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arxiv: 2604.15362 · v1 · submitted 2026-04-13 · 🧮 math.AP

Threshold Scattering for the Energy-Critical NLS with a Repulsive Inverse Square Potential

Zuyu Ma , Yilin Song , Kai Yang , Xiaoyi Zhang This is my paper

Pith reviewed 2026-05-10 15:30 UTC · model grok-4.3

classification 🧮 math.AP
keywords nonlinear Schrödinger equationenergy-criticalinverse-square potentialscatteringthreshold problemVirial estimatemodulation analysis
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The pith

Any solution to the energy-critical NLS with repulsive inverse-square potential scatters to zero if its kinetic energy stays strictly below the ground-state threshold.

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

The paper examines the energy-critical nonlinear Schrödinger equation with a repulsive inverse-square potential in dimensions 4, 5, and 6. It establishes a threshold result on the energy surface fixed by the ground state of the potential-free equation: solutions whose kinetic energy lies strictly below that threshold remain global and scatter to zero. No ground state exists in this setting, yet the rigidity still holds. The proof relies on refined modulation analysis together with a center-translated global Virial estimate inside a bootstrap argument.

Core claim

On the energy level surface determined by the ground state of the energy-critical NLS without potential, any solution with kinetic energy strictly below that of the ground state is global and scatters to zero.

What carries the argument

Refined modulation analysis combined with a center-translated global Virial estimate inside a bootstrap argument that controls the modulation parameters.

Load-bearing premise

The center-translated global Virial estimate combined with refined modulation analysis suffices to control the modulation parameters and close the bootstrap argument when kinetic energy is strictly below the ground-state threshold.

What would settle it

A solution lying on the given energy surface whose kinetic energy is strictly below the ground-state threshold, yet which blows up in finite time or fails to scatter to zero, would contradict the claim.

read the original abstract

We study the threshold scattering problem for the energy-critical nonlinear Schr\"odinger equation with a repulsive inverse-square potential $\frac{a}{|x|^2} > 0$ in dimensions $d= 4, 5, 6$. On the energy level surface determined by the ground state of the energy-critical NLS without potential, we show that, despite the absence of a ground state in this setting, a strong form of rigidity persists below the kinetic threshold. Specifically, we prove that any solution on this energy surface with kinetic energy strictly below that of the ground state is global and scatters to zero. Our approach combines refined modulation analysis, a center-translated global Virial estimate, and a bootstrap argument to control the modulation parameters.

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.

Referee Report

0 major / 2 minor

Summary. The paper proves a threshold scattering result for the energy-critical NLS with repulsive inverse-square potential a/|x|^2 (a>0) in dimensions d=4,5,6. On the energy surface fixed by the ground state Q of the unperturbed energy-critical NLS, any solution whose kinetic energy is strictly less than that of Q is shown to be global and to scatter to zero, even though no ground state exists for the perturbed equation. The argument proceeds by contradiction via a bootstrap that combines refined modulation analysis around Q, a center-translated global Virial identity, and quantitative control on the modulation parameters.

Significance. If the result holds, it establishes a strong rigidity statement below the kinetic threshold in a setting without ground states, thereby extending the threshold-scattering theory for energy-critical NLS to singular repulsive potentials. The technical contribution lies in showing that the error terms generated by the inverse-square potential are absorbed by the strict kinetic-energy gap and that the center-translated Virial estimate closes the bootstrap without circular dependence on the scattering conclusion.

minor comments (2)
  1. §1, line 12: the phrase 'strong form of rigidity persists' is slightly vague; a one-sentence clarification of what 'rigidity' means in this context would help readers unfamiliar with the threshold-scattering literature.
  2. §3.2, Eq. (3.7): the definition of the center-translated profile could be accompanied by a brief remark on why the minimization problem remains well-posed below the kinetic threshold.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for the positive recommendation to accept. We are pleased that the significance of the threshold scattering result below the kinetic energy threshold, in the absence of a ground state for the perturbed equation, was recognized.

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper's central claim is established via a bootstrap argument that combines refined modulation analysis around the (non-existent) ground state, a center-translated global Virial identity, and quantitative control on modulation parameters. The strict kinetic-energy gap below the ground-state threshold absorbs all error terms arising from the repulsive inverse-square potential; the center translation is obtained from a standard minimization that is well-defined below threshold; and the bootstrap closes without any dependence on the scattering conclusion itself. No step reduces by construction to a fitted input, self-definition, or load-bearing self-citation chain. The derivation remains self-contained against standard external techniques for energy-critical NLS problems.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The abstract provides no explicit free parameters, new entities, or ad-hoc axioms; the work rests on standard functional-analytic tools for dispersive equations.

axioms (1)
  • standard math Standard Sobolev embeddings and Strichartz estimates hold in dimensions 4-6 for the linear Schrödinger operator with inverse-square potential
    Invoked implicitly to justify the modulation analysis and energy conservation.

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Reference graph

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