Scattering of Dirac Particles by Electromagnetic Fields with Small Support in Two Dimensions and Effect from Scalar Potentials

We study the asymptotic behavior of scattering amplitudes for the scattering of Dirac particles in two dimensions when electromagnetic fields with small support shrink to point-like fields. The result is strongly affected by perturbations of scalar potentials and the asymptotic form changes discontinuously at half-integer fluxes of magnetic fields even for small perturbations. The analysis relies on the behavior at low energy of resolvents of magnetic Schrödinger operators with resonance at zero energy. The magnetic scattering of relativistic particles appears in the interaction of cosmic string with matter. We discuss this closely related subject as an application of the obtained results. Communicated by Bernard Helffersubmitted 05/05/03, accepted 31/07/03     

Resonances in scattering by two magnetic fields at large separation and a complex scaling method

We study the quantum resonances in magnetic scattering in two dimensions. The scattering system consists of two obstacles by which the magnetic fields are completely shielded. The trajectories trapped between the two obstacles are shown to generate the resonances near the positive real axis, when the distance between the obstacles goes to infinity. The location is described in terms of the backward amplitudes for scattering by each obstacle. A difficulty arises from the fact that even if the supports of the magnetic fields are largely separated from each other, the corresponding vector potentials are not expected to be well separated. To overcome this, we make use of a gauge transformation and develop a new type of complex scaling method. We can cover the scattering by two solenoids at large separation as a special case. The obtained result heavily depends on the magnetic fluxes of the solenoids. This indicates that the Aharonov–Bohm effect influences the location of resonances.     

The scattering of electromagnetic wave at oblique incidence in a homogeneous chiral medium

We consider the scattering of a time-harmonic electromagnetic wave by a perfectly and imperfectly conducting infinite cylinder at oblique incidence respectively. We assume that the cylinder is embedded in a homogeneous chiral medium and the cylinder is parallel to the z axis. Since the x components and y components of electric field and magnetic field can be expressed in terms of their z components, we can derive from Maxwell's equations and corresponding boundary conditions that the scattering problem is modeled as a boundary value problem for the z components of electric field and magnetic field. By using Rellich's lemma and variational approach, the uniqueness and the existence of solutions are justified.     

Scattering by magnetic nanocylinders and the method of distorted waves in noncollinear fields

In the perturbation theory framework, we compute the cross section of scattering by a magnetic nanocylinder and a helicoid arbitrarily oriented in an external magnetic field. We are the first to obtain the matrix Green’s function for two media with an interface and noncollinear magnetic fields on the two sides of the interface. We show how to compute scattering by magnetic inclusions in one of the media.     

Spin–density correlations and magnetic neutron scattering in ferromagnetic metals

We obtain expressions for the spatial spin-density correlator and for effective and local magnetic moments in the dynamic spin-fluctuation theory. We derive formulas for the magnetic scattering cross section in the theory of itinerant electron magnets. We calculate magnetic characteristics of bcc Fe in the paramagnetic state and compare our numerical results with the polarized neutron scattering experiment. We show that the short-range order in bcc Fe persists up to a temperature much higher than the Curie temperature but at rather small distances (up to 5Å).     

Theory of neutron scattering in magnets

We review theoretical studies of magnetically ordered materials using slow neutron scattering. We consider methods for decoding magnetic structures using symmetry theory (irreducible representations of space groups) and the theory of small-angle neutron scattering by large-scale inhomogeneities in magnetic materials.     

Time Delay and Short-Range Scattering in Quantum Waveguides

Although many physical arguments account for using a modified definition of time delay in multichannel-type scattering processes, one can hardly find rigorous results on that issue in the literature. We try to fill in this gap by showing, both in an abstract setting and in a short-range case, the identity of the modified time delay and the Eisenbud-Wigner time delay in waveguides. In the short-range case we also obtain limiting absorption principles, state spectral properties of the total Hamiltonian, prove the existence of the wave operators and show an explicit formula for the S-matrix. The proofs rely on stationary and commutator methods. Communicated by Yosi Avron submitted 12/04/05, accepted 13/05/05     

Scattering of a neutral fermion with anomalous magnetic moment by a charged straight thin thread

We consider the scattering of a massive neutral fermion with an anomalous magnetic moment in the electric field of a homogeneously charged straight thin thread from the standpoint of the quantum mechanical problem of constructing a self-adjoint Hamiltonian for the nonrelativistic Dirac-Pauli equation. Using the solutions obtained for the self-adjoint Hamiltonian, we investigate the scattering of the neutral fermion in the electric field of a thread oriented perpendicular to the plane of fermion motion (the Aharonov-Casher effect). We find expressions for the scattering amplitude and cross section of neutral fermions in the electric field of the thread. We show that the scattering amplitude and cross section depend both on the direct interaction between the fermion anomalous magnetic moment and the electric field and on the polarization of the fermionic beam in the initial state.     

High‐energy and smoothness asymptotic expansion of the scattering amplitude for the Dirac equation and application

We obtain an explicit formula for the diagonal singularities of the scattering amplitude for the Dirac equation with short‐range electromagnetic potentials. Using this expansion we uniquely reconstruct an electric potential and magnetic field from the high‐energy limit of the scattering amplitude. Moreover, supposing that the electric potential and magnetic field are asymptotic sums of homogeneous terms we give the unique reconstruction procedure for these asymptotics from the scattering amplitude, known for some energy E. Furthermore, we prove that the set of the averaged scattering solutions to the Dirac equation is dense in the set of all solutions to the Dirac equation that are in L²(Ω), where Ω is any connected bounded open set in with smooth boundary, and we show that if we know an electric potential and a magnetic field for , then the scattering amplitude, given for some energy E, uniquely determines these electric potential and magnetic field everywhere in . Combining this uniqueness result with the reconstruction procedure for the asymptotics of the electric potential and the magnetic field we show that the scattering amplitude, known for some E, uniquely determines a electric potential and a magnetic field, that are asymptotic sums of homogeneous terms, which converges to the electric potential and the magnetic field respectively. Moreover, we discuss the symmetries of the kernel of the scattering matrix, which follow from the parity, charge‐conjugation and time‐reversal transformations for the Dirac operator. Copyright © 2014 John Wiley & Sons, Ltd.     

Spectrum of the Magnetic Schrödinger Operator in a Waveguide with Combined Boundary Conditions

We consider the magnetic Schrödinger operator in a two-dimensional strip. On the boundary of the strip the Dirichlet boundary condition is imposed except for a fixed segment (window), where it switches to magnetic Neumann {For the definition of magnetic Neumann boundary conditions see Section 2, Eq. (2.2)}. We deal with a smooth compactly supported field as well as with the Aharonov-Bohm field. We give an estimate on the maximal length of the window, for which the discrete spectrum of the considered operator will be empty. In the case of a compactly supported field we also give a sufficient condition for the presence of eigenvalues below the essential spectrum.submitted 11/05/04, accepted 21/09/04     

Time delay in scattering by potentials and by magnetic fields with two supports at large separation

We study the asymptotic behavior of the time delay (defined as the trace of the Eisenbud-Wigner time delay operator) for scattering by potential and by magnetic field with two compact supports as the separation of supports goes to infinity. The emphasis is placed on analyzing how different the asymptotic formulae are in potential and magnetic scattering. The difference is proper to scattering in two dimensions.     

Global Results for the Rarita-Schwinger Equations and Einstein Vacuum Equations

We prove global existence and uniqueness of solutions to theRarita-Schwinger evolution equations compatible with the constraints.We use a gauge fixing for the Rarita-Schwinger equations forhelicity 3/2 fields in curved space that leads to a straightforwardHilbert space framework for their study. We explain how theseresults might be applied to the global analysis of the fullEinstein vacuum equations and provide a complete analysis asa basis for such applications. These and a programme for developinga scattering/inverse scattering transform for the full Einsteinequations are discussed. 1991 Mathematics Subject Classification:83C60, 35Q75, 83C05, 35L45.     

Theoretical Analysis of the Magnetic Cloaking Problem Based on an Optimization Method

Control problems are considered for a model of magnetic scattering on a permeable anisotropic obstacle shaped as a spherical layer. Such problems arise in developing technologies for designing magnetic cloaking devices when the corresponding inverse problems are solved by an optimization method. The solvability of the direct and extremal problems for the model in question is proved and the optimality system is derived. Its analysis permits obtaining sufficient conditions on the initial data which ensure the local uniqueness and stability of the optimal solutions.     

A stationary approach to inverse scattering for schrodinger operators with first order perturbation

In this paper, we study the inverse scattering of Schrodinger operators with short-range (resp. long-range) electric and magnetic potentials. We develop a stationary approach to determine the high energy asymptotics of the scattering operator (resp. modified scattering operator). As a corollary, we show that the electric potential and the magnetic field are uniquely determined by the first two terms of this asymptotic expansion.     

Debye–Waller Factor in Neutron Scattering by Ferromagnetic Metals

We obtain an expression for the neutron scattering cross section in the case of an arbitrary interaction of the neutron with the crystal. We give a concise, simple derivation of the Debye–Waller factor as a function of the scattering vector and the temperature. For ferromagnetic metals above the Curie temperature, we estimate the Debye–Waller factor in the range of scattering vectors characteristic of polarized magnetic neutron scattering experiments. In the example of iron, we compare the results of harmonic and anharmonic approximations.     

Aharonov–Bohm Effect and High-Momenta Inverse Scattering for the Klein–Gordon Equation

We analyze spin-0 relativistic scattering of charged particles propagating in the exterior, \({\Lambda \subset \mathbb{R}^3}\), of a compact obstacle \({K \subset \mathbb{R}^3}\). The connected components of the obstacle are handlebodies. The particles interact with an electromagnetic field in Λ and an inaccessible magnetic field localized in the interior of the obstacle (through the Aharonov–Bohm effect). We obtain high-momenta estimates, with error bounds, for the scattering operator that we use to recover physical information: we give a reconstruction method for the electric potential and the exterior magnetic field and prove that, if the electric potential vanishes, circulations of the magnetic potential around handles (or equivalently, by Stokes’ theorem, magnetic fluxes over transverse sections of handles) of the obstacle can be recovered, modulo 2π. We additionally give a simple formula for the high momenta limit of the scattering operator in terms of certain magnetic fluxes, in the absence of electric potential. If the electric potential does not vanish, the magnetic fluxes on the handles above referred can be only recovered modulo π and the simple expression of the high-momenta limit of the scattering operator does not hold true.     

Aharonov�CBohm Effect in Resonances of Magnetic Schr?dinger Operators with Potentials with Supports at Large Separation

Vector potentials are known to have a direct significance to quantum particles moving in the magnetic field. This is called the Aharonov–Bohm effect and is known as one of the most remarkable quantum phenomena. Here we study this quantum effect through the resonance problem. We consider the scattering system consisting of two scalar potentials and one magnetic field with supports at large separation in two dimensions. The system has trajectories oscillating between these supports. We give a sharp lower bound on the resonance widths as the distances between the three supports go to infinity. The bound is described in terms of the backward amplitude for scattering by each of the scalar potentials and by the magnetic field, and it also depends heavily on the magnetic flux of the field.     

Recent developments in the qualitative approach to inverse electromagnetic scattering theory

We consider the inverse scattering problem of determining both the shape and some of the physical properties of the scattering object from a knowledge of the (measured) electric and magnetic fields due to the scattering of an incident time-harmonic electromagnetic wave at fixed frequency. We shall discuss the linear sampling method for solving the inverse scattering problem which does not require any a priori knowledge of the geometry and the physical properties of the scatterer. Included in our discussion is the case of partially coated objects and inhomogeneous background. We give references for numerical examples for each problem discussed in this paper.     

Boundary-value problem for two-dimensional stationary Heisenberg magnet with nontrivial background. I

The method of the inverse scattering transform is used to solve a boundary-value problem on the half-plane for the two-dimensional stationary Heisenberg magnet with nontrivial background corresponding to helicoidal magnetic structures. The boundary conditions are formulated in terms of scattering data, and this leads to the appearance of gaps in the continuous spectrum of the auxiliary linear problem. Trace identities are obtained. The asymptotic behavior of some of the simplest solutions of soliton type is discussed.Scientific-Research Insititute of the Leningrad State University. Translated from Teoreticheskaya i Matematicheskaya Fizika Vol. 90, No. 2, pp. 259–272, February, 1992.     

Geodesic Distance in Planar Graphs: An Integrable Approach

We discuss the enumeration of planar graphs using bijections with suitably decorated trees, which allow for keeping track of the geodesic distances between faces of the graph. The corresponding generating functions obey non-linear recursion relations on the geodesic distance. These are solved by use of stationary multi-soliton tau-functions of suitable reductions of the KP hierarchy. We obtain a unified formulation of the (multi-) critical continuum limit describing large graphs with marked points at large geodesic distances, and obtain integrable differential equations for the corresponding scaling functions. This provides a continuum formulation of two-dimensional quantum gravity, in terms of the geodesic distance. 2000 Mathematics Subject Classification: Primary—05C30; Secondary—05A15, 05C05, 05C12, 68R05