Time splitting and error estimates for nonlinear Schrodinger equations with a potential

TL;DR

This paper develops low regularity error estimates for a Lie-Trotter splitting scheme applied to the nonlinear Schrödinger equation with a potential, using spectral theory and pseudodifferential calculus.

Contribution

It introduces a spectral localization approach to derive error estimates for time discretization of the Gross-Pitaevskii equation with low regularity assumptions.

Findings

Proves error estimates for the splitting scheme with spectral localization.

Establishes discrete Strichartz estimates supporting nonlinear analysis.

Provides a framework for low regularity analysis of Schrödinger equations.

Abstract

We consider the nonlinear Schr{\"o}dinger equation with a potential, also known as Gross-Pitaevskii equation. By introducing a suitable spectral localization, we prove low regularity error estimates for the time discretization corresponding to an adapted Lie-Trotter splitting scheme. The proof is based on tools from spectral theory and pseudodifferential calculus in order to obtain various estimates on the spectral localization, including discrete Strichartz estimates which support the nonlinear analysis.