Showing papers in "Communications in Mathematical Physics in 1980"
TL;DR: In this article, simple dissipative dynamical systems exhibiting a transition from a stable periodic behavior to a chaotic one were studied, where the inverse coherence time grows continuously from zero to zero due to the random occurrence of widely separated bursts in the time record.
Abstract: We study some simple dissipative dynamical systems exhibiting a transition from a stable periodic behavior to a chaotic one. At that transition, the inverse coherence time grows continuously from zero due to the random occurrence of widely separated bursts in the time record.
1,753 citations
TL;DR: In this article, the authors prove the statement in the title of the paper and prove it in the paper's Appendix A, Section 2, Section 3, Section 4, Section 5.
Abstract: We prove the statement in the title of the paper.
666 citations
TL;DR: In this paper, a method for solving nonlinear ordinary and partial differential equations with regular and irregular singular points is developed. But the main idea is to study monodromy preserving deformations of linear ordinary differential equations, and the connections with isospectral deformations are discussed.
Abstract: A method for solving certain nonlinear ordinary and partial differential equations is developed. The central idea is to study monodromy preserving deformations of linear ordinary differential equations with regular and irregular singular points. The connections with isospectral deformations and with classical and recent work on monodromy preserving deformations are discussed. Specific new results include the reduction of the general initial value problem for the Painleve equations of the second type and a special case of the third type to a system of linear singular integral equations. Several classes of solutions are discussed, and in particular the general expression for rational solutions for the second Painleve equation family is shown to be −d/dx ln(Δ+/Δ−), where Δ+ and Δ− are determinants. We also demonstrate that each of these equations is an exactly integrable Hamiltonian system. The basic ideas presented here are applicable to a broad class of ordinary and partial differential equations; additional results will be presented in a sequence of future papers.
621 citations
TL;DR: In this article, a general definition of symmetries of gauge fields is proposed and a method developed for constructing symmetric fields for an arbitrary gauge group, where scalar fields occur naturally in the formalism and pure gauge theory reduces to a Higgs model in lower dimensions.
Abstract: A general definition of symmetries of gauge fields is proposed and a method developed for constructing symmetric fields for an arbitrary gauge group. Scalar fields occur naturally in the formalism and the pure gauge theory reduces to a Higgs model in lower dimensions.
567 citations
TL;DR: In this paper, it is shown on examples that the distance between nearby states is related to quantum fluctuations; in particular, in the particular case of the harmonic oscillator group the condition of zero curvature appears to be identical to that of non dispersion of wave packets.
Abstract: A metric tensor is defined from the underlying Hilbert space structure for any submanifold of quantum states. The case where the manifold is generated by the action of a Lie group on a fixed state vector (generalized coherent states manifold hereafter noted G.C.S.M.) is studied in details; the geometrical properties of some wellknown G.C.S.M. are reviewed and an explicit expression for the scalar Riemannian curvature is given in the general case. The physical meaning of such Riemannian structures (which have been recently introduced to describe collective manifolds in nuclear physics) is discussed. It is shown on examples that the distance between nearby states is related to quantum fluctuations; in the particular case of the harmonic oscillator group the condition of zero curvature appears to be identical to that of non dispersion of wave packets.
520 citations
TL;DR: In this article, a class of random finite difference Schrodinger operators with a random potential was studied and the exact location of the spectrum was obtained with probability one, in various situations, and criterions for a given part in the spectrum to be pure point or purely continuous.
Abstract: We study a class of random finite difference operators, a typical example of which is the finite difference Schrodinger operator with a random potential which arises in solid state physics in the tight binding approximation. We obtain with probability one, in various situations, the exact location of the spectrum, and criterions for a given part in the spectrum to be pure point or purely continuous, or for the static electric conductivity to vanish. A general formalism is developped which transforms the study of these random operators into that of the asymptotics of a multiple integral constructed from a given recipe. Finally we apply our criterions and formalism to prove that, with probability one, the one-dimensional finite difference Schrodinger operator with a random potential has pure point spectrum and developps no static conductivity.
403 citations
TL;DR: In this article, it was shown that a set of not necessarily distinct points in the plane determine a unique analytic solution to the first order Ginzburg-Landau equations with vortex numberN. This solution has the property that the Higgs field vanishes only at the points in a set and the order of vanishing at a given point is determined by the multiplicity of that point in the set.
Abstract: We prove that a set ofN not necessarily distinct points in the plane determine a unique, real analytic solution to the first order Ginzburg-Landau equations with vortex numberN. This solution has the property that the Higgs field vanishes only at the points in the set and the order of vanishing at a given point is determined by the multiplicity of that point in the set. We prove further that these are the onlyC∞ solutions to the first order Ginzburg-Landau equations.
361 citations
TL;DR: In this paper, the authors considered the Schrodinger equation for a single particle and its discrete analogues and proved that the spectra of corresponding random operators and their point spectra are dense with probability 1 and that in the one-dimensional case they have no absolutely continuous component.
Abstract: The paper considers the Schrodinger equation for a single particle and its discrete analogues. Assuming that the coefficients of these equations are homogeneous and ergodic random fields, it is proved that the spectra of corresponding random operators and their point spectra are dense with probability 1 and that in the one-dimensional case they have no absolutely continuous component. Rather wide sufficient conditions of exponential growth of the Cauchy solutions of the one-dimensional equations considered are found.
315 citations
TL;DR: In this paper, a translation invariant pure phase of a ferromagnet, finite susceptibility and the FKG inequalities together imply convergence of the block spin scaling limit to the infinite temperature Gaussian fixed point.
Abstract: In a translation invariant pure phase of a ferromagnet, finite susceptibility and the FKG inequalities together imply convergence of the block spin scaling limit to the infinite temperature Gaussian fixed point. This result is presented in a rather general probabilistic context and is applicable to infinite cluster density fluctuations in percolation models and to boson field fluctuations in (Euclidean) Yukawa quantum field theory models as well as to magnetization fluctuations in Ising models.
310 citations
TL;DR: In this article, a generalization of the Haag-Kastler axioms for local observables to Lorentzian manifolds is proposed, which is intended to resolve ambiguities in the construction of quantum field theories on manifolds.
Abstract: We propose a generalization of the Haag-Kastler axioms for local observables to Lorentzian manifolds. The framework is intended to resolve ambiguities in the construction of quantum field theories on manifolds. As an example we study linear scalar fields for globally hyperbolic manifolds.
303 citations
TL;DR: In this article, a recursive method is presented that allows the explicit computation of this aperiodic behavior from the initial subharmonics, and results have a character universal over specific systems, so that all such transitions are characterized by noise of a universal internal similarity.
Abstract: Some systems achieve aperiodic temporal behavior through the production of successive half subharmonics. A recursive method is presented here that allows the explicit computation of this aperiodic behavior from the initial subharmonics. The results have a character universal over specific systems, so that all such transitions are characterized by noise of a universal internal similarity.
TL;DR: In this article, the authors considered the ℏ→0 limit of the quantum dynamics generated by the Hamiltonian and proved that the evolution of certain Gaussian states is determined asymptotically as ℎ→0 by classical mechanics.
Abstract: We consider the ℏ→0 limit of the quantum dynamics generated by the HamiltonianH(ℏ)=−(ℏ2/2m)Δ+V We prove that the evolution of certain Gaussian states is determined asymptotically as ℏ→0 by classical mechanics For suitable potentialsV inn≧3 dimensions, our estimates are uniform in time and our results hold for scattering theory
TL;DR: In this paper, the principal chiral fields on the symplectic, unitary and orthogonal Lie groups are shown to be integrable by means of the inverse scattering problem method.
Abstract: Well known classical spinor relativistic-invariant two-dimensional field theory models, including the Gross-Neveu, Vaks-Larkin-Nambu-Jona-Lasinio and some other models, are shown to be integrable by means of the inverse scattering problem method. These models are shown to be naturally connected with the principal chiral fields on the symplectic, unitary and orthogonal Lie groups. The respective technique for construction of the soliton-like solutions is developed.
TL;DR: In this article, the perturbation theory for H = p 2+x2+x 2+βx 2n+1,n=1, 2, 2.
Abstract: We study the perturbation theory forH=p
2+x
2+βx
2n+1,n=1, 2, .... It is proved that when Imβ≠0,H has discrete spectrum. Any eigenvalue is uniquely determined by the (divergent) Rayleigh-Schrodinger perturbation expansion, and admits an analytic continuation to Imβ=0 where it can be interpreted as a resonance of the problem.
TL;DR: In this paper, Markov Partitions for some classes of billiards in two-dimensional domains on ℝ2 or 2-dimensional torus are constructed and the microcanonical distribution of the corresponding dynamical system is represented in the form of a limit Gibbs state and investigated the character of its approximations by finite Markov chains.
Abstract: Markov Partitions for some classes of billiards in two-dimensional domains on ℝ2 or two-dimensional torus are constructed. Using these partitions we represent the microcanonical distribution of the corresponding dynamical system in the form of a limit Gibbs state and investigate the character of its approximations by finite Markov chains.
TL;DR: In this paper, a class of nonlinear Klein-Gordon systems which are soluble by means of a scattering transform is presented, and a Backlund transformation and superposition formula for the general system is presented.
Abstract: We present a class of nonlinear Klein-Gordon systems which are soluble by means of a scattering transform. More specifically, for eachN≧2 we present a system of (N−1) nonlinear Klein-Gordon equations, together with the correspondingN ×N matrix scattering problem which can be used to solve it. We illustrate these with some special examples. The general system is shown to be closely related to the equations of the periodic Toda lattice. We present a Backlund transformation and superposition formula for the general system.
TL;DR: In this article, the authors consider itcrates of maps of an interval to itself and their stable periodic orbits and investigate rigorously those aspects of these bifurcations which are universal, independent of the choice of a particular one-parameter family.
Abstract: We consider itcrates of maps of an interval to itself and their stable periodic orbits. When these maps depend on a parameter, one can observe period doubling bifurcations as the parameter is varied. We investigate rigorously those aspects of these bifurcations which are universal, i.e. independent of the choice of a particular one-parameter family. We point out that this universality extends to many other situations such as certain chaotic regimes. We describe the ergodic properties of the maps for which the parameter value equals the limit of the bifurcation points.
TL;DR: In this article, a variety of new correlation inequalities which bound intermediate distance correlations from below by long distance correlations are presented. But none of these inequalities hold for spin 1/2 nearest neighbor Ising ferromagnets.
Abstract: We prove a variety of new correlation inequalities which bound intermediate distance correlations from below by long distance correlations. Typical is the following which holds for spin 1/2 nearest neighbor Ising ferromagnets:
$$\langle S_\alpha S_\gamma \rangle \leqq \sum\limits_{\delta \in B} {\langle S_\alpha S_\delta \rangle } \langle S_\delta S_\gamma \rangle$$
whereB is any subset of the lattice whose removal divides the lattice into pieces with α,γ in distinct components. We describe various applications, e.g. the above inequality implies the critical exponent inequality η<1.
TL;DR: In this article, a constructive proof of complete integrability of spherically symmetric self-dual equations in Euclidean spaceR4 for an arbitrary embedding of SU(2) in an arbitrary gauge groupG is given on the base of Lax-type representation and representation theory.
Abstract: A constructive proof of complete integrability of spherically symmetric self-dual equations in Euclidean spaceR4 for an arbitrary embedding of SU(2) in an arbitrary gauge groupG is given on the base of Lax-type representation and representation theory. The equations are solved explicitly for the case of simple Lie groupsG.
TL;DR: In this paper, the authors extend Lieb's limit theorem to general compact Lie groups and prove that every bounded operator is an integral of projections onto coherent vectors (i.e. every operator has "diagonal form") and discuss the classical limit for various continuum systems.
Abstract: We extend Lieb's limit theorem [which asserts that SO(3) quantum spins approachS2 classical spins asL→∞] to general compact Lie groups. We also discuss the classical limit for various continuum systems. To control the compact group case, we discuss coherent states built up from a maximal weight vector in an irreducible representation and we prove that every bounded operator is an integral of projections onto coherent vectors (i.e. every operator has “diagonal form”).
TL;DR: In this article, it was shown that the initial value problem for the linearized spatially-homogeneous Boltzmann equation has the form ∂f/∂t+Lf=0 withf(ξ,t=0) given.
Abstract: The initial value problem for the linearized spatially-homogeneous Boltzmann equation has the form ∂f/∂t+Lf=0 withf(ξ,t=0) given. The linear operatorL operates only on the ξ variable and is non-negative, but, for the soft potentials considered here, its continuous spectrum extends to the origin. Thus one cannot expect exponential decay forf, but in this paper it is shown thatf decays likee−λtβ with β<1. This result will be used in Part II to show existence of solutions of the initial value problem for the full nonlinear, spatially dependent problem for initial data that is close to equilibrium.
TL;DR: In this paper, it was shown that any Gibbs state of the two-dimensional ferromagnetic Ising system is of the form λμ++(1−λ)μ−, with some λ∈ [0, 1].
Abstract: It is shown that any Gibbs state of the two dimensional ferromagnetic Ising system is of the form λμ++(1−λ)μ−, with some λ∈ [0, 1]. This excludes the possibility of a locally stable phase coexistence and of translation symmetry breaking, which are known to occur in higher dimensions. Use is made in the proof of the stochastic aspects of the geometry of the interface lines.
TL;DR: For the SU(2) Yang-Mills equation, the authors proved that every solution to the SU (2) problem is either self-dual or anti-self dual.
Abstract: We prove that every solution to the SU(2) Yang-Mills equations, invariant under the lifting to the principle bundle of the action of the group, O(3), of rotations about a fixed line in ℝ4, with locally bounded and globally square integrable curvature is either self-dual or anti-self dual. In other words we prove, under the above assumptions, that every critical point of the Yang-Mills functional is a global minimum.
TL;DR: In this paper, a weak solution to the Navier-Stokes equations of incompressible fluid flow in 3-space is proposed, such that the curl of the fluid flow is continuous outside a locally closed set whose 5/3 dimensional Hausdorff measure is finite.
Abstract: SupposeU is an open bounded subset of 3-space such that the boundary ofU has Lebesgue measure zero. Then for any initial condition with finite kinetic energy we can find a global (i.e. for all time) weak solutionu to the time dependent Navier-Stokes equations of incompressible fluid flow inU such that the curl ofu is continuous outside a locally closed set whose 5/3 dimensional Hausdorff measure is finite.
TL;DR: An existence theorem for a probability measure on continuous paths in space was proved in this paper, which is a stochastic model for the geometric properties of long polymer chains, and it was shown that the probability measure can be computed in a deterministic manner.
Abstract: An existence theorem is proved for a probability measure on continuous paths in space, proposed by Edwards as a stochastic model for the geometric properties of long polymer chains.
TL;DR: In this paper, the authors consider the motion of a particle in a weak mean zero random force field and show that under suitable mixing and moment conditions on the force field, the particle converges weakly to a diffusion Markov process.
Abstract: We consider the motion of a particle in a weak mean zero random force fieldF, which depends on the position,x(t), and the velocity,v(t)=\(\dot x\)(t). The equation of motion is\(\ddot x\)(t)=ɛF(x(t),v(t), ω), wherex(·) andv(·) take values in ℝd,d≧3, and ω ranges over some probability space. We show, under suitable mixing and moment conditions onF, that as ɛ→0,vɛ(t)≡v(t/ɛ2) converges weakly to a diffusion Markov processv(t), and ɛ2xɛ(t) converges weakly to\(\int\limits_0^t {v(s)ds + x} \), wherex=lim ɛ2xɛ(0).
TL;DR: In this paper, a renormalization group technique is used to study rigorously the λ(∇φ)4 perturbation of the massless lattice field φ in dimensionsd≧2.
Abstract: The renormalization group technique is used to study rigorously the λ(∇φ)4 perturbation of the massless lattice field φ in dimensionsd≧2. Asymptoticity of the perturbation expansion in powers of λ is established for the free energy density. This is achieved by using Kadanoff's block spin transformation successively to integrate out high momentum degrees of freedom and by applying ideas previously used by Gallavotti and Balaban in the context of the ultraviolet problems. The method works for arbitrary semibounded polynomials in ∇φ and △φ.
TL;DR: In this paper, it was shown that self-dual solutions of Einstein's equations, with cosmological constant λ, correspond to certain complex manifolds, and this result generalizes the work of Penrose [1], who dealt with the case λ=0.
Abstract: It is shown that self-dual solutions of Einstein's equations, with cosmological constant λ, correspond to certain complex manifolds. This result generalizes the work of Penrose [1], who dealt with the case λ=0.
TL;DR: In this article, it was shown that a certain four-dimensional field theory has powerful structures in common with the two-dimensional 0(1, 3) non-linear σ-model.
Abstract: We show that a certain four-dimensional field theory has powerful structures in common with the two-dimensional 0(1, 3) non-linear σ-model.
TL;DR: In this article, the dynamics of a quantum mechanical system which consists of some particles with large masses and some particles having small masses were considered and it was shown that the particles of smaller mass move adiabatically and determine an effective potential in which the heavier particles move semiclassically.
Abstract: We consider the dynamics of a quantum mechanical system which consists of some particles with large masses and some particles with small masses. As we increase the large masses to infinity we obtain the following results: The particles of smaller mass move adiabatically and determine an effective potential in which the heavier particles move semiclassically. Our methods can be applied to diatomic molecules with Coulomb forces.