Reflection positivity on real intervals

TL;DR

This paper develops integral representations for negative and reflection negative functions on intervals, generalizing classical results and providing new characterizations of reflection negativity in the context of positive definiteness.

Contribution

It introduces a Lévy–Khintchine formula for negative and reflection negative functions on arbitrary intervals, extending classical results and characterizing germs of reflection negative functions.

Findings

Lévy–Khintchine formula for negative definite functions on intervals

Characterization of reflection negative functions on R

Generalization of classical Bernstein and Horn results

Abstract

We study functions f : (a,b) ---> R on open intervals in R with respect to various kinds of positive and negative definiteness conditions. We say that f is positive definite if the kernel f((x + y)/2) is positive definite. We call f negative definite if, for every h > 0, the function e^{-hf} is positive definite. Our first main result is a L\'evy--Khintchine formula (an integral representation) for negative definite functions on arbitrary intervals. For (a,b) = (0,\infty) it generalizes classical results by Bernstein and Horn. On a symmetric interval (-a,a), we call f reflection positive if it is positive definite and, in addition, the kernel f((x - y)/2) is positive definite. We likewise define reflection negative functions and obtain a L\'evy--Khintchine formula for reflection negative functions on all of R. Finally, we obtain a characterization of germs of reflection negative functions on 0-neighborhoods in R.