In this paper we use a similarity transformation connecting some families of Nonlinear Schrödinger equations with time-varying coefficients with the autonomous cubic nonlinear Schrödinger equation. This transformation allows one to apply all known results for that equation to the original non-autonomous case with the additional dynamics introduced by the transformation itself. In particular, us...

The numerical approximation of the solution of the time-dependent Schrödinger equation arising in ultrafast laser dynamics is discussed. The linear Schrödinger equation is reduced to a computationally tractable, lower dimensional system of nonlinear partial differential equations by the multi-configuration time-dependent Hartree-Fock method. This method serves to approximate the original wave f...

A propagation method for the time dependent Schrödinger equation was studied leading to a general scheme of solving ode type equations. Standard space discretization of time-dependent pde’s usually results in system of ode’s of the form ut −Gu= s (0.1) where G is a operator (matrix) and u is a time-dependent solution vector. Highly accurate methods, based on polynomial approximation of a modifi...

This paper deals with two domain decomposition methods for two dimensional linear Schrödinger equation, the Schwarz waveform relaxation method and the domain decomposition in space method. After presenting the classical algorithms, we propose a new algorithm for the Schrödinger equation with constant potential and a preconditioned algorithm for the general Schrödinger equation. These algorithms...

We consider the cubic defocusing Schrödinger equation in two space dimen15 sions and prove that if the initial data are in H1/2, then there exists a global solution in time. The proof combines the argument from [5] with some new correlation estimates 17 for the Schrödinger equation.

Recent technological advance has prompted significant interest in developing the control theory of quantum systems. Following the increasing interest in the control of quantum dynamics, this paper examines the control problem of Schrödinger equation because quantum dynamics is basically governed by Schrödinger equation. From the practical point of view, stochastic disturbances cannot be avoided...