document: in recent years , the problem of anisotropic quantum scattering in two spatial dimensions ( 2d ) attracts increasing interest . 
   it is stimulated by the spectacular proposals for prospects to create exotic and highly correlated quantum systems with dipolar gases @xcite . 
   particularly , there were considered anisotropic superfluidity @xcite , 2d dipolar fermions @xcite , and few - body dipolar complexes @xcite . 
   the recent experimental production of ultracold polar molecules in the confined geometry of optical traps @xcite has opened up ways to realize these phenomena . 
   noteworthy also is a rather long history of research of 2d quantum effects in condensed matter physics . 
   one can note superfluid films @xcite , high - temperature superconductivity @xcite , 2d materials , such as graphene @xcite , and even possibilities for topological quantum computation @xcite . 
   unique opportunities for modeling these 2d effects in a highly controlled environment have recently appeared with the development of experimental techniques for creating quasi-2d bose and fermi ultracold gases @xcite . 
   interest in the processes and effects in 2d - geometry has stimulated the theory of elementary quantum two - body systems and processes in the plane . 
   special consideration should be given to the anisotropy and long - range character of the dipole - dipole interaction . 
   actually , usual partial - wave analysis becomes inefficient for describing the dipole - dipole scattering due to the strong anisotropic coupling of different partial - waves in the asymptotic region @xcite . 
   recently , considerable progress in the analysis of the 2d and quasi-2d ( q2d ) scattering of dipoles has been achieved  @xcite . 
   thus , the 2d dipolar scattering in the threshold and semiclassical regimes was studied in the case of the dipole polarization directed orthogonally to the scattering plane  @xcite . 
   an arbitrary angle of polarization was considered in @xcite .    in this work , 
   we develop a method for quantitative analysis of the 2d quantum scattering on a long - range strongly anisotropic scatterer . 
   particularly , it permits the description of the 2d collisions of unpolarized dipoles . 
   our approach is based on the method suggested in @xcite for the few - dimensional scattering which was successfully applied to the dipole - dipole scattering induced by an elliptically polarized laser field in the 3d free - space @xcite . 
   the key elements of the method are described in section ii . in section iii , we apply the method to the 2d scattering on the cylindrical potential with the elliptical base and the 2d dipole - dipole scattering of unpolarized dipoles . 
   we reproduce the threshold formula @xcite for the scattering amplitude on the cylinder potential with the circular base and the results of @xcite for the 2d scattering of polarized dipoles . 
   high efficiency of the method has been found in all problems being considered . 
   the last section contains the concluding remarks . 
   some important details of the computational scheme and illustration of the convergence are given in appendices . 
   the quantum scattering on the anisotropic potential @xmath0 in the plane is described by the 2d schrdinger equation in polar coordinates @xmath1 @xmath2 with the scattering boundary conditions @xmath3 in the asymptotic region @xmath4 and the hamiltonian of the system @xmath5 the unknown wave function @xmath6 and the scattering amplitude @xmath7 are searched for the fixed momentum @xmath8 defined by the colliding energy @xmath9 ( @xmath10 and the direction @xmath11 of the incident wave ( defined by the angle @xmath12 and for the scattering angle @xmath13 . here 
   @xmath14 is the reduced mass of the system . in the polar 
   coordinates , the angular part of the kinetic energy operator in @xmath15 has a simple form @xmath16 . 
   the interaction potential @xmath17 can be anisotropic in the general case , i.e. to be strongly dependent on @xmath13 . 
   it is clear that varying the direction of the incident wave @xmath11 can be replaced by the rotation @xmath18 of the interaction potential by the angle @xmath19 for the fixed direction of the incident wave , which we choose to be coincident with the x - axis . 
   thus , in the case of anisotropic potential @xmath17 the task is to solve the problem ( [ eq1 ] ) with the interaction potential @xmath20 for all possible @xmath19 and fixed @xmath9 with the scattering boundary conditions @xmath21 if the scattering amplitude @xmath7 is found , one can calculate the differential scattering cross section @xmath22 where @xmath23 , as well as the total cross section @xmath24 by averaging over all possible orientations @xmath19 of the scatterer and integration over the scattering angle @xmath13 .    to integrate the problem ( [ eq1]),([eq2 ] ) , we use 
   the method suggested in @xcite to solving a few - dimensional scattering problem and applied in @xcite for the dipole - dipole scattering in the 3d free - space . following the ideas of these works we choose the eigenfunctions @xmath25 of the operator @xmath26 as a fourier basis for the angular - grid representation of the searched wave - function @xmath27 . 
   we introduce the uniform grid @xmath28 ) over the @xmath13 and @xmath19-variables and search the wave function as expansion @xmath29 where @xmath30 is the inverse matrix to the @xmath31 square matrix @xmath32 defined on the angular grid , we use the completeness relation for the fourier basis @xmath33 , which in our grid representation reads @xmath34 . ] .    in the representation ( [ eq7 ] ) the unknown coefficients 
   @xmath35 are defined by the values of the searched wave function on the angular grid @xmath36 , any local interaction is diagonal    @xmath37    and the angular part @xmath38 of the kinetic energy operator has a simple form @xmath39 note that the presence in the interaction potential of the `` nonlocal '' angular part ( i.e. the integration or differentiation over angular variable ) leads to destroying the diagonal structure in ( [ eq8 ] ) . 
   thus , the 2d schrdinger equation ( [ eq1 ] ) is reduced in the angular - grid representation ( [ eq7 ] ) to the system of coupled ordinary differential equations of the second order : @xmath40 since the wave function @xmath41 must be finite at the origin @xmath42 , the `` left - side '' boundary condition for the functions @xmath35 reads as @xmath43 in the asymptotic region @xmath4 the scattering boundary condition ( [ eq3 ] ) accepts the form @xmath44    by using the fourier expansion for the plane wave @xmath45 and the scattering amplitude @xmath46 are the first kind bessel functions of integer order . 
   their asymptotic behavior @xcite : @xmath47{}\sqrt { \frac{2}{\pi z } } \cos \left ( { z-\frac{m\pi   } { 2}-\frac{\pi } { 4 } } \right)+e^{\left| { imz } \right|}{\rm o}\left ( { \left| z   \right|^{-1 } } \right),\\   \left ( { \left| { \arg ( z ) } \right|<\pi }   \right)\end{aligned}\ ] ] ] @xmath48 @xmath49 we eliminate the angular dependence from the asymptotic equation ( [ eq12 ] ) and represent the `` right - side '' boundary condition for the functions @xmath50 in the form @xmath51 to solve the boundary - value problem ( [ eq10]),([eq11 ] ) and ( [ eq15 ] ) , we introduce the grid over the @xmath52 and reduce the system of differential equations ( [ eq10 ] ) by using the finite - difference approximation of the sixth order to the system of @xmath53 algebraic equations    @xmath54    with the band - structure of the matrix @xmath55 with the width @xmath56 of the band . by using the asymptotic equations ( [ eq15 ] ) in the last two points @xmath57 and @xmath58 one can eliminate the unknown vector @xmath59 from equation ( [ eq15 ] ) and rewrite the `` right - side '' boundary condition in the form @xmath60 analogously 
   , one can eliminate unknown constant from expression ( [ eq11 ] ) by considering asymptotic equations ( [ eq11 ] ) at the first points @xmath61 and @xmath62 . 
   the acquired `` left - side '' boundary condition reads @xmath63    thus , the scattering problem is reduced to the boundary value problem ( [ my_eq15]-[my_eq17 ] ) @xmath64 which can be efficiently solved with standard computational techniques such as the sweeping method @xcite or the lu - decomposition @xcite . 
   the detailed structure of the matrix of the coefficients @xmath65 is discussed in appendix a. after the solving of eq.([my_eq19 ] ) and finding the wave function @xmath66 the scattering amplitude @xmath67 is constructed according to eqs.([eq15 ] ) and ( [ eq14 ] ) . 
   first , we have analyzed the 2d scattering on the cylindrical potential barrier with the elliptical base @xmath68    the case of the circular base @xmath69 was considered in @xcite , where analytic formula for the scattering amplitude @xmath70+i\frac{\pi } { 2}}\ ] ] was obtained at the zero - energy limit @xmath71 . here and @xmath72 is the euler constant . 
   we have analyzed the scattering on the potential barrier with circular base @xmath73 for arbitrary momentum @xmath74 . 
   the results of calculation presented in figs . 
   [ fig1 ] and [ fig3 ] confirm the convergence of the scattering amplitude @xmath67 to the analytical value ( [ eq17 ] ) at @xmath75 . in this subsection 
   all the calculations were performed in the units @xmath76 .    in the limiting case of the infinitely high potential barrier ( [ eq16 ] ) with the circular base @xmath77 
   the asymptotic formula ( [ eq17 ] ) becomes exact for arbitrary @xmath74 . 
   this is confirmed by investigation presented in table  [ tab1 ] which illustrates the convergence of the numerical values @xmath78 with increasing ( @xmath79 ) and narrowing ( @xmath80 ) of the potential barrier to the analytic result ( [ eq17 ] ) . in the limit case @xmath79 and @xmath81 
   we obtain @xmath82 for the scattering length @xmath83 extracted from the calculated amplitude @xmath84 by the formula ( [ eq17 ] ) , what is in agreement to the estimate given in @xcite . the range of applicability of eq . 
   ( [ eq17 ] ) was investigated recently in @xcite .    [ cols="^,^,^,^,^ " , ] 
   the boundary - value problem ( [ eq10]),([eq11 ] ) and ( [ eq15 ] ) , obtained in section ii in the angular - grid representation ( [ eq7 ] ) , reads in a matrix form as :    @xmath85\bm { \psi } ( \rho ) \,+\\   \hspace{5 cm } + \,\hat{h}^{(0 ) }   \bm { \psi } ( \rho ) = 0 \\    \bm { \psi } \xrightarrow[{\rho \to 0}]{}const\cdot \sqrt \rho \\    \frac{2\pi}{(2 m + 1)\sqrt \rho } \sum\limits_{j=0}^{2 m } { e ^{-im\phi_j } \psi _ j   ( \rho ) } = i^mj_m ( q\rho ) \sqrt { 2\pi } + \\ 
   \hspace{5 cm } + \,\frac{f_m(\phi_q)}{\sqrt { -i\rho }   } e^{iq\rho } \,,\\ 
   \end{array } } \right.\ ] ]    where and @xmath86 . in this representation 
   the angular dependence is built into the matrix @xmath87 and the interaction is included into the diagonal matrix @xmath88 of values of the potential in the angular grid nodes . 
   the constant matrix @xmath87 couples all equations in a system and does not depend on the radial variable . 
   there is no need to compute any matrix elements of the potential , what essentially minimizes the computational costs .    for solving boundary value problem ( [ eq23 ] ) the seven - point finite - difference approximation for second derivatives of sixth order @xmath89 
   is applied in the points @xmath90 of the radial grid , where @xmath91 . as a result 
   , the system ( [ eq23 ] ) reduces to the system of linear algebraic equations ( [ my_eq16 ] ) with the matrix @xmath55 whose band structure reads @xmath92 where the @xmath93 coefficients are the square @xmath94 matrices , and the elements @xmath95 of the right - side of ( [ eq25 ] ) are the @xmath96 dimensional vectors . after employing the `` right - side '' boundary condition in the form ( [ my_eq16 ] ) in the last three grid points @xmath97 and , analogously , the `` left - side '' boundary condition in the form ( [ my_eq17 ] ) in the first two points @xmath98 , the detailed structure of the matrix @xmath55 is represented as    @xmath99    the block structure of the system ( [ eq26 ] ) provides several significant advantages . 
   the block matrix can be stored in a packaged form , which allows the use of optimal resource . 
   the system ( [ eq26 ] ) can be efficiently solved by a fast implicit matrix algorithm based on the idea of the block sweep method @xcite . 
   in the table [ tab4 ] we illustrate the convergence of the calculated scattering amplitude @xmath67 over the number of angular grid points @xmath96 for the scatterers with weak and essential anisotropy at @xmath100 in the potential barrier ( [ eq16 ] ) . for the case @xmath101 we reach the accuracy of four significant digits in the scattering amplitude on the angular grids with @xmath102 . for stronger anisotropy @xmath103 the accuracy of two significant digits 
   was reached at @xmath104 . 
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summary: [ txt : abstract ] we study the quantum scattering in two spatial dimensions ( 2d ) without the usual partial - wave formalism . 
   the analysis beyond the partial - wave approximation allows a quantitative treatment of the anisotropic scattering with a strong coupling of different angular momenta nonvanishing even at the zero - energy limit . 
   high efficiency of our method is demonstrated for the 2d scattering on the cylindrical potential with the elliptical base and dipole - dipole collisions in the plane . 
   we reproduce the result for the 2d scattering of polarized dipoles in binary collisions obtained recently by ticknor [ phys . 
   rev . 
   a * 84 * , 032702 ( 2011 ) ] and explore the 2d collisions of unpolarized dipoles .