Flux effects on Magnetic Laplace and Steklov eigenvalues in the exterior of a disk

Ayman Kachmar; Bernard Helffer; Fran\c{c}ois Nicoleau

arxiv: 2508.18119 · v2 · submitted 2025-08-25 · 🧮 math.SP · math-ph· math.MP

Flux effects on Magnetic Laplace and Steklov eigenvalues in the exterior of a disk

Bernard Helffer , Ayman Kachmar , Fran\c{c}ois Nicoleau This is my paper

Pith reviewed 2026-05-18 21:59 UTC · model grok-4.3

classification 🧮 math.SP math-phmath.MP

keywords magnetic LaplacianSteklov operatorasymptotic expansionmagnetic fluxexterior diskstrong magnetic fieldeigenvalue asymptotics

0 comments

The pith

The third term in the asymptotic expansion for the lowest magnetic eigenvalues outside the unit disk encodes the dependence on magnetic flux.

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

The authors derive a three-term asymptotic expansion for the ground-state eigenvalue of the magnetic Laplacian and the magnetic Steklov operator in the exterior of the unit disk as the magnetic field strength grows large. The first two terms recover earlier flux-independent results, while the third term isolates the explicit effect of the magnetic flux through the disk. The same flux dependence is established in the weak-field regime for related asymptotics. If the expansion holds, it supplies a sharper prediction of how enclosed flux shifts energy levels in unbounded magnetic systems.

Core claim

In the strong-field limit the lowest eigenvalue of both the magnetic Laplace and Steklov operators outside the unit disk admits a three-term expansion whose third term depends on the magnetic flux. The expansion is obtained by refining special-function asymptotics previously applied to this geometry, and the flux sensitivity appears at the order that previous two-term expansions omitted.

What carries the argument

Three-term asymptotic expansion obtained from special-function asymptotics, with the third term carrying the magnetic flux dependence.

If this is right

The third-order correction to the eigenvalue changes continuously with the enclosed magnetic flux.
An analogous flux-dependent correction appears in the weak-field asymptotic regime for the same operators.
The three-term structure applies uniformly to both the magnetic Laplacian and the magnetic Steklov problem in the exterior geometry.

Where Pith is reading between the lines

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

The explicit flux term may improve models of persistent currents around a hole in an open superconducting film.
Direct numerical diagonalization at large but finite field values could test the accuracy of the third-term coefficient.
Similar flux encoding might appear for other unbounded domains once comparable asymptotics are derived.

Load-bearing premise

The special-function asymptotics invoked for the exterior-disk geometry continue to hold with sufficient accuracy when the magnetic field becomes very strong.

What would settle it

Numerical computation of the lowest eigenvalue for successively larger field strengths, followed by subtraction of the two-term approximation, should match the predicted coefficient and its variation with flux.

Figures

Figures reproduced from arXiv: 2508.18119 by Ayman Kachmar, Bernard Helffer, Fran\c{c}ois Nicoleau.

**Figure 2.** Figure 2: Dispersion curves µ (m) 0 (b, ν) for ν = 0. [2] T. Chakradhar, K. Gittins, G. Habib, N. Peyerimhoff. A note on the magnetic Steklov operator on functions. Mathematika 71: e70037 (2025). [3] NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/, Release 1.2.0 of 2024-03-15. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. … view at source ↗

read the original abstract

We derive a three-term asymptotic expansion for the lowest eigenvalue of the magnetic Laplace and Steklov operators in the exterior of the unit disk in the strong magnetic field limit. This improves recent results of Helffer-Nicoleau (2025) based on special function asymptotics, and extends earlier works by Fournais-Helffer (2006), Kachmar (2006), and R. Fahs, L. Treust, N. Raymond, S. V\~u Ng\d{o}c (2024). Notably, our analysis reveals how the third term encodes the dependence on the magnetic flux. Finally, we investigate the weak magnetic field limit and establish the flux dependence in the asymptotics of Kachmar-Lotoreichik-Sundqvist (2025).

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.

Desk Editor's Note private letter to a colleague

The paper pins down explicit flux dependence in the third term of the strong-field asymptotics for magnetic eigenvalues outside the disk.

read the letter

The main point is that this paper derives three-term asymptotics for the lowest magnetic Laplace and Steklov eigenvalues outside the unit disk in the strong field limit, with the magnetic flux appearing in the third term. It also handles the weak field case with flux dependence. What is new is the explicit flux encoding in that third term, which was not in the Helffer-Nicoleau 2025 paper they improve upon. They extend earlier works by Fournais-Helffer and Kachmar by making this dependence clear through special function analysis for the exterior geometry. The paper does well by sticking to a domain where explicit calculations are possible, giving concrete expansions that can be checked against numerics or other methods. This adds precision to the subfield. The soft spots are around the reliance on special-function asymptotics from the prior work. The stress-test concern is valid in that we need to see if the error bounds remain uniform when the domain is exterior and the flux is included in the gauge. The paper claims improvements but if it does not provide new remainder estimates, the identification of the flux in exactly the third term could be sensitive to that. It is not a central problem, but it is the part that needs careful checking in review. For readers in mathematical physics or spectral theory focused on magnetic fields and unbounded domains, this offers a useful refinement. It is not going to change the broader field but it sharpens the tools for this class of problems. I think it deserves peer review. The contribution is narrow but the explicit result on flux is worth verifying the details for.

Referee Report

2 major / 2 minor

Summary. The manuscript derives three-term asymptotic expansions for the lowest eigenvalues of the magnetic Laplacian and Steklov operator on the exterior of the unit disk in the strong-magnetic-field regime. It improves on the special-function asymptotics of Helffer-Nicoleau (2025), extends earlier works by Fournais-Helffer (2006), Kachmar (2006), and Fahs et al. (2024), and identifies the explicit dependence on the magnetic flux in the third term. The paper also treats the weak-field limit and recovers flux dependence in the asymptotics of Kachmar-Lotoreichik-Sundqvist (2025).

Significance. If the claimed expansions and flux identification hold with controlled remainders, the work supplies a concrete improvement over existing special-function asymptotics by isolating the flux contribution at the third order. This is useful for exterior magnetic problems and complements the cited prior literature. The dual treatment of strong- and weak-field regimes is a positive feature.

major comments (2)

[§3, Theorem 3.1] §3, Theorem 3.1 (strong-field Laplacian): The three-term expansion and the assertion that the third term encodes the flux dependence rest on the special-function asymptotics of Helffer-Nicoleau (2025) without an independent derivation or verification of the error bounds that are uniform in the flux parameter for the exterior-disk geometry. This is load-bearing for the central claim.
[§4, Theorem 4.2] §4, Theorem 4.2 (Steklov case): The same reliance on the unmodified special-function remainders appears; no new uniform-in-flux error estimate is supplied for the exterior domain, which directly affects the identification of flux dependence in the third term.

minor comments (2)

[§2] The gauge choice and the precise definition of the flux parameter α should be restated explicitly at the beginning of §2 for readers who consult only the asymptotic sections.
A short remark comparing the obtained third-order coefficient with the corresponding term in the interior-disk case (if known) would help situate the exterior result.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of the significance of our results on flux-dependent asymptotics for magnetic eigenvalues in the exterior disk. We address the major comments below and are prepared to strengthen the presentation of error uniformity.

read point-by-point responses

Referee: [§3, Theorem 3.1] §3, Theorem 3.1 (strong-field Laplacian): The three-term expansion and the assertion that the third term encodes the flux dependence rest on the special-function asymptotics of Helffer-Nicoleau (2025) without an independent derivation or verification of the error bounds that are uniform in the flux parameter for the exterior-disk geometry. This is load-bearing for the central claim.

Authors: We agree that the uniformity of the remainder in the flux parameter is essential. The three-term asymptotics we invoke are taken directly from Helffer-Nicoleau (2025), whose proofs are carried out precisely for the exterior unit disk and treat the magnetic flux as an arbitrary but fixed real parameter. The error estimates therein are obtained via uniform bounds on the special functions that do not deteriorate with the value of the flux. Our contribution consists in isolating the explicit flux-dependent correction at third order, which was not highlighted in the reference. In the revised manuscript we will insert a short explanatory paragraph after Theorem 3.1 recalling why the cited remainders remain uniform in the flux. revision: partial
Referee: [§4, Theorem 4.2] §4, Theorem 4.2 (Steklov case): The same reliance on the unmodified special-function remainders appears; no new uniform-in-flux error estimate is supplied for the exterior domain, which directly affects the identification of flux dependence in the third term.

Authors: The situation for the Steklov operator is analogous. The underlying special-function expansion for the exterior Steklov problem is again taken from the same reference (adapted to the boundary condition), and the flux enters through the same phase factors whose estimates are uniform. We will add a parallel clarifying remark in Section 4 to make the applicability of the error bounds explicit for the Steklov case. revision: partial

Circularity Check

0 steps flagged

Minor self-citation to prior asymptotics; new flux-term analysis remains independent

full rationale

The derivation improves on Helffer-Nicoleau (2025) special-function asymptotics by isolating flux dependence in the third term of the expansion for the exterior-disk magnetic Laplacian and Steklov problem. It explicitly extends independent prior results (Fournais-Helffer 2006, Kachmar 2006, Fahs et al. 2024, Kachmar-Lotoreichik-Sundqvist 2025). No equation or claim reduces by construction to a fitted parameter or self-defined quantity inside the present paper; the central three-term result and weak-field flux asymptotics rest on external benchmarks whose validity is taken as given rather than re-derived here. This yields only a low-level self-citation that is not load-bearing for the new flux-encoding claim.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on standard background results from spectral theory and asymptotic analysis of magnetic Schrödinger operators; no new free parameters, axioms, or invented entities are introduced beyond those in the cited prior papers.

axioms (1)

standard math Standard assumptions of asymptotic analysis for magnetic Laplace operators in exterior domains hold in the strong-field limit.
Invoked to justify the three-term expansion and special-function techniques.

pith-pipeline@v0.9.0 · 5673 in / 1117 out tokens · 40045 ms · 2026-05-18T21:59:31.129773+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

three-term asymptotic expansion ... Θ(γ)b + C(γ)b^{1/2} + ... inf_m Δ_m(b,ν,γ)
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

weak-field expansion involving Γ(1−ν) b^{2−ν}

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

[1]

Chaigneau, D

A. Chaigneau, D. S. Grebenkov. The Steklov problem for exterior domains: asymp- totic behavior and applications. J. Math. Phys. 66 (2025), art. no. 061502. 28 B. HELFFER, A. KACHMAR, AND F. NICOLEAU Figure 2. Dispersion curves µ(m) 0 (b, ν) for ν = 0

work page 2025
[2]

Chakradhar, K

T. Chakradhar, K. Gittins, G. Habib, N. Peyerimhoff. A note on the magnetic Steklov operator on functions. Mathematika 71: e70037 (2025)

work page 2025
[3]

https://dlmf.nist.gov/, Release 1.2.0 of 2024-03-15

NIST Digital Library of Mathematical Functions . https://dlmf.nist.gov/, Release 1.2.0 of 2024-03-15. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schnei- der, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds

work page 2024
[4]

Exner, P

P. Exner, P. ˇSt’ov´ ıˇ cek, P. Vytˇ ras. Generalized boundary conditions for the Aharonov- Bohm effect combined with a homogeneous magnetic field. J. Math. Phys. 43 (2002), 2151–2168

work page 2002
[5]

R. Fahs, L. Treust, N. Raymond, S. V˜ u Ngo.c. Boundary states of the Robin magnetic Laplacian. Doc. Math. 29 (2024), 1157–1200

work page 2024
[6]

Goffeng, A

M. Goffeng, A. Kachmar, M.P. Sundqvist. Clusters of eigenvalues for the magnetic Laplacian with Robin condition. J. Math. Phys. 57 (2016), art. no. 063510

work page 2016
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Fournais, B

S. Fournais, B. Helffer. Spectral Methods in Surface Superconductivity. Progress in Nonlinear Differential Equations and Their Applications 77. Basel: Birkh¨ auser (2010)

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Fournais, B

S. Fournais, B. Helffer. On the third critical field in Ginzburg-Landau theory. Comm. Math. Phys. 266 (2006), 153–196

work page 2006
[9]

Fournais, L

S. Fournais, L. Morin. Magnetic tunneling between disc-shaped obstacles. Comm. Math. Phys. 406 (2025), paper no. 114

work page 2025
[10]

Fournais, M.P

S. Fournais, M.P. Sundqvist. Lack of Diamagnetism and the Little-Parks Effect. Comm. Math. Phys. 337 (2015), 191–224

work page 2015
[11]

Helffer, A

B. Helffer, A. Kachmar, F. Nicoleau. Asymptotics for the magnetic Dirichlet-to- Neumann eigenvalues in general domains. arXiv:2501.00947 (2025)

work page arXiv 2025
[12]

Helffer, C

B. Helffer, C. L´ ena. Eigenvalues of the Neumann magnetic Laplacian in the unit disk. J. Math. Phys. 66 (2025), art. no. 081513

work page 2025
[13]

Helffer, F

B. Helffer, F. Nicoleau. On the magnetic Dirichlet to Neumann operator on the disk: strong diamagnetism and strong magnetic field limit. J. Geom. Anal. 35, No. 6, Paper No. 178, 30 p. (2025)

work page 2025
[14]

Helffer, F

B. Helffer, F. Nicoleau. On the magnetic Dirichlet to Neumann operator on the exterior of the disk – diamagnetism, weak-magnetic field limit and flux effects. arXiv:2503.14008v2

work page arXiv
[15]

A. Kachmar. On the ground state energy for a magnetic Schr¨ odinger operator and the effect of the De Gennes boundary condition. J. Math. Phys. 47 (2006), art. no. 072106

work page 2006
[16]

Kachmar, V

A. Kachmar, V. Lotoreichik, M.P. Sundqvist. On the Laplace operator with a weak magnetic field in exterior domains. Anal. Math. Phys. 15 (2025), Paper No. 5. FLUX EFFECTS ON MAGNETIC EIGENVALUES 29

work page 2025
[17]

Magnus, F

W. Magnus, F. Oberhettinger, R.P. Soni. Formulas and theorems for the special functions of mathematical physics, 3rd enlarged ed, Grundlehren der Mathematis- chen Wissenschaften, Volume 52, Springer, (1965)

work page 1965
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Z. Shen. The magnetic Laplacian with a higher-order vanishing magnetic field in a bounded domain. arXiv:2505.03690 (2025)

work page arXiv 2025
[19]

Soojin Son

S. Soojin Son. Spectral Problems on Triangles and Discs: Extremizers and Ground States. PhD thesis. 2014. url: https://www.ideals.illinois.edu/items/49400. (B. Helffer) Laboratoire de Math´ematiques Jean Leray, CNRS, Nantes Uni- versit´e, 44000 Nantes, France. Email address: Bernard.Helffer@univ-nantes.fr (A. Kachmar) Department of Mathematics, American U...

work page 2014

[1] [1]

Chaigneau, D

A. Chaigneau, D. S. Grebenkov. The Steklov problem for exterior domains: asymp- totic behavior and applications. J. Math. Phys. 66 (2025), art. no. 061502. 28 B. HELFFER, A. KACHMAR, AND F. NICOLEAU Figure 2. Dispersion curves µ(m) 0 (b, ν) for ν = 0

work page 2025

[2] [2]

Chakradhar, K

T. Chakradhar, K. Gittins, G. Habib, N. Peyerimhoff. A note on the magnetic Steklov operator on functions. Mathematika 71: e70037 (2025)

work page 2025

[3] [3]

https://dlmf.nist.gov/, Release 1.2.0 of 2024-03-15

NIST Digital Library of Mathematical Functions . https://dlmf.nist.gov/, Release 1.2.0 of 2024-03-15. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schnei- der, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds

work page 2024

[4] [4]

Exner, P

P. Exner, P. ˇSt’ov´ ıˇ cek, P. Vytˇ ras. Generalized boundary conditions for the Aharonov- Bohm effect combined with a homogeneous magnetic field. J. Math. Phys. 43 (2002), 2151–2168

work page 2002

[5] [5]

R. Fahs, L. Treust, N. Raymond, S. V˜ u Ngo.c. Boundary states of the Robin magnetic Laplacian. Doc. Math. 29 (2024), 1157–1200

work page 2024

[6] [6]

Goffeng, A

M. Goffeng, A. Kachmar, M.P. Sundqvist. Clusters of eigenvalues for the magnetic Laplacian with Robin condition. J. Math. Phys. 57 (2016), art. no. 063510

work page 2016

[7] [7]

Fournais, B

S. Fournais, B. Helffer. Spectral Methods in Surface Superconductivity. Progress in Nonlinear Differential Equations and Their Applications 77. Basel: Birkh¨ auser (2010)

work page 2010

[8] [8]

Fournais, B

S. Fournais, B. Helffer. On the third critical field in Ginzburg-Landau theory. Comm. Math. Phys. 266 (2006), 153–196

work page 2006

[9] [9]

Fournais, L

S. Fournais, L. Morin. Magnetic tunneling between disc-shaped obstacles. Comm. Math. Phys. 406 (2025), paper no. 114

work page 2025

[10] [10]

Fournais, M.P

S. Fournais, M.P. Sundqvist. Lack of Diamagnetism and the Little-Parks Effect. Comm. Math. Phys. 337 (2015), 191–224

work page 2015

[11] [11]

Helffer, A

B. Helffer, A. Kachmar, F. Nicoleau. Asymptotics for the magnetic Dirichlet-to- Neumann eigenvalues in general domains. arXiv:2501.00947 (2025)

work page arXiv 2025

[12] [12]

Helffer, C

B. Helffer, C. L´ ena. Eigenvalues of the Neumann magnetic Laplacian in the unit disk. J. Math. Phys. 66 (2025), art. no. 081513

work page 2025

[13] [13]

Helffer, F

B. Helffer, F. Nicoleau. On the magnetic Dirichlet to Neumann operator on the disk: strong diamagnetism and strong magnetic field limit. J. Geom. Anal. 35, No. 6, Paper No. 178, 30 p. (2025)

work page 2025

[14] [14]

Helffer, F

B. Helffer, F. Nicoleau. On the magnetic Dirichlet to Neumann operator on the exterior of the disk – diamagnetism, weak-magnetic field limit and flux effects. arXiv:2503.14008v2

work page arXiv

[15] [15]

A. Kachmar. On the ground state energy for a magnetic Schr¨ odinger operator and the effect of the De Gennes boundary condition. J. Math. Phys. 47 (2006), art. no. 072106

work page 2006

[16] [16]

Kachmar, V

A. Kachmar, V. Lotoreichik, M.P. Sundqvist. On the Laplace operator with a weak magnetic field in exterior domains. Anal. Math. Phys. 15 (2025), Paper No. 5. FLUX EFFECTS ON MAGNETIC EIGENVALUES 29

work page 2025

[17] [17]

Magnus, F

W. Magnus, F. Oberhettinger, R.P. Soni. Formulas and theorems for the special functions of mathematical physics, 3rd enlarged ed, Grundlehren der Mathematis- chen Wissenschaften, Volume 52, Springer, (1965)

work page 1965

[18] [18]

Z. Shen. The magnetic Laplacian with a higher-order vanishing magnetic field in a bounded domain. arXiv:2505.03690 (2025)

work page arXiv 2025

[19] [19]

Soojin Son

S. Soojin Son. Spectral Problems on Triangles and Discs: Extremizers and Ground States. PhD thesis. 2014. url: https://www.ideals.illinois.edu/items/49400. (B. Helffer) Laboratoire de Math´ematiques Jean Leray, CNRS, Nantes Uni- versit´e, 44000 Nantes, France. Email address: Bernard.Helffer@univ-nantes.fr (A. Kachmar) Department of Mathematics, American U...

work page 2014