Asymptotics of Determinants of Bessel Operators

 For aL ^∞ (ℝ₊)∩L ¹ (ℝ₊) the truncated Bessel operator B _τ (a) is the integral operator acting on L ² [0,τ] with the kernel

On the Absolutely Continuous Spectrum of Stark Operators

 The stability of the absolutely continuous spectrum of the one-dimensional Stark operator

Thermodynamical Limit for Correlated Gaussian Random Energy Models

 Let {E _Σ (N)} _ΣΣN be a family of |Σ _N |=2 ^N centered unit Gaussian random variables defined by the covariance matrix C _N of elements c _N (Σ,τ):=Av(E _Σ (N)E _τ (N)) and the corresponding random Hamiltonian. Then the quenched thermodynamical limit exists if, for every decomposition N=N ₁ +N ₂ , and all pairs (Σ,τ)Σ _N ×Σ _N :

Growth of Sobolev Norms in Linear Schrödinger Equations with Quasi-Periodic Potential

In this paper, we consider the following problem. Let iu _t +Δu+V(x,t)u= 0 be a linear Schr?dinger equation ( periodic boundary conditions) where V is a real, bounded, real analytic potential which is periodic in x and quasi periodic in t with diophantine frequency vector λ. Denote S(t) the corresponding flow map. Thus S(t) preserves the L ²-norm and our aim is to study its behaviour on H ^s (T ^D ), s> 0. Our main result is the growth in time is at most logarithmic; thus if φ∈H ^s , then

Spectral Properties of Hypoelliptic Operators

 We study hypoelliptic operators with polynomially bounded coefficients that are of the form K=∑ _i=1 ^m X _i ^T X _i +X ₀+f, where the X _j denote first order differential operators, f is a function with at most polynomial growth, and X _i ^T denotes the formal adjoint of X _i in L ². For any ɛ>0 we show that an inequality of the form ||u||_δ,δ≤C(||u||_0,ɛ+||(K+iy)u||_0,0) holds for suitable δ and C which are independent of yR, in weighted Sobolev spaces (the first index is the derivative, and the second the growth). We apply this result to the Fokker-Planck operator for an anharmonic chain of oscillators coupled to two heat baths. Using a method of Hérau and Nier [HN02], we conclude that its spectrum lies in a cusp {x+iy|x≥|y|^τ−c,τ(0,1],cR}. Received: 30 July 2002 / Accepted: 18 October 2002 Published online: 25 February 2003 RID="*" ID="*" Present address: Mathematics Research Centre of the University of Warwick Communicated by A. Kupiainen     

The trace identity and the planar Casimir effect

The familiar trace identity associated with the scale transformationx ^Μ → x′ ^Μ = e^-λ x ^Μ on the Lagrangian density for a noninteracting massive real scalar field in 2 + 1 dimensions is shown to be violated on a single plate on which the Dirichlet boundary condition Φ(t, x¹, x² = -a) = 0 is imposed. It is however respected in: (i) 1 + 1 dimensions in both free space and on a single plate on which the Dirichlet boundary condition Φ(t, x¹ = -a) = 0 holds and (ii) in 2 + 1 dimensions in free space, i.e. the unconstrained configuration. On the plate where Φ(t, x¹, x² = -a) = 0, the modified trace identity is shown to be anomalous with a numerical coefficient for the anomalous term equal to the canonical scale dimension, viz. 1/2. The technique of Bordaget al [Ann. Phys. (N.Y.),165, 162 (1985)] is used to incorporate the said boundary condition into the generating functional for the connected Green’s functions.     

New approach to the theory of spinodal decomposition

A new approach to the study of spinodal decomposition for a scalar field is proposed. The approach is based on treating this process as a relaxation of the one-time correlation function G(q,t)=∫d r<Φ (0, t)Φ (r,t)>exp(i q·r), which plays the role of an independent dynamical object (a unique two-point order parameter). The dynamical equation for G(q,t) (the Langevin equation in correlation-function space) is solved exactly in the one-loop approximation, which is the zeroth approximation in the approach proposed. This makes it possible to trace the asymptotic behavior of G(q,t) at long and intermediate times t (from the moment of onset of the spinodal decomposition). The values obtained for the power-law growth exponents for the height and position of the peak in G(q,t) at the intermediate stage is in satisfactory agreement with the data obtained by a number of authors through numerical simulation of the corresponding stochastic equations describing the relaxation of the local order parameter. Pis’ma Zh. éksp. Teor. Fiz. 66, No. 6, 432–437 (25 September 1997)     

Tails in harnesses

We consider space- and time-uniformd-dimensional random processes with linear local interaction, which we call harnesses and which may be used as discrete mathematical models of random interfaces. Their components are rea random variablesa _s ^t , wheres ∈ Z ^d andt=0, 1, 2.,... At every time step two events occur: first, every component turns into a linear combination of itsN neighbors, and second, a symmetric random i.i.d. “noise”v is added to every component. For any σ ∈Z _d ⁺ define Δ_σ a′ _s as follows. If σ=(0,...,0), σ=(0,...,0), Δ_σ a _s ^t =a _s ^t . Then by induction, wheree _i is thed-dimensional vector, whoseith component is one and other components are zeros. Denote |σ| the sum of components of σ. Call a real random variable ϕ symmetric if it is distributed as −ϕ. For any symmetric random variable ϕpower decay or P-decay is defined as the supremum of thoser for which therth absolute moment of ϕ is finite. Convergence a.s., in probability and in law whent→∞ is examined in terms of P-decay(v): Ifd=1, σ=0 ord=2, σ=(0,0), Δ_σ a _s ^t diverges. In all the other cases: If P-decay(v)<(d+2)/(d+|σ|), Δ_σ a _s ^t diverges; if P-decay(v)>(d+2)/(d+|σ|), Δ_σ a _s ^t , converges and P-decay(ν) For any symmetric random variable ϕexponential decay or E-decay is defined as the supremum of thoser for which the expectation of exp(|x|^r) is finite. Let E-decay(v)>0. Whenever Δ_σ a _s ^t converges (that is, ifd>2 or |σ|>0: Ifd>2, E-decay(lima _s ^t )=min(E-decay(v),d+2/2); if |σ|=1, E-decay (lim Δ_σ a _s ^t )=min(E-decay(ν),d+2); if |σ| ⩾, E-decay (lim Δ_σ a _s ^t )=E-decay(ν).     

Static charged spheres with anisotropic pressure in general relativity

We report a generalization of our earlier formalism [Pramana, 54, 663 (1998)] to obtain exact solutions of Einstein-Maxwell’s equations for static spheres filled with a charged fluid having anisotropic pressure and of null conductivity. Defining new variables: w=(4π/3)(ρ+ε)r ², u=4πξr ², v _r=4πp _r r ², v _⊥=4πp _⊥ r ²[ρ, ξ(=−(1/2)F ₁₄ F ¹⁴), p _r, p _⊥ being respectively the energy densities of matter and electrostatic fields, radial and transverse fluid pressures whereas ε denotes the eigenvalue of the conformal Weyl tensor and interpreted as the energy density of the free gravitational field], we have recast Einstein’s field equations into a form easy to integrate. Since the system is underdetermined we make the following assumptions to solve the field equations (i) u=v _r=(a ²/2κ)r ⁿ⁺², v _⊥=k ₁ v _r, w=k ₂ v _r; a ², n(>0), k ₁, k ₂ being constants with κ=((k ₁+2)/3+k ₂) and (ii) w+u=(b ²/2)r ⁿ⁺², u=v _r, v _⊥−v _r=k, with b and k as constants. In both cases the field equations are integrated completely. The first solution is regular in the metric as well as physical variables for all values of n>0. Even though the second solution contains terms like k/r ² since Q(0)=0 it is argued that the pressure anisotropy, caused by the electric flux near the centre, can be made to vanish reducing it to the generalized Cooperstock-de la Cruz solution given in [14]. The interior solutions are shown to match with the exterior Reissner-Nordstrom solution over a fixed boundary. Dedicated to Prof. F A E Pirani.     

Anomalous dynamical light scattering in soft glassy gels

We compute the dynamical structure factor S(q,τ) of an elastic medium where force dipoles appear at random in space and in time, due to “micro-collapses” of the structure. Various regimes are found, depending on the wave vector q and the collapse time θ. In an early time regime, the logarithm of the structure factor behaves as (qτ)^3/2, as predicted in (L. Cipelletti et al., Phys. Rev Lett. 84, 2275 (2000)) using heuristic arguments. However, in an intermediate-time regime we rather obtain a (qτ)^5/4 behaviour. Finally, the asymptotic long-time regime is found to behave as q ^3/2τ. We also give a plausible scenario for aging, in terms of a strain-dependent energy barrier for micro-collapses. The relaxation time is found to grow with the age t_w, quasi-exponentially at first, and then as t _w ^4/5 with logarithmic corrections. Received 15 April 2002     

Propagation of electromagnetic waves generated by moving sources in dispersive media

The propagation of electromagnetic waves issued by modulated moving sources of the form j( t,x ) = a( t )e ^{- iw₀ t} [(x)\dot]₀ ( t )d( x - x₀ ( t ) )j\left( {t,x} \right) = a\left( t \right)e^{ - i\omega _0 t} \dot x_0 \left( t \right)\delta \left( {x - x_0 \left( t \right)} \right) is considered, where j(t, x) stands for the current density vector, x = (x ₁, x ₂, x ₃) ∈ ℝ³ for the space variables, t ∈ ℝ for time, t → x ₀(t) ∈ ℝ³ for the vector function defining the motion of the source, ω ₀ for the eigenfrequency of the source, a(t) for a narrow-band amplitude, and δ for the standard δ function. Suppose that the media under consideration are dispersive. This means that the electric and magnetic permittivity ɛ(ω), μ(ω) depends on the frequency ω. We obtain a representation of electromagnetic fields in the form of time-frequency oscillating integrals whose phase contains a large parameter λ > 0 characterizing the slowness of the change of the amplitude a(t) and the velocity [(x)\dot]₀ ( t )\dot x_0 \left( t \right) and a large distance between positions of the source and the receiver. Applying the two-dimensional stationary phase method to the integrals, we obtain explicit formulas for the electromagnetic field and for the Doppler effects. As an application of our approach, we consider the propagation of electromagnetic waves produced by moving source in a cold nonmagnetized plasma and the Cherenkov radiation in dispersive media.     

Interactions of charmed mesons with nucleons in the ¯<Emphasis Type="Italic">p</Emphasis><Emphasis Type="Italic">d</Emphasis> reaction

We study the possibility to measure the elastic ΦN (Φ≡J/ψ,ψ(2S), ψ(3770), χ_2c) scattering cross section in the reaction ˉp+d→Φ+n _sp and the elastic D(ˉD)N scattering cross section in the reaction ˉp+d→D ⁻ D ⁰ p _sp. Our studies indicate that the elastic scattering cross sections can be determined for Φ momenta about 4–6 GeV/c and D/ˉD momenta 2–5 GeV/c by selecting events with p _t≥ 0.4 GeV/c for Φ's and p _t(p _sp) ≥ 0.5 GeV/c for D/ˉD-meson production. Received: 8 November 1999     

On the Spectrum of the Generator of an Infinite System of Interacting Diffusions

We study the spectrum of the operator

Construction of the Incipient Infinite Cluster for Spread-out Oriented Percolation Above 4 + 1 Dimensions

 We construct the incipient infinite cluster measure (IIC) for sufficiently spread-out oriented percolation on ℤ ^d × ℤ₊, for d +1 > 4+1. We consider two different constructions. For the first construction, we define ℙ ⁿ (E) by taking the probability of the intersection of an event E with the event that the origin is connected to (x,n)  ℤ ^d × ℤ₊, summing this probability over x  ℤ ^d , and normalising the sum to get a probability measure. We let n → ∞ and prove existence of a limiting measure ℙ_∞, the IIC. For the second construction, we condition the connected cluster of the origin in critical oriented percolation to survive to time n, and let n → ∞. Under the assumption that the critical survival probability is asymptotic to a multiple of n ⁻¹, we prove existence of a limiting measure ℚ_∞, with ℚ_∞ = ℙ_∞. In addition, we study the asymptotic behaviour of the size of the level set of the cluster of the origin, and the dimension of the cluster of the origin, under ℙ_∞. Our methods involve minor extensions of the lace expansion methods used in a previous paper to relate critical oriented percolation to super-Brownian motion, for d+1 > 4+1. Received: 13 December 2001 / Accepted: 11 July 2002 Published online: 29 October 2002 RID="*" ID="*" Present address: Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. E-mail: rhofstad@win.tue.nl     

Existence and Modulation of Uniform Sliding States in Driven and Overdamped Particle Chains

In this paper we are mainly concerned with existence and modulation of uniform sliding states for particle chains with damping γ and external driving force F. If the on-site potential vanishes, then for each F > 0 there exist trivial uniform sliding states x _n (t) = n ω + ν t + α for which the particles are uniformly spaced with spacing ω > 0, the sliding velocity of each particle is ν = F/γ, and the phase α is arbitrary. If the particle chain with convex interaction potential is placed in a periodic on-site potential, we show under some conditions the existence of modulated uniform sliding states of the form

Two-particle dispersion in model two-dimensional velocity fields

We consider two-particle dispersion in a velocity field, where the relative two-point velocity scales according to v ²(r) ∝r ^α and the corresponding correlation time scales as τ(r) ∝r ^β, and fix α = 2/3, as typical for turbulent flows. We show that two generic types of dispersion behavior arize: For α/2 + β < 1 the correlations in relative velocities decouple and the diffusion approximation holds. In the opposite case, α/2 + β > 1, the relative motion is strongly correlated. The case of Kolmogorov flows corresponds to a marginal, nongeneric situation. In this case, depending on the particular parameters of the flow, the dispersion behavior can be rather diffusive or rather ballistic. Received 13 March 2001     

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     

A Multiplicative Ergodic Theorem and Nonpositively Curved Spaces

We study integrable cocycles u(n,x) over an ergodic measure preserving transformation that take values in a semigroup of nonexpanding maps of a nonpositively curved space Y, e.g. a Cartan–Hadamard space or a uniformly convex Banach space. It is proved that for any y∈Y and almost all x, there exist A≥ 0 and a unique geodesic ray γ (t,x) in Y starting at y such that

The Fixed Point of a Generalization of the Renormalization Group Maps for Self-Avoiding Paths on Gaskets

Let W(x,y) = ax ³+ bx ⁴+ f ₅ x ⁵+ f ₆ x ⁶+ (3 ax ²)² y+ g ₅ x ⁵ y + h ₃ x ³ y ² + h ₄ x ⁴ y ² + n ₃ x ³ y ³+a ₂₄ x ² y ⁴+a ₀₅ y ⁵+a ₁₅ xy ⁵+a ₀₆ y ⁶, and X = , , where the coefficients are non-negative constants, with a > 0, such that X ²(x,x ²)−Y(x,x ²) is a polynomial of x with non-negative coefficients. Examples of the 2 dimensional map Φ: (x,y)↦ (X(x,y),Y(x,y)) satisfying the conditions are the renormalization group (RG) maps (modulo change of variables) for the restricted self-avoiding paths on the 3 and 4 dimensional pre-gaskets. We prove that there exists a unique fixed point (x _f ,y _f ) of Φ in the invariant set . 2000 Mathematics Subject Classification Numbers: 82B28; 60G99; 81T17; 82C41.     

Effect of carbon ion bombardment and surface oxidation on the twinning rate of single-crystal bismuth

A study is made of the change in the dependences of the normal velocity of twinning boundaries on the magnitude of shear stresses in the twinning plane v _n =v _n (τ) in bismuth crystals owing to ion-cluster doping and oxidation of the irradiated surface. Irradiation was by 25 keV carbonions at a dose of 10¹⁷ ion/cm². Twinning of the crystals took place under pulsed loading conditions with pulse durations of 10⁻⁵−10⁻⁴ s and stress amplitudes of (0.2–2.0)×10³ g/mm³. Carbon ion bombardment of single-crystal bismuth causes a shift in the v _n =v _n (τ) curve toward lower stresses. An oxide film slows down the motion of twinning dislocations. Zh. Tekh. Fiz. 69, 130–131 (May 1999)