Multi-Layer Schemes for Solving Time-Dependent Schrцdinger Equation by Finite Element Method

The symmetric implicit operator-difference multi-layer schemes for solving the timedependent Schrцdinger equation based on decomposition of the evolution operator via the explicit Magnus expansion up to the sixth order of accuracy with respect to the time step are presented. Reduced schemes for solving the Cauchy problem of a set of coupled timedependent Schrцdinger equations with respect to the hyperradial variable are devised by using the Kantorovich expansion of the wave packet over a set of appropriate parametric basis angular functions. The implicit algebraic schemes for numerical solving the problem with symmetric operators, using discretization of the component of the wave package by hyperradial variable by the high order finite-element method are formulated. The convergence and efficiency of the numerical schemes are demonstrated in numerical calculations of the exactly solvable models of one-dimensional oscillator with time-dependent frequency, two-dimensional oscillator in time-dependent external field by using the conventual angular basis.

нестационарное уравнение Шрёдингера, задача Коши, факторизация оператора эволюции, операторно-разностные многослойные схемы, метод Канторовича, метод конечных элементов