Nelson's symmetry and the infinite volume behavior of the vacuum inP(φ)2

LetH _l be the Hamiltonian in aP()₂ theory with sharp space cutoff in the interval (–l/2,l/2). LetE _l =inf(H _l ), (l)=–E _l /l, and let _l be the vacuum forH _l . discuss properties of (l) and _l . In particular, asl, there are finite constants <0 and such that (l), ((l)–)l, and hence (l)=+/l+o(l ^–1). Moreover exp(–c ₁ l) _{l 1}exp(–c ₂ l) forc ₁,c ₂ positive constants, where _{l 1} is theL ¹(Q, d₀) norm of ₁ with respect to the Fock vacuum measure. We also present a new proof of recent estimates of Glimm and Jaffe on local perturbations ofH _l in the infinite volume limit.Research sponsored by AFOSR under Contract No. F44620-71-C-0108.On leave from Istituto di Fisica Teorica, Universitá di Napoli and Istituto Nazionale di Fisica Nucleare, Sezione di Napoli.A. Sloan Foundation Fellow.     

The spectrum of a Schrödinger operator inL p ℝ v isp-independent

LetH _p =–1/2+V denote a Schrödinger operator, acting inL _p ^v , 1p. We show that (H _p )=(H ₂) for allp[1, ], for rather general potentialsV.     

On the bundle of connections and the gauge orbit manifold in Yang-Mills theory

In an appropriate mathematical framework we supply a simple proof that the quotienting of the space of connections by the group of gauge transformations (in Yang-Mills theory) is aC principal fibration. The underlying quotient space, the gauge orbit space, is seen explicitly to be aC manifold modelled on a Hilbert space.     

Anomalies and curvature ofW manifolds

We study the holomorphic structure of certain complex manifolds associated withW algebras, namely, the flag manifoldsW /T andW ₁₊/T ₁₊, and the spacesW /SL(),R) andW ₁₊/GL(,R), whereT andT ₁₊ are the maximal tori inW andW ₁₊. We compute their Ricci curvature and show how the results are related to the anomaly-freedom conditions forW andW ₁₊. We discuss the relation of these manifolds with extensions of universal Teichmüller space.Supported in part by the U.S. Department of Energy, under grant DE-AS05-81ER40039Supported in part by the U.S. Department of Energy, under grant DE-FG03-84ER40168     

Characteristic Properties of the Scattering Data for the mKdV Equation on the Half-Line

In this paper we describe characteristic properties of the scattering data of the compatible eigenvalue problem for the pair of differential equations related to the modified Korteweg-de Vries (mKdV) equation whose solution is defined in some half-strip or in the quarter plane (0<x<)×[0,T), T. We suppose that this solution has a C initial function vanishing as x, and C boundary values, vanishing as t when T=. We study the corresponding scattering problem for the compatible Zakharov-Shabat system of differential equations associated with the mKdV equation and obtain a representation of the solution of the mKdV equation through Marchenko integral equations of the inverse scattering method. The kernel of these equations is valid only for x0 and it takes into account all specific properties of the pair of compatible differential equations in the chosen half-strip or in the quarter plane. The main result of the paper is the collection A–B–C of characteristic properties of the scattering functions given below.     

Diffusion in a one-dimensional gas of hard point particles

We simulate the classical diffusion of a particle of massM in an infinite one-dimensional system of hard point particles of massm in equilibrium. Each computer run corresponds to about 10⁸ collisions of the diffusive particle. We find that (t) 1/t fort large enough, and a crossover from an M m regime where=2 to=3 forM=m. The diffusion constant has a sharp maximum atM=m. We study moments x(t)² and x(t)⁴, and examine the behavior ofq ² (t)=x(t)⁴/3x(t)²². We find thatq(t)1 (consistent with a normal distribution) in theM limit (for all timest) and in the t limit for allM. On sabbatical leave from IVIC-Instituto Venezolano de Investigaciones Cientificas.     

The infinite time limit for the time-dependent Born-Oppenheimer approximation

We analyze the Schrödinger equation , whereH() is the hamiltonian of the molecular system consisting of nuclei with masses proportional to ^–4 and electrons with masses of order 1. Using the Born-Oppenheimer method we construct the leading order asymptotic expansion to the exact solutions of the equation. We show that if the particles interact through smooth potentials decaying suitably as the distance between particles tends to , then the expansion holds uniformly for all timest[0,). By similar analysis one can show validity of the expansion fort(–,0], thus our results hold for scattering theory.The material in this paper is contained in a dissertation submitted to the faculty of VPI & SU in partial fulfillment of the requirements for the Ph.D. degree.     

Dispersive Estimates for Schrödinger Operators in Dimensions One and Three

We consider L¹L estimates for the time evolution of Hamiltonians H=–+V in dimensions d=1 and d=3 with bound We require decay of the potentials but no regularity. In d=1 the decay assumption is (1+|x|)|V(x)|dx<, whereas in d=3 it is |V(x)|C(1+|x|)^–3–.Supported by the NSF grant DMS-0070538 and a Sloan fellowship.     

Energy-level statistics of model quantum systems: Universality and scaling in a lattice-point problem

We investigate the statistics of the numberN(R, S) of lattice pointsnZ ², in an annular domain (R, w)=(R+w)A\RA, whereR, w>0. HereA is a fixed convex set with smooth boundary andw is chosen so that the area of (R, w) isS. The statistics comes fromR being taken as random (with a smooth density) in some interval [c ₁ T,c ₂,T],c ₂>c ₁>0. We find that in the limitT the variance and distribution of N=N(R; S)–S depend strongly on howS grows withT. There is a saturation regimeS/T, asT, in which the fluctuations in N coming from the two boundaries of are independent. Then there is a scaling regime,S/Tz, 0<z<, in which the distribution depends onz in an almost periodic way going to a Gaussian asz0. The variance in this limit approachesz for genericA, but can be larger for degenerate cases. The former behavior is what one would expect from the Poisson limit of a distribution for annuli of finite area.     

Electroweak interactions at an infinite sublayer quark level. II

It is shown that the standard model of the electroweak interactions holds at an infinite sublayer quark level, insofar as we consider the weak isospin doublet (u _{L,u L} ^cp , whereu is an infinite number of quarks at an infinite sublayer level.     

On a regularity problem occurring in connection with Anosov diffeomorphisms

Let be aC -manifold and _s and _u be two Hölder foliations, transverse, and with uniformlyC leaves. If a functionf is uniformlyC along the leaves of the two foliations, then it isC on . The proof is elementary.     

Exact exponent for the number of persistent spins in the zero-temperature dynamics of the one-dimensional Potts model

For the zero-temperature Glauber dynamics of theq-state Potts model, the fractionr(q, t) of spins which never flip up to timet decays like a power lawr(q, t)t ^–(q) when the initial condition is random. By mapping the problem onto an exactly soluble one-species coagulation model (A+AA) or alternatively by transforming the problem into a free-fermion model, we obtain the exact expression of (q) for all values ofq. The exponent (q) is in general irrational, (3)=0.53795082..., (4)=0.63151575..., ..., with the exception ofq=2 andq=, for which (2)=3/8 and ()=1.     

Zur Temperaturabhängigkeit einiger Kenngrössen der Verfestigungskurven von Zinkeinkristallen

Zusammenfassung Es werden Ergebnisse beschrieben über die Temperaturabhängigkeit: der kritischen Schubspannung ₀, des Verfestigungskoeffizienten _A , im BereichA der Verfestigungskurve; des Verfestigungskoeffizienten _B im BereichB, der Längea _A des BereichsA und der Schubspannung _A am Ende des BereichsA. Die kritische Schubspannung wächst im Temperaturbereich 225–344°K linear mit abnehmender Temperatur. Der Temperaturverlauf des Verfestigungskoeffizienten _A , stimmt überein mit älteren Messungen an h.k.p. Metallen. Ähnlich verhält sich der Verfestigungskoeffizient _B . Die Schubspannung am Ende des BereichsA wird durch thermisch aktivierte Vorgänge beeinflußt. Der BereichA verkürzt sich im Temperaturbereich 0–85°C linear mit zunehmender Temperatur. Temperaturwechselversuche zeigen, daß im BereichB thermisch irreversible Änderungen in der Versetzungsstruktur auftreten.

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: ₀, _A , _B , _{A A} . 226 344°K . _A . _B . . 0–85°C . , .

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Herrn Dr. F. Kroupa und Ing. B. esták bin ich für das Durchlesen des Manuskripts sowie wertvolle Hinweise zu Dank verpflichtet. Mein besonderer Dank für zahlreiche fruchtbare Diskusionen und Anregungen gilt Herrn P. Kratochvíl. Herrn Prof. Dr. A. Seeger danke ich ganz besonders für Kritik des Manuskripts und wertvolle Ratschläge.     

The Hamiltonian systems on the Poisson structure of the quasi-classical limit of GL q (∞)

We consider the Hamiltonian systems on the Poisson structure of GL() which is introduced from the quantum group GL _q () by the so-called quasi-classical limit of GL _q (). Furthermore, we show that the Toda lattice hierarchy is a Hamiltonian system of this structure.     

The Percolation Transition for the Zero-Temperature Stochastic Ising Model on the Hexagonal Lattice

On the planar hexagonal lattice , we analyze the Markov process whose state (t), in , updates each site v asynchronously in continuous time t0, so that _v (t) agrees with a majority of its (three) neighbors. The initial _v (0)'s are i.i.d. with P[ _v (0)=+1]=p[0,1]. We study, both rigorously and by Monte Carlo simulation, the existence and nature of the percolation transition as t and p1/2. Denoting by ⁺(t,p) the expected size of the plus cluster containing the origin, we (1) prove that ⁺(,1/2)= and (2) study numerically critical exponents associated with the divergence of ⁺(,p) as p1/2. A detailed finite-size scaling analysis suggests that the exponents and of this t= (dependent) percolation model have the same values, 4/3 and 43/18, as standard two-dimensional independent percolation. We also present numerical evidence that the rate at which (t)() as t is exponential.     

A note on the Lie algebras g(m∞)

The algebras g(m) are interpreted as realisations of the infinite rank affine Lie algebras g.     

Quantum Ergodicity of Boundary Values of Eigenfunctions

Suppose that ⁿ is a bounded, piecewise smooth domain. We prove that the boundary values (Cauchy data) of eigenfunctions of the Laplacian on with various boundary conditions are quantum ergodic if the classical billiard map on the ball bundle B^*() is ergodic. Our proof is based on the classical observation that the boundary values of an interior eigenfunction , =² is an eigenfunction of an operator F_h on the boundary of with h=^–1. In the case of the Neumann boundary condition, F_h is the boundary integral operator induced by the double layer potential. We show that F_h is a semiclassical Fourier integral operator quantizing the billiard map plus a small remainder; the quantum dynamics defined by F_h can be exploited on the boundary much as the quantum dynamics generated by the wave group were exploited in the interior of domains with corners and ergodic billiards in the work of Zelditch-Zworski (1996). Novelties include the facts that F_h is not unitary and (consequently) the boundary values are equidistributed by measures which are not invariant under and which depend on the boundary conditions. Ergodicity of boundary values of eigenfunctions on domains with ergodic billiards was conjectured by S. Ozawa (1993), and was almost simultaneously proved by Gerard-Leichtnam (1993) in the case of convex C^1,1 domains (with continuous tangent planes) and with Dirichlet boundary conditions. Our methods seem to be quite different. Motivation to study piecewise smooth domains comes from the fact that almost all known ergodic domains are of this form.The first author was partially supported by an Australian Research Council Fellowship.The second author was partially supported by NSF grant #DMS-0071358 and DMS-0302518.     

On Percolation with Fibers or Layers

We consider site percolation on Z ^d, directed edges going from any sZ ^d to s+A ₁,..., s+A _n, where A ₁,..., A _n are the same for all sites and at least two of them are noncollinear. A site is closed if it belongs to p+Block, where p is a point in a Poisson distribution in R ^dZ ^d with a density and Block={sL: |s|M}+{sR ^d: |s|}, where L is a linear subspace of R ^d, |·| is the Euclidean norm, =max(|A ₁|,..., |A _n|) and M is a parameter. We study the behavior of *, the critical value, and P _closed*, corresponding critical percentage of closed sites, when M. Denote R ^d/L the factor space. Call two nonzero vectors U, V codirected if U=kV, where k>0. Theorem. If there are A _i and A _j whose projections to R ^d/L are not codirected, then *1/M ^dim(L) and P _closed* remains separated both from 0 and 1 when M. If projections of all A ₁,..., A _n to R ^d/L are codirected, then *1/M ^dim(L)+1 and P _closed*1/M when M.     

The inverse scattering matrix for the Schrödinger equation when the potentialq(x) ∈L 1 1 with a singular term

When the potentialq(x) L ₁ ¹ with a singular term, the continuities of the scattering matrix of the Schrödinger equation are investigated. By means of the transformation approach, we arrive at the conclusion that the scattering matrix S(k) of such a potential is continuous for the wholek,- <k < .     

Invariants for smooth conjugacy of hyperbolic dynamical systems. IV

We show that if twoC transitive Anosov flows in a three-dimensional manifold are topologically conjugate and the Lyapunov exponents on corresponding periodic orbits agree, then the conjugating homeomorphism isC .Partially supported by NSF grant # DMS 85-04984