How the $\mu$-deformed Segal-Bargmann space gets two measures

TL;DR

This paper investigates the measures defining the $mbda$-deformed Segal-Bargmann space, showing they are solutions to a differential system involving Macdonald functions, and clarifies their unique and natural roles in the theory.

Contribution

It demonstrates that the measures' densities are solutions to a differential system, providing a new perspective on their origin and uniqueness in the $mbda$-deformed Segal-Bargmann space.

Findings

Densities are solutions to a differential system involving Macdonald functions.

Identifies a 'spurious' solution leading to a trivial space.

Provides a new elementary proof of the measures' naturalness and uniqueness.

Abstract

This note explains how the two measures used to define the $\mu$-deformed Segal-Bargmann space are natural and essentially unique structures. As is well known, the density with respect to Lebesgue measure of each of these measures involves a Macdonald function. Our primary result is that these densities are the solution of a system of ordinary differential equations which is naturally associated with this theory. We then solve this system and find the known densities as well as a "spurious" solution which only leads to a trivial holomorphic Hilbert space. This explains how the Macdonald functions arise in this theory. Also we comment on why it is plausible that only one measure will not work. We follow Bargmann's approach by imposing a condition sufficient for the $\mu$-deformed creation and annihilation operators to be adjoints of each other. While this note uses elementary techniques, it reveals in a new way basic aspects of the structure of the $\mu$-deformed Segal-Bargmann space.