The spectrum and the solutions of the one-dimensional Schrödinger equation with a quasiperiodic potential V are investigated numerically. V is taken as the sum of two incommensurate periodic δ-function series. Quasiperiodicity, in contrast to periodicity, makes possible the existence of localized modes. The typical spectra consist of bands corresponding to extended solutions and points corresponding to localized ones. Extended solutions (three-frequency quasiperiodic) are found from the initial-value problem. Localized solutions (exponential decay on both sides) cannot be found in this way owing to numerical instability. Instead, they are obtained from a boundary-value method. An alternative method to find the point spectrum for bounded potentials is proved to be also valid for δ-function potentials, as used here. Connections with the magnetohydrodynamic spectrum of toroidal plasmas are pointed out.

Development of solidification processes in rapidly supercooled one-component plasmas is studied by a Monte Carlo (MC) simulation method with a number of particles 1458. Evolutions of interparticle correlations and of local and extended bond-orientational orders are investigated microscopically. Layered structures emerge at prenucleation stages in the system and expedite a subsequent evolution into metastable states. The metastable states are bcc monocrystalline states with an admixture of a few defects in the form of intralayer interstitials, which differ significantly from the glassy states obtained in the previous simulations with a smaller number of particles. The role of MC periodic boundary conditions in the freezing transitions is discussed in conjunction with the number of MC particles.

For the description of the low-pressure Hg-Ar positive column a self-consistent collisional-radiative model has been used. Calculations of the population densities for the fundamental and the first five excited Hg levels, electron and rare-gas temperature, and densities have been carried out. All these quantities, except electron temperature, have been evaluated as functions of the radial position in the axially uniform positive column; no assumptions on the radial profiles have been made. Macroscopic quantities such as electric-field strength and radiative power output have been also calculated. A diffusion-controlled positive column has been assumed. The most important collisional processes in the plasma have been taken into account (elastic collisions, electron-atom and atom-atom inelastic collisions including excitation, direct and stepwise ionizations, and also superlastic collisions). Argon excitation and stepwise ionization have been included. Radiation trapping has been taken into account, including pressure broadening of resonance lines due to Ar. In our calculations a non-Maxwellian electron-energy distribution function has been used. Diffusion of the charged particles in the presence of ${\mathrm{Hg}}^{+}$ and ${\mathrm{Ar}}^{+}$ has been considered. Diffusion of the metastable Hg levels has been taken into account. The results obtained in this paper have been compared to corresponding values given in the literature and detailed discussion of the validity of several models has been carried out.

It is shown that a large-amplitude plasma wave is resonantly excited by the round-trip propagation of a short laser pulse in a plasma. The scheme results in a resonator of plasma waves that is useful for accelerating a charged particle beam. This method can be used in order to avoid wave saturation due to detuning resulting from large-amplitude nonlinear oscillation.

Beam-breakup (BBU) suppression due to BBU mode detuning is analyzed. The BBU growth equation with linear detuning is solved by the WKB approximation and saddle-point calculations, and an asymptotic formula for BBU growth amplitude is evaluated, which shows substantial saturation in BBU growth for appropriate detuning. Theoretical results are confirmed by comparison with numerical BBU simulations.

For free-electron laser (FEL) operating in the exponential regime before saturation, we present an analytic description of the effect on the gain of longitudinal velocity variations arising from wigger field errors. The average gain reduction and the width of the output power distribution are expressed in terms of the mean-square average of the ponderomotive phase shift per gain length. A scheme for correcting the electron trajectory using position monitors and dipole correctors is analyzed. Our work is directly applicable to the design of FEL amplifiers, and the results are encouraging for their feasibility.

A general formula of the energy-transfer rate of the transverse-electric-mode gyrotron, cyclotron-autoresonance-maser, and nonwiggler-free-electron-laser oscillators is rigorously derived within the Vlasov framework. The present result can be used to deal simultaneously with the influences of the finite cavity length, spreads, higher harmonics, and nonresonance, and to describe the situation of a large-orbit or a polyaxial relativistic electron beam. The general formula is examined for a cold electron beam and is found to be in agreement with the result obtained by the Madey theorem for these devices.

We have observed the spatial distribution of coherent or resonance transition radiation (RTR) in the soft-x-ray region of the spectrum (1–3 keV). Resonance transition radiators were constructed and tested at two accelerators using electron-beam energies ranging from 50 to 228 MeV. These radiators emitted soft x rays in a circularly symmetrical annulus with a half-angle divergence of 2.5–9.0 mrad. The angle of peak emission was found to increase with electron-beam energy, in contrast to the incoherent case, for which the angle of emission varied inversely with electron-beam energy. By careful selection of foil thickness and spacing, one may design radiators whose angle of emission varies over a range of charged-particle energies. A particular RTR mode (r=m=1) was found to give a sharp annular ring that becomes more accentuated as the number of foils is increased. The RTR effect has application in particle detection, beam diagnostics, x-ray source brightness enhancement, and x-ray free-electron-laser emission.

For near-integrable Hamiltonian systems with a nonintegrable piece of order ε≪1, we use Lie transforms to derive a generalized form of Madey’s theorem. Specifically, we find an expression for the average second-order (in ε) change of any function of momentum in terms of first-order quantities only. A formalism is given that makes gain calculations for devices like free-electron lasers and gyrotrons in complicated geometries tractable. An explicit expression is presented for the case where the nonintegrable part of the Hamiltonian is a harmonic function of the coordinates. As an example, the average change in particle kinetic energy is computed through second order in the field amplitude for gyrotrons in complicated geometries. The transform method is extended to non-Hamiltonian systems, and it is shown that there is a class of non-Hamiltonian differential equations to which Madey’s theorem applies.

We consider the parametrically driven discrete ${\mathrm{\phi }}^4$ model with loss. Using an anaytical approach and direct numerical simulations, which turn out to be in excellent agreement, we demonstrate that there is a range of driving parameters such that kinks cannot exist in the system. This effect may be understood with the help of an averaged external potential energy of the system, which has no double-well structure in this region.

We report on the experimental realization and theoretical analysis of four-level systems produced in passive optical ring cavities by introducing various intracavity elements. Birefringent elements couple waves with different polarizations and thus lift the polarization degeneracy of the cavity modes. Reflecting elements couple waves propagating in different directions and lift the propagation degeneracy. A combination of these elements leads, in general, to a splitting of every longitudinal cavity mode into a four-mode system. Under the restriction that the optical components are loss-free, the description of these ring cavities in a transmission-matrix formalism leads in a natural way to the study of the Lie group U(2,2) and its Lie algebra u(2,2). We associate each of the 16 generators of this algebra with a specific type of optical element, some of which are standard components, others not. From the commutation relations of the generators, we derive a recipe for the construction of the nonstandard components as a sequence of standard ones. It follows that the entire U(2,2) group can actually be realized. The number of independent parameters—16 for a general U(2,2) element—is shown to be reduced substantially if the optical components are selected out of a restricted number of types, provided that the corresponding generators define a subalgebra of u(2,2). In such cases, a subgroup of U(2,2) is realized and the number of independent parameters of the optical system is given by the number of generators of the subalgebra. A connection is established between the subalgebras and symmetry properties of the optical components in the cavity. We consider the influence of symmetries on the mode structure and consider our experimental results from this viewpoint. In particular, we discuss time-reversal invariance, leading to the symplectic group USp(2,2), and we identify antiunitary symmetries leading to Kramers’s degeneracy in the mode spectrum. We propose the application of the group U(2,2) in optics as a tool yielding direct practical results.

Statistical dynamical properties of the Frenkel-Kontorova model in an external field are studied with main emphasis put on kink-pair nucleation and kink movement at low temperatures. Our method of constraint(s) together with the transition state theory make it possible to determine the N-dimensional saddle-point configuration and to obtain velocity for kink movement, rate for kink-pair nucleation, and consequently velocity for chain propagation. Generalization of the method used in this paper is also presented.

The layered feedforward neural network is extended to a q-state Potts-glass model. The Potts-glass version of the network is realized by imposing local inhibition on a group of Ising spins and introducing competitive updating rules on them. The dynamics of such a system is solved exactly, and the storage capacity of the network is found to be proportional to ${\mathit{q}}^{\mathrm{\Delta }}$, with Δ≊1.85 in the case of storing unbiased patterns. For biased patterns, we obtain the phase diagram for q=3 as a function of the storage capacity and the bias parameters, which indicates that the storage capacity decreases with the bias.

We ‘‘derive’’ the traditional descriptions of Brownian motion from a microscopic process without introducing statistics. The macroscopic erratic motion is shown to be a consequence of a filtered process that is microscopically chaotic and yields a Langevin equation for the particle velocity and a corresponding Fokker-Planck equation in the appropriate parameter regime.

We point out that in certain classes of sandpile models, there exist conserved quantities, modulo the largest period of the system, under the toppling rule. In the deterministic case, explicit construction of such conserved quantities allows the exact solution of the models. With random seeding, the conservation laws also provide nontrivial informtion about the system.

We observe growth oscillations in our Monte Carlo simulations of kinetic antipercolation that have no analog in regular percolation. A mean-field theory that explains the existence of these oscillations is presented. The kinetic critical exponent obtained in our simulations is close to the corresponding exponent for regular percolation. This suggests that kinetic antipercolation may be in the same universality class as kinetic percolation.

For a differential-delay equation the time dependence of the variable is a function of the variable at a previous time. We consider a differential-delay equation with Gaussian nonlinearity that displays intermittent chaos. Although not the first example of a differential-delay equation that displays such behavior, for this example the intermittency is classified as type III, and the origin of the intermittent chaos may be qualitatively understood from the limiting forms of the equation for large and small variable magnitudes.

Recently, Stein [Phys. Rev. A 43, 768 (1991)] extended a method, originally given by Jeffreys, to calculate the stability boundaries of the conductive state in binary fluid mixtures. A part of this paper deals with a system we considered about two years ago [Phys. Rev. A 40, 7175 (1989)]. Stein gave a computer-assisted calculation of the onset of oscillatory convection for no-slip boundary conditions. This Brief Report presents a simple no-slip Galerkin approximation to support our previous statements on the role of no-slip boundary conditions and a comparison with Stein’s results. Beyond that simple scaling arguments will be given that relate the no-slip results to those of the free-slip case.

We report measurements of osmotic compressibility in a quasibinary liquid mixture of 3-methylpyridine, water, and heavy water. The measurements probed the lower consolute point (${\mathit{T}}_{\mathit{L}}$) for samples of varying loop sizes (ΔT). An exact doubling of the critical exponent (γ) was observed over the entire experimental temperature range (${\mathit{T}}_{\mathit{L}}$-T) for a sample with ΔT as small as 250 mK. For intermediate ΔT ’s (22≥ΔT≥1 °C), a smooth crossover from the double (2γ) to the single exponent (γ) was noticed, as t [=‖(${\mathit{T}}_{\mathit{L}}$-T)/${\mathit{T}}_{\mathit{L}}$‖]→0. The universal value of γ (=1.24) was recovered for experimen- tal paths terminating in ${\mathit{T}}_{\mathit{L}}$ for any ΔT, provided the field variable used was ${\mathit{t}}_{\mathit{U}\mathit{L}}$ [=‖(${\mathit{T}}_{\mathit{U}}$-T)(${\mathit{T}}_{\mathit{L}}$-T)/${\mathit{T}}_{\mathit{U}}$${\mathit{T}}_{\mathit{L}}$‖] instead of t. This field variable (${\mathit{t}}_{\mathit{U}\mathit{L}}$) follows naturally from the geometrical picture as well as the Landau-Ginzburg theory, as applied to the reentrant phase transitions.

Energy functionals of the momentum-space density are investigated. They are complementary supplements to the standard functionals of the configuration-space density and provide a self-consistent approach to computing momentum-space densities and quantities derived from these. We introduce the concept of an effective kinetic energy, analogous to the effective potential energy, and establish thereby an independent-particle picture in momentum space. In the new independent-particle energy, every electron pair is counted 5/3 times, not twice. The Thomas-Fermi functionals in momentum space are given and shown to be perfectly equivalent to the configuration-space ones.