Piecewise constant positive potentials with practically the same fixed energy phase shifts

It has recently been shown that spherically symmetric potentials of finite range are uniquely determined by the part of their phase shifts at a fixed energy level k² > 0. However, numerical experiments show that two quite different potentials can produce almost identical phase shifts. It has been guessed by physicists that such examples are possible only for “less physical” oscillating and changing sign potentials. In this note it is shown that the above guess is incorrect: we give examples of four positive spherically symmetric compactly supported quite different potentials having practically identical phase shifts. The note also describes a hybrid stochastic deterministic method for global minimization for the construction of these potentials.     

A Counterexample to a Uniqueness Result

A counterexample is given to the uniqueness result given in the article by Cox and Thompson (1970), "Note on the uniqueness of the solution of an equation of interest in inverse scattering problem." J. Math. Phys ., 11 , N3, 815-817.     

Partial Recovery of a Potential from Backscattering Data

ABSTRACT

We prove that the main singularities,measured in the scale of Sobolev spaces,of the potential q in the Schrödinger Hamiltonian ?Δ+q,in dimensions n=2,3,are contained in the Born approximation for backscattering data.     

Analysis of the Newton-Sabatier Scheme for Inverting Fixed-Energy Phase Shifts

Suppose that the inverse scattering problem is understood as follows: given fixed-energy phase shifts, corresponding to an unknown potential q = q ( r ) from a certain class, for example, q ] L 1,1 , recover this potential. Then it is proved that the Newton-Sabatier (NS) procedure does not solve the above problem. It is not a valid inversion method, in the following sense: (1) it is not possible to carry this procedure through for the phase shifts corresponding to a generic potential q ] L 1,1 , where $ L_{1,1} : = { q {:}, q = overline q, int ^infty _0 r |q(r)| dr lt infty } $ and recover the original potential: the basic integral equation, introduced by Newton without derivation, in general, may be not solvable for some r > 0, and if it is solvable for all r > 0, then the resulting potential is not equal to the original generic q ] L 1,1 . Here a generic q is any q which is not a restriction to (0, X ) of an analytic function. (2) the ansatz (*) $ K(r,s) = sum^infty _{l = 0} c_l varphi _l (r) u_l (s) $ , used by Newton, is incorrect: the transformation operator I m K , corresponding to a generic q ] L 1,1 , does not have K of the form (*), and (3) the set of potentials q ] L 1,1 , that can possibly be obtained by NS procedure, is not dense in the set of all L 1,1 potentials in the norm of L 1,1 . Therefore, one cannot justify NS procedure even for approximate solution of the inverse scattering problem with fixed-energy phase shifts as data. Thus, the NS procedure, if considered as a method for solving the above inverse scattering problem, is based on an incorrect ansatz, the basic integral equation of NS procedure is, in general, not solvable for some r > 0, and in this case this procedure breaks down, and NS procedure is not an inversion theory: it cannot recover generic potentials q ] L 1,1 from their fixed-energy phase shifts. Suppose now that one considers another problem: given fixed-energy phase shifts, corresponding to some potential, find a potential which generates the same phase shifts. Then NS procedure does not solve this problem either: the basic integral equation, in general, may be not solvable for some r > 0, and then NS procedure breaks down.     

Recovery of the singularities of a potential from backscattering data in general dimension

We prove that in dimension $n\ge 2$ the main singularities of a complex potential q having a certain a priori regularity are contained in the Born approximation $q_B$ constructed from backscattering data. This is archived using a new explicit formula for the multiple dispersion operators in the Fourier transform side. We also show that $q?q_B$ can be up to one derivative more regular than q in the Sobolev scale. On the other hand, we construct counterexamples showing that in general it is not possible to have more than one derivative gain, sometimes even strictly less, depending on the a priori regularity of q.     

Structure of the Semi-Classical Amplitude for General Scattering Relations

ABSTRACT

We consider scattering by general compactly supported semi-classical perturbations of the Euclidean Laplace-Beltrami operator. We show that if the suitably cut-off resolvent quantizes a Lagrangian relation on the product cotangent bundle, the scattering amplitude quantizes the natural scattering relation. In the case when the resolvent is tempered, which is true at non-trapping energies or at trapping energies under some non-resonance assumptions, and when we work microlocally near a non-trapped ray, our result implies that the scattering amplitude defines a semiclassical Fourier integral operator associated to the scattering relation in a neighborhood of that ray. Compared to previous work, we allow this relation to have more general geometric structure.     

一个声波散射区域的重建方法与唯一性定理

一个声波散射区域的重建方法与唯一性定理@王连堂...     

Uniqueness on identification of cubic convolution nonlinearity

We shall consider the inverse scattering problem for time dependent version of Hartree equation and nonlinear Klein-Gordon equation. The uniqueness theorem on identifying the cubic convolution nonlinearity from the knowledge of the scattering operator will be shown.     

Extensions and extremality of recursively generated weighted shifts

Given an -step extension of a recursively generated weight sequence , and if denotes the associated unilateral weighted shift, we prove that

1).\end{cases}\end{displaymath}">

In particular, the subnormality of an extension of a recursively generated weighted shift is independent of its length if the length is bigger than 1. As a consequence we see that if is a canonical rank-one perturbation of the recursive weight sequence , then subnormality and -hyponormality for eventually coincide. We then examine a converse--an ``extremality" problem: Let be a canonical rank-one perturbation of a weight sequence and assume that -hyponormality and -hyponormality for coincide. We show that is recursively generated, i.e., is recursive subnormal.

     

A formula for -hyponormality of backstep extensions of subnormal weighted shifts

Let be a weight sequence of positive real numbers and let be a subnormal weighted shift with a weight sequence . Consider an extended weight sequence with and let 0: W_{\alpha (x)} \text{is} k \text{-hyponormal}\}$">for , where is the set of natural numbers. We obtain a formula to find the interval , which provides several examples to distinguish the classes of -hyponormal operators from one another.

     

Electron in the Aharonov-Bohm potential and in the Coulomb field in 2+1 dimensions

We obtain exact solutions of the Dirac equation in 2+1 dimensions and the electron energy spectrum in the superposition of the Aharonov-Bohm and Coulomb potentials, which are used to study the Aharonov-Bohm effect for states with continuous and discrete energy spectra. We represent the total scattering amplitude as the sum of amplitudes of scattering by the Aharonov-Bohm and Coulomb potentials. We show that the gauge-invariant phase of the wave function or the energy of the electron bound state can be observed. We obtain a formula for the scattering cross section of spin-polarized electrons scattered by the Aharonov-Bohm potential. We discuss the problem of the appearance of a bound state if the interaction between the electron spin and the magnetic field is taken into account in the form of the two-dimensional Dirac delta function. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 149, No. 3, pp. 502–517, December, 2006. An erratum to this article is available at .     

Identification of Piecewise-Constant Potentials by Fixed-Energy Phase Shifts

The identification of a spherically symmetric potential by its phase shifts is an important physical problem. Recent theoretical results assure that such a potential is uniquely defined by a sufficiently large subset of its phase shifts at any one fixed energy level. However, two different potentials can produce almost identical phase shifts. To resolve this difficulty we suggest the use of phase shifts corresponding to several energy levels. The identification is done by a nonlinear minimization of the appropriate objective function. It is based on a combination of probabilistic global and deterministic local minimization methods. The Multilevel Single-Linkage Method (MSLM) is used for the global minimization. A specially designed Local Minimization Method (LMM) with a Reduction Procedure is used for the local searches. Numerical results show the effectiveness of this procedure for potentials composed of a small number of spherical layers. Accepted 2 February 2001. Online publication 11 May 2001.     

Inverse scattering with partial information on the potential

The one-dimensional Schrödinger equation is considered when the potential is real valued and integrable and has a finite first moment. The recovery of such a potential is analyzed in terms of the scattering data consisting of a reflection coefficient, all the bound-state energies, knowledge of the potential on a finite interval, and all of the bound-state norming constants except one. It is shown that a missing norming constant in the data can cause at most a double nonuniqueness in the recovery. In the particular case when the missing norming constant in the data corresponds to the lowest-energy bound state, the necessary and sufficient conditions are obtained for the nonuniqueness, and the two norming constants and the corresponding potentials are determined. Some explicit examples are provided to illustrate the nonuniqueness.     

Detection of multilayered media in the acoustic waveguide

This work is concerned with the inverse problem for ocean acoustics modeled by a multilayered waveguide with a finite depth. We provide explicit formulae to locate the layers, including the seabed, and reconstruct the speed of sound and the densities in each layer from measurements collected on the surface of the waveguide. We proceed in two steps. First, we use Gaussian type excitations on the upper surface of the waveguide and then from the corresponding scattered fields, collected on the same surface, we recover the boundary spectral data of the related 1D spectral problem. Second, from these spectral data, we reconstruct the values of the normal derivatives of the singular solutions, of the original waveguide problem, on that upper surface. Finally, we derive formulae to reconstruct the layers from these values based on the asymptotic expansion of these singular solutions in terms of the source points.     

Spectrum of bilateral shifts with operator-valued weights

We describe the spectrum of bilateral operator-weighted shifts.

     

Inverse Scattering Problem for the Schrödinger Operator with External Yang–Mills Potentials in Two Dimensions at Fixed Energy

ABSTRACT

We consider the Schrödinger equation in ?², with external Yang–Mills potentials that decay exponentially as |x| → ∞. We prove that the scattering amplitude at fixed positive energy determines the potentials uniquely modulo a gauge transformation, assuming that potentials are small.     

On perturbations of the group of shifts on the line by unitary cocycles

It is shown that the class of perturbations of the semigroup of shifts on by unitary cocycles with the property (where is the Hilbert-Schmidt class) contains strongly continuous semigroups of isometric operators, whose unitary parts possess spectral decompositions with the measure being singular with respect to the Lebesgue measure. Thus, we describe also the subclass of strongly continuous groups of unitary operators that are perturbations of the group of shifts on by Markovian cocycles with the property .

     

Existence of Gibbs measures for countable Markov shifts

We prove that a potential with summable variations and finite pressure on a topologically mixing countable Markov shift has a Gibbs measure iff the transition matrix satisfies the big images and preimages property. This strengthens a result of D. Mauldin and M. Urbanski (2001) who showed that this condition is sufficient.

     

Reconstruction of Scattered Field from Far-Field by Regularization

In this paper, we consider an inverse scattering problem for an obstacle D R^2 with Robin boundary condition. By applying the point source, we give a regularizing method to recover the scattered field from the far-field pattern. Numerical implementations are also presented.