Haag-Ruelle approximation of collision states

We investigate the rate of convergence of the Haag-Ruelle approximation Ψ(t) at large timest for arbitrary collision states Ψ with finite energy. An improved estimate of the norm distance ∥Ψ?Ψ(t)∥ is given. In particular for states Ψ with smooth asymptotic wave functions it turns out that ∥Ψ?Ψ(t)∥ approaches 0 almost liket ^?3/4.     

Threshold singularities of theS-matrix and convergence of Haag-Ruelle approximations

We establish a one-to-one correspondence between the continuity properties of theS-matrix at the 2-particle threshold and the rate of convergence of the Haag-Ruelle approximations ψ(t) for asymptotic 2-particle states ψ with smooth wavefunctions. It turns out that the norm distance ∥ψ?ψ(t)∥ approaches 0 liktt ^?5/4 if theS-matrix has the normal threshold singularities and liket ^?3/4 in the exceptional case where the threshold has “absorbed” a bound state. These connections are valid both in relativistic quantum field theory and in non-relativistic models with short range interaction.     

Universal power law for the energy spectrum of breaking Riemann waves

The universal power law for the spectrum of one-dimensional breaking Riemann waves is justified for the simple wave equation. The spectrum of spatial amplitudes at the breaking time t = t _b has an asymptotic decay of k ^?4/3, with corresponding energy spectrum decaying as k ^?8/3. This spectrum is formed by the singularity of the form (x ? x _b )^1/3 in the wave shape at the breaking time. This result remains valid for arbitrary nonlinear wave speed. In addition, we demonstrate numerically that the universal power law is observed for long time in the range of small wavenumbers if small dissipation or dispersion is taken into account in the viscous Burgers or Korteweg-de Vries equations.     

Trilinear mode effects on transport coefficients

The effects of tri-linear modes on transport coefficients is illustrated by explicit calculations for the self-diffusion constant. In two dimensions the form of the asymptotic time behavior of the velocity autocorrelation function is changed from t^-1 (bilinear modes) to (t log t)^-1 (trilinear modes). Thus, no finite mode calculation in two dimensions can yield the correct asymptotic form. A heuristic self-consistent calculation including all mode orders yields the form [t log^{$\text{1}\text{2}$} t]^-1. In three dimensions, the form of the asymptotic time behavior is not changed when higher-order modes are included.     

Scattering for the Zakharov System in 3 Dimensions

We prove global existence and scattering for small localized solutions of the Cauchy problem for the Zakharov system in 3 space dimensions. The wave component is shown to decay pointwise at the optimal rate of t ^?1, whereas the Schrödinger component decays almost at a rate of t ^?7/6.     

Nonlinear optical characteristics of a pseudoisocyanine solution

The saturated nonlinear absorption and Kerr nonlinearities of an aqueous pseudoisocyanine solution are investigated at the wavelengths of 532 and 529 nm with the use of pulsed laser radiation of different duration (8 ns and 475 fs). The measured values of the nonlinear refractive indices amount to ?6×10^?12 (t=8 ns) and ?8×10^?14 cm²W^?1 (t=475 fs). The change in self-action effects in pseudoisocyanine from self-defocusing to self-focusing revealed in the case of increasing intensity of femtosecond laser pulses is attributed to the effect of a fifth-order nonlinear optical process. The nonlinear refractive index responsible for this process amounts to 4×10^?24 cm⁴ W^?2. The imaginary part of the third-order nonlinear susceptibility, responsible for the induced transparency of the pseudoisocyanine solution, is Imχ ⁽³⁾=?2×10^?12 esu. Temporal changes in the shape of nanosecond laser pulses due to the nonlinear refraction induced by a thermal process are analyzed.     

Soft π−p and pp elastic scattering in the energy range 30 to 345 GeV

Differential cross sections for π^?p and pp elastic scattering have been measured at incident momenta ranging from 30 to 345 GeV and in the t range 0.002 (GeV/c)² ? |t| ? 0.04 (GeV/c)². From the analysis of the data, the ratio ? (t = 0) of the real to the imaginary parts of the forward scattering amplitude was determined together with the logarithmic slope b of the diffraction cone.The results on the real parts confirm the validity of the forward dispersion relations at high energies. Using the dispersion relations, it was shown that the experimental data on ?_{π?p(t = 0)} require a continuous rise of the total πp cross sections, at least up to the energy of 2000 GeV, thus revealing a close similarity in high-energy behaviour of πp and pp interactions.The results on the slope parameters from this experiment together with the analysis of the available world data demonstrate that the existing experimental data are consistent with the hypothesis of a universal shrinkage of the hadronic diffraction cone at high energies. The value of the asymptotic shrinkage parameter α_p^′ was found to be independent of the kind of the incident hadron and of the momentum transfer in the t range |t| ? 0.2 (GeV/c)² : 2α_{p^′ = ± 3 (GeV/c)^?2}.     

The long-time behaviour of the electric-field autocorrelation function in a finite photon gas

We investigate an autocorrelation function of a soluble three-dimensional system, namely the temporal coherence functionC _E(t)∝<E(0)E(t)> of the thermal radiation field in a cube-shaped cavity for the stochastic electrical fieldE. In the thermodynamic limit,C _E(t) relaxes exponentially at intermediate times, but a “long-tail” behaviourC ₀(t)=At^?4 withA<0 is predominant for long times. In the case of a finite, but not too small, cavity lengthL obeyingΛ=hc/k _BT?L and at timest withct?L, C _E(t) is described by an asymptotic expansion in powers ofL ^?1 using generalized Riemann zeta functions. Surface-and shape-effects enhance the long-tail. In the case of very small cavities withL«Λ, we calculate an expansion ofC _E(t) in terms of exp(?L ^?1) and cosines. An oscillatory, but not strictly periodic, long-time behaviour is observed in this case.     

ϱ-ω interference in π−p → π−π+ n at 17.2 GeV

Results are presented of a study of ?-ω interference effects in high statistics, good resolution, π^?p → π⁺π^?n data obtained at 17.2 GeV using the CERN-Munich spectrometer. We investigate the t dependence of ?-ω interference in the various ππ production amplitudes. While effects are observed in all the P-wave production amplitudes, the most pronounced structure occurs in the natural parity exchange amplitude in the t interval 0.1 ? ?t ? 0.4 GeV². Under certain assumptions we determine the relative magnitudes and phases of the ?- and ω-production amplitudes as functions of t.     

Axial vector meson production

Recently available differential cross section and density matrix information on π^?p → B^?p at 4 GeV/c are successfully described in terms of t channel exchanges and s channel absorptive effects. The s channel helicity amplitude which is dominant at small |t| (zero net helicity flip ω exchange) is found to have a zero at ?t = 0.2 GeV². SU(3) and higher symmetries are used to predict cross sections for further axial vector meson production processes, in particular non-diffractive A₁ production. The importance of these processes with regard to Regge phenomenology and meson spectroscopy is emphasized.     

The real part of the πN diffraction amplitude

The real part of the invariant isospin evenπ N amplitudeC ⁺≡A′ ⁺ at ¦t¦?0.4 (GeV/c)² is determined from the condition thatC ⁺ fulfills the fixed-t dispersion relation, is compatible with this amplitude as reconstructed from phase shifts up to 1.5 GeV/c and leads to differential cross sections above 3 GeV/c in agreement with the experimental data.-The result for ReC ⁺/ImC ⁺ is remarkably large (?0.5 att=?0.3 (GeV/c)² and 6 GeV/c) and up to 20 GeV/c its uncertainty is comparable with that in the upper part of the phase shift region.-Even at 40 GeV/c ReC ⁺ is far from being asymptotic because of large energy independent low energy contributions. The deviations from our simple high energy ansatz (k>4GeV/c): ImC ⁺(k, t)～(k + m)exp(9t/2) are almost comparable with the experimental errors.     

The long time behavior of the velocity autocorrelation function in a plasma

The utilization of the Landau-Placzek method results in a velocity autocorrelation function for a one component plasma having oscillatory behavior (at the plasma frequency) in addition to the t^?(d/2) (d = dimension) behavior previously found in neutral gas systems.     

Derivation of the Boltzmann equation for a Fermi gas

Point scatterers are placed on the real line such that the distances between scatterers are independent identically distributed random variables (stationary renewal process). For a fixed configuration of scatterers a particle performs the following random walk: The particle starts at the pointx with velocityυ, ¦υ¦=1. In between scatterers the particle moves freely. At a scatterer the particle is either transmitted or reflected, both with probability 1/2. For given initial conditions of the particle the velocity autocorrelation function is a random variable on the scatterer configurations. If this variable is averaged over the distribution of scatterers, it decays not faster thant ^?3/2.     

A−2 production in π−p interactions at 3.9 GeV/c

The reaction π^?p → A^?₂p at 3.9 GeV/c incident momentum is studied using data corresponding to the ?^°π^?, ηπ^? and K^δ_sK^? decay modes of the A^?₂. Unnatural parity exchange is found to be important at this energy. The natural parity exchange component of the differential cross section exhibits structure at t′ ≈ GeV².     

Determination of the nuclear vertex constants for the 7Be ↔ 3He4He vertex using theN/D equations and calculation of the astrophysical S factor for the 4He(3He, γ)7Be reaction

N/D equations taking into account the Coulomb interaction effects are used to consider the phase analysis data on ³He⁴He scattering and determine the nuclear vertex constants for the ⁷Be ? ³He⁴He vertex, where the beryllium nucleus is in the ground (3/2^?) or first excited (1/2^?) states. Information about the asymptotic normalization coefficients of the wave functions of the ⁷Be nucleus in the noted states is derived. The data obtained are used to calculate the astrophysical S factor for the ⁴He(³He, γ)⁷Be reaction.     

Long time behavior of structure function of alloy quenched to sufficiently low temperature

A slow asymptotic t^?a′ decay in the structure function of an unstable alloy quenched to a sufficiently low temperature is investigated. We find the exponent a′ = (d + 2)^?1, where d is the dimensionality, which is consistent with recent computer simulations.     

Polarization parameters in the reactions π− p ↑→ηn and π− p ↑→η′n at 40 GeV/c

The polarization parameter in the reactions π^? p→ηn and π^? p→η′n at 40 GeV/c with ?t ranging from 0 to 2.6 (GeV/c)² and 1.0 (GeV/c)², respectively, was measured in experiments using a propanediol polarized target. Only gamma particles from the neutral meson decay were detected. The polarization parameter of the first reaction turns out to be negative in a widet interval (0.05t<1.6 (GeV/c)²) and probably changes its sign at larger values oft. The average value of the polarization parameter of the second reaction is equal to ?17±8)% in the region 0.05t<0.5 (GeV/c)².     

The heat capacity of the layer compounds (CH3CH2CH2NH3)2MnCl4 and (CH3CH2CH2NH3)2CdCl4 from 10 K TO 300 K

The heat capacity of the layer compounds tetrachlorobis (n-propylammonium) manganese II and tetrachlorobis (n-propylammonium) cadmium II, (CH₃CH₂CH₂NH₃)₂MnCl₄ and (CH₃CH₂CH₂NH₃)₂CdCl₄ respectively, has been measured over the temperature range 10 K ?T ? 300 K.Two known structural phase transitions were observed for the Mn compound in this temperature region: at T = 112.8 ± 0.1 K (ΔH_t= 586 ± 2 J mol^?1; ΔS_t = 5.47 ± 0.02 J K^?1mol^?1) and at T =164.3 ± (ΔH_t = 496 ± 7 J mol^?1; ΔS_t =3.29 ± 0.05 J K^?1mol^?1). The lower transition is known to be from a monoclinic structure to a tetragonal structure, while the upper is from the tetragonal phase to an orthorhombic one. From comparison with the results for the corresponding methyl Mn compound it is deduced that the lower transition primarily involves changes in H-bonding while the upper transition involves motion in the propyl chain.A new structural phase transition was observed in the Cd compound at T= 105.5 ± 0.1 K (ΔH_t= 1472.3 ± 0.1 J mol^?1; ΔS_t = 13.956 ± 0.001 J K^?1mol^?1), in addition to two transitions that have been observed previously by other techniques. The higher of these transitions(T = 178.7 ± 0.3 K; ΔH_t = 982 ± 4 J mol^?1 ΔS_t = 6.16 ± 0.02 J K^? mol^?1) is known to be between two orthorhombic structures, while the structural changes at the lower transition (T= 156.8 ± 0.2 K; ΔH_t = 598 ± 5 J mol^?1, ΔS_t = 3.85 ± 0.03 J K^?1 mol^?1) and at the new transition are not known. It is proposed that these two transitions correspond respectively to the tetragonal to orthorhombic and monoclinic to tetragonal transitions in the propyl Mn compounds.In addition to the structural phase transitions (CH₃CH₂CH₂NH₃)₂MnCl₄ magnetically orders at t? 130 K. The magnetic contribution to the heat capacity is deduced from the heat capacity of the corresponding diamagnetic Cd compound and is of the form expected for a quasi 2-dimensional Heisenberg antiferromagnet.     

K−p and pp elastic scattering at 10.1 GeV/c

The differential cross sections for K^?p and $\text{p}$p elastic scattering have been measured over the range of four-momentum transfer squared 0.18<?t<3.3 (GeV/c)². The K^?p data decrease smoothly as a function of ?t, whereas, the $\text{p}\text{p}$ data shows a break at ?t = 0.6 (GeV/c)² followed by a fast drop to ?t ? 1.6 (GeV/c)² where the differential cross section levels off and stays constant out to ?t = 3 (GeV/c)².