Holomorphic Hermite functions in Segal-Bargmann spaces

TL;DR

This paper investigates holomorphic Hermite functions in Segal-Bargmann spaces, establishing conditions for their generators to satisfy canonical commutation relations and exploring their orthogonality, eigenvalues, and Rodrigues formulas.

Contribution

It provides necessary and sufficient conditions for entire functions to generate holomorphic Hermite functions with canonical commutation relations in Segal-Bargmann spaces.

Findings

Identified parameter conditions for generators satisfying CCR

Established orthogonality and eigenvalue properties

Derived Rodrigues formulas for these functions

Abstract

We study systems of holomorphic Hermite functions in the Segal-Bargmann spaces, which are Hilbert spaces of entire functions on the complex Euclidean space, and are determined by the Bargmann-type integral transform on the real Euclidean space. We prove that for any positive parameter which is strictly smaller than the minimum eigenvalue of the positive Hermitian matrix associated with the transform, one can find a generator of holomorphic Hermite functions whose anihilation and creation operators satisfy canonical commutation relations. In other words, we find the necessary and sufficient conditions so that some kinds of entire functions can be such generators. Moreover, we also study the complete orthogonality, the eigenvalue problems and the Rodrigues formulas.