Error estimates of an exponential wave integrator for the nonlinear Schrödinger equation with singular potential

TL;DR

This paper analyzes a first-order exponential wave integrator (EWI) for the nonlinear Schrödinger equation with a singular potential $V\in L^2(\mathbb{R}^d) + L^\infty(\mathbb{R}^d)$, including physically important Coulomb-type potentials. By introducing a time-filtered EWI and employing discrete Strichartz estimates, the authors overcome loss of integrability and establish rigorous $L^2$-norm error bounds that depend on dimension and the $L^p$-regularity of the potential. They prove optimal first-order convergence in $L^2$ for 1D and 2D, and a $3/4$-order rate in 3D under $L^2$-potential, with higher rates up to first-order in 3D when $p>12/5$, yielding first-order convergence for the 3D Coulomb potential. The results are complemented by extensive numerical experiments across 1D–3D that confirm the predicted rates and illustrate the practical relevance for simulations involving singular potentials.

Abstract

We analyze a first-order exponential wave integrator (EWI) for the nonlinear Schrödinger equation (NLSE) with a singular potential locally in $L^2$, which might be locally unbounded. The typical example is the inverse power potential such as the Coulomb potential, which is the most fundamental potential in quantum physics and chemistry. We prove that, under the assumption of $L^2$-potential and $H^2$-initial data, the $L^2$-norm convergence of the EWI is, roughly, first-order in one dimension (1D) and two dimensions (2D), and $\frac{3}{4}$-order in three dimensions (3D). In addition, under a stronger integrability assumption of $L^p$-potential for some $p>2$ in 3D, the $L^2$-norm convergence increases to almost ${\frac{3}{4}} + 3(\frac{1}{2} - \frac{1}{p})$ order if $p \leq \frac{12}{5}$ and becomes first-order if $p > \frac{12}{5}$. In particular, our results show, to the best of our knowledge for the first time, that first-order $L^2$-norm convergence can be achieved when solving the NLSE with the Coulomb potential in 3D. The key advancements are the use of discrete (in time) Strichartz estimates, which allow us to handle the loss of integrability due to the singular potential that does not belong to $L^\infty$, and the more favorable local truncation error of the EWI, which requires no (spatial) smoothness of the potential. Extensive numerical results in 1D, 2D, and 3D are reported to confirm our error estimates and to show the sharpness of our assumptions on the regularity of the singular potentials.

Figures (5)

• Figure 1: Errors in $L^2$- and $H^1$-norms of the EWI for the NLSE with singular potentials in 1D: (a) $\alpha = 0.51$ and (b) $\alpha = 0.76$
• Figure 2: Errors in $L^2$- and $H^1$-norms of the EWI for the NLSE with $L^2$-potential in 2D: (a) Coulomb potential and (b) random potential
• Figure 3: Errors in $L^2$- and $H^1$-norms of the EWI for the NLSE with singular potentials in 3D: (a) $L^{3^-}$-potential and (b) $L^2$-potential
• Figure 4: Illustration of the initial set up
• Figure 5: Plots of $|\psi(\cdot, t)|^2$ at different time $t$ for the dynamics under four Coulomb potentials in 2D

Theorems & Definitions (20)

• Theorem 2.1: Optimal convergence
• Theorem 2.2: Convergence for $L^p$-potential
• Corollary 2.3: Convergence for Coulomb potential
• Corollary 2.4
• Remark 2.5: $H^1$-norm convergence
• Lemma 3.1
• Proof 1
• Lemma 3.2
• Lemma 3.3
• Remark 3.4