Continuous sample paths in quantum field theory

Nelson's free Markoff field on ? ^l+1 is a natural generalization of the Ornstein-Uhlenbeck process on ?¹, mapping a class of distributions φ(x,t) on ? ^l ×?¹ to mean zero Gaussian random variables φ with covariance given by the inner product \(\left( {\left( {m^2 - \Delta - \frac{{\partial ^2 }}{{\partial t^2 }}} \right)^{ - 1} \cdot , \cdot } \right)_2 \) . The random variables φ can be considered functions φ〈q〉=∝ φ(x,t)q(x,t)d x dt on a space of functionsq(x,t). In the O.U. case,l=0, the classical Wiener theorem asserts that the underlying measure space can be taken as the space of continuous pathst →q(t). We find analogues of this, in the casesl>0, which assert that the underlying measure space of the random variables φ which have support in a bounded region of ? ^l+1 can be taken as a space of continuous pathst →q(·,t) taking values in certain Soboleff spaces.     

Theory of light scattering from a system of interacting Brownian particles

Starting from a N-particle diffusion equation for a system of N interacting spherical Brownian particles, a non-linear transport equation for concentration fluctuations δc(r, t) of the particles is derived. This dynamic equation is transformed into a hierarchy of equations for retarded propagators of increasing numbers of concentration fluctuations. A cluster expansion to lowest order in the average concentration results in a set of two coupled equations. The spectrum of light scattered by the interacting particles is in general not a Lorentzian, due to the non-linear term in the transport equation. For small scattering wave vectors k the width is D(ω)k², where ω is the transferred frequency. It is shown that D(0) = D_e, the effective diffusion coefficient. For a hardcore interaction potential the spectrum is Lorentzian and it is found that D_e = D₀(1 + φ), where D₀ is the diffusion constant for independent particles and φ the volume concentration of Brownian particles.     

On the temperature behavior of second sound and Poiseuille flow

The linearized Peierls equation for the phonon densityN (k λ,r t) is solved by replacing the collision operator in the subspace orthogonal to the collision invariants byk-dependent relaxation rates. For the normal process relaxation time the behaviorτ _N (k λ)∝|k|^?p for smallk is assumed. Taking into account thisk-dependence ofτ _N explicitly and avoiding an expansion with respect toΩτ _N (kλ) before performing the necessary integration overk yields new, non-analytic, terms in the hydrodynamic equations describing second sound and Poiseuille flow. It is shown that this may lead to a temperature dependence of second sound damping and thermal conductivity in the Poiseuille flow region differing from the usual theoretical predictions and in better agreement with experiments.     

A simple and direct periodic solution of the Korteweg-de vries equations

The solutionq(x, t) of one of the KdV hierarchy is assumed to be a potential in the Schrödinger equation as usual. We differentiate this equation with respect to the time variable and solve it with the aid of the Green function. The obtained equation relatesw _t (x, t, λ)=φ _t (x + c, x, t, λ) withq _t (x, t). The functionφ(x, x ₀,t, λ) obeys the Schrödinger equation and the boundary conditionsφ(x ₀,x ₀,t, λ)=0,φ _x (x ₀,x ₀;t, λ)=1. The shiftingc is equal to the period. We differentiatew _t (x, t, λ) three times with respect to thex coordinate and obtain the time derivative of the Milne equation. The integration of this equation with respect tox allows to solve simply the inverse problem. The reconstructed periodic potential is given by means of the well known formula for the root functions ofw(x, t, λ). The time behaviour of this function, i.e. the solution of the KdV equation, is obtained when one replacesq _t (x, t) by an expression of the KdV hiearchy in the relation betweenq _t (x, t) andw _t (x, t, λ) and transforms it. We estimated also the limit, whenc → ∞, i.e. the possible relation of the periodic solutions with the soliton ones.     

Critical Behavior for Maximal Flows on the Cubic Lattice

Let F ₀ and F _m be the top and bottom faces of the box [0, k]×[0, l]×[0, m] in Z ³. To each edge e in the box, we assign an i.i.d. nonnegative random variable t(e) representing the flow capacity of e. Denote by Φ _k, l, m the maximal flow from F ₀ to F _m in the box. Let p _c denote the critical value for bond percolation on Z ³. It is known that Φ _k, l, m is asymptotically proportional to the area of F ₀ as m, k, l→∞, when the probability that t(e)>0 exceeds p _c , but is of lower order if the probability is strictly less than p _c . Here we consider the critical case where the probability that t(e)>0 is exactly equal to p _c , and prove that $$\mathop {{\text{lim}}}\limits_{k,l,m \to \infty } \frac{1}{{kl}}\Phi _{k,l,m} = 0{\text{ a}}{\text{.s and in }}L_1 $$ The limiting behavior of related to surfaces on Z ³ are also considered in this paper.     

Dispersion induced splitting of the collective mode spectrum in A-phase of superfluid He

The whole collective mode spectrum in A-phase of superfluid ³He with dispersion corrections is calculated. The degeneracy of clapping-modes depends on the direction of the collective mode momentum k with respect to the vector l (mutual orbital moment of Cooper pairs), namely: the mode degeneracy remains the same as in case of zero momentum k for k∥l only. For any other directions there is a three-fold splitting of these modes, which reaches maximum for k⊥l. The obtained results means that new interesting features can be observed in ultrasound experiments in axial-phase: the change of the number of peaks in ultrasound absorption into clapping-mode. Single peak, observed for these modes in axial-phase by Ling et al. [R. Ling, W. Wojtanowski, J. Saunders, E.R. Dobbs, J. Low Temp. Phys. 78 (1990) 187] will split into three peaks under change the ultrasound direction with respect to the vector l.     

Waves in a superlattice with anisotropic inhomogeneities

Dependences of the dispersion laws and damping of waves in an initially sinusoidal superlattice on inhomogeneities with anisotropic correlation properties are studied for the first time. The period of the superlattice is modulated by the random function described by the anisotropic correlation function K_?(r) that has different correlation radii, k _∥ ^?1 and k _⊥ ^?1 , along the axis of the superlattice z and in the plane xy, respectively. The anisotropy of the correlation is characterized by the parameter λ=1?k_⊥/k_∥ that can change from λ=0 to λ=1 when the correlation wave number k⊥ changes from k_⊥=k_∥ (isotropic 3D inhomogeneities) to k_⊥=0 (1D inhomogeneities). The correlation function of the superlattice K(r) is developed. Its decreasing part goes to the asymptote L that divides the correlation volume into two parts, characterized by finite and infinite correlation radii. The dependences of the width of the gap in the spectrum at the boundary of the Brillouin zone δν and the damping of waves ξ on the value of λ are studied. It is shown that decreasing L leads to the decrease of δν, and increase of ξ, with the increase of λ.     

Spectral resonances which become eigenvalues

The stationary Schrödinger equation is ? _x ² φ + λV(x)φ=zφ for φ∈?²(R ⁺,dx). If the potential is bounded below, singular only atx=0, negative on some compact interval and behaves likeV(x)～1/x ^μ asx→∞ with 2≧μ>0, then the system admits shape resonances which continuously become eigenvalues as λ increases. Here λ>0 and for μ=2 a sufficiently large λ is required. Exponential bounds are obtained on Im(z) as λ approaches a threshold. The group velocity near threshold is also estimated.     

Four-wave mixing and hyper-Raman scattering in CuCl

Non-degenerate four-wave mixing using two non-collinear laser beams with frequencies (wavevectors) ω_p, ω_t (k_p, k_t) respectively is studied in CuCl. Two emission lines at frequencies ω⁽¹⁾=2ω_t-ω_p, and ω⁽²⁾=2ω_p-ω_t are observed. Their excitation spectrum is sharply peaked if the phase-match condition k⁽¹⁾=2k_t-k_p is fulfilled. This is the case, if ω_p coincides with the hyper-Raman lines (R⁺_T, R^-_T) of the laser labelled (t) in a well-defined geometrical configuration.     

Exact solutions for a forced Burgers equation with a linear external force

We investigate the solutions of the Burgers equation , where F(x,t) is an external force and Φ(x,t) represents a forcing term. This equation is first analyzed in the absence of the forcing term by taking F(x,t)=k₁(t)−k₂(t)x into account. For this case, the solution obtained extends the usual one present in the Ornstein-Uhlenbeck process and depending on the choice of k₁(t) and k₂(t) it can present a stationary state or an anomalous spreading. Afterwards, the forcing terms Φ(x,t)=Φ₁(t)+Φ₂(t)x and Φ(x,t)=Φ₃x−Φ₄/x³ are incorporated in the previous analysis and exact solutions are obtained for both cases.     

Study of two-phonon excitations in superfluid 4He

We explain the second branch of excitations in superfluid ⁴He observed by Cowley and Woods, by accounting for two-phonon contributions to the dynamic structure function, S(k, ω). Our theory gives a good fit with the experimental data in the high energy region for several values of momentum transfer. It is observed that the contribution to S(k, ω) due to two-phonon excitations is of the order of k² as against its k⁴ dependence reported in earlier theories.     

标定正交晶系粉末照相指数的计算方法及计算程序

本文讨论了标定正交晶系粉末照相指数的计算方法与计算程序。任何一条衍射线(h_i,k_i,l_i)在A-B-C空间可用一平面方程式h_i²A+k_i²B+l_i²C=sin²θ_i来表示。每三个联立方程式的共解代表三个平面在空间的交点。如这一交点能通过一等原子曲面ABC=(ρλ关键词：     

Diffusion in very chaotic hamiltonian systems

We study nonintegrable hamiltonian dynamics: H(I,θ) = H₀(I) + kH₁(I,θ), for large k, that is, far from integrability. An integral representation is given for the conditional probability P(I,θ, t¦I₀, θ₀, t₀) that the system is at I, θ at t, given it was at I₀, θ₀ at t₀. By discretizing time into steps of size ?, we show how to evaluate physical observables for large k, fixed ?. An explicit calculation of a diffusion coefficient in a two degrees of freedom problem is reported. Passage to ? = 0, the original hamiltonian flow, is discussed.     

Calculation of high-order rotational centrifugal-distortion matrix elements for a Hund's case (a) basis set

The derivation of explicit expressions for the Hund's case (a) matrix elements of R^2k is discussed, where R is the mechanical rotational angular momentum operator of the molecule. A recursion relation is developed that permits matrix elements of R^2k to be expressed in terms of those of R^2(k?1), thus affording a straightforward means of calculating the case (a) matrix elements of rotational centrifugal-distortion constants D_v, H_v, L_v, M_v, etc., to an arbitrarily high order. The explicit matrix elements of L_v are listed.     

The Rotational Spectrum of SiHF3 in the Vibrational State υ6=2: Observation of Direct l-Type Resonance Transitions

The υ₆=2 vibrational state of the main isotopomer of trifluorosilane, ²⁸SiHF₃, has been investigated in the centimeter- and millimeter-wave ranges. Rotational spectra following the Δ J=1, Δk=Δ l=0 selection rule have been measured up to J=24 and K=23 and for both values of ∣l∣. Two types of direct l-type resonance transitions induced by the (Δ l=Δk=±2) interaction could be observed by means of waveguide Fourier transform microwave spectroscopy in the range 8-26 GHz: 252 transitions following the Δ J=0, Δl=Δk=±2 selection rule covering values of J=7-39 and G=∣k-l∣ from 1 to 18 and 90 transitions following the Δ J=0, Δl=Δ k=±4 selection rule covering values of J=17-52 and G from 1 to 4. Due to the strong (2,2) resonance, 18 A₁-A₂ splittings of the k=l=±2 states from J=36-53 could also be observed. Accidental near-degeneracies lead to strong perturbations due to Δ (k-l)=±3 interactions, enabling the observation of perturbation-allowed transitions with selection rules k=±3(l=±2)↔±4(±2), ±2(±2), A₊↔ ±1(?2), A₋ and ± 1(0)↔± 6(±2). In a multiple-fit analysis the experimental data have been refined using five reduced forms of the effective Hamiltonian as proposed by Sarka and Harder [J. Mol. Spectrosc.197, 254-261 (1999)]. Parameters up to seventh order have been determined including the axial rotational constant C for both values of ∣l∣ and the vibrational separation of the ∣l∣=0 and 2 states. The unitary equivalence of the determined parameter sets has been demonstrated up to fifth order. Differences of the rotational constants in the various parameter sets have been explained by the theory of reduction. Sign relations of the fitted parameters and general features of the direct l-type resonance spectrum in a υ_t=2 level are discussed.     

An extension of the fourier convolution theorem in terms of poisson brackets

The Fourier convolution theorem is extended to cover nonstationary and inhomogeneous phenomena. The Fourier transforms of input and transfer functions, F and K, are assumed to be slowly varying functions of x and t. The first-order corrections to the usual convolution theorem are given by Poisson brackets of F and K. These are calculated over k, ω, x and t. The method is applied to study induced currents in a plasma.     

Lagrangian formulation of the Schrödinger equation

We recast the Schrödinger equation in a new Lagrangian formulation. The equation is —i?dψ (x,t)/dt = Lψ (x,t), whereL is the Lagrangian operator. Expressions forL and ford/dt — ⊥ are derived in terms of coordinate and momentum operators.     

Collapse of vortex lines in hydrodynamics

A new mechanism is proposed for collapse in hydrodynamics associated with the “breaking” of vortex lines. The collapse results in the formation of point singularities of the vorticity field, i.e., a generalized momentum curl. At the point of collapse the vorticity |Ω| increases as ((t ₀ ? t)^?1 and its spatial distribution for t → t ₀ approaches quasi-two-dimensional: in the “soft” direction contraction obeys the law l ₁ → (t ₀ ? t)^3/2 whereas in the other two “hard” directions it obeys l ₂ → (t ₀ ? t)^1/2. It has been shown that this collapse scenario takes place in the general case for three-dimensional integrable hydrodynamics with the Hamiltonian ? = ∫|Ω| d r.     

Stimulated Raman scattering of an extraordinary mode in a solid state plasma

This paper presents an investigation of stimulated Raman scattering of an extraordinary mode in a solid state plasma, n In Sb. As the pump wave (w₀, k₀) propagates in the semiconductor the electrons acquire an oscillatory drift velocity and the magnetic field of the pump interacts with a low frequency perturbation (w_l, k_l) to give rise to high frequency side bands (w_l ± w₀, k_l ± k₀). The side band (w_l − w₀, k_l − k₀) interacts with the pump to produce a low frequency ponderomotive force responsible for driving the original density perturbation. The expressions for the growth rate and threshold for the instability have been obtained. For typical plasma parameters of n In Sb and laser radiation of frequency 1.778 × 10¹⁴s⁻¹, the growth rate turns out to be ~ 10¹¹s⁻¹ for the scattering angle θ = 0°. The growth rate is found to reduce with increasing values of scattering angle. A magnetic field enhances the growth rate and tends to reduce the threshold for the instability. The present investigation may be used to obtain useful information about the nature of elementary excitations in solid state plasmas, and the estimate of the growth rate may help in diagnostics and in the characterization of semiconductors.     

Localization and transmission coefficient for two coupled metal chains with disorder

The equation for distribution of probabilities for the transmission coefficient T has been obtained for an electron passing through the finite section of the length L, consisting of two coupled disordered chains. The behavior of the mean 〈lnT〉 = - L/l_loc suggests localization of the electron on the length l, which depends on the coupling energy t between the chains. The ratio l_loc(t =)/l_loc(t?v_F/l) is found to be equal to 1 ? 1/π.