Surfaces and the Sklyanin Bracket

 We discuss the Lie Poisson group structures associated to splittings of the loop group LGL(N,ℂ), due to Sklyanin. Concentrating on the finite dimensional leaves of the associated Poisson structure, we show that the geometry of the leaves is intimately related to a complex algebraic ruled surface with a ℂ ^*-invariant Poisson structure. In particular, Sklyanin's Lie Poisson structure admits a suitable abelianisation, once one passes to an appropriate spectral curve. The Sklyanin structure is then equivalent to one considered by Mukai, Tyurin and Bottacin on a moduli space of sheaves on the Poisson surface. The abelianization procedure gives rise to natural Darboux coordinates for these leaves, as well as separation of variables for the integrable Hamiltonian systems associated to invariant functions on the group. Received: 8 August 2001/Accepted: 29 April 2002 Published online: 14 October 2002 RID="★" ID="★" The first author of this article would like to thank NSERC and FCAR for their support RID="★★" ID="★★" The second author was partially supported by NSF grant number DMS-9802532     

Torus Chiral <Emphasis Type="Italic">n</Emphasis>-Point Functions for Free Boson and Lattice Vertex Operator Algebras

 We obtain explicit expressions for all genus one chiral n-point functions for free bosonic and lattice vertex operator algebras. We also consider the elliptic properties of these functions. Received: 6 May 2002 / Accepted: 4 October 2002 Published online: 24 January 2003 RID="*" ID="*" Partial support provided by NSF DMS-9709820 and the Committee on Research, University of California, Santa Cruz RID="⋆⋆" ID="⋆⋆" Supported by an Enterprise Ireland Basic Research Grant and the Millenium Fund, National University of Ireland, Galway Communicated by L. Takhtajan     

Decay Rates and Probability Estimates¶for Massive Dirac Particles¶in the Kerr–Newman Black Hole Geometry

The Cauchy problem is considered for the massive Dirac equation in the non-extreme Kerr–Newman geometry, for smooth initial data with compact support outside the event horizon and bounded angular momentum. We prove that the Dirac wave function decays in at least at the rate t ^−5/6. For generic initial data, this rate of decay is sharp. We derive a formula for the probability p that the Dirac particle escapes to infinity. For various conditions on the initial data, we show that p = 0, 1 or 0 < p < 1. The proofs are based on a refined analysis of the Dirac propagator constructed in [4]. Received: 20 August 2001 / Accepted: 22 January 2002 RID="*" ID="*"Present address: NWF I – Mathematik, Universit?t Regensburg, 93040 Regensburg, Germany.?E-mail: felix.finster@mathematik.uni-regensburg.de RID="**" ID="**"Research supported by NSERC grant # RGPIN 105490-1998. RID="***" ID="***"Research supported in part by the NSF, Grant No. DMS-0103998. RID="****" ID="****"Research supported in part by the NSF, Grant No. 33-585-7510-2-30.     

In-beam γ-ray spectroscopy of 42Ca

High-spin states of the ⁴²Ca nucleus, populated in the 68 MeV ¹⁸O + ³⁰Si reaction, have been studied in a γ-γ-recoil coincidence experiment. The level scheme of ⁴²Ca has been extended up to 13.7 MeV. An elaborate decay pattern with various paths, together with high-quality DCO and polarization information assigns spins and parities for almost all observed levels. The sequence of non-yrast positive-parity states is discussed and compared with highly deformed bands in ³⁶Ar and ⁴⁰Ca. Received: 26 November 2002 / Accepted: 11 December 2002 / Published online: 18 February 2003 RID="a" ID="a"e-mail: Malgorzata.Lach@ifj.edu.pl RID="b" ID="b"Present address: TRIUMF, Vancouver, Canada. RID="†" ID="†"Deceased. Communicated by D. Schwalm     

2H( 3He, t)2p reaction at 2 GeV

The exclusive ²H( ³He, t)2p reaction has been studied at 2 GeV for energy transfers up to 500 MeV and triton angles up to 3.4^°. The protons were measured in the large acceptance magnetic detector DIOGENE, in coincidence with the forward tritons detected in a dedicated magnetic arm. The energy transfer spectra extend well above the pion threshold. However, in the region of Δ excitation, the yield is less than 10% of the inclusive ²H( ³He, t) cross-section, which indicates the small contribution of the ΔN ↦ NN process. The angular distributions of the two protons in their center of mass have been analysed as a function of energy transfer and triton angle and a Legendre polynomial decomposition has been achieved. These data have been compared to a model based on a coupled-channel approach for describing the NN and NΔ systems. Received: 21 October 2002 / Accepted: 15 November 2002 / Published online: 11 March 2003 RID="a" ID="a"e-mail: ramstein@ipno.in2p3.fr RID="†" ID="†"Deceased. RID="†" ID="†"Deceased. RID="d" ID="d"Present adress: School of Engineering, J?nk?ping University, P.O. Box 1026, S-551 11 J?nk?ping, Sweden. Communicated by M. Gar?on     

Atomic coherent states studied by virtue of the EPR entangled state and their Wigner functions

Based on the newly constructed Einstein, Podolsky and Rosen (EPR) entangled state representation we introduce macroscopic classical functions associated with atomic coherent state τ with angular momentum value j. These functions are proportional to the ordinary one-variable Hermite polynomials of order 2j. The corresponding Wigner quasiprobability function for τ in phase space is also derived which turns out to be a two-variable Hermite polynomial H _{2j, 2j}. In so doing, a new classical-quantum correspondence scheme for angular momentum system is established. Received 7 August 2002 / Received in final form 14 December 2002 Published online 24 April 2003 RID="a" ID="a"Work supported by the National Natural Science Foundation of China under grant 10175057. RID="b" ID="b"e-mail: fhym@sjtu.edu.en     

Power-Law Bounds on Transfer Matrices and Quantum Dynamics in One Dimension

 We present an approach to quantum dynamical lower bounds for discrete one-dimensional Schr?dinger operators which is based on power-law bounds on transfer matrices. It suffices to have such bounds for a nonempty set of energies. We apply this result to various models, including the Fibonacci Hamiltonian. Received: 5 June 2002 / Accepted: 20 January 2003 Published online: 28 March 2003 RID="⋆" ID="⋆" D.D. was supported in part by NSF Grant No. DMS–0227289 Communicated by M. Aizenman     

Hitchin–Kobayashi Correspondence, Quivers, and Vortices

 A twisted quiver bundle is a set of holomorphic vector bundles over a complex manifold, labelled by the vertices of a quiver, linked by a set of morphisms twisted by a fixed collection of holomorphic vector bundles, labelled by the arrows. When the manifold is K?hler, quiver bundles admit natural gauge-theoretic equations, which unify many known equations for bundles with extra structure. In this paper we prove a Hitchin–Kobayashi correspondence for twisted quiver bundles over a compact K?hler manifold, relating the existence of solutions to the gauge equations to a stability criterion, and consider its application to a number of situations related to Higgs bundles and dimensional reductions of the Hermitian–Einstein equations. Received: 10 December 2001 / Accepted: 10 November 2002 Published online: 28 May 2003 RID="⋆" ID="⋆" Current address: Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK. E-mail:L.Alvarez-Consul@maths.bath.ac.uk RID="⋆⋆" ID="⋆⋆" Current address: Instituto de Matemáticas y Física Fundamental, CSIC, Serrano 113 bis, 28006 Madrid, Spain. E-mail:oscar.garcia-prada@uam.es Communicated by R.H. Dijkgraaf     

New localized Superluminal solutions to the wave equations with finite total energies and arbitrary frequencies

By a generalized bidirectional decomposition method, we obtain new Superluminal localized solutions to the wave equation (for the electromagnetic case, in particular) which are suitable for arbitrary frequency bands; several of them being endowed with finite total energy. We construct, among the others, an infinite family of generalizations of the so-called “X-shaped" waves. Results of this kind may find application in the other fields in which an essential role is played by a wave-equation (like acoustics, seismology, geophysics, gravitation, elementary particle physics, etc.). Received 23 June 2002 Published online 24 September 2002 RID="a" ID="a"Work partially supported by MIUR and INFN (Italy), and by FAPESP (Brazil). This paper did first appear as e-print physics/0109062 [and as preprint INFN/FM-01/02 (I.N.F.N.; Frascati, 2001)]. RID="b" ID="b"e-mail: recami@mi.infn.it     

Algebraic Quantization of the Closed Bosonic String

 The gauge invariant observables of the closed bosonic string are quantized in four space-time dimensions by constructing their quantum algebra in a manifestly covariant approach, respecting all symmetries of the classical observables. The quantum algebra is the kernel of a derivation on the universal enveloping algebra of an infinite-dimensional Lie algebra. The search for Hilbert space representations of this algebra is separated from its construction, and postponed. Received: 26 February 2002 / Accepted: 5 September 2002 Published online: 28 March 2003 RID="⋆" ID="⋆" The article is based on the diploma thesis of the first author (C.M.) [1] under the supervision of K. Pohlmeyer, Universit?t Freiburg, completing a project by the second author (K.-H.R.) lying dormant since around 1987. RID="⋆⋆" ID="⋆⋆" Present address: Dept. of Mathematics, Heriot-Watt University, Riccarton, Edinburgh EH14 4AS, UK. E-mail:C.Meusburger@ma.hw.ac.uk Communicated by H. Araki, D. Buchholz and K. Fredenhagen     

Recursion Relations for Unitary Integrals,Combinatorics and the Toeplitz Lattice

 In a discussion in spring 2001, Alexei Borodin showed us recursion relations for the Toeplitz determinants going with the symbols e ^{t(z + z−1)} and \!. Borodin obtained these relations using Riemann-Hilbert methods; see the recent work of Borodin B and Baik Baik. The nature of Borodin's recursion relations pointed towards the Toeplitz lattice and its Virasoro algebra, introduced by us in AvM1. In this paper, we take the Toeplitz lattice and Virasoro algebra approach for a fairly large class of symbols, leading to a systematic way of generating recursion relations. The latter are very naturally expressed in terms of the L-matrices appearing in the Toeplitz lattice equations. As a surprise, we find, compared to Borodin's, a different set of relations, except for the 3-step relations associated with the symbol e ^{t(z + z−1)} . The Painlevé analysis of the Toeplitz lattice enables us to show the ``singularity confinement' for these recursion relations. Received: 30 January 2002 / Accepted: 6 January 2003 Published online: 19 May 2003 RID="⋆" ID="⋆" The support of a National Science Foundation grant DMS-01-00782 is gratefully acknowledged. RID="⋆⋆" ID="⋆⋆" The support of a National Science Foundation grant DMS-01-00782, a Nato, a FNRS and a Francqui Foundation grant is gratefully acknowledged. Communicated by L. Takhtajan     

Large Time Behavior for Vortex Evolution in the Half-Plane

 In this article we study the long-time behavior of incompressible ideal flow in a half plane from the point of view of vortex scattering. Our main result is that certain asymptotic states for half-plane vortex dynamics decompose naturally into a nonlinear superposition of soliton-like states. Our approach is to combine techniques developed in the study of vortex confinement with weak convergence tools in order to study the asymptotic behavior of a self-similar rescaling of a solution of the incompressible 2D Euler equations on a half plane with compactly supported, nonnegative initial vorticity. Received: 28 June 2002 / Accepted: 6 January 2003 Published online: 5 May 2003 RID="⋆" ID="⋆" Research supported in part by CNPq grant 300.962/91-6 RID="⋆⋆" ID="⋆⋆" Research supported in part by CNPq grant 300.158/93-9 Communicated by P. Constantin     

On Unitary Representations of Nilpotent Gauge Groups

 Groups of smooth maps from spheres to appropriate nilpotent Lie groups exhibit some peculiar properties of the unitary duals of infinite-dimensional groups. Received: 22 October 2000 / Accepted: 22 November 2002 Published online: 21 February 2003 RID="⋆" ID="⋆" This research was supported in part by CONICET, FONCYT, CONICOR and UNC. Communicated by H. Araki     

UCN production in superfluid helium

Ultra-cold neutrons (UCN) are produced in superfluid helium by single- and multi-phonon excitation. The UCN production rate density R _II via multiphonons can be larger than that by one-phonon excitation R _I being due to the dependence of the incident neutron spectral flux density dφ/dλ on the wavelength λ. Received: 28 March 2002 / Accepted: 19 December 2002 / Published online: 11 March 2003 RID="a" ID="a"e-mail: wschott@e18.physik.tu-muenchen.de RID="b" ID="b"On leave of absence from PNPI, Gatchina, Russia Communicated by T. Walcher     

Glauber Dynamics of the Random Energy Model

 We investigate the long-time behavior of the Glauber dynamics for the random energy model below the critical temperature. We give very precise estimates on the motion of the process to and between the states of extremal energies. We show that when disregarding time, the consecutive steps of the process on these states are governed by a Markov chain that jumps uniformly on all possible states. The mean times of these jumps are also computed very precisely and are seen to be asymptotically independent of the terminal point. A first indicator of aging is the observation that the mean time of arrival in the set of states that have waiting times of order T is itself of order T. The estimates proven in this paper will furnish crucial input for a follow-up paper where aging is analysed in full detail. Received: 9 October 2001 / Accepted: 17 October 2002 Published online: 28 February 2003 RID="*" ID="*" Work partially supported by the Swiss National Science Foundation under contract 21-65267.01 RID="⋆⋆" ID="⋆⋆" On leave from CPT-CNRS, Luminy, Case 907, 13288 Marseille Cedex 9, France. E-mail:veronique.gayrard@epfl.ch Communicated by M. Aizenman     

Long Range Scattering and Modified Wave Operators for the Maxwell-Schrödinger System I. The Case of Vanishing Asymptotic Magnetic Field

 We study the theory of scattering for the Maxwell-Schr?dinger system in space dimension 3, in the Coulomb gauge. In the special case of vanishing asymptotic magnetic field, we prove the existence of modified wave operators for that system with no size restriction on the Schr?dinger data and we determine the asymptotic behaviour in time of solutions in the range of the wave operators. The method consists in partially solving the Maxwell equations for the potentials, substituting the result into the Schr?dinger equation, which then becomes both nonlinear and nonlocal in time, and treating the latter by the method previously used for the Hartree equation and for the Wave-Schr?dinger system. Received: 1 August 2002 / Accepted: 9 December 2002 Published online: 14 March 2003 RID="⋆" ID="⋆" Unité Mixte de Recherche (CNRS) UMR 8627 Communicated by P. Constantin     

High-spin states in the <Superscript>97</Superscript>Tc nucleus

High-spin states in the ⁹⁷Tc nucleus have been studied by in-beam γ-ray spectroscopy with the reaction ⁸²Se(¹⁹F,4nγ) at 68 MeV incident energy. Excited states have been observed up to about 8 MeV excitation and spin 43/2. The observed level scheme is compared with results of shell model calculations. Received: 22 November 2002 / Accepted: 23 December 2002 / Published online: 18 March 2003 RID="a" ID="a"e-mail: bucurescu@tandem.nipne.ro RID="b" ID="b"Present address: INFN, Laboratori Nazionali di Legnaro, Legnaro, Italy. RID="b" ID="b"Present address: INFN, Laboratori Nazionali di Legnaro, Legnaro, Italy. RID="b" ID="b"Present address: INFN, Laboratori Nazionali di Legnaro, Legnaro, Italy. RID="c" ID="c"Present address: Università di Padova, Padova, Italy. Communicated by C. Signorini     

Potts Model on Infinite Graphs and the Limit of Chromatic Polynomials

 Given an infinite graph 𝔾 quasi-transitive and amenable with maximum degree Δ, we show that reduced ground state degeneracy per site W _r (𝔾, q) of the q-state antiferromagnetic Potts model at zero temperature on 𝔾 is analytic in the variable 1/q, whenever |2Δe ³ /q|<1. This result proves, in an even stronger formulation, a conjecture originally sketched in [12] and explicitly formulated in [16 and 19], based on which a sufficient condition for W _r (𝔾, q) to be analytic at 1/q=0 is that 𝔾 is a regular lattice. Received: 16 January 2002 / Accepted: 17 October 2002 Published online: 18 February 2003 RID="*" ID="*" Partially supported by CNPq (Brazil) RID="**" ID="**" Partially supported by CNR, G.N.F.M. (Italy) Communicated by H. Spohn     

Hedgehog production in spatially correlated noise

The production of topological defects during a quench in a φ⁴ model is investigated. The influence of a spatially correlated noise on defect production in two and three dimensions is demonstrated. Received 28 August 2001 / Received in final form 11 February 2002 Published online 2 October 2002 RID="a" ID="a"Paper supported in part by ESF “COSLAB” Programme RID="b" ID="b"e-mail: sfdobrow@kinga.cyf-kr.edu.pl     

A Stochastic Model Associated with Enskog Equation and Its Kinetic Limit

 We examine a system of particles in which the particles travel deterministically in between stochastic collisions. The collisions are elastic and occur with probability ɛ ^d when two particles are at a distance σ. When the number of particles N goes to infinity and Nɛ ^d goes to a nonzero constant, we show that the particle density converges to a solution of the Enskog Equation. Received: 29 January 2002 / Accepted: 30 July 2002 Published online: 14 November 2002 RID="*" ID="*" Research supported in part by NSF Grant DMS-0072666