Error estimates of an exponential wave integrator for the nonlinear Schr\"odinger equation with singular potential

Abstract

We analyze a first-order exponential wave integrator (EWI) for the nonlinear Schr\"odinger equation (NLSE) with a singular potential that is locally in $L^2$, which might be locally unbounded. A 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.