New, Highly Accurate Propagator for the Linear and Nonlinear Schr\"odinger Equation

TL;DR

This paper introduces a new, highly accurate propagator method for solving both linear and nonlinear time-dependent Schr"odinger equations, accommodating time-dependent right-hand sides and nonlinearities.

Contribution

The paper presents a novel algorithm that extends polynomial approximation techniques to handle time-dependent and nonlinear Schr"odinger equations.

Findings

Demonstrates high accuracy for time-dependent potentials

Effective for nonlinear Schr"odinger equations

Applicable to equations with time-dependent right-hand sides

Abstract

A propagation method for the time dependent Schr\"odinger 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 u_t -Gu = s where G is a operator (matrix) and u is a time-dependent solution vector. Highly accurate methods, based on polynomial approximation of a modified exponential evolution operator, had been developed already for this type of problems where G is a linear, time independent matrix and s is a constant vector. In this paper we will describe a new algorithm for the more general case where s is a time-dependent r.h.s vector. An iterative version of the new algorithm can be applied to the general case where G depends on t or u. Numerical results for Schr\"odinger equation with time-dependent potential and to non-linear Schr\"odinger equation will be presented.