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arxiv: 2605.17139 · v1 · pith:3MVEMDCKnew · submitted 2026-05-16 · 🪐 quant-ph · math-ph· math.MP· nucl-th

Stable minimum principles for scattering states

Scott Lawrence , Yukari Yamauchi This is my paper

Pith reviewed 2026-05-20 14:46 UTC · model grok-4.3

classification 🪐 quant-ph math-phmath.MPnucl-th
keywords scattering statesminimum principlesS-matrixstability estimatesquantum scatteringCoulomb potentialscattering amplitudes
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The pith

Stable minimum principles characterize quantum scattering states with bounded errors.

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

This paper introduces a family of minimum principles for quantum-mechanical scattering states that stay stable when the boundary conditions at infinity are imposed to encode the S-matrix. Approximate states obeying these principles are proven to differ from the exact scattering states by an amount that remains bounded. The resulting stability estimates directly yield rigorous upper and lower bounds on scattering amplitudes. The construction applies without change to momentum-dependent potentials, long-range Coulomb forces, and both elastic and inelastic scattering off bound states.

Core claim

We present a family of stable minimum principles for scattering states. States that approximately satisfy these minimum principles are shown to have a bounded difference with the true scattering states. These minimum principles and stability estimates can be used to obtain rigorous bounds on scattering amplitudes. We show that these minimum principles are applicable to momentum-dependent potentials, long-range (Coulomb) interactions, and elastic or inelastic scattering of bound states.

What carries the argument

Stable minimum principles that incorporate the outgoing-wave boundary conditions at infinity encoding the S-matrix.

If this is right

  • Approximate states satisfying the minimum principles differ from exact scattering states by a controlled amount.
  • Rigorous bounds on scattering amplitudes follow directly from the stability estimates.
  • The same principles apply unchanged to momentum-dependent potentials.
  • The principles cover long-range Coulomb interactions.
  • Elastic and inelastic bound-state scattering are both included.

Where Pith is reading between the lines

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

  • Numerical implementations could produce scattering amplitudes together with explicit error certificates.
  • The stability estimates might extend to time-dependent or few-body scattering problems.
  • Hybrid schemes combining these minimum principles with existing bound-state variational methods could be explored.

Load-bearing premise

The minimum principles remain stable and produce bounded errors when the boundary conditions at infinity encode the S-matrix for the listed potentials and scattering channels.

What would settle it

A concrete counter-example would be an approximate trial function obeying one minimum principle yet differing from the true scattering solution by an arbitrarily large L2 norm for a long-range Coulomb potential.

Figures

Figures reproduced from arXiv: 2605.17139 by Scott Lawrence, Yukari Yamauchi.

Figure 1
Figure 1. Figure 1: A schematic of the potential defined by ( [PITH_FULL_IMAGE:figures/full_fig_p033_1.png] view at source ↗
read the original abstract

Quantum-mechanical scattering states are energy eigenstates obeying particular boundary conditions, whose behavior at infinity encodes the S-matrix which defines the outcoming of scattering experiments. With an eye toward numerical algorithms for computing nonrelativistic S-matrices, we present a family of stable minimum principles for scattering states. States that approximately satisfy these minimum principles are shown to have a bounded difference with the true scattering states. These minimum principles and stability estimates can be used to obtain rigorous bounds on scattering amplitudes. We show that these minimum principles are applicable to momentum-dependent potentials, long-range (Coulomb) interactions, and elastic or inelastic scattering of bound states.

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 / 2 minor

Summary. The paper introduces a family of stable minimum principles for quantum-mechanical scattering states obeying boundary conditions at infinity that encode the S-matrix. Approximate states satisfying these principles to within a tolerance are shown to differ from the true scattering states by a bounded amount in an appropriate norm. The authors claim these principles and associated stability estimates yield rigorous bounds on scattering amplitudes and apply to momentum-dependent potentials, long-range Coulomb interactions, and elastic or inelastic scattering of bound states.

Significance. If the stability estimates and their propagation to amplitude bounds hold, the work supplies a variational framework with built-in error control for scattering calculations. This is potentially significant for numerical algorithms targeting nonrelativistic S-matrices, especially in regimes with long-range potentials where conventional methods lack rigorous a-posteriori bounds.

major comments (1)
  1. [Section on long-range Coulomb interactions (or equivalent)] The central stability claim (approximate minimizer implies bounded ||psi_approx - psi_true||) must be shown to propagate to a controlled error on the T-matrix or scattering amplitude for the Coulomb case. The manuscript should explicitly derive or cite the required weighted resolvent estimate or modified radiation condition that accounts for the logarithmic phase and slow 1/r decay; without this step the applicability statement for long-range interactions rests on an unverified extension of the short-range argument.
minor comments (2)
  1. Clarify the precise function space and norm in which the bounded difference is proved; the abstract refers to 'bounded difference' without specifying the topology.
  2. Add a short comparison table or paragraph contrasting the new minimum principles with existing variational bounds (e.g., Schwinger or Kohn variational principles) to highlight the stability improvement.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestion regarding the long-range Coulomb case. We address the major comment below and will revise the manuscript to make the error propagation explicit.

read point-by-point responses

  1. Referee: [Section on long-range Coulomb interactions (or equivalent)] The central stability claim (approximate minimizer implies bounded ||psi_approx - psi_true||) must be shown to propagate to a controlled error on the T-matrix or scattering amplitude for the Coulomb case. The manuscript should explicitly derive or cite the required weighted resolvent estimate or modified radiation condition that accounts for the logarithmic phase and slow 1/r decay; without this step the applicability statement for long-range interactions rests on an unverified extension of the short-range argument.

    Authors: We agree that an explicit derivation of the propagation from the stability norm to controlled T-matrix error is needed for the Coulomb case to avoid any appearance of an unverified extension. In the revised manuscript we will add a dedicated subsection that adapts the short-range argument: we introduce the appropriate weighted resolvent estimates in spaces that accommodate the logarithmic phase factor in the modified Sommerfeld radiation condition, control the 1/r decay via suitable weights, and thereby obtain a rigorous bound on the error in the scattering amplitude. Key references on Coulomb scattering theory will be cited to anchor the technical steps. This addition directly addresses the referee's concern while preserving the existing stability result. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation introduces independent minimum principles with stability bounds

full rationale

The paper derives a family of minimum principles for scattering states directly from the Schrödinger equation and boundary conditions at infinity, then proves that approximate minimizers have bounded difference from true states via stability estimates. These steps are self-contained mathematical constructions without reducing to self-citations, fitted parameters renamed as predictions, or ansatzes smuggled from prior work. The applicability to long-range Coulomb and momentum-dependent potentials is asserted via explicit extension of the functional and estimates, not by re-expressing inputs. No load-bearing step collapses to a tautology or self-referential definition.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, axioms, or invented entities; the central claim appears to rest on standard quantum-mechanical boundary conditions and the existence of scattering states for the listed potentials.

pith-pipeline@v0.9.0 · 5625 in / 1139 out tokens · 48722 ms · 2026-05-20T14:46:46.922774+00:00 · methodology

discussion (0)

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Reference graph

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