Algorithm for Computing Wave Functions, Reflection and Transmission Matrices of the Multichannel Scattering Problem in the Adiabatic Representation using the Finite Element Method

In adiabatic representation the multichannel scattering problem for a multidimensional Schr¨odinger equation is reduced to the boundary value problem (BVP) for a system of coupled self-adjoined second-order ordinary differential equations on a finite interval with homogeneous boundary conditions of the third type at the left and right boundary points in the framework of the Kantorovich method using adiabatic basis of surface functions depending on longitudinal variable as a parameter. The homogeneous third-type boundary conditions for the desirable wave functions of the BVP are formulated using the known set of linear independent regular and irregular asymptotic solutions in the open channels of the reduced multichannel scattering problem on an axis which involve the desirable reflection and transmission amplitude matrices, and the set of linear independent regular asymptotic solutions in the closed channels. The economical and stable algorithm for numerical calculation with given accuracy of reflection and transmission matrices, and the corresponding wave functions of the multichannel scattering problem for the system of equations containing potential matrix elements and first-derivative coupling terms is proposed using high-order accuracy approximations of the finite element method (FEM). The efficiency of the proposed algorithm is demonstrated by solving of the two-dimensional quantum transmittance problem for a pair of coupled particles with oscillator interaction potentials penetrating through repulsive Coulomb-type potentials and scattering problem of electron in a Coulomb field of proton and in the homogeneous magnetic field in the framework of the Kantorovich and Galerkin-type methods and studying their convergence.

multichannel scattering problem, reflection and transmission amplitude matrices, multidimensional Schr¨odinger equation, adiabatic representation, the Kantorovich method, boundary value problem, a system of coupled self-adjoined second-order ordinary differential equations, the finite element method