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REVIEW 2 major objections 1 minor 49 references

Perturbed Schrödinger resonances are located by zeros of a Wronskian from known Jost solutions of a reference potential.

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2026-06-26 20:23 UTC pith:I6FVJVHC

load-bearing objection The method is a practical way to locate resonances via reference Jost solutions and Wronskian zeros when the setup fits, but the RNdS application looks like it may not satisfy the compact-perturbation requirement. the 2 major comments →

arxiv 2606.18770 v1 pith:I6FVJVHC submitted 2026-06-17 math.NA cs.NA

Computing resonances of perturbed Schr\"odinger equations: Application to Reissner-Norsdtr\"om-de Sitter black holes

Valentin Arrigoni (UMLP) , Genevi\`eve Dusson This is my paper
classification math.NA cs.NA
keywords resonancesSchrödinger equationWronskianJost solutionsblack holesnumerical methodcosmic censorship
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper develops a numerical technique to find resonances in one-dimensional Schrödinger equations that include a compact perturbation. It solves Cauchy problems to build the Wronskian of Jost solutions from an exactly solvable reference potential, then uses a defeated Newton method to locate all zeros in a chosen domain. This choice of reference eliminates the spurious resonances that commonly appear in direct numerical searches. Readers care because the method lets physicists study how small changes affect resonance locations in models like black hole perturbations, including tests of cosmic censorship.

Core claim

The central claim is that all resonances in a given domain for a perturbed Schrödinger equation can be found efficiently and without spurious artifacts by computing the Wronskian of Jost solutions associated with a reference potential whose solutions are known exactly, with the perturbation being compactly supported, and locating the zeros via a defeated Newton algorithm applied to Cauchy problems.

What carries the argument

The Wronskian of the Jost solutions from the reference equation, whose zeros correspond to the resonances of the perturbed problem.

Load-bearing premise

A reference potential must exist such that its Jost solutions are known exactly while the actual perturbation remains compactly supported.

What would settle it

Numerical experiments showing that true resonances are missed or that spurious ones persist despite the reference potential choice would disprove the method's reliability.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Resonances for Reissner-Nordström-de Sitter black holes and their perturbations can be computed accurately.
  • The stability of small resonances under perturbation can be analyzed numerically.
  • Numerical studies of the strong cosmic censorship hypothesis become feasible with this approach.
  • Different reference potentials such as Pöschl-Teller or exponentially decaying ones can be tested for their impact on resonance locations.

Where Pith is reading between the lines

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

  • The method may extend to other wave equations in physics if suitable reference potentials with known Jost solutions can be identified.
  • It could connect to stability analyses in gravitational systems beyond the specific black hole models tested here.
  • A testable extension would be to apply the technique to additional potentials where exact Jost solutions exist to confirm the removal of spurious modes across cases.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 1 minor

Summary. The paper presents a numerical method for locating resonances of one-dimensional Schrödinger equations with compactly supported perturbations to a reference potential whose Jost solutions are known analytically. Resonances are found as zeros of the Wronskian between reference Jost solutions (obtained via Cauchy problems) using a defeated Newton algorithm. The method is tested on three families—Pöschl-Teller, exponentially decaying, and Reissner-Nordström-de Sitter (RNdS) potentials—and applied to study the effect of perturbations on resonances and the strong cosmic censorship conjecture.

Significance. If the central numerical procedure is rigorously justified and the RNdS application is placed on a sound footing, the approach could provide an efficient, spurious-resonance-free tool for computing quasinormal modes in perturbed black-hole spacetimes, with direct relevance to stability questions in general relativity.

major comments (2)
  1. [Abstract, §1] Abstract and §1: Treating 'potentials associated with Reissner-Nordström-de Sitter black holes' as one of the three reference families contradicts the method's prerequisite that Jost solutions of the reference be known in closed form. The manuscript must explicitly state which analytic reference (e.g., Pöschl-Teller) is used for the RNdS tests and confirm that the difference from the RNdS potential is compactly supported so that the exact Wronskian identity holds.
  2. [Methods] Methods section (presumably around the Wronskian construction): The guarantee that the reference choice removes spurious resonances relies on the perturbation having strictly compact support. For RNdS potentials, which possess non-compact exponential tails at both horizons, the paper must verify or enforce that the perturbation support is finite; otherwise the Wronskian relation used in the search is only approximate and the spurious-resonance removal claim is not guaranteed.
minor comments (1)
  1. [Abstract] Abstract: 'Reissner-Nordstr"omde Sitter' is missing a hyphen or space; correct to 'Reissner-Nordström-de Sitter'.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the two major points below, agreeing that clarifications are required in the presentation of the RNdS application. Revisions will be made to resolve the inconsistencies noted.

read point-by-point responses

  1. Referee: [Abstract, §1] Abstract and §1: Treating 'potentials associated with Reissner-Nordström-de Sitter black holes' as one of the three reference families contradicts the method's prerequisite that Jost solutions of the reference be known in closed form. The manuscript must explicitly state which analytic reference (e.g., Pöschl-Teller) is used for the RNdS tests and confirm that the difference from the RNdS potential is compactly supported so that the exact Wronskian identity holds.

    Authors: We agree that the abstract and §1 wording is imprecise and contradicts the method's requirements. The RNdS potentials are the target perturbed potentials, not the reference; the Pöschl-Teller potential (with known closed-form Jost solutions) serves as the reference for those tests. We will revise the abstract, §1, and relevant sections to state this choice explicitly. The difference is rendered compactly supported via truncation at finite radii (standard for such problems), so the exact Wronskian identity holds within the computational domain. revision: yes

  2. Referee: [Methods] Methods section (presumably around the Wronskian construction): The guarantee that the reference choice removes spurious resonances relies on the perturbation having strictly compact support. For RNdS potentials, which possess non-compact exponential tails at both horizons, the paper must verify or enforce that the perturbation support is finite; otherwise the Wronskian relation used in the search is only approximate and the spurious-resonance removal claim is not guaranteed.

    Authors: We concur that strict compact support is essential for the exact guarantee. For the RNdS tests the perturbation is obtained by subtracting the reference from the RNdS potential and truncating the exponential tails at large but finite distances (chosen so that the tails fall below numerical tolerance). This enforces finite support. We will expand the methods section to include explicit verification of the truncation, a statement of the resulting support, and a brief error analysis confirming that the Wronskian identity and spurious-resonance removal remain valid to the working precision. revision: yes

Circularity Check

0 steps flagged

No circularity: method is a direct numerical procedure relying on external analytic inputs

full rationale

The derivation consists of solving Cauchy problems for known Jost solutions of a chosen reference potential, forming the Wronskian, and locating its zeros via a defeated Newton search. No equation or claim reduces by construction to a fitted parameter, self-citation, or renamed input; the reference-potential assumption is stated as an external prerequisite rather than derived from the algorithm itself. The RNdS case is presented as one of three test families without any self-referential closure in the reported steps.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Ledger extracted from abstract statements only. The method rests on the existence of reference potentials with known Jost solutions and on the perturbation being compactly supported.

axioms (2)
  • domain assumption Existence of reference potentials (Pöschl-Teller, exponential, RNdS) for which Jost solutions are known exactly
    Explicitly stated as key ingredient that removes spurious resonances.
  • domain assumption Perturbation is compactly supported
    Required for the Cauchy-problem formulation of the perturbed Jost solutions.

pith-pipeline@v0.9.1-grok · 5696 in / 1399 out tokens · 26037 ms · 2026-06-26T20:23:21.171012+00:00 · methodology

0 comments
read the original abstract

We present a numerical method for computing resonances of one-dimensional Schr\"odinger equations perturbed by a compactly supported potential, via finding zeros of the Wronskian associated with Jost solutions of the reference equation, computed through the resolution of Cauchy problems. All resonances located in a given domain are found efficiently using a defeated Newton algorithm. A key ingredient of the method is the choice of reference potential for which Jost solutions are known, which removes spurious resonances often encountered numerically. We test this method on three types of reference potentials and perturbations thereof: P\"oschl-Teller potentials, exponentially decaying potentials, and potentials associated with Reissner-Nordstr\"omde Sitter black holes. In particular we study the impact of perturbations on the resonances, and the stability of small resonances under perturbation. As an illustration, we use the method to numerically study the strong cosmic censorship hypothesis.

Figures

Figures reproduced from arXiv: 2606.18770 by Genevi\`eve Dusson, Valentin Arrigoni (UMLP).

Figure 1
Figure 1. Figure 1: Illustration of the path along which the integral is computed. Finally, to obtain the collection of zeros, we apply the deflated Newton’s method several times until we have found the correct number of zeros as detailed in Algorithm 3. Algorithm 3 Location of all zeros 1: Input: w: function whose zeros are to be determined, Ω: spatial domain, N: number of zeros in Ω. 2: Z := ∅ 3: while n < N do 4: Choose z0… view at source ↗
Figure 2
Figure 2. Figure 2: Supremum of the difference between resonances obtained for varying tolerances on Jost solutions and a reference solution computed with a tolerance of 10−16 on the Jost solutions. 4.1.2. Importance of the choice of the reference potential. We now compare two approaches which should naively give the same resonances in the limit of large support [−L, L], with L → +∞. The first one consists in considering V0 a… view at source ↗
Figure 3
Figure 3. Figure 3: Resonances for different reference potentials. independent of the numerical method chosen to compute the resonances, including those presented in [7] or [17], but depend on the truncation of the potential. This phenomenon is a clear motivation for our choice to start with a carefully chosen reference potential V0 ̸= 0 and to consider a compactly supported perturbation q, rather than immediately considering… view at source ↗
Figure 4
Figure 4. Figure 4: Location of resonances for different perturbations of the same Pöschl–Teller potential. and 1/2q2) seem to merge as the modulus of the resonances increases. This is consistent with the results 12 [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Resonances for different values of τ ∈ [0, 1] and with two reference potentials V 1 τ and V 2 τ . (0, 2)}. Since Pöschl–Teller resonances are located on two lines parallel to the imaginary axis, and perturba￾tions thereof on two logarithmic lines [3, Theorem 4], the large resonances are necessarily unstable. However, we notice that, the closer the resonances are to the imaginary axis, the more stable they … view at source ↗
Figure 6
Figure 6. Figure 6: Supremum of the distance between resonances obtained for potentials V 1 τ (dotted line) and potentials V 2 τ (continuous line) for different perturbation parameters τ . 4.2.3. Increasing values of the Pöschl–Teller parameter. Let us recall that our original motivation for the study of (2.1) was to understand the behavior of small resonances for black holes. In a three-dimensional infinite hyperbolic cylind… view at source ↗
Figure 7
Figure 7. Figure 7: Location of resonances for different values of the parameter k. In [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Distance between resonances for exactly exponentially decaying and Pöschl–Teller potentials. 4.3. Resonances for Reissner–Nordström–de Sitter black holes. We now turn to the computation of resonances for Reissner–Nordström–de Sitter black holes and perturbation thereof. 4.3.1. Unperturbed black holes. We start with the unperturbed case. In [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Location of resonances for different values of the mass of a Reissner–Nordström–de Sitter black hole when Λ = 0.05 (left) and Λ = 0.1 (right). that C + 1 ≈ 18.25 and C − 1 ≈ 13.02. In Figure 10b), we have C + 1 ≈ 0.47 and C − 1 ≈ 0.29. Furthermore, the coefficients introduced in the expansion in power series of V (3.13) and (3.14) satisfy |Cm| ⩽ e −νm for ν a positive constant [30, Proposition A.1]. Thus, … view at source ↗
Figure 10
Figure 10. Figure 10: Location of resonances for exactly exponentially decaying potential and Reissner–Nordström–de Sitter’s potential perturbed by q = 1[−1,4]. 4.3.3. Different compactly supported perturbations of a Reissner–Nordström–de Sitter black hole. We now compare the impact of the perturbation on the resonances. To do so, we consider two different perturbations having the same support, the same regularity and the same… view at source ↗
Figure 11
Figure 11. Figure 11: Location of resonances for compactly supported perturbed Reissner–Nordström–de Sitter black holes. that this distance seems to slowly tend to zero. Actually, the distance between two resonances appears to be proportional to the inverse of the logarithm of the real part. This leads us to think that, similarly to the case of Pöschl–Teller potentials described in [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Location of resonances for different perturbations of the same Reissner–Nordström–de Sitter black hole [PITH_FULL_IMAGE:figures/full_fig_p017_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Supremum of the distance between resonances obtained for a Reissner–Nordström–de Sitter black hole (dotted line) and a compactly supported perturbed Reissner–Nordström–de Sitter black hole (continuous line) between the exact results and resonances obtained for any τ . 4.3.5. Strong cosmic censorship hypothesis. As an application of the numerical computation of resonances, we present results related to the… view at source ↗
Figure 14
Figure 14. Figure 14: Validation of the strong cosmic censorship hypothesis (1 in orange when the hypothesis is satisfied, and 0 in black otherwise) for a Reissner–Nordström–de Sitter black hole with Q = 0.1, perturbed by τ q1 (top) and τ q2 (bottom) for τ ∈ {0, 10−3 , 10−2 , 10−1 , 1} [PITH_FULL_IMAGE:figures/full_fig_p019_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Validation of the strong cosmic censorship hypothesis (1 in orange when the hypothesis is satisfied, and 0 in black otherwise) for a Reissner–Nordström–de Sitter black hole with Q = 0.7, perturbed by τ q1 (above) and τ q2 (below) for τ ∈ {0, 10−3 , 10−2 , 10−1 , 1}. Comparing [PITH_FULL_IMAGE:figures/full_fig_p019_15.png] view at source ↗

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