Density of States for Random Band Matrices

 By applying the supersymmetric approach we rigorously prove smoothness of the averaged density of states for a three dimensional random band matrix ensemble, in the limit of infinite volume and fixed band width. We also prove that the resulting expression for the density of states coincides with the Wigner semicircle with a precision 1/W ² , for W large but fixed. Received: 6 February 2002 / Accepted: 17 July 2002 Published online: 7 November 2002 RID="*" ID="*" Supported by NSF grant DMS 9729992     

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     

Twisted Orbifold K-Theory

 We use equivariant methods to define and study the orbifold K-theory of an orbifold X. Adapting techniques from equivariant K-theory, we construct a Chern character and exhibit a multiplicative decomposition for K ^* _orb (X)⊗ℚ, in particular showing that it is additively isomorphic to the orbifold cohomology of X. A number of examples are provided. We then use the theory of projective representations to define the notion of twisted orbifold K–theory in the presence of discrete torsion. An explicit expression for this is obtained in the case of a global quotient. Received: 21 August 2001 / Accepted: 27 January 2003 Published online: 13 May 2003 RID="*" ID="*" Both authors were partially supported by the NSF RID="*" ID="*" Both authors were partially supported by the NSF Communicated by R.H. Dijkgraaf     

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     

Construction of the Incipient Infinite Cluster for Spread-out Oriented Percolation Above 4 + 1 Dimensions

 We construct the incipient infinite cluster measure (IIC) for sufficiently spread-out oriented percolation on ℤ ^d × ℤ₊, for d +1 > 4+1. We consider two different constructions. For the first construction, we define ℙ ⁿ (E) by taking the probability of the intersection of an event E with the event that the origin is connected to (x,n)  ℤ ^d × ℤ₊, summing this probability over x  ℤ ^d , and normalising the sum to get a probability measure. We let n → ∞ and prove existence of a limiting measure ℙ_∞, the IIC. For the second construction, we condition the connected cluster of the origin in critical oriented percolation to survive to time n, and let n → ∞. Under the assumption that the critical survival probability is asymptotic to a multiple of n ⁻¹, we prove existence of a limiting measure ℚ_∞, with ℚ_∞ = ℙ_∞. In addition, we study the asymptotic behaviour of the size of the level set of the cluster of the origin, and the dimension of the cluster of the origin, under ℙ_∞. Our methods involve minor extensions of the lace expansion methods used in a previous paper to relate critical oriented percolation to super-Brownian motion, for d+1 > 4+1. Received: 13 December 2001 / Accepted: 11 July 2002 Published online: 29 October 2002 RID="*" ID="*" Present address: Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. E-mail: rhofstad@win.tue.nl     

On the Motion of a Charged Particle Interacting with an Infinitely Extended System

 We study the time evolution of a charged particle moving in a medium under the action of a constant electric field E. In the framework of fully Hamiltonian models, we discuss conditions on the particle/medium interaction which are necessary for the particle to reach a finite limit velocity. We first consider the case when the charged particle is confined in an unbounded tube of ℝ³. The electric field E is directed along the symmetry axis of the tube and the particle also interacts with an infinitely many particle system. The background system initial conditions are chosen in a set which is typical for any reasonable thermodynamic (equilibrium or non-equilibrium) state. We prove that, for large E and bounded interactions between the charged particle and the background, the velocity v(t) of the charged particle does not reach a finite limit velocity, but it increases to infinite as: |v(t)−Et|≤C ₀ (1+t), where C ₀ is a constant independent of E. As a corollary we obtain that, if the initial conditions of the background system are distributed according to any Gibbs state, then the average velocity of the charged particle diverges as time goes to infinite. This result is obtained for E large enough in comparison with the mean energy of the Gibbs state. We next study the one-dimensional case, in which the estimates can be improved. We finally discuss, at an heuristic level, the existence of a finite limit velocity for unbounded interactions, and give some suggestions about the case of small electric fields. Received: 7 March 2002 / Accepted: 23 September 2002 Published online: 8 January 2003 RID="*" ID="*" Work partially supported by the GNFM-INDAM and the Italian Ministry of the University. Communicated by J.L. Lebowitz     

Electron momentum density in Cu<Subscript>0.9</Subscript>Al<Subscript>0.1</Subscript>

A reconstruction technique based on the solution of the Radon transform in terms of Jacobi polynomials is used to obtain the 3D electron momentum density, ϱ(p), from nine high-resolution Compton profiles (CPs) for a Cu_0.9Al_0.1 disordered alloy single crystal. The method was also applied to theoretical CPs computed within the Korringa–Kohn–Rostoker coherent potential approximation (KKR-CPA) first-principles scheme for the same nine orientations of the crystal. The experimental ϱ(p) is in satisfactory agreement with the theoretical ϱ(p), shows most details of the Fermi surface (FS) and exhibits electron correlation effects. We comment on the map of the FS obtained by folding the reconstructed ϱ(p) into the first Brillouin zone, which yields the occupation number density, ϱ(k). A test of the validity of data via a consistency condition (within our reconstruction algorithm) as well as the propagation of experimental noise in the reconstruction of both ϱ(p) and ϱ(k) are investigated. Received: 24 October 2001 / Accepted: 20 January 2002 / Published online: 3 June 2002 RID="*" ID="*"Corresponding author. Fax: +48-71/344-10-29, E-mail: samsel@int.pan.wroc.pl     

Power-boosted difference-frequency source for high-resolution infrared spectroscopy

We analyzed the spectroscopic performance of a difference-frequency source that utilizes a 5-W Yb-fiber amplifier for the “signal” radiation in order to increase the “idler” power generated around 4.3 μm. The amplifier is seeded by a monolithic-cavity Nd:YAG laser at 1064 nm. The intensity noise spectral density of the “idler” radiation was characterized. Cavity-enhanced saturated-absorption spectroscopy was also performed to test the frequency resolution. In particular, we observed the Lamb-dip spectrum of the ro-vibrational (00⁰0-00⁰1)R(0)transition of ¹⁷O¹²C¹⁶O in natural abundance. To our knowledge, this is the first observation of that transition by the Lamb-dip technique. Received: 19 December 2002 / Revised version: 3 February 2003 / Published online: 16 April 2003 RID="*" ID="*"Corresponding author. Fax: +39-055/4572-451, E-mail: mazzotti@inoa.it     

Excitation of Tm<Superscript>+3</Superscript> at a wavelength of 1064 nm

We report on the experimental study of the excitation mechanism of fluorescence in a 10000-ppm wt. Tm⁺³-doped ZBLAN fiber. Visible (at 453 nm and 480 nm) and near-infrared (∼800 nm) radiations were studied under excitation at 1.064 μm. The up-conversion mechanisms responsible are investigated, based on experimental data. The results show that the³ H ₄ and the ¹ D ₂ levels are predominantly excited by ion-ion cross-relaxation processes. Received: 19 August 2002 / Published online: 8 January 2003 RID="*" ID="*"Corresponding author. Fax: +41/31-631-3765, E-mail: Reda.El-Agmy@iap.unibe.ch     

Connections on Naturally Reductive Spaces,Their Dirac Operator and Homogeneous Models in String Theory

 Given a reductive homogeneous space M=G/H endowed with a naturally reductive metric, we study the one-parameter family of connections ∇ ^t joining the canonical and the Levi-Civita connection (t=0, 1/2). We show that the Dirac operator D ^t corresponding to t=1/3 is the so-called ``cubic' Dirac operator recently introduced by B. Kostant, and derive the formula for its square for any t, thus generalizing the classical Parthasarathy formula on symmetric spaces. Applications include the existence of a new G-invariant first order differential operator on spinors and an eigenvalue estimate for the first eigenvalue of D <sup>1/3</sup>. This geometric situation can be used for constructing Riemannian manifolds which are Ricci flat and admit a parallel spinor with respect to some metric connection ∇ whose torsion T≠ 0 is a 3-form, the geometric model for the common sector of string theories. We present some results about solutions to the string equations and a detailed discussion of a 5-dimensional example. Received: 19 February 2002 / Accepted: 26 August 2002 Published online: 22 November 2002 RID="*" ID="*" This work was supported by the SFB 288 ``Differential geometry and quantum physics' of the Deutsche Forschungsgemeinschaft and the Max-Planck Society.     

Driven Tracer Particle in One Dimensional Symmetric Simple Exclusion

Consider an infinite system of particles evolving in a one dimensional lattice according to symmetric random walks with hard core interaction. We investigate the behavior of a tagged particle under the action of an external constant driving force. We prove that the diffusively rescaled position of the test particle εX(ε^-2 t), t > 0, converges in probability, as ε→ 0, to a deterministic function v(t). The function v(⋅) depends on the initial distribution of the random environment through a non-linear parabolic equation. This law of large numbers for the position of the tracer particle is deduced from the hydrodynamical limit of an inhomogeneous one dimensional symmetric zero range process with an asymmetry at the origin. An Einstein relation is satisfied asymptotically when the external force is small. Received: 5 December 1996 / Accepted: 30 June 1997     

Renormalized Kinetic Theory of Classical Fluids in and out of Equilibrium

We present a theory for the construction of renormalized kinetic equations to describe the dynamics of classical systems of particles in or out of equilibrium. A closed, self-consistent set of evolution equations is derived for the single-particle phase-space distribution function f, the correlation function C=〈δfδf〉, the retarded and advanced density response functions χ ^R,A =δf/δφ to an external potential φ, and the associated memory functions Σ ^R,A,C . The basis of the theory is an effective action functional Ω of external potentials φ that contains all information about the dynamical properties of the system. In particular, its functional derivatives generate successively the single-particle phase-space density f and all the correlation and density response functions, which are coupled through an infinite hierarchy of evolution equations. Traditional renormalization techniques (involving Legendre transform and vertex functions) are then used to perform the closure of the hierarchy through memory functions. The latter satisfy functional equations that can be used to devise systematic approximations that automatically imply the conservation laws of mass, momentum and energy. The present formulation can be equally regarded as (i) a generalization to dynamical problems of the density functional theory of fluids in equilibrium and (ii) as the classical mechanical counterpart of the theory of non-equilibrium Green’s functions in quantum field theory. It unifies and encompasses previous results for classical Hamiltonian systems with any initial conditions. For equilibrium states, the theory reduces to the equilibrium memory function approach used in the kinetic theory of fluids in thermal equilibrium. For non-equilibrium fluids, popular closures of the BBGKY hierarchy (e.g. Landau, Boltzmann, Lenard-Balescu-Guernsey) are simply recovered and we discuss the correspondence with the seminal approaches of Martin-Siggia-Rose and of Rose and we discuss the correspondence with the seminal approaches of Martin-Siggia-Rose and of Rose.     

Photoacoustic gas sensing employing fundamental and frequency-doubled radiation of a continuously tunable high-pressure CO2 laser

We present the first photoacoustic spectrometer for gas sensing employing both the fundamental and the frequency-doubled radiation of a continuously tunable high-pressure CO₂ laser with room temperature operation. A quasi-phase-matched diffusion-bonded GaAs crystal is used in the system for second-harmonic generation. A pulsed photoacoustic detection scheme with a non-resonant cell, equipped with an 80-microphone array, is employed. The wide continuous tuning range in the fundamental (9.2–10.7 μm) and the frequency-doubled (4.6–5.35 μm) regimes, together with the narrow linewidth of 540 MHz (0.018 cm^-1) for the 10-μm region and of 1050 MHz (0.0315 cm^-1) for the 5-μm region, allow the measurement of gas mixtures, individual species and isotope discrimination. This is illustrated with measurements on NO and CO₂. The measured isotope ratio ¹⁵ NO/¹⁴ NO=(3.58±0.55)×10^-3 agrees well with the literature (3.700×10^-3) and demonstrates the good selectivity of the system. Received: 30 April 2002 / Revised version: 10 June 2002 / Published online: 2 September 2002 RID="*" ID="*"Corresponding author. Fax: +41-1/633-1077, E-mail: sigrist@iqe.phys.ethz.ch     

Hitchin Systems – Symplectic Hecke Correspondence and Two-Dimensional Version

 The aim of this paper is two-fold. First, we define symplectic maps between Hitchin systems related to holomorphic bundles of different degrees. We call these maps the Symplectic Hecke Correspondence (SHC) of the corresponding Higgs bundles. They are constructed by means of the modification of the underlying holomorphic bundles. SHC allows to construct B?cklund transformations in the Hitchin systems defined over Riemann curves with marked points. We apply the general scheme to the elliptic Calogero-Moser (CM) system and construct SHC to an integrable SL(N,ℂ) Euler-Arnold top (the elliptic SL(N,ℂ)-rotator). Next, we propose a generalization of the Hitchin approach to 2d integrable theories related to the Higgs bundles of infinite rank. The main example is an integrable two-dimensional version of the two-body elliptic CM system. The previous construction allows us to define SHC between the two-dimensional elliptic CM system and the Landau-Lifshitz equation. Received: 19 April 2002 / Accepted: 14 November 2002 Published online: 18 February 2003 RID="*" ID="*" On leave from Institute of Oceanology, Moscow, Russia RID="**" ID="**" On leave from Institute of Theoretical and Experimental Physics, Moscow, Russia 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.     

A Dynamical Classification of the Range of Pair Interactions

We formalize a classification of pair interactions based on the convergence properties of the forces acting on particles as a function of system size. We do so by considering the behavior of the probability distribution function (PDF) P(F) of the force field F in a particle distribution in the limit that the size of the system is taken to infinity at constant particle density, i.e., in the “usual” thermodynamic limit. For a pair interaction potential V(r) with V(r→∞)∼1/r ^γ defining a bounded pair force, we show that P(F) converges continuously to a well-defined and rapidly decreasing PDF if and only if the pair force is absolutely integrable, i.e., for γ>d−1, where d is the spatial dimension. We refer to this case as dynamically short-range, because the dominant contribution to the force on a typical particle in this limit arises from particles in a finite neighborhood around it. For the dynamically long-range case, i.e., γ≤d−1, on the other hand, the dominant contribution to the force comes from the mean field due to the bulk, which becomes undefined in this limit. We discuss also how, for γ≤d−1 (and notably, for the case of gravity, γ=d−2) P(F) may, in some cases, be defined in a weaker sense. This involves a regularization of the force summation which is generalization of the procedure employed to define gravitational forces in an infinite static homogeneous universe. We explain that the relevant classification in this context is, however, that which divides pair forces with γ>d−2 (or γ<d−2), for which the PDF of the difference in forces is defined (or not defined) in the infinite system limit, without any regularization. In the former case dynamics can, as for the (marginal) case of gravity, be defined consistently in an infinite uniform system.     

Efficient generation of blue light by frequency doubling of a Nd:YAG laser operating on 4 F 3/24 I 9/2 transitions

Efficient room-temperature operation of ⁴ F _3/2⁴ I _9/2 transitions in diode-end-pumped Nd:YAG lasers at 946 nm and 938.5 nm is reported. 7.0-W continuous-wave output power at 946 nm and 3.9 W at 938.5 nm have been obtained. An analytical model has been developed for the quasi-three-level laser including the influence of energy-transfer upconversion. Frequency doubling of these transitions in periodically poled KTP generated blue light at 473 nm and 469 nm. Both single-pass extra-cavity as well as intracavity schemes have been investigated. Received: 31 July 2002 / Published online: 5 February 2003 RID="*" ID="*"Corresponding author. Fax: +46-8/750-5430, E-mail: stefan.bjurshagen@acreo.se     

Experimental investigation of non-linear selective reflection at the interface of Cs atomic vapor and quartz

We present an experimental study of non-linear selective reflection (SR) at a quartz–Cs-vapor interface in a V-type three-level scheme. The non-linear selective reflection at the Cs D₂ resonance line (⁶ S _1/2F=4→⁶ P _3/2) is monitored with and without pumping. The sub-Doppler reflection spectrum is observed and the effect of pumping on the signal of the selective reflection is investigated. The experimental result is in agreement with the theoretical calculation. Received: 16 April 2002 / Revised version: 12 June 2002 / Published online: 25 September 2002 RID="*" ID="*"Corresponding author. Fax: +86-351/701-1500, E-mail: zhaojm@sxu.edu.cn     

13CO2/12CO2 isotopic ratio measurements using a difference frequency-based sensor operating at 4.35 micrometers

A portable modular gas sensor for measuring the ¹³C/¹²C isotopic ratio in CO₂ with a precision of 0.8‰(±1σ) was developed for volcanic gas emission studies. This sensor employed a difference frequency generation (DFG)-based spectroscopic source operating at 4.35 μm (∼2300 cm^-1) in combination with a dual-chamber gas absorption cell. Direct absorption spectroscopy using this specially designed cell permitted rapid comparisons of isotopic ratios of a gas sample and a reference standard for appropriately selected CO₂ absorption lines. Special attention was given to minimizing undesirable precision degrading effects, in particular temperature and pressure fluctuations. Received: 16 April 2002 / Revised version: 28 May 2002 / Published online: 21 August 2002 RID="*" ID="*"Corresponding author. Fax: +1-713/5245237, E-mail: fkt@rice.edu     

Influence of storage time on laser cleaning of SiO<Subscript>2</Subscript> on Si

The influence of the ‘storage time’ τ_s on the threshold fluence φ_cl and the efficiency in dry laser cleaning is investigated. τ_s denotes the time between the deposition of particles and the cleaning. As a model system we employed silica spheres with diameters of 500 nm and 1500 nm on commercial silicon wafers and single-pulse KrF excimer laser radiation (τ_FWHM=28 ns). For the 1500-nm silica spheres, φ_cl was found to increase from about 65 mJ/cm² to 125 mJ/cm² for storage times of 4 h and 362 h, respectively. For 500-nm silica spheres the increase in the threshold fluence was less than 20% for storage times up to 386 h. Received: 12 July 2002 / Accepted: 12 July 2002 / Published online: 29 January 2003 RID="*" ID="*"Corresponding author. E-mail:dieter.baeurle@jku.at