We study spectral properties of the Fokker-Planck operator that describes particles diffusing in a quenched random velocity field. This random operator is non-Hermitian and has eigenvalues occupying a finite area in the complex plane. We calculate the eigenvalue density and averaged one-particle Green's function, compare our results with numerical simulations, and relate them to the time evolution of particle density. For strong disorder and short times, we find a novel time dependence of the mean-square displacement: $〈r^2〉\sim t^{2/d}$ in dimension $d>\mathrm{}2\mathrm{}$.

Single-photon states with arbitrarily fast asymptotic power-law falloff of energy density and photodetection rate are explicitly constructed. This goes beyond the recently discovered tenth-power law of the Hellwarth-Nouchi photon which itself superseded the long-standing seventh-power law of the Amrein photon.

Within an early Universe scenario, nonextensive thermostatistics is investigated on the basis of data concerning primordial helium abundance. We obtain first order corrections to the energy densities and weak interaction rates, and use them to compute the deviation in the primordial helium abundance. After comparing with observational results, we get $|q-1\mathrm{}|<2.08\mathrm{}\times {10}^{-5\mathrm{}}$ as a bound for the nonextensive parameter.

The effect of gravity on a two dimensional fluid, or Ising magnet, confined between opposing walls is analyzed by density matrix renormalization. Gravity restores two phase coexistence up to the bulk critical point, in agreement with mean field calculations. A detailed finite size scaling analysis of the critical point shift is performed. Density matrix renormalization results are very accurate and the technique is promising and best suited to study equilibrium properties of two dimensional classical systems in contact with walls or with free surfaces.

The stability problem for the $O\left(N\right)$ nonlinear sigma model in $2+\epsilon$ dimensions is considered. We present results for the $1/N^2$ order calculations of the critical exponents (in $2<d<\mathrm{}4\mathrm{}$ dimensions) of the composite operators relevant for this problem. Arguments in favor of the scenario with the conventional fixed point are given.

A novel, efficient optimization method for physical problems is presented. The method utilizes the noise inherent in stochastic functions. As an application, an algorithm for the variational optimization of quantum many-body wave functions is derived. The numerical results indicate superior performance when compared to traditional techniques.

We show that a high-frequency field may resonantly decrease the activation energy of escape from a potential well. For systems in spatially periodic potentials, the effect is different for the transitions in opposite directions, which gives rise to resonantly directed diffusion (DD). DD arises in both asymmetric and symmetric periodic potentials. It depends exponentially strongly on the field magnitude, and its direction can be controlled by varying the field spectrum.

A Comment on the Letter by Eric Lewin Altschuler et al., Phys. Rev. Lett. 78, 2681 (1997).

Starting from a quantum Langevin equation of a charged particle in a magnetic field we present a fully dynamical calculation of the orbital diamagnetism, from which the effect of dissipation on Landau diamagnetism can be assessed. The treatment throws light on subtle issues of confined boundaries and the approach to equilibrium of a quantum dissipative system. Additional results are presented for the diamagnetism in a confined parabolic potential.

A general discrete stochastic process involving random amplification with additive external noise is analyzed theoretically and numerically. Necessary and sufficient conditions to realize steady power law fluctuations with divergent variance are clarified. The power law exponent is determined by a statistical property of amplification independent of the external noise. By introducing a nonlinear effect a stretched exponential decay appears in the power law.

Lagrange distributed approximating functionals (LDAFs) are proposed as the basis for a new, collocation-type method for accurately approximating functions and their derivatives both on and off discrete grids. Example applications are presented to illustrate the use of LDAFs for solving the Schrödinger equation and Fokker-Planck equation. LDAFs are constructed by combining the DAF concept with the Lagrange interpolation scheme.

A quantum description for the phase difference between two harmonic oscillators is proposed. Being represented by a Hermitian operator, phase difference displays two striking quantum features. First, its eigenvalues form a new type of quantized value which is countable and dense in a $2\pi$ window. Second, because quantized phase difference cannot be defined as a difference of phases, it obeys a quantum addition rule which may not be commutative.

We report the development of an atom interferometer that uses optical standing waves as phase gratings and operates in the time domain. The observed signal is entirely caused by the wave nature of the atomic center-of-mass motion. The opportunities to measure recoil frequency and gravity acceleration are demonstrated.

X-ray bound states trapped on a curved surface are realized by coupling subangstrom photons at grazing angles into a bent silicon wafer with adjustable radius. At small radii of curvature the usual ray approximation breaks down and new phenomena associated with wave optical behavior of x-ray surface eigenmodes are revealed. For such samples, tunneling is shown to extinguish all but the ground state surface eigenmode, whose presence may be detected after extremely long distances of propagation along the surface.

The Balian-Low theorem is applied to the motion of an electron in the xy plane with a magnetic field B perpendicular to this plane. The energy spectrum for this problem is the Landau levels. It is shown that the eigenfunctions for the Landau levels cannot be chosen sufficiently localized in order to make both uncertainties $\Delta x$ and $\Delta y$ finite. A similar result holds for the coordinates of the orbit center. With this restriction on the localization, complete orthonormal sets are defined on von Neumann lattices.

Representation theory is shown to be incomplete in terms of enumerating all integrable limits of quantum systems. As a consequence, one can find exactly solvable Hamiltonians which have apparently strongly broken symmetry. The number of these hidden symmetries depends upon the realization of the Hamiltonian.

An exact inversion procedure for obtaining the central and spin-orbit potential from phase shifts at fixed energy is described. The method, based on Sabatier interpolation formulas, reduces the nonlinear problem to linear-algebraic equations. We have tested the method with a Woods-Saxon potential with a strong spin-orbit component.

The lowest radiative correction to the Casimir energy density between two parallel plates is calculated using effective field theory. Since the correlators of the electromagnetic field diverge near the plates, the regularized energy density is also divergent. However, the regularized integral of the energy density is finite and varies with the plate separation $L$ as ${1/L}^7$. This apparently paradoxical situation is analyzed in an equivalent but more transparent theory of a massless scalar field in $1+1$ dimensions confined to a line element of length $L$ and satisfying Dirichlet boundary conditions.

The ideal Bose gas in a highly anisotropic harmonic potential is studied. It is found that Bose-Einstein condensation occurs in two distinct steps as the temperature is lowered. In the first step the specific heat shows a sharp feature, but the system still occupies many one-dimensional quantum states. In the second step, at a significantly lower temperature, the ground state becomes macroscopically occupied. It should be possible to verify these predictions using present-day atom traps. The two-step behavior can occur in a rather general class of anisotropic traps, including the box potential.