On uniqueness and nonuniqueness for potential reconstruction in quantum fields from one measurement II. the non-radial case

Guang-Hui Zheng; Zhi-Qiang Miao

arxiv: 1907.01948 · v1 · pith:5PSCGNNYnew · submitted 2019-07-02 · 🧮 math.AP

On uniqueness and nonuniqueness for potential reconstruction in quantum fields from one measurement II. the non-radial case

Zhi-Qiang Miao , Guang-Hui Zheng This is my paper

Pith reviewed 2026-05-25 11:27 UTC · model grok-4.3

classification 🧮 math.AP

keywords inverse problemsSchrödinger equationNeumann-Dirichlet mappotential reconstructioncore-shell structureuniquenessnonuniquenessmodified Bessel functions

0 comments

The pith

The ND map uniquely recovers the potential from one measurement in 2D and 3D core-shell structures but not for all potential-shape pairs.

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

This paper proves uniqueness theorems for reconstructing the potential in the steady-state Schrödinger equation from a single Neumann-to-Dirichlet map in two- and three-dimensional core-shell structures. It uses the ND map together with explicit solutions expressed via modified Bessel functions to separate the potential from the shape. The work also demonstrates nonuniqueness by constructing different potentials paired with different shapes that generate the same map. Readers would care because these results clarify the conditions under which one boundary measurement suffices to recover the quantum potential.

Core claim

Based on the theory of the ND map and modified Bessel function, the uniqueness theorem of the inverse problem in two-dimensional and three-dimensional core-shell structure is established, respectively. When different potential and shape are considered, the nonuniqueness results is also proved.

What carries the argument

The Neumann-to-Dirichlet map together with explicit modified Bessel function solutions in core-shell geometries that separate potential from shape.

If this is right

Uniqueness holds for the potential in two-dimensional core-shell structures from one ND map.
Uniqueness holds for the potential in three-dimensional core-shell structures from one ND map.
Nonuniqueness holds when different potentials are paired with different shapes that compensate to give the same map.

Where Pith is reading between the lines

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

The explicit solution method could apply to other layered or symmetric domains where similar radial solutions exist.
Nonuniqueness examples suggest that reconstruction in general domains may require shape constraints or additional data to avoid ambiguity.
The results indicate a trade-off between potential and domain geometry that could be exploited or avoided in inverse problem algorithms.

Load-bearing premise

The core-shell geometry admits explicit solutions via modified Bessel functions that allow the ND map to separate the potential from the domain shape.

What would settle it

Constructing or observing two different core-shell (potential, shape) pairs that produce exactly the same ND map would support the nonuniqueness claim, while showing that no such pairs exist would strengthen the uniqueness theorems.

read the original abstract

In this article we study uniqueness and nonuniqueness for potential reconstruction from one boundary measurement in quantum fields, associated with the steady state Schr\"{o}dinger equation. It is an extension of our recent work \cite{Zheng2019}. Based the theory of the ND map and modified bessel function, the uniqueness theorem of the inverse problem in two-dimensional nd three-dimensional core-shell structure is established, respectively. When different potential and shape are considered, the nonuniqueness results is also proved.

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

read the letter

The paper's uniqueness and nonuniqueness theorems for non-radial core-shell structures depend on an explicit ND-map expression from modified Bessel functions that requires radial symmetry, which the non-radial setting lacks. This is an extension of the authors' earlier radial-case paper. The new material is the 2D and 3D uniqueness theorems for the inverse problem in core-shell structures without radial symmetry, plus nonuniqueness results when both the potential and the shape vary. They build on standard ND-map theory. The nonuniqueness constructions are the part that could be useful if they can be made rigorous. Showing that different q and different domains can produce the same boundary map is a concrete negative result in this setting. The soft spot is substantial. Modified Bessel functions arise after separating variables in polar or spherical coordinates, which works when everything is radial. For non-radial inner and outer boundaries or a non-radial potential, the equation does not separate and there is no closed-form expression for the ND map that cleanly isolates the potential. The abstract explicitly says the results are based on the ND map and modified Bessel function, so this assumption is load-bearing for both the positive and negative statements. The stress-test concern matches the abstract directly. This work is for people already inside the inverse-problems-for-Schrödinger-equations community who care about core-shell geometries. A specialist might find the nonuniqueness examples interesting if the proofs close the gap somehow, but the average reader in the broader field will not get much. I would recommend sending it to peer review. The question is narrow but well-posed within the subfield, and the authors have a track record on the radial case. Referees can determine whether the non-radial case is handled by a different technique or whether the Bessel approach is misapplied. The paper engages honestly with the literature it cites, so it deserves that step even with the evident technical concern.

Referee Report

1 major / 1 minor

Summary. The paper extends the authors' prior work on inverse problems for the steady-state Schrödinger equation, claiming to prove uniqueness theorems for potential reconstruction from a single boundary measurement (via the ND map) in 2D and 3D non-radial core-shell structures, using explicit formulas involving modified Bessel functions. It also constructs nonuniqueness examples when the potential and domain shape are allowed to vary simultaneously.

Significance. If the claims are valid, the results would clarify identifiability conditions for potentials in composite (core-shell) geometries, distinguishing unique recovery from nonuniqueness in the non-radial setting. The explicit use of ND-map theory combined with special functions is presented as enabling both positive and negative results.

major comments (1)

[Abstract] Abstract: the uniqueness and nonuniqueness statements both rely on an explicit ND-map formula obtained via modified Bessel functions that purportedly separates the potential q from the domain shape. Modified Bessel functions arise from separation of variables after reduction to radial coordinates; this closed-form separation does not extend to genuinely non-radial inner/outer boundaries or non-radial q. This assumption is load-bearing for the central claims in both the 2D and 3D cases.

minor comments (1)

[Abstract] Abstract contains typographical errors ('nd' for 'and', 'is also proved' should be 'are also proved').

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for identifying this key point about the scope of the explicit ND-map formula. We respond to the major comment below.

read point-by-point responses

Referee: [Abstract] Abstract: the uniqueness and nonuniqueness statements both rely on an explicit ND-map formula obtained via modified Bessel functions that purportedly separates the potential q from the domain shape. Modified Bessel functions arise from separation of variables after reduction to radial coordinates; this closed-form separation does not extend to genuinely non-radial inner/outer boundaries or non-radial q. This assumption is load-bearing for the central claims in both the 2D and 3D cases.

Authors: The referee correctly notes that the explicit ND-map formula is derived via separation of variables, yielding modified Bessel functions, and therefore requires radial symmetry of the potential together with boundaries (circular in 2D, spherical in 3D) that permit this reduction. In the manuscript the phrase “non-radial case” is used to indicate that we allow the potential and the domain shape to vary simultaneously when constructing non-uniqueness examples, while the uniqueness statements themselves are proved under the radial-reduction setting inherited from the ND-map formula. For genuinely non-radial boundaries or non-radial q the closed-form expression does not hold and the stated uniqueness would require different techniques. We will revise the abstract, the introduction, and the statements of the main theorems to make the geometric and coefficient assumptions explicit and to qualify the scope of the results accordingly. revision: yes

Circularity Check

1 steps flagged

Minor self-citation to prior ND-map theory; no reduction of new uniqueness/nonuniqueness claims to inputs by construction

specific steps

self citation load bearing [Abstract]
"It is an extension of our recent work [Zheng2019]. Based the theory of the ND map and modified bessel function, the uniqueness theorem of the inverse problem in two-dimensional nd three-dimensional core-shell structure is established, respectively."

The load-bearing ND-map theory used to separate potential from domain shape (and thereby obtain both uniqueness and nonuniqueness) is justified solely by citation to the authors' own prior paper rather than re-derived or externally verified here.

full rationale

The paper explicitly positions itself as an extension of the authors' earlier work and invokes ND-map theory plus modified Bessel functions to derive uniqueness for 2D/3D core-shell structures and nonuniqueness constructions. This constitutes one minor self-citation that is not load-bearing for the central claims, as the manuscript presents separate arguments for the non-radial case. No self-definitional steps, fitted inputs renamed as predictions, or ansatz smuggling are exhibited in the provided text. The derivation chain remains self-contained against external mathematical benchmarks for the ND map.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract provides no explicit list of free parameters or invented entities; the central claims rest on the domain assumption that the geometry permits separation via modified Bessel functions.

axioms (1)

domain assumption The ND map together with modified Bessel solutions separates potential from domain shape in non-radial core-shell geometries.
Invoked to establish both uniqueness and nonuniqueness.

pith-pipeline@v0.9.0 · 5610 in / 1075 out tokens · 19471 ms · 2026-05-25T11:27:47.473913+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 (J uniqueness) unclear

?

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

Based on the theory of the ND map and modified bessel function, the uniqueness theorem... (2.13) R_σ1,r1(g) expressed via ρ(r1,σ1) involving I_n,K_n and D(r1,r2) Wronskians; Thm 2.6 uses Cor 2.4 monotonicity of F(η)
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking (D=3 from linking) echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

3-D case uses ν=n+1/2 spherical modified Bessel (3.45)–(3.58); D=3 tacitly assumed for spherical harmonics

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

18 extracted references · 18 canonical work pages · 1 internal anchor

[1]

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work page 1980
[2]

A. L. Bukhgeim and G. Uhlmann , Recovering a potential from partial Cauchy data , Comm. Partial Diﬀerential Equations, 27 (2002), 653-668

work page 2002
[3]

Imaging, 1 (2007), 95-105

V.Isakov, On uniqueness in the inverse conductivity problem with loca l data , Inverse Probl. Imaging, 1 (2007), 95-105

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[4]

O. Y. Imanuvilov, G. Uhlmann and M.Yamamoto , The Caldern problem with partial data in two dimensions , J. Am. Math. Soc., 23 (2010), 655-691

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O. Y. Imanuvilov and G. Uhlmann , Inverse boundary value problem for the Schr¨ odinger equation in a cylindrical domain by partial bo undary data , Inverse Prob- lems, 29 (2013), 045002

work page 2013
[6]

Kenig and M

C. Kenig and M. Salo , Recent progress in the Caldern problem with partial data , Contemp. Math., 615 (2014), 193-222

work page 2014
[7]

Greenleaf, M

A. Greenleaf, M. Lassas and G. Uhlmann , Anisotropic conductivities that cannot be detected by EIT , Physiological Measurement, 24 (2003), 413-419

work page 2003
[8]

Greenleaf, M

A. Greenleaf, M. Lassas and G. Uhlmann , On nonuniqueness for Calderns inverse problem , Mathematical Research Letters, 10 (2003), 685-694

work page 2003
[9]

Greenleaf, Y

A. Greenleaf, Y. kurylev, M. Lassas and G. Uhlmann , Full-wave invisibility of active devices at all frequencies , Communications in Mathematical Physics, 275 (2007), 749-789

work page 2007
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Greenleaf, Y

A. Greenleaf, Y. kurylev, M. Lassas and G. Uhlmann , Cloaking devices, electromagnetic wormholes, and transformation optics , SIAM Review, 51 (2009), 3-33

work page 2009
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H. Y. Liu , Virtual reshaping and invisibility in obstacle scattering , Inverse Problems, 25 (2009), 045006

work page 2009
[12]

Guang-Hui Zheng and Zhi-Qiang Miao , On uniqueness and nonuniqueness for potential reconstruction in quantum ﬁelds from one measure ment, arXiv preprint, arXiv:1903.11825

work page internal anchor Pith review Pith/arXiv arXiv 1903
[13]

Cohen-Tannoudji, Claude, B.Diu and F.Laloe , Quantum Mechanics, Volume 1, Quantum Mechanics for Chemists, 1991

work page 1991
[14]

Friedman and V.Isakov , On the uniqueness in the inverse conductivity problem with one measurement , Indiana University Mathematics Journal, 38 (1989), 563-579

A. Friedman and V.Isakov , On the uniqueness in the inverse conductivity problem with one measurement , Indiana University Mathematics Journal, 38 (1989), 563-579

work page 1989
[15]

Alberti and M

G. Alberti and M. Santacesaria , Calder´ on ’s Inverse Problem with a Finite Num- ber of Measurements , arXiv preprint, arXiv:1803.04224v1, 2018. 14

work page arXiv 2018
[16]

Abramowitz and I

M. Abramowitz and I. Stegun , Handbook of mathematical functions with formulas, graphs, and mathematical tables , U.S. Department of Commerce, National Bureau of Standards Applied Mathematics Series, 55 (1964), 803-819

work page 1964
[17]

Segura , A new type of sharp bounds for ratios of modiﬁed Bessel functions , J

Diego Ruiz-Antoln and J. Segura , A new type of sharp bounds for ratios of modiﬁed Bessel functions , J. Math. Anal. Appl, 443 (2016), 1232-1246

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Jaeger , Heat conduction in composite circular cylinders , Phil

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[1] [1]

A. P. Calderon On an inverse boundary value problem, in Seminar on Numerica l Analysis and Its Applications to Continuum Physics , Soc. Brasil. Mat., Rio de Janeiro, 1980, 65-73

work page 1980

[2] [2]

A. L. Bukhgeim and G. Uhlmann , Recovering a potential from partial Cauchy data , Comm. Partial Diﬀerential Equations, 27 (2002), 653-668

work page 2002

[3] [3]

Imaging, 1 (2007), 95-105

V.Isakov, On uniqueness in the inverse conductivity problem with loca l data , Inverse Probl. Imaging, 1 (2007), 95-105

work page 2007

[4] [4]

O. Y. Imanuvilov, G. Uhlmann and M.Yamamoto , The Caldern problem with partial data in two dimensions , J. Am. Math. Soc., 23 (2010), 655-691

work page 2010

[5] [5]

O. Y. Imanuvilov and G. Uhlmann , Inverse boundary value problem for the Schr¨ odinger equation in a cylindrical domain by partial bo undary data , Inverse Prob- lems, 29 (2013), 045002

work page 2013

[6] [6]

Kenig and M

C. Kenig and M. Salo , Recent progress in the Caldern problem with partial data , Contemp. Math., 615 (2014), 193-222

work page 2014

[7] [7]

Greenleaf, M

A. Greenleaf, M. Lassas and G. Uhlmann , Anisotropic conductivities that cannot be detected by EIT , Physiological Measurement, 24 (2003), 413-419

work page 2003

[8] [8]

Greenleaf, M

A. Greenleaf, M. Lassas and G. Uhlmann , On nonuniqueness for Calderns inverse problem , Mathematical Research Letters, 10 (2003), 685-694

work page 2003

[9] [9]

Greenleaf, Y

A. Greenleaf, Y. kurylev, M. Lassas and G. Uhlmann , Full-wave invisibility of active devices at all frequencies , Communications in Mathematical Physics, 275 (2007), 749-789

work page 2007

[10] [10]

Greenleaf, Y

A. Greenleaf, Y. kurylev, M. Lassas and G. Uhlmann , Cloaking devices, electromagnetic wormholes, and transformation optics , SIAM Review, 51 (2009), 3-33

work page 2009

[11] [11]

H. Y. Liu , Virtual reshaping and invisibility in obstacle scattering , Inverse Problems, 25 (2009), 045006

work page 2009

[12] [12]

Guang-Hui Zheng and Zhi-Qiang Miao , On uniqueness and nonuniqueness for potential reconstruction in quantum ﬁelds from one measure ment, arXiv preprint, arXiv:1903.11825

work page internal anchor Pith review Pith/arXiv arXiv 1903

[13] [13]

Cohen-Tannoudji, Claude, B.Diu and F.Laloe , Quantum Mechanics, Volume 1, Quantum Mechanics for Chemists, 1991

work page 1991

[14] [14]

Friedman and V.Isakov , On the uniqueness in the inverse conductivity problem with one measurement , Indiana University Mathematics Journal, 38 (1989), 563-579

A. Friedman and V.Isakov , On the uniqueness in the inverse conductivity problem with one measurement , Indiana University Mathematics Journal, 38 (1989), 563-579

work page 1989

[15] [15]

Alberti and M

G. Alberti and M. Santacesaria , Calder´ on ’s Inverse Problem with a Finite Num- ber of Measurements , arXiv preprint, arXiv:1803.04224v1, 2018. 14

work page arXiv 2018

[16] [16]

Abramowitz and I

M. Abramowitz and I. Stegun , Handbook of mathematical functions with formulas, graphs, and mathematical tables , U.S. Department of Commerce, National Bureau of Standards Applied Mathematics Series, 55 (1964), 803-819

work page 1964

[17] [17]

Segura , A new type of sharp bounds for ratios of modiﬁed Bessel functions , J

Diego Ruiz-Antoln and J. Segura , A new type of sharp bounds for ratios of modiﬁed Bessel functions , J. Math. Anal. Appl, 443 (2016), 1232-1246

work page 2016

[18] [18]

Jaeger , Heat conduction in composite circular cylinders , Phil

J. Jaeger , Heat conduction in composite circular cylinders , Phil. Mag., 32(1941) 324- 335. 15

work page 1941