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arxiv: 2506.21457 · v2 · pith:VDW25ODLnew · submitted 2025-06-26 · 🧮 math-ph · math.MP· math.SP

The Born-Oppenheimer approximation for a 1D 2+1 particle system with zero-range interactions

Claudio Cacciapuoti , Andrea Posilicano , Hamidreza Saberbaghi This is my paper

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classification 🧮 math-ph math.MPmath.SP
keywords zero-range interactionsthree-body problemBorn-Oppenheimer approximationAiry functioneigenvalue asymptoticsmass ratioessential spectrumone-dimensional quantum mechanics
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The pith

In a 1D light-heavy-heavy system with zero-range attractions the nth eigenvalue behaves as E_n(ε) = -α² + |σ_n| α² ε^{2/3} + O(ε) where σ_n is drawn from the Airy function Ai.

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

The paper determines the low-lying eigenvalues of a self-adjoint Hamiltonian for a one-dimensional three-body quantum system in which one light particle interacts attractively with two heavy particles through zero-range forces. In the regime where the mass-ratio parameter ε is much smaller than one, the nth discrete eigenvalue below the essential spectrum takes the explicit asymptotic form E_n(ε) = -α² + |σ_n| α² ε^{2/3} + O(ε), with α a fixed negative number set by the interaction strength. The coefficient |σ_n| is the absolute value of the nth extremum of the Airy function Ai when the heavy particles are bosons and the nth zero when they are fermions. The essential spectrum is shown to occupy precisely the half-line beginning at -α²/(4 + ε²). A reader would care because the result supplies the first quantitative correction to the infinite-mass binding energy that arises once the heavy particles are allowed finite mass.

Core claim

For the self-adjoint Hamiltonian that encodes zero-range attractive interactions between a light particle and two heavy particles in one dimension, the eigenvalues E_n(ε) below the essential spectrum satisfy E_n(ε) = -α² + |σ_n| α² ε^{2/3} + O(ε) as ε → 0, where α < 0 is an explicit constant fixed by the physical parameters and σ_n is the nth extremum of the Airy function Ai for bosons or the nth zero of Ai for fermions. The essential spectrum coincides exactly with the half-line [-α²/(4 + ε²), +∞).

What carries the argument

Asymptotic reduction of the three-body eigenvalue problem for small mass ratio ε to a scaled Airy differential equation whose characteristic values fix the leading energy corrections.

Load-bearing premise

The zero-range interactions are attractive and the Hamiltonian is realized as a self-adjoint operator so that the effective-potential problem in the small-ε limit is well-defined and solvable by Airy asymptotics.

What would settle it

Numerical computation of the lowest few eigenvalues of the three-body Hamiltonian for a sequence of decreasing values of ε, followed by checking whether the differences from -α² scale as ε^{2/3} with coefficients matching the Airy extrema or zeros to within the stated O(ε) remainder.

Figures

Figures reproduced from arXiv: 2506.21457 by Andrea Posilicano, Claudio Cacciapuoti, Hamidreza Saberbaghi.

Figure 1
Figure 1. Figure 1: Plot of the eigenvalues of the light-particle Hamiltonian as functions of x: −λ0 is represented by the solid line; −λ1 by the dashed line. By Lemmata 3.1 and 3.2 there follows the following theorem Theorem 3.4. For all α ≥ 0 and x ∈ R, the quadratic form bx and the associated self-adjoint operator hx are non-negative. For all α < 0 and x ∈ R, −α 2 is a lower bound for the quadratic form bx and for the asso… view at source ↗
read the original abstract

We study the self-adjoint Hamiltonian that models the quantum dynamics of a one-dimensional (1D) three-body system consisting of a light particle interacting with two heavy ones through a zero-range force. For an attractive interaction we determine the behavior of the eigenvalues below the essential spectrum in the regime $\varepsilon\ll 1$, where $\varepsilon$ is proportional to the square root of the mass ratio. We show that the $n$-th eigenvalue behaves as $E_{n}(\varepsilon)=-\alpha^{2}+|\sigma_{n}|\alpha^{2}\varepsilon^{2/3}+O(\varepsilon)$, where $\alpha$ is a negative constant that explicitly relates to the physical parameters and $\sigma_{n}$ is either the $n$-th extremum or the $n$-th zero of the Airy function Ai, depending on the kind (respectively, bosons or fermions) of the two heavy particles. Additionally, we prove that the essential spectrum coincides with the half-line $[-\frac{\alpha^2}{4+\varepsilon^{2}},+\infty)$.

Editorial analysis

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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 analyzes the self-adjoint Hamiltonian for a one-dimensional three-body system consisting of one light particle and two heavy particles interacting via zero-range attractive forces. In the Born-Oppenheimer regime ε ≪ 1 (ε proportional to the square root of the mass ratio), it derives the asymptotic expansion of the discrete eigenvalues below the essential spectrum: the n-th eigenvalue satisfies E_n(ε) = −α² + |σ_n| α² ε^{2/3} + O(ε), where α < 0 is an explicit constant determined by the physical parameters of the model, and σ_n denotes either the n-th extremum or the n-th zero of the Airy function Ai according to whether the heavy particles are bosons or fermions. The paper additionally proves that the essential spectrum coincides with the half-line [−α²/(4 + ε²), +∞).

Significance. If the derivations hold, the work supplies a rigorous justification of the Born-Oppenheimer approximation for a singular-interaction three-body problem in one dimension. The reduction to an effective Schrödinger operator whose potential is approximated by a linear |R| term near the origin, yielding Airy eigenvalues with appropriate boundary conditions at R = 0, is a concrete and useful example. The explicit, parameter-free character of the leading correction (α determined directly from the model constants, no fitted quantities) and the precise location of the dissociation threshold (adjusted by the reduced-mass factor 1/(4 + ε²)) are strengths that distinguish the result within the literature on few-body systems with delta interactions.

minor comments (3)
  1. The abstract and introduction should explicitly state the precise definition of the constant α in terms of the interaction strength and masses, so that the claim of an 'explicit' relation can be verified without consulting later sections.
  2. Notation for the two cases (bosonic versus fermionic boundary conditions at R = 0) should be introduced uniformly; currently the distinction between 'extremum' and 'zero' of Ai is clear in the abstract but would benefit from a short table or sentence in the main text.
  3. The O(ε) remainder term is stated without an explicit constant or uniformity statement; adding a brief remark on the range of validity (e.g., for fixed n as ε → 0) would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the supportive summary of our manuscript and for the positive assessment of its significance. We are pleased that the recommendation is for minor revision and that the explicit, parameter-free nature of the leading correction term and the adjusted dissociation threshold are highlighted as strengths.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The claimed asymptotics E_n(ε) = -α² + |σ_n| α² ε^{2/3} + O(ε) follow from reducing the three-body Hamiltonian to an effective one-dimensional Schrödinger operator via the Born-Oppenheimer approximation in the small-mass-ratio regime. The constant α is defined explicitly from the physical parameters of the zero-range attractive interactions, and the Airy-function eigenvalues (with bosonic or fermionic boundary conditions) arise from the standard linear approximation to the effective potential near its minimum. The essential-spectrum threshold [-α²/(4+ε²), +∞) is obtained directly from the reduced-mass factor in the model definition. No step reduces by construction to a fitted parameter or self-citation chain; the derivation is independent of the target result and uses only the stated Hamiltonian and asymptotic regime.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard assumption that a self-adjoint Hamiltonian exists for the zero-range interaction model and on the mathematical properties of the Airy equation that arise from the effective Born-Oppenheimer potential.

free parameters (1)
  • α
    Negative constant explicitly related to the physical parameters (interaction strength and masses) that sets the leading energy scale.
axioms (1)
  • domain assumption The Hamiltonian that models the quantum dynamics is self-adjoint
    The abstract opens by studying the self-adjoint Hamiltonian for the 1D three-body system with zero-range forces.

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