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arxiv: 2604.17363 · v1 · submitted 2026-04-19 · 🧮 math.AP

Ground States of One-Dimensional Fermionic Schr\"{o}dinger Systems Near a Critical Exponent

Bin Chen , Yujin Guo , Yong Luo , Juncheng Wei This is my paper

Pith reviewed 2026-05-10 06:02 UTC · model grok-4.3

classification 🧮 math.AP
keywords fermionic nonlinear Schrödinger systemground statescritical exponentone dimensionnon-existencelimiting profileconcentration-compactness
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The pith

The one-dimensional fermionic nonlinear Schrödinger system J₂(p) has no ground states as p approaches 2 from above, while existing ground states develop two density bumps that separate to infinity as p approaches 2 from below.

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

The paper proves that the energy functional J₂(p) admits no ground states when the exponent p is slightly larger than 2. This settles the one-dimensional case of an earlier conjecture about the loss of minimizers near the critical value p=2. For p slightly smaller than 2, where ground states are known to exist, the paper derives a precise limiting profile: the density splits into exactly two bumps whose separation distance diverges to infinity. These two regimes together describe how the system transitions from existence to non-existence at the critical exponent. The results rest on variational methods and profile decomposition applied to the Sobolev space H¹(ℝ).

Core claim

We prove that there is no ground state of J₂(p) as p ↓ 2, which addresses the special case of Conjecture 5 in Gontier-Lewin-Nazar (ARMA 2021). The refined limiting profile of ground states for J₂(p) is also analyzed as p ↑ 2, which shows that the corresponding density admits exactly two bumps whose distance goes up to infinity as p ↑ 2.

What carries the argument

The energy functional J₂(p) on H¹(ℝ), analyzed via concentration-compactness and profile decomposition to control minimizing sequences near p=2.

If this is right

  • Ground states cease to exist for all p slightly larger than 2.
  • As p approaches 2 from below, ground-state densities consist of two bumps whose separation tends to infinity.
  • The transition at p=2 is marked by loss of compactness through separation of two components.
  • The one-dimensional case of the Gontier-Lewin-Nazar conjecture is confirmed.

Where Pith is reading between the lines

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

  • Similar two-bump separation may appear in multi-component or higher-dimensional fermionic systems near their own critical exponents.
  • Numerical minimization of J₂(p) for p close to 2 could measure the exact growth rate of bump separation.
  • The profile decomposition technique used here may extend to other nonlinear Schrödinger systems with polynomial nonlinearities.

Load-bearing premise

The analysis assumes the known existence of ground states for all 1 < p < 2 together with the validity of standard concentration-compactness arguments to rule out convergence of minimizing sequences when p exceeds 2.

What would settle it

An explicit construction of a ground state minimizer for some p > 2 close to 2, or a proof that every minimizing sequence for such p remains compact, would contradict the non-existence claim.

read the original abstract

We study ground states of the fermionic nonlinear Schr\"{o}dinger system $J_2(p)$ in $\R$, where $p>1$ denotes a polynomial exponent of the nonlinear term. It is known that the system $J_2(p)$ admits ground states for any $1<p<2$, while there is no ground state for $J_2(2)$. We prove that there is no ground state of $J_2(p)$ as $p\searrow 2$, which addresses the special case of Conjecture 5 in [D. Gontier, M. Lewin and F. Q. Nazar, ARMA, 2021]. The refined limiting profile of ground states for $J_2(p)$ is also analyzed as $p\nearrow 2$, which shows that the corresponding density admits exactly two bumps whose distance goes up to infinity as $p\nearrow 2$.

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 / 3 minor

Summary. The manuscript studies ground states of the one-dimensional fermionic nonlinear Schrödinger system J_2(p) for p > 1. It proves non-existence of ground states for p ↓ 2 (p > 2 sufficiently close to 2), resolving a special case of Conjecture 5 from Gontier-Lewin-Nazar (ARMA 2021). It also establishes that, as p ↑ 2, the density of ground states converges to a limiting profile consisting of exactly two bumps whose mutual distance tends to infinity.

Significance. If the proofs are correct, the work provides a concrete resolution to part of an open conjecture on existence thresholds and supplies a refined asymptotic description of the transition at the critical exponent p = 2. The combination of non-existence for p > 2 near 2 with the two-bump separation for p < 2 is consistent with the known existence regime 1 < p < 2 and adds quantitative information on concentration-compactness phenomena in fermionic systems.

minor comments (3)
  1. [Introduction] §1 (Introduction): the precise statement of the two main theorems (non-existence for p ↓ 2 and the two-bump profile for p ↑ 2) should be numbered and displayed separately from the surrounding discussion for easier reference.
  2. [Profile decomposition] §3 (or wherever the profile decomposition is carried out): the passage from the rescaled sequence to the exact two-bump limit would benefit from an explicit statement of the orthogonality conditions used to rule out additional bumps.
  3. [Notation] Notation: the functional J_2(p) is introduced without an immediate reminder of the precise form of the fermionic interaction term; a short displayed equation at first use would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, accurate summary of the main results, and recommendation for minor revision. No specific major comments or requests for changes were raised.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper establishes non-existence of ground states for the fermionic system J_2(p) as p ↓ 2 and derives the two-bump limiting profile as p ↑ 2 via standard variational arguments, concentration-compactness, and profile decomposition applied to the energy functional on H^1(ℝ). These steps rely on the independently known existence result for 1 < p < 2 and address an external conjecture from Gontier-Lewin-Nazar (ARMA 2021) with no overlapping authors. No step reduces by the paper's own equations to a fitted parameter, self-referential definition, or load-bearing self-citation; the central claims remain independent of the paper's inputs and are externally falsifiable.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The central claims rest on the standard definition of the energy functional for the two-component fermionic system and on classical 1D Sobolev embeddings and concentration-compactness principles; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

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Reference graph

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