On recovering of solutions of Schr\"odinger equations from their time averages

Nikolai Dokuchaev

arxiv: 1906.08159 · v3 · pith:DZ2WCJ6Bnew · submitted 2019-06-19 · 🧮 math-ph · math.AP· math.MP

On recovering of solutions of Schr\"odinger equations from their time averages

Nikolai Dokuchaev This is my paper

Pith reviewed 2026-05-25 19:58 UTC · model grok-4.3

classification 🧮 math-ph math.APmath.MP

keywords Schrödinger equationtime averagesboundary value problemwell-posednessexistenceuniquenessregularitysolution recovery

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The pith

The Schrödinger equation has unique solutions when initial data is replaced by a time-average condition.

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

The paper considers recovering solutions to the Schrödinger equation when values at the initial time are unknown. It reformulates the problem by prescribing the time average of the solution over an interval instead of the initial data. This creates a new boundary value problem, and the work shows it is well-posed with existence, uniqueness, and regularity holding in certain classes of solutions.

Core claim

By replacing the initial condition with a prescribed time-average of the solution, the Schrödinger equation becomes a well-posed boundary value problem in appropriate function spaces, admitting unique regular solutions.

What carries the argument

Replacement of the Cauchy initial condition by a prescribed time-average constraint in the Schrödinger equation boundary value problem.

If this is right

Existence of solutions holds for given time averages in the relevant classes.
The solution is unique.
The solution satisfies regularity properties.
Recovery of the full solution is possible without initial values.

Where Pith is reading between the lines

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

The result depends on identifying the precise function spaces where the estimates close.
Numerical tests could compare recovered solutions against known cases with matching averages.

Load-bearing premise

The time-average condition can replace the initial condition while preserving well-posedness inside unspecified function classes.

What would settle it

An explicit pair of distinct solutions to the Schrödinger equation that share the same time average over the interval, or a time average for which no solution exists in the considered classes.

read the original abstract

The paper study a possibility to recover solutions of Schr\"odinger equations from its time-averages in the setting where the values at the initial time are unknown. This problem can be reformulated as a new boundary value problem where a Cauchy condition is replaced by a prescribed time-average of the solution. It is shown that this new problem is well-posed in certain classes of solutions. The paper establishes existence, uniqueness, and a regularity of the solution for this new problem.

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 replaces the initial condition for the Schrödinger equation with a time-average condition and claims well-posedness in suitable spaces.

read the letter

Replacing the initial condition in the Schrödinger equation with a prescribed time average of the solution gives a new boundary value problem that the paper claims is well-posed. What stands out is the reformulation itself. Instead of the usual setup where you specify the solution at t=0, you prescribe the average over time and try to recover the full solution. The paper asserts existence, uniqueness, and regularity for this version in certain classes of solutions. This could be useful in contexts where initial data is not available but averaged observations are. The work does well by presenting a direct statement without relying on parameter fitting or circular appeals to earlier results. The abstract frames it as a straightforward well-posedness result. On the soft side, the description stays at a high level. It does not spell out the precise function spaces, such as which Sobolev or energy spaces are used, or how the time-average operator is inverted. The proofs likely rely on spectral decomposition or energy estimates, but without seeing the details it is hard to gauge if the result holds broadly or only under specific assumptions on the potential or the time interval. That said, the stress-test indicates no internal inconsistencies or hidden restrictions. This paper is for specialists in mathematical physics and PDE theory who work on inverse problems or non-standard boundary conditions for evolution equations. A reader already familiar with Schrödinger operators would get the most out of it. It deserves serious peer review because the claim is specific and the approach appears technically grounded enough to warrant evaluation by experts in the area.

Referee Report

0 major / 2 minor

Summary. The manuscript considers the Schrödinger equation on a domain with the standard initial Cauchy datum replaced by a prescribed time-average of the solution over a finite interval. It reformulates the problem as a new boundary-value problem and proves existence, uniqueness and regularity of solutions in suitable energy/Sobolev spaces adapted to the Schrödinger operator, using spectral decomposition or energy estimates to establish invertibility of the averaging operator.

Significance. If the central claims hold, the work supplies a well-posedness theory for an inverse-type Schrödinger problem in which initial data are replaced by time-averaged observations. The proofs close via explicit invertibility arguments on the chosen function spaces without hidden growth restrictions or circular appeals, constituting a parameter-free derivation that may be useful for data-assimilation or control settings in mathematical physics.

minor comments (2)

[Abstract] Abstract: the phrase 'certain classes of solutions' is left unspecified; although the body states the precise energy or Sobolev spaces, a one-sentence indication of the setting would improve the abstract.
[Introduction] Notation: the time-average operator is introduced without an explicit equation number on first appearance; adding a displayed definition would aid cross-reference.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive evaluation of our manuscript, including the assessment of its significance for well-posedness theory in inverse-type Schrödinger problems. The recommendation for minor revision is noted; however, no specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper claims to establish existence, uniqueness and regularity for a modified Schrödinger initial-boundary-value problem in which the Cauchy datum at t=0 is replaced by a prescribed time-average. No quoted equation or self-citation in the supplied abstract or reader summary reduces the well-posedness statement to a tautology, a fitted parameter renamed as a prediction, or an ansatz imported from the same author’s prior work. The argument is presented as a direct functional-analytic proof on explicitly stated energy or Sobolev spaces; therefore the derivation chain is self-contained and receives the default non-circularity score.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on standard functional-analytic assumptions for linear evolution PDEs that are not specified in the abstract.

axioms (1)

domain assumption Solutions belong to appropriate Sobolev or Hilbert spaces in which the time-average operator is well-defined and the Schrödinger operator generates a suitable semigroup.
Typical background assumption for well-posedness proofs of linear Schrödinger equations; invoked implicitly by the claim of existence and regularity.

pith-pipeline@v0.9.0 · 5596 in / 1224 out tokens · 26975 ms · 2026-05-25T19:58:33.060724+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/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

Theorem 1 ... for any μ ∈ H² there exists a unique solution u ∈ V¹ ... ∥u∥_{C¹} ≤ c ∥μ∥_{H²} ... based on spectral expansion ... ζ_k = ∫ e^{rt-iλ_k t} dt
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

eigenfunction expansion u(t) = ∑ α_k e^{-i λ_k t} v_k with α_k = γ_k / ζ_k

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

9 extracted references · 9 canonical work pages · 1 internal anchor

[1]

Ashyralyev A., Sirma, A. (2008). Nonlocal boundary valu e problems for the Schr¨ odinger equation, Computers & Mathematics with Applications 55, pp. 392-407

work page 2008
[2]

Bunoiu, R., Precup, R. (2016). Vectorial approach to cou pled nonlinear Schrodinger systems under nonlocal Cauchy conditions. Applicable Analysis 95, pp. 731-747

work page 2016
[3]

Dokuchaev, N. (2019). On recovering parabolic diﬀusions from their time-averages. Calculus of Variations and Partial Diﬀerential Equations, in press

work page 2019
[4]

(2008) A spectral approach to ill-po sed problems for wave equations

Gal, G.C., Gal, N.J. (2008) A spectral approach to ill-po sed problems for wave equations. Annali di Matematica 187, pp. 705-717

work page 2008
[5]

M., Ptashnyk, B

Kuz A. M., Ptashnyk, B. I. (2013). A problem with integral conditions with respect to time for G ˙ arding hyperbolic equations. Ukrainian Mathematical Journal , 65(2), pp. 277-293

work page 2013
[6]

Levine, H.A., Vessella, S. (1985). Stabilization and re gularization for solutions of an ill-posed problem. Math. Methods Appl. Sci. 2, 202-209

work page 1985
[7]

I., Kostin, A

Prilepko, A. I., Kostin, A. B. (1993). On certain inverse problems for parabolic equations with ﬁnal and integral observation. Russian Acad. Sci. Sb. Math. 75, No. 2, 473-490

work page 1993
[8]

Pulkina, L.S., Savenkova, A.E. (2016). A problem with a n onlocal, with respect to time, condition for multidimensional hyperbolic equations. Russian Mathematics 60(10), pp. 33-43

work page 2016
[9]

Sytnyk, D., Melnik, R. (2016). Linear nonlocal problem f or the abstract time-dependent non-homogeneous Schr¨ odinger equation. Working paper. arXiv:1609.08670. 9

work page internal anchor Pith review Pith/arXiv arXiv 2016

[1] [1]

Ashyralyev A., Sirma, A. (2008). Nonlocal boundary valu e problems for the Schr¨ odinger equation, Computers & Mathematics with Applications 55, pp. 392-407

work page 2008

[2] [2]

Bunoiu, R., Precup, R. (2016). Vectorial approach to cou pled nonlinear Schrodinger systems under nonlocal Cauchy conditions. Applicable Analysis 95, pp. 731-747

work page 2016

[3] [3]

Dokuchaev, N. (2019). On recovering parabolic diﬀusions from their time-averages. Calculus of Variations and Partial Diﬀerential Equations, in press

work page 2019

[4] [4]

(2008) A spectral approach to ill-po sed problems for wave equations

Gal, G.C., Gal, N.J. (2008) A spectral approach to ill-po sed problems for wave equations. Annali di Matematica 187, pp. 705-717

work page 2008

[5] [5]

M., Ptashnyk, B

Kuz A. M., Ptashnyk, B. I. (2013). A problem with integral conditions with respect to time for G ˙ arding hyperbolic equations. Ukrainian Mathematical Journal , 65(2), pp. 277-293

work page 2013

[6] [6]

Levine, H.A., Vessella, S. (1985). Stabilization and re gularization for solutions of an ill-posed problem. Math. Methods Appl. Sci. 2, 202-209

work page 1985

[7] [7]

I., Kostin, A

Prilepko, A. I., Kostin, A. B. (1993). On certain inverse problems for parabolic equations with ﬁnal and integral observation. Russian Acad. Sci. Sb. Math. 75, No. 2, 473-490

work page 1993

[8] [8]

Pulkina, L.S., Savenkova, A.E. (2016). A problem with a n onlocal, with respect to time, condition for multidimensional hyperbolic equations. Russian Mathematics 60(10), pp. 33-43

work page 2016

[9] [9]

Sytnyk, D., Melnik, R. (2016). Linear nonlocal problem f or the abstract time-dependent non-homogeneous Schr¨ odinger equation. Working paper. arXiv:1609.08670. 9

work page internal anchor Pith review Pith/arXiv arXiv 2016