New Method for Solving Three-Dimensional Schroedinger Equation

Abstract

A new method is developed for solving the multidimensional Schroedinger equation without the variable separation. To solve the Schroedinger equation in a multidimensional coordinate space X, a difference grid Ω _i (i=1,2,...,N) for some of variables ,Ω, from X = {R, Ω} is introduced and the initial partial-differential equation is reduced to a system of N differential-difference equations in terms of one of the variables R. The arising multi-channel scattering (or eigen-value) problem is solved by the algorithm based on a continuous analog of the Newton method. The approach has been successfully tested for several two-dimensional problems (scattering on a nonspherical potential well and ”dipole” scatterer, a hydrogen atom in a homogeneous magnetic field) and for a three-dimensional problem of the helium-atom bound states.