On the Weyl transverse frames in type I spacetimes

We apply a covariant and generic procedure to obtain explicit expressions of the transverse frames that a type I spacetime admits in terms of an arbitrary initial frame. We also present a simple and general algorithm to obtain the Weyl scalars ₂^T, ₀^T and ₄^T associated with these transverse frames. In both cases it is only necessary to choose a particular root of a cubic expression.     

Discrete Painlevé equations and their appearance in quantum gravity

We discuss an algorithmic approach for both deriving discrete analogues of Painlevé equations as well as using such equations to characterize similarity reductions of spatially discrete integrable evolution equations. As a concrete example we show that a discrete analogue of Painlevé I can be used to characterize similarity solutions of the Kac-Moerbeke equation. It turns out that these similarity solutions also satisfy a special case of Painlevé IV equation. In addition we discuss a methodology for obtaining the relevant continuous limits not only at the level of equations but also at the level of solutions. As an example we use the WKB method in the presence of two turning points of the third order to parametrize (at the continuous limit) the solution of Painlevé I in terms of the solution of discrete Painlevé I. Finally we show that these results are useful for investigating the partition function of the matrix model in 2D quantum gravity associated with the measure exp [–t ₁ z ² –t ₂ z ⁴ –t ₃ z ⁶].     

L'asymptotique de Weyl pour les bouteilles magnétiques

Nous prouvons une formule pour le comportement asymptotique de la fonctionN() de dénombrement des valeurs propres de l'opérateur de Schrödinger avec un champ magnétique qui tend vers l'infini `a l'infini de ^d . La preuve utilise un résultat précis sur l'estimation des valeurs propres pour un champ magnétique constant dans un cube de ^d.     

Castaing’s instability in a trapped ultra-cold gas

We consider a trapped ultra-cold gas of (non-condensed) bosons with two internal states (described by a pseudo spin) and study the stability of a longitudinal pseudo spin polarization gradient. For this purpose, we numerically solve a kinetic equation corresponding to a situation close to the experiment at JILA [1]. It shows the presence of Castaings instability of transverse spin polarization fluctuations at long wavelengths. This phenomenon could be used to create spontaneous transverse spin waves.Received: 1st October 2002, Published online: 15 July 2003PACS: 03.75.Nt Other Bose-Einstein condensation phenomena - 51.10.+y Kinetic and transport theory of gases - 75.30.Ds Spin wavesO. Prévoté: Present address: Université de Cergy-Pontoise, Département de Physique, 5 mall Gay Lussac, Neuville-sur-Oise, 95031 Cergy-Pontoise, France.     

The transverse correlation length for randomly rough surfaces

It is shown by numerical simulations for a random, one-dimensional surface defined by the equationx ₃=(x ₁), where the surface profile function (x ₁) is a stationary, stochastic, Gaussian process, that the transverse correlation lengtha of the surface roughness is a good measure of the mean distance d between consecutive peaks and valleys on the surface. In the case that the surface height correlation function (x ₁)(x ₁)/²(x ₁)=W (|x ₁–x ₁|) has the Lorentzian formW(|x ₁|)=a ²/(x ₁ ² +a ²) we find that d=0.9080a; when it has the Gaussian formW(|x ₁|)=exp(–x ₁ ² /a ²), we find that d=1.2837a; and when it has the nonmonotonic formW(|x ₁|)=sin(x ₁/a)/(x ₁/a), we find that d=1.2883a. These results suggest that d is larger, the faster the surface structure factorg(|Q|) [the Fourier transform ofW(|x ₁|)] decays to zero with increasing |Q|. We also obtain the functionP(itx ₁), which is defined in such a way that, ifx ₁=0 is a zero of (x ₁),P(x ₁)dx ₁ is the probability that the nearest zero of (x ₁) for positivex ₁ lies betweenx ₁ andx ₁+dx ₁.     

Nonlinear evolution equations associated with ‘enegry-dependent Schrödinger potentials’

The Inverse Scattering Transform is used to solve a class of nonlinear equations associated with the inverse problem for the one-dimensional Schrödinger equation with the energy-dependent potential V(k,x)=U(x)+kQ(x).Physique Mathématique et Théorique, Equipe de recherche associée au N.C.R.S. n⁰ 154.This work has been done as part of the program Recherche Coopérative sur Programme n⁰ 264: Etude interdisciplinaire des problèmes inverses.     

A Mathematical Representation of Quantizing the Motion of Relativistic Electrons in a Magnetic Field

It is shown that the quadratic component of the kinetic energy of continuous longitudinal motion of relativistic electrons in the external magnetic field is varied continuously between 0 and 2(2m _e c ²_B H) within each Landau energy level, undergoing an abrupt change at the boundaries of the levels. This results in the fact that in the quantum limit of a superstrong magnetic field where all electrons are at the zero Landau level, the maximum quadratic component of the kinetic energy of free longitudinal electron motion along the direction of the magnetic field is twice as high as the maximum quadratic component of the kinetic energy of its bound transverse motion.     

Do we understand the single-spin asymmetry for $\pi^0$ inclusive production in pp collisions?

The cross section data for inclusive production in pp collisions is considered in a rather broad kinematic region in energy , Feynman variable x _F and transverse momentum p _T. The analysis of these data is done in the perturbative QCD framework at the next-to-leading order. We find that they cannot be correctly described in the entire kinematic domain and this leads us to conclude that the single-spin asymmetry, A _N for this process, observed several years ago at FNAL by the experiment E704 and the recent result obtained at BNL-RHIC by STAR, are two different phenomena.Received: 10 March 2004, Revised: 19 May 2004, Published online: 14 July 2004preprint CPT-2003/P.4581(UMR 6207 is Unité Mixte de Recherche du CNRS and of Universités Aix-Marseille I and Aix-Marseille II and of Université du Sud Toulon-Var, Laboratoire affilié á la FRUMAM)     

Non-linear multi-plane wave solutions of self-dual Yang-Mills theory

New solutions of self-dual Yang-Mills (SDYM) equations are constructed in Minkowski space-time for the gauge groupSL(2, ). After proposing a Lorentz covariant formulation of Yang's equations, a set of Ansätze for exact non-linear multiplane wave solutions are proposed. The gauge fields are rational functions ofe ^x·ki (K _i ² =0, 1iN) for these Ansätze. At least, three families of multisoliton type solutions are derived explicitly. Their asymptotic behaviour shows that non-linear waves scatter non-trivially in Minkowski SDYM.On leave from LPTHE Université Paris VI, 4, Place Jussieu, Tour 16, ler étage, F-75230 Paris Cedex 05, France     

SL(n+1, C) strata and orbits in the solution space of Euclidean CP n models

We further study the action of SL(n+1, C) on the space of finite action solutions of the bidimensional Euclidean CP ⁿ models. We decompose the space of k-instantons into strata. Each stratum in characterized by an integer m with 0mmin(k, n) which can be calculated from the instanton by purely algebraic means. The k-instantons with m=n are called generic. Their stratum is shown to be dense in the space of k-instantons when kn. The isotropy subgroups for each stratum are identified.

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Résumé Nous poursuivons l'étude de l'action de SL(n+1, C) sur l'espace des solutions à action finie du modèle CP ⁿ sur l'espace euclidien bi-dimensionnel. L'espace des k-instantons est décomposé en strates. Chaque strate est caractérisée par un entier m tel que 0 mmin(k, n) et qui peut être calculé à partir de l'instanton par des méthodes purement algébriques. Les k-instantons avec m=n sont dits génériques. Leur strate est dense dans l'espace des k-instantons (lorsque k n). Les sous-groupes d'isotropie de chacune des strates sont identifiés.

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Supported in part by the Natural Sciences and Engineering Research Council of Canada and by the Fonds FCAR pour l'aide et le soutien à la recherche.     

Complex parabolic subgroups of G 2 and nonlinear differential equations

Nonlinear ordinary differential equations with superposition formulas corresponding to the exceptional Lie group G ₂ and its two maximal (complex) parabolic subgroups are determined. The G ₂-invariance of a third-order skewsymmetric tensor is exploited. The obtained ODEs have polynomial nonlinearities of order 2 in one case and of order 4 in the other.Supported in part by Les accords culturels Québec-Belgique 1985.Chargé de recherches FNRS.     

The exact phase diagram of the quantumXY spin glass model in a transverse field

The infinite rangeXY spin glass model in a transverse field is investigated by means of the Trotter-Suzuki approach. The exact phase diagram is obtained showing that a spin glass transition takes place for non-zero values of the transverse field up to a critical value _c =1.440.01. The present numerically exact calculations are in good agreement with our previous approximate results and they clear remaining discrepancies from previous work.     

Spectral fluctuations and zeta functions

We study theoretically and numerically the role of the fluctuations of eigenvalue spectra {it_n in a particular analytical continuation process applied to the (generalized) zeta functionZ(s)= _{n n} ^–s fors large and positive. A particularly interesting example is the spectrum of the Laplacian on a triangular domain which tessellates a compact surface of constant negative curvature (of genus two). We indeed find that the fluctuations restrict the abscissa of convergence, and also affect the rate of convergence. This then initiates a new approach to the exploration of spectral fluctuations through the convergence of analytical continuation processes.work done while at the Service de Physique Théorique, Saclay, and Institut de Physique Nucléaire, Orsay.     

A discrete spectrum of solutions of the wave equation with strong cubic nonlinearity

The nonlinear wave equation, _tt –+³=0, has many solutions that are periodic in time and localized in space, all with infinte energies. The search for spherically symmetric solutions that are well represented by the simple approximation, (r, t)A(r) sin t, leads to a discrete spectrum of solutions{ _N (r, t; )}. The solutions are nonlinear wavepackets, and they can be regarded as particles. The asymptotic theory () of the motion of the guiding center of theNth wavepacket, in the presence of a specified potential, is characterized by an infinite mechanical mass and an infinte interaction mass, and they are compatible. The rest mass in the classical relativistic mechanics of guiding centers ism ₀ c ²= _N ; i.e. the spectrum { _N } determines a spectrum of Planck's constants.On leave (1972–73) Université de Paris VI, Département de Mécanique, 75 Paris 5^e, France.     

Re-entrant ferromagnetism in a two-dimensional mixed spin Ising model with random nearest-neighbour interactions

Using the finite cluster approximation we study a mixed spin model with random interactions on a ferrimagnetic square lattice (with spins =1/2 andS=1). The interactions are assumed to be independent random variables with distribution,P(K _ij )=p(K _ij –K)+(1–p) (K _ij –K), whereK>0 and ||1. In a certain range of negative values of the phase diagrams exhibit re-entrant behaviour. This indicates that re-entrance seems to be a characteristic feature of systems in which both frustration and disorder are present.On leave from Faculté des Sciences I, Université Hassan II, Casablanca, Morocco; and Laboratoire de Magnétisme Université de Rabat, MoroccoSupported by the agreement of cooperation between the DFG and the CNR-Maroc.     

Coupling of the zero sound to transverse spin waves for neutral and normal Fermi liquids in dc magnetic field at precessing magnetization

Following the idea by Ketterson, this author showed that in neutral and normal Fermi liquids, the polarization effects lead to coupling of transverse spin waves to zero sound at precessing magnetization of the system. The observable effects, at the range of dc fields applied to³He, can appear only if the zero sound is degenerate with transverse spin waves excited in the reference frame rotating with the Larmor frequency about theH _dc-axis. This possibility is investigated at Landau parameters vanishing atl>1 and nonzero, though small, coupling of the density to the spin density and the particle current to the spin current, resulting from the polarization of the system byH _dc. The degeneration is, in general, possible but, at large values of the zero sound velocity with respect to the Fermi velocity, caused by large values of the Landau parameterF ₀ ^s and alsoF ₁ ^s for³He, atF ₀ ^a –0.7, it appears only at positiveF ₁ ^a . All experimental estimations of this parameter agree in its negativity excluding the degeneration of the zero sound and transverse spin waves for³He.     

The role of strange sea quarks in chiral extrapolations on the lattice

Since the strange quark has a light mass of order , fluctuations of sea pairs may play a special role in the low-energy dynamics of QCD by inducing significantly different patterns of chiral symmetry breaking in the chiral limits N _f = 2 (m _u = m _d = 0, m _s physical) and N _f = 3 (m _u = m _d = m _s = 0). This effect of vacuum fluctuations of pairs is related to the violation of the Zweig rule in the scalar sector, described through the two O(p ⁴) low-energy constants L ₄ and L ₆ of the three-flavour strong chiral lagrangian. In the case of significant vacuum fluctuations, three-flavour chiral expansions might exhibit numerical competition between leading- and next-to-leading-order terms according to the chiral counting, and chiral extrapolations should be handled with special care. We investigate the impact of the fluctuations of pairs on chiral extrapolations in the case of lattice simulations with three dynamical flavours in the isospin limit. Information on the size of the vacuum fluctuations can be obtained from the dependence of the masses and decay constants of pions and kaons on the light quark masses. Even in the case of large fluctuations, corrections due to the finite size of spatial dimensions can be kept under control for large enough boxes ( fm).Received: 20 October 2004, Revised: 28 December 2004, Published online: 9 February 2005Laboratoire de Physique Théorique: LPT is an Unité Mixte de Recherche du CNRS et de lUniversité Paris-Sud 11 (UMR 8627).     

Localization for some continuous random Schrödinger operators

We study the spectrum of random Schrödinger operators acting onL ²(R ^d ) of the following type . The are i.i.d. random variables. Under weak assumptions onV, we prove exponential localization forH at the lower edge of its spectrum. In order to do this, we give a new proof of the Wegner estimate that works without sign assumptions onV.

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Résumé Dans ce travail, nous étudions le spectre d'opérateurs de Schrödinger aléatoires agissant surL ²(R ^d ) du type suivant . Les sont des variables aléatoires i.i.d. Sous de faibles hypothèses surV, nous démontrons que le bord inférieur du spectre deH n'est composé que de spectre purement ponctuel et, que les fonctions propres associées sont exponentiellement décroissantes. Pour ce faire nous donnons une nouvelle preuve de l'estimée de Wegner valable sans hypothèses de signe surV.

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U.R.A. 760 C.N.R.S.     

Two-dimensional mixed spin Ising models with bond dilution and random ±J interactions

Using the finite cluster approximation we study a mixed spin model (spins =1/2 andS=1) on a square lattice with nearest-neighbour and crystal field interactions. The nearest-neighbour couplingsK _ij are assumed to be independent random variables with distribution,P(K _ij )=p(K _ij –K)+(1–p)(K _ij –K), whereK>0. We investigate the cases =0 and =–1 corresponding to bond dilution and to random ±J interactions, respectively. In certain ranges ofp the phase diagrams exhibit tricritical behaviour and reentrance.Supported by the agreement of cooperation between the DFG and the CNR-MarocOn leave from Faculté des Sciences I, Université Hassan II, Casablanca, Morocco; and Laboratoire de Magnétisme Université de Rabat, Morocco     

Dynamics of the generalized Glauber-Ising chain in a magnetic field

A one-dimensional kinetic Ising model with nearest neighbor interactionJ and magnetic fieldH 0 is treated in both linear and nonlinear response, using the most general single spin-flip transition probabilities that depend on nearest neighbor states only. The dynamics is reformulated in terms of kinetic equations for the concentration n_l ⁺(t) [@#@ n_l(t) of clusters containingl up- [or down-] spins, which is exact in the homogeneous case. The initial relaxation time ^* of the magnetization is obtained rigorously for arbitraryJ, H, and temperatureT. The relaxation function is found by numerical integration forJ/T < 2. It is shown that coagulation of minus-clusters becomes negligible for bothJ/T andH/T large, and the resulting set of equations is solved exactly in terms of an eigenvalue problem. A perturbation theory is developed to take into account the neglected coagulation terms. The relaxation function is found to be non-Lorentzian in general, in contrast to the Glauber results atH = 0, which are recovered as a special case. In addition, nonlinear and linear relaxation functions differ forH 0. Consequences for the application to biopolymers are briefly mentioned.Supported in part by the Deutsche Forschungsgemeinschaft (SFB 130).