Hölder-Banach space analysis of the Bethe-Salpeter equation

We find a Hölder Banach space in which the Bethe-Salpeter equation is a compact integral equation as it stands. We study the properties of the solution in preparation for an analysis of linear field theory models of 3-body amplitudes. In particular we study the properties of the Regge poles of the solution and prove the existence and uniqueness of on mass shell scattering amplitudes.     

On the description of transport phenomena

The linear response of a quantity like the electric current to a time and space dependent field can be described by Kubo's formula, and this again can be written as a resolvent. The expansion of this resolvent yields exactly a transport equation of the structure of the Boltzmann equation. Perturbation theory — the only practical way to deal with it — gives back the usual Boltzmann equation. This derivation has advantages like yielding a symmetric collision operator. Also including time and space dependence with this approach is very easy. The purpose of this paper is to show the closeness of linear response theory and kinetic equations. We also discuss merits and drawbacks of the resolvent method.     

A dynamical formulation of one-dimensional scattering theory and its applications in optics

We develop a dynamical formulation of one-dimensional scattering theory where the reflection and transmission amplitudes for a general, possibly complex and energy-dependent, scattering potential are given as solutions of a set of dynamical equations. By decoupling and partially integrating these equations, we reduce the scattering problem to a second order linear differential equation with universal initial conditions that is equivalent to an initial-value time-independent Schrödinger equation. We give explicit formulas for the reflection and transmission amplitudes in terms of the solution of either of these equations and employ them to outline an inverse-scattering method for constructing finite-range potentials with desirable scattering properties at any prescribed wavelength. In particular, we construct optical potentials displaying threshold lasing, antilasing, and unidirectional invisibility.     

Scattering theory of gilbert damping

The magnetization dynamics of a single domain ferromagnet in contact with a thermal bath is studied by scattering theory. We recover the Landau-Liftshitz-Gilbert equation and express the effective fields and Gilbert damping tensor in terms of the scattering matrix. Dissipation of magnetic energy equals energy current pumped out of the system by the time-dependent magnetization, with separable spin-relaxation induced bulk and spin-pumping generated interface contributions. In linear response, our scattering theory for the Gilbert damping tensor is equivalent with the Kubo formalism.     

Inverse scattering transform for the time dependent Schrödinger equation with applications to the KPI equation

For the direct-inverse scattering transform of the time dependent Schrödinger equation, rigorous results are obtained based on an opertor-triangular-factorization approach. By viewing the equation as a first order operator equation, similar results as for the first ordern x n matrix system are obtained. The nonlocal Riemann-Hilbert problem for inverse scattering is shown to have solution.     

Physical meaning of the stretched exponential Kohlrausch function

In this paper the physical meaning of the empirical Kohlrausch-Williams-Watts (KWW) function is explained in terms of the linear oscillator theory. It is shown that the KWW function is a solution of the non-autonomous linear first order equation for an overdamped linear oscillator. From the linear oscillator model it follows that the KWW-type relaxation is the linear relaxation with a time (coordinate, stress, voltage, etc.) dependent dissipation of energy. The theoretical results are validated by measurements. A method for modeling KWW-type relaxation using simple electrical circuits is proposed.     

Linear magnetoresistance and magnetic field dependent scattering probability

Within the framework of the Boltzmann equation it is possible to find a linear behaviour for the Magnetoresistance. Instead of using a purely random impurity scattering model, one makes this model slightly magnetic field dependent. Keeping only the linear term of the power expansion in the magnetic field, one finds a linear dependence of the Magnetoresistance.     

Master equation for a quantum particle in a gas

The equation for the quantum motion of a Brownian particle in a gaseous environment is derived by means of S-matrix theory. This quantum version of the linear Boltzmann equation accounts nonperturbatively for the quantum effects of the scattering dynamics and describes decoherence and dissipation in a unified framework. As a completely positive master equation it incorporates both the known equation for an infinitely massive Brownian particle and the classical linear Boltzmann equation as limiting cases.     

Solitary wave dynamics in shallow water over periodic topography

The problem of long-wave scattering by piecewise-constant periodic topography is studied both for a linear solitary-like wave pulse, and for a weakly nonlinear solitary wave [Korteweg-de Vries (KdV) soliton]. If the characteristic length of the topographic irregularities is larger than the pulse length, the solution of the scattering problem is obtained analytically for a leading wave in the framework of linear shallow-water theory. The wave decrement in the case of the small height of the topographic irregularities is proportional to delta2, where delta is the relative height of the topographic obstacles. An analytical approximate solution is also obtained for the weakly nonlinear problem when the length of the irregularities is larger than the characteristic nonlinear length scale. In this case, the Korteweg-de Vries equation is solved for each piece of constant depth by using the inverse scattering technique; the solutions are matched at each step by using linear shallow-water theory. The weakly nonlinear solitary wave decays more significantly than the linear solitary pulse. Solitary wave dynamics above a random seabed is also discussed, and the results obtained for random topography (including experimental data) are in reasonable agreement with the calculations for piecewise topography.     

Transport equation for a gas of bosons

A kinetic equation for the Wigner density function of a Bose gas of quasiparticles (magnons) is derived in the region of linear response. The method uses a simple factorization procedure to transform the Hierarchy equations into a system which can be solved iteratively. The first nontrivial approximation is examined in the limit of slow time and space variation. It gives the linearized Boltzmann equation in the form assumed in the Landau theory. The correction of lowest order in (coupling constant × density) is shown to be a correction to the Born approximation of the scattering cross section.Work supported by the Fonds zur Förderung der wissenschaftlichen Forschung.     

Exact solution of the inverse acoustic scattering problem for the one-dimensional density profile in cylindrical geometry

The inhomogeneous acoustic wave equation in cylindrically symmetric geometry, is reduced to a one-dimensional Schrödinger equation for time harmonic waves. The scattering potential so formed is uniquely reconstructed by the Gelfand-Levitan theory. In physical terms, the scattering potential is a function of the impedance profile, which is then recovered. If the velocity profile is known (or constant) the density profile, as a function of the radial coordinate, can be deduced. A practical experiment is envisaged whereby the back-scattered time sequence is measured in a pulse-echo mode, and the data are used to reconstruct the density profile.     

PROLONGATION STRUCTURE APPROACH TO THE GROUP-THEORETICAL TECHNIQUE FOR GENERATING STATIONARY AXISYMMETRIC SPACE-TIMES

The group-theoreti cal technique for generating stationary axisymmetric gravitational fields is approached by means of the prolongation structure theory for soliton systems. An sp(2)xc(t) structure is obtained via solving the fundamental equation for prolongation structures and the F-equation for Kinnersley-Chitre's generating function is naturally introduced as an inverse scattering equation. A homogeneous Hilbert problem(HRP) associated with the Geroch group K and a corresponding linear singular integral equation are derived based upon a general condition satisfied by the auto-Bäcklund transformations in the sense of prolongation structure theory.     

Canonical transformations and exact invariants for time‐dependent Hamiltonian systems

An exact invariant is derived for n‐degree‐of‐freedom non‐relativistic Hamiltonian systems with general time‐dependent potentials. To work out the invariant, an infinitesimalcanonical transformation is performed in the framework of the extended phase‐space. We apply this approach to derive the invariant for a specific class of Hamiltonian systems. For the considered class of Hamiltonian systems, the invariant is obtained equivalently performing in the extended phase‐space a finitecanonical transformation of the initially time‐dependent Hamiltonian to a time‐independent one. It is furthermore shown that the invariant can be expressed as an integral of an energy balance equation. The invariant itself contains a time‐dependent auxiliary function ξ (t) that represents a solution of a linear third‐order differential equation, referred to as the auxiliary equation. The coefficients of the auxiliary equation depend in general on the explicitly known configuration space trajectory defined by the system's time evolution. This complexity of the auxiliary equation reflects the generally involved phase‐space symmetry associated with the conserved quantity of a time‐dependent non‐linear Hamiltonian system. Our results are applied to three examples of time‐dependent damped and undamped oscillators. The known invariants for time‐dependent and time‐independent harmonic oscillators are shown to follow directly from our generalized formulation.     

Curle's equation and acoustic scattering by a sphere

Recent papers have initiated interesting comparisons between aeroacoustic theory and the results of acoustic scattering problems. In this paper, we consider some aspects of these comparisons for acoustic scattering by a sphere. We give a derivation of Curle's equation for a specific class of linear acoustic scattering problems, and, in response to previous claims to the contrary, give an explicit confirmation of Curle's equation for plane wave scattering by a stationary rigid sphere of arbitrary size in an inviscid fluid. We construct the complete solution for scattering by a rigid sphere in a viscous fluid, and show that the neglect of viscous terms in Curie's equation yields an incomplete prediction of the far field dipole pressure. We also consider the null field solution of the sphere scattering problem, and give its extension to the vorticity modes associated with viscosity. Finally, we construct a solution for an elastic sphere in a viscous fluid, and show that the rigid sphere/null field solution is recovered from the limit of infinite longitudinal and shear wave speeds in the elastic solid.     

Fractal Characteristics Investigation on Light Scattering from Two Dimensional Rough Surface

A normalized two dimensional band-limited Weierstrass fractal function is used for modeling the rough surface. Based on Kirchhoff theory, an analytic solution of the average scattering field and the variance of scattering intensity are derived with emphasis on examining the relation of fractal dimension with the scattering pattern. The important conclusion is obtained that the diffracted envelope slopes of scattering pattern can be approximated to a slope of linear equation.     

Thermoelectric power of magnetic metals: II. Anisotropic metals

We present a theory of the thermoelectric power tensor of anisotropic ferromagnetic metals with localized magnetic moments starting from the Boltzmann equation and incorporating anisotropy effects due to the lattice structure through a parameter measuring the anisotropy in the sound velocity. Elastic and inelastic phonon and spin scattering contributions are taken into account through a linear superposition of scattering cross sections. A mean field approximation is used to describe the ordered magnetic phase. Spin wave and impurity scattering, phonon and magnon drag are not included. In a range encompassing the Curie temperature, i.e. at “moderate temperatures”, the theory quantitatively reproduces observed features except for specific details (e.g. rounding near T_c) needing other physical input. We compare our theory to data on single crystals of Gd and Tb₇₅Gd₂₅. The c-axis thermoelectric power is well recovered for very reasonable values of anisotropy and scattering strength parameters. A conjecture is given to explain the basal plane thermoelectric power positive slope at high temperature.     

Linear alternative to the Boltzmann equation without linearization: III. Impurity scattering and electric field

Our previously reported projection formalism leading to a linear alternative to the Boltzmann (-Uehling-Uhlenbeck) kinetic equation is here applied to standard elastic scattering on impurities in solids and external electric field. In the former case in which the Boltzmann equation is linear, we prove that our formalism reproduces this equation. In the dc as well as ac electric field and for any scattering mechanism, it is verified that inclusion of the field reduces (upon neglecting of the field dependence of scattering processes) in the lowest order to inclusion of just the standard field driving term.     

Solvability of linear kinetic equations with multi-energetic inelastic scattering

In this paper we shall analyse the linear Boltzmann equation describing the motion of test particles through a background of heavy field particles that can appear in several energy states. The inelastic scattering process consists in the exchange of quanta of energy between the field and test particles. The well-posedness of the problem is investigated by means of the substochastic semigroup theory and the conditions on the scattering collision frequencies are given for the evolution to keep the collision rate finite and to preserve the total number of particles.     

Boundedness and stability of impulsively perturbed systems in a Banach space

We consider the influence of impulsive perturbations of a linear impulsive equation in a Banach space on the existence of bounded solutions and the exponential stability of the equation.