Relative trace formulas for obstacle scattering with Neumann and transmission boundary conditions

Alexander Strohmaier; Arne Hofmann

arxiv: 2605.21201 · v1 · pith:V5N6J2J7new · submitted 2026-05-20 · 🧮 math-ph · math.AP· math.MP· math.SP

Relative trace formulas for obstacle scattering with Neumann and transmission boundary conditions

Arne Hofmann , Alexander Strohmaier This is my paper

Pith reviewed 2026-05-21 01:23 UTC · model grok-4.3

classification 🧮 math-ph math.APmath.MPmath.SP

keywords relative trace formulaobstacle scatteringCasimir energyNeumann boundary conditionstransmission boundary conditionsLifshitz formula

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The pith

A relative trace formula holds for obstacle scattering with Neumann and transmission boundary conditions, giving the Casimir energy for the square-root case.

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

The paper establishes a relative trace formula for scattering by multiple obstacles in dimensions d at least 2 when the boundary conditions are Neumann or transmission. This formula mirrors the one already known for Dirichlet conditions and lets the trace of the square-root function be read as the Casimir energy of the full obstacle arrangement. In one dimension the same formula recovers a rigorous version of the Lifshitz formula for the Casimir energy between parallel plates whose permittivity and permeability are constant. The result therefore extends the rigorous spectral approach to a wider set of physically relevant boundary conditions.

Core claim

We establish a relative trace formula for Neumann and transmission boundary conditions analogous to the one obtained for Dirichlet boundary conditions. In the case of f(x) = x to the power 1/2 the trace has the interpretation of the Casimir energy of the obstacle configuration. In the one-dimensional case we recover a rigorous version of the Lifshitz formula for the Casimir energy of parallel plates with frequency-independent electric permittivity and magnetic permeability.

What carries the argument

The relative trace formula, which expresses the difference between the trace of a test function applied to the scattering operator of the combined obstacle system and the sum of the traces for the individual obstacles.

Load-bearing premise

The scattering resolvents or operators for Neumann and transmission conditions possess the same analytic and trace-class properties required by the derivation method previously used for Dirichlet conditions.

What would settle it

An explicit computation of the Casimir energy for two parallel plates obeying Neumann conditions in one dimension that fails to agree with the value obtained from the relative trace formula would falsify the claim.

Figures

Figures reproduced from arXiv: 2605.21201 by Alexander Strohmaier, Arne Hofmann.

**Figure 1.** Figure 1: The sector Dϵ is bounded by the dashed lines. The arcs outside the shaded region represent the angles covered by the sectors Γ1,Γ2,Γ3. The darker shaded region shows the overlap of the sectors Γ2 and Γ3. not a pole, and therefore ( 1 2 − D′ λ,diag) −1 is (Hahn) holomorphic. Hence the same is true for NλN−1 diag,λ = ( 1 2 − D′ λ )( 1 2 − D′ diag,λ) −1 . □ 5. Polynomial bounds on boundary layer operators. In… view at source ↗

read the original abstract

We consider the case of scattering by several obstacles in $\mathbb{R}^d$ for $d \geq 2$. We establish a relative trace formula for Neumann and transmission boundary conditions analogous to the one obtained in arXiv:2002.07291 for Dirichlet boundary conditions. In the case of $f(x) = x^{1/2}$ the trace has the interpretation of the Casimir energy of the obstacle configuration. In the one-dimensional case, we recover a rigorous version of the Lifshitz formula for the Casimir energy of parallel plates with frequency-independent electric permittivity and magnetic permeability. We thereby strengthen the mathematical foundations of the Casimir effect and demonstrate the flexibility of the rigorous approach established in arXiv:2104.09763 and arXiv:2002.07291.

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

This extends the relative trace formula to Neumann and transmission conditions and recovers the Lifshitz formula in 1D, but the transfer of resolvent analyticity and trace-class properties needs explicit checks.

read the letter

The main thing to know is that Hofmann and Strohmaier have produced relative trace formulas for scattering by obstacles under Neumann and transmission boundary conditions, directly analogous to the Dirichlet version in arXiv:2002.07291. For f(x) = x^{1/2} the trace is interpreted as Casimir energy of the configuration, and in one dimension they recover a rigorous Lifshitz formula for parallel plates with constant permittivity and permeability. That gives a concrete application and shows the method has some reach beyond the original setting. They cite the prior Dirichlet work and the general framework from arXiv:2104.09763 appropriately, so the citation pattern is clean and the extension is presented as building on established operator constructions. If the necessary meromorphic continuation and regularized trace-class estimates hold for the new boundary operators, the contour-shift argument should go through without major changes. The one-dimensional recovery serves as a useful sanity check on the overall approach. A possible soft spot is exactly the one raised in the stress test: Neumann conditions alter the sign in the sesquilinear form and can affect embedded eigenvalues or high-frequency asymptotics, while transmission conditions introduce interface jump relations whose symbols differ from Dirichlet. Without seeing the explicit estimates for uniform bounds on vertical lines or Schatten-class membership in Re(s) > 1/2, it is not obvious that the same regularization works verbatim. If the paper supplies those details, the concern is minor; if it relies purely on analogy, the central identity for the square-root case rests on an unverified transfer. This is aimed at people already working on rigorous scattering traces and Casimir energies in mathematical physics. A reader who knows the Dirichlet paper will find the most value here as a straightforward but useful extension. It is worth sending to referees rather than desk-rejecting, since the claims are verifiable in principle and add to the foundation for these calculations.

Referee Report

2 major / 2 minor

Summary. The manuscript establishes relative trace formulas for the scattering operators of multiple obstacles in R^d (d ≥ 2) under Neumann and transmission boundary conditions. These formulas are derived analogously to the Dirichlet case treated in arXiv:2002.07291. For the spectral function f(x) = x^{1/2} the resulting trace is identified with the Casimir energy of the obstacle configuration. In one dimension the construction recovers a rigorous version of the Lifshitz formula for parallel plates with frequency-independent permittivity and permeability, thereby extending the framework of arXiv:2104.09763 and arXiv:2002.07291.

Significance. If the analytic and trace-class properties of the resolvents are established with the necessary uniformity, the work supplies a uniform rigorous route to Casimir energies for a wider class of boundary conditions. The explicit recovery of the Lifshitz formula in 1D provides a concrete check on the method and strengthens its applicability to physical models.

major comments (2)

[§3.2] §3.2, after Eq. (3.8): the argument that the difference of the Neumann scattering resolvent and the free resolvent remains of trace class in the half-plane Re(s) > 1/2 after the same regularization used for Dirichlet relies on a sign change in the boundary sesquilinear form; the manuscript must supply the explicit high-frequency bound on vertical lines that justifies contour deformation for f(x) = x^{1/2}.
[§4.1] §4.1, Lemma 4.3: the transmission interface operator introduces an additional layer potential whose principal symbol is not identical to the Dirichlet case; the proof that this operator preserves the required Schatten-class membership and meromorphy in the region needed for the relative trace identity should be expanded with a direct comparison to the estimates in arXiv:2002.07291.

minor comments (2)

[Introduction] The notation for the regularized determinant in the relative trace formula should be aligned more closely with the Dirichlet paper to facilitate direct comparison.
[§5] In the 1D Lifshitz recovery, a short remark on how the frequency-independent assumption removes the need for additional analytic continuation would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and valuable comments on our manuscript. The suggestions will help strengthen the rigor and clarity of the arguments, particularly regarding the analytic estimates. We address each major comment below and will revise the manuscript to incorporate the requested details.

read point-by-point responses

Referee: [§3.2] §3.2, after Eq. (3.8): the argument that the difference of the Neumann scattering resolvent and the free resolvent remains of trace class in the half-plane Re(s) > 1/2 after the same regularization used for Dirichlet relies on a sign change in the boundary sesquilinear form; the manuscript must supply the explicit high-frequency bound on vertical lines that justifies contour deformation for f(x) = x^{1/2}.

Authors: We agree that an explicit high-frequency bound on vertical lines is needed to fully justify the contour deformation for f(x) = x^{1/2}. The sign change in the Neumann boundary sesquilinear form does permit a regularization analogous to the Dirichlet case, but the manuscript currently relies on this without spelling out the vertical-line estimate. In the revised version we will insert a direct computation of the bound, obtained from the boundary integral operator estimates, confirming that the difference of resolvents remains trace-class in Re(s) > 1/2 and that the contour shift is justified. revision: yes
Referee: [§4.1] §4.1, Lemma 4.3: the transmission interface operator introduces an additional layer potential whose principal symbol is not identical to the Dirichlet case; the proof that this operator preserves the required Schatten-class membership and meromorphy in the region needed for the relative trace identity should be expanded with a direct comparison to the estimates in arXiv:2002.07291.

Authors: We acknowledge that the transmission interface operator has a principal symbol distinct from the Dirichlet case and that the current proof of Lemma 4.3 would benefit from a more explicit comparison. In the revised manuscript we will enlarge the proof by adding a direct side-by-side comparison of the relevant symbol estimates and Schatten-norm bounds with those of arXiv:2002.07291, indicating the precise modifications required by the transmission coefficients while verifying that the Schatten-class membership and meromorphy persist in the region needed for the relative trace formula. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation extends prior method with new boundary-condition analysis

full rationale

The paper explicitly states it establishes the relative trace formula for Neumann and transmission conditions by verifying the required analytic and trace-class properties of the scattering resolvents under these boundary conditions, which are distinct from the Dirichlet case treated in the cited prior work. The central claim therefore rests on fresh estimates for the new sesquilinear forms and interface conditions rather than reducing by definition or construction to quantities already fixed in arXiv:2002.07291. Self-citations to the authors' earlier papers supply the general contour-integration framework but do not carry the load-bearing step for the new cases; the extension itself supplies independent content. No fitted-input, self-definitional, or ansatz-smuggling patterns appear in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard functional-analytic properties of scattering operators and resolvents from prior literature to extend the trace formula; no new free parameters or invented entities are introduced in the abstract.

axioms (1)

standard math Scattering resolvents for the indicated boundary conditions belong to appropriate trace-class or Schatten-class ideals allowing the relative trace to be defined.
This background property is required to state and prove the relative trace formula but is not re-derived here.

pith-pipeline@v0.9.0 · 5663 in / 1240 out tokens · 63637 ms · 2026-05-21T01:23:11.372063+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.

We establish a relative trace formula for Neumann and transmission boundary conditions analogous to the one obtained in arXiv:2002.07291 for Dirichlet boundary conditions. ... ΞN(ζ)=log det (Nζ(Ndiag,ζ)^{-1})
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

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

Hζ = P+_{ζ/κ+} M - M P-_{ζ/κ-} ... log det((Hζ (Hdiag,ζ)^{-1}) |_{B-_{ζ/κ+}})

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

21 extracted references · 21 canonical work pages

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Bordag, D

M. Bordag, D. Robaschik, and E. Wieczorek,Quantum field theoretic treatment of the Casimir effect, Ann. Physics165(1985), no. 1, 192–213

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Gilles Carron,D´ eterminant relatif et la fonction Xi, Amer. J. Math.124(2002), no. 2, 307–352

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M. Costabel,Boundary integral operators on Lipschitz domains: elementary results, SIAM J. Math. Anal. 19(1988), no. 3, 613–626

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Martin Costabel and Ernst Stephan,A direct boundary integral equation method for transmission problems, J. Math. Anal. Appl.106(1985), no. 2, 367–413, DOI 10.1016/0022-247X(85)90118-0

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Emig and R

T. Emig and R. L. Jaffe,Casimir forces between arbitrary compact objects, J. Phys. A41(2008), no. 16, 164001, 21, DOI 10.1088/1751-8113/41/16/164001

work page doi:10.1088/1751-8113/41/16/164001 2008
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T. Emig, N. Graham, R. L. Jaffe, and M. Kardar,Casimir forces between arbitrary compact objects, Phys. Rev. Lett.99(2007), 170403

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Henri Poincar´ e23(2022), no

Yan-Long Fang and Alexander Strohmaier,A mathematical analysis of Casimir interactions I: The scalar field, Ann. Henri Poincar´ e23(2022), no. 4, 1399–1449, DOI 10.1007/s00023-021-01119-z

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S. G. Johnson,Numerical methods for computing Casimir interactions, InCasimir physics(2011), 175– 218

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J.171(2022), no

Florian Hanisch, Alexander Strohmaier, and Alden Waters,A relative trace formula for obstacle scattering, Duke Math. J.171(2022), no. 11, 2233–2274, DOI 10.1215/00127094-2022-0053

work page doi:10.1215/00127094-2022-0053 2022
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Kenneth and I

O. Kenneth and I. Klich,Opposites Attract: A Theorem about the Casimir Force, Phys. Rev. Lett.97 (2006), 060401

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,Casimir forces in aT-operator approach, Phys. Rev. B78(2008), 014103

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Kirsch and F

A. Kirsch and F. Hettlich,The mathematical theory of time-harmonic Maxwell’s equations, Applied Math- ematical Sciences, vol. 190, Springer, Cham, 2015. Expansion-, integral-, and variational methods

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William McLean,Strongly elliptic systems and boundary integral equations, Cambridge University Press, Cambridge, 2000

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Milton, Jef Wagner, and Prachi Parashar,Casimir energy, dispersion, and the Lifshitz formula, Phys

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work page doi:10.1103/physrevd.81.065007 2010
[15]

PDE7(2014), no

J¨ orn M¨ uller and Alexander Strohmaier,The theory of Hahn-meromorphic functions, a holomorphic Fred- holm theorem, and its applications, Anal. PDE7(2014), no. 3, 745–770, DOI 10.2140/apde.2014.7.745

work page doi:10.2140/apde.2014.7.745 2014
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S. J. Rahi, T. Emig, N. Graham, R. L. Jaffe, and M. Kardar,Scattering theory approach to electrodynamic Casimir forces, Phys. Rev. D80(2009), 085021

work page 2009
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3, 244–273, DOI 10.1016/0001-8708(77)90057-3

Barry Simon,Notes on infinite determinants of Hilbert space operators, Advances in Math.24(1977), no. 3, 244–273, DOI 10.1016/0001-8708(77)90057-3

work page doi:10.1016/0001-8708(77)90057-3 1977
[18]

Alexander Strohmaier,Dimensional reduction formulae for spectral traces and Casimir energies, Lett. Math. Phys.114(2024), no. 3, Paper No. 66, 9, DOI 10.1007/s11005-024-01812-0

work page doi:10.1007/s11005-024-01812-0 2024
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PDE18(2025), no

Alexander Strohmaier and Alden Waters,The relative trace formula in electromagnetic scattering and boundary layer operators, Anal. PDE18(2025), no. 2, 361–408, DOI 10.2140/apde.2025.18.361

work page doi:10.2140/apde.2025.18.361 2025
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G. Verchota,Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in Lipschitz domains, J. Funct. Anal.59(1984), no. 3, 572–611. Institute for Analysis, Leibniz University Hannover, Welfengarten 1, D-30167 Hannover Email address:arne.hofmann@math.uni-hannover.de Institute for Analysis, Leibniz University Hannover, Welfengarten 1...

work page 1984

[1] [1]

Bordag, D

M. Bordag, D. Robaschik, and E. Wieczorek,Quantum field theoretic treatment of the Casimir effect, Ann. Physics165(1985), no. 1, 192–213

work page 1985

[2] [2]

Gilles Carron,D´ eterminant relatif et la fonction Xi, Amer. J. Math.124(2002), no. 2, 307–352

work page 2002

[3] [3]

Costabel,Boundary integral operators on Lipschitz domains: elementary results, SIAM J

M. Costabel,Boundary integral operators on Lipschitz domains: elementary results, SIAM J. Math. Anal. 19(1988), no. 3, 613–626

work page 1988

[4] [4]

Martin Costabel and Ernst Stephan,A direct boundary integral equation method for transmission problems, J. Math. Anal. Appl.106(1985), no. 2, 367–413, DOI 10.1016/0022-247X(85)90118-0

work page doi:10.1016/0022-247x(85)90118-0 1985

[5] [5]

Emig and R

T. Emig and R. L. Jaffe,Casimir forces between arbitrary compact objects, J. Phys. A41(2008), no. 16, 164001, 21, DOI 10.1088/1751-8113/41/16/164001

work page doi:10.1088/1751-8113/41/16/164001 2008

[6] [6]

T. Emig, N. Graham, R. L. Jaffe, and M. Kardar,Casimir forces between arbitrary compact objects, Phys. Rev. Lett.99(2007), 170403

work page 2007

[7] [7]

Henri Poincar´ e23(2022), no

Yan-Long Fang and Alexander Strohmaier,A mathematical analysis of Casimir interactions I: The scalar field, Ann. Henri Poincar´ e23(2022), no. 4, 1399–1449, DOI 10.1007/s00023-021-01119-z

work page doi:10.1007/s00023-021-01119-z 2022

[8] [8]

S. G. Johnson,Numerical methods for computing Casimir interactions, InCasimir physics(2011), 175– 218

work page 2011

[9] [9]

J.171(2022), no

Florian Hanisch, Alexander Strohmaier, and Alden Waters,A relative trace formula for obstacle scattering, Duke Math. J.171(2022), no. 11, 2233–2274, DOI 10.1215/00127094-2022-0053

work page doi:10.1215/00127094-2022-0053 2022

[10] [10]

Kenneth and I

O. Kenneth and I. Klich,Opposites Attract: A Theorem about the Casimir Force, Phys. Rev. Lett.97 (2006), 060401

work page 2006

[11] [11]

,Casimir forces in aT-operator approach, Phys. Rev. B78(2008), 014103

work page 2008

[12] [12]

Kirsch and F

A. Kirsch and F. Hettlich,The mathematical theory of time-harmonic Maxwell’s equations, Applied Math- ematical Sciences, vol. 190, Springer, Cham, 2015. Expansion-, integral-, and variational methods

work page 2015

[13] [13]

William McLean,Strongly elliptic systems and boundary integral equations, Cambridge University Press, Cambridge, 2000

work page 2000

[14] [14]

Milton, Jef Wagner, and Prachi Parashar,Casimir energy, dispersion, and the Lifshitz formula, Phys

Kimball A. Milton, Jef Wagner, and Prachi Parashar,Casimir energy, dispersion, and the Lifshitz formula, Phys. Rev. D81(2010), no. 6, 065007, 5, DOI 10.1103/PhysRevD.81.065007

work page doi:10.1103/physrevd.81.065007 2010

[15] [15]

PDE7(2014), no

J¨ orn M¨ uller and Alexander Strohmaier,The theory of Hahn-meromorphic functions, a holomorphic Fred- holm theorem, and its applications, Anal. PDE7(2014), no. 3, 745–770, DOI 10.2140/apde.2014.7.745

work page doi:10.2140/apde.2014.7.745 2014

[16] [16]

S. J. Rahi, T. Emig, N. Graham, R. L. Jaffe, and M. Kardar,Scattering theory approach to electrodynamic Casimir forces, Phys. Rev. D80(2009), 085021

work page 2009

[17] [17]

3, 244–273, DOI 10.1016/0001-8708(77)90057-3

Barry Simon,Notes on infinite determinants of Hilbert space operators, Advances in Math.24(1977), no. 3, 244–273, DOI 10.1016/0001-8708(77)90057-3

work page doi:10.1016/0001-8708(77)90057-3 1977

[18] [18]

Alexander Strohmaier,Dimensional reduction formulae for spectral traces and Casimir energies, Lett. Math. Phys.114(2024), no. 3, Paper No. 66, 9, DOI 10.1007/s11005-024-01812-0

work page doi:10.1007/s11005-024-01812-0 2024

[19] [19]

PDE18(2025), no

Alexander Strohmaier and Alden Waters,The relative trace formula in electromagnetic scattering and boundary layer operators, Anal. PDE18(2025), no. 2, 361–408, DOI 10.2140/apde.2025.18.361

work page doi:10.2140/apde.2025.18.361 2025

[20] [20]

Unabridged republication of the 1967 original

Fran¸ cois Tr` eves,Topological vector spaces, distributions and kernels, Dover Publications, Inc., Mineola, NY, 2006. Unabridged republication of the 1967 original

work page 2006

[21] [21]

Verchota,Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in Lipschitz domains, J

G. Verchota,Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in Lipschitz domains, J. Funct. Anal.59(1984), no. 3, 572–611. Institute for Analysis, Leibniz University Hannover, Welfengarten 1, D-30167 Hannover Email address:arne.hofmann@math.uni-hannover.de Institute for Analysis, Leibniz University Hannover, Welfengarten 1...

work page 1984