The group of automorphisms of the galilei group

The group of automorphisms of the Galilei groupG: Aut(G) is calculated. It is shown that Aut(G) has the structure of a semi-direct product byG of the group _m ^* ×_m where _m is the group of reals noted multiplicatively and _m ^* <_m is the subgroup of positive reals.     

Monopoles,braid groups,and the dirac operator

Using the relation between the space of rational functions on , the space ofSU(2)-monopoles on ³, and the classifying space of the braid group, see [10], we show how the index bundle of the family of real Dirac operators coupled toSU(2)-monopoles can be described using permutation representations of Artin's braid groups. We also show how this implies the existence of a pair consisting of a gauge fieldA and a Higgs field on ³ whose corresponding Dirac equation has an arbitrarily large dimensional space of solutions.The first author was supported by a grant from the NSF     

Strong electric field eigenvalue asymptotics for the Schrödinger operator with electromagnetic potential

We consider the Schrödinger operator with electric potential V, which decays at infinity, and magnetic potential A. We study the asymptotic behaviour for large values of the electric field coupling constant of the eigenvalues situated under the essential-spectrum lower bound. We concentrate on the cases of rapidly decaying V (e.g. V L ^m/2( ^m ) for m 3) and arbitrary A, or slowly decaying V (i.e. V(x |x|^– , (0,2), as |x| ) and magnetic potentials A corresponding to constant magnetic fields B = curl A.Partially supported by the Ministry of Culture, Science and Education under Grant No. 52.     

A New Estimate for the Groundstate Energy of Schrödinger Operators

A new estimate for the groundstate energy of Schrödinger operators on L²(ⁿ) (n 1) is presented. As a corollary, it is shown that the groundstate energy of the Schrödinger operator with a scalar potential V is more than the classical lower bound ess.inf_x__V(x) if V is essentially bounded from below in a certain manner (enhancement of the groundstate energy due to quantization). As an application, it is proven that the groundstate energy of the Hamiltonian of the hydrogen-like atom is enhanced under a class of perturbations given by scalar potentials (vanishing at infinity).     

Integration on then th power of a hyperbolic space in terms of invariants under diagonal action of isometries (Lorentz transformations)

The integral of a function over then'th power of hyperbolicd-dimensional spaceH is decomposed into integration along each orbit under diagonal action onH ⁿ of the isometry groupG onH, followed by integration over the orbit space, parametrized in terms of a complete set of invariants. The Jacobian entering in this last integral is expressed explicitly in terms of certain determinants. When viewingH as a half-hyperboloid in _d+1 ,G is induced by the homogeneous Lorentz groupO (1,d) acting on _d+1 .     

Lorentzian Warped Products and Singularity

We show that to any convex function f: ⁿ there correspondinfinitely many geodesically complete metricsds² such that Ric() 0 for anynonspacelike vector . These metrics are constructedas the warped products of the natural metric in and the inner metric of a convexhyperface (the graph of f) in ^{n + 1}.     

Hyper-Kähler metrics and a generalization of the Bogomolny equations

We generalize the Bogomolny equations to field equations on ^{3 n} and describe a twistor correspondence. We consider a general hyper-Kähler metric in dimension 4n with an action of the torusT ⁿ compatible with the hyper-Kähler structure. We prove that such a metric can be described in terms of theT ⁿ -solution of the field equations coming from the twistor space of the metric.     

Stochastic coupling and thermodynamic inequalities

Let ₁ and ₂ be thermodynamic Gibbs measures on ^m and ⁿ , respectively. Diffusions are constructed having ₁, and ₂ as invariant measures. These diffusions are then coupled; inequalities between expectations of certain random variables on the two spaces result.Partially supported by NSF-MCS 74-07313-A03     

Poisson–Lie Diffeomorphism Groups

We explicitly construct a class of coboundary Poisson–Lie structures on the group of formal diffeomorphisms of ⁿ . Equivalently, these give rise to a class of coboundary triangular Lie bialgebra structures on the Lie algebra W _n of formal vector fields on ⁿ . We conjecture that this class accounts for all such coboundary structures. The natural action of the constructed Poisson–Lie diffeomorphism groups gives rise to large classes of compatible Poisson structures on ⁿ , thus making it a Poisson space. Moreover, the canonical action of the Poisson–Lie groups FDiff( ^m ) × FDiff ⁿ ) gives rise to classes of compatible Poisson structures on the space J ( ^m , ⁿ ) of infinite jets of smooth maps ^{m n} , which makes it also a Poisson space for this action. Poisson modules of generalized densities are also constructed. Initial steps towards a classification of these structures are taken.     

Translation group and spectrum condition

Let {A, ^d ,} be aC*-dynamical system, where ^d is thed-dimensional vector group. LetV be a convex cone in ^d and its dual cone. We will characterize those representations ofA with the properties (i) _a ,a ^d is weakly inner, (ii) the corresponding unitary representationU(a) is continuous, and (iii) the spectrum ofU(a) is contained in .     

Local theory of solutions for the 0(2k+1) σ-model

We develop a theory of solutionsn for the Euclidean nonlinear 0(2k+1)-model for arbitraryk and for a regionG². We consider a subclass of solutions characterized by a condition which is fulfilled, forG=², by thosen that live on the Riemann sphere S²². We are able to characterize this class completely in terms ofk meromorphic functions, and we give an explicit inductive procedure which allows us to calculate all 0(2k+1) solutions from the trivial 0(1) solutions.     

Anisotropy of the low temperature resistivity of hexagonal single crystalline spin glass systems

The impurity contribution to the resistivity in zero field (T) of dilute hexagonal single crystals of ZnMn, CdMn and MgMn has been studied in the mK range on samples cut parallel () and perpendicular () to thec-axis, using a SQUID technique for the measurements. Typical spin glass behavior is found in (T) as well as (T) for all alloys, with Kondo like logarithmic increases at higher temperatures and maxima atT _m at lower temperatures, indicating the influence of impurity interactions. The differences in the corresponding isotropic resistivity _poly(T) between the three systems can qualitatively be understood within the framework of a theoretical model by Larsen, describing (T) as a function of universal quantitiesT/T _K and _RKKY/T _K , where _RKKY is the RKKY-interaction strength andT _K the Kondo temperature. With respect to the two lattice directions studied, the behavior of (T and (T is anisotropic in the Kondo regime as well as in the range where ordering becomes important. While the anisotropy in the Kondo slope can be understood by an anisotropic unitarity limit, the understanding of the anisotropy in region where impurity interactions are important remains problematic.Dedicated to Prof. Dr. S. Methfessel on the occasion of his 60th birthday     

Classical solutions of the chiral model,unitons, and holomorphic vector bundles

This paper deals with classical solutions of theSU(2) chiral model on ², and of a generalized chiral model on ²⁺¹. Such solutions are shown to correspond to certain holomorphic vector bundles over minitwistor space. With an appropriate boundary condition, the solutions (called 1-unitons in [9]) correspond to bundles over a compact 2-dimensional complex manifold, and the problem becomes one of algebraic geometry.     

Schrödinger Operators with Empty Singularly Continuous Spectra

Let H be a semibounded perturbation of the Laplacian H ₀ in L ²( ^d ). For an admissible function sufficient conditions are given for the completeness of the scattering system (H), (H ₀). If is the exponential function and if e^{– H} is an integral operator we denote the kernel of the difference D = e^{– H} – e^{– H} ₀ by D (x, y), > 0. The singularly continuous spectrum of H is empty if^d dx ^d dy |D(x,y)| (1 + |y|²)< for some > 1. This result is applied to potential perturbations and to perturbations by imposing Dirichlet boundary conditions.     

The Broadwell model for initial values inL + 1 (ℝ)

The Cauchy problem for the Broadwell model is shown to have a global mild solution for initial data inL ₊ ¹ () with smallL ¹-norm, and a local solution for arbitrary initial data inL ₊ ¹ (). For data which are small inL ¹(), the asymptotic behaviour of the solutions ast is determined. Moreover, it is shown that a global solution exists for all initial values inL ₊ ¹ () with finite entropy if theH-Theorem holds.     

Semi-global solutions of Einstein equations,minkowskian near past infinity

On a universe homeomorphic toV ^T =]– ,T[x³, we prove the existence of solutions of Einstein equations, minkowskian near past infinity, if the sources are small enough for some norms. We prove that some of these solutions verify at least the positivity condition (Weak energy condition) on some domains homeomorphic toV ^T .     

The shape resonance

For a class of Schrödinger operatorsH:=–(²/2m)+V onL ²( ⁿ ), with potentials having minima embedded in the continuum of the spectrum and non-trapping tails, we show the existence of shape resonances exponentially close to the real axis as 0. The resonant energies are given by a convergent perturbation expansion in powers of a parameter exhibiting the expected exponentially small behaviour for tunneling.     

Bianchi identities,electromagnetic waves,and charge conservation in theP(4) theory of gravitation and electromagnetism

The Bianchi identities for theP(4)=O(1, 3) ^4* theory of gravitation and electromagnetism are decomposed into the standardO(1, 3) Riemannian Bianchi identity plus an additional ^4* component. When combined with the Einstein-Maxwell affine field equations the ^4* components of theP(4) Bianchi identities imply conservation of magnetic charge and the wave equation for the Maxwell field strength tensor. These results are analyzed in light of the special geometrical postulates of theP(4) theory. We show that our development is the analog of the manner in which the Riemannian Bianchi identities, when combined with Einstein's field equations, imply conservation of stress-energy-momentum and the wave equation for the LanczosH-tensor.     

The λϕ 3 4 field theory in a finite volume

The unnormalized doubly cutoff Schwinger functions converge as the ultraviolet cutoff is removed. The limits, the finite volume unnormalized Schwinger functions, are tempered distributions and areC in the coupling constant. They have asymptotic expansions given by perturbation theory. For sufficiently small they can be normalized and then they are the moments of a measure onI (IR³).Supported in part by the National Science Foundation under Grant GP 40354X.     

Un phénomène de concentration évanescente pour des flots non-stationnaires incompressibles en dimension deux

We consider a sequencev of non-stationary solutions of the incompressible 2D-Euler equation, locally bounded inL ². We prove that if the defect measure is supported in a one-dimensional set (³) of some special type (which we call finite type), the weak limitv ofv is a solution of the Euler equations: our theorem is of the type concentration-cancellation.