Inverse of the flow and moments of the free Jacobi process associated with a single projection

TL;DR

This paper investigates the inverse flow and moments of the free Jacobi process linked to a single projection, revealing connections to L"owner equations, Aleksandrov-Clark measures, and spectral properties of associated operators.

Contribution

It establishes the univalence of the flow via L"owner equations and provides explicit expressions for moments of the free Jacobi process, connecting spectral and measure-theoretic aspects.

Findings

Flow solves a radial L"owner equation and is univalent.

Inverse flow defines the Aleksandrov-Clark measure at z=1.

z=1 belongs only to the discrete spectrum of the related unitary operator.

Abstract

This paper is a companion to a series of papers devoted to the study of the spectral distribution of the free Jacobi process associated with a single projection. Actually, we notice that the flow solves a radial L\"owner equation and as such, the general theory of L\"owner equations implies that it is univalent in some connected region in the open unit disc. We also prove that its inverse defines the Aleksandrov-Clark measure at $z=1$ of some Herglotz function which is absolutely-continuous with an essentially bounded density. As a by-product, we deduce that $z=1$ belongs only to the discrete spectrum of the unitary operator whose spectral dynamics are governed by the flow. Moreover, we use a previous result due to the first author in order to derive an explicit, yet complicated, expression of the moments of both the unitary and the free Jacobi processes. The paper is closed with some remarks on the boundary behavior of the flow's inverse.