The decomposition method which makes the parallel solution of the block-tridiagonal matrix systems possible is presented. The performance of the method is analytically estimated based on the number of elementary multiplicative operations for its para
llel and serial parts. The computational speedup with respect to the conventional sequential Thomas algorithm is assessed for various types of the application of the method. It is observed that the maximum of the analytical speedup for a given number of blocks on the diagonal is achieved at some finite number of parallel processors. The values of the parameters required to reach the maximum computational speedup are obtained. The benchmark calculations show a good agreement of analytical estimations of the computational speedup and practically achieved results. The application of the method is illustrated by employing the decomposition method to the matrix system originated from a boundary value problem for the two-dimensional integro-differential Faddeev equations. The block-tridiagonal structure of the matrix arises from the proper discretization scheme including the finite-differences over the first coordinate and spline approximation over the second one. The application of the decomposition method for parallelization of solving the matrix system reduces the overall time of calculation up to 10 times.
The zero range potential is constructed for a system of two particles interacting via the Coulomb potential. The singular part of the asymptote of the wave function at the origin which is caused by the common effect of the zero range potential singul
arity and of the Coulomb potential is explicitly calculated by using the Lippmann-Schwinger type integral equation. The singular pseudo potential is constructed from the requirement that it enforces the solution to the Coulomb Schrodinger equation to possess the calculated asymptotic behavior at the origin. This pseudo potential is then used for constructing a model of the imaginary absorbing potential which allows to treat the annihilation process in positron electron collisions on the basis of the non relativistic Schrodinger equation. The functional form of the pseudo potential constructed in this paper is analogous to the well known Fermi-Breit-Huang pseudo potential. The generalization of the optical theorem on the case of the imaginary absorbing potential in presence of the Coulomb force is given in terms of the partial wave series.
