Splitting Analysis for Yukawa Potential

Abstract

Splitting methods are among the most classical and fundamental tools for the simulation of quantum dynamics, and their importance has grown further with the rise of quantum computing. In this work, we analyze the Schr\"odinger equation with Yukawa potential, a physically relevant and widely used model potential. It may be viewed as a Coulomb interaction with exponential decay at spatial infinity, preserving the Coulomb singularity at the origin while removing the long-range Coulomb tail. We prove that the operator splitting for this unbounded Hamiltonian achieves a global $1/4$-order convergence rate in the time step for many-body Yukawa interactions, with explicit polynomial dependence on the number of particles. The result holds for all initial wavefunctions in $H^2(\mathbb R^{3N})$, the natural domain of the Hamiltonian, and our numerical experiments are consistent with the theoretical estimates. To identify the sharp obstruction behind this rate, we prove a short-time lower bound in the one-body setting of order $t^{5/4}$ for the one-step error, which rules out any uniform global estimate of order better than $1/4$ in general. This agreement with the optimal $1/4$ rate in the Coulomb case is particularly interesting, as Yukawa potential is short-ranged compared to Coulomb potential. For the many-body upper bound, one of the new technical ingredients is the explicit polynomial-in-system-size Sobolev estimates of many-body Yukawa systems. These estimates are crucial for obtaining fully a priori bounds that depend only on the norms of the initial states, rather than on the solution at time $t$. For the one-body lower bound, we leverage a new analysis argument based on Fourier analysis and Kato smoothing.