General features and contact modeling of $N$-body isolated resonances near threshold

The author considers some general features of N-body resonances in D spatial dimensions, in the absence of the Efimov effect. There is an effective repulsive potential that is inversely proportional to the square of the hyperradius rho, with a strength characterized by the parameter s. This problem is of broad interest in both ultracold atoms physics and nuclear physics. The method the author presents is interesting, although I did not check for myself whether it is accurate (in particular the approximation made in Eq.(15)).
This work seems to be an extension of Ref.[26] by the author.
Because of the importance of the topic and because of the author's expertise on
the subject, I support the publication of the paper.

Some comments and questions which I hope the author can address:
1. Some grammatical changes are desired. It may be helpful to ask a native English
speaker to make such changes.
Some necessary changes, not a complete list:
A. in the introduction,
"Although the huge different scales"
->
"In spite of the enormously different scales"
B. "ultracold atoms experiments gives the marvelous opportunity"
-> "ultracold atoms experiments provide marvelous opportunities"
C. "Efimov states, can be observed"
-> "Efimov states can be observed"
D. "s wave"
-> "s-wave" (where the letter "s" should be in italic face)
E. "At the heart of this universality, is the scale"
-> "At the heart of this universality is the scale"
F. "The beauty of this effects"
-> "The beauty of this effect"
G. "no more separable" (several places in the paper)
-> "no longer separable"
H. "First studies of this regime using a contact model, was done"
-> "First studies of this regime using a contact model were done"
I. "verify" or "verifying" (on pages 6,7,10,11,12,13,15)
-> "satisfy" or "satisfying".
J. "verified" (pages 7, 11, 12, 17)
-> "satisfied"
K. "traduce" (page 3)
-> "replace"
L. "traduces" (page 16)
-> "models"
M. "and effective angular momentum" (page 3)
-> "an effective angular momentum"

2. Around Eq.(4), the hyperradius rho is introduced.
I suggest that the author define rho in terms of the position vectors of the N particles
or in terms of the Jacobi coordinates.

3. Immediately after Eq.(11), it may be helpful to show the explicit formula for s:
s=|K+\frac{d}{2}-1|.

4. In the Conclusions section, the author writes
"The identification of possible long-lived quasi-bound states in this last regime for small but finite detuning, when \frac{1}{2} \leq s < 1, is perhaps the most important physical result of this paper."
Could the author show a possible physical system in which this effect may be realized?

Ask for minor revision

Referee report
"General features and contact modeling of N-body isolated resonances near
threshold"
by L. Pricoupenko

The author investigates the emergence of N-body bound and quasi-bound
states (collectively denoted as resonances) in systems with short-range
interactions in the non-Efimovian regime. In this case the long-range
potential in the hyperradial equation is repulsive. Furthermore, it is
assumed that no M-body resonances with M<N occur.

The author generalizes his previous work for 3d systems close to the unitary
limit in [26] to system away from the unitary limit and to 1d and 2d systems.
A contact model with two parameters is constructed that reproduces the
main features of a finite-range reference model for the considered states.
The methodology is reminiscent of description of a scattering state in a
channel with a shallow two-body state in terms of a pole position (binding
energy) and an asymptotic normalization constant. Here the formalism is
extended to N-body resonances by using hyperspherical coordinates.

The paper discusses a new class of universal of few-body states,
an important problem in few-body physics.
It is carefully written and it is obvious that the author has
made some effort to present the results in a comprehensive way,
for example by including a summary of the main results in the introduction.
Despite this, the paper is still rather technical and the connection to
experiment is not obvious. The paper could be published in SciPost after
the author has adressed the comments below.

1) p.4, 1st column, end of first paragraph:
The last sentence implies that the scenario of Ref. [8,9] is common
in two-neutron halo nuclei. However, most two-neutron halos have
shallow resonances (using the terminology of the current paper)
in the neutron-core and neutron-neutron channels. They are bound by the
Efimov effect, even though there might be only one state in the universal
region (see the discussion in [4]). The scenario of Ref. [8,9] (no
neutron-core resonance) may be applicable to 22C but there is conflicting
information from experiment.

2) Sec. IIC:
The results on the occupation probability for the small hyperradius region
are interesting. In [Hammer, Lee, Phys.Lett.B 681 (2009) 500-503]
causality bounds on the occupation probability for large distances in the
limit E -> 0 were derived. Are these bounds consistent with the results
on the occupation probability for small hyperradii in the paper?

3) While some references to similar results obtained using other
methods are made in the conclusions no connections to actual physical
systems or experimental results are made. Are the states discussed in the
manuscript theoretical artifacts or can they be observed in experiment?
A brief discussion of experimental prospects for the observation of the
N-body resonances in the conclusions would be useful.