Critical Slowing Down in One-Dimensional Maps and Beyond

This is a brief review on critical slowing down near the Feigenbaum period-doubling bifurcation points and its consequences. The slowing down of numerical convergence leads to an “operational” fractal dimension D=2/3 at a finite order bifurcation point. There is a cross-over to D ₀=0.538... when the order goes to infinity, i.e., to the Feigenbaum accumulation point. The problem of whether there exists a “super-scaling” for the dimension spectrum D _q ^W that does not depend on the primitive word W underlying the period-n-tupling sequence seems to remain open     

On Convergence to Equilibrium Distribution,I.¶The Klein–Gordon Equation with Mixing

Consider the Klein–Gordon equation (KGE) in ℝ ⁿ , n≥ 2, with constant or variable coefficients. We study the distribution μ _t of the random solution at time t∈ℝ. We assume that the initial probability measure μ₀ has zero mean, a translation-invariant covariance, and a finite mean energy density. We also assume that μ₀ satisfies a Rosenblatt- or Ibragimov–Linnik-type mixing condition. The main result is the convergence of μ _t to a Gaussian probability measure as t→∞ which gives a Central Limit Theorem for the KGE. The proof for the case of constant coefficients is based on an analysis of long time asymptotics of the solution in the Fourier representation and Bernstein's “room-corridor” argument. The case of variable coefficients is treated by using an “averaged” version ofthe scattering theory for infinite energy solutions, based on Vainberg's results on local energy decay. Received: 4 January 2001 / Accepted: 2 July 2001     

Three-Point Functions for MN/SN Orbifolds¶with?= 4 Supersymmetry

The D1–D5 system is believed to have an “orbifold point” in its moduli space where its low energy theory is a ?=4 supersymmetric sigma model with target space M ^N /S _N , where M is T ⁴ or K3. We study correlation functions of chiral operators in CFTs arising from such a theory. We construct a basic class of chiral operators from twist fields of the symmetric group and the generators of the superconformal algebra. We find explicitly the 3-point functions for these chiral fields at large N; these expressions are “universal” in that they are independent of the choice of M. We observe that the result is a significantly simpler expression than the corresponding expression for the bosonic theory based on the same orbifold target space. Received: 29 March 2001 / Accepted: 20 January 2002     

Orthosymplectic Lie Superalgebras in Superspace Analogues of Quantum Kepler Problems

A Schrödinger type equation on the superspace $\mathbb {R}^{D|2n}A Schr?dinger type equation on the superspace is studied, which involves a potential inversely proportional to the negative of the osp(D|2n) invariant “distance” away from the origin. An osp(2, D + 1|2n) dynamical supersymmetry for the system is explicitly constructed, and the bound states of the system are shown to form an irreducible highest weight module for this superalgebra. A thorough understanding of the structure of the irreducible module is obtained. This in particular enables the determination of the energy eigenvalues and the corresponding eigenspaces as well as their respective dimensions.     

Medium effects in the production and π0γ decay of ω-mesons in pA collisions in the GeV region

The ω resonance production and its π⁰γ decay in pA reactions close to threshold is considered within the Intranuclear Cascade (INC) model. The π⁰γ invariant-mass distribution shows two components which correspond to the ω decay “inside” and “outside” the nucleus, respectively. The “inside” component is distorted by medium effects, which introduce a mass shift as well as collisional broadening for the ω-meson and its decaying pion. The relative contribution of the “inside” component is analyzed in detail for different kinematical conditions and nuclear targets. It is demonstrated that a measurement of the correlation in azimuthal angle between the π⁰ and γ momenta allows to separate events related to the “inside”ω decay from different sources of background when uncorrelated π⁰'s and γ's are produced. Received: 2 April 2001 / Accepted: 5 June 2001     

Statistical mechanics of braided Markov chains: I. Analytic methods and numerical simulations

We investigate numerically and analytically the statistics of Markov chains on so-called braid (B _n ) and locally free (ℒℱ _n ) groups. Namely, we compute the mean length 〈μ〉 and the variance 〈μ²〉−〈μ〉² of the shortest word which remains after applying of all group relations to the randomly generatedN-letter word (Markov chain). We express the conjecture (numerically justified) that the mean value 〈μ〉 for the random walk on the groupB _n (n≫1) coincides with high accuracy with the same value for the random walk on the “locally free group weth errors” if the number of errors is of order of 20%.     

Paraboson coherent states

It is known that the defining relations of the orthosymplectic Lie superalgebra osp(1|2n) are equivalent to the defining (triple) relations of n pairs of paraboson operators b _i ^±. In particular, the “parabosons of order p” correspond to a unitary irreducible (infinite-dimensional) lowest weight representation V(p) of osp(1|2n). Recently we constructed these representations V(p) giving the explicit actions of the osp(1|2n) generators. We apply these results for the n = 2 case in order to obtain “coherent state” representations of the paraboson operators.     

Generalized q-Hermite Polynomials

We consider two operators A and A ⁺ in a Hilbert space of functions on the exponential lattice , where 0<q<1. The operators are formal adjoints of each other and depend on a real parameter . We show how these operators lead to an essentially unique symmetric ground state ψ₀ and that A and A ⁺ are ladder operators for the sequence . The sequence (ψ _n /ψ₀) is shown to be a family of orthogonal polynomials, which we identify as symmetrized q-Laguerre polynomials. We obtain in this way a new proof of the orthogonality for these polynomials. When γ=0 the polynomials are the discrete q-Hermite polynomials of type II, studied in several papers on q-quantum mechanics. Received: 6 December 1999 / Accepted: 21 May 2001     

Riemannian Geometry of Quantum Groups¶and Finite Groups with Nonuniversal Differentials

We construct noncommutative “Riemannian manifold” structures on dual quasitriangular Hopf algebras such as ℂ _q [SU ₂] with its standard bicovariant differential calculus, using the quantum frame bundle approach introduced previously. The metric is provided by the braided-Killing form on the braided-Lie algebra on the tangent space and the n-bein by the Maurer–Cartan form. We also apply the theory to finite sets and in particular to finite group function algebras ℂ[G] with differential calculi and Killing forms determined by a conjugacy class. The case of the permutation group ℂ[S ₃] is worked out in full detail and a unique torsion free and cotorsion free or “Levi–Civita” connection is obtained with noncommutative Ricci curvature essentially proportional to the metric (an Einstein space). We also construct Dirac operators in the metric background, including on finite groups such as S ₃. In the process we clarify the construction of connections from gauge fields with nonuniversal calculi on quantum principal bundles of tensor product form. Received: 22 June 2000 / Accepted: 26 August 2001     

Critical Coupling Constants and Eigenvalue Asymptotics of Perturbed Periodic Sturm–Liouville Operators

Perturbations of asymptotic decay c/r ² arise in the partial-wave analysis of rotationally symmetric partial differential operators. We show that for each end-point λ₀ of the spectral bands of a perturbed periodic Sturm–Liouville operator, there is a critical coupling constant c _crit such that eigenvalues in the spectral gap accumulate at λ₀ if and only if c/c _crit>1. The oscillation theoretic method used in the proof also yields the asymptotic distribution of the eigenvalues near λ₀. Received: 23 September 1999 / Accepted: 21 December 1999     

On the static permittivity of the Coulomb system in the long-wavelength limit

The general expression for the static permittivity ε(q, 0) of the Coulomb system in the region of small wave vectors was derived based on exact limit relations. The relation obtained describes the function ε(q, 0) in both “metal” and “dielectric” states of the Coulomb system. On this basis, the concept of the “true” dielectric is introduced and the definition of the “true” screening length was discussed. Exact relations were derived for the function ε(q, 0) in the region of small wave vectors q within the random phase approximation at an arbitrary degeneracy.     

Zeros of the Jones Polynomial for Multiple Crossing-Twisted Links

Let D be a general connected reduced alternating link diagram, C be the set of crossings of D and C′ be the nonempty subset of C. In this paper we first define a multiple crossing-twisted link family {D ⁿ (C′)|n=1,2,…} based on D and C′, which produces (2,2n+1)-torus knot family, the link family A _n defined in Chang and Shrock (Physica A 301:196–218, 2001) and the pretzel link family P(n,n,n) as special cases. Then by applying Beraha-Kahane-Weiss’s Theorem we prove that limits of zeros of Jones polynomials of {D ⁿ (C′)|n=1,2,…} are the unit circle |z|=1 (It is independent of the selections of D and C′) and several isolated limits, which can be determined by computing flow polynomials of subgraphs of G corresponding to D. Furthermore, we use the method of Brown and Hickman (Discrete Math. 242:17–30, 2002) to prove that, for any ε>0, all zeros of Jones polynomial of the link D ⁿ (C) lie inside the circle |z|=1+ε, provided that n is large enough. Our results extend results of F.Y. Wu, J. Wang, S.-C. Chang, R. Shrock and the present authors and refine partial result of A. Champanerkar and L. Kofman.     

Dual Forms on Supermanifolds and Cartan Calculus

We introduce and study the complex of “stable forms” on supermanifolds. Stable forms on a supermanifold M are represented by Lagrangians of “copaths” (formal systems of equations, which may or may not specify actual surfaces) on M×ℝ ^D . Changes of D give rise to stability isomorphisms. The resulting (direct limit) {Cartan-de Rham} complex made of stable forms extends both in positive and negative degree. Its positive half is isomorphic to the complex of forms defined as Lagrangians of paths, studied earlier. Including the negative half is crucial, in particular, for homotopy invariance. For stable forms we introduce (non-obvious) analogs of exterior multiplication by covectors and contraction with vectors and find the anticommutation relations that they obey. Remarkably, the version of the Clifford algebra so obtained is based on the super anticommutators rather than the commutators and (before stabilization) it includes some central element σ. An analog of Cartan's homotopy identity is proved, which also contains this “stability operator”σ. Received: 3 January 2000 / Accepted: 15 September 2001     

On the Energy Growth of Some Periodically Driven Quantum Systems with Shrinking Gaps in the Spectrum

We consider quantum Hamiltonians of the form H(t)=H+V(t) where the spectrum of H is semibounded and discrete, and the eigenvalues behave as E _n ∼n ^α , with 0<α<1. In particular, the gaps between successive eigenvalues decay as n ^α−1. V(t) is supposed to be periodic, bounded, continuously differentiable in the strong sense and such that the matrix entries with respect to the spectral decomposition of H obey the estimate ‖V(t) _m,n ‖≤ε|m−n|^−p max {m,n}^−2γ for m≠n, where ε>0, p≥1 and γ=(1−α)/2. We show that the energy diffusion exponent can be arbitrarily small provided p is sufficiently large and ε is small enough. More precisely, for any initial condition Ψ∈Dom(H ^1/2), the diffusion of energy is bounded from above as 〈H〉 _Ψ (t)=O(t ^σ ), where . As an application we consider the Hamiltonian H(t)=|p| ^α +ε v(θ,t) on L ²(S ¹,dθ) which was discussed earlier in the literature by Howland.     

Periodic Orbits Contribution to the 2-Point Correlation Form Factor for Pseudo-Integrable Systems

Using heuristic arguments based on the trace formulas we relate the 2-point correlation form factor, K ₂(τ), at small values of τ with sums over classical periodic orbits for typical examples of pseudo-integrable systems. The later sums have been explicitly calculated for the following models: (i) plane billiards in the form of right triangles with one angle π/n and (ii) rectangular billiards with the Aharonov-Bohm flux line. In the first model, using the properties of the Veech structure, it is shown that K ₂(0)=(n+ε(n))/(3(n−2)), where ε(n)= 0 for odd n, ε(n)= 2 for even n not divisible by 3, and ε(n)=6 for even n divisible by 3. For completeness we also recall informally the main features of the Veech construction. In the second model the answer depends on arithmetical properties of ratios of flux line coordinates to the corresponding sides of the rectangle. When these ratios are non-commensurable irrational numbers, K ₂(0)=1−3 , where is the fractional part of the flux through the rectangle when and it is symmetric with respect to the line when . The comparison of these results with numerical calculations of the form factor is discussed in detail. The above values of K ₂(0) differ from all known examples of spectral statistics, thus confirming analytically the peculiarities of statistical properties of the energy levels in pseudo-integrable systems. Received: 10 January 2000 / Accepted: 18 May 2001     

A Kohno–Drinfeld Theorem for the Monodromy of Cyclotomic KZ Connections

We compute explicitly the monodromy representations of “cyclotomic” analogs of the Knizhnik–Zamolodchikov differential system. These are representations of the type B braid group B_n¹{B_n^1} . We show how the representations of the braid group B _n obtained using quantum groups and universal R-matrices may be enhanced to representations of B_n¹{B_n^1} using dynamical twists. Then, we show how these “algebraic” representations may be identified with the above “analytic” monodromy representations.     

Atomic parity-violation and precision physics

The atomic parity-violation (APV) parameter for a nucleus with n neutrons and z protons has been included in the list of pseudo-observables accessible with the codes TOPAZ0 and ZFITTER. In this way one can add the APV results in the LEP EWWG “global” electroweak fits, checking the corresponding effect when added to the existing precision measurements. Received: 8 March 2001 / Published online: 21 September 2001     

Specific heat of the harmonic oscillator within generalized equilibrium statistics

Summary Within the generalized equilibrium statistics recently introduced by Tsallis (p _n ∝[1−β(q−-1) _εn ]^1/(q−)), we calculate the thermal dependence of the specific heat corresponding to a harmonic-oscillator-like spectrum, namely ε _n ω(n−α) (∀ω>0,n=0,1,2,...). The influences ofq and α are exhibited. Physically inaccessible and/or thermally frozen gaps are obtained in the low-temperature region, and, forq>1, oscillations are observed in the high-temperature region. The specific heat of the two-level system is also shown.     

Deposition of mass-selected cluster ions using a pulsed arc cluster-ion source

A PACIS (pulsed arc cluster-ion source) developed for high average cluster-ion currents is presented. The performance of the PACIS at different operational modes is described, and the suitability for cluster-deposition experiments is discussed in comparison with other cluster-ion sources. Maximum currents of mass-selected cluster ions of 3–6 nA of small Si_n ^- (n=4–10) clusters and 0.3–0.5 nA of large Al_n ^+/- (n=20–70) clusters are achieved. The mass-selected cluster ions are soft-landed on a substrate at residual kinetic energies lower than 1 eV/atom, and the samples are characterized by X-ray photoelectron spectroscopy and scanning tunneling microscopy. First results on the soft landing of “magic” Si₄ ^- clusters on graphite are presented. Received: 30 May 2001 / Accepted: 14 June 2001 / Published online: 2 October 2001     

On Minimal Eigenvalues¶of Schrödinger Operators on Manifolds

We consider the problem of minimizing the eigenvalues of the Schr?dinger operator H=−Δ+αF(κ) (α>0) on a compact n-manifold subject to the restriction that κ has a given fixed average κ₀. In the one-dimensional case our results imply in particular that for F(κ)=κ² the constant potential fails to minimize the principal eigenvalue for α>α_c=μ₁/(4κ₀ ²), where μ₁ is the first nonzero eigenvalue of −Δ. This complements a result by Exner, Harrell and Loss, showing that the critical value where the constant potential stops being a minimizer for a class of Schr?dinger operators penalized by curvature is given by α _c . Furthermore, we show that the value of μ₁/4 remains the infimum for all α >α _c . Using these results, we obtain a sharp lower bound for the principal eigenvalue for a general potential. In higher dimensions we prove a (weak) local version of these results for a general class of potentials F(κ), and then show that globally the infimum for the first and also for higher eigenvalues is actually given by the corresponding eigenvalues of the Laplace–Beltrami operator and is never attained. Received: 17 July 2000 / Accepted: 11 October 2000