Asymptotic stability of the degree-one vortex in the abelian Yang-Mills-Higgs model: Spectral Theory and Numerics

Fabio Pusateri; Jonas Luhrmann; Jos\'e M. Palacios; Sohrab Shahshahani; Wilhelm Schlag

REVIEW 3 major objections 5 minor 17 references

Asymptotic stability of the degree-one vortex in the abelian Yang-Mills-Higgs model: Spectral Theory and Numerics

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper proves that the linearized operator around the degree-one vortex has one internal mode at a certified frequency and that the Fermi Golden Rule coefficients are negative, so the mode is radiatively damped.

desk verdict Solid certified spectral analysis for the degree-one vortex, but the headline spectral and damping claims rest on two companion papers; referee the in-paper work and the certificates. read the letter →

arxiv 2608.10608 v1 pith:VZPBJZMN submitted 2026-08-11 math.AP math-phmath.MP

Jonas Luhrmann , José M. Palacios , Fabio Pusateri , Wilhelm Schlag , Sohrab Shahshahani This is my paper · ORCID

classification math.APmath-phmath.MP MSC 35P0535P1535Q5135B3547E0565G40

keywords abelianYang-Mills-Higgsdegree-onevortexinternalmodeFermiGoldenRuledistortedFouriertransformthresholdresonanceintervalarithmeticcomputer-assistedproof

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves the spectral facts that a planned stability proof for the degree-one vortex in the abelian Yang-Mills-Higgs model needs. It shows that the linearized operator around the vortex has purely continuous spectrum from $1$ upward, with no threshold resonance, and exactly one discrete gap eigenvalue $\lambda^2$, enclosed in $[0.777471875, 0.77747375]$, with a two-dimensional eigenspace called an internal mode. It also certifies by interval arithmetic that the three Fermi Golden Rule coefficients $D_1, D_2, D_{12}$ are strictly negative, the sign condition under which the internal mode is nonlinearly damped rather than persistent. These are the first of three papers; the companion papers supply the distorted Fourier basis and the normal-form estimates that complete the argument. If the conclusion holds, the internal mode is unique, nondegenerate, and radiatively damped, which is the configuration the companion papers need for asymptotic stability.

What carries the argument

The central object is the supersymmetric factorization $\mathbf L = B^*B$ and its partner $\widetilde{\mathbf L} = BB^* = \operatorname{diag}(L_1, L_2)$, obtained from the Bogomolny equations at self-dual coupling. This identity uncouples the two-component linearized problem into two scalar radial Schrödinger operators, which after conjugation become half-line operators $H_1$ and $H_2$. The proof runs on a division of labor: analytic Frobenius series near $r=0$, certified outward-rounded interval arithmetic on compact intervals, and Volterra, barrier, and tail estimates at infinity, all producing enclosures of the vortex profile, the eigenpair, the distorted Fourier basis, and the Fermi Golden Rule integrals. The sign of those integrals is the mechanism: negative coupling coefficients route energy from the bound state into radiation.

What would settle it

Run the threshold Wronskian and the three Fermi Golden Rule radial integrals with an independent validated interval-arithmetic solver: if the Wronskian interval at $r=10$ contains $0$, or if any of $D_1, D_2, D_{12}$ is certified nonnegative, the paper's central claim is refuted.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is Theorem 1.1 together with Proposition 4.3: the linearized operator $\mathbf M$ has purely absolutely continuous spectrum $[1,\infty)$ on its continuous subspace, the threshold is not resonant, and the unique positive gap eigenvalue $\lambda^2$ lies in $[0.777471875, 0.77747375]$, with eigenspace spanned by $Y_1 = \lambda^{-1}(U\psi,-\psi',0,0)^t$ and $Y_2 = \lambda^{-1}(0,0,U\psi,-\psi')^t$, where $\psi$ is the $L^2_{rdr}$-normalized ground state of $-\Delta + U^2$. The Fermi Golden Rule coefficients $D_1, D_2, D_{12}$, formed from the distorted Fourier transform of the quadratic interaction $Q_s(Y_i,Y_j)$ at the resonant frequency $k_\lambda = \sqrt{4\lambda^2-1}$, are certified negative; the code proves the equivalent positive lower bounds $b^{(0)}_1 > 0.017297889$, $b^{(0)}_2 > 0.047248801$, and $b^{(0)}_{12} > 0.022940050$. Because $C_{12}$ is controlled by Cauchy-Schwarz by $D_1 D_2$, the three negative signs imply positive damping constants $\Gamma_0, \Gamma_1$ in the differential inequality for $|Z_1|^2 + |Z_2|^2$.

Load-bearing premise

The central claim collapses if the companion papers' distorted Fourier basis and normal-form remainder estimates fail, or if the 220-bit interval arithmetic certificate is wrong.

Editorial extensions

If this is right

The linearized flow near the vortex has no trapped state at the bottom of the continuum: the continuous spectrum starts at $1$ and is purely absolutely continuous, so radiation disperses rather than accumulating.
There is exactly one internal frequency, so any equivariant perturbation has a two-dimensional bound-state component whose squared frequency is known to within $2\cdot 10^{-6}$.
With $D_1, D_2, D_{12} < 0$, the leading-order cubic terms in the internal-mode ODE are dissipative, so $|Z_1|^2 + |Z_2|^2$ satisfies a differential inequality with positive damping rates, the mechanism that makes the internal mode decay instead of surviving as a periodic state.
No separate numerical certification of the mixed coefficient $C_{12}$ is needed, since $|C_{12}|^2 \le D_1 D_2$ follows from the positive spectral measure.
These spectral and Fermi Golden Rule inputs are exactly what the companion papers use to construct the linear dispersive estimates and the nonlinear normal-form stability argument.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the companion nonlinear analysis succeeds, the decay rate of the internal-mode amplitude should be governed by the certified quantities $b^{(0)}_1, b^{(0)}_2, b^{(0)}_{12}$, so an independent time-dependent simulation of small equivariant perturbations could test the predicted slow radiative damping rate.
The same hybrid analytic-interval-arithmetic pipeline is portable to $n$-vortices and to non-self-dual couplings, where the internal mode count is not known a priori; the crossing dichotomy and Setô bound suggest the eigenvalue count is the key quantity to certify next.
The paper's theorem is exactly as strong as the underlying interval-arithmetic transcripts and the deferred companion estimates, so an independent implementation of the same validated computations would be a decisive check of the spectral and damping claims.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The reader's take is about right, and I largely agree with it. This paper proves, with a careful mix of analysis and interval arithmetic, the spectral reduction for the linearized operator around the degree-one vortex, the absence of threshold resonances, the uniqueness and sharp enclosure of the internal mode, and certified negativity of the Fermi Golden Rule coefficients. The in-paper work is thorough and internally consistent: the vortex profile estimates, the Seto-count argument, the Wronskian matching, and the tail bounds all hang together. The archived CAPD certificates and transcripts are a real reproducibility artifact, and the fact that no free parameters are fitted is a genuine strength.

Now the soft spots, in proportion. Theorem 1.1(i) — the purely absolutely continuous spectrum on [1,∞) — and the effective nonlinear damping conclusion are not proved in this manuscript. They depend on Proposition 3.7 (the distorted Fourier transform for the matrix operator) and Proposition 4.2 (the normal-form ODE with remainder estimates), both deferred to the not-yet-available companion papers [Luh+26b, Luh+26c]. The paper flags these deferrals explicitly, so it is not hiding anything, but it does mean the headline claims are conditional. The in-paper results are solid, but they are statements about the spectral reduction and the FGR sign assuming those deferred constructions exist and have the stated properties.

The circularity burden is low: the certified enclosures are model-derived, not fitted. The main engineering assumption is that CAPD's 220-bit interval arithmetic and the supplied transcripts are correct; I did not independently re-run the code, and I would not expect a referee to do so either, but spot-checking the transcript format and a few of the stated enclosures is reasonable.

Who is this for? Specialists in soliton stability, spectral theory, and computer-assisted proof. It deserves a serious referee, not a desk rejection. My recommendation: send it to review, with the expectation that the referee checks the in-paper proofs and samples the certificates. Final acceptance should hinge on the companion papers being public and correct, or on the authors agreeing to state Theorem 1.1(i) and the damping claim as conditional on those papers.

Referee Report

3 major / 5 minor

Summary. The paper studies the spectral theory of the linearized operator M around the degree-one vortex in the abelian Yang-Mills-Higgs model at self-dual coupling, in the context of equivariant perturbations. Using the Bogomolny factorization, the authors reduce the spectral analysis of the matrix operator M to two scalar strongly singular half-line operators L1 and L2. They combine analytic estimates (Frobenius expansions, comparison and Volterra arguments, Seto's bound) with CAPD-based interval arithmetic to prove several spectral statements: absence of discrete spectrum and threshold resonance for L1, absence of threshold resonance for L2, and existence of exactly one gap eigenvalue of L2 (and hence of M) with a two-dimensional eigenspace, with certified enclosure λ^2 in [0.777471875,0.77747375]. The paper also certifies strict negativity of the three Fermi Golden Rule coefficients D1,D2,D12, which is the non-degeneracy condition needed for nonlinear damping of the internal mode. A substantial part of the argument, however, is deferred: the distorted Fourier transform for M (Proposition 3.7), the asymptotic and symbol bounds for the generalized eigenfunctions (Proposition 3.5), and the normal-form ODE and remainder estimates connecting the signs of D1,D2,D12 to effective damping (Proposition 4.2) are all stated as results of the companion papers [Luh+26b] and [Luh+26c]. Thus the paper's in-house contributions are Propositions 3.3 and 4.3, conditional on those deferred inputs.

Significance. If the deferred companion papers are valid, this manuscript provides a carefully executed hybrid analytic-numerical proof of a nontrivial spectral statement for a gauge-theory soliton: the linearized vortex has a unique internal mode, no threshold resonance, and a negative definite Fermi Golden Rule form. The methodology is transparent: the interval enclosures are derived from the model equations rather than fitted to the desired output; the analytic-numerical split is clearly documented; and the repository with CAPD code and 220-bit transcripts is a strong reproducibility feature. The proof of the eigenvalue count via Seto's bound, the threshold-resonance exclusion via Wronskian matching, and the certified tail estimates are all substantial in-paper contributions. However, the headline claims in Theorem 1.1(i) and in the abstract's 'effective nonlinear damping' are not proved in this manuscript: they depend on the distorted Fourier construction and on the normal-form analysis that are explicitly deferred to the two companion papers.

major comments (3)

[Section 3.2 and proof of Theorem 1.1] Theorem 1.1(i), the claim that the restriction of M to the continuous spectral subspace has purely absolutely continuous spectrum [1,∞), is not established within this manuscript. The proof of Theorem 1.1 invokes Proposition 3.7, which is introduced with 'We now state, without proof' and is deferred to [Luh+26b, Theorem 1.2], together with Proposition 3.5, whose asymptotic and symbol bounds are also deferred. The purely a.c. statement is stronger than the absence of eigenvalues and threshold resonances proved in Sections 6 and 7; it requires the distorted Fourier basis and its completeness, which are not proved here. The authors should either include the construction and proof of Proposition 3.7 or explicitly restate Theorem 1.1(i) as a conditional theorem dependent on the companion paper.
[Section 4.1, Proposition 4.2 and the abstract] The abstract's assertion that the certified signs yield 'effective nonlinear damping of the internal mode' is not proved in this paper. Proposition 4.2, which contains the normal-form ODE, the remainder estimates, and the step from D1,D2,D12<0 to the differential inequality (4.8), is explicitly deferred to [Luh+26c, Sections 9 and 10]; Remark 4.1 and the paragraph before Proposition 4.2 say exactly this. What the present manuscript proves is Proposition 4.3, namely the strict negativity of the dissipative coefficients, assuming the distorted Fourier representation of Proposition 3.7. The wording of the abstract and of Section 1.4 should be revised so that the nonlinear damping conclusion is presented as conditional on the normal-form analysis in the companion paper.
[Sections 7-9, numerical certification] The central quantitative claims, including the eigenvalue enclosure in Lemma 7.3 and the Fermi Golden Rule lower bounds in (4.9), are certified by CAPD interval arithmetic at 220-bit precision and are reported through transcripts in an external repository. This is a standard and welcome practice in computer-assisted proofs, and the analytic-numerical division is clearly explained. Nevertheless, the correctness of these claims depends on the executing code and environment, which are not independently verified in the manuscript. The authors should state more prominently that the proofs are hybrid and that the numerical certificate is machine-checked only by running the provided code; if a simpler independent verifier for the final inequalities exists, it would reduce this risk.

minor comments (5)

[Throughout] The symbol λ denotes the coupling constant in Section 1.1 and then is redefined as the internal frequency after Theorem 1.1. The redefinition is stated explicitly, but a separate symbol (e.g., ω) for the internal frequency would avoid confusion for readers scanning the paper.
[Section 4.1] The notation H2 is used both for the scalar operator in (3.6) and for the cubic Hamiltonian in (4.2). The authors note this conflict, but renaming the Hamiltonian in (4.2) (e.g., H_cubic) would make the text clearer.
[References] The companion papers [Luh+26b], [Luh+26c], and the certificate repository [Luh+26a2] are cited as preprints without stable identifiers. Since the present paper's main claims depend on the first two, the authors should add arXiv numbers or other permanent identifiers once available, and archive the repository in a persistent service.
[Paragraph before Lemma 5.9] The sentence 'The interval J0 has width 9.0·10^{-13}, whereas that lemma reduces this to 1.263·10^{-16}' could be misleading: Lemma 5.9 actually provides an interval Icert of width 1.57·10^{-14} around the refined value, and the stated 1.263·10^{-16} is the error bound around the midpoint. Please reword to distinguish the two quantities.
[Section 9.2] The sentences defining L1,L2,L12 appear twice, once in the completion of the proof of Proposition 4.3 and once at the beginning of Section 9.2. One of the repetitions should be removed.

Circularity Check

2 steps flagged · score 4.0 of 10

The certified interval computations are not fitted or circular, but the paper's headline spectral and nonlinear-damping conclusions are partly imported from same-author companion papers via load-bearing self-citations.

self citation load bearing [Section 3.2, Proposition 3.7; proof of Theorem 1.1(i)]
"We now state, without proof, the distorted Fourier transform associated withM, whose construction is given in [Luh+26b, Theorem 1.2]. ... Finally, the purely absolutely continuous nature of the spectrum ofM on [1,∞) follows from the distorted Fourier basis in Proposition 3.7, combined with [RS78, Theorem XIII.19]."

Theorem 1.1(i) is a headline spectral conclusion of this paper, but its proof is not contained here: it invokes Proposition 3.7, which the manuscript itself introduces as stated 'without proof' and attributes to the same authors' companion [Luh+26b, Theorem 1.2], together with a resolvent-kernel representation from [Luh+26b, Section 2.4.4]. The absolutely continuous spectrum statement is therefore carried entirely by a same-author citation, not by the in-paper Sections 5-9. Moreover, this same Proposition 3.7 kernel E(r,k) is the object used to rewrite and certify the Fermi Golden Rule coefficients in Section 8, so the deferral is load-bearing for more than one main result.
self citation load bearing [Section 4.1, Proposition 4.2 and abstract]
"In [Luh+26c], following similar ideas to [SW99] (see also [LP]), we prove the following result, which is the leading-order ODE for the internal mode. We refer to [Luh+26c, Sections 9, 10] for a detailed proof and estimates of the “remainders” in (4.6), as well as a more precise statement of the following important proposition."

The abstract advertises that certifying the Fermi Golden Rule coefficients yields 'effective nonlinear damping of the internal mode'. But the step from strict negativity of D1, D2, D12 to the differential inequality (4.8) with Γ0, Γ1 > 0 is Proposition 4.2, whose proof and remainder estimates are deferred to [Luh+26c, Sections 9, 10], another companion by the same authors. Remark 4.1 makes the same deferral explicit. Thus the nonlinear-damping outcome is not derived in this paper from the certified signs; it is imported from a same-author citation whose contents are not presented or independently verified here.

full rationale

The in-paper numerical and analytical work is largely self-contained and not fitted: the vortex profile, the eigenvalue enclosure, the threshold-Wronskian checks, and the strict lower bounds for the FGR coefficients are obtained by outward-rounded interval arithmetic and analytic estimates from the model equations, with no parameter adjusted to force the stated signs or enclosures. No prediction is equal by construction to a fitted input. The circularity burden is moderate and comes exclusively from two load-bearing self-citations: (i) Theorem 1.1(i) rests on Proposition 3.7, which is explicitly stated 'without proof' and deferred to [Luh+26b, Theorem 1.2]; and (ii) the advertised effective nonlinear damping rests on Proposition 4.2, deferred to [Luh+26c, Sections 9, 10]. These are not independent, machine-checked, or externally verifiable within the present manuscript, so they raise the score. The eigenvalue count, absence of threshold resonances, and the FGR sign certification have substantial independent content, so the score is 4 rather than higher.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No physical constants are fitted. The vortex profile, shooting parameter c*, eigenvalue lambda^2, and Fermi Golden Rule coefficients are determined by the model equations and certified by interval enclosures; numerical truncation parameters are computational choices, not free parameters of the central claim. The load-bearing external dependencies are the deferred companion-paper results and the trust assumption in CAPD.

assumptions (7)

ad hoc to paper The distorted Fourier transform for M and the associated asymptotic/symbol bounds of Propositions 3.5 and 3.7 are valid as stated.
Stated without proof in Section 3.2 and deferred to the authors' companion preprint [Luh+26b]. It is used to prove purely absolutely continuous spectrum and to define the Fermi Golden Rule coefficients.
ad hoc to paper The leading-order internal-mode ODE and remainder control of Proposition 4.2 are valid.
Deferred to [Luh+26c, Sections 9 and 10]; the negativity of D1, D2, and D12 implies nonlinear damping only through this external result.
domain assumption CAPD and the multiprecision interval transcripts correctly implement the certified computations.
The numerical proof relies on the CAPD library, 220-bit MpInterval arithmetic, and the supplied transcripts; this is an engineering trust assumption not verifiable by reading.
standard math Reed-Simon spectral theorem XIII.19 applies via the distorted Fourier basis to yield purely absolutely continuous spectrum.
Invoked in the proof of Theorem 1.1; the theorem is standard, but its applicability depends on the deferred distorted Fourier transform construction.
standard math Seto's two-dimensional eigenvalue bound (6.10) is valid for the radial operator L2 minus 1.
External bound from [Set73] used in Proposition 6.4; standard in the field.
standard math The Yang-Chu ratio bound K1/K0 < 1 + 1/(2x) holds.
Used in Lemmas 7.5 and 8.1; standard external estimate.
domain assumption Stuart's gauge (1.10) fixes the gauge and yields the linearized system (1.11).
The analysis is confined to this gauge; the spectral operator M describes the linearized flow in that gauge.

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Pith. "Pith review of Asymptotic stability of the degree-one vortex in the abelian Yang-Mills-Higgs model: Spectral Theory and Numerics." pith.science (2026). https://pith.science/paper/VZPBJZMN

@misc{pith2026260810608,
  author       = {Pith},
  title        = {Pith review of: Asymptotic stability of the degree-one vortex in the abelian Yang-Mills-Higgs model: Spectral Theory and Numerics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VZPBJZMN}},
  note         = {Machine review of arXiv:2608.10608}
}

abstract

This is the first of three papers proving asymptotic stability of the degree-one vortex under equivariant perturbations in the $(1+2)$-dimensional abelian Yang-Mills-Higgs model at self-dual coupling. In the orthogonal gauge, the linearized dynamics are governed by a selfadjoint matrix Schr\"odinger operator $\mathbf{M}$. The super-symmetric partner operator is a diagonal matrix whose diagonal entries are strongly singular radial Schr\"odinger operators on $\mathbb{R}^2$. After a conjugation, this reduces the spectral problem to the analysis of two strongly singular scalar half-line operators. Combining analysis with rigorous interval arithmetic, we prove absence of threshold resonances and show that the discrete spectrum of $\mathbf{M}$ consists of exactly one positive gap eigenvalue (internal mode) with a two-dimensional eigenspace. We also certify that the relevant nonlinear Fermi Golden Rule coefficients form a definite quadratic form, yielding effective nonlinear damping of the internal mode. These spectral inputs form the basis of the stability analysis in the subsequent papers.

Figures

Figures reproduced from arXiv: 2608.10608 by the authors.

**Figure 1.1.** Magnetic vortex (U, a) in the case n = 1. For all values λ > 0 of the coupling constant, the degree-one vortex is (spectrally and variationally) stable, while for n ≥ 2 the n-vortices are stable if λ < 1 and unstable if λ > 1 [GS00]. The case λ = 1 is called the self-dual case. In that case, as observed by Bogomolny [BC89; Bog76]bookJT, the energy functional can be rewritten as a constant plus a sum of squares. Set… view at source ↗

**Figure 5.1.** Illustration of Lemma 5.7. The solid curves show the vortex profile (U, a), and the dashed and dotted curves show the comparison functions defined in (5.22). The boundary inequalities (5.23) are verified by interval arithmetic. These enclosures become highly accurate for large r. last zero of Xs before Y first reaches zero, the exact difference equations give Y (r∗∗) ≥ Ge(r∗∗); integration then gives the same contra… view at source ↗

**Figure 6.1.** Graph of V1(r) on (0, 12) and (10, 25) respectively. 6.1. Absence of discrete spectrum and of a threshold resonance for L1. Next, we give a rigorous computer-assisted proof of the fact that V1 > 1. The argument is analytic on (0, √ 6) and (ρ0, ∞); interval arithmetic is needed only on the compact interval [√ 6, ρ0]. Lemma 6.2. The potential V1 satisfies V1(r) − 1 = (2 − a(r))2 r 2 − 1 2 [PITH_FULL_IMAGE:figures/ful… view at source ↗

Figures from the paper (2 more)

**Figure 7.1.** Figure 7.1: Visualization of the internal mode associated with L2. Lemma 7.4. Let ψ0 = ψ0(r) be the unique radial eigenfunction14 characterized by (L2ψ0)(r) = −ψ ′′ 0 (r) − 1 r ψ ′ 0 (r) + U 2 (r)ψ0(r) = λ 2ψ0(r), r > 0, (7.40) with λ 2 ∈ (0, 1), ψ0(0) = 1, ψ ′ 0 (0) = 0, and ψ0…

**Figure 7.2.** Figure 7.2: Graph of ψ0(r)/K0(κλr) with δ0 = 0.0005. Proof. The pointwise comparison is Lemma 7.4, with the certified endpoint check made at ϱ0 = 8.001. For δ0 = 0.0005, the interval certificate proves U(ϱ0) 2 − (1 − δ0) > 3.8992 · 10−7 . Since U is increasing, the third inequal…

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

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