Moments of spectral functions: Monte Carlo evaluation and verification

TL;DR

This paper presents a novel Monte Carlo method using pseudospectral techniques to accurately evaluate moments of spectral functions from path integrals, ensuring numerical stability and verifying moment sequence conditions.

Contribution

It introduces a pseudospectral differentiation approach for spectral moments, utilizing the entire interval and path integral interpolation of the action for improved numerical stability.

Findings

Pseudospectral differentiation enhances moment evaluation accuracy.

Hamburger's conditions verify the consistency of computed moments.

The method mitigates the sign problem in Monte Carlo spectral calculations.

Abstract

The subject of the present study is the Monte Carlo path-integral evaluation of the moments of spectral functions. Such moments can be computed by formal differentiation of certain estimating functionals that are infinitely-differentiable against time whenever the potential function is arbitrarily smooth. Here, I demonstrate that the numerical differentiation of the estimating functionals can be more successfully implemented by means of pseudospectral methods (e.g., exact differentiation of a Chebyshev polynomial interpolant), which utilize information from the entire interval $(-\beta \hbar / 2, \beta \hbar/2)$. The algorithmic detail that leads to robust numerical approximations is the fact that the path integral action and not the actual estimating functional are interpolated. Although the resulting approximation to the estimating functional is non-linear, the derivatives can be computed from it in a fast and stable way by contour integration in the complex plane, with the help of the Cauchy integral formula (e.g., by Lyness' method). An interesting aspect of the present development is that Hamburger's conditions for a finite sequence of numbers to be a moment sequence provide the necessary and sufficient criteria for the computed data to be compatible with the existence of an inversion algorithm. Finally, the issue of appearance of the sign problem in the computation of moments, albeit in a milder form than for other quantities, is addressed.