Quantizations from reproducing kernel spaces

Abstract

The purpose of this work is to explore the existence and properties of reproducing kernel Hilbert subspaces of $L^2(\C, \, d^2z/π)$ 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.

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• Non linear pseudo-bosons
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• $(p,q)$-complex Itô--Hermite polynomials
• On the range of weighted planar Cauchy transform
• Wave equation for 2D Landau Hamiltonian