KANTBP 3.1: A program for computing energy levels, reflection and transmission matrices, and corresponding wave functions in the coupled-channel and adiabatic approaches

Abstract

A FORTRAN program for calculating energy values, reflection and transmission matrices, and corresponding wave functions in a coupled-channel approximation of the adiabatic approach is presented. In this approach, a multidimensional Schrödinger equation is reduced to a system of the coupled second-order ordinary differential equations on a finite interval with the homogeneous boundary conditions of the third type at left- and right-boundary points for the discrete spectrum and scattering problems. The resulting system of such equations, containing potential matrix elements and first-derivative coupling terms is solved using high-order accuracy approximations of the finite element method. The scattering problem is solved with non-diagonal potential matrix elements in the left and/or right asymptotic regions and different left and right threshold values. Benchmark calculations for the fusion cross sections of 36S+48Ca, 64Ni+100Mo reactions are presented. As a test desk, the program is applied to the calculation of the reflection and transmission matrices and corresponding wave functions of the exact solvable wave-guide model, and also the fusion cross sections and mean angular momenta of the 16O+144Sm reaction. Program summaryProgram Title: KANTBPCPC Library link to program files:https://doi.org/10.17632/4vm9fhyvh3.1Licensing provisions: CC BY NC 3.0Programming language: FORTRANNature of problem: In the adiabatic approach [1], a multidimensional Schrödinger equation for quantum reflection [2], the photoionization and recombination of a hydrogen atom in a homogeneous magnetic field [3–6], the three-dimensional tunneling of a diatomic molecule incident upon a potential barrier [7], wave-guide models [8], the fusion model of the collision of heavy ions [9–11], and low-energy fusion reactions of light- and medium mass nuclei [12] is reduced by separating the longitudinal coordinate, labeled as z, from transversal variables to a system of second-order ordinary differential equations containing the potential matrix elements and first-derivative coupling terms. The purpose of this paper is to present a program based on the use of high-order accuracy approximations of the finite element method (FEM) for calculating energy levels, reflection and transmission matrices and wave functions for such systems of coupled-channel second order differential equations (CCSODEs) on finite intervals of the variable z∈[zmin,zmax] with homogeneous boundary conditions of the third-type at the left- and right-boundary points, which follow from the discrete spectrum and scattering problems.Solution method: The boundary-value problems for the system of CCSODEs are solved by the FEM using high-order accuracy approximations [13,14]. The generalized algebraic eigenvalue problem AF=EBF with respect to pair unknowns (E,F), arising after the replacement of the differential eigenvalue problem by the finite-element approximation, is solved by the subspace iteration method [14]. The generalized algebraic eigenvalue problem of a special form (A−EB)F=DF with respect to pair unknowns (D,F) arising after the corresponding replacement of the scattering boundary problem in open channels at fixed energy value, E, is solved by the L DLT factorization of the symmetric matrix and back-substitution methods [14].Additional comments including restrictions and unusual features: The user must supply subroutine POTCAL for evaluating potential matrix elements. The user should also supply subroutines ASYMEV (when solving the eigenvalue problem) or ASYMSL and ASYMSR (when solving the scattering problem) which evaluate asymptotics of the wave functions at boundary points in the case of a boundary conditions of the third-type for the above problems.