Wavelet-based multiresolution analysis of quantum fractals in confined dynamics

TL;DR

This paper introduces a wavelet-based multiresolution method for directly quantifying quantum fractals in confined systems, avoiding prior assumptions and providing robust, unified analysis of space, time, and space-time fractal structures.

Contribution

It presents a novel, assumption-free wavelet framework for quantifying quantum fractality across space, time, and space-time, improving upon previous spectral and geometric methods.

Findings

The method accurately extracts fractal dimensions from wavelet energy distributions.

Fractal dimensions are consistent across different wavelet families and system parameters.

The approach validates Berry's predictions for space--time quantum fractals.

Abstract

Fractal structures naturally emerge in quantum systems whose initial states exhibit spatial discontinuities, a phenomenon first identified by Berry in the paradigmatic case of a particle confined in an infinite potential well. While previous analyses of quantum fractals have mainly relied on spectral decompositions and geometric scaling arguments, their quantitative characterization often depends on scale choices and truncation effects. Here we present a wavelet-based multiresolution framework that enables a direct and assumption-free quantification of quantum fractality. Fractal dimensions are extracted from the scale-dependent distribution of wavelet energies, without invoking prior power-law hypotheses. The method is applied to space and time quantum fractals arising in confined dynamics, as well as to dynamical curves generated by the associated quantum probability flux. These flux-driven trajectories provide a natural space--time parametrization of the underlying fractal structure and yield scaling properties fully consistent with Berry's predictions for space--time fractals. The resulting fractal dimensions are shown to be robust with respect to the choice of wavelet family, numerical cutoffs, and system parameters. Beyond validating earlier conjectures, the present framework offers a unified and computationally efficient tool for the multiscale analysis of quantum fractality in confined and interference-driven quantum dynamics. That is, it provides an operational, scale-adaptive criterion that unifies the characterization of space, time, and space--time quantum fractals within a single, hypothesis-free approach.