scholarly journals The inverse scattering problem at fixed energy for the three-dimensional Schrödinger equation with an exponentially decreasing potential

1994 ◽  
Vol 161 (3) ◽  
pp. 569-595 ◽  
Author(s):  
Roman G. Novikov
Keyword(s):  
Schrödinger Equation ◽  
Inverse Scattering ◽  
Schrodinger Equation ◽  
Scattering Problem ◽  
Three Dimensional ◽  
Inverse Scattering Problem ◽  
Fixed Energy

The first solution of a long standing problem: Reconstruction formula for a 3-d phaseless inverse scattering problem for the Schrödinger equation

2015 ◽  
Vol 23 (4) ◽  
Author(s):  
Michael V. Klibanov ◽  
Vladimir G. Romanov
Keyword(s):  
New York ◽  
Schrödinger Equation ◽  
Inverse Scattering ◽  
Schrodinger Equation ◽  
Scattering Problem ◽  
Three Dimensional ◽  
Inverse Scattering Problem ◽  
Potential Applications ◽  
Inverse Radon Transform ◽  
Standing Problem

AbstractA long standing problem is completely solved here for the first time. This problem was posed by K. Chadan and P. C. Sabatier in their classical book “Inverse Problems in Quantum Scattering Theory”, Springer, New York, 1977. The inverse scattering problem of the reconstruction of the unknown potential with a compact support in the three-dimensional Schrödinger equation is considered. Only the modulus of the scattering complex-valued wave field is known, whereas the phase is unknown. It is shown that the unknown potential can be reconstructed via the inverse Radon transform. This solution has potential applications in imaging of nanostructures.

scholarly journals USING MIXED DATA IN THE INVERSE SCATTERING PROBLEM

2008 ◽  
Vol 22 (23) ◽  
pp. 2181-2189 ◽  
Author(s):  
M. LASSAUT ◽  
S. Y. LARSEN ◽  
S. A. SOFIANOS ◽  
J. C. WALLET
Keyword(s):  
Schrödinger Equation ◽  
Inverse Scattering ◽  
Regular Solution ◽  
Schrodinger Equation ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Phase Shifts ◽  
Mixed Data ◽  
Monotonic Functions ◽  
Unique Potential

Consider the fixed-ℓ inverse scattering problem. We show that the zeros of the regular solution of the Schrödinger equation, rn(E), which are monotonic functions of the energy, determine a unique potential when the domain of the energy is such that the rn(E) range from zero to infinity. This suggests that the use of the mixed data of phase-shifts {δ(ℓ0, k), k ≥ k0} ∪ {δ(ℓ, k0), ℓ ≥ ℓ0}, for which the zeros of the regular solution are monotonic in both domains, and range from zero to infinity, offers the possibility of determining the potential in a unique way.

AN INVERSE SCATTERING PROBLEM FOR NONLOCAL POTENTIALS I: THE FAMILY OF PHASE EQUIVALENT WAVEFUNCTIONS

1986 ◽  
Vol 01 (07) ◽  
pp. 449-454 ◽  
Author(s):  
V.M. MUZAFAROV
Keyword(s):  
Integral Equation ◽  
Schrödinger Equation ◽  
Inverse Scattering ◽  
Schrodinger Equation ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Scattering Data ◽  
The Family ◽  
Consistent Approach ◽  
Nonlocal Potentials

We develop a consistent approach to an inverse scattering problem for the Schrodinger equation with nonlocal potentials. The main result presented in this paper is that for the two-body scattering data, given the problem of reconstructing both the family of phase equivalent two-body wavefunctions and the corresponding family of phase equivalent half-off-shell t-matrices, is reduced to solving a regular integral equation. This equation may be regarded as a generalization of the Gel’fand-Levitan equation.

A Sampling Method to an Inverse Scattering Problem for Stationary Schrödinger Equation

2013 ◽  
Vol 275-277 ◽  
pp. 1585-1589
Author(s):  
Yuan Li ◽  
Ming Chen Yao ◽  
Chun Mei Wang ◽  
Fa Yong Zhang
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Sampling Method ◽  
Near Field ◽  
Scattering Problem ◽  
Three Dimensional ◽  
Inverse Scattering Problem ◽  
Potential Scattering ◽  
Stationary Schrödinger Equation ◽  
Direct Sampling

For an inverse potential scattering problem of stationary Schrödinger equation, we employ a direct sampling method to reconstruct the support of the potential. Compared with the general sampling method, the method we adopt is applicable even when the measured data (near-field data) are only available for one or several incident directions, and has the advantages of simple computation and insensitivity to noises. By the mathematical derivations, we conclude theoretically that for both two dimensional and three dimensional cases, this direct sampling method is feasible and efficient.

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