Narrow Results

In this paper, we study the inverse scattering of massive charged Dirac fields in the exterior region of (de Sitter)–Reissner–Nordström black holes. Firstly, we obtain a precise high-energy asymptotic expansion of the diagonal elements of the scattering matrix (i.e. of the transmission coefficients) and we show that the leading terms of this expansion allow to recover uniquely the mass, the charge and the cosmological constant of the black hole. Secondly, in the case of nonzero cosmological constant, we show that the knowledge of the reflection coefficients of the scattering matrix on any interval of energy also permits to recover uniquely these parameters.

We introduce a general class of long-range magnetic potentials and derive high velocity limits for the corresponding scattering operators in quantum mechanics, in the case of two dimensions. We analyze the high velocity limits that we obtain in the presence of an obstacle and we uniquely reconstruct from them the electric potential and the magnetic field outside the obstacle, that are accessible to the particles. We additionally reconstruct the inaccessible fluxes (magnetic fluxes produced by fields inside the obstacle) modulo 2π, which give a proof of the Aharonov–Bohm effect. For every magnetic potential A in our class, we prove that its behavior at infinity can be characterized in a natural way; we call it the long-range part of the magnetic potential. Under very general assumptions, we prove that can be uniquely reconstructed for every . We characterize properties of the support of the magnetic field outside the obstacle that permit us to uniquely reconstruct either for all or for in a subset of 𝕊¹. We also give a wide class of magnetic fields outside the obstacle allowing us to uniquely reconstruct the total magnetic flux (and for all ). This is relevant because, as it is well-known, in general the scattering operator (even if it is known for all velocities or energies) does not define uniquely the total magnetic flux (and ). We analyze additionally injectivity (i.e. uniqueness without giving a method for reconstruction) of the high velocity limits of the scattering operator with respect to . Assuming that the magnetic field outside the obstacle is not identically zero, we provide a class of magnetic potentials for which injectivity is valid.

The large-N limit of fermionic vectors models is studied using bilocal variables in the framework of a collective field theory approach. The large-N configuration is determined completely using only classical solutions of the model. Further, the Bethe–Salpeter equations of the model are cast as a Green's function problem. One of the main results of this work is to show that this Green's function is in fact the large-N bilocal itself.

We investigate the solvability of an integrable nonlinear nonlocal reverse-time six-component fourth-order AKNS system generated from a reduced coupled AKNS hierarchy under a reverse-time reduction. Riemann–Hilbert problems will be formulated by using the associated matrix spectral problems, and exact soliton solutions will be derived from the reflectionless case corresponding to an identity jump matrix.

The paper aims to discuss nonlocal reverse-space multicomponent nonlinear Schrödinger equations and their inverse scattering transforms. The inverse scattering problems are analyzed by means of Riemann–Hilbert problems, and Gelfand–Levitan–Marchenko-type integral equations for generalized matrix Jost solutions are determined by the Sokhotski–Plemelj formula. Soliton solutions are generated from the reflectionless transforms associated with zeros of the Riemann–Hilbert problems.

We consider three-dimensional inverse scattering with fixed energy for which the spherically symmetrical potential is nonvanishing only in a ball. We give exact upper and lower bounds for the phase shifts. We provide a variational formula for the Weyl–Titchmarsh m-function of the one-dimensional Schrödinger operator defined on the half-line.

A review of some of the author's results in the area of inverse scattering is given. The following topics are discussed: (1) Property C and applications, (2) Stable inversion of fixed-energy 3D scattering data and its error estimate, (3) Inverse scattering with “incomplete” data, (4) Inverse scattering for inhomogeneous Schrödinger equation, (5) Krein's inverse scattering method, (6) Invertibility of the steps in Gel'fand–Levitan, Marchenko, and Krein inversion methods, (7) The Newton–Sabatier and Cox–Thompson procedures are not inversion methods, (8) Resonances: existence, location, perturbation theory, 9) Born inversion as an ill-posed problem, (10) Inverse obstacle scattering with fixed-frequency data, (11) Inverse scattering with data at a fixed energy and a fixed incident direction, (12) Creating materials with a desired refraction coefficient and wave-focusing properties.

We consider the inverse problem of determining the shape and location of planar screens inside a 3D body through acoustic or electromagnetic imaging. In order to determine the flaws we perform acoustic measurements with prescribed over-determined boundary data. The reciprocity gap concept is exploited to determine sound-hard planar screens of general shapes.

We discuss the inverse problem associated with the identification of the location and shape of a scatterer fully embedded in a homogeneous halfplane, using scant surficial measurements of its response to probing scalar waves. The typical applications arise in soils under shear (SH) waves (antiplane motion), or in acoustic fluids under pressure waves. Accordingly, we use measurements of either the Dirichlet-type (displacements), or of the Neumann-type (fluid velocities), to steer the localization and detection processes, targeting rigid and sound-hard objects, respectively. The computational approach for localizing single targets is based on partial-differential-equation-constrained optimization ideas, extending our recent work from the full-¹ to the half-plane case. To improve on the ability of the optimizer to converge to the true shape and location we employ an amplitude-based misfit functional, and embed the inversion process within a frequency- and directionality-continuation scheme, which seem to alleviate solution multiplicity. We use the apparatus of total differentiation to resolve the target's evolving shape during inversion iterations over the shape parameters, à la.^2,3 We report numerical results betraying algorithmic robustness for both the SH and acoustic cases, and for a variety of targets, ranging from circular and elliptical, to potato-, and kite-shaped scatterers.

We consider the Cauchy problem for the cubic focusing nonlinear Schrödinger (NLS) equation on the line with initial datum close to an $N$-soliton. Using the inverse scattering method and the $\overline{\partial }$-method, we establish the decay of the $L^{\infty }(ℝ)$-norm of the residual term in time.

The authors prove the uniqueness in the inverse acoustic scattering problem within convex polygonal domains by a single incident direction in the sound-soft case and the sound-hard case, and by two incident directions in the case of the impedance boundary condition. The proof is based on analytic continuation on a straight line.

Using the Inverse Scattering Method with a nonvanishing boundary condition, we obtain an explicit breather solution with nonzero vacuum parameter b of the focusing modified Korteweg–de Vries (mKdV) equation. Moreover, taking the limiting case of zero frequency, we obtain a generalization of the double pole solution introduced by M. Wadati et al.

The paper is devoted to the inverse scattering problem in order to study the complex media whose electrical and magnetic properties are the tensor functions. The tensor Fourier diffraction theorem, derived in the paper, gives the relations connecting the scattered field and the inherent properties of the scattering medium. The method is based on the first Born approximation. The tensor (dyadic) analysis for derivation of the tensor Fourier diffraction theorem in the coordinate-free form is used.

In Linear Sampling the profile of an unknown object can be visualized from far-field scattering data by plotting the norm of the Tikhonov regularized solution of a set of discretized far-field equations sampled over a computational grid. Here we present a new 3D formulation of the Linear Sampling, in which the set of far-field equations for all sampling points and all polarizations is replaced by a single functional equation, whose Tikhonov regularized solution can be analytically determined. Such an implementation is computationally much faster than the traditional one and does not reduce the quality of the reconstructions.