Absence of symmetry breaking for systems of rotors with random interactions

We prove that Gibbs states for the Hamiltonian , with thes _x varying on theN-dimensional unit sphere, obtained with nonrandom boundary conditions (in a suitable sense), are almost surely rotationally invariant if withJ _xy i.i.d. bounded random variables with zero average, 1 in one dimension, and 2 in two dimensions.     

Transport Properties of Kicked and Quasiperiodic Hamiltonians

We study transport properties of Schrödinger operators depending on one or more parameters. Examples include the kicked rotor and operators with quasi-periodic potentials. We show that the mean growth exponent of the kinetic energy in the kicked rotor and of the mean square displacement in quasiperiodic potentials is generically equal to 2: this means that the motion remains ballistic, at least in a weak sense, even away from the resonances of the models. Stronger results are obtained for a class of tight-binding Hamiltonians with an electric field E(t) = E ₀+ E ₁cos t. For with 3/2)$$ " align="middle" border="0"> we show that the mean square displacement satisfies for suitable choices of , E ₀, and E ₁. We relate this behavior to the spectral properties of the Floquet operator of the problem.     

Invariant measures for the2D-defocusing nonlinear Schrödinger equation

Consider the2D defocusing cubic NLSiu _t+u–u|u|²=0 with Hamiltonian . It is shown that the Gibbs measure constructed from the Wick ordered Hamiltonian, i.e. replacing ||⁴ by ||⁴ :, is an invariant measure for the appropriately modified equationiu _t + u‒ [u|u ²–2(|u|² dx)u]=0. There is a well defined flow on thesupport of the measure. In fact, it is shown that for almost all data the solutionu, u(0)=, satisfiesu(t)–e ^it C _Hs (), for somes>0. First a result local in time is established and next measure invariance considerations are used to extend the local result to a global one (cf. [B2]).     

On a limiting procedure for obtaining physical states for an infinite Fermi system

We propose a limiting procedure for obtaining physical states for an infinite non-relativistic Fermi system. We take the thermodynamic limit of vector states in the Fock representation of the C.A.R. algebra, representing a condensate state of atoms each of which is formed by 4 fermions. In a simplified example considered in detail, the limit state has a simple decomposition into the product of two B.C.S. states. IfB ⁺ is the operator creating the atom from the vacuum |_0F , it is proved that the states obtained by taking the thermodynamic limit of the vector states corresponding to (B ⁺) ⁿ |_0F and respectively, coincide on the gauge-invariant elements of the algebra for a suitable value ofz.Partially supported by C.N.R.     

A nonlinear instability for 3×3 systems of conservation laws

The phenomenon of nonlinear resonance provides a mechanism for the unbounded amplification of small solutions of systems of conservation laws. We construct spatially 2-periodic solutionsu ^N C ([0,t _N ] × witht _N bounded, satisfying

On the asymptotics of nodes ofL 2-solutions of Schrödinger equations in dimensions ≧ 3

Let _R = ⁿ B _R, wheren3 andB _R={x ⁿ :|x|R}. We investigate the asymptotics of real valued solutions L ²( _R ) of the Schrödinger equation (–+V–E)=0, whereE<0>V(x)0 for |x|: LetD denote an unbounded nodal domain of (i.e. a component of _R {x:(x)=0}), and letS(r)={yS ^n–1:ryD} withS ^n–1 the unit sphere in ⁿ . Under suitable assumptions onV it is shown that for some >0,

Phonon-Exchange Attraction. Optical Phonons and Type II Superconductivity, and d-Wave Cooper Pairs

The electron-pair density matrix (k ₁, k ₂; k ₃, k ₄, t) =: (1, 2; 3, 4, t) changes in time, following a quantum Liouville equation \rho (5,6;3,4,t) - < 5,6|v_c |3,4 >\rho (1,2,5,6,t)} \right]} } $$ " align="middle" border="0"> with : = 4\pi e^2 k_0 (\Omega q^2 )^{ - 1} \delta _{k_1 + k_2 ,k_3 + k_4 } \delta _{k_s - k_1 ,q} $$ " align="middle" border="0"> in the presence of a Coulomb interaction _c, where is the volume. If the virtual phonon exchange is in action, the density matrix is shown to change similarly with an effective interaction _e, \hfill \\ = |V_q |^2 \bar h\omega _q |(\varepsilon _3 - \varepsilon _1 )^2 - \bar h^2 \omega _q^2 |^{ - 1} \delta _{k_1 + k_2 ,k_3 + k_4 } \delta _{k_3 - k_1 ,q} \hfill \\ \end{gathered} $$ " align="middle" border="0"> , by using a time-dependent perturbation theory and a Markoffian approximation. The dominant longitudinal-acoustic-(optical)-phonon-exchange attraction at 0K is shown to be q-independent (-dependent). The results are used to discuss the Cooper pair size, the origin of type II superconductivity and the formation of d-wave Cooper pairs in the cuprates.     

The Navier Wall Law at a Boundary with Random Roughness

We consider the Navier-Stokes equation in a domain with irregular boundaries. The irregularity is modeled by a spatially homogeneous random process, with typical size . In the parent paper [8], we derived a homogenized boundary condition of Navier type as . We show here that for a large class of boundaries, this Navier condition provides a approximation in L ², instead of for periodic irregularities. Our result relies on the study of an auxiliary boundary layer system. Decay properties of this boundary layer are deduced from a central limit theorem for dependent variables.     

Decay of correlations in the one dimensional Ising model withJ ij =|i−j|−2

A low temperature expansion is constructed for the one dimensional Ising model with Hamiltonian . It is shown that the two point function _i ; _j obeys upper and lower bounds of the formf()|i–j|^–2 for inverse temperature sufficiently large.Junior Fellow, Harvard University Society of Fellows. Supported in part by the National Science Foundation under Grant No. PHY79-16812.     

Quantitative universality for a class of nonlinear transformations

A large class of recursion relationsx _{n + 1} = f(x_n) exhibiting infinite bifurcation is shown to possess a rich quantitative structure essentially independent of the recursion function. The functions considered all have a unique differentiable maximum . With sufficiently small),z > 1, the universal details depend only uponz. In particular, the local structure of high-order stability sets is shown to approach universality, rescaling in successive bifurcations, asymptotically by the ratio ( = 2.5029078750957... forz = 2). This structure is determined by a universal functiong ^*(x), where the 2ⁿth iterate off,f ⁽ⁿ⁾, converges locally to ^–n g ^*( ⁿ x) for largen. For the class off's considered, there exists a _n such that a 2ⁿ-point stable limit cycle including exists; – _n R~ ^–n ( = 4.669201609103... forz = 2). The numbers and have been computationally determined for a range ofz through their definitions, for a variety off's for eachz. We present a recursive mechanism that explains these results by determiningg ^* as the fixed-point (function) of a transformation on the class off's. At present our treatment is heuristic. In a sequel, an exact theory is formulated and specific problems of rigor isolated.Research performed under the auspices of the U.S. Energy Research and Development Administration.     

Power law scaling of the top Lyapunov exponent of a Product of Random Matrices

A sequence of i.i.d. matrix-valued random variables with probabilityp and with probability 1–p is considered. Leta() = a ₀ + O(), c() = c ₀ + O() lim ₀ b() = Oa ₀,c ₀, >0, andb()>0 for all >0. It is shown show that the top Lyapunov exponent of the matrix productX _n X _n-1...X ₁, = lim_n (1/n) n X _n X _n-1...X ₁ satisfies a power law with an exponent 1/2. That is, lim ₀(ln /ln ) = 1/2.     

Spectral and scattering theory for a Schrödinger operator with a class of momentum-dependent long-range potentials

Let be the selfadjoint operator for the static electromagnetic field where W _j for 0, 1, 2, ..., n is a sum of (i) a short-range potential and (ii) a smooth long-range potential decreasing at as |x|^- with in (0, 1]. Then for >1/2, asymptotic completeness holds for the scattering system (H, H ₀).     

The mean spherical model in a random external field and the replica method

We provide a quick elementary solution of the mean spherical model in a random external field. This also allows an immediate proof of the self-averaging property of the free energy. We calculate the free energy by means of the replica method, i.e., for any (not necessarily integer) replica numbern, and show that when a phase transition occurs the limits andn 0 are not interchangeable.     

Evaluation ofS,T, U parameters, triple gauge boson vertices and some oblique corrections in extraU(1) superstring inspired model

Explicit evaluation of the following parameters has been carried out in the extraU (1) superstring inspired model: (i) As Mz₂ varies from 555 GeV to 620 GeV and (m _t) CDF = 175.6 ± 5.7 GeV (Table 1): (a) S^New varies from -0.100 ± 0.089 to -0.130 ± 0.090, (b) T^New varies from -0.098 ± 0.097 to -0.129 ± 0.098, (c) U^New varies from -0.229 ± 0.177 to -0.253 ± 0.206, (d) Τ_z varies from 2.487 ± 0.027 to 2.486 ± 0.027, (e) A_LR varies from 0.0125 ± 0.0003 to 0.0126 ± 0.0003, (f) A _FB ^b remains constant at 0.0080 ± 0.0007. Almost identical values are obtained for (m _t)D0 = 169 GeV (see table 2). (ii) Triple gauge boson vertices (TGV) contributions: AsMz ₂ varies from 555 GeV to 620 GeV and (m _t) CDF = 175.6 ±5.7 GeV. (a)√s = 500 GeV, asymptotic case: varies from -0.301 to -0.179; varies from -0.622 to -0.379; varies from +0.0061 to 0.0056; varies from -3.691 to -2.186. varies from +0.270 to +0.118; varies from +0.552 to 0.238; varies from +0.0004 to +0.0002; remains constant at -0.110. (b)√s = 700 GeV, asymptotic case: varies from -0.297 to -0.176; varies from -0.609 to -0.370; varies from -0.0082 to -0.0078; varies from -3.680 to -2.171.√s = 700 GeV, nonasymptotic case: varies from -0.173 to -0.299; varies from-0.343 to -0.591; varies from -0.005 to -0.011; remains constant at -0.110. The pattern of form factors values for√s = 1000, 1200 GeV is almost identical to that of√s= 700 GeV. Further the values of the form factors for (m _t)D0 (=169 GeV) follow identical pattern as that of (m _t) CDF form factors values (see tables 5, 6, 9, 10). We conclude that the values of all the form factors with the exception of these of , are comparable or larger than theS, T values and therefore the TGV contributions are important while deciding the use of extraU (1) model for doing physics beyond standard model.     

Limiting behavior of asymptotically flat gravitational fields

Couch and Torrence suggest that the vacuum Einstein equations admit a larger class of asymptotically flat solutions than those exhibiting the peeling property. Starting with the assumption that , (d/dr) and (/x ^A ) , wherex ^A (A = 2, 3) are angular coordinates, they show that , where ₁ 2 and ₁<₀; , where ₂ 1 and ₁< ₁; and ₄ and ₃ peel as they would under the stronger peeling conditions. The Winicour-Tamburino energy-momentun and angular momentum integrals for these solutions, in general, diverge. In fact, since Couch and Torrence determine only the radial dependence of the solution, it is not clear that the solutions are well defined. We find that the stronger assumption , (d/dr) , and (/x ^A ) does result in well-defined solutions for which both the energy-momentum and angular momentum intergrals are not only finite but result in the same expressions as are obtained for peeling space-times. This assumption appears to be the minimal assumption that is necessary for investigating outgoing radiation at null infinity.In part based on a dissertation by Stephanie Novak and submitted to Syracuse University in partial fulfillment of the requirement for the Ph.D. degree.     

The Homogeneous Hamilton–Jacobi and Bernoulli Equations Revisited

The one-dimensional case of the homogeneous Hamilton–Jacobi and Bernoulli equations S_t S _x ² =0, where S(x, t) is Hamilton's principal function of a free particle and also Bernoulli's momentum potential of a perfect liquid, is considered. Non-elementary solutions are looked for in terms of odd power series in t with x-dependent coefficients and even power series in x with t-dependent coefficients. In both cases, and depending upon initial conditions, unexpected regularities are observed in the terms of these expansions and this suggests that S(^x, ^t) should be written as a product of the elementary solution x²/(2^t) and a function f=f() where =(x, |t|) owing to the symmetry property which is that S(x, –t)=–S(x, t). Requiring that this Ansatz satisfies the said equation and choosing the simplest realization of (x, |t|)=₀ |t/t₀| (x/x ₀)0 with , results in a soluble ordinary differential equation, of first order in u=ln and quadratic in f. This ODE has two fixed points: f=1, obviously, and f=0, a new fixed point which is often called trivial. The phase plane (f_u, f) consists of a family of parabolas, all of which pass through the two fixed points. Explicit solutions of the general case are given close to these fixed points. A one-parameter family of solution is found to emerge from the trivial fixed point with non-trivial initial values S(x, 0). Detailed analyses of these findings will be reported elsewhere, bearing in mind that Bernoulli's equation has to be supplemented by the continuity equation satisfied by the density of the liquid.     

Quenched Averages for Self-Avoiding Walks and Polygons on Deterministic Fractals

We study rooted self avoiding polygons and self avoiding walks on deterministic fractal lattices of finite ramification index. Different sites on such lattices are not equivalent, and the number of rooted open walks W _n (S), and rooted self-avoiding polygons P _n (S) of n steps depend on the root S. We use exact recursion equations on the fractal to determine the generating functions for P _n (S), and W _n(S) for an arbitrary point S on the lattice. These are used to compute the averages ,, and over different positions of S. We find that the connectivity constant μ, and the radius of gyration exponent are the same for the annealed and quenched averages. However, , and , where the exponents and , take values different from the annealed case. These are expressed as the Lyapunov exponents of random product of finite-dimensional matrices. For the 3-simplex lattice, our numerical estimation gives and , to be compared with the known annealed values and .     

Ergodic states in a non commutative ergodic theory

Using the Godement mean of positive-type functions over a groupG, we study -abelian systems { , } of aC*-algebra and a homomorphic mapping of a groupG into the homomorphism group of . Consideration of the Godement mean off(g)U _g withf a positive-type function overG andU a unitary representation ofG first yields a generalized mean-ergodic theorem. We then define the Godement mean off(g) ( _g (A)) withA and a covariant representation of the system { , } for which theG-invariant Hilbert space vectors are cyclic and study its properties, notably in relation with ergodic and weakly mixing states over . Finally we investigate the discrete spectrum of covariant representations of { , } (i.e. the direct sum of the finite-dimensional subrepresentations of the associated representations ofG).On leave of absence from Istituto di Fisica G. Marconi Piazzale delle Scienze 5 — Roma.     

Distribution of the error term for the number of lattice points inside a shifted circle

We investigate the fluctuations inN (R), the number of lattice pointsnZ ² inside a circle of radiusR centered at a fixed point [0, 1)². Assuming thatR is smoothly (e.g., uniformly) distributed on a segment 0RT, we prove that the random variable has a limit distribution asT (independent of the distribution ofR), which is absolutely continuous with respect to Lebesgue measure. The densityp (x) is an entire function ofx which decays, for realx, faster than exp(–|x|^4–). We also obtain a lower bound on the distribution function which shows thatP (–x) and 1–P (x) decay whenx not faster than exp(–x ⁴⁺). Numerical studies show that the profile of the densityp (x) can be very different for different . For instance, it can be both unimodal and bimodal. We show that , and the variance depends continuously on . However, the partial derivatives ofD are infinite at every rational point Q ², soD is analytic nowhere.     

Test of quantum mechanics by neutron interferometry

Measurements of the π⁰ and direct photon nuclear modification factors in p+p and A+A collisions in the PHENIX experiment at RHIC will be reviewed and recent results at  GeV will be presented. Using for the first time the p+p reference measured in the same experiment instead of averaging world data the π⁰ suppression turns out to be almost as large at  GeV as at  GeV, implying gluon densities dN^g/dy>800. Possible origins of photon suppression at high p_T in  GeV are discussed in light of the new results on photon R_AA at  GeV.