Study of a Class of Regularizations of 1/|x| using Gaussian Integrals

TL;DR

This paper investigates a class of regularized functions approximating 1/|x| using Gaussian integrals, analyzing their properties, inequalities, and related polynomials, with applications in atomic physics under strong magnetic fields.

Contribution

It introduces new regularizations of 1/|x| via Gaussian integrals, explores their properties, inequalities, differential equations, and polynomial relations, and discusses their mathematical and physical implications.

Findings

Derived tight bounds for the regularized functions.

Established differential equations and recursion relations.

Showed convexity of the inverse functions and triangle inequality analogues.

Abstract

This paper presents a comprehensive study of a class of functions which approximate 1/|x| for large x but which are finite at the origin. These functions arise naturally in the study of atoms in strong magnetic fields where the so-called "Landau states" give rise to Gaussian integrals. Generalizations in which e^{-x^2} is replaced by e^{-x^p} are also considered and approximate x^{1-p} for large x. The limiting behavior and monotonicity properties of these functions are discussed in terms of parameters which arise in the approximations as well as x. Several classes of inequalities, some of which provide tight bounds, are established. Some differential equations and recursion relations satisfied by these functions are given. The recursion relations give rise to two classes of polynomials, one of which is related to confluent hypergeometric functions. Finally, it is shown that the inverse of the approximating function is convex and this implies an analogue of the triangle inequality. Some comments are made about possible extensions and several open questions are raised.