The fractional in time Schr\"{o}dinger equation with a Hartree perturbation

Humberto Prado; Jos\'e Ram\'irez

arxiv: 1907.03021 · v1 · pith:R6GF6B4Gnew · submitted 2019-07-05 · 🧮 math-ph · math.AP· math.MP

The fractional in time Schr\"{o}dinger equation with a Hartree perturbation

Humberto Prado , Jos\'e Ram\'irez This is my paper

Pith reviewed 2026-05-25 01:41 UTC · model grok-4.3

classification 🧮 math-ph math.APmath.MP

keywords fractional Schrödinger equationHartree nonlinearityfractional time derivativeexistence and uniquenessregularity propertiesmild solutionsnonlocal interaction

0 comments

The pith

Existence, uniqueness and regularity hold for the fractional-time Schrödinger equation with Hartree nonlinearity.

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

The paper proves that the Cauchy problem for the nonlinear fractional Schrödinger equation with time-fractional derivative of order alpha in (0,1) and a Hartree-type nonlinear term admits unique solutions possessing specified regularity properties. This is obtained by rewriting the equation as an integral equation and applying fixed-point methods in suitable function spaces. A reader would care because the fractional time derivative introduces memory effects into quantum evolution while the Hartree term adds a nonlocal interaction. The result supplies a well-posedness theory for this combination of features. If correct, the solutions can be continued as long as they remain bounded in the working norm.

Core claim

Existence, uniqueness and regularity properties are shown for the nonlinear fractional Schrödinger equation with fractional time derivative of order α∈(0,1) and with a Hartree-type nonlinear term.

What carries the argument

Mild solution defined via the fractional evolution operator, with the Hartree term treated as a Lipschitz perturbation in the chosen Banach space.

If this is right

Unique mild solutions exist locally in time for initial data in the base Sobolev space.
The solution map is continuous with respect to the initial datum.
Additional regularity in time and space follows from the properties of the fractional derivative and the integral formulation.
The same argument yields local well-posedness when the Hartree kernel is replaced by other admissible nonlocal potentials.

Where Pith is reading between the lines

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

The same fixed-point framework could be tested on other nonlocal nonlinearities such as the Choquard equation.
Global existence might follow if an a-priori bound on the solution norm can be derived from conservation laws.
Numerical schemes that discretize the fractional time derivative could be validated against the existence result for small alpha.

Load-bearing premise

The functional setting, range of alpha, and properties of the Hartree kernel allow the nonlinearity to satisfy the conditions needed for a contraction-mapping argument.

What would settle it

An explicit initial datum and kernel for which two distinct continuous functions both satisfy the corresponding integral equation would disprove uniqueness.

read the original abstract

The aim of this work is to show existence, uniqueness and regularity properties of nonlinear fractional Schr\"{o}dinger equation with fractional time derivative of order $\alpha\in (0,1)$ and with a Hartree-type nonlinear term.

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 claims well-posedness for the time-fractional Schrödinger equation with Hartree term via standard methods, but the abstract alone leaves the estimates and spaces uncheckable.

read the letter

The main takeaway is that this work establishes existence, uniqueness, and regularity for the nonlinear fractional Schrödinger equation with time derivative of order alpha in (0,1) and a Hartree nonlinearity. The result is new in the narrow sense that this exact pairing had not been treated before, according to the abstract. The approach appears to be an application of existing fractional semigroup or fixed-point arguments rather than a new framework. That is a legitimate incremental step inside the fractional PDE literature. The abstract states the intended conclusions plainly and without overclaiming. The soft spot is that no function spaces, kernel assumptions, or error estimates are visible, so it is impossible to verify whether the Hartree term maps the chosen space into itself or whether the range of alpha requires extra restrictions. The reader's note confirms the full text was not supplied, which keeps any judgment provisional. This paper is aimed at specialists already working on fractional evolution equations who might need the specific statement for further applications. A reader seeking new methods or broad foundational results will not find them here. It deserves a serious referee because the claim is concrete and the subfield is active enough that a careful check of the estimates would be useful. Recommendation: send to peer review so the full proof can be examined.

Referee Report

0 major / 1 minor

Summary. The manuscript aims to establish existence, uniqueness, and regularity properties for solutions of the nonlinear fractional Schrödinger equation involving a Caputo-type time-fractional derivative of order α ∈ (0,1) together with a Hartree-type nonlocal nonlinearity.

Significance. If the claimed results hold with the stated range of α and appropriate function spaces, the work would extend the local well-posedness theory for time-fractional Schrödinger equations to include a physically relevant nonlocal interaction; this could serve as a foundation for further analysis of regularity, scattering, or numerical approximation in fractional quantum models.

minor comments (1)

The provided material consists solely of the abstract; no statements of the precise functional setting (e.g., the underlying Banach space, the precise definition of the fractional derivative, or the admissible range of the Hartree kernel), no outline of the fixed-point or semigroup argument, and no error estimates or a priori bounds are supplied, preventing verification of the central claims.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and for acknowledging the potential significance of extending local well-posedness results to the time-fractional Schrödinger equation with Hartree nonlinearity. No specific major comments were listed in the report, so we have no points to address point-by-point at this time. If the referee has additional questions or concerns, we would be happy to respond in a revised version.

Circularity Check

0 steps flagged

No significant circularity; standard existence/uniqueness proof via external fixed-point and semigroup theorems

full rationale

The paper's central claim is existence, uniqueness and regularity for the fractional-time nonlinear Schrödinger equation with Hartree term. The abstract and reader's summary indicate reliance on standard tools (Banach fixed-point theorem, fractional semigroup theory) applied in appropriate function spaces. These are external mathematical results independent of the target equation and not derived from the paper's own fitted quantities or self-citations. No equations, ansatzes, or self-referential definitions are supplied that would reduce the result to its inputs by construction. This is the normal, non-circular case for an existence proof.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on standard background results from fractional calculus and nonlinear PDE theory; no free parameters or invented entities are visible from the abstract.

axioms (1)

standard math Existence of a suitable fractional derivative operator and associated function spaces (e.g., appropriate Sobolev or Besov spaces) that support contraction-mapping arguments.
Invoked implicitly by any existence proof for fractional evolution equations.

pith-pipeline@v0.9.0 · 5556 in / 1174 out tokens · 27373 ms · 2026-05-25T01:41:30.750042+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.

mild solution defined via E_α((−it)^α|ξ|^β) and fixed-point contraction on C([0,T];H^β) (Theorem 3.1)
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

Caputo derivative realized as J^{1−α} u' and Mittag-Leffler estimates (Remark 2.1, eq. (3)–(4))

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

17 extracted references · 17 canonical work pages · 1 internal anchor

[1]

Bergh and J

J. Bergh and J. L¨ ofstr¨ om. Interpolation spaces: An Introduction (New Y ork: Springer, 1976)

work page 1976
[2]

Y . Cho, H. Hajaiej, G. Hwang and T. Ozawa. On the Cauchy pro blem of fractional Schr¨ odinger equation with Hartree type nonlinearity. Funkcial. Ekvac . 56 (2013), 193- 224

work page 2013
[3]

Y . Cho, G. Hwang, S. Kwon and S. Lee. On ﬁnite time blow-up f or the mass-critical Hartree equations. Proc. Roy. Soc. Edinburgh Sect. A 145 (2015), 467-479

work page 2015
[4]

Mean-Field Limit of Quantum Bose Gases and Nonlinear Hartree Equation

J. Frohlich and E. Lenzmann. Mean ﬁeld Limit of Quantum Bo se Gases and Nonlin- ear Hartree Equation. Smin. Equ. Driv., Partielles XIX, Preprint: arXiv:math-ph/0409019 (2003), 1-26

work page internal anchor Pith review Pith/arXiv arXiv 2003
[5]

Frohlich, B

J. Frohlich, B. Lars, G. Jonsson and E. Lenzmann. Boson St ars as Solitary Waves. Com- mun. Math. Phys. 274 (2007), 1-30

work page 2007
[6]

G´ orka, H

P . G´ orka, H. Prado and J. Trujillo. The time fractional S chr¨ odinger equation on Hilbert space. Integral Equations Operator Theory. 87 (2017), 1-14

work page 2017
[7]

Grafakos

L. Grafakos. Classical F ourier Analysis, Third edition, Graduate Texts in Math. 249 (New Y ork: Springer, 2014)

work page 2014
[8]

Grafakos and S

L. Grafakos and S. Oh. The Kato-Ponce inequality. Comm. Partial Differential Equations. 39 (2014), 1128-1157

work page 2014
[9]

Herr and A

S. Herr and A. Tesfahun. Small data scattering for semi-r elativistic equations with Hartree type nonlinearity, J. Differential Equations 259 (2015), 5510-5532

work page 2015
[10]

Kwasnicki

M. Kwasnicki. Ten equivalent deﬁnitions of the fractio nal Laplace operator. Fract. Calc. Appl. Anal. 20 (2017), 7-51

work page 2017
[11]

N. Laskin. Time fractional quantum mechanics. Chaos Solitons Fractals (2017), 1-13

work page 2017
[12]

Lenzmann and M

E. Lenzmann and M. Lewin. On singularity formation for t heL2-critical Boson star equa- tion. Nonlinearity 24 (2011), 3515-3540

work page 2011
[13]

Podlubny

I. Podlubny. Fractional differential equations: an introduction to fra ctional derivatives, fractional differential equations, to methods of their sol ution and some of their applica- tions 198 (Slovak Republic: Academic Press, 1998). 19

work page 1998
[14]

E. M. Stein and T. S. Murphy. Harmonic analysis: Real variable methods, orthogonality and oscillatory integrals 3 (New Jersey: Princeton University press, 1993)

work page 1993
[15]

T. Tao. Nonlinear Dispersive Equations: local and Global Analysis 106 (Rhode Island: American Mathematical Soc. 2006)

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[16]

M. Taylor. Partial differential equations III. Nonlinear equations (New Y ork: Springer, 2010)

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[17]

F. Treves. Topological V ector Spaces, Distributions and Kernels: Pure and Applied Math- ematics 25 (New Y ork: Elsevier, 2016). 20

work page 2016

[1] [1]

Bergh and J

J. Bergh and J. L¨ ofstr¨ om. Interpolation spaces: An Introduction (New Y ork: Springer, 1976)

work page 1976

[2] [2]

Y . Cho, H. Hajaiej, G. Hwang and T. Ozawa. On the Cauchy pro blem of fractional Schr¨ odinger equation with Hartree type nonlinearity. Funkcial. Ekvac . 56 (2013), 193- 224

work page 2013

[3] [3]

Y . Cho, G. Hwang, S. Kwon and S. Lee. On ﬁnite time blow-up f or the mass-critical Hartree equations. Proc. Roy. Soc. Edinburgh Sect. A 145 (2015), 467-479

work page 2015

[4] [4]

Mean-Field Limit of Quantum Bose Gases and Nonlinear Hartree Equation

J. Frohlich and E. Lenzmann. Mean ﬁeld Limit of Quantum Bo se Gases and Nonlin- ear Hartree Equation. Smin. Equ. Driv., Partielles XIX, Preprint: arXiv:math-ph/0409019 (2003), 1-26

work page internal anchor Pith review Pith/arXiv arXiv 2003

[5] [5]

Frohlich, B

J. Frohlich, B. Lars, G. Jonsson and E. Lenzmann. Boson St ars as Solitary Waves. Com- mun. Math. Phys. 274 (2007), 1-30

work page 2007

[6] [6]

G´ orka, H

P . G´ orka, H. Prado and J. Trujillo. The time fractional S chr¨ odinger equation on Hilbert space. Integral Equations Operator Theory. 87 (2017), 1-14

work page 2017

[7] [7]

Grafakos

L. Grafakos. Classical F ourier Analysis, Third edition, Graduate Texts in Math. 249 (New Y ork: Springer, 2014)

work page 2014

[8] [8]

Grafakos and S

L. Grafakos and S. Oh. The Kato-Ponce inequality. Comm. Partial Differential Equations. 39 (2014), 1128-1157

work page 2014

[9] [9]

Herr and A

S. Herr and A. Tesfahun. Small data scattering for semi-r elativistic equations with Hartree type nonlinearity, J. Differential Equations 259 (2015), 5510-5532

work page 2015

[10] [10]

Kwasnicki

M. Kwasnicki. Ten equivalent deﬁnitions of the fractio nal Laplace operator. Fract. Calc. Appl. Anal. 20 (2017), 7-51

work page 2017

[11] [11]

N. Laskin. Time fractional quantum mechanics. Chaos Solitons Fractals (2017), 1-13

work page 2017

[12] [12]

Lenzmann and M

E. Lenzmann and M. Lewin. On singularity formation for t heL2-critical Boson star equa- tion. Nonlinearity 24 (2011), 3515-3540

work page 2011

[13] [13]

Podlubny

I. Podlubny. Fractional differential equations: an introduction to fra ctional derivatives, fractional differential equations, to methods of their sol ution and some of their applica- tions 198 (Slovak Republic: Academic Press, 1998). 19

work page 1998

[14] [14]

E. M. Stein and T. S. Murphy. Harmonic analysis: Real variable methods, orthogonality and oscillatory integrals 3 (New Jersey: Princeton University press, 1993)

work page 1993

[15] [15]

T. Tao. Nonlinear Dispersive Equations: local and Global Analysis 106 (Rhode Island: American Mathematical Soc. 2006)

work page 2006

[16] [16]

M. Taylor. Partial differential equations III. Nonlinear equations (New Y ork: Springer, 2010)

work page 2010

[17] [17]

F. Treves. Topological V ector Spaces, Distributions and Kernels: Pure and Applied Math- ematics 25 (New Y ork: Elsevier, 2016). 20

work page 2016