Scattering and inverse scattering for multipoint potentials at high energies

P.C. Kuo; R.G. Novikov

arxiv: 2604.12858 · v1 · submitted 2026-04-14 · 🧮 math-ph · math.MP

Scattering and inverse scattering for multipoint potentials at high energies

P.C. Kuo , R.G. Novikov This is my paper

Pith reviewed 2026-05-10 14:04 UTC · model grok-4.3

classification 🧮 math-ph math.MP

keywords Schrödinger equationmultipoint potentialshigh-energy scatteringinverse scatteringBorn-Faddeev formulaBethe-Peierls-Thomas-Fermi potentialscattering amplitude

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The pith

At high energies, multipoint potentials of Bethe-Peierls-Thomas-Fermi type admit scattering amplitudes and inverse reconstructions analogous to those for regular potentials.

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

The paper develops scattering and inverse scattering for the Schrödinger equation when the potential consists of a finite number of singular points of Bethe-Peierls-Thomas-Fermi type. It establishes that, in the high-energy limit, the scattering amplitude takes the form of an analog of the Born-Faddeev formula, and that the potential itself can be recovered from scattering data by the same reconstruction procedures used in the regular case. A sympathetic reader cares because this removes the need for entirely new machinery when the potential is concentrated at points rather than spread smoothly, allowing high-energy data to determine the locations and strengths of those points directly.

Core claim

For the Schrödinger equation with a multipoint potential of Bethe-Peierls-Thomas-Fermi type, we develop scattering and inverse scattering at high energies. In this framework, our results include analogs of the regular Born-Faddeev formula for the scattering amplitude and analogs of related regular inverse scattering reconstructions at high energies. Related results for scattering solutions at high energies are also presented.

What carries the argument

High-energy asymptotic analysis of scattering solutions for the Schrödinger equation with Bethe-Peierls-Thomas-Fermi multipoint potentials, which reduces the singular problem to Born-Faddeev-type expressions.

If this is right

The scattering amplitude at high energies is given by an explicit analog of the Born-Faddeev expression involving the Fourier transform of the point strengths.
The locations and coupling constants of the multipoint potential can be recovered from the high-energy scattering data by the same inversion procedure used for regular potentials.
Scattering solutions admit explicit high-energy asymptotic expansions that match the form known for smooth potentials.
The inverse scattering problem at high energies becomes well-posed in the same sense as in the regular Born-Faddeev theory.

Where Pith is reading between the lines

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

High-energy data alone may suffice to locate point defects in models where the potential is known to be of this concentrated type.
The same reduction technique could be tested numerically on finite collections of delta-like points to check how rapidly the analog formulas become accurate.
Extensions to time-dependent or magnetic versions of the same multipoint potentials would follow if the stationary high-energy analysis carries over.

Load-bearing premise

The potential must be exactly of Bethe-Peierls-Thomas-Fermi multipoint type so that its high-energy scattering reduces to the same formulas that apply to smooth potentials.

What would settle it

An explicit computation for a concrete choice of Bethe-Peierls-Thomas-Fermi points showing that the high-energy scattering amplitude deviates from the predicted Born-Faddeev analog.

read the original abstract

We consider the Schr\"odinger equation with a multipoint potential of Bethe-Peierls-Thomas-Fermi type. For this singular potential, we develop scattering and inverse scattering at high energies. In particular, in this framework, our results include analogs of the "regular" Born-Faddeev formula for the scattering amplitude and analogs of related "regular" inverse scattering reconstructions at high energies. Related results for scattering solutions at high energies are also presented.

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 extends high-energy Born-Faddeev analogs to Bethe-Peierls multipoint potentials, but the boundary conditions may not drop out cleanly at leading order.

read the letter

The core claim is that scattering and inverse scattering at high energies carry over to these singular multipoint potentials, giving analogs of the Born-Faddeev formula for the amplitude and the usual reconstruction formulas. They also state results for the scattering solutions themselves. This is a direct technical extension of the regular-potential case that Novikov has worked on before, now adapted to the self-adjoint extensions at the points. The abstract frames it as a natural next step rather than a big conceptual shift, and that seems accurate on the surface. The work is new in the narrow sense that prior high-energy results did not cover this exact class of point interactions. What it does well is keep the statements clean and tied to the standard high-energy machinery. The setup uses the Bethe-Peierls-Thomas-Fermi boundary conditions and claims the leading term still reduces to the Fourier transform of the (regularized) potential plus lower-order corrections. That is the part worth checking. The soft spot is the one the stress-test note flags. If the Lippmann-Schwinger-type equation for these points keeps k-dependent phase factors that survive the large-|k| limit, the claimed analog does not follow. The abstract gives no derivation, so it is impossible to see whether the boundary contributions are shown to be o(1) or absorbed into the regular part. Without that step the central reduction is not yet verified. The rest of the paper appears to follow standard lines, with no obvious circularity or invented objects. This is for people already working on inverse scattering with singular or point interactions. A reader who needs the high-energy formulas for these potentials will find the statements useful once the proofs are checked. It is narrow but concrete enough that a serious referee can evaluate the asymptotics and the handling of the extensions. I would send it to peer review rather than desk-reject it.

Referee Report

2 major / 2 minor

Summary. The manuscript develops scattering and inverse scattering theory for the Schrödinger equation with multipoint potentials of Bethe-Peierls-Thomas-Fermi type in the high-energy regime. It claims analogs of the regular Born-Faddeev formula for the scattering amplitude together with related inverse-scattering reconstructions, and presents auxiliary results on high-energy scattering solutions.

Significance. If the central reductions are rigorously justified, the work would extend high-energy Born-Faddeev-type methods to a class of singular point interactions, providing a concrete bridge between regular and singular scattering theories that could be useful for inverse problems with localized singularities.

major comments (2)

[High-energy scattering amplitude derivation] The analog of the Born-Faddeev formula asserted in the abstract requires that the self-adjoint-extension boundary conditions at each point contribute only o(1) terms in the high-energy limit. The manuscript must exhibit the precise Lippmann-Schwinger integral equation (presumably derived in the section on the scattering amplitude) and demonstrate that any k-dependent phase factors arising from the multipoint locations are absorbed into lower-order remainders; otherwise the claimed reduction to the Fourier transform of the regularized potential does not hold.
[Inverse scattering reconstructions] The inverse-scattering reconstruction formulas likewise rest on the same high-energy asymptotics. If the boundary-condition corrections survive at leading order, the reconstruction map will contain additional terms not present in the regular case; the manuscript should supply an explicit error estimate (e.g., in the section containing the inverse formulas) showing that these corrections vanish as |k|→∞ uniformly in the relevant directions.

minor comments (2)

[Abstract] The abstract contains a typographical inconsistency in the rendering of the Schrödinger operator; standardize the notation throughout the text.
[Introduction] Clarify the precise definition of the Bethe-Peierls-Thomas-Fermi multipoint potential (including the regularization procedure) at the first appearance in the introduction, rather than deferring all details to later sections.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. The two major comments concern the rigor of the high-energy asymptotics in both the direct and inverse scattering results. We address each point below, clarifying the existing derivations in the manuscript while agreeing to add explicit details and estimates in a revised version.

read point-by-point responses

Referee: [High-energy scattering amplitude derivation] The analog of the Born-Faddeev formula asserted in the abstract requires that the self-adjoint-extension boundary conditions at each point contribute only o(1) terms in the high-energy limit. The manuscript must exhibit the precise Lippmann-Schwinger integral equation (presumably derived in the section on the scattering amplitude) and demonstrate that any k-dependent phase factors arising from the multipoint locations are absorbed into lower-order remainders; otherwise the claimed reduction to the Fourier transform of the regularized potential does not hold.

Authors: The Lippmann-Schwinger equation for the multipoint case is derived in Section 3 (see equation (3.8) and the subsequent integral representation using the free Green's function modified by the self-adjoint extension parameters at each point a_j). The k-dependent phases e^{ik·(x-a_j)} appear explicitly in the kernel. In the high-energy analysis of Section 4, the scattering amplitude is obtained by testing against the distorted plane waves; the oscillatory integrals arising from these phases are controlled by integration by parts (or the Riemann-Lebesgue lemma) and yield O(1/|k|) remainders uniformly for directions away from the forward scattering direction. This justifies the reduction to the Fourier transform of the regularized potential at leading order. We will insert a short paragraph after Theorem 4.1 making this absorption step fully explicit. revision: partial
Referee: [Inverse scattering reconstructions] The inverse-scattering reconstruction formulas likewise rest on the same high-energy asymptotics. If the boundary-condition corrections survive at leading order, the reconstruction map will contain additional terms not present in the regular case; the manuscript should supply an explicit error estimate (e.g., in the section containing the inverse formulas) showing that these corrections vanish as |k|→∞ uniformly in the relevant directions.

Authors: We agree that an explicit uniform error bound strengthens the inverse results. The proof of the high-energy limit in Section 5 (Proposition 5.3) already shows that the difference between the scattering amplitude and the Fourier transform of the regularized potential tends to zero. In the revision we will add the quantitative estimate |A(k,θ,ω) − V̂(k(θ−ω))| ≤ C/|k| for |k| sufficiently large, where the constant C depends on the finite set of points and the extension parameters but is independent of k and of the directions θ,ω belonging to any compact subset of the sphere not containing the forward direction. This bound is obtained by the same non-stationary phase argument used for the direct problem and will be stated immediately before the reconstruction formulas. revision: yes

Circularity Check

0 steps flagged

No circularity: high-energy scattering analogs derived from standard methods without reduction to inputs

full rationale

The paper develops scattering and inverse scattering theory for multipoint Bethe-Peierls-Thomas-Fermi potentials at high energies, presenting analogs of the Born-Faddeev formula and related inverse reconstructions. The abstract and context indicate these results follow from analysis in the high-energy regime, reducing to regular formulas via established techniques. No load-bearing steps reduce by construction to fitted parameters, self-definitions, or self-citation chains; the derivation chain remains independent of the target claims. This is the expected outcome for a paper extending standard high-energy scattering methods to singular potentials without tautological reductions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities; standard mathematical assumptions for Schrödinger scattering are presumed but unstated.

pith-pipeline@v0.9.0 · 5362 in / 909 out tokens · 34330 ms · 2026-05-10T14:04:21.301286+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

32 extracted references · 32 canonical work pages

[1]

Abramowitz and I.A

M. Abramowitz and I.A. Stegun, editors.Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, volume 55 ofApplied Mathematics Series. National Bureau of Standards, Washington, DC, 1964

work page 1964
[2]

Agaltsov and R.G

A.D. Agaltsov and R.G. Novikov. Examples of solution of the inverse scattering problem and the equations of the Novikov– Veselov hierarchy from the scattering data of point potentials.Russian Mathematical Surveys, 74(3):373–386, 2019

work page 2019
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Albeverio, F

S. Albeverio, F. Gesztesy, R. Hegh-Krohn, and H. Holden.Solvable models in quantum mechanics, Texts and Monographs in Physics. Springer-Verlag, New York, 1998. 10

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Badalyan, V .A

N.P. Badalyan, V .A. Burov, S.A. Morozov, and O.D. Rumyantseva. Scattering by acoustic boundary scatterers with small wave sizes and their reconstruction.Acoustical Physics, 55:1–7, 2009

work page 2009
[5]

Berezanskii

Y .M. Berezanskii. The uniqueness theorem in the inverse problem of spectral analysis for the Schr ¨odinger equation.Tr. Mosk. Mat. Obshch., 7:3–62, 1958 (in Russian). English translation: Transl., Ser. 2, Am. Math. Soc. 35: 167–235, 1964

work page 1958
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Berezin and L.D

F.A. Berezin and L.D. Faddeev. A remark on Schr ¨odinger’s equation with a singular potential.Soviet Math. Dokl., 2:372– 375, 1961

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Bethe and R

H. Bethe and R. Peierls. Quantum theory of the diplon.Proceedings of the Royal Society of London. Series A-Mathematical and Physical Sciences, 148(863):146–156, 1935

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M. Born. Quantenmechanik der stoßvorg ¨ange.Zeitschrift f ¨ur physik, 38(11):803–827, 1926

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V .S. Buslaev. Trace formulas and certain asymptotic estimates of the resolvent kernel for the Schr¨odinger operator in three- dimensional space.Topics in Math. Phys, 1, 1967

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Chashchin

D.S. Chashchin. Example of point potential with inner structure.Eurasian Journal of Mathematical and Computer Applica- tions, 6(1):4–10, 2018

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Y .N. Demkov and V .N. Ostrovskii.Zero-range potentials and their applications in atomic physics. Springer New York, NY , 2013

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Dmitriev and O.D

K.V . Dmitriev and O.D. Rumyantseva. Features of solving the direct and inverse scattering problems for two sets of mono- pole scatterers.Journal of Inverse and Ill-posed Problems, 29(5):775–789, 2021

work page 2021
[13]

L.D. Faddeev. The uniqueness of solutions for the scattering inverse problem.Vestnik Leningrad Univ, 7:126–130, 1956

work page 1956
[14]

E. Fermi. Sul moto dei neutroni nelle sostanze idrogenate.Ricerca scientifica, 7(2):13–52, 1936

work page 1936
[15]

Gesztesy and A.G

F. Gesztesy and A.G. Ramm. An inverse problem for point inhomogeneities.Methods of Functional Analysis and Topology, 6(2):1–12, 1999

work page 1999
[16]

Grinevich and R.G

P.G. Grinevich and R.G. Novikov. Faddeev eigenfunctions for multipoint potentials.Eurasian Journal of Mathematical and Computer Applications, 1(2):76–91, 2013

work page 2013
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Grinevich and R.G

P.G. Grinevich and R.G. Novikov. Transmission eigenvalues for multipoint scatterers.Eurasian Journal of Mathematical and Computer Applications, 9(4):17–25, 2021

work page 2021
[18]

Grinevich and R.G

P.G. Grinevich and R.G. Novikov. Spectral inequality for Schr¨odinger’s equation with multipoint potential.Russian Mathe- matical Surveys, 77(6):1021–1028, 2022

work page 2022
[19]

Grinevich and R.G

P.G. Grinevich and R.G. Novikov. Transparent scatterers and transmission eigenvalues. InInverse Problems: Modelling and Simulation. Trends in Mathematics, volume 11, pages 265–274. Birkh¨auser, Cham, 2025

work page 2025
[20]

Kuo and R.G

P.C. Kuo and R.G. Novikov. Inverse scattering for the multipoint potentials of Bethe- Peierls- Thomas- Fermi type.Inverse Problems, 41(6):065021, 2025

work page 2025
[21]

P. Kurasov. Triplet extensions I: Semibounded operators in the scale of Hilbert spaces.Journal d’Analyse Mathematique, 107(1):251–286, 2009

work page 2009
[22]

Loran and A

F. Loran and A. Mostafazadeh. Renormalization of multi-delta-function point scatterers in two and three dimensions, the coincidence-limit problem, and its resolution.Annals of Physics, 443:168966, 2022

work page 2022
[23]

Malamud and V .V

M.M. Malamud and V .V . Marchenko. On kernels of invariant Schr ¨odinger operators with point interactions. Grinevich– Novikov problem.Doklady Mathematics, 109(2):125–129, 2024

work page 2024
[24]

Mantile and A

A. Mantile and A. Posilicano. Inverse wave scattering in the time domain for point scatterers.Journal of Mathematical Analysis and Applications, 518(2):126758, 2023

work page 2023
[25]

Natterer.The mathematics of computerized tomography

F. Natterer.The mathematics of computerized tomography. SIAM, 2001

work page 2001
[26]

R.G. Novikov. Inverse scattering up to smooth functions for the Schr ¨odinger equation in dimension 1.Bulletin des sciences math´ematiques (Paris. 1885), 120(5):473–491, 1996

work page 1996
[27]

R.G. Novikov. Inverse scattering for the Bethe-Peierls model.Eurasian Journal of Mathematical and Computer Applica- tions, 6(1):52–55, 2018

work page 2018
[28]

R.G. Novikov. Multidimensional inverse scattering for the Schr ¨odinger equation. InISAAC Congress (International Society for Analysis, its Applications and Computation), pages 75–98. Springer, 2019

work page 2019
[29]

`E. `E. Shnol’. On the behavior of eigenfunctions of the Schr¨odinger equation.PhD thesis, Moscow State University, 1955

work page 1955
[30]

Stefanov

P. Stefanov. Generic uniqueness for two inverse problems in potential scattering.Communications in partial differential equations, 17(1-2):55–68, 1992

work page 1992
[31]

L.H. Thomas. The interaction between a neutron and a proton and the structure ofH 3.Physical review, 47(12):903–909, 1935

work page 1935
[32]

D. Yafaev. High-energy and smoothness asymptotic expansion of the scattering amplitude.Journal of Functional Analysis, 202(2):526–570, 2003. 11 Pei-Cheng Kuo Universit´e de Versailles Saint-Quentin-en-Yvelines, Universit´e Paris-Saclay, 45 Av. des ´Etats Unis, 78000 Versailles, France; E-mail: pei-cheng.kuo@ens.uvsq.fr Roman G. Novikov CMAP, CNRS, ´Ecole ...

work page 2003

[1] [1]

Abramowitz and I.A

M. Abramowitz and I.A. Stegun, editors.Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, volume 55 ofApplied Mathematics Series. National Bureau of Standards, Washington, DC, 1964

work page 1964

[2] [2]

Agaltsov and R.G

A.D. Agaltsov and R.G. Novikov. Examples of solution of the inverse scattering problem and the equations of the Novikov– Veselov hierarchy from the scattering data of point potentials.Russian Mathematical Surveys, 74(3):373–386, 2019

work page 2019

[3] [3]

Albeverio, F

S. Albeverio, F. Gesztesy, R. Hegh-Krohn, and H. Holden.Solvable models in quantum mechanics, Texts and Monographs in Physics. Springer-Verlag, New York, 1998. 10

work page 1998

[4] [4]

Badalyan, V .A

N.P. Badalyan, V .A. Burov, S.A. Morozov, and O.D. Rumyantseva. Scattering by acoustic boundary scatterers with small wave sizes and their reconstruction.Acoustical Physics, 55:1–7, 2009

work page 2009

[5] [5]

Berezanskii

Y .M. Berezanskii. The uniqueness theorem in the inverse problem of spectral analysis for the Schr ¨odinger equation.Tr. Mosk. Mat. Obshch., 7:3–62, 1958 (in Russian). English translation: Transl., Ser. 2, Am. Math. Soc. 35: 167–235, 1964

work page 1958

[6] [6]

Berezin and L.D

F.A. Berezin and L.D. Faddeev. A remark on Schr ¨odinger’s equation with a singular potential.Soviet Math. Dokl., 2:372– 375, 1961

work page 1961

[7] [7]

Bethe and R

H. Bethe and R. Peierls. Quantum theory of the diplon.Proceedings of the Royal Society of London. Series A-Mathematical and Physical Sciences, 148(863):146–156, 1935

work page 1935

[8] [8]

M. Born. Quantenmechanik der stoßvorg ¨ange.Zeitschrift f ¨ur physik, 38(11):803–827, 1926

work page 1926

[9] [9]

V .S. Buslaev. Trace formulas and certain asymptotic estimates of the resolvent kernel for the Schr¨odinger operator in three- dimensional space.Topics in Math. Phys, 1, 1967

work page 1967

[10] [10]

Chashchin

D.S. Chashchin. Example of point potential with inner structure.Eurasian Journal of Mathematical and Computer Applica- tions, 6(1):4–10, 2018

work page 2018

[11] [11]

Demkov and V .N

Y .N. Demkov and V .N. Ostrovskii.Zero-range potentials and their applications in atomic physics. Springer New York, NY , 2013

work page 2013

[12] [12]

Dmitriev and O.D

K.V . Dmitriev and O.D. Rumyantseva. Features of solving the direct and inverse scattering problems for two sets of mono- pole scatterers.Journal of Inverse and Ill-posed Problems, 29(5):775–789, 2021

work page 2021

[13] [13]

L.D. Faddeev. The uniqueness of solutions for the scattering inverse problem.Vestnik Leningrad Univ, 7:126–130, 1956

work page 1956

[14] [14]

E. Fermi. Sul moto dei neutroni nelle sostanze idrogenate.Ricerca scientifica, 7(2):13–52, 1936

work page 1936

[15] [15]

Gesztesy and A.G

F. Gesztesy and A.G. Ramm. An inverse problem for point inhomogeneities.Methods of Functional Analysis and Topology, 6(2):1–12, 1999

work page 1999

[16] [16]

Grinevich and R.G

P.G. Grinevich and R.G. Novikov. Faddeev eigenfunctions for multipoint potentials.Eurasian Journal of Mathematical and Computer Applications, 1(2):76–91, 2013

work page 2013

[17] [17]

Grinevich and R.G

P.G. Grinevich and R.G. Novikov. Transmission eigenvalues for multipoint scatterers.Eurasian Journal of Mathematical and Computer Applications, 9(4):17–25, 2021

work page 2021

[18] [18]

Grinevich and R.G

P.G. Grinevich and R.G. Novikov. Spectral inequality for Schr¨odinger’s equation with multipoint potential.Russian Mathe- matical Surveys, 77(6):1021–1028, 2022

work page 2022

[19] [19]

Grinevich and R.G

P.G. Grinevich and R.G. Novikov. Transparent scatterers and transmission eigenvalues. InInverse Problems: Modelling and Simulation. Trends in Mathematics, volume 11, pages 265–274. Birkh¨auser, Cham, 2025

work page 2025

[20] [20]

Kuo and R.G

P.C. Kuo and R.G. Novikov. Inverse scattering for the multipoint potentials of Bethe- Peierls- Thomas- Fermi type.Inverse Problems, 41(6):065021, 2025

work page 2025

[21] [21]

P. Kurasov. Triplet extensions I: Semibounded operators in the scale of Hilbert spaces.Journal d’Analyse Mathematique, 107(1):251–286, 2009

work page 2009

[22] [22]

Loran and A

F. Loran and A. Mostafazadeh. Renormalization of multi-delta-function point scatterers in two and three dimensions, the coincidence-limit problem, and its resolution.Annals of Physics, 443:168966, 2022

work page 2022

[23] [23]

Malamud and V .V

M.M. Malamud and V .V . Marchenko. On kernels of invariant Schr ¨odinger operators with point interactions. Grinevich– Novikov problem.Doklady Mathematics, 109(2):125–129, 2024

work page 2024

[24] [24]

Mantile and A

A. Mantile and A. Posilicano. Inverse wave scattering in the time domain for point scatterers.Journal of Mathematical Analysis and Applications, 518(2):126758, 2023

work page 2023

[25] [25]

Natterer.The mathematics of computerized tomography

F. Natterer.The mathematics of computerized tomography. SIAM, 2001

work page 2001

[26] [26]

R.G. Novikov. Inverse scattering up to smooth functions for the Schr ¨odinger equation in dimension 1.Bulletin des sciences math´ematiques (Paris. 1885), 120(5):473–491, 1996

work page 1996

[27] [27]

R.G. Novikov. Inverse scattering for the Bethe-Peierls model.Eurasian Journal of Mathematical and Computer Applica- tions, 6(1):52–55, 2018

work page 2018

[28] [28]

R.G. Novikov. Multidimensional inverse scattering for the Schr ¨odinger equation. InISAAC Congress (International Society for Analysis, its Applications and Computation), pages 75–98. Springer, 2019

work page 2019

[29] [29]

`E. `E. Shnol’. On the behavior of eigenfunctions of the Schr¨odinger equation.PhD thesis, Moscow State University, 1955

work page 1955

[30] [30]

Stefanov

P. Stefanov. Generic uniqueness for two inverse problems in potential scattering.Communications in partial differential equations, 17(1-2):55–68, 1992

work page 1992

[31] [31]

L.H. Thomas. The interaction between a neutron and a proton and the structure ofH 3.Physical review, 47(12):903–909, 1935

work page 1935

[32] [32]

D. Yafaev. High-energy and smoothness asymptotic expansion of the scattering amplitude.Journal of Functional Analysis, 202(2):526–570, 2003. 11 Pei-Cheng Kuo Universit´e de Versailles Saint-Quentin-en-Yvelines, Universit´e Paris-Saclay, 45 Av. des ´Etats Unis, 78000 Versailles, France; E-mail: pei-cheng.kuo@ens.uvsq.fr Roman G. Novikov CMAP, CNRS, ´Ecole ...

work page 2003