Reaction functions for weakly bound systems

We propose a new technique to analyze total reaction cross sections. In this technique, which has been previously applied to fusion reactions, the experimental data are used to build a dimensionless reaction function, which does not depend on the system size or details of the optical potential. In this way, total reaction cross sections for different systems can be directly compared. We employ this technique to perform a systematic study of reaction cross sections of weakly bound systems in different mass ranges, and compare their reaction functions with the ones of tightly bound systems with similar masses. We show that breakup reactions and neutron transfers in halo systems lead to large reaction functions, well above the ones of typical tightly or weakly bound stable systems.     

A convolution theorem for weakly non-stationary and weakly inhomogeneous systems

It is shown how one can generalize the convolution theorem to the case where the input signal varies on two time-scales - a fast and a slow one - as well as being weakly space-dependent, and where the transfer function is weakly non-stationary as well as weakly space-dependent.     

Metrizability of spaces of homomorphisms between metric vector spaces

This note tries to give an answer to the following question: Is there a sufficiently rich class of metric vector spaces such that sufficiently large spaces of continuous linear maps between them are metrizable?     

Evolution of weakly correlated systems

A propagator with an effective action is constructed for a quantum many-particle system in the approximation taking into account only pair correlations between particles. The theory can easily be generalized to the case of two species of particles. In the case of a rarefied plasma and a gas, this propagator for a one-particle density matrix provides a solution to the kinetic problem over time intervals longer than the relaxation time.     

Superconductivity in weakly correlated electron systems

We study the influence of the short-ranged Hubbard correlation U between the conduction electrons on the Cooper pair formation in normal (s-wave) superconductors. The Coulomb correlation is considered within the standard second order perturbation theory, which becomes exact in the weak coupling limit but goes beyond the simple Hartree-Fock treatment by yielding a finite lifetime of the quasiparticles at finite temperature. An attractive pairing interaction V, which may be mediated by the standard electron-phonon mechanism, is considered between nearest neighbor sites. A critical value for the attractive interaction is required to obtain a superconducting state. For finite temperature a gapless superconductivity is obtained due to the finite lifetime of the quasiparticles, i.e. the Coulomb correlation has a pair-breaking influence. The energy gap and depend very sensitively on U, V and band filling n and develop a maximum away from half filling as function of n. The ratio varies with n, being higher than the BCS value near half filling and reaching the BCS value for lower n. Received 17 February 1999     

Kramers time in weakly nonpotential systems

We consider a bistable Fokker-Planck system with a known stationary distribution and a small nonpotential part in the drift force. We perform a perturbation calculation of its Kramers time, _K, and compare it with the corresponding time, _K ⁽⁰⁾ , for the potential system which has the same stationary distribution. We show that _K/ _K ⁽⁰⁾ depends only on the properties of the drift force close to the saddle-point.The authors would like to dedicate this work to their colleagues Y. Orlov, R. Nazarian, and V. Brailovski.     

Yano vector fields and conformally related spaces

We discuss some geometrical properties of Killing forms of order three and exhibit a procedure for constructing spaces admitting such tensors. As an example we prove that all conformally flat spaces admitting such a tensor are related to spaces of constant curvature.On leave April 1979 from the Service de Mécanique et Relativité Générales, Université de l'Etat à Mons, Belgium.     

Quantum electrodynamics of weakly bound systems

An overview of quantum electrodynamic effects in two-body systems is presented. Recent advances in the calculation of the hydrogen Lamb shift, muonium hyperfine structure, and positronium energy levels are described in detail. The comparison of experimental results to current theoretical predictions gives agreement in most cases. However, a few significant discrepancies remain, which indicates the necessity for further refined calculations and measurements. This revised version was published online in July 2006 with corrections to the Cover Date.     

Localized chaotic patterns in weakly dissipative systems

A generalized parametrically driven damped nonlinear Schrödinger equation is used to describe, close to the resonance, the dynamics of weakly dissipative systems, like a harmonically coupled pendula chain or an easy-plane magnetic wire. The combined effects of parametric forcing, spatial coupling, and dissipation allows for the existence of stable non-trivial uniform states as well as homogeneous pattern states. The latter can be regular or chaotic. A new family of localized states that connect asymptotically a non-trivial uniform state with a spatio-temporal chaotic pattern is numerically found. We discuss the parameter range, where these localized structures exist. This article is dedicated to Prof. Helmut R. Brand on the occasion of his 60th birthday.     

Approaching universality in weakly bound three-body systems

Atom-dimer scattering below the three-body breakup threshold is studied for a system of three identical bosons. The atom-dimer scattering length and the energy of the most weakly bound three-body state are shown to be strongly correlated. An appropriate rescaling of the observables reveals the subtlety of the correlation and serves to identify universal trends in the unitary limit of divergent two-body scattering length. The correlation provides a new quantitative measure of the degree of universality in three-body systems with short-ranged interactions, as well as a consistency check of effective field theories and other theoretical models.     

Semilinear boundary value problems for degenerate pseudodifferential operators in spaces of Sobolev type

Boundary value problems for degenerate semilinear elliptic pseudodifferential operators are considered. Using the apparatus of the theory of pseudodifferential operators, function spaces introduced in [4–8], and the Rabinowitz construction [12] based on the Borsuk theorem (see [11]), we prove the existence of solutions of the problem in suitable function spaces. To the memory of Leonid Romanovich Volevich     

Self-consistent theory of localization in weakly disordered systems

An analytic treatment of localization in a weakly disordered system is presented for the case where the real lattice is approximated by a Cayley tree. Contrary to a recent assertion we find that the mobility edge moves inwards into the band as disorder increases from zero.     

Chaotic and Arnold stripes in weakly chaotic Hamiltonian systems

The dynamics in weakly chaotic Hamiltonian systems strongly depends on initial conditions (ICs) and little can be affirmed about generic behaviors. Using two distinct Hamiltonian systems, namely one particle in an open rectangular billiard and four particles globally coupled on a discrete lattice, we show that in these models, the transition from integrable motion to weak chaos emerges via chaotic stripes as the nonlinear parameter is increased. The stripes represent intervals of initial conditions which generate chaotic trajectories and increase with the nonlinear parameter of the system. In the billiard case, the initial conditions are the injection angles. For higher-dimensional systems and small nonlinearities, the chaotic stripes are the initial condition inside which Arnold diffusion occurs.     

On extrema of functionals on vector spaces

A functional on a vector space is constructed which does not attain a minimum at the origin although its first derivative vanishes and its second derivative is strictly positive definite at 0.     

Integration by parts formulas and rotationally invariant Sobolev calculus on free loop spaces

We give formulas for integration by parts over the path space and over the loop space of a manifold. We define Sobolev spaces and an Ornstein-Uhlenbeck operator on the loop space. We find some functionals which belong to all the Sobolev spaces.     

Spinor two-point functions in maximally symmetric spaces

The two-point function for spinors on maximally symmetric four-dimensional spaces is obtained in terms of intrinsic geometric objects. In the massless case, Weyl spinors in anti de Sitter space can not satisfy boundary conditions appropriate to the supersymmetric models. This is because these boundary conditions break chiral symmetry, which is proven by showing that the order parameter for a massless Dirac spinor is nonzero. We also give a coordinate-independent formula for the bispinor introduced by Breitenlohner and Freedman [1], and establish the precise connection between our results and those of Burges, Davis, Freedman and Gibbons [2].     

Vector two-point functions in maximally symmetric spaces

We obtain massive and massless vector two-point functions in maximally symmetric spaces (and vacua) of any number of dimensions. These include de Sitter space and anti-de Sitter space, and their Euclidean analogsS ⁿ andH ⁿ. Our method is based on a simple way of constructing every possible maximally symmetric bitensorT _a...bc...d(x, x) which carries tangent-space indicesa...b atx andc...d atx.     

Relaxation properties of weakly coupled classical systems

The relaxation properties of a small classical system weakly coupled to a large classical system which acts as a heat bath are described using a generalized Fokker-Planck equation. The Fokker-Planck equation is derived in general using a modification of the elimination of fast variables techniques previously described. The specific example in which the small system is a harmonic oscillator linearly coupled to the heat bath is treated in detail and it is demonstrated that there is a dynamic frequency shift as well as a statistical shift of the oscillator frequency.     

Exchange interaction of weakly delocalized multielectron systems

The polar model of a metal developed by Bogolyubov [1] is extended to the case of a group of equivalent electrons on each of the atoms that form a crystal. A procedure is suggested for treating the second-quantized Hamiltonian that includes the interconfigurational interaction with the help of the double tensor operators W _q . The relationship between these operators and the unitary operators ^a , which act separately in the space of the spin and orbital functions, is pointed out. The conditions for the applicability of the Heisenberg model are analyzed.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 80–87, May, 1973.In conclusion I wish to thank B. V. Karpenko for the statement of this problem and for numerous consultations and discussions.