Search results for "Inverse scattering problem"
showing 10 items of 19 documents
Determining a Random Schrödinger Operator : Both Potential and Source are Random
2020
We study an inverse scattering problem associated with a Schr\"odinger system where both the potential and source terms are random and unknown. The well-posedness of the forward scattering problem is first established in a proper sense. We then derive two unique recovery results in determining the rough strengths of the random source and the random potential, by using the corresponding far-field data. The first recovery result shows that a single realization of the passive scattering measurements uniquely recovers the rough strength of the random source. The second one shows that, by a single realization of the backscattering data, the rough strength of the random potential can be recovered…
Spectral approach to the scattering map for the semi-classical defocusing Davey–Stewartson II equation
2019
International audience; The inverse scattering approach for the defocusing Davey–Stewartson II equation is given by a system of D-bar equations. We present a numerical approach to semi-classical D-bar problems for real analytic rapidly decreasing potentials. We treat the D-bar problem as a complex linear second order integral equation which is solved with discrete Fourier transforms complemented by a regularization of the singular parts by explicit analytic computation. The resulting algebraic equation is solved either by fixed point iterations or GMRES. Several examples for small values of the semi-classical parameter in the system are discussed.
Explicit Characterization of Inclusions in Electrical Impedance Tomography
2001
In electrical impedance tomography one seeks to recover the spatial conductivity distribution inside a body from knowledge of the Neumann--Dirichlet map. In many practically relevant situations the conductivity is smooth apart from some inhomogeneities where the conductivity jumps to a higher or lower value. An explicit characterization of these inclusions is developed in this paper. To this end a class of dipole-like indicator functions is introduced, for which one has to check whether their boundary values are contained in the range of an operator determined by the measured Neumann--Dirichlet map. It is shown that this holds true if and only if the dipole singularity lies inside the inhom…
Role of Levinson’s theorem in neutron-deuteron quartetS-wave scattering
1990
The real part of the phase shift for elastic neutron-deuteron scattering in the quartet {ital S} wave channel, as calculated with the exact three-body theory, assumes at threshold the value {pi} if normalized to zero at infinity; that is, it does not comply with the expectations raised by a naive application of Levinson's theorem since no bound state exists in this channel. A description of this situation on an equivalent two-body level via a potential, constructed by means of the Marchenko inverse scattering theory, necessitates the introduction of a fictitious bound state. This predominantly attractive, equivalent local potential can be related via supersymmetry to a strictly phase equiva…
Soliton Solutions with Real Poles in the Alekseev formulation of the Inverse-Scattering method
1999
A new approach to the inverse-scattering technique of Alekseev is presented which permits real-pole soliton solutions of the Ernst equations to be considered. This is achieved by adopting distinct real poles in the scattering matrix and its inverse. For the case in which the electromagnetic field vanishes, some explicit solutions are given using a Minkowski seed metric. The relation with the corresponding soliton solutions that can be constructed using the Belinskii-Zakharov inverse-scattering technique is determined.
Free boundary methods and non-scattering phenomena
2021
We study a question arising in inverse scattering theory: given a penetrable obstacle, does there exist an incident wave that does not scatter? We show that every penetrable obstacle with real-analytic boundary admits such an incident wave. At zero frequency, we use quadrature domains to show that there are also obstacles with inward cusps having this property. In the converse direction, under a nonvanishing condition for the incident wave, we show that there is a dichotomy for boundary points of any penetrable obstacle having this property: either the boundary is regular, or the complement of the obstacle has to be very thin near the point. These facts are proved by invoking results from t…
Nonrecursive multiple shock formation via four-wave mixing: theory and experiment
2002
We show theoretically and experimentally that a beat signal propagating along a normally dispersive fiber can trigger the formation of multiple shocks. This phenomenon critically depends on the input frequency separation and power of the beat signal.
The Factorization Method for Electrical Impedance Tomography in the Half-Space
2008
We consider the inverse problem of electrical impedance tomography in a conducting half-space, given electrostatic measurements on its boundary, i.e., a hyperplane. We first provide a rigorous weak analysis of the corresponding forward problem and then develop a numerical algorithm to solve an associated inverse problem. This inverse problem consists of the reconstruction of certain inclusions within the half-space which have a different conductivity than the background. To solve the inverse problem we employ the so-called factorization method of Kirsch, which so far has only been considered for the impedance tomography problem in bounded domains. Our analysis of the forward problem makes u…
Complex, energy-independent, local potential reproducing an absorptive phase shift and a bound state
1994
The triton binding energy, and the partly real and partly complex neutron-deuteron doublet channel elastic scattering phase shifts, calculated by means of the exact three-body theory, are used as input in the fixed-[ital l] inverse scattering theory of Marchenko. The local potentials obtained hereby are independent of energy, and complex. Their strong imaginary part reflects the strong absorption in the doublet channel arising from the opening of the deuteron breakup channel. For total orbital angular momentum [ital l] different from zero the potentials are unique, reproducing the input phase shift in the whole energy region. For [ital l]=0 where there exists, in addition, a bound state we …
Fixed angle inverse scattering in the presence of a Riemannian metric
2020
We consider a fixed angle inverse scattering problem in the presence of a known Riemannian metric. First, assuming a no caustics condition, we study the direct problem by utilizing the progressing wave expansion. Under a symmetry assumption on the metric, we obtain uniqueness and stability results in the inverse scattering problem for a potential with data generated by two incident waves from opposite directions. Further, similar results are given using one measurement provided the potential also satisfies a symmetry assumption. This work extends the results of [23,24] from the Euclidean case to certain Riemannian metrics.