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     

Scattering theory for the Schrödinger equation in some external time-dependent magnetic fields

We study the theory of scattering for a Schrödinger equation in an external time-dependent magnetic field in the Coulomb gauge, in space dimension 3. The magnetic vector potential is assumed to satisfy decay properties in time that are typical of solutions of the free wave equation, and even in some cases to be actually a solution of that equation. That problem appears as an intermediate step in the theory of scattering for the Maxwell-Schrödinger (MS) system. We prove in particular the existence of wave operators and their asymptotic completeness in spaces of relatively low regularity. We also prove their existence or at least asymptotic results going in that direction in spaces of higher regularity. The latter results are relevant for the MS system. As a preliminary step, we study the Cauchy problem for the original equation by energy methods, using as far as possible time derivatives instead of space derivatives.     

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.     

Aharonov-Bohm Effect in Scattering by Point-like Magnetic Fields at Large Separation

The aim is to study the Aharonov–Bohm e.ect in the scattering by two point–like magnetic .elds at large separation in two dimensions. We analyze the asymptotic behavior of scattering amplitude when the distance between the centers of two .elds goes to in.nity. The obtained result heavily depends on the .uxes of .elds and on incident and .nal directions.     

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.     

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.     

Semiclassical Analysis for Magnetic Scattering by Two Solenoidal Fields: Total Cross Sections

The fact that vector potentials have a direct significance to quantum particles moving in magnetic fields is known as the Aharonov–Bohm effect (A–B effect). We study this quantum effect through the semiclassical analysis on total cross sections in the magnetic scattering by two solenoidal (point-like) fields with total flux vanishing in two dimensions. We derive the asymptotic formula with first three terms. The system with two parallel fields seems to be important in practical aspects as well as in theoretical aspects, because it may be thought of a toroidal solenoid with zero cross section in three dimensions under the idealization that the two fields connect at infinity in their direction. The corresponding classical mechanical system has the trajectory oscillating between two centers of fields. The special emphasis is placed on analyzing how the trapping effect from classical mechanics is related to the A–B quantum effect in the semiclassical asymptotic formula. Submitted: September 3, 2006. Accepted: January 10, 2007.     

Dispersion of internal waves by an obstacle floating on the boundary separating two liquids

The problem of the scattering of a wave, that propagates along the boundary between two liquids, by a semi-infinite obstacle floating on this boundary is solved in a two-dimensional formulation. The solution is constructed using the Wiener-Hopf method interpreted by Jones in the framework of linear potential theory /1/. The fundamental properties of the processes of scattering and reflection of a wave by the obstacle are stated and an asymptotic analysis of the field in a far zone is presented.     

Quantum scattering near the lowest Landau threshold for a Schrödinger operator with a constant magnetic field

For fixed magnetic quantum number m results on spectral properties and scattering theory are given for the three-dimensional Schrödinger operator with a constant magnetic field and an axisymmetrical electric potential V. In various, mostly singular settings, asymptotic expansions for the resolvent of the Hamiltonian H _m+H_om+V are deduced as the spectral parameter tends to the lowest Landau threshold. Furthermore, scattering theory for the pair (H _m, H _om) is established and asymptotic expansions of the scattering matrix are derived as the energy parameter tends to the lowest Landau threshold.     

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.     

Dynamics of the density dependent predator-prey system with Beddington-DeAngelis functional response

Two models of a density dependent predator-prey system with Beddington-DeAngelis functional response are systematically considered. One includes the time delay in the functional response and the other does not. The explorations involve the permanence, local asymptotic stability and global asymptotic stability of the positive equilibrium for the models by using stability theory of differential equations and Lyapunov functions. For the permanence, the density dependence for predators is shown to give some negative effect for the two models. Further the permanence implies the local asymptotic stability for a positive equilibrium point of the model without delay. Also the global asymptotic stability condition, which can be easily checked for the model is obtained. For the model with time delay, local and global asymptotic stability conditions are obtained.     

Two-dimensional nuclear Coulomb scattering of a slow quantum particle

We study two-dimensional scattering of a quantum particle by the superposition of a Coulomb potential and a central short-range potential. We analyze the low-energy asymptotic behavior of all radial wave functions, partial phases, and scattering cross sections of such a particle. We propose two approaches for evaluating the scattering length and the effective radius.     

Wave function and the probability current distribution for a bound electron moving in a uniform magnetic field

We study the effects of electromagnetic fields on nonrelativistic charged spinning particles bound by a short-range potential. We analyze the exact solution of the Pauli equation for an electron moving in the potential field determined by the three-dimensional δ-well in the presence of a strong magnetic field. We obtain asymptotic expressions for this solution for different values of the problem parameters. In addition, we consider electron probability currents and their dependence on the magnetic field. We show that including the spin in the framework of the nonrelativistic approach allows correctly taking the effect of the magnetic field on the electric current into account. The obtained dependences of the current distribution, which is an experimentally observable quantity, can be manifested directly in scattering processes, for example.     

Interaction of Two Charges in a Uniform Magnetic Field: II. Spatial Problem

The interaction of two charges moving in ℝ³ in a magnetic field B can be formulated as a Hamiltonian system with six degrees of freedom. Assuming that the magnetic field is uniform and the interaction potential has rotation symmetry, we reduce this system to one with three degrees of freedom. For special values of the conserved quantities, choices of parameters or restriction to the coplanar case, we obtain systems with two degrees of freedom. Specialising to the case of Coulomb interaction, these reductions enable us to obtain many qualitative features of the dynamics. For charges of the same sign, the gyrohelices either “bounce-back”, “pass-through”, or exceptionally converge to coplanar solutions. For charges of opposite signs, we decompose the state space into “free” and “trapped” parts with transitions only when the particles are coplanar. A scattering map is defined for those trajectories that come from and go to infinite separation along the field direction. It determines the asymptotic parallel velocities, guiding centre field lines, magnetic moments and gyrophases for large positive time from those for large negative time. In regimes where gyrophase averaging is appropriate, the scattering map has a simple form, conserving the magnetic moments and parallel kinetic energies (in a frame moving along the field with the centre of mass) and rotating or translating the guiding centre field lines. When the gyrofrequencies are in low-order resonance, however, gyrophase averaging is not justified and transfer of perpendicular kinetic energy is shown to occur. In the extreme case of equal gyrofrequencies, an additional integral helps us to analyse further and prove that there is typically also transfer between perpendicular and parallel kinetic energy.      

Electromagnetic scattering by small dielectric inhomogeneities

In this work we carefully derive accurate asymptotic expansions of the electric and magnetic fields, the resonant frequencies, and the scattering amplitude in the practically interesting situation, where a number of dielectric objects of small diameter and with different material characteristics are imbedded in an otherwise smooth medium.     

Asymptotic and numerical study of electron flow spin polarization in 2D waveguides of variable cross‐section in the presence of magnetic field

We consider an infinite two‐dimensional waveguide that, far from the coordinate origin, coincides with a strip. The waveguide has two narrows of diameter ?. The narrows play the role of effective potential barriers for the longitudinal electron motion. The part of the waveguide between the narrows becomes a ‘resonator’, and there can arise conditions for electron resonant tunneling. A magnetic field in the resonator can change the basic characteristics of this phenomenon. In the presence of a magnetic field, the tunneling phenomenon is feasible for producing spin‐polarized electron flows consisting of electrons with spins of the same direction. We assume that the whole domain occupied by a magnetic field is in the resonator. An electron wave function satisfies the Pauli equation in the waveguide and vanishes at its boundary. Taking ? as a small parameter, we derive asymptotics for the probability T(E) of an electron with energy E to pass through the resonator, for the ‘resonant energy’ E_res, where T(E) takes its maximal value and for some other resonant tunneling characteristics. The asymptotic formulas contain some unknown constants. We find them by solving several auxiliary boundary value problems (independent of ?) in unbounded domains. Having the asymptotics with calculated constants, we can take it as numerical approximation to the resonant tunneling characteristics. Independently, we compute numerically the scattering matrix and compare the asymptotic and numerical results. Copyright © 2013 John Wiley & Sons, Ltd.     

Dynamical behavior of a delay virus dynamics model with CTL immune response

In this paper, the dynamical behavior of a virus dynamics model with CTL immune response and time delay is studied. Time delay is used to describe the time between the infected cell and the emission of viral particles on a cellular level. The effect of time delay on stability of the equilibria of the CTL immune response model has been studied and sufficient criteria for local asymptotic stability of the disease-free equilibrium, immune-free equilibrium and endemic equilibrium and global asymptotic stability of the disease-free equilibrium are given. Some conditions for Hopf bifurcation around immune-free equilibrium and endemic equilibrium to occur are also obtained by using the time delay as a bifurcation parameter. Numerical simulation with some hypothetical sets of data has been done to support the analytical findings.     

The scattering of a plane wave by a trap in the critical case

The scattering of a plane wave by a resonator with a narrow coupling channel is considered. The velocity potential of the scattered wave in this resonator has two series of poles with small imaginary parts, corresponding to the main trap and the coupling channel, the effect of which inside the trap differs by an order of magnitude. The critical case, when the limiting value for the poles from both series is the same, is investigated. It is shown that in this case two poles exist, which converge to this limiting value, and they both inherit resonance properties, characteristic for poles generated by the main trap. The principal terms of the asymptotic forms of the poles and the scattered wave are constructed.     

Inverse scattering transform for the two-component Gerdjikov–Ivanov equation with nonzero boundary conditions: Dark-dark solitons

The two-component Gerdjikov–Ivanov equation with nonzero boundary conditions is studied by the inverse scattering transform. A fundamental set of analytic eigenfunctions is obtained with the aid of the associated adjoint problem. Three symmetry conditions are discussed to curb the scattering data. The behavior of the Jost functions and the scattering matrix at the branch points is discussed. The inverse scattering problem is formulated by a matrix Riemann–Hilbert problem. The trace formula in terms of the scattering data and the so-called asymptotic phase difference for the potential are obtained. The solitons classification is described in detail. When the discrete eigenvalues lie on the circle, the dark-dark soliton is obtained for the first time in this work. And the discrete eigenvalues off the circle generate the dark-bright, bright-bright, breather-breather, M(-type)-W(-type) solitons, and their interactions.     

On generalized Eisenstein series and Ramanujan’s formula for periodic zeta-functions

We construct the asymptotic formulas for solutions of a certain linear second-order delay differential equation as independent variable tends to infinity. When the delay equals zero this equation turns into the so-called one-dimensional Schrödinger equation at energy zero with Wigner–von Neumann type potential. The question of interest is how the behaviour of solutions changes qualitatively and quantitatively when the delay is introduced in this dynamical model. We apply the method of asymptotic integration that is based on the ideas of the centre manifold theory in its presentation with respect to the systems of functional differential equations with oscillatory decreasing coefficients.