AI 中文总结
研究含汤川势的薛定谔方程，通过算子分裂方法，证明多体相互作用时间步长有全局1/4阶收敛率，给出与粒子数的多项式关系，数值实验相符，还给出单体短时误差下界及多体上界相关技术成分和论证。
AI 中文摘要
分裂方法是模拟量子动力学最经典和基本的工具之一，随着量子计算的兴起其重要性进一步提升。本文分析了含汤川势的薛定谔方程，它是一个物理相关且广泛使用的模型势。证明了该无界哈密顿量的算子分裂在多体汤川相互作用的时间步长上实现了全局1/4阶收敛率，并给出了与粒子数的显式多项式依赖关系。数值实验与理论估计一致。还证明了单体设置下单步误差的短时下界为t^{5/4}阶，排除了一般情况下优于1/4阶的任何统一全局估计。多体上界的新技术成分是多体汤川系统的显式多项式系统大小Sobolev估计。单体下界则利用了基于傅里叶分析和加藤平滑的新分析论证。
英文摘要
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 Schrödinger 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.
