On a Fractional Schr\"odinger equation in the presence of Harmonic potential

Hichem Hajaiej; Zhiyan Ding

REVIEW 2 major objections 4 minor 20 references

On a Fractional Schr\"odinger equation in the presence of Harmonic potential

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For 0<σ<2s/N, the fractional Schrödinger equation with a harmonic trap has nonnegative, radial, decreasing ground-state standing waves, orbitally stable when the Cauchy problem is unique.

desk verdict A plausible extension of ground-state theory to fractional NLS with a harmonic trap, but the proof of the main existence theorem rests on a faulty compactness lemma. read the letter →

arxiv 1908.05719 v2 pith:UMAFPWSV submitted 2019-08-15 math.AP

Zhiyan Ding , Hichem Hajaiej This is my paper · ORCID

classification math.AP MSC 35Q5535J6047J30

keywords fractionalSchrödingerequationLaplacianharmonicpotentialstandingwavesgroundstatesolutionsorbitalstabilityconstrainedminimizationnormalizedgradientflow

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper establishes that the fractional nonlinear Schrödinger equation with a harmonic trapping potential, $i\psi_t=(-\Delta)^s\psi+|x|^2\psi-|\psi|^{2\sigma}\psi$, has standing-wave solutions when the nonlinearity is subcritical, $0<\sigma<2s/N$. The proof works by minimizing the associated energy on the sphere of fixed $L^2$ mass within the weighted space $\Sigma_s(\mathbb{R}^N)$, where the potential restores compactness lost on the whole space. The resulting minimizer is nonnegative, radial, and radially decreasing, and it gives a ground-state solution through a Lagrange multiplier $\lambda$. Under an additional assumption that the initial-value problem has unique solutions conserving mass and energy, these standing waves are orbitally stable. The paper also supplies numerical evidence for ground states, stability, and dynamics, including the critical and supercritical ranges via a second constrained problem.

What carries the argument

The central object is the weighted Sobolev space $\Sigma_s(\mathbb{R}^N)=\{u\in H^s(\mathbb{R}^N): \|u\|_{L^2}+\|\nabla^s u\|_{L^2}+\|xu\|_{L^2}<\infty\}$, whose norm includes the harmonic potential. Its compact embedding into $L^p$ for $2\le p<2N/(N-2s)$ is what lets a minimizing sequence for $I_c$ converge strongly, overcoming the lack of compactness on $\mathbb{R}^N$. The fractional Gagliardo\textendash Nirenberg inequality supplies the $L^{2\sigma+2}$ bound that keeps the energy finite below the threshold $\sigma<2s/N$. Schwarz symmetrization then converts any minimizer into a radial, radially decreasing one without raising the energy. For stability, the machinery is a standard compactness-based variational argument: the set of minimizers at fixed mass is shown to be stable under the flow whenever the Cauchy problem is well posed with conserved mass and energy.

What would settle it

Translate a fixed compactly supported bump: $u_n(x)=\varphi(x-ne_1)$ is bounded in $H^s(\mathbb{R}^N)$ but has no convergent subsequence in $L^p(\mathbb{R}^N)$, so the proof's appeal to compactness of $H^s$ on the whole space is demonstrably false; a correct proof must use the $\|xu\|_{L^2}$ weight, for example by showing that $\|u_n\|_{\Sigma_s}\to\infty$ for such translations.

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Extended reading notes

Core claim

The central claim is Theorem 2.1: for $0<\sigma<2s/N$, the constrained minimization problem $I_c=\inf\{J(u): u\in\Sigma_s(\mathbb{R}^N),\ \|u\|_{L^2}=c\}$ admits a minimizer, and that minimizer is a nonnegative, radial, radially decreasing ground state solution of $(-\Delta)^s u+|x|^2u-|u|^{2\sigma}u=\lambda u$. The mechanism is that the harmonic potential makes the embedding $\Sigma_s(\mathbb{R}^N)\hookrightarrow L^p(\mathbb{R}^N)$ compact for $2\le p<2N/(N-2s)$, so minimizing sequences converge; the fractional Gagliardo\textendash Nirenberg inequality bounds the energy below precisely when $\sigma<2s/N$. Theorem 2.2 then asserts orbital stability of the set of minimizers, conditional on uniqueness and on conservation of mass and energy for the Cauchy problem. Numerical sections complement the existence result by solving the constrained problems with normalized gradient flow and split-step Fourier methods, and by probing the critical range $2s/N\le\sigma<2s/(N-2s)$ through an alternative minimization with fixed $L^{2\sigma+2}$ norm.

Load-bearing premise

The argument's load-bearing premise is Lemma 3.1's claim that the weighted space $\Sigma_s(\mathbb{R}^N)$ sits compactly inside the Lebesgue spaces $L^p(\mathbb{R}^N)$; as written, the proof of that lemma appeals to a compactness of $H^s$ on the whole space that is false, so the extraction of a convergent minimizing sequence is not justified until that point is repaired.

Editorial extensions

If this is right

For every mass $c>0$ and exponent $0<\sigma<2s/N$, the constrained problem (8) has a minimizer, so the elliptic equation (6) has a nonnegative, radial, radially decreasing ground-state solution for some Lagrange multiplier $\lambda$.
Because the minimizer set is compact in $\Sigma_s(\mathbb{R}^N)$, the standing waves $e^{-i\lambda t}u(x)$ form an orbitally stable family whenever the Cauchy problem is well posed with conserved mass and energy.
The threshold $\sigma=2s/N$ is sharp for the original minimization: above it the energy is unbounded below, and the paper's alternative fixed-$L^{2\sigma+2}$ formulation gives standing waves numerically up to the $L^2$-supercritical threshold $\sigma<2s/(N-2s)$.
Numerical experiments indicate that as $s$ decreases toward $\sigma N/2$, ground states become more peaked, the energy diverges to $-\infty$ at fixed mass, and orbital stability degrades.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

An editorial repair of Lemma 3.1: a correct compactness proof should split $\mathbb{R}^N$ into a ball and its complement, use the tail bound $\|u\|_{L^2(|x|>R)}\le R^{-1}\|xu\|_{L^2}$, then take a diagonal subsequence; the false step in the paper is a proof gap, not evidence against the theorem.
The convergence $\lambda_c\to\lambda_0$ as $c\to0$ seen numerically is consistent with $\lambda_0$ being the lowest eigenvalue of the linear operator $(-\Delta)^s+|x|^2$; proving this would connect the nonlinear ground states to the linear spectrum.
The non-radial numerical ground states for asymmetric potentials suggest the radial symmetry theorem depends essentially on the potential being radial; extending existence to general trapping potentials would require new rearrangement tools.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

Short version: this paper fills a natural gap—fractional NLS with a harmonic trap—and the main results are probably true, but the proof of Theorem 2.1 has a real gap in its compactness lemma, and the critical-range existence is only asserted. I'd accept it for peer review, but revision is required.

What's genuinely new is the combination of [9] (s=1 with trap) and [15] (fractional without trap). The weighted space Σ_s is the right setting, and the existence/stability structure follows the standard variational pattern. The numerical work is the most original part: it explores ground states for different s, non-radial potentials, the critical case, and dynamics, and the figures show the expected behavior.

The stress-test concern about Lemma 3.1 lands. The proof says H^s(R^N) compacts into L^p(R^N) for 22 case is missing, and for s≥N/2 the exponent 2N/(N−2s) is meaningless. As written, the strong convergence of a minimizing subsequence in L^{2σ+2} is unsupported. This is repairable via standard compactness/interpolation arguments for weighted spaces, but it's a genuine gap, not a typo.

Second, Section 6 introduces the constrained problem (62)-(63) and claims existence of a local minimizer 'similar to Lemma 3.2 and Theorem 2.1,' without proof. That's a separate unproven assertion, especially since the functional K and the projection step differ from the subcritical case.

Third, the numerics have no code, data, or error analysis. The stability measure D(s,t) in Figure 9(b) isn't obviously connected to the orbital stability norm, so the conclusion that smaller s gives worse stability is informal.

On citations: the paper leans on several co-author papers for the variational framework and symmetrization inequalities. That's legitimate; there's no circularity. The stability theorem is conditional on uniqueness and conservation, which is standard in this literature but not proved here.

The core message: the main ideas are sound and the theorems are likely correct, but the write-up is not. A serious referee should ask for a corrected Lemma 3.1 and either a full proof or a clearly labeled conjecture for the critical range. I wouldn't cite it in its current form; after careful repair I might. It's a good reading-group example of a false Sobolev embedding claim slipping into a proof.

I'd send it to review, with the expectation of major revision.

Referee Report

2 major / 4 minor

Summary. The paper studies the fractional nonlinear Schrödinger equation with a harmonic potential, Eq. (1). It introduces the weighted Hilbert space Σ_s(R^N), Eq. (10), and studies the constrained minimization of the energy functional J on the mass sphere S_c, Eq. (8). Theorem 2.1 claims that for 0<σ<2s/N this problem admits a nonnegative, radial, radially decreasing minimizer; Theorem 2.2 claims orbital stability of the associated ground states under a uniqueness/well-posedness assumption on the Cauchy problem. The second half of the paper develops split-step Fourier spectral and normalized-gradient-flow numerical methods, reports numerical ground states over a wider parameter range, and studies dynamics and stability numerically.

Significance. If the existence and stability theorems are established, the paper would provide a natural variational treatment of fractional NLS with a harmonic trap, extending known results for the Gross-Pitaevskii equation (s=1) and for fractional NLS without potential. The numerical section is broad and gives useful heuristics on the role of s, σ, and the trapping potential. The central existence proof, however, has a gap in Lemma 3.1: the compactness argument rests on a false Sobolev embedding statement, and no alternative proof is supplied. The result is likely repairable, and the stability theorem is explicitly conditional on an unproved well-posedness assumption, so the manuscript's claims are defensible as conditional statements but not yet fully proved as written.

major comments (2)

[Section 3, Lemma 3.1] The proof of compactness for p>2 relies on the assertion that H^s(R^N) is compactly embedded into L^p(R^N) for 2<p<2N/(N-2s). This is false on the unbounded domain R^N, where translation invariance prevents compactness. Moreover, the stated range 2≤p<2N/(N-2s) is empty when N≤2s, while Theorem 2.1 allows such values (e.g., N=1, s=0.8, σ=1). The lemma is used in the proof of Theorem 2.1 to extract a subsequence converging strongly in L^{2σ+2}, and again in Theorem 4.1. As written, these subsequence extractions are unsupported. The compactness of the weighted space Σ_s(R^N) is likely true, but a correct proof must be supplied, and the statement must be modified to handle the case N≤2s.
[Section 6, Eqs. (62)-(63)] The paper claims existence of ground states for the range 2s/N ≤ σ < 2s/(N-2s) via the constrained problem (62)-(63), and states 'Similar to Lemma 3.2 and Theorem 2.1, there exists a local minimizer'. This is not a proof; no rigorous argument is given that the auxiliary problem has a minimizer or that it yields a solution to (65). If the paper intends to claim existence in this range, a complete proof is needed. As it stands, those results are numerical evidence only and should be labeled as such.

minor comments (4)

[Section 3, Lemma 3.2, Eq. (15)] The exponent θ in the fractional Gagliardo-Nirenberg inequality appears to be misprinted as θ = Ns/(2s(σ+1)), which simplifies to N/(2(σ+1)). The subsequent definitions p = 1/(θ(1+σ)) and q = 1/(1-θ(1+σ)) are consistent only if θ = Nσ/(2s(σ+1)). Please correct this typo.
[Theorems 2.2 and 4.1] The orbital-stability theorem is conditional on an assumed uniqueness and well-posedness result for the Cauchy problem (1). The paper is explicit about this, but the abstract and introduction should state clearly that stability is proved only under this assumption, and it would be helpful to cite or prove local well-posedness in Σ_s(R^N).
[Section 2, Eqs. (11)-(13)] The Lagrange multiplier argument is presented formally. Since Theorem 2.1 only asserts existence of a minimizer of (8), this is not a flaw for the theorem as stated, but the paper repeatedly calls these minimizers 'ground state solutions'; a precise statement connecting minimizers to solutions of (6) would strengthen the paper.
[Throughout] There are numerous typographical and notational errors: in Lemma 3.2 the integral set is written as Ω but should be R^N; 'Figure 7.2' in Section 7.2 should presumably be 'Figure 10'; the caption of Figure 10 says 'δ = 1' but should be 'σ = 1'; and 'Abosolute value' should be 'Absolute value'. The manuscript would benefit from a careful proofreading pass.

Circularity Check

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No significant circularity: the variational existence proof is self-contained up to cited external inequalities; no prediction reduces by construction to a fitted input.

full rationale

The paper's central claim is Theorem 2.1: existence of a nonnegative, radial, radially decreasing minimizer for the constrained problem (8). The proof proceeds by standard direct methods: Lemma 3.2 bounds minimizing sequences via the fractional Gagliardo-Nirenberg inequality cited from [12]; Lemma 3.1 supplies compact embedding of the weighted space; and the minimizer is then symmetrized using rearrangement results from [14] and equality cases quoted from [9]. None of these cited results is the target theorem, and the paper does not fit any parameter and then 'predict' a quantity determined by that fit. The stability result, Theorem 2.2/4.1, is explicitly conditional on a uniqueness-and-conservation assumption for the Cauchy problem, so its conclusion is not hidden in the hypothesis. The numerical section is presented as illustration and comparison, not as the derivation of the existence theorem. The proof of Lemma 3.1 as written invokes a false statement about compactness of H^s on all of R^N, and Remark 3.1 over-claims convergence in the full Σ_s norm; these are correctness gaps, not circularity. The self-citations (e.g., [9], [12], [13], [14]) are external published results used as lemmas with their own stated assumptions, and they do not redefine the main conclusion. Therefore no circular step meeting the evidentiary standard is present.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper does not fit any free parameters to data; c and ω are problem parameters, not empirical fits. The main mathematical inputs are standard inequalities and a well-posedness assumption that is not proved. The single most fragile input is the false compact-embedding assertion used in Lemma 3.1.

assumptions (4)

standard math Fractional Gagliardo-Nirenberg inequality (15)
Invoked in Lemma 3.2 to bound the L^{2σ+2} norm of functions in S_c; cited from [12].
standard math Symmetrization inequalities for the fractional Laplacian and the harmonic potential (23)-(24)
Used in Theorem 2.1 to build a nonnegative, radial, radially decreasing minimizer; cited from [9,1,14].
ad hoc to paper Compact embedding of H^s(R^N) into L^p(R^N) for 2<p<2N/(N-2s)
Asserted in the proof of Lemma 3.1 as a known Sobolev fact. It is false on unbounded domains and is the main gap in the proof.
domain assumption Global well-posedness and uniqueness of the Cauchy problem (1) with conserved mass (3) and energy (4)
Assumed in Theorem 2.2 and Theorem 4.1; the paper does not prove well-posedness, making the stability result conditional.

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Pith. "Pith review of On a Fractional Schr\"odinger equation in the presence of Harmonic potential." pith.science (2026). https://pith.science/paper/UMAFPWSV

@misc{pith2026190805719,
  author       = {Pith},
  title        = {Pith review of: On a Fractional Schr\"odinger equation in the presence of Harmonic potential},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UMAFPWSV}},
  note         = {Machine review of arXiv:1908.05719}
}

read the original abstract

In this paper, we establish the existence of ground state solutions for a fractional Schr\"odinger equation in the presence of a harmonic trapping potential. We also address the orbital stability of standing waves. Additionally, we provide interesting numerical results about the dynamics and compare them with other types of Schr\"odinger equations. Our results explain the effect of each term of the Schr\"odinger equation : The fractional power, the power of the nonlinearity and the harmonic potential.

Figures

Figures reproduced from arXiv: 1908.05719 by the authors.

**Figure 2.** Ground state solutions with σ = 1 and different s From (2(a)) and (2(b)), it seems ground state solutions change continuously with s. We use L 2 distance between u s 0 (x) − u 1 (x) to check and see the convergence of ground state solutions in L 2 space with s → 1. ( [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗

**Figure 3.** L 2 distance between ground state solutions of s < 1 and s = 1 when σ = 1 Then, we test another two things. The first is the relation between the constrained minimal energy in (8) and s. We calculate the discrete energy by E(s) = h X J−1 j=0   J/ X 2−1 l=−J/2 |µl | 2s |ubl | 2 + |xj | 2 |uj | 2 − 1 σ + 1 |uj | 2(σ+1)  . (69) From figure 4(a), we find the energy’s dependence (E(s)) on s is monotonic. When s appro… view at source ↗

Figures from the paper (8 more)

**Figure 4.** Figure 4: Energy and λc Up to now, we only consider the case with radial symmetrical potential. However, when potential is not radially symmetric, we can still find standing waves to (1) using (8). We tried the case where potential is |x| 2 + a sin(2πx) with a = 1 and a = 5. Fro…

**Figure 5.** Figure 5: Ground state solutions with non-symmetric potential [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]

**Figure 6.** Figure 6: Energy and stability check with s = 0.8, σ = 1 [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]

**Figure 7.** Figure 7: kψ s (x, t) − u s (x, t)kΣes when s = 0.8 and σ = 1 We can see from figure 2(b) and theoretic results that when s → σN 2 , the regularity of ground state solutions becomes worse. This inspires us to investigate its stability relationship with s. By Theorem 2.2, Def 4.1…

**Figure 8.** Figure 8: Abosolute of solution with different s (a) D(0.8, t) vs D(0.6, t) (b) D(1, t) with different s [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]

**Figure 9.** Figure 9: Stability test with ψ s 0 (x) = 0.9 ∗ u s 0 (x) Second, we try to obtain some numerical result when we touch the critical point s = σN 2 . In this case, we can’t find the ground state solution through (8) because Ic = −∞. However, as we discussed before, we can find a …

**Figure 10.** Figure 10: Standing wave and ground state solution when [PITH_FULL_IMAGE:figures/full_fig_p020_10.png]

**Figure 11.** Figure 11: Dynamics of FNLS [2] W. Bao and Q. Du, Computing the ground state solution of bose–einstein condensates by a normalized gradient flow, SIAM J. Sci. Comput., 25 (2004), pp. 1674–1697. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_11.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [9]

Hadj Selem, H

F. Hadj Selem, H. Hajaiej, P. Markowich, and S. Trabelsi , Variational approach to the orbital stability of standing waves of the gross-pitaevskii equation , Milan Journal of Mathematics, 82 (2014)

work page 2014

[2] [15]

Klein, C

C. Klein, C. Sparber, and P. Markowich , Numerical study of fractional nonlinear schrodinger equations, Proceedings. Mathematical, physical, and engineering sciences / the Royal Society, 470 (2014), p. 20140364

work page 2014

[3] [1]

F. J. Almgren and E. H. Lieb , Symmetric decreasing rearrangement is sometimes continuous, Journal of the American Mathematical Society, 2 (1989), pp. 683–773. 19 (a) Standing waves when s = 0.5 (b) Absolute value of ground state solution w s = 0.5 (c) Time evolution of Re(ψ(0,t )) (d) Time evolution of Im(ψ(0,t )) (e) Blow up of solution with ψ0(x,t ) = ...

work page 1989

[4] [2]

Bao and Q

W. Bao and Q. Du , Computing the ground state solution of bose–einstein condensates by a normalized gradient ﬂow , SIAM J. Sci. Comput., 25 (2004), pp. 1674–1697. 20

work page 2004

[5] [3]

Carles and H

R. Carles and H. Hajaiej , Complementary study of the standing wave solutions of the gross-pitaevskii equation in dipolar quantum gases , Bulletin of the London Mathematical Society, 47 (2014)

work page 2014

[6] [4]

Cazenave and P.-L

T. Cazenave and P.-L. Lions , Orbital stability of standing waves for some nonlinear schr¨ odinger equations, Communications in Mathematical Physics, 85 (1982)

work page 1982

[7] [5]

Chang , Ground state solutions of asymptotically linear fractional schrdinger equation , Journal of Mathematical Physics, 54 (2013)

X. Chang , Ground state solutions of asymptotically linear fractional schrdinger equation , Journal of Mathematical Physics, 54 (2013)

work page 2013

[8] [6]

Cheng , Bound state for the fractional schrdinger equation with unbounded potential , Journal of Mathematical Physics, 53 (2012)

M. Cheng , Bound state for the fractional schrdinger equation with unbounded potential , Journal of Mathematical Physics, 53 (2012)

work page 2012