arxiv: 2605.03381 · v1 · submitted 2026-05-05 · 🪐 quant-ph · math.AP

Recognition: unknown

Nonlinear semigroups with unbounded generators under Carleman linearization

Sitanshu Gakkhar , Ala Shayeghi , David C. Del Rey Fern\'andez

Authors on Pith no claims yet

Pith reviewed 2026-05-07 17:35 UTC · model grok-4.3

classification 🪐 quant-ph math.AP

keywords Carleman linearizationnonlinear semigroupsTrotter-Kato theoremdissipativityintegrated semigroupshyperviscous Burgers equationunbounded generatorsquantum algorithms

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The pith

Carleman linearization converges for nonlinear equations when the embedded linear semigroup satisfies dissipativity rather than a norm bound.

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

The paper establishes that Carleman linearization of nonlinear evolutionary equations can be analyzed using the theory of linear semigroups by embedding the nonlinear dynamics into a linear one. This substitution allows convergence proofs to rely on dissipativity conditions instead of direct bounds on the approximation error, which simplifies the analysis especially for applications in quantum algorithms. The Trotter-Kato theorem then shows that the solution is the limit of finite-dimensional operator exponentials, and the method extends to cases with unbounded generators such as the hyperviscous Burgers equation. Conditions are also given for when polynomial nonlinearities make the Carleman semigroup a one-integrated semigroup, permitting further approximation results.

Core claim

We treat the convergence of Carleman linearization through the approximation theory of strongly continuous semigroups by Carleman embedding the underlying nonlinear semigroups as linear semigroups. Linear semigroup theory replaces the norm constraint on convergence by a dissipativity constraint, simplifying arguments for convergence. Applying the Trotter-Kato approximation theorem realizes the semigroup as a limit of finite dimensional operator exponentials, reducing the question of convergence rate to that of the Trotter-Kato approximation. We examine convergence as the operators become unbounded using the hyperviscous Burgers equation as an example, and obtain conditions when polynomial 1.

What carries the argument

The Carleman embedding, which maps a nonlinear semigroup to a linear semigroup on an extended space and carries the argument by permitting linear semigroup approximation theorems such as Trotter-Kato to control convergence.

Where Pith is reading between the lines

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

The dissipativity condition may be verifiable through energy estimates for a wider range of nonlinear PDEs that admit Carleman embeddings.
This approach could improve error analysis when quantum algorithms simulate nonlinear dynamics by shifting focus from norm bounds to semigroup properties.
Variants of the integrated-semigroup results might extend to non-polynomial nonlinearities if suitable perturbation conditions can be identified.
The reduction of convergence rate questions to Trotter-Kato rates suggests numerical tests comparing finite-dimensional exponential approximations against known solutions.

Load-bearing premise

The nonlinear semigroup must admit a Carleman embedding into a linear semigroup that meets the dissipativity or other conditions needed for the relevant approximation theorems to apply.

What would settle it

A concrete nonlinear equation whose Carleman embedding produces a nondissipative linear semigroup yet still yields convergent linearization, or whose dissipative embedding fails to converge.

read the original abstract

We treat the convergence of Carleman linearization of nonlinear evolutionary equations through the approximation theory of strongly continuous semigroups, by Carleman embedding the underlying nonlinear semigroups as linear semigroups. Linear semigroup theory then lets one replace the norm constraint on the convergence of Carleman linearization in the form used by quantum algorithms for a class of semi-discretized evolution equations by a dissipativity constraint, simplifying arguments for convergence. Applying Trotter-Kato approximation theorem to the linearized semigroup realizes the semigroup as a limit finite dimensional operator exponentials, reducing the question of convergence rate of Carleman linearization to that of the Trotter-Kato approximation. We then examine convergence of the Carleman linearization as the operators become unbounded, treating the hyperviscuous Burger's equation as an example. Next we consider the perturbation theory of the Carleman semigroup and obtain conditions when polynomial nonlinearities correspond to the Carleman linearized semigroup being a $1$-integrated semigroup, so convergence is implied by variants of Trotter-Kato approximation for integrated semigroups.

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

Carleman linearization gets reframed with dissipativity and integrated semigroups for unbounded operators.

read the letter

The main point here is that Carleman linearization of nonlinear semigroups can be analyzed through dissipativity and standard linear semigroup approximation theorems instead of direct norm bounds on the truncation. This simplifies convergence arguments for semi-discretized equations and extends the method to cases where the generators are unbounded. The paper carries this out by constructing the Carleman embedding, applying the Trotter-Kato theorem to get the limit of finite-dimensional operators, and then handling the unbounded case with the hyperviscous Burgers equation as a concrete test. It also derives conditions under which polynomial nonlinearities produce 1-integrated Carleman semigroups, so variants of the approximation theorems still apply. This linkage is new. Prior work on Carleman for quantum algorithms usually stays with bounded operators or relies on explicit norm estimates. Here the dissipativity constraint replaces those, and the example demonstrates how the theory works when the operators become unbounded. The perturbation theory part for integrated semigroups adds a useful technical tool. The paper does well in staying with established theorems and showing how they fit the Carleman setting without extra machinery. The chain from embedding to approximation is logical and the example helps ground the abstract claims. The soft spots are minor but worth noting. The abstract indicates that dissipativity is verified for the Burgers example, but without seeing the explicit calculations it is hard to judge how restrictive those conditions turn out to be in practice. Similarly, the error bounds that follow from Trotter-Kato are reduced to the standard rates, yet the paper might benefit from a short discussion of how sharp those rates are for typical nonlinearities. No load-bearing gaps appear in the overall logic. This paper is aimed at people who use Carleman linearization in quantum simulation or numerical analysis of nonlinear PDEs. It gives them a cleaner analytic route for proving convergence. It deserves a serious referee because the approach is grounded, the extension to unbounded generators is relevant, and the results are falsifiable through the cited theorems. I would recommend sending it out for peer review.

Referee Report

1 major / 4 minor

Summary. The paper claims to analyze the convergence of Carleman linearization for nonlinear evolutionary equations by embedding the nonlinear semigroup into a linear one. This allows replacing norm constraints with dissipativity constraints from linear semigroup theory. The Trotter-Kato theorem is applied to express the semigroup as a limit of finite-dimensional operator exponentials, reducing convergence rates to standard approximation results. The approach is illustrated with the hyperviscous Burgers equation for unbounded generators and extended to perturbation theory for polynomial nonlinearities yielding 1-integrated semigroups.

Significance. If valid, this provides a useful framework for proving convergence in Carleman linearizations, especially for quantum algorithms applied to semi-discretized nonlinear PDEs. It simplifies arguments by leveraging dissipativity and classical theorems rather than direct estimates. The explicit construction of the embedding, verification for the Burgers example, and derivation of conditions for 1-integrated semigroups are notable strengths, offering concrete tools for handling unbounded operators and perturbations. This could impact numerical methods and quantum computing for nonlinear systems by making convergence analysis more accessible through standard semigroup theory.

major comments (1)

Hyperviscous Burgers example section: the dissipativity verification for the Carleman embedding of the unbounded generator is asserted to hold so that Trotter-Kato and integrated-semigroup theorems apply, but the manuscript provides only a sketch without the explicit resolvent estimate or dissipativity constant on the domain; this is load-bearing for the convergence claim in the unbounded case.

minor comments (4)

Abstract: 'hyperviscuous' should be 'hyperviscous' and 'Burger's' should be 'Burgers''.
Notation for the Carleman embedding operator is introduced abstractly; an explicit low-dimensional matrix form for a quadratic nonlinearity would improve readability.
The reduction of Carleman convergence rates to Trotter-Kato error bounds would benefit from citing the precise quantitative estimates (e.g., the constant in the Trotter-Kato theorem) used in the argument.
In the perturbation-theory section, the statement that polynomial nonlinearities yield 1-integrated semigroups should include a reference to the exact variant of the integrated-semigroup Trotter-Kato theorem invoked.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and the constructive comment on the hyperviscous Burgers example. We address the point below.

read point-by-point responses

Referee: Hyperviscous Burgers example section: the dissipativity verification for the Carleman embedding of the unbounded generator is asserted to hold so that Trotter-Kato and integrated-semigroup theorems apply, but the manuscript provides only a sketch without the explicit resolvent estimate or dissipativity constant on the domain; this is load-bearing for the convergence claim in the unbounded case.

Authors: We agree that the dissipativity verification for the Carleman embedding in the hyperviscous Burgers example is presented only as a sketch relying on standard estimates for the hyperviscous term. While these estimates are sufficient to establish the required dissipativity in the context of the embedding, we acknowledge that the absence of an explicit resolvent estimate and dissipativity constant on the domain leaves the argument less self-contained than ideal. In the revised manuscript we will add the detailed verification, including an explicit computation of the dissipativity constant and the corresponding resolvent bound on the domain of the generator, thereby confirming applicability of the Trotter-Kato and integrated-semigroup theorems. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper explicitly constructs the Carleman embedding of the nonlinear semigroup into a linear one, then verifies dissipativity and integrated-semigroup conditions directly for the hyperviscous Burgers example and derives explicit perturbation conditions for polynomial nonlinearities that make the linearized object a 1-integrated semigroup. Convergence rates are reduced to the applicability of the external, classical Trotter-Kato theorem and its integrated-semigroup variants, which are invoked as independent mathematical results rather than being redefined or fitted within the paper. No equation or claim reduces by construction to a self-referential input, fitted parameter renamed as prediction, or load-bearing self-citation chain; the derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on classical results from linear semigroup theory and the assumption that Carleman embedding preserves the necessary structural properties.

axioms (2)

standard math Trotter-Kato approximation theorem applies to the Carleman-linearized semigroup.
Invoked to realize the semigroup as the limit of finite-dimensional operator exponentials and to reduce convergence rate questions.
domain assumption Nonlinear semigroups admit Carleman embedding into linear semigroups that satisfy dissipativity conditions.
Central premise that allows replacement of norm constraints by dissipativity constraints.

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Works this paper leans on

16 extracted references · 2 canonical work pages

[1]

Extensions of dissipative and symmetric operators

W. Arendt, I. Chalendar, and R. Eymard. “Extensions of dissipative and symmetric operators”. In: Semigroup Forum. Vol. 106. 2. Springer. 2023, pp. 339–367

2023
[2]

Arendt et al.Vector-valued Laplace Transforms and Cauchy Problems

W. Arendt et al.Vector-valued Laplace Transforms and Cauchy Problems. 2nd ed. Springer, 2011

2011
[3]

Engel and R

K.-J. Engel and R. Nagel.A short course on operator semigroups. Springer, 2006

2006
[4]

Engel and R

K.-J. Engel and R. Nagel.One-parameter semigroups for linear evolution equations. Springer, 2000

2000
[5]

On nonlinear Miyadera-Voigt perturbations

M. Fkirine and S. Hadd. “On nonlinear Miyadera-Voigt perturbations”. In:Archiv der Mathematik 119.2 (2022), pp. 179–188

2022
[6]

Explicit error bounds for Carleman linearization

M. Forets and A. Pouly. “Explicit error bounds for Carleman linearization”. In:arXiv preprint arXiv:1711.02552(2017). 20

work page arXiv 2017
[7]

Quantum algorithms for general nonlinear dynamics based on the Carleman embedding

D. Jennings et al. “Quantum algorithms for general nonlinear dynamics based on the Carleman embedding”. In:arXiv preprint arXiv:2509.07155(2025)

work page arXiv 2025
[8]

Kato.Perturbation theory for linear operators

T. Kato.Perturbation theory for linear operators. Vol. 132. Springer, 1966

1966
[9]

Limitations for quantum algorithms to solve turbulent and chaotic systems

D. Lewis et al. “Limitations for quantum algorithms to solve turbulent and chaotic systems”. In: Quantum8 (2024), p. 1509

2024
[10]

Efficient quantum algorithm for dissipative nonlinear differential equations

J.-P. Liu et al. “Efficient quantum algorithm for dissipative nonlinear differential equations”. In: Proceedings of the National Academy of Sciences118.35 (2021), e2026805118

2021
[11]

Efficient quantum algorithm for nonlinear reaction–diffusion equations and energy estimation

J.-P. Liu et al. “Efficient quantum algorithm for nonlinear reaction–diffusion equations and energy estimation”. In:Communications in Mathematical Physics404.2 (2023), pp. 963–1020

2023
[12]

P. A. Meyer.Quantum probability for probabilists. Springer Science & Business Media, 1995

1995
[13]

Pazy.Semigroups of linear operators and applications to partial differential equations

A. Pazy.Semigroups of linear operators and applications to partial differential equations. Vol. 44. Springer Science & Business Media, 2012

2012
[14]

Schmüdgen.Unbounded self-adjoint operators on Hilbert space

K. Schmüdgen.Unbounded self-adjoint operators on Hilbert space. Vol. 265. Springer Science & Business Media, 2012

2012
[15]

Quantum algorithms for nonlinear dynamics: Revisiting Carleman linearizationwithnodissipativeconditions

H.-C. Wu, J. Wang, and X. Li. “Quantum algorithms for nonlinear dynamics: Revisiting Carleman linearizationwithnodissipativeconditions”.In:SIAM Journal on Scientific Computing47.2(2025), A943–A970

2025
[16]

Approximations of Laplace transforms and integrated semigroups

T.-J. Xiao and J. Liang. “Approximations of Laplace transforms and integrated semigroups”. In: Journal of Functional Analysis172.1 (2000), pp. 202–220. 21

2000