Lattice Dislocations in a 1-Dimensional Model

The spectral properties of the Schr?dinger operator T(t)=−d ²/dx ²+q(x,t) in L ²(ℝ) are studied, where the potential q is defined by q=p(x+t), x>0, and q=p(x), x<0; p is a 1-periodic potential and t∈ℝ is the dislocation parameter. For each t the absolutely continuous spectrum σ _ac (T(t))=σ _ac (T(0)) consists of intervals, which are separated by the gaps γ _n (T(t))=γ _n (T(0))=(α _n ⁻,α _n ⁺), n≥1. We prove: in each gap γ _n ≠?, n≥ 1 there exist two unique “states” (an eigenvalue and a resonance) λ _n ^±(t) of the dislocation operator, such that λ _n ^±(0)=α _n ^± and the point λ _n ^±(t) runs clockwise around the gap γ _n changing the energy sheet whenever it hits α _n ^±, making n/2 complete revolutions in unit time. On the first sheet λ _n ^±(t) is an eigenvalue and on the second sheet λ _n ^±(t) is a resonance. In general, these motions are not monotonic. There exists a unique state λ₀(t) in the basic gap γ₀(T(t))=γ₀(T(0))=(−∞ ,α₀ ⁺). The asymptotics of λ _n ^±(t) as n→∞ is determined. Received: 5 April 1999 / Accepted: 3 March 2000     

An exactly solvable linear model giving indication of stochastic resonance

The linear stochastic equation dx _β /dt+[1+f _β (t)]x _β (t)=A sin (Ωt) is discussed. The functionƒ _β (t) is defined as a Poissonian noise dependent on a parameterβ>0,ƒ _β (t)=β Σ _j [δ(t − t _j ⁺ ) −δ (t − t _j ⁻ )]. The mean frequency of the delta-pulses is chosen asβ-dependent in the formλ(β)=2γ(β ⁻² + 1) exp(−β) whereγ is a constant from the interval (0, 0.974). With the stochastic functionƒ _β (t) defined in this way, attention is paid on the oscillational term of the averaged function 〈x(t)〉, 〈x(t)〉_osc=Āsin(Ωt − α). It is found that the dependenceĀ=Ā(β) exhibits one maximum and one minimum. The occurrence of these extrema seems to affirm the presence of stochastic resonance. This work has been supported by the Slovak Grant Agency VEGA under contract No. 1/4319/97.     

Regge behavior of DIS structure functions in scalar theories

The Bethe-Salpeter formalism is used to incorporate the valence Regge behavior into the total DIS amplitude. For a special case of scalar quarks with massless scalar exchange, the model is solved both analytically and numerically and exact scaling is found for the valence quark contribution F ₂(x) ∼ (1/x) ^l(0)−1 which mimicks the ρ-trajectory term. The solution solves a long-standing problem by showing that the coefficient in the Regge pole expansion is indeed fine-tuned to give the expected scaling. The method allows for generalization to the region of nonzero momentum transfer and calculation of the DVCS amplitude. The text was submitted by the authors in English.     

Time Quasi-Periodic Unbounded Perturbations¶of Schrödinger Operators and KAM Methods

We eliminate by KAM methods the time dependence in a class of linear differential equations in ℓ² subject to an unbounded, quasi-periodic forcing. This entails the pure-point nature of the Floquet spectrum of the operator H ₀+εP(ωt) for ε small. Here H ₀ is the one-dimensional Schr?dinger operator p ²+V, V(x)∼|x|^α, α <2 for |x|→∞, the time quasi-periodic perturbation P may grow as |x|^β, β <(α−2)/2, and the frequency vector ω is non resonant. The proof extends to infinite dimensional spaces the result valid for quasiperiodically forced linear differential equations and is based on Kuksin's estimate of solutions of homological equations with non-constant coefficients. Received: 3 October 2000 / Accepted: 20 December 2000     

Generalized holographic and Ricci dark energy models

In this paper, we consider generalized holographic and Ricci dark energy models where the energy densities are given as ρ _R =3c ² M _pl² Rf(H ²/R) and ρ _h =3c ² M _pl² H ² g(R/H ²), respectively; here f(x), g(y) are positive defined functions of the dimensionless variables H ²/R or R/H ². It is interesting that holographic and Ricci dark energy densities are recovered or recovered interchangeably when the function f(x)=g(y)≡1 or f(x)=Id and g(y)=Id are taken, respectively (for example f(x),g(x)=1−ε(1−x), ε=0or1, respectively). Also, when f(x)≡xg(1/x) is taken, the Ricci and holographic dark energy models are equivalent to a generalized one. When the simple forms f(x)=1−ε(1−x) and g(y)=1−η(1−y) are taken as examples, by using current cosmic observational data, generalized dark energy models are considered. As expected, in these cases, the results show that they are equivalent (ε=1−η=1.312), and Ricci-like dark energy is more favored relative to the holographic one where the Hubble horizon was taken as an IR cut-off. And the suggested combination of holographic and Ricci dark energy components would be 1.312R−0.312H ², which is 2.312H²+1.312[(H)\dot]2.312H^{2}+1.312\dot{H} in terms of H ² and [(H)\dot]\dot{H} .     

Asymptotic behavior of the β function in the ϕ<Superscript>4</Superscript> theory: A scheme without complex parameters

The previously-obtained analytical asymptotic expressions for the Gell-Mann-Low function β(g) and anomalous dimensions in the ϕ⁴ theory in the limit g → ∞ are based on the parametric representation of the form g = f(t), β(g) = f ₁(t) (where t ∝ g ₀^−1/2 is the running parameter related to the bare charge g ₀), which is simplified in the complex t plane near a zero of one of the functional integrals. In this work, it has been shown that the parametric representation has a singularity at t → 0; for this reason, similar results can be obtained for real g ₀ values. The problem of the correct transition to the strong-coupling regime is simultaneously solved; in particular, the constancy of the bare or renormalized mass is not a correct condition of this transition. A partial proof has been given for the theorem of the renormalizability in the strong-coupling region.     

Absorption spectra and magneto-optic figures of merit in the Bi x Sm3-x Fe5-y Ga y O12 system

The absorption (α) and Faraday rotation (θ) spectra of 14 garnets belonging to the series Bi _x Sm_3-x Fe_5-y Ga _y O₁₂ (0<x<1.05, 0.8<y<1.15) have been measured between 15 000 cm⁻¹ and 19 000 cm⁻¹. The figure of merit (θ/α) at 17 850 cm⁻¹ (560 nm) increases linearly with increasing bismuth concentration up tox∼0.6 where it begins to increase less rapidly. For operation of magneto-optic display devices at 17 850 cm⁻¹ there is no advantage in using garnets in this series withx>0.8. The Faraday rotation at 17 850 cm⁻¹ increases linearly with bismuth concentration whereas the absorption coefficient increases more rapidly. The presence of Bi³⁺ increases the intensity of all Fe³⁺ pair transitions in the garnet system as a result of the increased superexchange induced by Bi³⁺. This is in keeping with the observation that the intensity of the⁶A_1g (S)→⁴T_1g (G) transition in (RE)₃Fe₅O₁₂ (RE=Er, Y, Dy, Gd, Eu) increases on traversing the above RE series as do the Curie temperatures of these iron garnets.     

Electrical and magnetic properties of [(CoFeZr)x(Al2O3)1 ? x/(α-SiH)]n multilayer structures

The concentration dependences of the electrical resistivity and complex permeability of [“(Co₄₅Fe₄₅Zr₁₀) _x (Al₂O₃)_{100 − x} ”/“α-Si: H”] _n multilayer structures and (Co₄₅Fe₄₅Zr₁₀) _x (Al₂O₃)_{100 − x} composites have been studied. It has been established that introduction of a semiconductor interlayer into the (Co₄₅Fe₄₅Zr₁₀) _x (Al₂O₃)_{100 − x} composites substantially decreases the electrical resistivity of [“(Co₄₅Fe₄₅Zr₁₀) _x (Al₂O₃)_{100 − x} ”/“α-Si: H”] _n multilayer structures. The concentration dependences of the real and imaginary parts of the complex permeability of the [“(Co₄₅Fe₄₅Zr₁₀) _x (Al₂O₃)_{100 − x} ”/“α-Si: H”] _n nanomultilayer structures substantially differ from those of the (Co₄₅Fe₄₅Zr₁₀) _x (Al₂O₃)_{100 − x} composites. The real part of the complex permeability of the [“(Co₄₅Fe₄₅Zr₁₀) _x (Al₂O₃)_{100 − x} ”/“α-Si: H”] _n nanomultilayer structures follows the curve with a minimum near the percolation threshold of the composite, and the imaginary part smoothly decreases as the ferromagnetic phase concentration increases. The results obtained are explained by the increase in the bifurcation temperature due to the conduction electrons of the semiconductor interlayer, which favor magnetic ordering of ferromagnetic grains.     

Internal waves in a compressible two-layer model atmosphere: Hamiltonian description

We consider slow, compared to the speed of sound, motions of an ideal compressible fluid (gas) in a gravitational field in the presence of two isentropic layers with a small specific-entropy difference between them. Assuming the flow to be potential in each of the layers (v _{1, 2} = ▿ϕ_{1, 2}) and neglecting the acoustic degrees of freedom (div($ \bar \rho $ \bar \rho (z)▿ϕ_{1, 2}) ≈ 0, where $ \bar \rho $ \bar \rho (z) is the average equilibrium density), we derive the equations of motion for the boundary in terms of the shape of the surface z = η(x, y, t) itself and the difference between the boundary values of the two velocity field potentials: ψ(x, y, t) = ψ₁ − ψ₂. We prove the Hamilto nian structure of the derived equations specified by a Lagrangian of the form ℒ = ∫$ \bar \rho $ \bar \rho (η)η _t ψdxdy − ℋ{η, ψ}. The system under consideration is the simplest theoretical model for studying internal waves in a sharply stratified atmosphere in which the decrease in equilibrium gas density due to gas compressibility with increasing height is essentially taken into account. For plane flows, we make a generalization to the case where each of the layers has its own constant potential vorticity. We investigate a system with a model dependence $ \bar \rho $ \bar \rho (z) ∝ e ^−2αz with which the Hamiltonian ℋ{η, ψ} can be represented explicitly. We consider a long-wavelength dynamic regime with dispersion corrections and derive an approximate nonlinear equation of the form u _t + auu _x − b[−$ \hat \partial _x^2 $ \hat \partial _x^2 + α²]^1/2 u _x = 0 (Smith’s equation) for the slow evolution of a traveling wave.     

Cylindrically Symmetric Inhomogeneous Universes with?a?Cloud of Strings

Cylindrically symmetric inhomogeneous string cosmological models are investigated in presence of string fluid as a source of matter. To get the three types of exact solutions of Einstein’s field equations we assume A=f(x)k(t), B=g(x)ℓ(t) and C=h(x)ℓ(t). Some physical and geometric aspects of the models are discussed.     

The Asymptotic Limits of Zero Modes of Massless Dirac Operators

Asymptotic behaviors of zero modes of the massless Dirac operator H = α · D + Q(x) are discussed, where α = (α₁, α₂, α₃) is the triple of 4 × 4 Dirac matrices, , and Q(x) = (q _jk (x)) is a 4 × 4 Hermitian matrix-valued function with | q _jk (x) | ≤ C 〈x〉^−ρ, ρ > 1. We shall show that for every zero mode f, the asymptotic limit of |x|² f (x) as |x| → + ∞ exists. The limit is expressed in terms of the Dirac matrices and an integral of Q(x) f (x).      

Distribution of a Particle’s Position in the ASEP with the Alternating Initial Condition

In this paper we give the distribution of the position of a particle in the asymmetric simple exclusion process (ASEP) with the alternating initial condition. That is, we find ℙ(X _m (t)≤x) where X _m (t) is the position of the particle at time t which was at m=2k−1, k∈ℤ at t=0. As in the ASEP with step initial condition, there arises a new combinatorial identity for the alternating initial condition, and this identity relates the integrand of the integral formula for ℙ(X _m (t)≤x) to a determinantal form together with an extra product.     

A Construction of Blow up Solutions for Co-rotational Wave Maps

The existence of co-rotational finite time blow up solutions to the wave map problem from ${\mathbb{R}^{2+1} \to N}The existence of co-rotational finite time blow up solutions to the wave map problem from \mathbbR²⁺¹ ? N{\mathbb{R}^{2+1} \to N} , where N is a surface of revolution with metric d ρ ² + g(ρ)² dθ², g an entire function, is proven. These are of the form u(t,r)=Q(l(t)t)+R(t,r){u(t,r)=Q(\lambda(t)t)+\mathcal{R}(t,r)} , where Q is a time independent solution of the co-rotational wave map equation −u _tt  + u _rr  + r ⁻¹ u _r  = r ⁻² g(u)g′(u), λ(t) = t ^−1-ν, ν > 1/2 is arbitrary, and R{\mathcal{R}} is a term whose local energy goes to zero as t → 0.     

A Note on Classical Ground State Energies

The pair-specific ground state energy ε _g (N):=ℰ _g (N)/(N(N−1)) of Newtonian N body systems grows monotonically in N. This furnishes a whole family of simple new tests for minimality of putative ground state energies ℰ _g ^x (N) obtained through computer experiments. Inspection of several publicly available lists of such computer-experimentally obtained putative ground state energies ℰ _g ^x (N) has yielded several dozen instances of ℰ _g ^x (N) which failed one of these tests; i.e., for those N one concludes that ℰ _g ^x (N)>ℰ _g (N) strictly. Although the correct ℰ _g (N) is not revealed by this method, it does yield a better upper bound on ℰ _g (N) than ℰ _g ^x (N) whenever ℰ _g ^x (N) fails a monotonicity test. The surveyed N-body systems include in particular N point charges with 2- or 3-dimensional Coulomb pair interactions, placed either on the unit 2-sphere or on a 2-torus (a.k.a. Thomson, Fekete, or Riesz problems).     

Exact Analysis of Level-Crossing Statistics for (d+1)-Dimensional Fluctuating Surfaces

We carry out an exact analysis of the average frequency ν⁺ _αxi in the direction x _i of positiveslope crossing of a given level α such that, h(x, t) − = α, of growing surfaces in spatial dimension d. Here, h(x, t) is the surface height at time t, and is its mean value. We analyze the problem when the surface growth dynamics is governed by the Kardar-Parisi-Zhang (KPZ) equation without surface tension, in the time regime prior to appearance of cusp singularities (sharp valleys), as well as in the random deposition (RD) model. The total number N ⁺ of such level-crossings with positive slope in all the directions is then shown to scale with time as t ^d/2 for both the KPZ equation and the RD model. PACS number(s): 52.75.Rx, 68.35.Ct     

Universal behavior of CePd<Subscript>1−<Emphasis Type="Italic">x</Emphasis></Subscript>Rh<Subscript><Emphasis Type="Italic">x</Emphasis></Subscript> ferromagnet at the quantum critical point

The heavy-fermion metal CePd_1−x Rh _x can be tuned from ferromagnetism at x = 0 to the nonmagnetic state at some critical concentration x _c . The non-Fermi liquid behavior (NFL) at x ≃ x _c is recognized by the power-law dependence of the specific heat C(T) given by the electronic contribution susceptibility X(T) and volume expansion coefficient α(T) at low temperatures: C/T ∝ X(T) ∝ α(T)/T∝ 1/ √T. We also demonstrate that the behavior of the normalized effective mass M _N ^* observed in CePd_1−x Rh _x at x ≃ 0.8 agrees with that of M _N ^* observed in paramagnetic CeRu₂Si₂ and conclude that these alloys exhibit the universal NFL thermodynamic behavior at their quantum critical points. We show that the NFL behavior of CePd_1−x Rh _x can be accounted for within the frameworks of the quasiparticle picture and fermion condensation quantum phase transition, while this alloy exhibits a universal thermodynamic NFL behavior that is independent of the characteristic features of the given alloy such as its lattice structure, magnetic ground state, dimension, etc. The text was submitted by the authors in English.     

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.     

Superconductivity and magnetic ordering in YBa2(Cu1−x Fe x )3O7

Transverse-and zero-field μSR measurements have been made for YBa₂(Cu_1−x Fe _x )₃O₇ withx=0.04, 0.08 and 0.12. The temperature range studied was from approximately 7.5 K to 100 K. The onset of magnetic ordering commences at about 7.5 K forx=0.04, 10 K forx=0.08 and 20 K forx=0.12. The Gaussian depolarization parameter, σ ofG _x (t) = exp(−σ² t ²/2), is depressed by a factor of about 0.6 forx=0.04, but for thex=0.08 sample σ is depressed by a factor of 10 and increasing suppression is seen as the temperature is lowered below 45 K. This decrease in σ is interpreted in terms of decreasing electronic mean free paths.     

The Behaviour of Aging Functions in One-Dimensional Bouchaud's Trap Model

Let τ_i be a collection of i.i.d. positive random variables with distribution in the domain of attraction of an α-stable law with α<1. The symmetric Bouchaud's trap model on ℤ is a Markov chain X(t) whose transition rates are given by w_xy=(2τ_x)⁻¹ if x, y are neighbours in ℤ. We study the behaviour of two correlation functions: ℙ[X(t_w+t)=X(t_w)] and It is well known that for any of these correlation functions a time-scale t=f(t_w) such that aging occurs can be found. We study these correlation functions on time-scales different from f(t_w), and we describe more precisely the behaviour of a singular diffusion obtained as the scaling limit of Bouchaud's trap model. Work supported by DFG Research Center Matheon ``Mathematics for key technologies'     

The Discrete Spectrum in the Gaps of the Continuous One for Non-Signdefinite Perturbations with a Large Coupling Constant

Given two selfadjoint operators A and V=V ₊ -V _-, we study the motion of the eigenvalues of the operator A(t)=A-tV as t increases. Let α>0 and let λ be a regular point for A. We consider the quantities N ₊(λ,α), N _-(λ,α), N ₀(λ,α) defined as the number of the eigenvalues of the operator A(t) that pass point λ from the right to the left, from the left to the right or change the direction of their motion exactly at point λ, respectively, as t increases from 0 to α>0. An abstract theorem on the asymptotics for these quantities is presented. Applications to Schr?dinger operators and its generalizations are given. Received: 9 April 1997 / Accepted: 26 August 1997