P-polarized nonlinear surface polaritons near the surface of a saturable self-defocusing medium: Threshold case

We have obtained the solution of the stationary problem for P-polarized nonlinear surface polaritons (NLSP) near the interface of two media, one of which is optically linear and the other having an optical nonlinearity with saturation and a dielectric function of the form: =_xx=_zz=₀+a|E|²/(1+b|E|²), where E is the amplitude of the electric field. On this basis the NLSP energy and propagation properties, the polarization structure and the localization depth of the NLSP field as well as the physically allowed region in the parameter space have been investigated for the case in which the parameter ₀ is positive, b is non-negative, and a and ₁ are negative (₁ is the dielectric function of the linear medium); ₀>|₁|.     

Kinetic limits of the HPP cellular automaton

We study the Boltzmann-Grad limit in various versions of the two-dimensional HPP cellular automaton. In the completely deterministic case we prove convergence to an evolution that is not of kinetic type, a well-known phenomenon after Uchyiama's paper on the Broadwell gas, whereas the limiting equation becomes of kinetic type in the model with random collisions. The main part of the paper concerns the case where the collisions are deterministic and the randomness comes from inserting, between any two successive HPP updatings, ^- stirring updatings, <1 being any fixed positive number and a parameter which tends to 0. The initial measure is a product measure with average occupation numbers of the order of (low-density limit) and varying on distances of the order of ^–1. The limit as 0 of the system evolved for times of the order of ^-1- corresponds to the Boltzmann-Grad limit. We prove propagation of chaos and that the renormalized average occupation numbers (i.e., divided by) converge to the solution of the Broadwell equation. Convergence is proven at all times for which the solution of the Broadwell equation is bounded.     

Static and dynamic properties of XY systems with extended defects in cubic anisotropic crystallines

Static and dynamic critical behavior ofXY systems in cubic anisotropic crystallines, with extended defects (or quenched nonmagnetic impurities) strongly correlated along _d -dimensional space and randomly distributed ind – _d dimensions, were studied. These extended defects make the systems coordinate anisotropic, resulting in unique critical behavior due to competition between the cubic anisotropy and the coordinate anisotropy. The systems were analyzed by an ^1/2 (4 – d) type of expansion with double expansion parameters based on a renormalization-group (RG) approach. Critical exponents were calculated near the second-order phase transition point and the behavior of the first-order transition was evaluated near the tricritical point.     

Accurate calculation of functional determinants

A dependence of the functional determinant of the operator from the family D() satisfying the condition D()=fD()+D()f on the parameter m² of infrared regularization is found in the regularization method using a generalized function.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 27–32, June, 1988.     

Scattering states and bound states in λ℘(φ)2

By analyzing the Bethe-Salpeter equation for even ()₂ models we show that for weak coupling the mass spectrum is discrete and of finite multiplicity below 2m. Moreover on even states of energy less than 4(m–) we show that theS matrix is unitary. Herem is the physical mass and =()0 as 0. Our results rely essentially only on a simple assumption about the analyticity of the Bethe-Salpeter kernel which has been verified for weak coupling. For the interaction ⁴, (/m _o ² 1) we show that there are no even bound states of energy less than 4(m–).Work supported in part by NSF, Grant MPS 74-13252     

Natural boundaries for area-preserving twist maps

We consider KAM invariant curves for generalizations of the standard map of the form (x, y)=(x+y, y+f(x)), wheref(x) is an odd trigonometric polynomial. We study numerically their analytic properties by a Padé approximant method applied to the function which conjugates the dynamics to a rotation +. In the complex plane, natural boundaries of different shapes are found. In the complex plane the analyticity region appears to be a strip bounded by a natural boundary, whose width tends linearly to 0 as tends to the critical value.     

Spectral behavior of quasi periodic potentials

Spectral properties of Schrödinger operators of the typeH =–+V, where is the Laplacian,V a quasiperiodic potential and a coupling constant, are developed.V is taken to be finite sum of exponentials with generic frequencies. For small a strong stability is shown. On the other hand, examples (in the finite diffeence case) are given, for which a transition in the type of spectrum occurs, as is increased.Research supported in part by grant NSF MCS 7900813     

Retardation of electromagnetic waves by helices of large diameters

This paper is devoted to the study of the wave retardation by helices of large diameters, when the lengthL of one turn of the helix is larger than the vacuum wavelength, i.e.L/>1. The dispersion equation of the tape helix wound up on a dielectric cylinder was derived and the dependence of the phase velocity of slowed-down waves on the diameter of helix was calculated. The theory was verified experimentally at a frequencyf=2·385 GHz. Helices were wound up on polyfoam (=1·04) and plexiglass (=2·55) cylinders of various diameters. For the helix wound up on plexiglass cylinders it was found that the wave retardation depends on the form of conductor from which the helix is made. The retardation of waves is determined by an effective dielectric constant _ef which is equal either ( _i + _e)/2 for the tape helix or ( _{i e}) for the helix made from a wire of the circular cross-section; here _i is the dielectric constant of inner dielectric rod and _e is that of outer space.     

The Statistics of the Trajectory of a Certain Billiard in a Flat Two-Torus

We consider a billiard in the punctured torus obtained by removing a small disk of radius >0 from the flat torus ², with trajectory starting from the center of the puncture. In this case the phase space is given by the range of the velocity only. Let (), and respectively R(), denote the first exit time (length of the trajectory), and respectively the number of collisions with the side cushions when ² is being identified with [0,1)². We prove that the probability measures on [0, ) associated with the random variables and R are weakly convergent as 0⁺ and explicitly compute the densities of the limits. Research partially supported by ANSTI grant C6189/2000.     

One-dimensional harmonic lattice caricature of hydrodynamics

We derive the hydrodynamic (Euler) approximation for the harmonic time evolution of infinite classical oscillator system on one-dimensional lattice ¹ It is known that equilibrium (i.e., time-invariant attractive) states for this model are translationally invariant Gaussian ones, with the mean 0, which satisfy some linear relations involving the interaction quadratic form. The natural parameter characterizing equilibrium states is the spectral density matrix function (SDMF)F(), [– , ). Time evolution of a space profile of local equilibrium parameters is described by a space-time SDMFF(t;x, ) t, xR ¹. The hydrodynamic equation forF(t; x, ) which we derive in this paper means that the normal mode profiles indexed by are moving according to linear laws and are mutually independent. The procedure of deriving the hydrodynamic equation is the following: We fix an initial SDMF profileF(x, ) and a familyP ,>0 of mean 0 states which satisfy the two conditions imposed on the covariance of spins at various lattice points: (a) the covariance at points close to the value ^–1 x in the stateP is approximately described by the SDMFF(x, ); (b) The covariance (on large distances) decreases with distance quickly enough and uniformly in. Given nonzerotR ¹, we consider the states P _–1 ,>0, describing the system at the time moments ^–1 t during its harmonic time evolution. We check that the covariance at lattice points close to ^–1 x in the state P _–1 is approximately described by a SDMFF(t;x, ) and establish the connection betweenF(t; x, ) andF(x,).     

Dielectric properties of NaCl1−xBr and KCl1−xBrx mixed ionic solution

In the present paper a three-body potential model (TBPM) has been employed for the analysis of dielectric behaviour of NaCl-NaBr and and KCl-KBr mixed crystals with varying compositions. The physical properties like dielectric constants ( ₀ and ), optic mode frequencies ( _OLO and _TO), effective charge parameter (e _s ^* ), optic mode Grüneisen parameters and strain derivatives of ₀ and dielectric constants have been calculated. The results achieved in the present study are found in fairly good agreement with the available experimental data. The results obtained by previous investigators are also shown for the sake of comparison.     

О взаимодействии экс итонов с колебаниями кристаллической реш етки

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Asymptotic theory of multidimensional chaos

A delay-differential equationu(t)+u(t)=f(u(t–1)), 0t < , and its generalization are investigated in the limit 0, when the attractor's dimension increases infinitely. It is shown that a number of statistical characteristics are asymptotically independent of. As for the attractor, it can be regarded as a direct product ofO(1/) equivalent subattractors, their statistical characteristics being asymptotically independent of . The results enable one to predict some characteristics of the attractor with fractal dimensionD 1 for the case 1, when they are inaccessible numerically. The approach developed seems to be applicable for a wide class of spatiotemporal systems.     

Determination of electrodynamical parameters of dielectric pipe-shaped materials using millimeter wave cavity

Employment ofE _0np modes of cylindrical cavity resonator for measurement of electrodynamical parameters of dielectric pipe-shaped materials in millimeter wavelength band is suggested. Dielectric permittivity and conductivity of a sample can be found by measuring the frequency shift and theQ-factor change of the resonator.Method of mode basis is used for the calculation of sample permittivity and conductivity . Measurements were carried out at 8mm wavelength and related calculations have demonstrated use and possibilities of the method in studying the dielectric properties of samples of cylindrical and other forms.     

Simulation calculation of dielectric constants: Comparison of methods on an exactly solvable model

A mean spherical model of classical dipoles on a simple cubic lattice of sideM=2N+1 sites is considered. Exact results are obtained for finite systems using periodic boundary conditions with an external dielectric constant and using reaction field boundary conditions with a cutoff radiusR _c N and an external dielectric constant. The dielectric constant in the disordered phase is calculated using a variety of fluctuation formulas commonly implemented in Monte Carlo and molecular dynamics simulations of dipolar systems. The coupling in the system is measured by the parametery=4 ²/9kT, where ² is the fixed mean square value of the dipole moments on the lattice. The system undergoes a phase transition aty2.8, so that very high dielectric constants cannot be obtained in the disordered phase. The results show clearly the effects of system size, cutoff radius, external dielectric constant, and different measuring techniques on a dielectric constant estimate. It is concluded that with periodic boundary conditions, the rate of approach of the dielectric constant estimate to its thermodynamic limit is asN ^–2/3 and depends only weakly on. Methods of implementing reaction field boundary conditions to give rapid convergence to the thermodynamic limit are discussed.     

Self-focusing of laser beams in inhomogeneous dielectrics

This paper presents a theoretical investigation of the propagation and focusing of an electromagnetic wave in an inhomogeneous dielectric whose dielectric constant is given by= ₀(z)+(r, z), whereZ-axis is the direction of propagation of the beam. The quantity (r, z) may either be due to non-linear effects or due to inhomogeneity in the actual fabrication of the guide.Numerical calculations show that for a cylindrically symmetrical Gaussian beam, even a small axial inhomogeneity has appreciable effect on self-focusing; if the real part ofd ₀ (z)/dz is larger than zero the focusing effect increases withz, and vice-versa.Work supported by Institute of Telecomm. Science, Boulder, Coloado and NSF, USA.     

Lattice and continuum wavelets and the block renormalization group

We obtain a resolution of the identity operator, for functions on a latticeZ ^d, which is derived from the block renormalization group. We use eigenfunctions of the terms of the decomposition to form a basis forl ₂(Z^d) and show how the basis is generated from lattice wavelets. The lattice spacing is taken to zero and continuum wavelets are obtained.     

Effective Dynamics for Bloch Electrons: Peierls Substitution and Beyond

We consider an electron moving in a periodic potential and subject to an additional slowly varying external electrostatic potential, (x), and vector potential A(x), with x ^d and 1. We prove that associated to an isolated family of Bloch bands there exists an almost invariant subspace of L ² ( ^d ) and an effective Hamiltonian governing the evolution inside this subspace to all orders in . To leading order the effective Hamiltonian is given through the Peierls substitution. We explicitly compute the first order correction. From a semiclassical analysis of this effective quantum Hamiltonian we establish the first order correction to the standard semiclassical model of solid state physics.     

к ВОпРОсУ ОБ ИжлУЧЕНИИ ВАВИлОВА-ЧЕРЕНкОВА В ОДНООсНых кРИстАллАх

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On the Vavilov-erenkov radiation in uniaxial crystals

An investigation is made of the Vavilov-erenkov radiation energy in uniaxial magnetic crystals and in uniaxial crystals with double anisotropy.

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A limit theorem for stochastic acceleration

We consider the motion of a particle in a weak mean zero random force fieldF, which depends on the position,x(t), and the velocity,v(t)= (t). The equation of motion is (t)=F(x(t),v(t), ), wherex(·) andv(·) take values in ^d ,d3, and ranges over some probability space. We show, under suitable mixing and moment conditions onF, that as 0,v (t)v(t/²) converges weakly to a diffusion Markov processv(t), and ² x (t) converges weakly to , wherex=lim ² x (0).