The purpose of the present article is to establish the properties of a class of radially confined relativistic-electron-beam equilibria for longitudinal wiggler free-electron-laser applications. The theoretical model is based on the steady-state ($\frac{\partial }{\partial t}=0\mathrm{}$) Vlasov equation, assuming a thin, tenuous electron beam with $k_0^2{R}_b^2\ll 1$ and negligibly small equilibrium self fields. For the approximate magnetic field configuration, $B_r^0=0\mathrm{}$ and $B_z^0=B_0[1+(\frac{\delta B}{B_0}\left)sin\right(k_{0}z\left)\right]$, the single-particle constants of the motion are the axial momentum $p_z$, perpendicular momentum $p_{\perp }={(p_r^2+p_{\theta }^2)}^{\frac{1}{2}}$, energy $\gamma {\mathrm{mc}}^2={(m^2{c}^4+c^2{p}_z^2+c^2{p}_{\perp }^2)}^{\frac{1}{2}}$, and canonical angular momentum $P_{\theta }=r[p_{\theta }-(\frac{e}{c}\left)A_{\theta }^0\mathrm{}\right(r,z\left)\right]$, where $A_{\theta }^0\mathrm{}(r,z)=\left(\frac{r{B}_0}{2}\right)[1+(\frac{\delta B}{B_0}\left)sin\right(k_{0}z\left)\right]$. Beam equilibrium properties are investigated for the class of self-consistent Vlasov equilibria $f_b^0(\overrightarrow{\mathrm{x}},\overrightarrow{\mathrm{p}})=F(p_{\perp }^2-{2\gamma }_{b}m{\omega }_b{P}_{\theta })G\left(p_z\right)$, where ${\omega }_b=\mathrm{const}$ is related to the mean rotation of the electron beam, and ${\gamma }_b{\mathrm{mc}}^2=\mathrm{const}$ is the characteristic energy of a beam electron. Specific examples of sharpboundary equilibria and diffuse equilibria are analyzed in detail, including the $r-z$ modulation of the density profile $n_b^0\mathrm{}(r,z)\mathrm{}$ by the longitudinal wiggler field.

We consider open reactive systems driven far from equilibrium sustaining a hard-mode instability. We derive the analytic form of the coefficients in the Ginzburg-Landau potential functional by first reducing the problem to Poincaré normal form and then restricting the equations to the locally attractive manifold of states emerging beyond the instability. We conclude that the Hopf transversality conditions for bifurcation are not only sufficient but also necessary for the application of the Ginzburg-Landau equation. The Poincaré normal form reduction yields additional restrictions besides the Hopf hypotheses.

A stochastic approach to Brownian motion of a quantum system is presented. The theory is based on physical considerations of conservation of probability and detailed balance of transitions at finite temperature.

We present a simplified derivation of the Yamada-Kawasaki formula for the nonlinear adiabatic response of the stress tensor to planar Couette flow. This formally exact expression is then used to prove the validity of two nonequilibrium molecular-dynamics algorithms that have been used to study fluids undergoing planar Couette flow, very far from equilibrium.

The nonlinear properties of long-wavelength ion acoustic surface modes in a semi-infinite plasma are studied. It is shown that finite-amplitude surface-charge layers can propagate along the plasma boundary with a velocity that exceeds the ion sound speed by a small term which is proportional to the squared inverse width of the solitary layer.

It is shown that electron inertia removes the ideal magnetohydrodynamic Alfvén-wave continuum, leaving discrete modes which are damped.

Using a method introduced by Zwanzig, we study the power spectrum of a system described by a simple nonlinear Fokker-Planck equation. The high-frequency region is analyzed, and the asymptotic behavior is established.

A momentum representation of the Dirac relativistic wave functions for a hydrogenlike atom is obtained in a simple form by means of Fourier transformation. The expressions given are convenient for analytic as well as numerical applications.

We report a revised interpretation of the photoionization cross section of C1 I between the lowest ionization threshold ${}_{}{}^3{}_{}{}^{}P$ and the ${}_{}{}^1{}_{}{}^{}D$ threshold. The two series of narrow resonances which have been observed in the recent measurement by Ruščić and Berkowitz are interpreted as the $\left({}_{}{}^1{}_{}{}^{}D\right)\mathrm{ns}{}_{}{}^2{}_{}{}^{}D$ and $\left({}_{}{}^1{}_{}{}^{}D\right)\mathrm{nd}{}_{}{}^2{}_{}{}^{}S$ series, where the latter is allowed (and narrow) due to a (small) admixture of $\left({}_{}{}^1{}_{}{}^{}D\right)\mathrm{nd}{}_{}{}^2{}_{}{}^{}P$ into the ${}_{}{}^2{}_{}{}^{}S$ wave function. The broad resonances are identified as a superposition of $\left({}_{}{}^1{}_{}{}^{}D\right)\mathrm{nd}{}_{}{}^2{}_{}{}^{}P$ and $\left({}_{}{}^1{}_{}{}^{}D\right)\mathrm{nd}{}_{}{}^2{}_{}{}^{}D$.

In a recent paper by Compagno and Persico a factorization for three-operator expectation values was used. In connection with this factorization the result obtained in that paper is discussed for the relative variance of the population-inversion operator in the pure superfluorescence, and we propose a new approximation procedure.

In relation to Seke's Comment, some remarks are presented concerning the qualitative agreement between the results obtained by the simple decoupling technique used by the authors and previous numerical calculation.

Relative intensities for different fine-structure components of the two-photon transitions $2p {}_{}{}^3{}_{}{}^{}P\to 3p {}_{}{}^3{}_{}{}^{}P$ in oxygen and $2p^3{}_{}{}^4{}_{}{}^{}S_{}^o{}_{}{}^{}\to 2p^{2}3p^4{D}^o$ in nitrogen had been measured and found to agree well with calculations involving a single virtual intermediate level. They are in direct conflict, however, with a two-photn selection rule $\Delta J\ne 1$ derived in an earlier but erroneous purely theoretical treatment of these transitions. Five other experiments are also briefly examined, with the conclusion that relative fine-structure intensities in two-photon transitions are well understood as straightforward extensions of angular momentum coupling in single-photon cases, in accordance with allowed $\Delta J=0\mathrm{}$, ±1, and ±2 transitions.

Continued-fraction representations of the Rayleigh-Schrödinger perturbation series for several levels $E_K\left(\lambda \right)$ of the quartic anharmonic oscillator are calculated accurately to large order. A general formula for the asymptotic behavior of the continued-fraction coefficients $c_n^K$ is given. This behavior is then related to (1) the corresponding behavior of the Rayleigh-Schrödinger coefficients ${E_K}^{\mathrm{}\left(n\right)\mathrm{}}$, and (2) the known asymptotic limit $E_K\left(\lambda \right)\sim F_K^{\mathrm{}\left(0\right)\mathrm{}}{\lambda }^{\frac{1}{3}}$, $\left|\lambda \right|\to \infty$.

The central problem of perturbational approaches is that of convergence. For example, although offering an elegant method for computing atomic properties, atomic $\frac{1}{Z}$ expansions are plagued with poor convergence for low $Z$. Because of computational difficulties, analytic continuation of perturbation expansions by the fundamental procedure of reexpanding the series about a shifted origin has not been previously employed. We show here that analytic continuation of atomic $\frac{1}{Z}$ or other perturbation expansions via such shifted-origin series can be simply and accurately implemented in a novel manner by means of the flexible large-order perturbational-variational Rayleigh-Ritz formalism, combined with the recently introduced method for obtaining good estimates of radii of convergence. We illustrate our procedure by applying it to the calculation of the radial limit of the $1s^2{}_{}{}^1{}_{}{}^{}S$ state of the helium isoelectronic sequence via shifted-origin $\frac{1}{Z}$ expansions, thus achieving, for the first time, perturbational convergence to high accuracy for the difficult case of ${\mathrm{H}}^{-}$.

The dielectronic recombination cross section for the Ca II ion is calculated for the process in which the $4s$ electron is excited to the $4p$ state as the projectile electron is captured to a high Rydberg state ($\mathrm{nl}$). The cross section, averaged over a bin size of 0.3 eV, exhibits a peak of 1.8 × ${10}^{-17\mathrm{}}$ ${\mathrm{cm}}^2$, while a cutoff for $n<~80$ reduces this to 2.3 × ${10}^{-18\mathrm{}}$ ${\mathrm{cm}}^2$. The recent crossed-beam experiment gives a cross section of 1.8 × ${10}^{-17\mathrm{}}$ ${\mathrm{cm}}^2$ which is at least seven times larger than the theoretical value.

The effects of core relaxation on the shape-resonant $K$-shell photoionization cross section in ${\mathrm{N}}_2$ are shown to be very significant. Cross relaxation produces substantial broadening and shifting of the shape resonance to higher energy. The results suggest that the frozen-core approximation may not be an appropriate model for $K$-shell photoionization involving a low-energy shape resonance.