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Inhaltsverzeichnis

Recent Progress in Classical Mechanics

The goal of this lecture is to review several developments in classical mechanics that have taken place in the last years, that will fit in the time of the talk and that I have become aware of. Unfortunately, the latter is a constraint more severe than what I would like and I apologize to the authors and the audience for many things that have been left out. In particular, I have left out topics such as “twist mappings”, “geometric quantization” or “geometric phases” and “quantum chaos” that are generating a great deal of activity in the literature. I would also like to recommend the excellent set of reviews [AANS] for the many developments that took place up to the time the books were published.

R. de la Llave

Introduction to the Differential Geometry of Quantum Groups

An introduction to the noncommutative differential calculus on quantum groups. The invariant group average is also discussed.

Bruno Zumino

Chaotic Quantum Systems

New ideas concerning the peculiar phenomenon of quantum chaos are presented with special emphasis on a number of unsolved problems and current apparent contradictions.

Boris V. Chirikov

Dynamical Zeta Functions: Where Do They Come from and What Are They Good for ?

The properties and usefulness of dynamical zeta functions associated with maps and flows are discussed, and they are compared with the more traditional number-theoretic zeta functions.

David Ruelle

Billiard-Type Systems with Chaotic Behaviour and Space-Time Chaos

In the last two decades the subject of the branch of mathematics that is called Dynamical Systems is almost identified with the study of chaotic motion of such systems. However until recently there were very few examples of in a sense realistic systems that have been proven to be chaotic. The first found models of that type were geodesic flows on manifolds of negative curvature (see [Ha], [He], [H1]). The general ideas of these papers were developed by D. V. Anosov, Ya. G. Sinai and S. Smale (see [An], [AS], [S2], [Sm]) and lead to the concept of hyperbolicity as the basic mechanism of chaos in classical dynamical systems. This concept together with the closely related notion of Lyapunov exponents (see [O], [P]) serve as the foundation of the modern theory of nonuniformly hyperbolic dynamical systems.

Leonid A. Bunimovich

Statistical Physics and Spectral Theory of Disordered Systems: Some Recent Developments

Disorder is one of the paradigms of modern physics. It suffices to look through issues of leading physical journals and conference proceedings to see how numerous the papers on the disordered systems are, treating an extremely wide variety of problems, from highly conceptual questions of quantum physics and neurobiology to crack propagation in concrete and spray painting. This certainly reflects the richness and freshness of the field and the continuous “production” of new problems and results.

L. A. Pastur

Low Temperature Stochastic Spin Dynamics: Metastability, Convergence to Equilibrium and Phase Segregation

I will report on recent progresses made in the rigorous study of the long time behaviour of random dynamics of physical and mathematical interest for ferromagnetic spin systems in the low temperature regime. In particular I will discuss metastability and its connection with the problem of the convergence to equilibrium for a standard Glauber dynamics for the Ising model, dynamical phase transition, dynamics of the interface and phase segregation for conservative spin flips dynamics.

Fabio Martinelli

Entropy Methods in Hydrodynamical Scaling

We shall describe these methods by examining very closely an explicit model. Consider N lattice sites arranged periodically in one dimension with a lattice spacing of 1/N. We have spin variables xj attached to each site j/N, the sites being viewed as equally spaced points on the circle of unit circumference. The spins xj vary in time in such a manner that they undergo a diffusion on IRN denoted by {x1(t),..., x N (t)}. The diffusion process is described by $$d{x_{i}}(t) = s{z_{{i - 1,i}}}(t) - d{z_{{i,i + 1}}}(t),\,d{z_{{i,i + 1}}}(t) = \frac{{{N^{2}}}}{2}\left[ {\phi '({x_{i}}(t)) - \phi '({x_{{i + 1}}}(t))} \right]dt + Nd{\beta _{{i,i + 1}}}(t)$$.

Quantum Groups and Non-commutative Differential Geometry

Recently it became common to identify the notion of a quantum group with that of a Hopf algebra. However, this does not quite agree with the experience gained in classical group theory. In fact, classically, Hopf algebras arise in the following framework. One starts with a category S of “spaces” (finite sets, schemes, differential manifolds, topological spaces ...) and a linear functor ℱ on it, covariant or contravariant. The functor must satisfy certain formal properties, in particular, transform direct products into tensor products. The values of this functor upon group objects in S will then be Hopf algebras, commutative in the contravariant case, cocommutative in the covariant case.

Yu. I. Manin

Hamiltonian Methods in Conformal Field Theory

Conformai Field Theory (CFT) existed for a long time on the periphery of Quantum Field Theory (QFT). It got a very strong boost by the bootstrap programm of Belavin, Polyakov and Zamolodchikov [1]. Hundreds of papers were published since. CFT has conscripted new young and revolutionary devoters (call them the conformists to distinguish from die konformisten). New terminology and methodology, e.g. primary fields, vertex operators, operator expansion, mixing of states and operators is indispensable for the paper on CFT.

On the General Theory of Quantized Fields

The term “General Theory of Quantized Fields”, replacing the synonymous but somewhat misleading term “Axiomatic Field Theory”, is to my knowledge due to Res Jost. He was one of the great pioneers in our field, and I dedicate this lecture to his memory. What is the aim of the general theory of quantized fields?

Klaus Fredenhagen

Atomic and Molecular Structure

— A Renormalized Picture —

It is my goal here to review some recent developments in the rigorous mathematical study of atomic and molecular structure.

Jan Philip Solovej

Gauge Invariance in Non-Relativistic Many-Body Theory

We review some recent results on the physics of two-dimensional, incompressible electron- and spin liquids. These results follow from Ward identities reflecting the U(1)em × SU(2)spin-gauge invariance of non-relativistic quantum mechanics. They describe a variety of generalized quantized Hall effects.

J. Fröhlich, U. M. Studer

Asymptotic Completeness for N-Body Quantum Systems

We give a sketch of a geometrical proof of asymptotic completeness for an arbitrary number of quantum particles interacting through short-range pair potentials.

Gian Michele Graf

Self-Dual Chern—Simons Solitons

These lectures are devoted to the two-year old topic of Chern-Simons solitons. Solitons have interested particle theorists since the mid-1970’s[1]; Chern-Simons dynamics, since the early 1980’s [2]. Now in the decade of the 1990’s, we are combining the two subjects.

R. Jackiw

Mathematical Theory of Classical Fields and General Relativity

The subject of Classical Field Theory has received relatively little attention from both mathematicians and theoretical physicists. The former look at it primarily from the perspective of Quantum Mechanics, namely, as subject of the elusive task of quantization. Mathematicians, on the other hand, never quite took the underlying geometric structure of Classical Field Theory, the space-time, seriously. The geometers, though greatly emboldened by the success of Riemann’s visionary ideas in the formulation of General Relativity, have stayed away, with few notable exceptions, from the fundamental new twist given to them by Einstein who replaced the positive definite metric of Riemannian Geometry by a Lorentzian, or more appropriate, Einsteinian metric.

Sergiu Klainerman

Representations of Quantized Differential Forms in Non-Commutative Geometry

Besides giving a survey of some basic structures and ideas in K-theory and cyclic cohomology for non-commutative algebras, we describe a new way to realize algebras of abstract differential forms, over a given algebra A, and their “quantum” deformations. For this we use subalgebras and quotients of an algebra A[D, F] obtained from A by adjoining two additional elements D, F. This is closely related to the notion of a Fredholm module.

Joachim Cuntz

Solutions of Nonlinear Wave Equations and Localization Theory

In this note we describe a new method for constructing time periodic solutions of nonlinear wave equations of the form (1)$${u_{{tt}}} = {u_{{xx}}} - g\left( {x,u} \right)\quad 0 < x < \pi$$, with either periodic or Dirichlet boundary conditions. For a complete description of the results and methods with detailed proofs, see [2]. This article will attempt only to explain the general ideas behind this method and some of its possible extensions.

Walter Craig, C. Eugene Wayne

Connections on Symplectic Manifolds and Characteristic Classes of Hamiltonian System

We present a method for constructing the generalization of Maslow-Arnold classes to a certain class of symplectic manifolds.

Valerii V. Trofimov

Symplectic Twist Maps and the Theorem of Conley-Zehnder for General Cotangent Bundles

We announce the existence and multiplicity of periodic orbits for time dependent Hamiltonian systems on cotangent bundle of arbitrary compact manifolds. We discretize the variational problem by decomposing the time 1 map into a product of symplectic twist maps.

Christophe Golé

Floer Homology for Mapping Cylinders

The symplectic and instanton Floer homologies are presented. When applied to a symplectomorphism on the space of flat connections on a Riemann surface and and for a 3-dimensional mapping cylinder of the Riemann surface, respectively, the resulting homologies are naturally isomorphic.

Stamatis Dostoglou, Dietmar Salamon

κ-Deformation of (Super)Poincaré Algebra

The notion of quantum groups and quantum algebras (see e.g. ref. [1]-[6]) can be used in order to study the deformations of space-time symmetries as well as their supersymmetric extensions. In order to obtain the quantum deformation of semisimple Lie algebras describing Minkowski or Euclidean group of motions mostly the contraction techniques have been used. In particular there were obtained: a)quantum deformation of D = 2 and D = 3 Euclidean and Minkowski geometries, described as quantum Lie algebra or quantum Lie group [7], [8]b)quantum deformation of D = 4 Poincaré algebra [9], [10]c)quantum deformations of D = 2 supersymmetry algebra in its Minkowski as well as its Euclidean version [11]–[13].

J. Lukierski

Quantum Physics as Non-Commutative Geometry

For a person in mathematical physics, notions of non-commutative geometry (NCG) seem very natural. Related ideas to those in NCG occur in quantum theory — especially supersymmetric quantum theory — and also in statistical mechanics. One can interpret NCG as a quantization of geometry, in the sense that quantum theory is a quantization of classical physics. Many basic notions of non-commutative geometry can be understood by thinking of NCG as a way to define and to integrate differentials a0da1 ··· da n in a framework more general than that of differential forms on manifolds. The quantum functions a0,..., an are operators; their integrals can be thought of as quantum mechanical expectation values. What results is a theory in which classical notions of geometry carry over. In particular, there is a natural interpretation of NCG in terms of a cohomology theory. This cohomology reduces to de Rham theory in the usual commutative case.

Arthur Jaffe

An Operator Algebraic Framework for the Duality of Quantum Groups

The duality for a locally compact group established by Pontrjagin, Tannaka, Krein, Steinspring, Eymard and Tatsuuma is an important theoretical basis for the harmonic analysis. At the formal level of pure algebras, the notion of Hopf algebras is used to deal with the algebraic groups, discrete groups, or their dual objects at the same time. In order to control the infinite dimensional unitary representations, functional analysis is necessarily combined with the algebraic framework of Hopf algebras. This consideration suggests us to introduce the notion of Kac-algebras in the language of the Neumann algebras, in with the above mentioned duality is generalized by Kac [5], Takesaki [9] and Enock and Schwartz [2].

Tetsuya Masuda, Yoshiomi Nakagami

Distribution of Energy Levels in Quantum Systems with Integrable Classical Counterpart. Rigorous Results

Let E0 ≤ E1 ≤ E2 ≤... be the energy levels (eigenvalues) of the Schrödinger operator H = -1/2Δ + U(q) on a closed d-dimensional Riemannian manifold Md. Here (1)$$- \Delta = - \frac{1}{{\sqrt {g} }}\frac{\partial }{{\partial {q^{i}}}}(\sqrt {g} {g^{{ij}}}\frac{\partial }{{\partial {g^{i}}}})]$$ is the Laplace-Beltrami operator and to ensure the discreteness of the spectrum of H we assume, in the case of a non-compact Md, that limq→∞U(q) = ∞. For simplicity we assume also that Md has no boundary. Otherwise it is neccessary to supply H with Dirichlet (or some other) boundary conditions.

P. M. Bleher

Entropy-Invariants of Dynamical Systems and Perturbations of Operators

In [12] we proved a sharp extension to the case of commuting n-tuples, of the well-known invariance of absolutely continuous spectra in trace-class scattering theory. Our proof relied on invariants k (τ)(ℐ a normed ideal, τ an n-tuple of operators), playing the role of a ”size ℐ”-dimensional measure of τ.We recently found the entropy of dynamical systems is related to k(τ) with ℐ = C- the Macaev ideal. This report is about k (τ) and its use in constructing entropy-line invariants. We also explain the connection between k1(τ) and the entropy of Bogoliubov automorphisms ([11]).

Dan Voiculescu

Chaotic Motion in Coulombic Potentials

The classical motion of a particle in an attracting Coulomb (-1/r) potential is described by conic sections, whereas the equations of motion in the field of two fixed centres have been solved by Euler and Jacobi.

Markus Klein, Andreas Knauf

Generalized Floquet Operator for Quasiperiodically Driven Quantum Systems

The behaviour of systems with time dependent perturbations can be qualitatively different from the one of isolated quantum systems, e.g. in a bounded spatial domain, or confined by a potential going rapidly to infinity. Since the energy levels of such a system are discrete, the wave function and thus all expectation values and correlation functions will of necessity be almost period in time [1]. This rules out any good ergodic properties or other forms of classical chaoticity, which require decay of correlation functions. Of course as the size of the system becomes large and the spacing between levels becomes very small, the ensemble averages of same classes of quantum variables may exhibit good decay properties over long time periods. These can become, in suitable limits, indistinguishable from those given by chaotic dynamics.

H. R. Jauslin, J. L. Lebowitz

Magnetization and Slow Decay of Correlations in Continuum 1/r 2 Ising Models

The continuum Ising model is a (one dimensional) spin system defined on the real line R instead of on the discrete lattice ℤ. It arises in the study of a quantum mechanical model of the motion of a particle in a double well potential, subjected to a field. Localization in this model is equivalent to magnetization in the continuum Ising model (see [1]).

Luiz R. G. Fontes

Existence of the Kosterlitz-Thouless Phase for Two-Dimensional Coulomb Gases at Inverse Temperatures above 8π

A two-dimensional (lattice) Coulomb gas is a system of classical particles with electric charges ±1, whose possible positions range over a finite array of sites Λ ⊂ ℤ2, interacting via a two-body Coulomb potential.

Abel Klein

Large Deviation Behavior of Statistical Mechanics Models in the Multiphase Regime

The theory of large deviations was initiated by the study of the asymptotical behavior of probabilities of large deviations of sums of n identically distributed independent random variables. In this case the logarithms of the probabilities are asymptotically equal to -nI, where I is a constant which can be defined by a minimization of the so-called action functional. The theory of large deviations is now a well-developed branch of the probability theory (see the books [1], [2] for example) devoted mainly to generalization of the mentioned asymptotics to wide classes of random processes. Also these types of results were recently generalized on a wide class of Gibbsian random fields (see [3], [4]).

R. L. Dobrushin, S. B. Shlosman

A Variational Approach to the Random Diffeomorphisms Type Perturbations of a Hyperbolic Diffeomorphism

Let f be a diffeomorphism of a compact Riemannian manifold M. The standard model of random perturbations of f is generated by a particle which jumps from x to fx and smears with some distribution close to the δ — function at fx. This model has the continuous time version where a flow is perturbed by a small diffusion. This approach leads to a Markov chain X n ε (or diffusion X t ε, in the continuous time case) with a small parameter ε > 0 and one is interested whether invariant measures of X n ε converge as ε → 0 to a particular invariant measure of the diffeomorphism f. To describe limiting measures I employed in [5] the Donsker-Varadhan variational formula (1)$${\lambda ^\varepsilon }(V) = \mathop {\sup }\limits_{\mu \in p(M)} (\int {Vd\mu } - {I^\varepsilon }(\mu ))$$ for the principal eigenvalues λε(V) of the operators PVεg = Pε(eVg) where Pε is the transition operator of the Markov chain Xnε, V is a continuous function, and I(μ) is certain lower semicontinuous convex functional on the space P(M) of probability measures on M. It turns out that if f is a hyperbolic diffeomorphism then λε(V) converges as ε → 0 to the topological pressure Q(V + φu) of f corresponding to the function V + φu where φu = -log Ju(x) and Ju(x) is the Jacobian of the differential Df restricted to the unstable subbundle.

Yuri Kifer

Chaotic Mappings and Stochastic Markov Chains

The main aim of this paper is to investigate some unusual properties of such well known examples of chaotic deterministic dynamical systems as piecewise expanding maps and to discuss the peculiarities of the so called variational approach proposed in the one-dimensional case by A. Lasota and J.A. Yorke in [1].

M. L. Blank

Quantum Spin Systems in a Random Environment

Quantum spin systems with random parameters have been introduced to study the effect of impurities in several physical systems (e.g., [1],[2], where models related to superfluidity and superconductivity are discussed). The finite volume Hamiltonian is typically of the form H Λ = JH 0,Λ + V Λ where H 0,Λ favors long-range order and V Λ has discrete spectrum with a unique ground state in which the elementary excitations of the system are perfectly localized, so for J = 0 the ground state correlation function is a δ-function. One or both of H0,Λ and V Λ may contain random parameters. J is a coupling constant.

Abel Klein

Rounding Effects in Systems with Static Disorder

Condensed matter physics often has to consider systems with static disorder [6], [7], i.e. with impurities, dislocations, substitutions etc. which vary from sample to sample (thus introducing disorder) but which do not exhibit thermal fluctuations on relevant time scales (hence the word static). To account for such disorder mathematically one often uses lattice spin systems with random parameters in the interaction (e.g. random magnetic fields or random coupling constants). For each fixed realization of these parameters one then obtains a spin system in which the usual quantities of physical interest — magnetization, free energy etc. — can be calculated. Random parameters of this type are often called quenched, to stress the fact that they remain constant during the calculation of spin averages — corresponding to the static nature of the disorder in the modelled physical system.

Jan Wehr

Derivation of the Euler Equation from Hamiltonian Systems with Negligible Random Noise

The Euler equation of conservation law has been one of the fundamental equations in fluid dynamics since its discovery two centuries ago. Although there are disputes regarding its maximum range of applicability, it has been firmly established in suitable region. In principle, it is a logical consequence of Newtonian mechanics and a rigorous derivation of it from Newton’s equation should be possible. Certainly some scaling has to be chosen and Euler equation is exact only in the scaling limit. This problem indeed is much harder than it appears, as Euler equation involves thermodynamical quantities such as pressure and temperature while classical Hamiltonian systems are characterized by the pair potential. So there is a link via classical statistical mechanics which does not enter explicitly in the classical Hamiltonain systems. In other words, to prove Euler equation from classical Hamiltonian systems one in a certain sense justifies Boltzmann’s principle from classical mechanics. So far no one knows how to achieve this except in some artificial systems. The known results in this direction include: a heuristic derivation of Euler equation by assuming some strong ergodic property and all equilibrium states being Gibbs; one dimensional hard rod systems; Lanford’s theorem in the small time and low density limit. (For a review see [3].) In this note, I shall report some recent progress in this direction done in collaboration with S. Olla and S. R. S. Varadhan [2].

Horng-Tzer Yau

The Spatial Structure in Diffusion Limited Two-Particle Reactions

We analyze the limiting behavior of the densities ρ A (t) and ρ B (t) and the random spatial structure ξ(t) = (ξ A (t), (ξ B (t)) for the diffusion controlled chemical reaction A + B → inert with equal initial densities. There is a change in behavior from d ≤ 4, where ρ A (t) = ρ B (t) ≈ c d /td/4, to d ≥ 4, where ρ A (t) = ρ B (t) ≈ c d /t as t → ∞. There is a corresponding change in the spatial structure. In d < 4, the particle types separate with only one type present locally, and ξ, after suitable rescaling, tends to a random Gaussian process. In d > 4, both particle types are, after large times, present locally in concentrations not depending on type or location. In d = 4, both particle types are present locally but with random concentrations, and the process tends to a limit.

Maury Bramson, Joel L. Lebowitz

Multilevel Models of Interacting Diffusions and Large Deviations

As compared with short range models, hierarchical models of interaction show a more realistic behavior than mean field models concerning both equilibrium and nonequilibrium phenomena. One of the remarkable features of such models is the observation that for large, but finite, system size the cooperative effects are organized in multiple time scales and in fact this provides a caricature of the behavior of short range systems at successively larger spatial scales, cf. [5].

Donald A. Dawson, Jürgen Gärtner

Fields, Particles and Analyticity: Recent Results or 30 Goldberg ...(er) Variations on B.A.C.H. where B=Bethe Salpeter Equation, A=Axiomatic Field Theory C=Complex Variables and H=Harmonic Analysis

As it is known, Axiomatic Field Theory (A) implies “double analyticity” of the n-point functions in space-time and energy-momentum Complex Variables (C), with various interconnections by Fourier-Laplace analysis. When the latter is replaced by Harmonic Analysis (H) on spheres and hyperboloids, a new kind of double analyticity results from (A) (i.e. from locality, spectral condition, temperateness and invariance): complex angular momentum is thereby introduced (a missing chapter in (A)). Exploitation of Asymptotic Completeness via Bethe-Salpeter-type equations (B) leads to new developments of the previous theme on (A,C,H) (complex angular momentum) and of other themes on (A, C) (crossing, Haag-Swieca property etc...). Various aspects of (A) + (B) have been implemented in Constructive Field Theory (composite spectrum, asymptotic properties etc...) by a combination of specific techniques and of model-independent methods.

J. Bros

Quantum Field Theory in Curved Spacetime

In this talk, I discuss the theory of the covariant Klein Gordon equation, (□ g + m2)∅ = 0, on the class of globally hyperbolic spacetimes (M, g). (Here globally hyperbolic means there exists a diffeomorphism δ : M → R × C where C is a three manifold and each of the surfaces δ-1({t} × C) is a Cauchy surface i.e. an achronal 3-surface which is cut precisely once by every inextendible causal curve in M.) My aim is to give an impression of what has been achieved in clarifying the conceptual and mathematical foundations of quantum field theory in curved spacetime and to raise some open issues of importance for the further development of the subject. Lack of space enforces the omission of many important topics (especially, the back-reaction problem and the definition of a quantum energy momentum tensor T μν , higher spin fields, interacting fields, and non-globally hyperbolic spacetimes) and there will be space neither for an adequate bibliography nor to always provide adequate motivation and details. The remedy for some of these deficiencies will be found in the references (see especially Section 3 in [1]).

Bernard S. Kay

Charges in Quantum Field Theory

In quantum field theory one has to deal with superselection sectors, i.e. inequivalent representations πα of the algebra of observables A. It is then desirable to have a unified description, i.e. a Hilbert space which is a direct sum of all sectors together with an algebra of charged fields interpolating among them, and a gauge principle, i.e. a transformation law for the fields to single out the observables as the gauge-invariant quantities.

Karl-Henning Rehren

Three Exactly Soluble Quantum Field Theory Models in 2-,3- and 4-Dimensional Space Time and Some General Questions They Suggest

As a result of three decades of hard work on the local algebra formalism, we now have a general theory of quantized fields that provides a satisfactory framework for field theory. On the other hand, constructive quantum field theory has made rather limited progress toward the objective of characterizing and constructing all field theories that satisfy the axioms of the general theory. We do have the nontrivial examples P(ϕ)2,Y234,Y3,Higgs2,Higgs3 and fragments of Y M3 and Y M4. However, these examples do not provide enough information to suggest reasonable guesses for the answers to general questions. For example, how do the perturbatively non-renormalizable theories fit into the general picture? The recent results of da Veiga and coworkers establishing the existence of tempered solutions of the Gross-Neveu model in three dimensional Euclidean space-time, show that a non-perturbative treatment of a perturbatively non-renormalizable theory is possible using rigorous renormalization group methods. What distinguishes such theories? Is the applicability of renormalization group methods to be regarded as always a reliable guide to the existence of non-trivial solutions of theories?

Arthur S. Wightman

CPT- and Lorentz-Transformations in Two-Dimensional Q.F.T.

The C.P.T.-theorem plays an important role in relativistic quantum field theory. In 1957 R.Jost [3] gave a proof of the C.P.T.-theorem in the frame of Wightman-field theory in which he revealed the connection of the C.P.T.-symmetry with the assumptions of positivity of the energy, Lorentz-invariance, and the standard locality assumptions. In this proof the existence of a vacuum state was essential. But up to now there is no proof of the C.P.T.-theorem in the theory of local observables in the sense of Araki, Haag, and Kastler.

H. J. Borchers

The Concept of Spontaneously Broken Symmetry and a New Approach to Goldstone’s Theorem

The work reported on here forms part of investigations on spontaneously broken symmetries done in collaboration with D. Buchholz, S. Doplicher and R. Longo.

John Roberts

On the Atomic Energy Asymptotics

Consider an atom consisting of N quantized electrons at positions xi and a nucleus fixed at the origin. The Schrödinger Hamiltonian of such a system is given by $${H_{Z,N}} = \sum\limits_{i = 1}^N {\left( { - {\Delta _{{x_i}}} - \frac{Z} {{\left| {{x_i}} \right|}}} \right)} + \frac{1} {2}\sum\limits_{i \ne j} {\frac{1} {{\left| {{x_i} - {x_i}} \right|}}}$$ acting on $$= \wedge _{i = 1}^N{L^2}$$ (R3) (in this exposition, in order to simplify notation, we neglect spin.) Define the ground state of an atom of charge Z by $$E\left( Z \right) = {\kern 1pt} \mathop {\inf }\limits_N \mathop {\inf }\limits_{\mathop {\left\| \Psi \right\| = 1}\limits_{\Psi \in } } \;\left\langle {{H_{Z,N\Psi ,\Psi }}} \right\rangle$$.

C. L. Fefferman, L. A. Seco

Existence of the Schwinger Functions of the Three-Dimensional Gross-Neveu Model

One of the conclusions of previous rigorous studies on renormalization theory is that the concept of perturbative renormalizability shall not be considered as a fundamental requirement for a quantum field model to exist. Indeed, to solve the question of the ultraviolet (UV) limit, ρ → ∞, of a model presenting a set of parameters Ω ρ [ρ labelling an UV cutoff], in most cases one must drop out completely the traditional intermediate step involving a Taylor expansion in the coupling constant and properly ask whether or not one can prescribe functions for the parameters appearing in Ω ρ , in terms of the variable ρ and the set Ω ren of (finite and often positive) renormalized parameters, such that the n-point Schwinger (correlation) functions S n,ρ exist when ρ → ∞ and describe a non-trivial (interacting) system.

P. A. Faria da Veiga

Progress in Completely Integrable Models of Quantum Field Theory

Recently a number of subjects became interrelated. For a long time field theory and statistical mechanics developed separately. From the first subject we mention as a cornerstone the Thirring model. Classical field theoretic models like KdV, NSE, KP, Sine-Gordon and Toda models were seen to be integrable and led to the Lax pair, the AKNS and Zhakarov-Shabat scheme.

H. Grosse

Inverse Scattering at Fixed Energy

Let - Δ + V be a quantum mechanical two-body Hamiltonian in L2(Rn), n ≥ 3, and let S(k) be the corresponding scattering matrix at energy k2. We consider the classical problem of recovering V from knowledge of S(k) at one energy. The potential V(x) is not assumed to have any spherical symmetry. (The spherically symmetric case, including the non-uniqueness which arises if one allows potentials with reasonably mild decay at infinity, has been extensively studied—see [3] and references given there.) We show (Theorem 3.1) that if V has compact support and is in Ln/2 then it is uniquely determined by S(k); the proof gives a method to reconstruct the potential from the scattering matrix.

Semi-Classical and High Energy Asymptotics of the Scattering Phase for Perturbations of Elliptic Operators

For usual Schrödinger operators the scattering phase can be defined as follows: Let us consider the free Hamiltonian $${H_0} = - \frac{{{\hbar ^2}}} {{2m}}\Delta$$ and a smooth perturbation H = H0 + V of H0 such that V(x) = O(|x|-ρ) as |x| → + ∞, x ∈Rn, the configuration space. If ρ > n then it is well known that the scattering matrix S(λ), λ > 0, for the system (H, H0) is a trace-class perturbation of the identity on L2(Sn - 1). The scattering phase: s(λ), is then defined by the equality: (1)$$detS(\lambda ) = \exp \left( { - 2i\pi s\left( \lambda \right)} \right)$$s(λ) and its derivatives have many interesting properties as well as from physical and mathematical point of view. This notion was first introduced for central potentials. In that case S(λ) is diagonalized on the spherical harmonics: S(λ)f = (exp (2iδ(λ))f ∀f ∈ ϰ;(spherical harmonics of degree ℓ) [11]. We have clearly: $$s\left( \lambda \right) = - \frac{1} {\pi }\sum\nolimits_{\ell \geqslant 0} {\left( {2\ell + 1} \right)} {\delta _\ell }\left( \lambda \right)$$.

Didier Robert

On the Mapping Properties of the Wave Operators in Scattering Theory

In this note we describe some recent results on the mapping properties of the wave operators for two-body Schrödinger operators with long range potentials. We show that the wave operators and adjoint wave operators, localized in energy, map a weighted L2-space into a slightly larger weighted L2-space. We also give some preliminary results on the mapping properties between the Lp-spaces. The precise statement of the results requires some preparation.

Arne Jensen, Shu Nakamura

Reduction of Kac-Moody Systems

The general structure of reductions of the Wess-Zumino-Novikov-Witten theory by first class Kac-Moody constraints is analysed. Algebraic conditions are derived which ensure exact integrability, conformal invariance and TV-symmetry for the reduced effective theories. Solutions to the general algebraic conditions are given leading to new generalized conformal Toda theories. A Lagrangean implementation of the reduction is established in general and is used for setting up the framework for quantizing the effective theories.

Andreas W. Wipf

Quantum Symmetry of Rational Conformal Models

Recent progress in understanding the Uq(sl2) symmetry of sû(2) k current algebra and of“thermal” minimal conformal models is previewed. New features include the introduction of a pair of regular bases of 4-point invariants in the space of conformal blocks and in the quantum group space as well as a derivation of the results for minimal models from the current algebra results for a fractional level k.

Ludmil K. Hadjiivanov, Yassen S. Stanev, Ivan T. Todorov

Quasi-Normal Modes of the Schwarzschild Space-Time

”Loosely speaking, the black hole vibrates around spherical symmetry in a quasi-normal mode, and the mode is slowly damped by gravitational radiation.” With this sentence W.Press [1] introduced 1971 the notion of “quasi-normal modes” into the physics of black holes. The purpose of this note is to clarify the mathematical meaning of that notion. 1 This seems to be useful, because in the literature there is some confusion about which properties of normal modes are shared by quasi-normal ones, and which are not. An example of this confusion is the question about the completeness of the quasi-normal modes.

Bernd G. Schmidt

The Initial Value Problem for Self-Gravitating Fluid Bodies

Surprisingly little is known concerning the initial value problem for material bodies within the Newtonian and Einsteinian theories of gravity. An understanding of this question has an obvious significance for the study of the problem of motion in these two theories. The following is a summary of some new results concerning the case where the matter model chosen is a perfect fluid. The fact that fluid bodies are being described will be encoded by the assumption that the matter density has spatially compact support; it is important to note that replacing this by the requirement that the density tends to zero at infinity at each fixed time does not lead to a significant simplification. In this section results concerning local in time existence will be discussed while in the following section some global statements will be collected.

Alan D. Rendall

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