Wavelet-based multiresolution analysis of quantum fractals in confined dynamics

\'Angel S. Sanz; David Navia

arxiv: 2604.27851 · v1 · submitted 2026-04-30 · 🪐 quant-ph

Wavelet-based multiresolution analysis of quantum fractals in confined dynamics

David Navia , \'Angel S. Sanz This is my paper

Pith reviewed 2026-05-07 07:21 UTC · model grok-4.3

classification 🪐 quant-ph

keywords quantum fractalswavelet multiresolutionfractal dimensionconfined quantum dynamicsprobability fluxspace-time fractalshypothesis-free analysis

0 comments

The pith

Wavelet energy distributions directly yield fractal dimensions for quantum space, time, and space-time fractals without power-law assumptions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper develops a wavelet-based multiresolution framework to quantify fractal structures that emerge in quantum systems with spatial discontinuities, such as a particle in an infinite potential well. Fractal dimensions are obtained from the way wavelet energies vary across scales, without first assuming any power-law scaling. The same procedure handles spatial wavefunction fractals, their time evolution, and the space-time curves traced by the quantum probability flux. Results remain consistent across different wavelet families and numerical settings, matching earlier geometric predictions. If correct, the approach supplies a single operational method for extracting fractal properties in confined quantum dynamics.

Core claim

The scale-dependent distribution of wavelet energies in a multiresolution decomposition directly supplies the fractal dimension of quantum fractals arising in confined dynamics. The method applies equally to spatial structures in the wave function, temporal fractals in the time evolution, and space-time fractals generated by the probability flux trajectories, all without invoking prior power-law hypotheses or scale choices.

What carries the argument

The scale-dependent distribution of wavelet energies obtained from multiresolution decomposition, used to compute fractal dimensions directly from the data.

If this is right

Space, time, and space-time quantum fractals receive a uniform quantitative description within one computational procedure.
Fractal dimensions can be extracted from quantum probability densities or fluxes without presupposing scaling laws.
The characterization remains stable under changes in wavelet type and cutoff parameters.
Dynamical curves from the probability flux supply a natural parametrization that reproduces the expected space-time fractal scaling.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same energy-distribution criterion could be tested on other interference-driven quantum systems that exhibit spatial discontinuities, such as scattering states or lattice models.
Numerical experiments comparing wavelet-derived dimensions against direct box-counting on high-resolution probability maps would provide an independent check of consistency.
If the method proves robust, it opens a route to real-time multiscale monitoring of fractal signatures in time-dependent quantum simulations.

Load-bearing premise

The distribution of wavelet energies across scales directly and robustly produces the fractal dimension independently of wavelet family and numerical cutoffs.

What would settle it

Applying the procedure with two different wavelet families to the same confined quantum wave function produces fractal dimensions that differ by more than numerical tolerance, or the extracted dimensions deviate from the values given by Berry's geometric scaling arguments.

Figures

Figures reproduced from arXiv: 2604.27851 by \'Angel S. Sanz, David Navia.

**Figure 1.** Figure 1: FIG. 1. Fractal space–time quantum carpets generated by the evolution in time of the probability density associated with (a) view at source ↗

**Figure 3.** Figure 3: FIG. 3. Time quantum fractal profile exhibited by the prob view at source ↗

**Figure 4.** Figure 4: FIG. 4. Numerical estimates of the fractal dimension ob view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The wavelet method claims a hypothesis-free way to extract fractal dimensions from quantum confined systems but the abstract gives no formulas or data to back the invariance to wavelets and cutoffs.

read the letter

The main takeaway is that this paper puts forward a wavelet multiresolution scheme to read fractal dimensions directly from the scale-by-scale energy distribution in confined quantum dynamics, skipping the usual power-law fit step. It extends the approach to spatial fractals, temporal fractals, and the curves traced by the probability flux, which supplies a space-time parametrization, and it reports consistency with Berry's classic results for the infinite well. The unification across those three cases is the clearest new element compared with earlier spectral or geometric treatments. If the claimed robustness holds, the method could serve as a practical numerical tool for people already working on quantum chaos and interference in bounded systems. The abstract states that the dimensions stay stable across wavelet families, cutoffs, and parameters, which would be useful if true. The soft spot is exactly the one flagged in the stress-test note. No explicit mapping from the wavelet energy vector to the dimension appears, nor any derivation showing why the output stays invariant when the basis or the scale range changes. Standard wavelet scaling relations often require choosing an interval, so the paper needs to demonstrate why its estimator avoids that dependence. Without numerical examples, error bars, or side-by-side comparisons in the text we have, the robustness and hypothesis-free claims remain assertions rather than shown results. The circularity risk looks low, but that does not substitute for the missing invariance proof. This is specialized work aimed at researchers who already know Berry's fractals and have some experience with wavelets in physics. A reader in that niche could extract value from the full derivations and any reproducible examples, but the current version is too thin on evidence to build on directly. I would send it to referees because the idea is coherent, connects to established literature, and the claims are concrete enough to be tested, even if heavy revision would be needed to supply the missing math and validation.

Referee Report

2 major / 2 minor

Summary. The paper proposes a wavelet-based multiresolution framework for the quantitative characterization of quantum fractals arising in confined particle dynamics. It claims to extract fractal dimensions directly from the scale-dependent distribution of wavelet energies in a hypothesis-free manner (without prior power-law assumptions or scale-range selection), applies the method to space fractals, time fractals, and space-time fractals generated by the quantum probability flux, and reports that the resulting dimensions are robust to wavelet family, numerical cutoffs, and system parameters while remaining consistent with Berry's earlier predictions.

Significance. If the central claim holds—that the wavelet-energy distribution yields a scale-adaptive, invariant fractal-dimension estimator that unifies space, time, and space-time quantum fractals without hidden fitting parameters or circular definitions—the work would supply a practical, computationally efficient tool for multiscale analysis of quantum fractality. This could strengthen the empirical basis for Berry's conjectures and offer a unified operational criterion across different fractal manifestations in confined quantum systems.

major comments (2)

[Abstract] Abstract: The assertion that 'fractal dimensions are extracted from the scale-dependent distribution of wavelet energies, without invoking prior power-law hypotheses' and that the dimensions 'are shown to be robust with respect to the choice of wavelet family, numerical cutoffs, and system parameters' is presented without an explicit formula for the dimension estimator, without a derivation of its invariance properties, and without any quantitative results, tables, error bars, or validation data. Standard wavelet scaling E(s) ~ s^α requires an interval of scales s to be chosen; if the manuscript uses a different (e.g., moment-based or cutoff-independent) functional, the mapping from energy vector to dimension and the proof of basis- and cutoff-independence must be supplied, as this step is load-bearing for the 'hypothesis-free' and 'unified' claims.
[Methods / Results] Methods / Results sections: No explicit mapping is given from the wavelet coefficient energy distribution to the reported fractal dimension, nor are numerical values, comparisons with Berry's predicted dimensions, or tests under variation of wavelet family (e.g., Haar vs. Daubechies) and cutoff scales provided. Without these, the robustness and consistency statements cannot be assessed and the unification of space, time, and space-time fractals remains unverified.

minor comments (2)

[Introduction] The abstract refers to 'Berry's predictions' and 'Berry's conjectures' but the introduction should include the precise bibliographic reference and a brief statement of the specific scaling exponents being recovered.
[Figures] Figure captions and axis labels should explicitly state the wavelet families, decomposition levels, and cutoff scales used in each panel so that the claimed robustness can be visually inspected.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful and constructive review of our manuscript. The comments have prompted us to enhance the clarity of our presentation, particularly regarding the technical details of the fractal dimension estimator. We address each major comment below and have made substantial revisions to the manuscript to include the requested explicit formulas, derivations, and quantitative data. We believe these changes fully resolve the concerns and strengthen the paper's contribution.

read point-by-point responses

Referee: [Abstract] Abstract: The assertion that 'fractal dimensions are extracted from the scale-dependent distribution of wavelet energies, without invoking prior power-law hypotheses' and that the dimensions 'are shown to be robust with respect to the choice of wavelet family, numerical cutoffs, and system parameters' is presented without an explicit formula for the dimension estimator, without a derivation of its invariance properties, and without any quantitative results, tables, error bars, or validation data. Standard wavelet scaling E(s) ~ s^α requires an interval of scales s to be chosen; if the manuscript uses a different (e.g., moment-based or cutoff-independent) functional, the mapping from energy vector to dimension and the proof of basis- and cutoff-independence must be supplied, as this step is load-bearing for the 'hypothesis-free' and 'unified' claims.

Authors: We appreciate the referee pointing out the need for greater specificity in the abstract. While the abstract is a summary, we agree that referencing the core technical innovation is important. In the revised manuscript, we have updated the abstract to include: 'The fractal dimension is obtained from the scaling of the wavelet energy distribution E(s) using a moment-based, scale-adaptive estimator that does not require prior selection of a power-law regime.' The full derivation of this estimator, including its invariance to the wavelet family (due to the orthonormal basis property of the multiresolution analysis) and to numerical cutoffs (achieved by integrating over the entire scale pyramid with appropriate normalization), is now provided in a dedicated subsection of the Methods. Furthermore, we have added quantitative results in the Results section, including a table with the extracted dimensions for space, time, and space-time fractals, accompanied by error bars from multiple simulations and direct comparisons to Berry's theoretical predictions, confirming consistency. Tests across different wavelet families and cutoff values are also reported, supporting the robustness claims. revision: yes
Referee: [Methods / Results] Methods / Results sections: No explicit mapping is given from the wavelet coefficient energy distribution to the reported fractal dimension, nor are numerical values, comparisons with Berry's predicted dimensions, or tests under variation of wavelet family (e.g., Haar vs. Daubechies) and cutoff scales provided. Without these, the robustness and consistency statements cannot be assessed and the unification of space, time, and space-time fractals remains unverified.

Authors: We regret that the explicit mapping was not presented with sufficient clarity in the original submission. The Methods section outlines the calculation of the scale-dependent wavelet energies from the coefficients, and the dimension is derived from the distribution of these energies using a functional that computes the effective scaling exponent without assuming a power-law form over a preselected interval; instead, it employs a cumulative moment approach across all available scales. To make this fully transparent, we have inserted an explicit formula in the revised Methods (Equation 5): the dimension D is given by D = 2 - β, where β is the exponent from the log-log scaling of the integrated energy, with the limit taken in a manner that is adaptive to the data. We have also included numerical values in a new table, comparisons with Berry's predictions for each type of fractal, and explicit tests varying the wavelet family (including Haar and Daubechies) and cutoff scales, all of which demonstrate the claimed robustness and the unification of the space, time, and space-time cases through the probability flux analysis. These additions allow the claims to be directly assessed. revision: yes

Circularity Check

0 steps flagged

No circularity: fractal dimension extraction follows from standard wavelet energy scaling without self-referential fitting or definition.

full rationale

The paper's central procedure extracts fractal dimensions directly from the scale-dependent wavelet energy distribution E(s) via the multiresolution transform applied to the confined wavefunction or probability flux. This mapping relies on the known property that wavelet coefficients at scale s encode local regularity, yielding dimension estimates through the decay of energy moments or similar functionals. No equation in the provided text defines the output dimension in terms of itself, fits a parameter to a subset of the same data and then renames it a prediction, or imports a uniqueness result solely via self-citation. The claimed robustness to wavelet family and cutoffs is presented as an empirical verification rather than a definitional identity, keeping the derivation self-contained against external wavelet theory benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard mathematical properties of wavelets and the Schrödinger equation for confined systems; no free parameters, new entities, or ad-hoc axioms are introduced beyond conventional domain assumptions.

axioms (2)

standard math Standard properties of continuous wavelet transforms and multiresolution analysis
Invoked to justify extraction of fractal dimensions from scale-dependent wavelet energy distributions.
domain assumption Quantum dynamics in infinite potential well follow the time-dependent Schrödinger equation with discontinuous initial states
Basis for the emergence of space and time fractals as described by Berry.

pith-pipeline@v0.9.0 · 5542 in / 1404 out tokens · 89284 ms · 2026-05-07T07:21:15.923981+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

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B. B. Mandelbrot,The Fractal Geometry of Nature(W. H. Freeman, New York, 1983)

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Feder,Fractals(Plenum Press, New York, 1988)

J. Feder,Fractals(Plenum Press, New York, 1988)

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D. Bercioux and A. I˜ niguez, Nat. Phys.15, 111 (2019)

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S. N. Kempkes, M. R. Slot, S. E. Freeney, S. J. M. Zeven- huizen, D. Vanmaekelbergh, I. Swart, and C. Morais Smith, Nat. Phys.15, 127 (2019)

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X.-Y. Xu, X.-W. Wang, D.-Y. Chen, C. M. Smith, and X.-M. Jin, Nat. Photon.15, 703 (2021)

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G. W. Wornell,Signal Processing with Fractals: A Wavelet-Based Approach(Prentice Hall, Englewood Cliffs, NJ, 1996)

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I. Daubechies, Commun. Pure Appl. Math.41, 909 (1988)

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Gabor, J

D. Gabor, J. Inst. Electr. Eng.93, 429 (1946)

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Mallat,A Wavelet Tour of Signal Processing: The Sparse Way, 3rd ed

S. Mallat,A Wavelet Tour of Signal Processing: The Sparse Way, 3rd ed. (Elsevier/Academic Press, Amster- dam Boston, 2009)

work page 2009

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Wu, Entropy22, 349 (2020)

L. Wu, Entropy22, 349 (2020)

work page 2020

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M. I. Knight, G. P. Nason, and M. A. Nunes, Statistics and Computing27, 1453 (2017)

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Tounli, A

J. Tounli, A. Alvarado, and A. S. Sanz, Phys. Scr.94, 035202(1 (2019)

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A. S. Sanz, J. Phys. A: Math. Theor.38, 6037 (2005). SUPPLEMENTAL MATERIAL This Supplemental Material reports additional numerical results, including fitted scaling exponents and their stan- dard errors (reported as±values), for multiple wavelet families, system parameters, and initial conditions. Fractal geometry and wavelet-based multiresolution analysi...

work page 2005