Algorithms for Solving the Boundary-Value Problems for Atomic Trimers in Collinear Configuration using the Kantorovich Method

The model of atomic trimers with molecular pair interactions for collinear configuration is formulated as a 2D boundary-value problem (BVP) in the Jacobi and polar coordinates. The latter is reduced to a 1D BVP for a system of second-order ordinary differential equations (ODEs) by means of the Kantorovich method using the expansion of the desired solutions over a set of angular basis functions, parametrically dependent on the (hyper)radial variable. The algorithms for solving the 1D parametric BVP by means of the finite element method (FEM) and calculating the asymptotes of the parametric angular functions and effective potentials of the system of ODEs at large values of the parameter are presented. The efficiency of the algorithms is confirmed by comparing the calculated asymptotic solutions and effective potentials with those of the parametric eigenvalue problem obtained by applying the FEM at large values of the parameter. The applicability of the algorithms is demonstrated by calculating the asymptotic expansions of the parametric BVP solution, effective potentials and sets of binding energies for the beryllium trimer in the collinear configuration.

boundary-value problems, the Kantorovich method systems of second-order ordinary differential equations, finite element method