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arxiv: 1907.03333 · v1 · pith:PA46PCDWnew · submitted 2019-07-07 · 🧮 math-ph · math.MP· math.SP

Resonances in the one dimensional Stark effect in the limit of small field

Richard Froese , Ira Herbst This is my paper

Pith reviewed 2026-05-25 01:16 UTC · model grok-4.3

classification 🧮 math-ph math.MPmath.SP
keywords resonancesStark effectone dimensionsmall field limitreflection coefficientanalytic continuationasymptoticspotential scattering
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The pith

For small electric fields in one dimension, resonances lie near the real axis at the zeros of the analytic continuation of the scattering reflection coefficient and near the line with argument -2π/3.

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

The paper examines how resonances behave in the one-dimensional Stark Hamiltonian when the electric field strength approaches zero. It shows that these resonances cluster near points on or close to the real line that correspond to the zeros of an analytically continued reflection coefficient from the zero-field scattering problem. Additional resonances appear near a specific ray in the complex plane at angle -2π/3. A reader would care because this describes the transition from bound states or scattering states to resonances under a weak electric field perturbation, providing precise asymptotic locations and behaviors.

Core claim

In the small-field limit, the resonances of the one-dimensional Stark Hamiltonian occur near the real axis, specifically near the zeros of the analytic continuation of the reflection coefficient associated with potential scattering, and also near the ray in the complex plane with argument -2π/3. The paper calculates the asymptotics of these resonances.

What carries the argument

The analytic continuation of the reflection coefficient for the potential scattering problem, which locates some of the resonances in the small electric field limit.

If this is right

  • Resonances approach the real axis at specific locations determined by the scattering data of the potential.
  • Their positions and widths have explicit asymptotic expansions as the field strength tends to zero.
  • Some resonances lie along the direction arg z = -2π/3 with particular asymptotic behavior.
  • The results include remarks on possible differences or extensions in higher dimensions.

Where Pith is reading between the lines

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

  • These asymptotic locations could be used to estimate lifetimes or decay rates of states in the presence of a weak field by linking them directly to scattering data.
  • The clustering near reflection zeros might extend to other one-dimensional perturbation problems where analytic continuations of scattering quantities are available.
  • Numerical verification for concrete potentials would test the predicted approach rates to the real axis.

Load-bearing premise

The potential is such that the reflection coefficient for scattering can be analytically continued to find its zeros in the complex plane.

What would settle it

For a specific potential like a finite well, compute the complex eigenvalues of the Stark Hamiltonian for successively smaller field values and check if they approach the predicted zeros of the reflection coefficient or the -2π/3 ray.

read the original abstract

We discuss the resonances of Hamiltonians with constant electric field in one dimension in the limit of small field. These resonances occur near the real axis, near zeros of the analytic continuation of a reflection coefficient for potential scattering, and near the line arg z = -2\pi/3. We calculate their asymptotics. In conclusion we make some remarks about the higher dimensional problem.

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 / 0 minor

Summary. The manuscript analyzes resonances of one-dimensional Schrödinger operators with a constant electric field (Stark Hamiltonian) in the small-field limit. It claims that these resonances occur near the real axis, near zeros of the analytic continuation of the reflection coefficient from the zero-field scattering problem, and near the ray arg z = -2π/3, and derives asymptotic expansions for their locations. Brief remarks on the higher-dimensional case are included.

Significance. If the analytic-continuation hypothesis is justified and the asymptotics are derived rigorously, the work would supply concrete asymptotic formulas linking Stark resonances to zero-field scattering data. Such results could be useful for analyzing tunneling and resonance phenomena under weak electric fields.

major comments (1)
  1. [Abstract] Abstract (and presumably the main text): the identification of resonances with zeros of the analytic continuation of the reflection coefficient r(z) is load-bearing for the central claim. The abstract refers only to 'the potential V' without stating decay, regularity, or support conditions on V that would guarantee the required analytic continuation of r(z) into a region containing those zeros. If the manuscript neither imposes such conditions nor proves the continuation, the location statement rests on an unverified hypothesis rather than a derived property.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need to make the assumptions on V explicit. We address the single major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract (and presumably the main text): the identification of resonances with zeros of the analytic continuation of the reflection coefficient r(z) is load-bearing for the central claim. The abstract refers only to 'the potential V' without stating decay, regularity, or support conditions on V that would guarantee the required analytic continuation of r(z) into a region containing those zeros. If the manuscript neither imposes such conditions nor proves the continuation, the location statement rests on an unverified hypothesis rather than a derived property.

    Authors: We agree that the abstract (and opening paragraphs) should state the hypotheses on V explicitly rather than leaving them implicit. The manuscript works throughout with smooth, compactly supported potentials; under these conditions the reflection coefficient r(z) admits analytic continuation to a neighborhood of the real axis and into the sector containing the zeros under discussion, by standard results on the Jost solutions for compactly supported potentials. We will revise the abstract to read 'for smooth, compactly supported potentials V' and add a short paragraph in the introduction recalling the relevant analyticity statement with a reference to the scattering literature. This makes the central claim rest on stated assumptions rather than an unverified hypothesis. revision: yes

Circularity Check

0 steps flagged

No circularity; claims rest on external scattering theory without self-referential reduction

full rationale

The abstract states resonances occur near zeros of the analytic continuation of a reflection coefficient and near arg z = -2π/3, with asymptotics calculated. No equations or steps are shown that define a quantity in terms of itself, rename a fit as a prediction, or rely on load-bearing self-citations whose content reduces to the present result. The identification with reflection-coefficient zeros is an assumption about V (as noted by the skeptic), but this is an external hypothesis rather than a circular derivation. The paper's central claims therefore remain independent of their own inputs on the basis of the provided text.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete. The central claim rests on the existence of an analytically continuable reflection coefficient for the unperturbed potential.

axioms (1)
  • domain assumption The potential V admits analytic continuation of its reflection coefficient whose zeros locate some resonances.
    Stated in the abstract as the mechanism placing resonances near those zeros.

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Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

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Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages · 2 internal anchors

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