On Evaluation of Riesz Constants for Systems of Shifted Gaussians

TL;DR

This paper analyzes the Riesz constants for systems of shifted Gaussian and Lorenz functions, deriving explicit formulas, limit behaviors, and a key monotonicity property with applications to coherent states and atomic spectra.

Contribution

It provides explicit calculations of Riesz constants and nod functions for shifted Gaussian and Lorenz systems, including limit behaviors and a novel monotonicity property.

Findings

Explicit Riesz constants for Gaussian and Lorenz systems

Limit behavior of Riesz constants as parameters vary

A sharp monotonicity property of Jacobi theta-functions

Abstract

In this paper we study single-parametric systems of integer shifts of Gauss and Lorenz functions. In case of Cauchy--Lorenz system we explicitly calculate nod functions and prove that it tends to sinc function in limit. For both Gauss and Cauchy--Lorenz systems and corresponding nod functions we explicitly calculate Riesz constants via trigonometric, hyperbolic and Jacobi theta-functions, also limit behavior of this values is found depending on parameters. A special result is a sharp monotonicity property proved for a special ratio of Jacobi theta-functions which is important in many areas. Brief discussion of numerical methods is included with additional references. Different applications are outlined to coherent states and atomic spectra. Some unsolved problems and comments are added.