Bounded perturbations of dynamics

If and are one-parameter automorphism groups of a von Neumann algebraM is said to be a bounded perturbation of if _t – _t 0 ast0. We give a complete characterization of the bounded perturbations of . In particular, we show that if can be implemented by a strongly continuous one-parameter group with self-adjoint generator (Hamiltonian)H, then can be implemented in the same way and the corresponding HamiltonianH can be chosen to be of the formH=VHV ^–1+h, whereV is a unitary ofM andh=h*M.On leave of absence from II. Institut für Theoretische Physik, Universität Hamburg, D-2000 Hamburg 50, Federal Republic of Germany     

Upper bounds onT c for one-dimensional Ising systems

We present upper bounds on the critical temperature of one-dimensional Ising models with long-range,l/n interactions, where 1<2. In particular for the often studied case of =2 we have an upper bound onT _c which is less than theT _c found by a number of approximation techniques. Also for the case where is small, such as =1.1, we obtain rigorous bounds which are extremely close, within 1.0%, to those found by approximation methods.     

Lattice vibrations in ordered and disordered thiourea

The critical dynamics of the Syozi model for dilute ferromagnetism is considered by the use of master equations. The dynamics is soluble as it is assumed that the time scale of motion on the sublattice on which the impurities move is so much faster than on the other sublattice that fast relaxing variables may be adiabatically eliminated, leaving a new soluble master equation. It is found that the linear and non-linear relaxation of magnetization exponents ^(l) and ^(nl) increase on dilution to ^(l)/(1–) and ^(nl)/(1–) respectively ( is the specific heat exponent for the pure system, which itself changes on dilution to –/(1–)). Thus if the exponents for the pure system obey the scaling law of Rácz and Fisher ^(nl)= ^(l)– ( is the magnetization exponent which changes on dilution to /(1–)) then so do the exponents for the diluted system. Similarly the exponent for spin diffusion changes on dilution to /(1–).     

Perturbations of flows on Banach spaces and operator algebras

For automorphism groups of operator algebras we show how properties of the difference _t – ' _t are reflected in relations between the generators , . Indeed for a von Neumann algebraM with separable predual we show that if _t – '_t 0.28 for smallt, then = 0(+)°^-1 where is an inner automorphism ofM and is a bounded derivation ofM. If the difference _t – ' _t =O(t) ast ; 0, then = + and if _t – ' _t 0.28 for allt then =. We prove analogous results for unitary groups on a Hilbert space andC ₀,C ₀ ^* groups on a Banach space.This paper subsumes an earlier work of the same title which appeared as a report from Z.I.F. der Universität BielefeldWith partial support of the U.S. National Science Foundation     

Some results for the exponential interaction in two or more dimensions

We show that for the regularized exponential interaction :e : ind space-time dimensions the Schwinger functions converge to the Schwinger functions for the free field ifd>2 for all or ifd=2 for all such that ||>₀.Partially sponsored by the I.H.E.S. through the Stiftung Volkswagenwerk     

Solution of coupled differential equations of soliton theory

We present the complete set of solutions of the coupled differential equations of the form ()²=(), ² =(). Equations of this form appear in several physical situations.     

InGaAs/GaAs multiple-quantum-well modulators and switches

The design, fabrication and characterization of electrooptical modulators and switches based on pseudomorphic InGaAs/GaAs multiple-quantum-well (MQW) structures is presented. The absorption and refractive index changes (, n) of In_0.2Ga_0.8As/GaAs MQW structures due to the quantum-confined Stark effect are examined in detail. The figures of merit /₀ and n/₀ give information on the design of modulation and switching devices. Based on these results, we develop two types of efficient and high-speed modulators, vertical and waveguide modulators, and for the first time an InGaAs/GaAs intersectional X-type switch. Recent experimental results for each device are presented.     

Schrödinger Operators with Empty Singularly Continuous Spectra

Let H be a semibounded perturbation of the Laplacian H ₀ in L ²( ^d ). For an admissible function sufficient conditions are given for the completeness of the scattering system (H), (H ₀). If is the exponential function and if e^{– H} is an integral operator we denote the kernel of the difference D = e^{– H} – e^{– H} ₀ by D (x, y), > 0. The singularly continuous spectrum of H is empty if^d dx ^d dy |D(x,y)| (1 + |y|²)< for some > 1. This result is applied to potential perturbations and to perturbations by imposing Dirichlet boundary conditions.     

Power law growth for the resistance in the Fibonacci model

Many one-dimensional quasiperiodic systems based on the Fibonacci rule, such as the tight-binding HamiltonianH(n)=(n+1)+(n–1)+v(n) (n),n,l ²(),, wherev(n)=[(n+1)]–[n],[x] denoting the integer part ofx and the golden mean , give rise to the same recursion relation for the transfer matrices. It is proved that the wave functions and the norm of transfer matrices are polynomially bounded (critical regime) if and only if the energy is in the spectrum of the Hamiltonian. This solves a conjecture of Kohmoto and Sutherland on the power-law growth of the resistance in a one-dimensional quasicrystal.     

Low-temperature and long-time anomalies of a damped quantum particle

The time evolution of a damped quantum particle is discussed. Dissipation is modeled by the bilinear coupling to a set of harmonic oscillators. Using a functional integral technique that accounts for initial correlations between the particle and the reservoir, one can express the dynamics of the damped particle entirely in terms of equilibrium correlation functions. The long-time behavior of these correlations is determined for memory damping arising from the coupling to a reservoir with spectral densityI() at low frequencies, where > 0. The time evolution of nonequilibrium initial states of the damped particle is discussed. At finite temperatures an initially localized state is found to spread subdiffusively or superdiffusively, depending on . For > 2 the damping becomes ineffective for long times, and the width of a state grows kinematically. At zero temperature and for < 1, an initially localized state remains localized for all times. For 1 the state spreads, but with a slower rate than at finite temperatures. Study of arbitrary initial states indicates that the process is ergodic at finite temperatures only for 2 and at zero temperature for 1 2.     

Quantum Observables,Lie Algebra Homology and TQFT

Let us consider a Lie (super)algebra G spanned by T where T are quantum observables in BV formalism. It is proved that for every tensor c^... that determines a homology class of the Lie algebra G the expression c^...T...T is again a quantum observable. This theorem is used to construct quantum observables in the BV sigma model. We apply this construction to explain Kontsevich's results about the relation between homology of the Lie algebra of Hamiltonian vector fields and topological invariants of manifolds.     

A Note on Anisotropic Percolation

We consider an anisotropic independent bond percolation model on , i.e. we suppose that the vertical edges of are open with probability p and closed with probability 1–p, while the horizontal edges of are open with probability p and closed with probability 1– p, with 0 < p, < 1. Let , with x₁ < x₂, and . It is natural to ask how the two point connectivity function P_p,({0 x}) behaves, and whether anisotropy in percolation probabilities implies the strict inequality P_p,({0 x})> P_p,({0 x}). In this note we give affirmative answer at least for some regions of the parameters involved.Mathematics Subject Classifications (2000). 82B20, 82B41, 82B43.     

Structure factor of substitutional sequences

We study the structure factor for a large class of sequences of two elementsa andb such that longer sequences are generated from shorter ones by a simple substitution rulea ₁(a, b) andb ₂(a, b), where the's are some sequences ofa's andb's. Such sequences include periodic and quasiperiodic systems (e.g., the Fibonacci sequence), as well as systems such as the Thue-Morse sequence, which are neither. We show that there are values of the frequency at which the structure factors of these sequences have peaks that scale withL, the size of the system likeL ⁽⁾. For a given sequence a simple one- or two-dimensional dynamical iterative map of the variable can easily be abstracted from the substitution algorithm. The basin of attraction of a given fixed point or limit cycle of this map is a set of values of at which there are peaks of the structure factor all of which share the same value of. Furthermore, only those values of which are in the basin of attraction of the origin can have()=2. All other peaks will grow less rapidly withL. We show how to construct many sequences which, like the Thue-Morse sequence, have noL ² peaks. Other qualitative features of the structure factors are presented. Our approach unifies the treatment of a large class of apparently very diverse systems. Implications for the band structure of these systems as well as for the analysis of sequences with more than two elements are discussed.     

Hamiltonian formulation of general relativity in terms of Dirac spinors

The Dirac spinors and matrices are used in combination with the Arnowitt-Deser-Misner formalism in order to obtain yet another formulation of Hamiltonian general relativity, together with a new form of the Gauss-Codazzi equations. The relation with Ashtekar's variables is analyzed; it is shown, for instance, that the matrices are equivalent to the electric field variable. The electric and magnetic decomposition of the gravitational field is also studie using Dirac matrices.     

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.     

Growth of correlation in the Hopfield model

Introducing a finite correlation ₀ between any two learned patterns (others remaining uncorrelated), we observe in a numerical simulation that the Hopfield model stores these two patterns with correlation _f such that _f0 for any loading capacity. The patterns are memorized perfectly (with _f= ₀) up to -0.05 for finite correlations ₀ not exceeding a value _c(), where _c() decreases continuously to zero at -0.05.     

Nonlinear superposition for Liouville's equation in three spatial dimensions

We derive in 3+0 dimensions exact solutions of Liouville's equation ²=exp , by applying the Bäcklund transformation technique in conjunction with the principle of nonlinear superposition. The procedure, which is later extended to 3+1 dimensions, yield, as a byproduct, particular solutions of ² and ² =exp (² +² ).     

Multifractality and scaling in disordered mesoscopic systems

We suggest a new method for investigating scaling properties of mesoscopic observables and their distributions in disordered systems showing metal-insulator transition. In such systems quantum interference effects lead to multifractal structure of eigenstates on scales much smaller than the correlation length of the transition which can be described by a set of exponents, thef() spectrum. The analysis off() spectra can be extended to any scaling variable. Multifractality is an indication for broad distributions of these variables. If the transition is governed by one correlation length only then thef() spectra of normalized scaling variables must be universal. The critical exponentv of the correlation length is determined by the value (0) wheref() takes its maximum and the scaling exponent of normalizationxv ^–1=(0)+x. As an illustrative example we calculate numerically thef() spectra of eigenstates in the critical regime of 2d disordered electron systems in high magnetic fields. We find similarf() spectra indicating universal log-normal distributions of scaling variables.Work performed within the research program of the Sonderforschungsbereich 341, Köln-Aachen-Jülich     

Massive spin-two particle in a gravitational field

The spin-two particle is described by a symmetric tensorh subject to the subsidiary conditionsh = h =0. Their covariant generalization and the wave equation have been obtained directly from the Eulerian variational equations by algebraic methods only. In addition to the tensor fieldh a symmetric third-rank tensor = as well as a vector fieldA have been added, neither of which enter in the final result. The Lagrangian function is taken as a linear sum of all combinations which can be constructed from these functions, as well as terms involving the curvature tensor and its two possible contractions. Variation with respect toh , andA independently gives the Euler equations. Combining the various trace equations and choice of arbitrary constants yields the subsidiary conditions, while the Euler equations themselves give the connection between the auxiliary functions and the tensorh as well as the generalization of the wave equationD D h + 2R h -R h -R h +g R h +Rh =m ² h Finally, variation with respect tog yields the energy-momentum tensor.