Complex and real Hermite polynomials and related quantizations

Abstract

It is known that the anti-Wick (or standard coherent state) quantization of the complex plane produces both canonical commutation rule and quantum spectrum of the harmonic oscillator (up to the addition of a constant). In the present work, we show that these two issues are not necessarily coupled: there exists a family of separable Hilbert spaces, including the usual Fock-Bargmann space, and in each element in this family there exists an overcomplete set of unit-norm states resolving the unity. With the exception of the Fock-Bargmann case, they all produce non-canonical commutation relation whereas the quantum spectrum of the harmonic oscillator remains the same up to the addition of a constant. The statistical aspects of these non-equivalent coherent states quantizations are investigated. We also explore the localization aspects in the real line yielded by similar quantizations based on real Hermite polynomials.

Very weak solutions of wave equation for Landau Hamiltonian with irregular electromagnetic field

“…35]), the relations to complex Hermite polynomials [18], to quantization questions [3,4,9], to timefrequency analysis, partial differential equations and planar point processes, see [1,16,17], respectively.…”

Pisot -coherent states quantization of the harmonic oscillator

“…The change of the frame family C produces another quantization, another point of view, possibly equivalent to the previous one, possibly not. The present study lies in the continuity of a series of such explorations, which were already present in the first works by Klauder at the beginning of the sixties of the past century (see for instance [16,33] and references therein), pursued by Berezin [15] in his famous paper of 1975, and more recently extended to various measure sets (see for instance [34][35][36][37][38][39]). …”

Wave Equation for Operators with Discrete Spectrum and Irregular Propagation Speed

“…This basis appears also in many related subjects, such as complex Hermite polynomials [34], in quantization [3,5,9], time-frequency analysis [1], partial differential equations [27], planar point processes [32], as well as in the Feynman-Schwinger displacement operator [46]. Their perturbations have been investigated in [35,44], and the asymptotic behaviour of the eigenvalues was analysed in [37,40,41,[48][49][50]57].…”