INVERSE BORN APPROXIMATION FOR THE GENERALIZED NONLINEAR SCHRÖDINGER OPERATOR IN TWO DIMENSIONS

2008 ◽  
Vol 22 (23) ◽  
pp. 2257-2275 ◽  
Author(s):  
VALERIY SEROV
Keyword(s):  
Born Approximation ◽  
Representation Formula ◽  
Uniqueness Result ◽  
Two Dimensions ◽  
Scattering Data ◽  
Leading Order ◽  
Scattering Problems ◽  
Smooth Bounded Domain ◽  
Special Behaviour ◽  
Nonlinear Schrödinger Operator

This work deals with the inverse scattering problems for the two-dimensional Schrödinger equation [Formula: see text] with a power-like nonlinearity, where the real-valued unknown functions αl on belong to [Formula: see text] with certain special behaviour at infinity. We prove Saito's formula which implies the uniqueness result and a representation formula for a sum of the functions αl in the sense of tempered distributions. What is more, we prove that the leading order singularities of this sum can be obtained exactly by the inverse Born approximation method from general scattering data at arbitrarily large energies. Especially, we show for the functions in Lp, for certain values of p, that the approximation agrees with the true sum up to the functions from the Sobolev spaces. In particular, for the sum being the characteristic function of a smooth bounded domain this domain is uniquely determined by this scattering data.

scholarly journals Inverse scattering problem for Sturm-Liouville operator on non-compact A-graph. Uniqueness result.

2015 ◽  
Vol 46 (4) ◽  
pp. 401-422 ◽  
Author(s):  
Mikhail Ignatyev
Keyword(s):  
Inverse Problem ◽  
Differential Operators ◽  
Scattering Problem ◽  
Spectral Characteristics ◽  
Inverse Scattering Problem ◽  
Uniqueness Result ◽  
Scattering Data ◽  
Scattering Problems ◽  
Sturm Liouville ◽  
Liouville Operators

We consider a connected metric graph with the following property: each two cycles can have at most one common point. Such graphs are called A-graphs. On noncompact A-graph we consider a scattering problem for Sturm--Liouville differential operator with standard matching conditions in the internal vertices. Transport, spectral and scattering problems for differential operators on graphs appear frequently in mathematics, natural sciences and engineering. In particular, direct and inverse problems for such operators are used to construct and study models in mechanics, nano-electronics, quantum computing and waveguides. The most complete results on (both direct and inverse) spectral problems were achieved in the case of Sturm--Liouville operators on compact graphs, in the noncompact case there are no similar general results. In this paper, we establish some properties of the spectral characteristics and investigate the inverse problem of recovering the operator from the scattering data. A uniqueness theorem for such inverse problem is proved.

Numerical Computation of the Inverse Born Approximation for the Nonlinear Schrödinger Equation in Two Dimensions

2016 ◽  
Vol 16 (1) ◽  
pp. 133-143 ◽  
Author(s):  
Markus Harju
Keyword(s):  
Schrödinger Equation ◽  
Numerical Computation ◽  
Nonlinear Schrödinger Equation ◽  
Born Approximation ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Two Dimensions ◽  
Scattering Problems ◽  
Nonlinear Schrödinger ◽  
Angle Data

AbstractThis work deals with the numerical computation of the inverse Born approximation associated with inverse scattering problems for the nonlinear Schrödinger equation in two space dimensions. We consider both backscattering and fixed angle data. The problem of computing the Born approximation is formulated as a linear inverse problem which is solved using a regularization method. Numerical examples with noisy data are given to illustrate the effectiveness of this method.

scholarly journals Determining the helicity structure of the nucleon at the Electron Ion Collider in China

2021 ◽  
Vol 2021 (8) ◽  
Author(s):  
Daniele Paolo Anderle ◽  
Tie-Jiun Hou ◽  
Hongxi Xing ◽  
Mengshi Yan ◽  
C.-P. Yuan ◽  
...  
Keyword(s):  
Electron Beam ◽  
Inelastic Scattering ◽  
Scattering Data ◽  
Leading Order ◽  
Specific Topic ◽  
Good Separation ◽  
Effective Electron ◽  
Deep Inelastic Scattering Data ◽  
Polarized Proton ◽  
Helicity Structure

Abstract Understanding how sea quarks behave inside a nucleon is one of the most important physics goals of the proposed Electron-Ion Collider in China (EicC), which is designed to have a 3.5 GeV polarized electron beam (80% polarization) colliding with a 20 GeV polarized proton beam (70% polarization) at instantaneous luminosity of 2 × 1033cm−2s−1. A specific topic at EicC is to understand the polarization of individual quarks inside a longitudinally polarized nucleon. The potential of various future EicC data, including the inclusive and semi-inclusive deep inelastic scattering data from both doubly polarized electron-proton and electron-3He collisions, to reduce the uncertainties of parton helicity distributions is explored at the next-to-leading order in QCD, using the Error PDF Updating Method Package (ePump) which is based on the Hessian profiling method. We show that the semi-inclusive data are well able to provide good separation between flavour distributions, and to constrain their uncertainties in the x > 0.005 region, especially when electron-3He collisions, acting as effective electron-neutron collisions, are taken into account. To enable this study, we have generated a Hessian representation of the DSSV14 set of PDF replicas, named DSSV14H PDFs.

scholarly journals Cramér-Rao Bound Study of Multiple Scattering Effects in Target Separation Estimation

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Edwin A. Marengo ◽  
Paul Berestesky
Keyword(s):  
Multiple Scattering ◽  
Born Approximation ◽  
Single Scattering ◽  
Scattering Data ◽  
Approximation Model ◽  
Scattering Parameters ◽  
Helmholtz Operator ◽  
Scattering Effects ◽  
Cramer Rao Bound ◽  
Multiple Scattering Effects

The information about the distance of separation between two-point targets that is contained in scattering data is explored in the context of the scalar Helmholtz operator via the Fisher information and associated Cramér-Rao bound (CRB) relevant to unbiased target separation estimation. The CRB results are obtained for the exact multiple scattering model and, for reference, also for the single scattering or Born approximation model applicable to weak scatterers. The effects of the sensing configuration and the scattering parameters in target separation estimation are analyzed. Conditions under which the targets' separation cannot be estimated are discussed for both models. Conditions for multiple scattering to be useful or detrimental to target separation estimation are discussed and illustrated.

scholarly journals Nondestructive Testing of Metallic Cables Based on a Homogenized Model and Global Measurements

2010 ◽  
Vol 2010 ◽  
pp. 1-21 ◽  
Author(s):  
Valdemar Melicher ◽  
Peter Sergeant
Keyword(s):  
Cross Section ◽  
Low Frequency ◽  
Real Data ◽  
Cost Effective ◽  
Uniqueness Result ◽  
Two Dimensions ◽  
Cost Effective Method ◽  
Single Inclusion ◽  
Homogenized Model ◽  
The Cross

We propose a simple, quick, and cost-effective method for nondestructive eddy-current testing of metallic cables. Inclusions in the cross section of the cable are detected on the basis of certain global data: hysteresis loop measurements for different frequencies. We detect air-gap inclusions inside the cross section using a homogenized model. The problem, which can be understood as an inverse spectral problem, is posed in two dimensions. We consider its reduction to one dimension. The identifiability is studied. We obtain a uniqueness result for a single inclusion in 1D by two measurements for sufficiently low frequency. We study the sensibility of the inverse problem numerically. A study case with real data is performed to confirm the usefulness.

scholarly journals A uniqueness result for the continuity equation in two dimensions

2014 ◽  
Vol 16 (2) ◽  
pp. 201-234 ◽  
Author(s):  
Giovanni Alberti ◽  
Stefano Bianchini ◽  
Gianluca Crippa
Keyword(s):  
Continuity Equation ◽  
Uniqueness Result ◽  
Two Dimensions
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