Non-commutative reading of the complex plane through Delone sequences

TL;DR

This paper explores a generalized non-commutative framework for the complex plane using sequences close to natural numbers, extending the Berezin-Klauder-Toeplitz quantization by analyzing perturbations of the harmonic oscillator spectrum.

Contribution

It introduces a novel approach to non-commutative geometry on the complex plane using Delone sequences, expanding the standard coherent states framework beyond natural numbers.

Findings

Perturbations of the harmonic oscillator spectrum affect localization properties.

The probabilistic and functional aspects of the non-commutative reading are influenced by the chosen sequences.

The approach provides new insights into non-commutative geometry and quantum state localization.

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.