies . in the limiting case @xmath63 
   dt induces the long - range forces @xmath173 depending on angular momentum and leads to singularity in the cross section at zero energy . 
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summary: a dressing of a nonspherical potential , which includes @xmath0 zero range potentials , is considered . 
   the dressing technique is used to improve zrp model . 
   concepts of the partial waves and partial phases for non - spherical potential are used in order to perform darboux transformation . 
   the problem of scattering on the regular @xmath1 and @xmath2 structures is studied . 
   the possibilities of dressed zrp are illustrated by model calculation of the low - energy electron - silane ( @xmath3 ) scattering . 
   the results are discussed .