Moment problems and bounds for matrix-valued smeared spectral functions

TL;DR

This paper establishes a rigorous mathematical framework for bounding matrix-valued spectral functions in lattice field theory, leveraging moment problems and implementing numerical bounds using correlation data.

Contribution

It introduces a novel connection between moment problems and spectral function bounds, applying mathematical results to practical lattice QCD data analysis.

Findings

First numerical implementation of rigorous spectral bounds

Connection between moment problems and Rayleigh-Ritz method

Discussion of finite precision limitations

Abstract

Numerical analytic continuation arises frequently in lattice field theory, particularly in spectroscopy problems. This work shows the equivalence of common spectroscopic problems to certain classes of moment problems that have been studied thoroughly in the mathematical literature. Mathematical results due to Kovalishina enable rigorous bounds on smeared matrix-valued spectral functions, which are implemented numerically for the first time. The required input is a positive-definite matrix of Euclidean-time correlation functions; such matrices are routinely computed in variational spectrum studies using lattice quantum chromodynamics. This work connects the moment-problem perspective to recent developments using the Rayleigh--Ritz method and Lanczos algorithm. Possible limitations due to finite numerical precision are discussed.