Pisot q-Coherent states quantization of the harmonic oscillator

TL;DR

This paper explores a q-dependent coherent state quantization of the harmonic oscillator, focusing on cases where q relates to quadratic Pisot numbers, revealing unique phase space and statistical properties.

Contribution

It introduces a novel q-deformed quantization framework for the harmonic oscillator using Pisot number-based coherent states, analyzing their phase space and dynamical features.

Findings

Coherent states form an overcomplete set resolving the identity.

Phase space localization and semi-classical trajectories are characterized.

Unique statistical and dynamical properties emerge for Pisot number q-values.

Abstract

We revisit the quantized version of the harmonic oscillator obtained through a q-dependent family of coherent states. For each q, 0< q < 1, these normalized states form an overcomplete set that resolves the unity with respect to an explicit measure. We restrict our study to the case in which 1/q is a quadratic unit Pisot number: the q-deformed integers form Fibonacci-like sequences of integers. We then examine the main characteristics of the corresponding quantum oscillator: localization in the configuration and in the phase spaces, angle operator, probability distributions and related statistical features, time evolution and semi-classical phase space trajectories.