Review Reports

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

In the paper, the quantum properties of atoms in fractal spaces have been explored, both as a theoretical extension of integer-dimensional Euclidean spaces and as an experimentally realizable scenario. The threshold of fractality at which Ehrenfest atomic instability has been identified, where the Schrödinger equation for a single electron orbiting an atom becomes scale-free. Rydberg states of stable atoms have been studied using the Wentzel-Kramers-Brillouin approximation, with a proposed extension for the Langer modification in general fractal dimensionalities. It has been shown that atoms near instability can expand dramatically even at low-number excited states, suggesting potential for strong entanglement and long-range many-body interactions.

However, in Eq. (2), ordinary derivatives have been used on fractal spaces. As is well known, ordinary derivatives cannot be directly applied to fractals due to their non-integer measure and complex geometry. I suggest that the authors compare their approach with established works in fractal analysis, such as:

Kigami, J.: Analysis on Fractals, vol. 143, Cambridge University Press, Cambridge (2001).

Golmankhaneh, A.K., Pellis, S., Zingales, M.: Fractal Schrödinger Equation: Implications for Fractal Sets, J. Phys. A: Math. Theor. 57(18), 185201 (2024).

Freiberg, U., Zahle, M.: Harmonic Calculus on Fractals – A Measure Geometric Approach I, Potential Anal. 16(3), 265–277 (2002).

By considering these references, a more rigorous formulation using fractal derivatives could be implemented, strengthening the theoretical foundation and ensuring consistency with fractal measures.

Author Response

We thank the referee for this important remark. We agree that on genuinely fractal sets a fully rigorous formulation typically requires an intrinsic Laplacian/derivative (e.g. via Dirichlet forms, harmonic calculus, or measure–geometric constructions), and that ordinary derivatives are not directly applicable in that setting. Our intent in Eq. (2) is not to claim a complete intrinsic theory on an arbitrary fractal geometry, but to work within an effective fractional-dimensional continuum model (dimensionally continued Euclidean description) for media with fractal-like scaling. We have now revised the end of Section 2 (where we introduce the mathematics behind our work) to make this assumption explicit, to delineate the scope of our results, and to cite representative rigorous approaches on fractals, including Kigami, Freiberg & Zahle, and the recent discussion by Golmankhaneh et al. We view extending our analysis to a fully intrinsic fractal Laplacian framework as an interesting direction for future work. We hope for your understanding.

Reviewer 2 Report

Comments and Suggestions for Authors

The paper presents studies of the physical quantum properties of atoms in fractal spaces. The stability of atoms for various fractalities is investigated. Scaled energies and sizes of Rydberg states of stable atoms are presented for different fractalities. The research is interesting and has potential applications to quantum computing. I recommend publications after the authors will have addressed the following questions/comments:

line 133:
"Ueff(r) in Eq. (7) is monotonic for D>=4 and there is no minimum"
I agree that the potential does not have a minimum, but it has a maximum in this case, and therefore is *not* monotonic.

lines 149-155
The authors conclude that the QM atom in Euclidean is unstable for D=4 only whereas classically it is unstable for D>=4. Why the difference? 

Line 169. I don't understand this statement: (Dv,Ds)=(1.79,1.48) satisfies neither (11), nor (12).

Why the centrifugal term is missing in eq. (14)? Is the present study limited to spherically-symmetric states? If so, the authors should state this explicitly.

Line 202: the choice of the Maslov index should be explained.

Section 5 looks more like a Conclusion rather than Discussion.

 

Author Response

See attached file

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

I recommend the revised version of the manuscript for publication, as it satisfactorily addresses the reviewers’ comments.