Error Estimates of an Exponential Wave Integrator for the Nonlinear Schrödinger Equation with Singular Potential

Abstract.

In this paper we analyze a first-order exponential wave integrator (EWI) for the nonlinear Schrödinger equation (NLSE) with a singular potential that is locally in [Formula: see text], 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 [Formula: see text]-potential and [Formula: see text]-initial data, the [Formula: see text]-norm convergence of the EWI is, roughly, first-order in one dimension (1D) and two dimensions (2D), and [Formula: see text]-order in three dimensions (3D). In addition, under a stronger integrability assumption of [Formula: see text]-potential for some [Formula: see text] in 3D, the [Formula: see text]-norm convergence increases to almost [Formula: see text]-order if [Formula: see text] and becomes first-order if [Formula: see text]. In particular, our results show, to the best of our knowledge for the first time, that first-order [Formula: see text]-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 [Formula: see text], 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.