Results concerning the rigorous justification of the effective Hamiltonians for band electrons in the presence of weak homogeneous electric and magnetic fields are reviewed. In the electric-field case the existence, in the sense of spectral concentration, of the Stark-Wannier resonances is proved. In the magnetic-field case, the existence of exponentially localized magnetic Wannier functions is established. As a consequence the Peierls-Onsager effective Hamiltonian is obtained.

For more than 40 years it was thought that polaron- and exciton-phonon systems exhibited unexpected localization properties. Particular attention was paid to the so-called phonon-induced self-trapping transition, which, it was believed, should manifest itself as a point of nonanalyticity in the ground-state energy as a function of the electron-phonon coupling parameter. It will be demonstrated for a large class of (generalized Fröhlich) models that no such transition exists. The dimensionality of space has no qualitative influence; insofar, an application of the authors' results to problems in lower dimensions (e.g., polarons in quantum wells) is straightforward. The same holds true if homogeneous external fields are involved; for example, a discontinuous mass stripping for magnetopolarons can be excluded. On the other hand, a phase-transition-like behavior will be found, if a polaron or exciton is exposed to a short-range potential, allowing a so-called pinning transition. The authors emphasize, however, that even in this case the transition is only modified, and not induced, by phonons.

In the more than half century since the semiclassical Thomas-Fermi theory of the atom was introduced, there have been literally thousands of publications based on that theory; they encompass a broad range of atomic bound-state and scattering problems. (The theory has also been applied to nuclear physics and solid-state problems.) We will concentrate here on the essence of the theory, namely, its implementation of the uncertainty and exclusion principles and of the Coulomb or Newton force law. Since we are often far more interested in physical concepts than in numerical accuracy or rigor, we will sometimes consider the implementation in a qualitative rather than quantitative fashion. The theory is then capable of giving only qualitative information about a system—one obtains the dependence of the total ground-state binding energy $E$ and radius $R$ of an atom on the nuclear charge $Z$, for example, but one obtains only rough estimates of the numerical coefficients; in compensation, the calculations are often literally trivial, very much simpler than the already simple Thomas-Fermi calculations. A point to be emphasized is that in the course of obtaining an estimate of $E$ and $R$ of an atom in a Thomas-Fermi approach, one also obtains an estimate of the electronic density, and, particularly if the analysis is more than simply qualitative, an electronic-density estimate can be very useful in a wide variety of problems. We include a short comment on alternative formulations of Thomas-Fermi theory in a $D$-dimensional space. We will review the applications of the theory, from both qualitative and (Thomas-Fermi) quantitative viewpoints, to heavy atoms, where we are concerned with a Coulomb interaction, and to neutron stars and white dwarfs, where we are concerned with a gravitational interaction and with gravitational-plus-Coulomb interactions, respectively. In the latter case, the first two Coulomb corrections are evaluated. Very rough (relativistic) estimates are made of the conditions under which heavy atoms, neutron stars, and white dwarfs collapse. A one-dimensional Thomas-Fermi-like theory also exists for heavy atoms in a uniform strong magnetic field $B$, of the order of the field believed to exist at the surface of a neutron star. Here, too, the qualitative picture immediately gives some of the main results, namely, the dependence of $E$ and $R$ on $B$ and $Z$. We also comment briefly on some relatively recent and very recent developments in Thomas-Fermi theory. These include a proof of the stability of matter. Though it was first proved by Dyson and Lenard, we consider the Lieb-Thirring proof, both because it is much simpler and because it makes extensive use of Thomas-Fermi theory, including a no-molecular-binding theorem that follows in the Thomas-Fermi approximation: Teller proved that, in that approximation, atoms could not form molecular bound states. These developments also include (a) the Lieb-Simon proof that the prediction of the theory that $E=-c_7{Z}^{\frac{7}{3}}$, with $c_7$ a specified coefficient, becomes exact at $Z\sim \infty$, (b) the Scott-$c_6{Z}^{\frac{6}{3}}$ correction term, with $c_6$ specified and now known also to be exact, and (c) the Schwinger estimate of the coefficient $c_5$ of the $Z^{\frac{5}{3}}$ term, which there is good reason to believe is exact. The many digressions include comments on QED, on lower bounds on the ground-state energy of a system, and on mini-boson stars.

This article investigates in a systematic way the properties of the classical continuous mean-field theory governed by the generalized Poisson-Boltzmann-Emden equation $\rho \mathrm{}\left(x\right)=A\exp [-\beta \int {\Lambda }^{}\mathrm{dy}\rho \mathrm{}\left(y\right)V(x-y)\mathrm{}]$ together with the associated variational problem ${infop}_{\rho }\frac{1}{2}\int {\Lambda }^{}\mathrm{dx}\int {\Lambda }^{}\mathrm{dy}\rho \mathrm{}\left(x\right)\mathrm{}\rho \mathrm{}\left(y\right)V(x-y)\mathrm{}+kT\int {\Lambda }^{}\mathrm{dx}\rho \mathrm{}\left(x\right)\mathrm{}\ln \rho \mathrm{}\left(x\right)\mathrm{}$. Origins of the theory are traced back. Past studies (freezing theories, electrostatic and self-gravitating systems) are relocated in a broader framework. New results concerning the thermodynamic limit, phase transitions, metastability, and the shape of density profiles are provided. In particular, the question of ground states (in relationship to condensation and wetting phenomena) is illustrated by numerous explicit solutions.

The spin-½ antiferromagnetic Heisenberg model on a square lattice is used to describe the dynamics of the spin degrees of freedom of undoped copper oxides. Even though the model lacks an exact solution, a solid, accurate, and rather conventional picture emerges from a number of techniques—analytical (spin-wave theory, Schwinger boson mean-field theory, renormalization-group calculations), semianalytical (variational theory, series expansions), and numerical (quantum Monte Carlo, exact diagonalization, etc.). At zero temperature, the effect of the zero-point fluctuations is not strong enough to destroy the antiferromagnetic long-range order, despite the fact that we are dealing with a low-spin low-dimensional system. The corrections to the spin-wave theory, which treats perturbatively the effect of such fluctuations around the classical Néel ground state, appear to be small. At any nonzero temperature the order disappears and the correlation length at low temperature $T(k_B\frac{T}{J}\ll 1$, where $J$ is the antiferromagnetic coupling) follows the singular form $\xi \mathrm{}\left(T\right)=C\exp \left(\alpha \frac{J}{k_B}T\right)$. In the long-wavelength limit and at low $T$, the model has the same behavior as the quantum nonlinear $\sigma$ model in two spatial dimensions and one Euclidean time dimension, which we also study with available analytical and Monte Carlo techniques. The quasiparticles of the theory are bosons; at low $T$ and for wavelengths shorter than the correlation length they are welldefined spin-wave excitations. The spectrum of such excitations and the temperature-dependent correlation length have been determined by neutron and Raman scattering experiments done on ${\mathrm{La}}_2$Cu${\mathrm{O}}_4$. The good agreement of the experimental data with the predictions of this theory suggests that the magnetic state of the undoped materials is the conventional ordered state. We discuss, within a simple mean-field theory, the effect of weak three-dimensional antiferromagnetic coupling and the role of an antisymmetric term, introduced to explain a hidden ferromagnetic behavior of the uniform susceptibility. We find that understanding the copper-oxide antiferromagnetic insulator is only the first essential step towards the development of a theory of the superconductor created upon doping such materials.

This review covers recent experimental and theoretical investigations into cooperative phenomena in crystals containing off-center ions. These phenomena have attracted much attention in recent years because of a general interest in disordered systems, in particular in spin glasses, whose electrical analog is the dipole glass. Specific features of the dipole glass state in alkali halide crystals with off-center ions are discussed and compared with spin glasses. Experimental studies performed in recent years have demonstrated that off-center ions in highly polarizable crystals can at certain concentrations induce ferroelectric domains with regions of macroscopic spontaneous polarization. The physical causes of this phenomenon are examined and some physical properties of crystals exhibiting such an impurity-induced phase transition are analyzed. Primary emphasis is placed on the range of low impurity concentrations, where system properties differ substantially from predictions of mean-field theory.

In the first half of this article, theoretical treatments of the infrared divergence involved in the edge problem of soft-x-ray absorption, emission, and photoemission spectra of simple metals are reviewed historically. In the second half, recent developments in the work of the present authors using the Fermi golden rule are described to show that the method permits an analytical treatment and provides exact results for various aspects of the edge problem.

In this review, a general algorithm for constructing coherent states of dynamical groups for a given quantum physical system is presented. The result is that, for a given dynamical group, the coherent states are isomorphic to a coset space of group geometrical space. Thus the topological and algebraic structure of the coherent states as well as the associated dynamical system can be extensively discussed. In addition, a quantum-mechanical phase-space representation is constructed via the coherent-state theory. Several useful methods for employing the coherent states to study the physical phenomena of quantum-dynamic systems, such as the path integral, variational principle, classical limit, and thermodynamic limit of quantum mechanics, are described.

Supernovae of Type II occur at the end of the evolution of massive stars. The phenomenon begins when the iron core of the star exceeds a Chandrasekhar mass. The collapse of that core under gravity is well understood and takes a fraction of a second. To understand the phenomenon, a detailed knowledge of the equation of state at the relevant densities and temperatures is required. After collapse, the shock wave moves outward, but probably does not succeed in expelling the mass of the star. The most likely mechanism to do so is the absorption of neutrinos from the core by the material at medium distances. Observations and theory connected with SN 1987A are discussed, as are the conditions just before collapse and the emission of neutrinos by the collapsed core.

The author reviews some of the important successes achieved by Eliashberg theory in describing the observed superconducting properties of many conventional superconductors. Functional derivative techniques are found to help greatly in understanding the observed deviations from BCS laws. Approximate analytic formulas with simple correction factors for strong-coupling corrections embodied in the single parameter $\frac{T_c}{{\omega }_{\ln }}$ are also found to be very helpful. Here $T_c$ is the critical temperature and ${\omega }_{\ln }$ is an average boson energy mediating the pairing potential in Eliashberg theory. In view of the discovery of high-$T_c$ superconductivity in the copper oxides, results in the very strong coupling limit of $\frac{T_c}{{\omega }_{\ln }}\sim 1$ are also considered, as is the asymptotic limit when $\frac{T_c}{{\omega }_{\ln }}\to \infty$. This case is of theoretical interest only, but it is nevertheless important because simple analytic results apply that give insight into the more realistic strong-coupling regime. A discussion more specific to the oxides is included in which it is concluded that some high-energy boson-exchange mechanism must be operative, with, possibly, some important phonon contribution in some cases. A more definitive application of boson-exchange models to the oxides awaits better experimental results.

This is a study of simple kinetic models of open systems, in the sense of systems that can exchange conserved particles with their environment. The system is assumed to be one dimensional and situated between two particle reservoirs. Such a system is readily driven far from equilibrium if the chemical potentials of the reservoirs differ appreciably. The openness of the system modifies the spatial boundary conditions on the single-particle Liouville—von Neumann equation, leading to a non-Hermitian Liouville operator. If the open-system boundary conditions are time reversible, exponentially growing (unphysical) solutions are introduced into the time dependence of the density matrix. This problem is avoided by applying time-irreversible boundary conditions to the Wigner distribution function. These boundary conditions model the external environment as ideal particle reservoirs with properties analogous to those of a blackbody. This time-irreversible model may be numerically evaluated in a discrete approximation and has been applied to the study of a resonant-tunneling semiconductor diode. The physical and mathematical properties of the irreversible kinetic model, in both its discrete and its continuum formulations, are examined in detail. The model demonstrates the distinction in kinetic theory between commutator superoperators, which may become non-Hermitian to describe irreversible behavior, and anticommutator superoperators, which remain Hermitian and are used to evaluate physical observables.

This review contains a comprehensive examination of all modern measurements and calculations of the temperature-dependent electrical resistivity $\rho \mathrm{}\left(T\right)\mathrm{}$ for the alkali metals—and especially potassium (K)—from their melting points down to below 0.1 K. The simplicity of the electronic structures of these metals makes them unique for testing our fundamental understanding of $\rho \mathrm{}\left(T\right)\mathrm{}$. At all temperatures down to a few K, $\rho \mathrm{}\left(T\right)\mathrm{}$ is dominated by its electron-phonon scattering component, ${\rho }_{\mathrm{ep}}\mathrm{}\left(T\right)\mathrm{}$. Current quantitative understanding of ${\rho }_{\mathrm{ep}}\mathrm{}\left(T\right)\mathrm{}$ in the alkali metals is examined in detail, including effects of phonon drag at temperatures below ≃ 10 K. In the vicinity of 1 K, $\rho \mathrm{}\left(T\right)\mathrm{}$ in pure, unperturbed, bulk alkali metals is predicted to be dominated by electron-electron scattering. ${\rho }_{\mathrm{ee}}\mathrm{}\left(T\right)=A_{\mathrm{ee}}{T}^2$. In disagreement with previous reviews, the authors argue that $A_{\mathrm{ee}}$ is nearly constant for each alkali metal and—at least for K—also in quantitative agreement with calculation. Below 1 K, alloys based on K and lithium display both previously predicted and completely unexpected effects. Perturbations such as deformation and thinning of K wires induce unusual and interesting behaviors. An unexpected Kondo-like effect appears when K contacts polyethylene. Charge-density-wave-based predictions of contributions to $\rho \mathrm{}\left(T\right)\mathrm{}$ in the alkali metals are also considered. Three appendices examine (a) what is involved in a realistic calculation of $\rho \mathrm{}\left(T\right)\mathrm{}$; (b) the experimental problems encountered in high-precision measurements of $\rho \mathrm{}\left(T\right)\mathrm{}$ at low temperatures and how they are solved; and (c) the most recent experimental data concerning charge-density waves in the alkali metals.

The interaction of resonant radiation with atoms or molecules present in a discharge can induce variation in the electrical properties of the discharge. This effect, known as the optogalvanic effect, has been shown to be a powerful and inexpensive technique for the investigation of atomic and molecular species. Many theoretical and experimental articles have been published in the last ten years on this subject. In this review the different types of optogalvanic techniques that have been developed are reviewed and the principal theoretical models are analyzed. The spectroscopic applications are examined. An extensive reference list is included, providing an overview of the literature.

The intensity interferometry technique, commonly referred to as the Hanbury-Brown/Twiss effect, has been applied to nuclear and elementary-particle collisions as a method of investigating their space-time evolution. In this review the theoretical framework of the technique is presented, describing the formulations in common use. A survey is made of its application to subatomic collisions, ranging from high-energy elementary-particle reactions to low-energy nuclear reactions. Results derived from experimental data analysis are compiled and discussed. A critique is made of the interpretational difficulties associated with the use of the technique in reaction studies.