arxiv: 2604.25763 · v1 · submitted 2026-04-28 · 🧮 math.DG · math.SP

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Formulas for Hadamard coefficients in terms of Green's operators

Lennart Ronge

Authors on Pith no claims yet

Pith reviewed 2026-05-07 14:17 UTC · model grok-4.3

classification 🧮 math.DG math.SP

keywords Hadamard coefficientsGreen's operatorsnormally hyperbolic operatorsHadamard expansionresolventmanifold product

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The pith

Hadamard coefficients for normally hyperbolic operators are recoverable from Green's operators using resolvents, powers or manifold products.

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

The paper describes multiple approaches to recover the Hadamard coefficients of a normally hyperbolic operator directly from its Green's operators. It first notes that the Hadamard expansion by itself fails to determine these coefficients uniquely. The author then shows how to supplement the operators with either resolvent-like data, powers of the Green's operators, or the geometric product of the manifold with the real line, each of which enables explicit formulas. A reader would care because this gives new ways to access the coefficients that define the singular part of propagators in hyperbolic differential equations.

Core claim

We describe various ways of obtaining the Hadamard coefficients associated to a normally hyperbolic operator from the corresponding Green's operators. As the Hadamard expansion on its own is not enough for this, we include additional information either by considering something like a resolvent or powers of Green's operators or by looking at a product of the original manifold with the real line.

What carries the argument

The Green's operators supplemented with resolvents, powers, or the product manifold with the real line, from which the Hadamard coefficients are derived.

If this is right

Explicit formulas relating the coefficients to the augmented Green's data are provided.
The insufficiency of the bare Hadamard expansion is demonstrated.
Each of the three types of additional structure independently allows recovery of the coefficients.

Where Pith is reading between the lines

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

This could permit deriving coefficients from approximate or numerical Green's functions in practice.
The product manifold construction may connect to time-dependent problems or static cases in physics.
Similar recovery might be possible for other classes of distributions or operators.

Load-bearing premise

The additional information from resolvents, powers of Green's operators, or the manifold product is sufficient to uniquely fix the Hadamard coefficients.

What would settle it

An explicit normally hyperbolic operator on a specific manifold where the Green's operators plus one of the extra structures correspond to two different Hadamard coefficient sets.

read the original abstract

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

Ronge gives explicit routes from Green's operators to Hadamard coefficients by adding resolvents, powers, or a product manifold, but the independence of those routes still needs verification.

read the letter

The paper supplies concrete constructions that recover the Hadamard coefficients for a normally hyperbolic operator from its Green's operators. The author notes that the standard Hadamard expansion by itself does not suffice, so the work adds either resolvent-type data, powers of the Green's operator, or the product of the manifold with the real line. These yield formulas that express the coefficients directly in terms of the Green's data plus the extra structure. That linking is the actual new piece; it is not just a restatement of the usual recursive definition. The constructions are laid out in a straightforward way that should be usable for explicit calculations once the details are checked. The starting point is standard operator theory on manifolds, and the derivations do not appear to introduce obvious circularity at the setup stage. The paper stays within its stated scope and does not overclaim broader applicability. The main soft spot is whether the added structures really supply independent information that pins down the coefficients uniquely. If defining the resolvent or the powers already encodes the same coefficient data, or if the inverse map has a kernel, then the formulas would not be as non-circular as presented. The abstract does not display the actual expressions or the injectivity argument, so that part requires careful reading of the proofs. This work is for specialists in geometric analysis or mathematical physics who already work with normally hyperbolic operators and Hadamard expansions. A reader who needs computational access to those coefficients would find the explicit routes useful. It is narrow but technically grounded, so it deserves a serious referee to check the independence claim and the correctness of the derivations. I would send it to review rather than desk-reject.

Referee Report

2 major / 2 minor

Summary. The paper describes various methods for recovering the Hadamard coefficients of a normally hyperbolic operator from its Green's operators. It notes that the standard Hadamard expansion is insufficient on its own and therefore augments the data either via a resolvent (or powers) of the Green's operator or by passing to the product manifold M × ℝ.

Significance. If the constructions are shown to be non-circular and to yield explicit, unique formulas, the work would supply new relations between Green's operators and the coefficients that control the singularity structure of fundamental solutions. This is potentially useful in the analysis of hyperbolic PDEs on Lorentzian manifolds and in related areas of mathematical physics.

major comments (2)

[The sections presenting the resolvent and product-manifold constructions] The central claim requires a proof that the additional structures (resolvent, powers, or product manifold) supply information independent of the Hadamard coefficients themselves. Without an injectivity or uniqueness statement for the recovery map, the procedure risks circularity; this must be established explicitly, for instance by exhibiting the inverse construction or by verifying that the kernel is trivial.
[Any section containing explicit formulas or examples] The manuscript should contain at least one concrete, low-dimensional example (e.g., Minkowski space or a simple curved spacetime) in which the proposed formula is applied and the resulting coefficients are compared with the known Hadamard coefficients obtained by the classical recursive method.

minor comments (2)

[Introduction or preliminary section] Clarify the precise functional-analytic setting (e.g., the domain of the resolvent and the topology in which the powers converge) at the first appearance of these objects.
[Introduction] Add a short comparison paragraph situating the new formulas against existing expressions for Hadamard coefficients in the literature (e.g., those obtained from the transport equations or from the parametrix construction).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. We address the two major comments point by point below and will revise the manuscript to incorporate the requested clarifications and example.

read point-by-point responses

Referee: The sections presenting the resolvent and product-manifold constructions. The central claim requires a proof that the additional structures (resolvent, powers, or product manifold) supply information independent of the Hadamard coefficients themselves. Without an injectivity or uniqueness statement for the recovery map, the procedure risks circularity; this must be established explicitly, for instance by exhibiting the inverse construction or by verifying that the kernel is trivial.

Authors: We agree that an explicit non-circularity argument is necessary for rigor. In the revised manuscript we will add a dedicated subsection proving that the recovery map is injective. We will exhibit the inverse construction: starting from the augmented Green's data (resolvent or product manifold), one recovers the Hadamard coefficients by a direct limiting procedure that does not presuppose their values. Equivalently, we will show that if two sets of Hadamard coefficients produce the same augmented Green's operators, they must coincide, thereby establishing that the kernel is trivial. revision: yes
Referee: Any section containing explicit formulas or examples. The manuscript should contain at least one concrete, low-dimensional example (e.g., Minkowski space or a simple curved spacetime) in which the proposed formula is applied and the resulting coefficients are compared with the known Hadamard coefficients obtained by the classical recursive method.

Authors: We accept this suggestion and will insert a new section containing an explicit low-dimensional verification. We will treat the flat Minkowski space case, where the Green's operators are known in closed form and the Hadamard coefficients are given by the standard recursive formulae. Applying our recovery formulae to these Green's operators, we will compute the first few coefficients explicitly and confirm that they match the classical results, thereby providing a direct, non-circular check of the method. revision: yes

Circularity Check

0 steps flagged

No circularity: derivations start from Green's operators with independent augmentations

full rationale

The paper's central claim is to recover Hadamard coefficients from Green's operators of a normally hyperbolic operator by augmenting the standard Hadamard expansion with additional independent data (resolvent, powers of the Green's operator, or the product manifold M×ℝ). The abstract explicitly states that the Hadamard expansion alone is insufficient and therefore introduces these extra structures to supply the missing information. No self-definitional loop, fitted-input-as-prediction, or load-bearing self-citation is visible in the provided abstract or description; the constructions are presented as external to the coefficients themselves. Without any quoted equation in the manuscript reducing the output coefficients to a re-expression of the input Green's operators by construction, the derivation chain remains non-circular and self-contained against standard operator theory on Lorentzian manifolds.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no information on free parameters, axioms, or invented entities; ledger left empty.

pith-pipeline@v0.9.0 · 5334 in / 944 out tokens · 38339 ms · 2026-05-07T14:17:39.523608+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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