KANTBP 4M Program for Solving the Scattering Problem for a System of Ordinary Second-Order Differential Equations

We report an upgrade of the program KANTBP 4M implemented in the computer algebra system MAPLE for solving, with a given accuracy, the multichannel scattering problem, which is reduced to a boundary-value problem for a system of ordinary differential equations of the second order with continuous or piecewise continuous real or complex-valued coefficients. The solution over a finite interval is subject to mixed homogeneous boundary conditions: Dirichlet and/or Neumann, and/or of the third kind. The discretization of the boundary problem is implemented by means of the finite element method with the Lagrange or Hermite interpolation polynomials. The efficiency of the proposed algorithm is demonstrated by solving a multichannel scattering problem with coupling of channels in both the reaction region and the asymptotic one.


Introduction
A wide class of quantum mechanical problems such as tunneling, ionization, recombination problem, are formulated as boundary-value problems (BVPs) for elliptic differential equations. One of the ways to solve these problems is to reduce them to a multichannel scattering problem for a set of secondorder ordinary differential equations (SOODEs) in an infinite region (on axis or semiaxis), which can be performed by different methods. Further reduction of the problem to a finite interval implies the formulation of asymptotic boundary conditions depending upon the kind of the physical problem considered.
For the general multichannel scattering problem, the nonsymmetry in the coefficient functions leads to the different numbers of open channels in the left-hand and right-hand asymptotic regions, i.e., a scattering matrix composed of square matrices of reflection amplitudes and rectangular matrices of transmission amplitudes [1]. To solve this problem, as well as the problems of bound or metastable states [2], we elaborated a new version of the program KANTBP 4M [3] implementing a finite element method (FEM) with Lagrange or Hermite interpolation polynomials [4] in the computer algebra system (CAS) MAPLE.
In the present paper we modify the algorithms to solve the scattering problem with coupled ODEs in both the reaction regions and the asymptotic ones, implemented in an upgraded version of the program KANTBP 4M. The efficiency of the proposed algorithm is demonstrated by solving a multichannel scattering problem with coupling of channels in the asymptotic region similar to the one present in sub-barrier fusion reactions [6].

The multichannel scattering problem
The program KANTBP 4M is intended for solving the BVPs for the system of SOODEs with respect to the unknown functions Φ(z) = (Φ 1 (z), . . . , Φ N (z)) T of the independent variable z ∈ Ω = (z min , z max ): Here f B (z) > 0 and f A (z) > 0 are continuous or piecewise continuous positive functions, I is the unit matrix, V(z) is a symmetric matrix, V i j (z) = V ji (z), and Q(z) is an antisymmetric matrix, Q i j (z) = −Q ji (z), of the effective potentials having the dimension N × N. The elements of these matrices are continuous or piecewise continuous real or complex-valued coefficients from the Sobolev space H s≥1 2 (Ω), providing the existence of nontrivial solutions subject to mixed homogeneous boundary conditions: Dirichlet and/or Neumann, and/or of the third kind at the boundary points of the interval z ∈ [z min , z max ] at given values of the elements of the real or complex-valued For the multichannel scattering problem at fixed energy E ≡ E the desired matrix of solutions Φ(z) is calculated with the asymptotic form "incident wave + outgoing waves" The Here the label t marks the asymptotic region: t = L at z ≤ z min and t = R at z ≥ z max . In this case the expressions (3) -(4) are written in the explicit form Here The details of the FEM solution of the above problems are given in Refs. [2,3].

A nontrivial case study
We consider the multichannel scattering problem for the Schrödinger equation in the 2D domain where the square R → , R ← and rectangular T → , T ← submatrices are separated by lines. The corresponding wave functions of the multichannel scattering problem are presented in Fig. 1. The FEM calculations were performed in intervals z ∈ [−2, 2] and z ∈ [−10, 10] with seventh order Hermite elements (p = 7, p = 3, κ max = 2) on the finite element meshes Ω z = {−2(4)2} and Ω z = {−10(6)−2(4)2(6)10}, respectively, where in the parentheses the number of elements is indicated. The results obtained by FEM in the above intervals and MMFS coincide within 10 −10 .

Conclusions
The KANTBP 4M code and the test examples of solving eigenvalue and scattering problems of quantum mechanics are available in the program library JINRLIB [3]. The upgrade of KANTBP 4M program implemented in the CAS MAPLE for solving with a given accuracy the boundary value problems for systems of coupled SOODEs, in particular, the class of scattering problem for the system of coupled SOODEs in both the reaction regions and the asymptotic ones with different number of open channels in the left-hand and right-hand coupled asymptotic regions is approbated. The Fortran version of KANTBP 4M program that allows one to perform large scale calculations of the sub-barrier fusion reactions of heavy ions will be published elsewhere.