? expansions in semiclassical theories for systems with smooth potentials and discrete symmetries

We extend a theory of first order ? corrections to Gutzwiller’s trace formula for systems with a smooth potential to systems with discrete symmetries and, as an example, apply the method to the two-dimensional hydrogen atom in a uniform magnetic field. We exploit the C_4v-symmetry of the system in the calculation of the correction terms. The numerical results for the semiclassical values will be compared with values extracted from exact quantum mechanical calculations. The comparison shows an excellent agreement and demonstrates the power of the ? expansion method.     

Statistical analysis of discrete control systems with varying parameters using the method of orthogonal expansions. II

The determination of the mathematical expectation and correlation function of the output signal of discrete automatic control systems with varying parameters is considered. The statistical characteristics of the output signal are determined by means of an expansion in a certain specially formulated orthonormalized system of functions. The computing algorithm may be realized on a digital computer.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 15, No. 3, pp. 438–447, March, 1972.     

Analysis of discrete automatic control systems with varying parameters using the method of orthogonal expansions. I

An analysis of discrete automatic control systems with varying parameters is considered when the coefficients of the difference equation of the system satisfy specific conditions. The analysis is carried out by expanding the output signal in a specially formulated orthonormalized system of functions. The computation algorithm may be realized on a digital computer.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 15, No. 3, pp. 427–437, March, 1972.     

Calculation of the coupling constants of many-electron atoms with external fields using expansions in a discrete basis of sturmian-type virtual orbitals

Based on a generalization of the method of Sturmian expansions to the case of many-electron systems, a fundamentally new approach was developed for the construction of a complete set of intermediate states in many-body perturbation theory thereby avoiding integration over states of the continuous spectrum. A complete system of eigenfunctions of the generalized eigenvalue problem for the one-electron Hartree-Fock equation with a factor before the energy operator and fixed values of the Lagrange multipliers is used as a basis of single-particle virtual orbitals for an initial approximation to the eigenfunctions of the Hartree-Fock self-consistent field method. In the case of bound states, the spectrum of the effective operator is purely discrete, thus permitting one to completely eliminate integration over continuum virtual orbitals in applying methods of many-body perturbation theory to atomic and molecular calculations for a number of cases. As a consequence, one manages to avoid a number of the technical difficulties associated with inclusion of such states. The possibilities of the developed approach are illustrated in application to the investigation of the interaction of many-electron atoms with external fields.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 73–85, August, 1990.     

Multiscale Thermodynamics

Multiscale thermodynamics is a theory of the relations among the levels of investigation of complex systems. It includes the classical equilibrium thermodynamics as a special case, but it is applicable to both static and time evolving processes in externally and internally driven macroscopic systems that are far from equilibrium and are investigated at the microscopic, mesoscopic, and macroscopic levels. In this paper we formulate multiscale thermodynamics, explain its origin, and illustrate it in mesoscopic dynamics that combines levels.     

Resonance expansions in quantum mechanics

The goal of this contribution is to discuss various resonance expansions that have been proposed in the literature. Presented at the 3rd International Workshop “Pseudo-Hermitian Hamiltonians in Quantum Physics”, Istanbul, Turkey, June 20–22, 2005.     

Generalized Newman-Unti expansions

Newman-Unti type expansions for the metric, the spin coefficients and the Riemann tensor components are exhibited for spacetimes which are not necessarily vacuum and in a frame which is not necessarily a Bondi frame. These expansions are given for both the physical spacetime and for the corresponding conformally rescaled (“unphysical”) space.     

Multiscale SOC in turbulent convection

Using data obtained in a laboratory thermal convection experiment at high Rayleigh numbers, it is shown that the multiscaling properties of the observed mean wind reversals are quantitatively consistent with analogous multiscaling properties of the Bak–Tang–Wiesenfeld prototype model of self-organized criticality in two dimensions.     

Multiscale cosmological dynamics

The recently developed mean field theory of relativistic gravitation predicts the emergence of an “apparent matter” field at large scales describing the net effect of small-scale fluctuations on the large-scale dynamics of the universe. It is found that this so-called back reaction effect is much stronger for gravitational waves than for matter density fluctuations. At large scales, gravitational waves behave like radiation and, for them, the perturbative effect scales as the squared relative amplitude times squared frequency. In particular, a bath of gravitational waves of relative amplitude 10⁻⁵ and frequency would not be directly detectable by today’s technology but would generate an effective large-scale radiation of amplitude comparable to the unperturbed matter density of the universe.     

Asymptotic expansions in nonlinear regression theory

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

Mixing of operators in Wilson expansions

The tree-loop theorem of 't Hooft and Veltman is used in Fermi type gauges to show that gauge invariant operators in Wilson expansions mix in general with non-gauge invariant operators. Background field gauges are proposed in which this does not occur. Calculations of anomalous dimensions in these gauges are discussed.     

Numerical singularities in many-body moments expansions

It is well known that connected moments expansions when used to approximate the ground-state energy of Hamiltonian systems may suffer from extraneous singularities in various regions of parameter space. In this brief Letter a numerical investigation is presented to study an approximation scheme developed by Ullah to avoid the singularities found in moments expansions. Specifically, we study the single-impurity Wolff model as well as the 32-site Anderson lattice.     

Generalized Wilson expansions in dimensional renormalization

Operator product expansions in the framework of dimensional regularization and renormalization are discussed. Following the definition of a subtraction operator for dimensionally regularized and points-split Green's functions, a generalized Wilson expansion pansion is proved. The terms of the expansion are normal products defined via dimensional renormalization, and the coefficients are doubly regularized with singularities in the physical dimension as the spacial separations of the product fields vanish, or at zero separations as the dimension of space-time becomes physical.     

Variational localized-site cluster expansions

The simplest localized-site cluster expansion for a system of doublet sites described within the context of a valence bond or Heisenberg model is considered. Exact matrix element formulae are obtained for a variety of systems, including some with fused rings. Numerical results for the linear chain case suggest that this simple ansatz improves over the common resonance theory presumption of Kekulé structures only.     

Convergence of Mayer expansions

The tree graph bound of Battle and Federbush is extended and used to provide a simple criterion for the convergence of (iterated) Mayer expansions. As an application estimates on the radius of convergence of the Mayer expansion for the two-dimensional Yukawa gas (nonstable interaction) are obtained.     

Separable expansions of thet-matrix

A recent result of Osborn on the non-convergence of separable expansions of the t-matrix $\text{(}\text{p}\text{,}\text{p}\text{′;z)}$ is discussed. It is shown that the theoretical difficulties arise from large values of $\text{p}$ and $\text{p}\text{′}$, and cause no difficulties in practical situations.     

Convergent expansions for tunneling

A new method to compute effects of tunneling in one-dimensional multiple well is developed. A tunneling parameter built with physical quantities is introduced to measure the splitting between eigenvalues due to tunneling. These splittings are given by convergent series in term of this tunneling parameter for a wide class of double well.