Volume dependence of the energy spectrum in massive quantum field theories

The low-lying energy values associated to energy eigenstates describing two stable particles enclosed in a (space-like) box of sizeL are shown to be expandable in an asymptotic power series of 1/L. The coefficients in these expansions are related to the appropriate elastic scattering amplitude in a simple and apparently universal manner. At low energies, the scattering amplitude can thus be determined, if an accurate calculation of two-particle energy values is possible (by numerical simulation, for example).     

The effect of high momentum transfer on scattering from oscillators and crystals

We examine two aspects of scattering at high energy and momentum transfer; asymptotic forms of scattering laws and comparison of approximate and exact results for a particular system, a harmonic lattice.By treating model systems of harmonically bound particles, an asymptotic analysis shows that, under conditions of high momentum transfer the particle looks as though it is unbound, in the sense that the weighting (envelope) of the various inelastic channels is that of a free particle, although the discreteness of the allowed energy transfers reflects the fact that the particles are, in reality, bound. The discreteness of the energy transfers requires an extension of the usual impulse approximation, which had been realised by Wick for a total cross section calculation, and this is treated in an appendix.An exact numerical technique for handling multiphonon effects directly, that is, without expansions, for the case of scattering from a lattice is used as a check of the Sjölander approximation. The latter treats elastic and one-phonon processes exactly, the remainder in a Gaussian-like approximation. As expected this scheme works well at low and high momentum transfer. At intermediate values the scattering law turns out to have a lot of structure which isnot reproduced. Simple interpretations would then give split peaks and such effects which in reality are simply manifestations of multiphonon processes.     

Scaling and decay in periodically driven scattering systems

We investigate irregular scattering in a periodically driven Hamiltonian system of one degree of freedom. The potential is asymptotically attracting, so there exist parabolically escaping scattering orbits, i.e. orbits with asymptotic energy E(out)=0. The scattering functions (i.e. the asymptotic out-variables as functions of an asymptotic in-variable) show a characteristic algebraic scaling in the vicinity of these orbits. This behavior is explained by asymptotic properties of the interaction. As a consequence, the number N(Deltat) of temporarily bound particles decays algebraically with the delay time Deltat, although no KAM scenario can be found in phase space. On the other hand, we find the number N(n) of temporarily bound particles to decay exponentially with the number n of zeros of x(t).     

A non-relativistic model of two-particle decay IV. Relation to the scattering theory,spectral concentration,and bound states

The present paper which represents the fourth part of the series devoted to analysis of a simple Lee-type model of two-particle decay deals with three problems. The first one concerns relation of the model to the scattering theory. We prove asymptotic completeness for the elastic scattering of the two light particles and show that for a sufficiently weak coupling this system has just one resonance whose position is the same as that of the pole which yield the main contribution to the decay law. The second problem concerns spectral concentration; we prove its occurrence for families of intervals aroundE that shrink slower than quadratically ing. Finally, necessary and sufficient conditions for the existence of bound states are discussed.The authors are indebted to Dr. M. Sotona for Remark 5.7.     

New Computational Techniques to Simulate Light Scattering from Arbitrary Particles

The coupled dipole method, as originally formulated by Purcell and Pennypacker [3], is a very powerful method to simulate the elastic light scattering from arbitrary particles. This method, however, has one major drawback: if the size of the particles grows, or if scattering from an ensemble of randomly oriented particles has to be simulated, the computational demands of the coupled dipole method soon become too high. This paper presents two new computational techniques to resolve this problem. First the coupled dipole method was implemented on a massively parallel computer. The parallel efficiency can be very close to 1, implying that the attained computational speed scales perfectly with the number of processors. Second, it is proposed to reduce the computational complexity of the coupled dipole method by including ideas from the so-called fast multipole methods (hierarchical algorithms) into the coupled dipole method. In this way calculation time can be decreased by orders of magnitude.     

Allowing for Coulomb effects in the effective range expansion for two coupled channels

A Coulomb-modified matrix of scattering amplitudes (an [(F)\tilde]\tilde F matrix) is considered for the case of two coupled channels of elastic scattering of charged particles with different orbital angular momenta (l ₁ and l ₂ = l ₁ + 2). Matrix elements of the [(F)\tilde]\tilde F matrix are expressed in terms of the matrix elements of a [(K)\tilde] ^{- 1}\tilde K^{ - 1} matrix inverse to a modified reaction K matrix. The elements of the [(K)\tilde] ^{- 1}\tilde K^{ - 1} matrix are written as expansions that are generalizations of single-channel effective range expansion with allowance for the Coulomb interaction. If a system of colliding particles involves a bound state, the analytic continuation of these expansions into the region of negative energies makes it possible to obtain both the position of the pole corresponding to the bound state and the scattering amplitude residues in this pole, in terms of which the corresponding vertex constants and asymptotic normalization coefficients are expressed.     

Coulomb wave function corrections for n-particle Bose–Einstein correlations

The effect of multi-particle Coulomb final state interactions on higher-order intensity correlations is determined in general, based on a scattering wave function which is a solution of the n-body Coulomb Schr?dinger equation in (a large part of) the asymptotic region of n-body configuration space. In particular, we study Coulomb effects on the n-particle Bose–Einstein correlation functions of similarly charged particles and remove a systematic error as big as 100% from higher-order multi-particle Bose–Einstein correlation functions. Received: 24 November 1999 / Published online: 17 March 2000     

Time asymmetric quantum theory – II. Relativistic resonances from S‐matrix poles

Relativistic resonances and decaying states are described by representations of Poincaré transformations, similar to Wigner's definition of stable particles. To associate decaying state vectors to resonance poles of the S‐matrix, the conventional Hilbert space assumption (or asymptotic completeness) is replaced by a new hypothesis that associates different dense Hardy subspaces to the in‐ and out‐scattering states. Then one can separate the scattering amplitude into a background amplitude and one or several “relativistic Breit‐Wigner” amplitudes, which represent the resonances per se. These Breit‐Wigner amplitudes have a precisely defined lineshape and are associated to exponentially decaying Gamow vectors which furnish the irreducible representation spaces of causal Poincaré transformations into the forward light cone.     

Additive conservation laws in local relativistic quantum field theory

We consider generatorsQ of symmetry transformations acting additively on asymptotic particle states according to (1.1). [This equation can be derived forQ defined as integral over a conserved local current!]. For simplicity, we consider only the case that all asymptotic fields are scalar. Assuming that elastic scattering occurs at least in an open subset of the scattering manifold we show thatQ is at most alinear combination of generators of the Poincaré group and internal symmetries.     

Applicability of the T-matrix method and its modifications

The range of applicability of the T-matrix method and its modifications for solving the problem if the scattering of electromagnetic radiation by nonspherical, axially symmetric particles is investigated analytically and numerically. The use of this method for calculating the characteristics of scattered radiation in the farfield region (the extinction and scattering cross sections, the scattering indicatrix, etc.) is shown to be appropriate for “weak-Rayleigh-type” particles. This condition is met when the intersection of the analytic continuations of the scattered and internal fields contains a ring with the center at the origin of coordinates. For a reliable calculation of the scattered field in the near-field region, it is necessary that a particle be a “Rayleigh-type” one (i.e., the Rayleigh hypothesis be valid for it). In this case, the singularities of the scattered field must occur inside a sphere lying inside a scatterer. Spheroidal particles are weak-Rayleigh-type ones if their semiaxes ratio is $a/b < (\sqrt {2 + 1} )The range of applicability of the T-matrix method and its modifications for solving the problem if the scattering of electromagnetic radiation by nonspherical, axially symmetric particles is investigated analytically and numerically. The use of this method for calculating the characteristics of scattered radiation in the farfield region (the extinction and scattering cross sections, the scattering indicatrix, etc.) is shown to be appropriate for “weak-Rayleigh-type” particles. This condition is met when the intersection of the analytic continuations of the scattered and internal fields contains a ring with the center at the origin of coordinates. For a reliable calculation of the scattered field in the near-field region, it is necessary that a particle be a “Rayleigh-type” one (i.e., the Rayleigh hypothesis be valid for it). In this case, the singularities of the scattered field must occur inside a sphere lying inside a scatterer. Spheroidal particles are weak-Rayleigh-type ones if their semiaxes ratio is , and they are Rayleigh-type ones if . Numerical calculations for spheroids and Chebyshev particles corroborate these conclusions. However, the indicated boundaries are “ spread” (toward the expansion), because the expansion coefficients for the fields are determined with the use of the reduced (i.e., finite) systems. The limiting sizes of the particles for which the T-matrix method gives plausible results are primarily determined by their geometry (shape). __________ Translated from Optika i Spektroskopiya, Vol. 92, No. 5, 2002, pp. 813–825. Original Russian Text Copyright ? 2002 by Farafonov.     

On the asymptotic behavior in the scattering problem for several charged quantum particles interacting via repulsive pair potentials

The ansatz proposed earlier by Buslaev and Levin [Funct. Anal. Appl. 46, 147 (2012)] for describing the leading order in the asymptotic behavior of continuum eigenfunctions in the scattering problem for three charged quantum particles is extended to the system of n charged quantum particles of identical mass and identical charge. It is shown that the proposed ansatz generates a fast decreasing (faster than the potential) discrepancy of the Schrödinger equation for a set of asymptotic configurations.     

Propagation of circularly polarized light in media with large-scale inhomogeneities

Small-angle multiple scattering of circularly polarized waves in disordered systems composed of large (larger than the light wavelength) spherical particles is discussed. The equation for Stokes’s fourth parameter V — the difference between the intensities of the left-and right-hand polarized light — is shown to have the form similar to that of the scalar transport equation for intensity I, the only difference being the presence of an additional “non-small-angle” term responsible for depolarization. In the case of small-angle scattering, depolarizing collisions are relatively rare and, in contrast to the scalar case, the problem contains an additional spatial scale, namely the depolarization depth. The polarization degree and helicity of the scattered light are calculated for the case of purely elastic scattering and in the presence of absorption in the medium. For strong absorption, depolarization is shown to follow the transition to the asymptotic regime of wave propagation. The features appearing in strong (non-Born) single scattering are also discussed. Zh. éksp. Teor. Fiz. 115, 769–790 (March 1999)     

Channeling problem for charged particles produced by confining environment

Channeling problem produced by confining environment that leads to resonance scattering of charged particles via quasistationary states imbedded in the continuum is examined. Nonmonotonic dependence of physical parameters on collision energy and/or confining environment due to resonance transmission and total reflection effects is confirmed that can increase the rate of recombination processes. The reduction of the model for two identical charged ions to a boundary problem is considered together with the asymptotic behavior of the solution in the vicinity of pair-collision point and the results of R-matrix calculations. Tentative estimations of the enhancement factor and the total reflection effect are discussed. The text was submitted by the authors in English.     

Pulse propagation of acoustic waves scattered in a three-dimensional channel

Abstract

In a previous paper (Whitman et al 1999 Waves Random Media 9 1–11) we discussed the scattering of acoustic waves by random sound-speed fluctuations in a two-dimensional channel and presented an asymptotic form for an acoustic pulse propagating in the channel. Here we include the three-dimensional effect of transverse scattering. We find an asymptotic solution in which initially the two-dimensional mode-transfer effect is more important than the transverse scattering effect. However, for large enough propagation distances the transverse scattering effect dominates the pulse spread. In this paper we shall show the form of the pulse shape in both propagation ranges as well as in the transition regime. We shall begin with a discussion of the physics of the problem and then present a mathematical discussion.     

Scattering of a Light Particle by a System of Harmonic Oscillators

We discuss the asymptotic wave function of a quantum system in ?³ composed by heavy and light particles, in the case where the light particles are in scattering states and no interaction is assumed among particles of the same kind. We first review a recent result concerning the case of K heavy and N light particles, where the one-particle potential acting on each heavy particle decays at infinity. Then we consider the case of one light particle interacting with a system of harmonic oscillators and prove the same kind of result following, with some modification, the proof of the previous case. A possible application to the analysis of the scattering of a light particle from condensed matter is also outlined.     

Calculation of the energy density function by the method of steepest descent

We present a unified theory of diatomic molecules which reconciles bound state spectroscopy and atomic scattering theory. The total wave-function is expanded in a complete set of atomic channel states which is entirely equivalent to an expansion in Hund's case (e) electronic-rotational states. An analysis of the coupled radial, that is vibrational, functions places strong constraints on the asymptotic properties of the molecular wave-functions. These are presented in terms of the reactance K and scattering S matrices of atomic scattering theory which offers a uniform treatment for open channels (inelastic scattering and continuum spectroscopy), closed channels (bound state spectroscopy) and mixtures of both (predissociation). The normalization of the total wavefunction is derived and related to the asymptotic boundary conditions both for continuum and bound states.     

Light Propagation through Weakly Absorbing Layers of Dielectric Cube-Shaped Particles

Light propagation through ensembles of dielectric particles with various packing density is studied by the method of physical modeling in the SHF range. Monolayers and multilayers of weakly absorbing cube-shaped particles with the refractive index n = 1.5 are investigated. Particle dimensions are comparable to the incident radiation wavelength. Dependences of the transmittance T and the scattering efficiency factor Q on the diffraction parameter are obtained for ensembles of particles with various overlap factors . For monolayers of particles, the effect of an enhanced transmittance with increasing packing density of particles is observed. Conditions are specified at which the monolayer transmittance is equal to zero. It is shown that the Mie formulas, derived for spherical particles, can be used to calculate the optical characteristics of an ensemble of particles with chaotic orientation of particle sides, if the diameter of an equivalent sphere is set equal to the cube edge length multiplied by a correction factor of 1.15. Transition to an ordered orientation of cube-shaped particle sides leads to increase in the scattering efficiency factor for > 2.     

Impact parameter representation from the Watson-Sommerfeld transform

Using the Watson-Sommerfeld transform the elastic scattering amplitude of two spinless particles is shown to have an exact and unique impact parameter, or Fourier-Bessel (FB) representation. The representation is valid for all physical energies and scattering angles. Wallace's recent work is found to be an asymptotic expansion of the FB amplitude obtained from the partial-wave expansion. The way singularities of the partial-wave amplitude in the l-plane enter in the FB amplitude is also explicitly shown.     

The asymptotic form of three-particle wavefunctions and the cross sections

The asymptotic behavior of three-particle wavefunctions in the coordinate representation is examined as all three particles move towards infinity. We investigate both the wave-function for a particle incident on a bound state, and that for three free incident particles. The results are expressed in terms of cross sections, and they lead to a clean separation of the double-scattering terms from the “true” three-particle scattering amplitude. We also derive the relevant reciprocity theorems from time-reversal invariance.     

Using the T‐Matrix Method for Light Scattering Computations by Non‐axisymmetric Particles: Superellipsoids and Realistically Shaped Particles

Light scattering by non‐axisymmetric particles is commonly needed in particle characterization and other fields. After much work devoted to volume discretization methods to compute scattering by such particles, there is renewed interest in the T‐matrix method. We extended the null‐field method with discrete sources for T‐matrix computation and implemented the superellipsoid shape using an implicit equation. Additionally, a triangular surface patch model of a realistically shaped particle can be used for scattering computations. In this paper some exemplary results of scattering by non‐axisymmetric particles are presented.