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arxiv: 2604.23507 · v1 · submitted 2026-04-26 · 🪐 quant-ph

Partial solvability induced by dark states in a box trap with decentered two-body interaction

Hossein Abedi , Nathan L. Harshman , Peter Schmelcher This is my paper

Pith reviewed 2026-05-08 06:11 UTC · model grok-4.3

classification 🪐 quant-ph
keywords dark statesdecentered interactionone-dimensional boxpartial solvabilitytwo-body interactionnonintegrable systemsbosonic statesfermionic states
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The pith

Decentered two-body interaction in a one-dimensional box creates dark states that form exactly solvable subspaces.

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

The paper studies two particles confined in a one-dimensional box where they interact only when separated by a precise distance c. This decentered interaction renders the overall system nonintegrable, unlike the case with harmonic confinement. Nevertheless, certain bosonic and fermionic states remain completely unaffected by the interaction because their wave functions produce no overlap with the interaction points. These dark states form closed subspaces whose energies and wave functions can be solved exactly using the noninteracting box solutions. The result provides analytical access to parts of the spectrum in a model that would otherwise demand fully numerical treatment.

Core claim

The central claim is that a decentered two-body interaction in a one-dimensional box, active only at particle separation c, produces dark states unaffected by the interaction. These dark states constitute exactly solvable subspaces inside an otherwise nonintegrable spectrum, allowing exact determination of certain energies and wave functions.

What carries the argument

Dark states: bosonic or fermionic eigenstates of the full Hamiltonian that remain identical to noninteracting box states because the decentered interaction vanishes on their wave-function support.

If this is right

  • Dark states appear only for specific values of the decentering distance c that align with the nodal structure of box eigenfunctions.
  • The full spectrum splits into exactly solvable dark sectors and sectors that remain interacting and require numerical solution.
  • Stationary properties such as energies, densities, and correlation functions are available in closed form inside the dark subspaces.
  • The partial solvability delineates interacting from noninteracting sectors in the energy spectrum.

Where Pith is reading between the lines

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

  • Varying the decentering parameter c offers a tunable knob to enlarge or shrink the dimension of the solvable subspace without changing the overall nonintegrability.
  • The same nodal-avoidance mechanism that protects dark states may appear in other position-dependent interaction models, suggesting a broader route to partial analytic control in few-body systems.

Load-bearing premise

The interaction occurs strictly only when the particles are separated by exactly distance c inside a strictly one-dimensional hard-wall box.

What would settle it

Numerical diagonalization of the two-particle Hamiltonian for a chosen value of c should yield a subset of eigenenergies and wave functions that exactly match the noninteracting particle-in-a-box solutions; absence of such states would disprove the claim.

Figures

Figures reproduced from arXiv: 2604.23507 by Hossein Abedi, Nathan L. Harshman, Peter Schmelcher.

Figure 1
Figure 1. Figure 1: FIG. 1. Configuration space for two particles confined in a view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The ground-state wave function view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The six lowest-lying eigenfunctions Ψ view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Energy levels at view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Wavefunctions of the lowest-lying view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Energy vs view at source ↗
Figure 7
Figure 7. Figure 7: displays all pairs (n, m) which give rise to a dark state of the interaction, as a function of c up to ε/π2 = 200, grouped by q − p where c = p/q. Note that as c deviates from 1/2, the corresponding dark states move higher in the spectrum. For each value of c in the spectrum, there exists a close rational value in the spectrum that corresponds, at sufficiently high energies, to a dark state. We discussed i… view at source ↗
read the original abstract

We consider a generalization of the two-body contact interaction for nonrelativistic particles confined to a one-dimensional box, in which the interaction is decentered, i.e., the particles interact only when they are separated by a distance c. In contrast to the harmonically trapped system, this model is nonintegrable. Despite this, we demonstrate that the system exhibits partial solvability due to the presence of dark states, i.e., bosonic or fermionic states unaffected by the interaction. These states form exactly solvable subspaces embedded within an interacting spectrum. We characterize the stationary properties of the system, identify the conditions for the appearance of dark states, and show how they structure the spectrum and delineate interacting and noninteracting sectors.

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

1 major / 3 minor

Summary. The manuscript considers two particles confined to a one-dimensional infinite square well subject to a decentered contact interaction that acts only at interparticle separation exactly equal to a fixed distance c. It identifies a class of eigenstates of the non-interacting Hamiltonian (dark states) that vanish identically on the locus |x1−x2|=c and therefore remain exact eigenstates of the full interacting Hamiltonian for arbitrary interaction strength. These states form solvable subspaces embedded in an otherwise non-integrable spectrum; the authors derive the algebraic conditions on the ratio c/L under which such states exist for both bosonic and fermionic statistics and characterize the resulting stationary properties and spectral structure.

Significance. If the construction is correct, the work supplies a transparent, analytically tractable example of partial exact solvability induced by dark states in a non-integrable few-body quantum system. The explicit conditions on c/L, the contrast with the integrable harmonic-trap analog, and the separation into interacting and non-interacting sectors are useful for studies of integrability breaking and the emergence of solvable sectors in confined geometries.

major comments (1)
  1. [Introduction and discussion of integrability] The non-integrability claim is asserted by noting the absence of the additional conserved quantities present in the harmonic-trap version, but no explicit verification (e.g., via level-spacing statistics or direct search for further integrals of motion) is supplied; this leaves open whether the dark-state subspaces are embedded in a spectrum that is genuinely non-integrable or merely appears so at the two-body level.
minor comments (3)
  1. [Model definition] The definition of the interaction potential (Eq. (2) or equivalent) should be written with an explicit indicator function or delta-function form to avoid ambiguity about whether the interaction is strictly on the line |x1−x2|=c or in a neighborhood.
  2. [Numerical results and figures] Several figures lack explicit labels for the values of c/L used in the plotted spectra; adding these labels would make the comparison between dark-state and interacting sectors immediate.
  3. [Conditions for dark states] The manuscript would benefit from a short table summarizing the allowed c/L ratios that produce dark states for N=2 bosons and fermions, together with the corresponding quantum numbers.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript, the positive assessment of its significance, and the recommendation for minor revision. We address the single major comment below.

read point-by-point responses

  1. Referee: The non-integrability claim is asserted by noting the absence of the additional conserved quantities present in the harmonic-trap version, but no explicit verification (e.g., via level-spacing statistics or direct search for further integrals of motion) is supplied; this leaves open whether the dark-state subspaces are embedded in a spectrum that is genuinely non-integrable or merely appears so at the two-body level.

    Authors: We appreciate the referee highlighting this point. The non-integrability of the model follows directly from the structure of the Hamiltonian: the infinite square well breaks translational invariance (preventing center-of-mass separation), and the decentered interaction acts only at a fixed separation c rather than depending solely on the relative coordinate in a manner that would preserve additional integrals of motion. This contrasts with the harmonic-trap case, where the quadratic potential allows exact separation and yields extra conserved quantities. At the two-body level, the system is fully described by this Hamiltonian, and the absence of further symmetries implies that the spectrum (outside the dark-state subspaces) is not analytically solvable in closed form. We will revise the introduction and discussion sections to articulate this structural argument more explicitly, thereby clarifying why the dark states are embedded in a genuinely non-integrable spectrum. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The central construction defines dark states as eigenstates of the non-interacting box Hamiltonian (products of sine functions) that additionally vanish identically on the interaction locus |x1-x2|=c. Because the interaction term is supported only on that locus, it annihilates these states, so they remain exact eigenstates of the full Hamiltonian for arbitrary interaction strength. This is a direct algebraic consequence of the Hamiltonian definition rather than a self-referential fit or imported uniqueness theorem. Conditions on c/L for the existence of such states follow from the explicit nodal structure of the free solutions and are stated as algebraic constraints without parameter fitting. Non-integrability is asserted by explicit contrast with the harmonic-trap case (absence of extra conserved quantities) and does not rely on self-citation chains. No step renames a fitted quantity as a prediction or reduces the solvability claim to its own inputs by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

The claim rests on the standard non-relativistic 1D Schrödinger equation with hard walls plus the new assumption of a strictly distance-c interaction; no additional fitted constants or new particles are introduced.

free parameters (1)
  • decentering distance c
    Model parameter that sets the separation at which the two-body interaction is nonzero; its value is chosen by hand for the study.
axioms (2)
  • domain assumption Particles obey the non-relativistic Schrödinger equation in one dimension with infinite walls.
    Standard setup stated in the abstract for the box trap.
  • domain assumption The two-body interaction is exactly zero except at separation c.
    The defining generalization of the contact interaction introduced in the abstract.
invented entities (1)
  • dark states no independent evidence
    purpose: Bosonic or fermionic eigenstates that remain unaffected by the decentered interaction.
    Newly identified states that form the solvable subspaces; no independent experimental signature is supplied in the abstract.

pith-pipeline@v0.9.0 · 5424 in / 1373 out tokens · 54421 ms · 2026-05-08T06:11:32.754156+00:00 · methodology

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