Quantum Hermite functions and Fourier Transform of operators

TL;DR

This paper develops operator analogues of Hermite functions to create an orthonormal basis for Hilbert-Schmidt operators, enabling the definition and analysis of a Fourier transform on this operator space.

Contribution

It introduces a novel basis of Hermite function analogues for operators, facilitating the extension of Fourier analysis to the space of Hilbert-Schmidt operators.

Findings

Constructed operator Hermite functions forming an orthonormal basis.

Defined Fourier transform on the space of Hilbert-Schmidt operators.

Analyzed basic properties of this operator Fourier transform.

Abstract

We construct operator analogues of Hermite functions which form an orthonormal basis for the Hilbert space $ \mathcal{S}_2$ of Hilbert-Schmidt operators on $ L^2(\R^n).$ We use this orthonormal basis to define Fourier transform on $ \mathcal{S}_2 $ and study some of its basic properties.