Nonlocalization of singular potentials in quantum dynamics

TL;DR

This paper demonstrates that the nonlocal Wigner function model effectively handles singular potentials in quantum dynamics, enabling accurate numerical solutions and revealing interesting quantum behaviors.

Contribution

It introduces a nonlocal Wigner function approach that converts singular potentials into weakly singular kernels, improving numerical accuracy in quantum simulations.

Findings

Successfully computed Wigner functions for various singular potentials.

The method achieves high numerical accuracy and convergence.

Revealed interesting quantum behaviors under singular potentials.

Abstract

Nonlocal modeling has drawn more and more attention and becomes steadily more powerful in scientific computing. In this paper, we demonstrate the superiority of a first-principle nonlocal model -- Wigner function -- in treating singular potentials which are often used to model the interaction between point charges in quantum science. The nonlocal nature of the Wigner equation is fully exploited to convert the singular potential into the Wigner kernel with weak or even no singularity, and thus highly accurate numerical approximations are achievable, which are hardly designed when the singular potential is taken into account in the local Schr\"odinger equation. The Dirac delta function, the logarithmic, and the inverse power potentials are considered. Numerically converged Wigner functions under all these singular potentials are obtained with an operator splitting spectral method, and display many interesting quantum behaviors as well.