Quantizations from reproducing kernel spaces

TL;DR

This paper investigates reproducing kernel Hilbert subspaces derived from complex Hermite polynomials, exploring their properties, associated coherent states, and implications for quantization and spectral analysis.

Contribution

It introduces a new class of reproducing kernel Hilbert spaces based on complex Hermite polynomials and analyzes their coherent states and quantization properties.

Findings

Existence of harmonic oscillator spectrum without canonical commutation rules

Analysis of $s$-dependent quantizations and their equivalences

Insights into CS quantization in non-standard Hilbert spaces

Abstract

The purpose of this work is to explore the existence and properties of reproducing kernel Hilbert subspaces of $L^2(\C, \, d^2z/\pi)$ based on subsets of complex Hermite polynomials. The resulting coherent states (CS) form a family depending on a nonnegative parameter $s$. We examine some interesting issues, mainly related to CS quantization, like the existence of the usual harmonic oscillator spectrum despite the absence of canonical commutation rules. The question of mathematical and physical equivalences between the $s$-dependent quantizations is also considered.