Symbolic-numeric algorithms for solving BVPs for a system of ODEs of the second order: Multichannel scattering and eigenvalue problems

Abstract

Symbolic-numeric algorithms for solving multichannel scattering and eigenvalue problems of the waveguide or tunneling type for systems of ODEs of the second order with continuous and piecewise continuous coefficients on an axis are presented. The boundary-value problems are formulated and discretized using the FEM on a finite interval with interpolating Hermite polynomials that provide the required continuity of the derivatives of the approximated solutions. The accuracy of the approximate solutions of the boundary-value problems, reduced to a finite interval, is checked by comparing them with the solutions of the original boundary-value problems on the entire axis, which are calculated by matching the fundamental solutions of the ODE system. The efficiency of the algorithms implemented in the computer algebra system Maple is demonstrated by calculating the resonance states of a multichannel scattering problem on the axis for clusters of a few identical particles tunneling through Gaussian barriers.