Inverse scattering problem in nuclear physics—Optical model

Isozaki, Hiroshi; Nakazawa, Hideo; Uhlmann, Günther

doi:10.1063/1.1753665

Cited by 6 publications

(6 citation statements)

References 16 publications

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“…Since (8) implies that K(x, ξ) is contained in a ball of radius 3M/2, we obtain the desired estimate for u from (7) in the case…”

Section: Estimates For ∂ Equationsmentioning

confidence: 81%

“…Lemma 4.1 is a global nonsmooth version of the pseudodifferential conjugation technique in [16] (see also [17]). Similar ideas have been used in inverse scattering [5], [8], nonlinear Schrödinger equations [26], [9] and periodic Schrödinger operators [22]. The problem in extending the method to the global case is seen in (15), where the derivatives in ξ of the symbol grow in x.…”

Section: Proof Of Theorem 11mentioning

confidence: 98%

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Semiclassical Pseudodifferential Calculus and the Reconstruction of a Magnetic Field

Salo

2006

Communications in Partial Differential Equations

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We give a procedure for reconstructing a magnetic field and electric potential from boundary measurements given by the Dirichlet to Neumann map for the magnetic Schrödinger operator in R n , n ≥ 3. The magnetic potential is assumed to be continuous with L ∞ divergence and zero boundary values. The method is based on semiclassical pseudodifferential calculus and the construction of complex geometrical optics solutions in weighted Sobolev spaces.

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“…Since (8) implies that K(x, ξ) is contained in a ball of radius 3M/2, we obtain the desired estimate for u from (7) in the case…”

Section: Estimates For ∂ Equationsmentioning

confidence: 81%

Section: Proof Of Theorem 11mentioning

confidence: 98%

Semiclassical Pseudodifferential Calculus and the Reconstruction of a Magnetic Field

Salo

2006

Communications in Partial Differential Equations

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“…Some other examples of complex potentials which enjoy existence of scattering solution for large energy are studied in [9,16,18,37] and specially in [24].…”

Section: Definitions Preliminaries and Main Resultsmentioning

confidence: 99%

“…However, many new difficulties arise, starting from the proof of the existence of the scattering solutions (see [9,16,18,24,37] for some results concerning the complex case), which often can be just proved for large wave number k. Since large values of k control the values for large frequencies of the Fourier transform of q, it is then a natural question whether the singularities of the actual complex potential q are the same as the singularities of the Born approximation (mathematical basis for diffraction tomography, see [1]). The question of recovery of singularities using the approach of [27][28][29][30] has not been addressed for complex potentials.…”

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confidence: 99%

Reconstruction of singularities from full scattering data by new estimates of bilinear Fourier multipliers

et al. 2009

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We prove that the singularities of a complex valued potential q in the Schrödinger hamiltonian + q can be reconstructed from the linear Born approximation for full scattering data by averaging in the extra variables. We prove that, with this procedure, the accuracy in the reconstruction improves the previously known accuracy obtained from fixed angle or backscattering data. In particular, for q ∈ W α,2 for α ≥ 0, in 2D we recover the main singularity of q with an accuracy of one derivative; in 3D the accuracy is > 1/2, increasing with α. This gives a mathematical basis for diffraction tomography. The proof is based on some new estimates for multidimensional bilinear Fourier multipliers of independent interest.

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“…Therefore, methods of approximation in understanding the many-body problem in quantum mechanics have most often been proposed. A result on inverse scattering problems in nuclear physics by using the optical model, which is one of the method of approximation in the many-body problems, was reported by Isozaki-Nakazawa and Uhlmann [5].…”

Section: Introductionmentioning

confidence: 99%

Time-dependent methods in inverse scattering problems for the Hartree-Fock equation

Watanabe

2019

Journal of Mathematical Physics

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The inverse scattering theory for many-body systems in quantum mechanics is an important and difficult issue not only in physicsatomic physics, molecular physics and nuclear physics-but also mathematics. The major purpose in this paper is to establish a reconstruction procedure of two-body interactions from scattering solutions for a Hartree-Fock equation. More precisely, this paper gives a uniqueness theorem and proposes a new reconstruction procedure of the short-range and two-body interactions from a high-velocity limit of the scattering operator for the Hartree-Fock equation. Moreover, it will be found that the high-velocity limit of the scattering operator is equal to a small-amplitude limit of it. The main ingredients of mathematical analysis in this paper are based on the theory of integral equations of the first kind and a Strichartz type estimates on a solution to the free Schrödinger equation.

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Inverse scattering problem in nuclear physics—Optical model

Cited by 6 publications

References 16 publications

Semiclassical Pseudodifferential Calculus and the Reconstruction of a Magnetic Field

Semiclassical Pseudodifferential Calculus and the Reconstruction of a Magnetic Field

Reconstruction of singularities from full scattering data by new estimates of bilinear Fourier multipliers

Time-dependent methods in inverse scattering problems for the Hartree-Fock equation

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