Eigenfunction expansions in the imaginary Lobachevsky space

Eigenfunctions of the Schrödinger equation with the Coulomb potential in the imaginary Lobachevsky space are studied in two coordinate systems admitting solutions in terms of hypergeometric functions. Normalization and coefficients of mutual expansions for some sets of solutions are found.     

Eigenfunction expansions for Schrödinger and Dirac operators with singular potentials

The spectral theory (eigenfunction expansion theorem) is developed for Schrödinger and Dirac operators with classes of potentials includingg exp(?μ∣x∣)/∣x∣² in the Schrödinger case andg exp(?μ∣x∣)/∣x∣ in the Dirac case.     

Cumulants in perturbation expansions for non-equilibrium field theory

The formulation of perturbation expansions for a quantum field theory of strongly interacting systems in a general non-equilibrium state is discussed. Non-vanishing initial correlations are included in the formulation of the perturbation expansion in terms of cumulants. The cumulants are shown to be the suitable candidate for summing up the perturbation expansion. Also a linked-cluster theorem for the perturbation series with cumulants is presented. Finally, a generating functional of the perturbation series with initial correlations is studied. We apply the methods to a simple model of a fermion-boson system.     

On negative norm states in supersymmetric theories

We study the effective kinetic energy of scalar fields for two classes fo supersymmetric theories. In theories with very large VEVs of scalar fields, as proposed by Witten, the use of the renormalization group improved effective action prevents the appearance of negative norm states. For simpler theories a general criterium for the absence of negative norm states is given, which is violated in a model withO(N)—symmetry proposed recently.     

Asymptotic expansions in nonlinear regression theory

Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 20–25, September, 1995.     

Casimir effect for gauge fields in spaces with negative constant curvature

We consider gauge theories based on abelian p-forms on real compact hyperbolic spaces. Using the zeta-function regularization method and the trace tensor kernel formula, we determine explicitly an expression for the vacuum energy (Casimir energy) corresponding to skew-symmetric tensor fields. It is shown that the topological component of the Casimir energy for co-exact forms on even-dimensional spaces, associated with the trivial character, is always negative. We infer on the possible cosmological consequences of this result.Received: 14 September 2004, Revised: 14 November 2004, Published online: 23 December 2004PACS: 04.70.Dy, 11.25.Mj     

A note on possibility of existence of the stationary state scattering theory for the density operators

The aim of this paper is an attempt to construct the scattering theory in terms of density operators. The wave operators in the space of trace class linear operators are defined. It is shown that the stationary state theory may be formulated in this space for the H-bounded perturbations.     

Necessary stationary condition for polarized radiation in scattering media

The stationary condition is derived taking into account the polarization of radiation in the general case of a scattering inhomogeneous medium in an arbitrary-shape emitter. The necessary stationary condition for an emitter in which radiation is emitted and extinguished simultaneously is complete extinction of the entire emitted radiation. Radiation extinction as a result of absorption by the medium and the emergence of radiation from the emitter is analyzed. The stationary condition is an analytical form of writing that extinction of radiation is a sure event whose probability is equal to unity. The passage of radiation through the medium is described on the basis of the linear transport theory with the help of the matrices of the Green functions. The stationary condition includes the characteristics of polarized radiation extinction of which is analyzed, the absorption coefficients of the medium, and the elements of the matrices of the Green functions, which are determined by optical and geometrical parameters of the emitter. The stationary condition obtained is used for deriving the relations between the components of scalar intensity observed in an arbitrary region of the emitter. These relations include, in addition to the absorption coefficients and the matrix elements of the Green functions, the powers of the primary radiation. Possible applications of the stationary condition and the relations between intensity components in computations and experimental studies are considered.     

Convergent perturbation expansions for Euclidean quantum field theory

Mayer perturbation theory is designed to provide computable convergent expansions which permit calculation of Greens functions in Euclidean quantum field theory to arbitrary accuracy, including nonper-turbative contributions from large field fluctuations. Here we describe the expansions at the example of 3-dimensional ⁴-theory (in continuous space). They are not essentially more complicated than standard perturbation theory. Then ^th order term is expressed in terms ofO(n)-dimensional integrals, and is of order ^k if 4k–3n4k.Dedicated to the memory of Kurt Symanzik     

A field expansions method for scattering by periodic multilayered media

The interaction of acoustic and electromagnetic waves with periodic structures plays an important role in a wide range of problems of scientific and technological interest. This contribution focuses upon the robust and high-order numerical simulation of a model for the interaction of pressure waves generated within the earth incident upon layers of sediment near the surface. Herein described is a boundary perturbation method for the numerical simulation of scattering returns from irregularly shaped periodic layered media. The method requires only the discretization of the layer interfaces (so that the number of unknowns is an order of magnitude smaller than finite difference and finite element simulations), while it avoids not only the need for specialized quadrature rules but also the dense linear systems characteristic of boundary integral/element methods. The approach is a generalization to multiple layers of Bruno and Reitich's "Method of Field Expansions" for dielectric structures with two layers. By simply considering the entire structure simultaneously, rather than solving in individual layers separately, the full field can be recovered in time proportional to the number of interfaces. As with the original field expansions method, this approach is extremely efficient and spectrally accurate.     

High-temperature expansions in statistical theory of atomic ordering

In the present article a high-temperature series for the statistical sum of a binary alloy is considered. The study is carried out within a statistical ensemble, corresponding to fixing atomic concentration in certain chosen groups of lattice sites. Diagrammatic technique for calculating the statistical sum of the ensemble indicated is introduced.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 104–108, February, 1982.     

Numerical simulation of stationary negative corona in air

A negative corona in air is numerically simulated for the coaxial cylindrical geometry of electrodes. The goal of the investigation is to gain insight into physical processes and physical conditions in the gap that favor the corona-arc transition. An experiment aimed at contrasting calculated and real characteristics is described.     

Light scattering from a fluid with a stationary temperature gradient

The spectrum of light scattered from a system in which a stationary temperature gradient is maintained is calculated on the basis of fluctuating hydrodynamics. Explicit expressions are obtained for the frequency dependence of the intensity of the modified Brillouin lines.     

Light scattering from a fluid with a stationary temperature gradient

The spectrum of light, scattered from a fluid with a stationary temperature gradient, is calculated on the basis of fluctuating hydrodynamics. Explicit expressions are obtained for the spectrum of the scattered light which is no longer symmetric around the frequency of the incident light. In particular the difference in height and intensity of the Brillouin lines is given. Furthermore the shift in the position of the maximum of the Rayleigh line is calculated.     

Eigenstate expansions in a soluble model of identical particle scattering

Solutions to the Schrödinger equation and the inhomogeneous equation for the case of two identical particles interacting with a center of force are studied. Eigenstate expansions for solving each equation are explicitly introduced and their properties discussed. The case when the interparticle interaction v₁₂ is zero is then examined; this is a completely soluble problem. The eigenstate expansion solutions for the Schrödinger and inhomogeneous equations are used to explore the means by which the correct solution is obtained. Finally, approximate solutions, obtained by truncating the eigenfunction expansions, are introduced. It is seen that both methods lead to the correct amplitude when τ₁₂ = 0, even though the approximate solution to the inhomogeneous equation does not lead, in the end, to an antisymmetric solution.     

The CCGM theory applied to the scattering of atoms by a one-dimensional stationary sinusoidal hard wall

The CCGM theory of elastic scattering of atoms by surfaces was proposed by Cabrera, Celli, Goodman and Manson (Surface Sci. 19 (1970) 67) and the first calculations with CCGM were made by Goodman (Surface Sci. 19 (1970) 93) and Goodman and Tan (J. Chem. Phys. 59 (1973) 1805). Although clearly the forerunner of much of the more recent work in this field, CCGM has not been more widely used because it is generally thought to have several deficiencies: for example, it is thought that it cannot be meaningfully used in the context of the nowpopular stationary sinusoidal hard wall model of the scattering; furthermore, the completeness of the basis set of states in the present form of the CCGM theory has been questioned recently. We show how CCGM may, in fact, be successfully used in the context of that model, and we compare our results with corresponding “exact” results. We conclude that CCGM works well provided that neither the dimensionless wavenumber of the scattering atom nor the dimensionless corrugation amplitude of the surface is too large.