标题： 期望值的Herman--Kluk近似中的抽样策略 显示英文标题

标题： Sampling strategies for expectation values within the Herman--Kluk approximation

摘要： 在计算量子力学可观测量时，"维度灾难"限制了使用量子力学波函数的朴素方法。半经典Herman--Kluk传播子通过采用无网格假设来评估这些可观测量的期望值，从而缓解了这一问题。在此，我们研究了这种高维且高度振荡的传播子的求积技术。特别是，我们分析了三种不同初始采样方法的蒙特卡洛求积。前两种方法分别基于Husimi密度或其平方根，与可观测量无关，而第三种方法是新的，它在采样中引入了可观测量，以在初始时间最小化蒙特卡洛被积函数的方差。我们证明了蒙特卡洛估计量收敛的充分条件，并提供了收敛误差估计。分析结果通过在谐振子和具有逐渐增加的非简谐度的Henon-Heiles势能上的各种维度数值实验得到了验证。 显示英文摘要

摘要： When computing quantum-mechanical observables, the ``curse of dimensionality'' limits the naive approach that uses the quantum-mechanical wavefunction. The semiclassical Herman--Kluk propagator mitigates this curse by employing a grid-free ansatz to evaluate the expectation values of these observables. Here, we investigate quadrature techniques for this high-dimensional and highly oscillatory propagator. In particular, we analyze Monte Carlo quadratures using three different initial sampling approaches. The first two, based either on the Husimi density or its square root, are independent of the observable whereas the third approach, which is new, incorporates the observable in the sampling to minimize the variance of the Monte Carlo integrand at the initial time. We prove sufficient conditions for the convergence of the Monte Carlo estimators and provide convergence error estimates. The analytical results are validated by numerical experiments in various dimensions on a harmonic oscillator and on a Henon-Heiles potential with an increasing degree of anharmonicity.