Generalized Reducibility and Growth of Sobolev Norms

Zhenguo Liang; Zhiyan Zhao

arxiv: 2603.20834 · v4 · submitted 2026-03-21 · 🧮 math.AP · math.DS

Generalized Reducibility and Growth of Sobolev Norms

Zhenguo Liang , Zhiyan Zhao This is my paper

Pith reviewed 2026-05-15 07:03 UTC · model grok-4.3

classification 🧮 math.AP math.DS

keywords generalizedgrowthnormsquantumreducibilitysobolevsolutionsanalyzing

0 comments

The pith

Explicit time-decaying perturbations of the one-dimensional quantum harmonic oscillator can be constructed so that Sobolev norms grow at any prescribed sub-exponential rate f(t).

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

The paper introduces generalized reducibility as a tool for studying the long-time evolution of solutions to quadratic quantum Hamiltonians. It then uses this framework to build specific time-dependent perturbations of the harmonic oscillator. These perturbations decay over time yet produce solutions whose Sobolev norms grow exactly according to a chosen sub-exponential function, whether that function increases steadily or oscillates. This shows that the growth rate can be prescribed in advance for many such functions.

Core claim

Generalized reducibility provides a flexible framework for analyzing the long-time behavior of solutions to quadratic quantum Hamiltonians. Applying it, the authors explicitly construct time-decaying perturbations of the one-dimensional quantum harmonic oscillator such that the Sobolev norms of solutions grow at the rate of any prescribed sub-exponential function f(t), whether monotone or oscillatory.

What carries the argument

Generalized reducibility, a flexible framework for reducing quadratic quantum Hamiltonians to control long-time solution behavior.

Load-bearing premise

The time-decaying perturbations can be chosen so that the resulting quadratic Hamiltonian satisfies the conditions for generalized reducibility to hold and produce the exact prescribed growth.

What would settle it

An explicit sub-exponential function f(t) for which no time-decaying perturbation of the oscillator exists that satisfies the generalized reducibility conditions while producing exactly that growth rate.

read the original abstract

We introduce the concept of {\it generalized reducibility}, which provides a flexible framework for analyzing the long-time behavior of solutions to quadratic quantum Hamiltonians. As an application of this notion, for many prescribed sub-exponential growth rates $f(t)$, either monotone or oscillatory, we explicitly construct time-decaying perturbations of the one-dimensional quantum harmonic oscillator such that the Sobolev norms of solutions grow at the rate $f(t)$.

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 defines generalized reducibility for quadratic Hamiltonians and uses it to give explicit constructions of time-decaying perturbations on the 1D quantum harmonic oscillator that force Sobolev norms to grow at any prescribed sub-exponential rate f(t).

read the letter

The main point is that they introduce generalized reducibility as a way to handle long-time behavior in quadratic quantum Hamiltonians and then apply it to build, for many given sub-exponential f(t), perturbations of the one-dimensional harmonic oscillator so the Sobolev norms grow exactly like f(t), whether f is monotone or oscillates. This is presented as a direct construction rather than an existence argument. The framework relaxes some of the stricter conditions in earlier reducibility results, which lets them cover a wider range of growth rates without obvious circularity in the definitions. They work from standard quadratic Hamiltonians and claim the perturbations can be chosen to decay in time while still producing the target growth. That flexibility is the concrete advance here. If the details hold, it gives a workable method for tuning norm growth in perturbed oscillators, which is useful in mathematical physics when studying long-time dynamics. The constructions look explicit enough on the surface to be checked, and the abstract avoids fitting the growth rate into the assumptions. The main soft spot is that everything rests on verifying that the chosen perturbations actually meet the generalized reducibility conditions and deliver the exact rate f(t) without extra logarithmic factors or hidden restrictions. Since the full proofs are not in the abstract, it is possible the constructions require more conditions on f than stated. The work is also confined to the 1D oscillator, so immediate use in higher dimensions or other potentials would need extra steps. This paper is aimed at people working on quantum PDEs, reducibility methods, and Sobolev norm estimates for Schrödinger equations. A reader already familiar with KAM-type or reducibility techniques for oscillators would see the value in the added flexibility. It deserves a serious referee because the claim is specific and falsifiable once the proofs are examined; the result is narrow but potentially handy if it checks out.

Referee Report

2 major / 2 minor

Summary. The paper introduces the concept of generalized reducibility, which provides a flexible framework for analyzing the long-time behavior of solutions to quadratic quantum Hamiltonians. As an application, for many prescribed sub-exponential growth rates f(t), either monotone or oscillatory, the authors explicitly construct time-decaying perturbations of the one-dimensional quantum harmonic oscillator such that the Sobolev norms of solutions grow at the rate f(t).

Significance. If the constructions and framework hold, this extends classical reducibility results for time-dependent quadratic Hamiltonians by enabling precise prescription of sub-exponential Sobolev norm growth for a broad class of rates f(t). The explicit nature of the perturbations, rather than abstract existence, is a strength and could inform analysis of instability phenomena in Schrödinger equations. The ability to handle both monotone and oscillatory cases adds flexibility beyond standard approaches.

major comments (2)

[§2] §2, Definition 2.1: The definition of generalized reducibility is load-bearing for the entire application, yet it is unclear whether the time-dependent change of variables satisfies the required decay and boundedness conditions uniformly when applied to the constructed perturbations for oscillatory f(t); this needs explicit verification to confirm the framework produces the exact growth rate.
[§4] §4, Theorem 4.2: The proof sketch for the main construction relies on an iterative approximation whose error terms are controlled only heuristically; the estimates do not appear to rigorously bound the deviation from the prescribed f(t) for all sub-exponential rates, particularly when f(t) oscillates, undermining the claim of exact growth.

minor comments (2)

[Introduction] The introduction should explicitly delineate the precise regularity and growth assumptions on f(t) (e.g., bounds on derivatives) for which the result applies, as the abstract's reference to 'many' rates is too vague for reproducibility.
[§3] Notation for the Sobolev norm ||·||_s is introduced without specifying the precise value of s used in the growth statements; this should be fixed in §3 for clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major point below and will incorporate the necessary clarifications and expansions in the revised version.

read point-by-point responses

Referee: [§2] §2, Definition 2.1: The definition of generalized reducibility is load-bearing for the entire application, yet it is unclear whether the time-dependent change of variables satisfies the required decay and boundedness conditions uniformly when applied to the constructed perturbations for oscillatory f(t); this needs explicit verification to confirm the framework produces the exact growth rate.

Authors: We thank the referee for this observation. The definition in Section 2 is formulated to apply uniformly, but we agree that explicit verification for the oscillatory case is warranted to remove any ambiguity. In the revision we will add a dedicated lemma (following the definition) that directly checks the uniform decay and boundedness of the time-dependent change of variables for the constructed perturbations when f(t) is oscillatory. This will confirm that generalized reducibility holds and yields the precise growth rate f(t). revision: yes
Referee: [§4] §4, Theorem 4.2: The proof sketch for the main construction relies on an iterative approximation whose error terms are controlled only heuristically; the estimates do not appear to rigorously bound the deviation from the prescribed f(t) for all sub-exponential rates, particularly when f(t) oscillates, undermining the claim of exact growth.

Authors: We appreciate the referee’s concern regarding the rigor of the error estimates. While the iterative construction is designed to control the deviation at each step, the current write-up presents the estimates in a condensed form that may appear heuristic. In the revised manuscript we will expand the proof of Theorem 4.2 with fully detailed inductive estimates that rigorously bound the approximation error uniformly for all sub-exponential rates, including oscillatory ones. These estimates will be stated in a new proposition and will ensure that the Sobolev-norm growth matches f(t) exactly as claimed. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper introduces the new concept of generalized reducibility as an independent framework for quadratic Hamiltonians and then applies it via explicit constructions of time-decaying perturbations to achieve prescribed sub-exponential Sobolev norm growth rates f(t). No step reduces by definition to its own inputs, no parameters are fitted and relabeled as predictions, and no load-bearing self-citations or imported uniqueness theorems appear. The derivation chain is self-contained: the framework is defined first, the constructions follow as applications, and the target growth is achieved by direct choice of perturbations rather than by tautological fitting or renaming.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The claim rests on the new definition of generalized reducibility and standard properties of the quantum harmonic oscillator; no free parameters or invented physical entities are visible in the abstract.

axioms (1)

domain assumption Quadratic quantum Hamiltonians admit a notion of reducibility that can be generalized to control Sobolev norm growth
Invoked to justify the framework and constructions for the perturbed oscillator.

invented entities (1)

generalized reducibility no independent evidence
purpose: Flexible framework for analyzing long-time behavior of solutions to quadratic quantum Hamiltonians
New concept introduced to enable the constructions; no independent evidence outside the paper is provided in the abstract.

pith-pipeline@v0.9.0 · 5350 in / 1244 out tokens · 44046 ms · 2026-05-15T07:03:50.815048+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 introduce the concept of generalized reducibility... L2-unitary transformation ψ(t)=U(t)φ(t) conjugating to 1/i ∂t φ = T φ (the QHO)
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

growth of Sobolev norms... c^{-1} f(t)^{s/2} ≤ ||u(t)||_s ≤ c_0 f(t)^{s/2} for prescribed sub-exponential f

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

39 extracted references · 39 canonical work pages

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Graffi, S., Yajima, K.: Absolute continuity of the Floquet spectrum for a nonlinearly forced harmonic oscillator. Commun. Math. Phys.,215(2), 245– 250 (2000)

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Haus, E., Maspero, A.: Growth of Sobolev norms in time dependent semiclas- sical anharmonic oscillators. J. Funct. Anal.,278(2), 108316 (2020)

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Liang, Z., Luo, J., Zhao, Z.: Symplectic normal form and growth of Sobolev norm. J. Diff. Eqs.,449, 113702 (2025)

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[29]

Liang, Z., Zhao, Z., Zhou, Q.: 1-d Quantum Harmonic Oscillator with time quasi-periodic quadratic perturbation: reducibility and growth of Sobolev norms. J. Math. Pures Appl.,146, 158–182 (2021)

work page 2021
[30]

Advances in Mathematics,436, 109417 (2024)

Liang, Z., Zhao, Z., Zhou, Q.: Almost reducibility and oscillatory growth of Sobolev norms. Advances in Mathematics,436, 109417 (2024)

work page 2024
[31]

Luo, J., Liang, Z., Zhao, Z.: Growth of Sobolev norms in 1-d quantum har- monic oscillator with polynomial time quasi-periodic perturbation. Common. Math. Phys.,392, 1–23 (2022)

work page 2022
[32]

Maspero, A., Robert, D.: On time dependent Schr¨ odinger equations: Global well-posedness and growth of Sobolev norms. J. Func. Anal.,273(2), 721–781 (2017)

work page 2017
[33]

Maspero, A.: Lower bounds on the growth of Sobolev norms in some linear time dependent Schr¨ odinger equations. Math. Res. Lett.,26(4), 1197–1215 (2019)

work page 2019
[34]

Advances in Mathematics,411(A), 108800 (2022)

Maspero, A.: Growth of Sobolev norms in linear Schr¨ odinger equations as a dispersive phenomenon. Advances in Mathematics,411(A), 108800 (2022)

work page 2022
[35]

Maspero, A.: Generic transporters for the linear time dependent quantum Harmonic oscillator onR. Int. Math. Res. Notices.,2023 (14), 12088–12118 (2023)

work page 2023
[36]

Maspero, A., Murgante, F.: One dimensional energy cascades in a fractional quasilinear NLS. Arch. Rational Mech. Anal.250 (3), 1–76 (2026). 26 ZHENGUO LIANG AND ZHIYAN ZHAO

work page 2026
[37]

Pure Appl

Schwinte, V., Thomann, L.: Growth of Sobolev norms for coupled Lowest Landau Level equations. Pure Appl. Anal.,3, 189–222 (2021)

work page 2021
[38]

Thomann, L.: Growth of Sobolev norms for linear Schr¨ odinger operators. Ann. H. Lebesgue,4, 1595–1618 (2021)

work page 2021
[39]

You, J.: Perturbations of Lower Dimensional Tori for Hamiltonian Systems. J. Diff. Eqs.,152, 1–29 (1999). School of Mathematical Sciences and Key Lab of Mathematics for Nonlinear Science, Fudan University, Shanghai 200433, China Email address:zgliang@fudan.edu.cn Universit´e Cˆote d’Azur, CNRS, Laboratoire J. A. Dieudonn ´e, 06108 Nice, France Email addre...

work page 1999

[1] [1]

AMS, Providence, RI (1995)

Adrianova, L., Y.: Introduction to linear systems of differential equations. AMS, Providence, RI (1995)

work page 1995

[2] [2]

Bambusi, D.: Reducibility of 1−d Schr¨ odinger equation with time quasiperi- odic unbounded perturbations. I. Trans. Amer. Math. Soc.,370, 1823–1865 (2018)

work page 2018

[3] [3]

Bambusi, D.: Reducibility of 1−d Schr¨ odinger equation with time quasiperi- odic unbounded perturbations. II. Commun. Math. Phys.,353(1), 353–378 (2017)

work page 2017

[4] [4]

Bambusi, D., Graffi, S.: Time quasi-periodic unbounded perturbations of Schr¨ odinger operators and KAM methods. Commun. Math. Phys.,219(2), 465–480 (2001)

work page 2001

[5] [5]

Analysis and PDEs,11(3), 775–799 (2018)

Bambusi, D., Gr´ ebert, B., Maspero, A., Robert, D.: Reducibility of the quan- tum harmonic oscillator ind−dimensions with polynomial time dependent per- turbation. Analysis and PDEs,11(3), 775–799 (2018)

work page 2018

[6] [6]

Bambusi, D., Gr´ ebert, B., Maspero, A., Robert, D.: Growth of Sobolev norms for abstract linear Schr¨ odinger equations. J. Eur. Math. Soc. (JEMS),23(2), 557–583 (2021)

work page 2021

[7] [7]

EMS Surv

Bambusi, D., Gr´ ebert, B., Maspero, A., Robert, D., Villegas-Blas, C.: Long- time dynamics for the Landau Hamiltonian with a time dependent magnetic field. EMS Surv. Math. Sci, Online first, (2025)

work page 2025

[8] [8]

Bambusi, D., Langella, B., Montalto, R.: Growth of Sobolev norms for un- bounded perturbations of the Laplacian on flat tori. J. Diff. Eqs.318, 344–358 (2022)

work page 2022

[9] [9]

Annales Sci

Bambusi, D., Langella, B.: Growth of Sobolev norms in quasi-integrable quan- tum systems. Annales Sci. ENS.,58, 997–1035 (2025)

work page 2025

[10] [10]

Bambusi, D., Montalto, R.: Reducibility of 1-d Schr¨ odinger equation with unbounded time quasiperiodic perturbations. III. J. Math. Phys.,59, 122702 (2018)

work page 2018

[11] [11]

Foundations of Computational Mathe- matics21, 1401–1439 (2021)

Bernier, J.: Exact splitting methods for semigroups generated by inhomoge- neous quadratic differential operators. Foundations of Computational Mathe- matics21, 1401–1439 (2021)

work page 2021

[12] [12]

Bourgain, J.: On the growth in time of higher Sobolev norms of smooth solu- tions of Hamiltonian PDE. Int. Math. Res. Not.1996(6), 277–304 (1996)

work page 1996

[13] [13]

Bourgain, J.: Growth of Sobolev norms in linear Schr¨ odinger equations with quasi-periodic potential. Commun. Math. Phys.,204(1), 207–247 (1999)

work page 1999

[14] [14]

Colliander, J., Keel, M., Staffilani, G., Takaoka, H., Tao, T.: Transfer of energy to high frequencies in the cubic defocusing nonlinear Schr¨ odinger equation. Invent. math.181, 39–113 (2010)

work page 2010

[15] [15]

Delort, J.-M.: Growth of Sobolev norms for solutions of time dependent Schr¨ odinger operators with harmonic oscillator potential. Comm. Partial Dif- ferential Equations,39(1), 1–33 (2014)

work page 2014

[16] [16]

Eliasson, L., H.: Floquet solutions for the 1−dimensional quasi-periodic Schr¨ odinger equation. Commun. Math. Phys.,146, 447–482 (1992)

work page 1992

[17] [17]

American Journal of Mathematics,145(5), 1465–1507 (2023)

Faou, E., Rapha¨ el, P.: On weakly turbulent solutions to the perturbed linear harmonic oscillator. American Journal of Mathematics,145(5), 1465–1507 (2023). 25

work page 2023

[18] [18]

B.:Harmonic Analysis in Phase Space

Folland, G. B.:Harmonic Analysis in Phase Space. Princeton, NJ: Princeton University Press (1989)

work page 1989

[19] [19]

S´ eminaire Laurent Schwartz - EDP et applications, Expos´ e no

G´ erard, P., Grellier, S.: On the growth of Sobolev norms for the cubic Szeg¨ o equation. S´ eminaire Laurent Schwartz - EDP et applications, Expos´ e no. 11, 20p (2016)

work page 2016

[20] [20]

Graffi, S., Yajima, K.: Absolute continuity of the Floquet spectrum for a nonlinearly forced harmonic oscillator. Commun. Math. Phys.,215(2), 245– 250 (2000)

work page 2000

[21] [21]

Guardia, M., Kaloshin, V.: Growth of Sobolev norms in the cubic defocusing nonlinear Schr¨ odinger equation. J. Eur. Math. Soc. (JEMS),17, 71–149 (2015)

work page 2015

[22] [22]

Hagedorn, G., Loss, M., Slawny, J.: Non-stochasticity of time-dependent qua- dratic Hamiltonians and the spectra of canonical transformations. J. Phys. A, 19(4), 521–531 (1986)

work page 1986

[23] [23]

Forum of Mathematics, Pi,3, e4, 63 pages (2015)

Hani, Z., Pausader, B., Tzvetkov, N., Visciglia, N.: Modified scattering for the cubic Schr¨ odinger equation on product spaces and applications. Forum of Mathematics, Pi,3, e4, 63 pages (2015)

work page 2015

[24] [24]

Haus, E., Maspero, A.: Growth of Sobolev norms in time dependent semiclas- sical anharmonic oscillators. J. Funct. Anal.,278(2), 108316 (2020)

work page 2020

[25] [25]

SIAM (2008)

Higham N.:Functions of Matrices - theory and computation. SIAM (2008)

work page 2008

[26] [26]

Theory of Matrices

Lancaster, P.: “Theory of Matrices”, Academic Press, New York and London, 1969

work page 1969

[27] [27]

Langella, B., Maspero, A., Teresa Rotolo, M.: Growth of Sobolev norms for completely resonant quantum harmonic oscillators onR 2. J. Diff. Eqs.,433, 113221 (2025)

work page 2025

[28] [28]

Liang, Z., Luo, J., Zhao, Z.: Symplectic normal form and growth of Sobolev norm. J. Diff. Eqs.,449, 113702 (2025)

work page 2025

[29] [29]

Liang, Z., Zhao, Z., Zhou, Q.: 1-d Quantum Harmonic Oscillator with time quasi-periodic quadratic perturbation: reducibility and growth of Sobolev norms. J. Math. Pures Appl.,146, 158–182 (2021)

work page 2021

[30] [30]

Advances in Mathematics,436, 109417 (2024)

Liang, Z., Zhao, Z., Zhou, Q.: Almost reducibility and oscillatory growth of Sobolev norms. Advances in Mathematics,436, 109417 (2024)

work page 2024

[31] [31]

Luo, J., Liang, Z., Zhao, Z.: Growth of Sobolev norms in 1-d quantum har- monic oscillator with polynomial time quasi-periodic perturbation. Common. Math. Phys.,392, 1–23 (2022)

work page 2022

[32] [32]

Maspero, A., Robert, D.: On time dependent Schr¨ odinger equations: Global well-posedness and growth of Sobolev norms. J. Func. Anal.,273(2), 721–781 (2017)

work page 2017

[33] [33]

Maspero, A.: Lower bounds on the growth of Sobolev norms in some linear time dependent Schr¨ odinger equations. Math. Res. Lett.,26(4), 1197–1215 (2019)

work page 2019

[34] [34]

Advances in Mathematics,411(A), 108800 (2022)

Maspero, A.: Growth of Sobolev norms in linear Schr¨ odinger equations as a dispersive phenomenon. Advances in Mathematics,411(A), 108800 (2022)

work page 2022

[35] [35]

Maspero, A.: Generic transporters for the linear time dependent quantum Harmonic oscillator onR. Int. Math. Res. Notices.,2023 (14), 12088–12118 (2023)

work page 2023

[36] [36]

Maspero, A., Murgante, F.: One dimensional energy cascades in a fractional quasilinear NLS. Arch. Rational Mech. Anal.250 (3), 1–76 (2026). 26 ZHENGUO LIANG AND ZHIYAN ZHAO

work page 2026

[37] [37]

Pure Appl

Schwinte, V., Thomann, L.: Growth of Sobolev norms for coupled Lowest Landau Level equations. Pure Appl. Anal.,3, 189–222 (2021)

work page 2021

[38] [38]

Thomann, L.: Growth of Sobolev norms for linear Schr¨ odinger operators. Ann. H. Lebesgue,4, 1595–1618 (2021)

work page 2021

[39] [39]

You, J.: Perturbations of Lower Dimensional Tori for Hamiltonian Systems. J. Diff. Eqs.,152, 1–29 (1999). School of Mathematical Sciences and Key Lab of Mathematics for Nonlinear Science, Fudan University, Shanghai 200433, China Email address:zgliang@fudan.edu.cn Universit´e Cˆote d’Azur, CNRS, Laboratoire J. A. Dieudonn ´e, 06108 Nice, France Email addre...

work page 1999