Quantum Hermite functions and Fourier Transform of operators

TL;DR

We introduce operator Hermite functions on the Hilbert-Schmidt space $\mathcal{S}_2$ by transporting a dilated Hermite basis on $L^2(\mathbb{R}^{2n})$ through the Weyl transform, yielding $S_\mu^\lambda$ and their Weyl-pulled partners $\Psi_\mu^\lambda$. A Gauss-Bargmann framework defines a unitary Fourier transform on $\mathcal{S}_2$ with $\widehat{S_\mu^\lambda}=(-i)^{|\mu|}S_\mu^\lambda$, mirroring the classical Hermite-Fourier relation; the construction is tied to twisted Fock spaces via the twisted Bargmann transform and a deformation map $D_\lambda$. The paper develops Sobolev and Schwartz operator spaces, analyzes operator multipliers, and studies radial (Laguerre-type) operators, including a Hardy-type uncertainty principle for operator transforms. Collectively, these results unify Hermite analysis, noncommutative harmonic analysis on the Heisenberg group, and operator-valued Fourier theory, providing a versatile framework for spectral multipliers and deformation-quantization in $\mathcal{S}_2$.

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.

Theorems & Definitions (38)

• Theorem 1.1
• Proposition 2.1
• Proposition 2.2
• Remark 2.3
• Lemma 2.4
• proof
• Proposition 3.1
• proof
• Remark 3.2
• Remark 3.3