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Moment Problems and Spectral Functions

Ryan Abbott, William Jay, Patrick Oare

TL;DR

The paper addresses reconstructing smeared spectral functions from Euclidean lattice data using causality-based analytic frameworks. It surveys Nevanlinna-Pick interpolation and Hamburger moment problems, showing how positivity constraints on the Stieltjes transform $G(z)$ via the Pick and Hankel matrices yield rigorous bounds on spectral densities. A key result is a simple, general convexity proof for the space of causal data, clarifying the geometry of feasible interpolation data and its extremal cases. The outlook envisions practical, uncertainty-quantified bounds for lattice QFT calculations, potentially enhanced by matrix correlators and multi-operator analyses to improve precision and enable new spectral reconstructions.

Abstract

Nevanlinna-Pick interpolation and moment problems use the analytic structures provided by causality in order to provide rigorous bounds on smeared spectral functions. This proceedings discusses Nevanlinna-Pick interpolation and moment problems and reviews some useful results, including a simple proof that the space of causal data in Nevanlinna--Pick interpolation is convex.

Moment Problems and Spectral Functions

TL;DR

The paper addresses reconstructing smeared spectral functions from Euclidean lattice data using causality-based analytic frameworks. It surveys Nevanlinna-Pick interpolation and Hamburger moment problems, showing how positivity constraints on the Stieltjes transform via the Pick and Hankel matrices yield rigorous bounds on spectral densities. A key result is a simple, general convexity proof for the space of causal data, clarifying the geometry of feasible interpolation data and its extremal cases. The outlook envisions practical, uncertainty-quantified bounds for lattice QFT calculations, potentially enhanced by matrix correlators and multi-operator analyses to improve precision and enable new spectral reconstructions.

Abstract

Nevanlinna-Pick interpolation and moment problems use the analytic structures provided by causality in order to provide rigorous bounds on smeared spectral functions. This proceedings discusses Nevanlinna-Pick interpolation and moment problems and reviews some useful results, including a simple proof that the space of causal data in Nevanlinna--Pick interpolation is convex.
Paper Structure (4 sections, 3 theorems, 16 equations)

This paper contains 4 sections, 3 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. Review of Analyticity Methods
  3. Geometry of Causal Subsets
  4. Outlook

Key Result

Theorem 1

Let $z_1, \dots, z_n, w_1, \dots, w_n \in \mathbb{D}$. Then there exists $f : \mathbb{D} \to \mathbb{D}$ such that $f(z_i) = w_i$ for all $i \in \{1, \dots, n\}$ if and only if the Pick matrix is positive-definite.

Theorems & Definitions (4)

  • Theorem 1: Pick Pick1915
  • Theorem 2: Hamburger Hamburger:1920
  • Lemma 1
  • proof