Noncommutative reading of the complex plane through Delone sequences

Abstract

The Berezin-Klauder-Toeplitz ("anti-Wick") quantization or "non-commutative reading" of the complex plane, viewed as the phase space of a particle moving on the line, is derived from the resolution of the unity provided by the standard (or gaussian) coherent states. The construction properties of these states and their attractive properties are essentially based on the energy spectrum of the harmonic oscillator, that is on the natural numbers. This work is an attempt for following the same path by considering sequences of non-negative numbers which are not "too far" from the natural numbers. In particular, we examine the consequences of such perturbations on the non-commutative reading of the complex plane in terms of its probabilistic, functional, and localization aspects.

On a Generalization of the Binomial Distribution and Its Poisson-like Limit

“…The proof is obtained by recurrence. It is obviously true for n = 1, since p 1 (η) = (1 − η) obeys (6). Suppose that the assertion is true for all k ≤ n − 1, i.e.…”

“…. which makes compatible (6) and (37) is the power sequence a n = q n , where q = a 1 , and for which the corresponding sequence X is precisely X q .…”

“…An important example of sequences in the class S are the Delone sequences [6], which are infinite strictly increasing sequences of nonnegative real numbers…”

Quantizations from reproducing kernel spaces

“…In this section, we analyze in more details, like in [19], the localization properties of the "almost" canonical operators (55) and (56). Actually, it is enough to study the quantum position operator Q ǫ s .…”

“…Coherent state quantization: a short review. The resolution of the identity (33) allows to implement a coherent state or frame quantization [14,15,16,17,18,19,12] of the set of parameters Σ by associating to a function Σ ∋ α → f (α) that satisfies appropriate conditions the following operator in H:…”