Operator splitting for non-autonomous evolution equations

TL;DR

This paper develops product formulas for solving non-autonomous evolution equations using semigroup methods, demonstrating their application with a time-dependent Schrödinger equation and analyzing Strang-splitting convergence.

Contribution

It introduces general product formulas for non-autonomous problems via evolution semigroup techniques, extending autonomous results to non-autonomous cases.

Findings

Derived convergence rates for Strang-splitting method.

Applied results to a time-dependent Schrödinger equation.

Provided a framework for non-autonomous evolution equations.

Abstract

We provide general product formulas for the solutions of non-autonomous abstract Cauchy problems. The main technical tool is the application of evolution semigroup methods, allowing the direct application of existing results on autonomous problems. The results are then illustrated by the example of a imaginary time Schr\"odinger equation with time dependent potential. We also obtain convergence rates for the Strang-splitting applied to this problem.