Algebraic integrability and generalized symmetries of dynamical systems

It has been shown that when an n-dimensional dynamical system admits a generalized symmetry vector field which involves a divergence-free Liouville vector field, then it possesses n−1 independent first integrals (i.e., it is algebraically integrable). Furthermore, the Liouville vector field can be employed for the classification of algebraically integrable dynamical systems. The results have been discussed on examples which arise in physics.     

Symmetries,integrating factors and Nambu mechanics

We show that if an n-dimensional autonomous dynamical system (DS) with a vector field (VF) which has constant divergence possesses n − 1 independent first integrals, then it admits a symmetry VF which involves Nambu mechanics (NM). If the DS is conservative, then the Nambu VF happens to be a symmetry VF of the DS. We also show that the integrating factors can be constructed via NM. We illustrate our results on the Lotka-Volterra DS.     

Generalized Landau-Lifshitz models on the interval

We study the classical generalized gln Landau-Lifshitz (L-L) model with special boundary conditions that preserve integrability. We explicitly derive the first non-trivial local integral of motion, which corresponds to the boundary Hamiltonian for the sl₂ L-L model. Novel expressions of the modified Lax pairs associated to the integrals of motion are also extracted. The relevant equations of motion with the corresponding boundary conditions are determined. Dynamical integrable boundary conditions are also examined within this spirit. Then the generalized isotropic and anisotropic gln Landau-Lifshitz models are considered, and novel expressions of the boundary Hamiltonians and the relevant equations of motion and boundary conditions are derived.     

Reduction method for dimensionally regulatedone-loop <Emphasis Type="Italic">N</Emphasis>-point Feynman integrals

We present a systematic method for reducing an arbitrary one-loop N-point massless Feynman integral with generic 4-dimensional momenta to a set comprised of eight fundamental scalar integrals: six box integrals in D = 6, a triangle integral in D = 4, and a general two-point integral in D space-time dimensions. All the divergences present in the original integral are contained in the general two-point integral and associated coefficients. The problem of vanishing of the kinematic determinants has been solved in an elegant and transparent manner. Being derived with no restrictions regarding the external momenta, the method is completely general and applicable for arbitrary kinematics. In particular, it applies to the integrals in which the set of external momenta contains subsets comprised of two or more collinear momenta, which are unavoidable when calculating one-loop contributions to the hard-scattering amplitude for exclusive hadronic processes at large-momentum transfer in PQCD. The iterative structure makes it easy to implement the formalism in an algebraic computer program.Received: 18 August 2003, Revised: 6 February 2004, Published online: 23 April 2004     

Quantum-mechanical confinement with the robin condition

The energy spectrum of a nonrelativistic quantum particle in the confinement state in a closed spatial volume at general boundary confinement conditions (the Robin conditions) is investigated. It is shown that the properties of such a state are substantially more nontrivial compared with particle confinement using the potential barrier. It is also shown for a hydrogen-like atom arranged in a spherical cavity with radius R that if the surface layer with nonzero depth d plays the role of the boundary of the confinement region, all the energy levels of a discrete spectrum of the atom have a finite limit at R → 0, while the R-dependence of the lower layer at physically substantial parameters of the surface layer contains a deep well-pronounced minimum, in which the binding energy is considerably higher than for the lower 1s level of a free atom.     

Experimental study of proton-emulsion nuclei interactions at high energy

Proton-emulsion nuclei interactions at 200 and 400 GeV have been studied. The results on the energy dependence of <n _s(n _s-1)> and <n _s>/D, multiplicity scaling, multiplicity density scaling in the central region, integral angular distribution, rapidity dispersion and maximum rapidity gap are presented and compared with those of proton-nucleon interactions at the corresponding energies.     

Quantum brownian motion on a triangular lattice and c=2 boundary conformal field theory

We study a single particle diffusing on a triangular lattice and interacting with a heat bath, using boundary conformal field theory (CFT) and exact integrability techniques. We derive a correspondence between the phase diagram of this problem and that recently obtained for the 2-dimensional 3-state Potts model with a boundary. Exact results are obtained on phases with intermediate mobilities. These correspond to nontrivial boundary states in a conformal field theory with 2 free bosons which we explicitly construct for the first time. These conformally invariant boundary conditions are not simply products of Dirichlet and Neumann ones and unlike those trivial boundary conditions, are not invariant under a Heisenberg algebra.     

Normal forms for pseudo-Riemannian 2-dimensional metrics whose geodesic flows admit integrals quadratic in momenta

We discuss pseudo-Riemannian metrics on 2-dimensional manifolds such that the geodesic flow admits a nontrivial integral quadratic in velocities. We construct local normal forms of such metrics. We show that these metrics have certain useful properties similar to those of Riemannian Liouville metrics, namely:

•

they admit geodesically equivalent metrics;     

Quantum spring from the Casimir effect

The Casimir effect arises not only in the presence of material boundaries but also in space with nontrivial topology. In this Letter, we choose a topology of the flat (D+1)$(D+1)$-dimensional spacetime, which causes the helix boundary condition for a Hermitian massless scalar field. Especially, Casimir effect for a massless scalar field on the helix boundary condition is investigated in two and three dimensions by using the zeta function techniques. The Casimir force parallel to the axis of the helix behaves very much like the force on a spring that obeys the Hooke's law when the ratio r of the pitch to the circumference of the helix is small, but in this case, the force comes from a quantum effect, so we would like to call it quantum spring. When r is large, this force behaves like the Newton's law of universal gravitation in the leading order. On the other hand, the force perpendicular to the axis decreases monotonously with the increasing of the ratio r. Both forces are attractive and their behaviors are the same in two and three dimensions.     

On the riemannian metrics in which admit all hyperplanes as minimal hypersurfaces

Inspired by a result of Bekkar (1991), Robert Lutz raised the following problem: determine the riemannian metrics in domains of ⁿ which admit all hyperplanes as minimal hypersurfaces. We solve the problem giving a formula which expresses its solutions in terms of the non-degenerate quadratic first integrals of the geodesic motion in the euclidean space (second-order Killing tensor fields). Then, we prove that for n = 3 the non-flat polynomial solutions of the problem are the left invariant riemannian metrics on the Heisenberg group.     

Adsorption of gas mixtures on heterogeneous surfaces: The integral representation for a monolayer total adsorption isotherm

The authors extended the integral equation for a one-component adsorption isotherm on to n-component adsorption from gaseous mixtures. It appears that the n-component adsorption isotherm on heterogeneous surfaces can be presented as a multiple integral. Two methods were used for evaluation of the surface heterogeneity with regard to a gas mixture. The first method is based on the n-dimensional gaussian energy distribution. The second one concerns the extension of our method for studying adsorption systems of high surface heterogeneity. Two-component adsorption systems have been analyzed in details.     

The influence of Bose-Einstein correlations on intermittency inp\bar p collisions at\sqrt s = 630 GeV

The influence of Bose-Einstein correlations on the rise of factorial moments is small in the 1-dimensional phase space given by the pseudorapidity η, where the 2-body correlation function is dominated by unlike-sign particle correlations. Contraily, the influence is dominant in the higher dimensional phase space. This is shown by using correlation integrals. They exhibit clear power law dependences on the four-momentum transferQ ² for all orders investigated (i=2–5). When searching for the origin of this behaviour, we found that the Bose-Einstein ratio itself shows a steep rise forQ ²→0, compatible with a power law.     

Upon the utility of spherical harmonic expansions in the evaluation of cluster integrals

A novel procedure for the analytic evaluation of cluster integrals is given. By means of a result of Silverstone and Moats which transforms the spherical harmonic expansion of a function around a given point into a new spherical harmonic expansion around a displaced point, a 3N-dimensional cluster integral forN point particles (N > 2) may be reduced to 2N+1 trivial integrals andN– 1 interesting integrals, an improvement over the usual reduction to six trivial integrals and3N–6 nontrivial integrals. For hard spheres, theN–1 integrals involve only a series of simple polynomials taken between linear algebraic bounds.This work was supported in part by the National Science Foundation under Grant No. CHE79-20389.     

关于线性阻尼振子第一积分的研究

通过引入一维线性阻尼振子基本积分来构造其他第一积分, 包括不含时的积分. 将这种方法推广到多维情形, 构造二维和n维线性阻尼振子不同形式的第一积分; 证明不同类型的二维线性阻尼振子都存在三个独立的不含时的第一积分, n维线性阻尼振子存在2n-1个独立的不含时的第一积分. 利用变量变换将线性阻尼振子的第一积分变换成简谐振子形式的第一积分. 关键词： 线性阻尼振子 第一积分 基本积分 简谐振子     

Bound States for a Klein–Gordon Particle in Vector Plus Scalar Generalized Hulthén Potentials in D Dimensions

The Green’s function associated with a Klein–Gordon particle moving in a D-dimensional space under the action of vector plus scalar q-deformed Hulthén potentials is constructed by path integration for \({q \geq 1}\) and \({\frac{1}{\alpha} \ln q < r < \infty}\). An appropriate approximation of the centrifugal potential term and the technique of space-time transformation are used to reduce the path integral for the generalized Hulthén potentials into a path integral for q-deformed Rosen–Morse potential. Explicit path integration leads to the radial Green’s function for any l state in closed form. The energy spectrum and the correctly normalized wave functions, for a state of orbital quantum number \({l \geq 0}\), are obtained. Eventually, the vector q-deformed Hulthén potential and the Coulomb potentials in D dimensions are considered as special cases.     

Lorentz-covariant deformed algebra with minimal length

TheD-dimensional two-parameter deformed algebra with minimal length introduced by Kempf is generalized to a Lorentz-covariant algebra describing a (D + 1)-dimensional quantized space-time. ForD=3, it includes Snyder algebra as a special case. The deformed Poincaré transformations leaving the algebra invariant are identified. Uncertainty relations are studied. In the case ofD=1 and one nonvanishing parameter, the bound-state energy spectrum and wavefunctions of the Dirac oscillator are exactly obtained.