On solving the low-dimensional boundary value problems of quantum mechanics by Kantorovich method - reduction to ordinary differential equations

The calculation scheme for solving the elliptic boundary problem by Kantorovich method for impurity states in models of quantum dots, wires and wells in the effective mass approximation with parabolic confinement potential of harmonic oscillator and infinitely-high walls is presented. The rate of convergence of the method and the efficiency of the proposed program complex, realized by the finite element method, is demonstrated on examples of calculation of spectral characteristics of the models and new effects of resonance transmission and total reflection for the Coulomb scattering, induced by axial homogeneous magnetic field, crystal lattice, or quantum wire.