Bound states and critical behavior of the Yukawa potential

We investigate the bound states of the Yukawa potential V (r)=−λexp(−αr)/r, using different algorithms: solving the Schr?dinger equation numerically and our Monte Carlo Hamiltonian approach. There is a critical α = α_C, above which no bound state exists. We study the relation between α_C and λ for various angular momentum quantum number l, and find in atomic units, α_C(l) = λ[A ₁ exp(−l/B ₁) + A ₂ exp(−l/B ₂)], with A ₁ = 1.020(18), B ₁ = 0.443(14), A ₂ = 0.170(17), and B ₂ = 2.490(180).     

Trend to equilibrium of weak solutions of the Boltzmann equation in a slab with diffusive boundary conditions

Recently R. Illner and the author proved that, under a physically realistic truncation on the collision kernel, the Boltzmann equation in the one-dimensional slab [0, 1] with general diffusive boundary conditions at 0 and 1 has a global weak solution in the traditional sense. Here it is proved that when the Maxwellians associated with the boundary conditions atx=0 andx=1 are the same MaxwellianM _w , then the solution is uniformly bounded and tends toM _w fort.     

Distribution function for large velocities of a two-dimensional gas under shear flow

The high-velocity distribution of a two-dimensional dilute gas of Maxwell molecules under uniform shear flow is studied. First we analyze the shear-rate dependence of the eigenvalues governing the time evolution of the velocity moments derived from the Boltzmann equation. As in the three-dimensional case discussed by us previously, all the moments of degreek⩾4 diverge for shear rates larger than a critical valuea _c ^(k) , which behaves for largek asa _c ^(k) ∼k ⁻¹. This divergence is consistent with an algebraic tail of the formf(V) ∼V ^−4-σ(a), where σ is a decreasing function of the shear rate. This expectation is confirmed by a Monte Carlo simulation of the Boltzmann equation far from equilibrium.     

A maxwellian lower bound for solutions to the Boltzmann equation

We prove that the solution of the spatially homogeneous Boltzmann equation is bounded pointwise from below by a Maxwellian, i.e. a function of the formc ₁ exp(-c ₂ v ²). This holds for any initial data with bounded mass, energy and entropy, and for any positive timet≧t ₀. The constantsc ₁, andc ₂, depend on the mass, energy and entropy of the initial data, and ont ₀>0 only. A similar result is obtained for the Kac caricature of the Boltzmann equation, where the proof is easier.     

An improved model for band tail states inn-type degenerate semiconductors

Summary Using two screened donor potential energy models and a wave vectork-positionr uncertainty relation, the results of the exponential band tail states, inn-type degenerate semiconductors, obtained in our previous paper, are improved. The second-order cumulant or the correlation functionW (ϱ,E) is expressed as a function of the total donor concentration ϱ and total carrier energyE. Near band edges, the conduction and valence band tails are, respectively, proportional to exp [E/E ₀(ϱ)] and exp [−E/E ₀(ϱ)], whereE ₀(ϱ) is the energy characteristic of the appropriate band tail, in good accordance with those experimentally obtained by Pankove. Far below the conduction band edge, our result of the conduction band tail is proportional to exp [−BE ² exp [A ]], whereA andB are the functions of ϱ, which is reduced to zero more rapidly in comparison with the corresponding result obtained by Halperin and Lax (i.e. exp [−|E| ⁿ , wheren varies between 1/2 and 2). To speed up publication, the authors of this paper have agreed to not receive the proofs for correction.     

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.     

Flexible-to-semiflexible chain crossover on the pressure-area isotherm of a lipid bilayer

We find theoretically that competition between ∼K _f q ⁴ and ∼Qq ² terms in the Fourier-transformed conformational energy of a single-lipid chain, in combination with interchain entropic repulsion in the hydrophobic part of the lipid (bi)layer, may cause a crossover on the bilayer pressure-area isotherm P(A)∼(A−A ₀)^−α. The crossover manifests itself in the transition from α = 5/3 to α = 3. Our microscopic model represents a single-lipid molecule as a worm-like chain with a finite irreducible cross-section area A ₀, a flexural rigidity K _f , and a stretching modulus Q in a parabolic potential with the self-consistent curvature B(A) formed by entropic interactions between hydrocarbon chains in the lipid layer. The crossover area A* obeys the relation Q/√K _f B(A*) ≈ 2. We predict a peculiar possibility of deducing the effective elastic moduli K _f and Q of an individual hydrocarbon chain from the analysis of the isotherm with such a crossover. Also calculated is the crossover-related behavior of the area compressibility modulus K _A , the equilibrium area per lipid A _t , and the chain order parameter S(θ). The text was submitted by the authors in English.     

Electroluminescence in the phosphor anthracene doped by 9-vinylanthracene

Electroluminescence has been excited in the phosphor anthracene doped by 9-vinylanthracene (10⁻⁴ mole %) at room temperature (300°K). Variation of light output with voltage is governed by the relationB=B ₀ exp [−(C/V ^1/2)] as in the case of inorganic phosphors. HereB ₀ andC are constants andV is the rms value of the voltage applied to the phosphor. Frequency dependence of the electroluminescence has also been studied at different voltages of the applied electric field.     

Sum-of-Squares Results for Polynomials Related to the Bessis–Moussa–Villani Conjecture

We show that the polynomial S _m,k (A,B), that is the sum of all words in noncommuting variables A and B having length m and exactly k letters equal to B, is not equal to a sum of commutators and Hermitian squares in the algebra R〈X,Y〉, where X ²=A and Y ²=B, for all even values of m and k with 6≤k≤m−10, and also for (m,k)=(12,6). This leaves only the case (m,k)=(16,8) open. This topic is of interest in connection with the Lieb–Seiringer formulation of the Bessis–Moussa–Villani conjecture, which asks whether Tr (S _m,k (A,B))≥0 holds for all positive semidefinite matrices A and B. These results eliminate the possibility of using “descent + sum-of-squares” to prove the BMV conjecture.     

Diffusing wave spectroscopy of dense colloids: Liquid,crystal and glassy states

Using intensity autocorrelation of multiply scattered light, we show that the increase in interparticle interaction in dense, binary colloidal fluid mixtures of particle diameters 0.115μm and 0.089μm results in freezing into a crystalline phase at volume fractionφ of 0.1 and into a glassy state atφ=0.2. The functional form of the field autocorrelation functiong ⁽¹⁾(t) for the binary fluid phase is fitted to exp[−γ(6k ₀ ² D _eff t)^1/2] wherek ₀ is the magnitude of the incident light wavevector andγ is a parameter inversely proportional to the photon transport mean free pathl*. TheD _eff is thel* weighted average of the individual diffusion coefficients of the pure species. Thel* used in calculatingD _eff was computed using the Mie theory. In the solid (crystal or glass) phase, theg ⁽¹⁾(t) is fitted (only with a moderate success) to exp[−γ(6k ₀ ² W(t))^1/2] where the mean-squared displacementW(t) is evaluated for a harmonically bound overdamped Brownian oscillator. It is found that the fitted parameterγ for both the binary and monodisperse suspensions decreases significantly with the increase of interparticle interactions. This has been justified by showing that the calculated values ofl* in a monodisperse suspension using Mie theory increase very significantly with the interactions incorporated inl* via the static structure factor.     

Dual analytic model of generalized parton distributions

A model for generalized parton distributions (GPDs) in the form of ∼(x/g ₀)^{(1−x)ᾶ(t)}, where ᾶ(t) = α(t) − α(0) is the nonlinear part of the Regge trajectory and g ₀ is a parameter, g ₀ > 1, is presented. For linear trajectories, it reduces to earlier proposals. We compare the calculated moments of these GPDs with the experimental data on form factors and find that the effects from the nonlinearity of Regge trajectories are large. By Fourier transforming the obtained GPDs, we access the spatial distribution of protons in the transverse plane. The relation between dual amplitudes with Mandelstam analyticity and composite models in the infinite-momentum frame is discussed, the integration variable in dual models being associated with the quark longitudinal-momentum fraction x in the nucleon. The text was submitted by the authors in English.     

High resolution optogalvanic study in nitrogen discharge

Doppler limited high resolution spectrum in the wavelength region 17224 to 17236 cm⁻¹ of the first positive system (B ³Π _g −A ³Σ _u ⁺ ) of the N₂ molecule is recorded by optogalvanic spectroscopic technique using a single mode ring dye laser. It is observed that the intensity and line width of the rotational line increase with the discharge current. Dependence of the collision broadening coefficient on the current was also evaluated.     

Conservation laws for classical and relativistic collisions. II

On the basis of elementary symmetry arguments it is shown that (1) if in classical mechanics there exists a quantity λ+Σ_iμ_iυ_i+1/2νυ ² that is conserved, where λ,μ _i, andν are particle parameters, then theμ _i andν are all proportional to a single parameterμ and the quantityAμ+Σ_iB_iμυ_i+C(λ+ 1/2Dμυ ²), whereD ≡ν/μ, is conserved for all values ofA, B _i, andC; (2) if in relativistic mechanics there exists a quantity λ+Σ_iμ_iυ_i[1−(υ ²/c ²)]^−1/2+νc[1−(υ ²/c ²)]^−1/2 that is conserved, then theμ _i andν are all proportional to a single parameterμ and the quantityAλ+Σ_iB_iμν_i[1−(υ ²/c ²)]^−1/2+Cμc [1−(υ ²/c ²)]^−1/2 is conserved for all values ofA, B _i, andC.     

Polarization of leading Λ hyperons produced by neutrons on target nuclei

The polarization of the leading Λ hyperons produced on carbon and lead target nuclei by 4–10 GeV neutrons in the angle range Θ<8.5° with respect to the beam and with only neutral-particle accompaniment is measured: 〈Π〉=−0.096±0.018 for C and 〈Π〉=− 0.128±0.047 for Pb. The dependence of the polarization on the transverse momentum and the Feynman variable is measured. The normalized invariant cross section as a function of p _⊥ ² is found to be approximated by the function A exp(−Bp _⊥ ² ), where the parameter B is independent of the kind of nucleus (B=8.71±0.09 (GeV/c)⁻² for carbon and B=8.83±0.18 (GeV/c)⁻² for lead). Pis’ma Zh. éksp. Teor. Fiz. 64, No. 4, 237–240 (25 August 1996)     

Magnetic hyperfine field of arsenic in nickel by NMR/ON of71,72As

The magnetic hyperfine splitting frequencies of⁷¹AsNi and⁷²AsNi in a 0.11 Tesla external magnetic field have been determined by NMR/ON method as 66.00(6) MHz and 106.17(13) MHz respectively. Using the known magnetic moments of μ(⁷¹As)=1.6735(18) and μ(⁷²As)=−2.1566(3), the hyperfine fields were deduced asB _hf(⁷¹AsNi)=12.824(19) Tesla andB _hf(⁷²AsNi)=12.807(16) Tesla.     

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.     

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.     

e + e − pair production in ultrarelativistic heavy-ion collisions at intermediate impact parameters

Using the semiclassical Green’s function in the Coulomb field, we analyze the probabilities of single and multiple e ⁺ e ⁻ pair production at a fixed impact parameter b between colliding ultrarelativistic heavy nuclei. We perform calculations in the Born approximation with respect to the parameter Z _Bα and exactly in Z _Aα, where Z _A and Z _B are the charge numbers of the corresponding nuclei. We also obtain the approximate formulas for the probabilities valid for Z _Aα, Z _Bα ≲ 1. The text was submitted by the authors in English.     

On Strong Convergence to Equilibrium for the Boltzmann Equation with Soft Potentials

The paper concerns L ¹-convergence to equilibrium for weak solutions of the spatially homogeneous Boltzmann Equation for soft potentials (−4≤γ<0), with and without angular cutoff. We prove the time-averaged L ¹-convergence to equilibrium for all weak solutions whose initial data have finite entropy and finite moments up to order greater than 2+|γ|. For the usual L ¹-convergence we prove that the convergence rate can be controlled from below by the initial energy tails, and hence, for initial data with long energy tails, the convergence can be arbitrarily slow. We also show that under the integrable angular cutoff on the collision kernel with −1≤γ<0, there are algebraic upper and lower bounds on the rate of L ¹-convergence to equilibrium. Our methods of proof are based on entropy inequalities and moment estimates. E.A. Carlen work partially supported by US National Science Foundation grant DMS 06-00037. M.C. Carvalho work partially supported by POCI/MAT/61931/2004. X. Lu work partially supported by NSF of China grant 10571101.     

The Parameter Planes of λz<Superscript><Emphasis Type="Italic">m</Emphasis></Superscript> exp(z) for <Emphasis Type="Italic">m</Emphasis> ≥ 2*

We consider the families of entire transcendental maps given by F _λ,m (z) = λz ^m exp(z), where m ≥ 2. All functions F _λ,m have a superattracting fixed point at z = 0, and a critical point at z = −m. In the parameter planes we focus on the capture zones, i.e., λ values for which the critical point belongs to the basin of attraction of z = 0, denoted by A(0). In particular, we study the main capture zone (parameter values for which the critical point lies in the immediate basin, A ^*(0)) and prove that is bounded, connected and simply connected. All other capture zones are unbounded and simply connected. For each parameter λ in the main capture zone, A(0) consists of a single connected component with non-locally connected boundary. For all remaining values of λ, A ^*(0) is a quasidisk. On a different approach, we introduce some families of holomorphic maps of which serve as a model for F _λ,m , in the sense that they are related by means of quasiconformal surgery to F _λ,m . Both authors were supported by MTM2005-02139/Consolider (including a FEDER contribution) and CIRIT 2005 SGR01028. The first author was also supported by MTM2006-05849/Consolider (including a FEDER contribution).