Exact hierarchical solution of the damped oscillator problem

As a alternative to other exact approaches we present a procedure, in which the eigenvectors of a coupled oscillatory system are calculated in a hierarchically exact (N) manner. In the system a single oscillator is coupled to a dense sequence of background oscillators, which among themselves are not mutually coupled (Oscillatory Fano problem). The eigenvectors beeing known, all decay laws become computable. Two of them are given. It is strongly emphasized thatquite different decay laws evolve fordifferent initial conditions.Dedicated to B. Mühlschlegel on the occasion of his 60th birthday     

Exact solution of the standard transfer problem in a stellar atmosphere

We come back to the analytical solution of the standard transfer problem in a stellar atmosphere. It consists in solving the radiative transfer equation in a homogeneous and isothermal plane-parallel atmosphere, with light scattering taken as isotropic and monochromatic. The literature on the subject is reviewed and the existing solution in a finite slab is improved thanks to the introduction of non classical auxiliary functions. Eleven-figure tables of the solution are given for typical values of the input parameters currently met in stellar atmospheres.     

Exact solution to the averaging problem in cosmology

The exact solution of a two-scale Buchert average of the Einstein equations is derived for an inhomogeneous universe that represents a close approximation to the observed universe. The two scales represent voids, and the bubble walls surrounding them within which clusters of galaxies are located. As described elsewhere [New J. Phys. 9, 377 (2007)10.1088/1367-2630/9/10/377], apparent cosmic acceleration can be recognized as a consequence of quasilocal gravitational energy gradients between observers in bound systems and the volume-average position in freely expanding space. With this interpretation, the new solution presented here replaces the Friedmann solutions, in representing the average evolution of a matter-dominated universe without exotic dark energy, while being observationally viable.     

Exact solution of a modified periodic Anderson model in one dimension

A periodic Anderson model for electron correlation in a localized-electron band mixed with a conduction band is exactly solved in one dimension, by a Bethe Ansatz, in a modified version with interaction only between right- or left-going wavenumbers. The singlet-ground-state energy for the symmetric model is obtained for the half-filled case N/N_a=2.     

X-ray photoelectron spectroscopy: Exact solution to the problem with internal sources

A fundamentally new model of multiple photoelectron scattering in solids is described in this paper. It takes into account the nonuniform depth distribution of photoelectron sources and the anisotropic character of single scattering and yields an exact solution to the problem of multiple elastic scattering. Highly accurate and fast algorithms for calculating X-ray photoelectron spectroscopy signals can be developed on the basis of this model.     

Exact solution of the multichannel Kondo problem,scaling, and integrability

In this work we give the exact solution of the model describing the scattering of conduction electrons by an impurity in the orbital singlet state (so-calledn- channel Kondo problem). Depending on the relation between the impurity spinS and the number of electron scattering channelsn, the model behaves differently at low energies. At 2S the effective charge increases to infinity at low energies, whereas atn > 2S it tends to a finite fixed point. The model under study is the first example of the one-dimensional quantum field theory exhibiting scaling behavior.     

Exact solution of the Boltzmann equation in the homogeneous color conductivity problem

An exact solution of the Boltzmann equation for a binary mixture of colored Maxwell molecules is found. The solution corresponds to a nonequilibrium homogeneous steady state created by a nonconservative external force. Explicit expressions for the moments of the distribution function are obtained. By using information theory, an approximate velocity distribution function is constructed, which is exact in the limits of small and large field strengths. Comparison is made between the exact energy flux and the one obtained from the information theory distribution.     

Exact solution of Kramers' problem for piecewise parabolic potential profiles

This paper presents a general procedure for deducing exact time characteristics of Brownian diffusion in the high damping limit in piecewise parabolic pontential profiles.Our approach to solve this problem is based on the Laplace transform method for obtaining the desired time characteristics. For some concrete examples of piecewise parabolic profiles' exact values of metastable state decay times and bistable system relaxation times are found. These results are valid for any height of the potential barrier and coincide with Kramer's results for a high potential barrier. Conditions for the validity of Kramer's formula are also derived.     

Exact solution of the problem of laser focusing of an atomic beam

The exact solution (Green’s function) of a parabolic equation describing the motion of neutral atoms in the field of a hollow TEM₀₁* mode of laser radiation is found. The positions of the focus and the width of the focal spot are found on the basis of this solution for various configurations of the laser atomic beam. Pis’ma Zh. éksp. Teor. Fiz. 70, No. 10, 654–659 (25 November 1999)     

Exact solution of the problem of the equilibrium configuration of the charged surface of a liquid metal

A broad class of exact solutions is obtained for the problem of the equilibrium configuration of the charged surface of a conducting liquid allowing for capillary forces. An analysis of the solutions showed that when the amplitudes of the perturbations reached certain critical values, the region occupied by the liquid ceases to be singly connected, which corresponds to the formation of liquid metal droplets. It is shown that a steady-state liquid metal profile may exist for which appreciable local amplification of the electric field can be achieved. Zh. éksp. Teor. Fiz. 116, 1990–2005 (December 1999)     

Exact solution to the cauchy problem for a generalized “linear” vectorial Fokker-Planck equation: Algebraic approach

The exact solution to the Cauchy problem for a generalized “linear” vectorial Fokker-Planck equation is found by using the disentangling techniques of Feynman and algebraic (operational) methods.     

Exact L series solution of the Dirac-Coulomb problem for all energies

We obtain exact solution of the Dirac equation with the Coulomb potential as an infinite series of square integrable functions. This solution is for all energies, the discrete as well as the continuous. The spinor basis elements are written in terms of the confluent hypergeometric functions and chosen such that the matrix representation of the Dirac-Coulomb operator is tridiagonal. The wave equation results in a three-term recursion relation for the expansion coefficients of the wavefunction which is solved in terms of the Meixner-Pollaczek polynomials.     

High-density percolation: Exact solution on a Bethe lattice

A new percolation problem is posed where the sites on a lattice are randomly occupied but where only those occupied sites with at least a given numberm of occupied neighbors are included in the clusters. This problem, which has applications in magnetic and other systems, is solved exactly on a Bethe lattice. The classical percolation critical exponents=gg=1 are found. The percolation thresholds vary between the ordinary percolation thresholdp _c (m=1)=l/(z – 1) andp _c(m=z) =[l/(z – 1)]¹/(z–1). The cluster size distribution asymptotically decays exponentially withn, for largen, p p _c .Supported in part by National Science Foundation grant DMR78-10813.