Three-dimensional Direct and Inverse Scattering for the Schrödinger Equation with a General Nonlinearity

2013 ◽  
pp. 257-273 ◽  
Author(s):  
Markus Harju ◽  
Valery Serov
Keyword(s):  
Schrödinger Equation ◽  
Inverse Scattering ◽  
Schrodinger Equation ◽  
Three Dimensional ◽  
General Nonlinearity

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.

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