Analytical solution of the double-logarithmic integral equation with QCD running coupling

The analytical solution of the double-logarithmic integral equation with QCD running coupling describing small-x behaviour of the non-singlet structure function ? _NS(x,Q ²) has been found for any cut-off parameter μ. Analytical properties of the solution and a position of the right-most singularity in the complex ρ-plane which determines the asymptotics of ? _NS(x,Q ²) at small x have been studied. The asymptotical formula ? _NS(x,Q ²) = C ₁ x ^-λ1{ln^κ1(Q ²/Λ²) —ln^κ1 (μ ²/Λ²) + κ ₁ ln^κ1-1(Q ²/Λ²)[ψ(1) - ψ(λ₁)]} valid if x ? 1 and ln(Q ²/Λ²) ? 1 has been obtained where C ₁, λ₁ are constants, κ ₁ = g/λ₁, λ₁ < g = 8/(33 - 2gh _f), gh _f is a number of active flavours and ψ(ξ) denotes the digamma function.     

Effects of cross-correlation between the real and imaginary parts of the quantum noise in a two-mode saturation laser system

A two-mode saturation laser model with cross-correlation between the real and imaginary parts of the quantum noise is considered. The laser intensity Langevin equation and corresponding Fokker-Planck equation are derived by the phase-locking method. The effects of the cross-correlation strength λ between the real and imaginary parts of quantum noise and the cavity decay constant K on the steady-state intensity distribution Q(I₁,I₂), the mean light intensity 〈I〉, the normalization autocorrelation λ₁₁(0) and cross correlation λ₁₂(0) are studied by numerical calculation. The results show that as λ increases the Qs(I₁,I₂) show two extrema, and λ almost does not affect the 〈I〉, λ₁₁(0) and λ₁₂(0) when the laser system is operated far above threshold. Nevertheless, when the laser system is operated at and below threshold, λ makes the curves of Qs(I₁,I₂) have the higher peak and drop faster. Furthermore, it enhances the deviation of λ₁₁(0) and λ₁₂(0) and lessens the mean light intensity 〈I〉 when the laser system is operated at and below threshold.     

Semiclassical analysis of low lying elgenvalues. III. Width of the ground state band in strongly coupled solids

A smooth periodic potential, V, with one minima per unit cell, is considered. Let Δ(λ) be the width of the ground state band for ?Δ + λ²V. It is rigorously proved that lim_λ→∞ ? λ^?1 ln Δ(λ) is given by the minimum action among all instantons connecting two distinct minima of V.     

The twistM λ,T=0 giant modes in spherical nuclei

The theory of twisting nuclear vibrations developed by Holzwarth and Eckart to elucidate the nature of the 2^? giant mode is extended to high-multipole, spin-independentM λ,T=0 resonances. Within the framework of the distorted Fermi-surface model the parameters of inertia and stiffness against multipolarity for the nuclear incompressible Fermi-drop are derived. The energies and probabilities of the twistingM λ,T=0 modes calculated in fact without adjustable constants readE(M λ)=?ω _F \(\left[ {\frac{{(2\lambda + 3)(\lambda - 1)}}{5}} \right]^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}} \) andB(M λ) = γ_λ Z² A ^(2λ-4)/3 μ² fm^(2λ-2)respectively. The comparison with data for the 2^? resonance and predictions for higher multipoles are presented.     

Intensity correlation function of an optical bistable system subjected to cross-correlated noises

The normalized correlation function C(s) of an optical bistable system driven by cross-correlated noises is investigated. Based on the numerical computation and simulation results, it is found that: (1) The intensity of multiplicative noise D and the intensity of additive noise Q play opposite roles on C(s), i.e., D enhances the rate of fluctuation decay of transmitted light intensity and Q slows down the rate of fluctuation decay of the transmitted light intensity; (2) The strength of correlations between the multiplicative and additive noises λ slows down the rate of decay of intensity fluctuation; (3) For the case of positive correlated noises (λ>0), the increasing cooperativity parameter C slows down the rate of decay of intensity fluctuation firstly and then enhances it; (4) For the case of negative correlated noises (λ<0), the increasing C almost does not affect the rate of decay of intensity fluctuation firstly and then enhances it.     

Solitons and poles in the nonlinearity parameter

The KdVN-soliton solutions are analysed in terms of the perturbation parameter λ which governs the nonlinearity. They are generated by rational Stieltjes functionsQ ^(N) (λ), each pole of which can be associated with a soliton. The asymptotic emergence of the separate solitons follows at once from the motion of the poles along the negative real λ-axis. Successive diagonal Padé approximants ofQ ^(N) (λ) are considered. They provide a class of approximate solutions with a striking semisoliton like behaviour.     

On the basic states of one-dimensional disordered structures

The purpose of this paper is to study a limit probability distribution of the set of the first κ eigenvalues λ₁(?)<λ₂(?)<...<λ_κ(?) (with a fixed κ and ?→∞) of the boundary problem on the interval [0, ?] wherea(t, ω),q(t, ω) are the random stationary processes. Particularly the question of the repulsion between the first eigenvalues (small energetic levels) is studied. It has been proved that in the “divergent” case (q(t, ω)=0,a(t, ω)≠0) levels repulsion exists. As for the “potential” case (a(t, ω)≡1,q(t, ω)≠0) there is not any repulsion at all. This is one of the main differences between these two cases.     

Verification of the laws of luminescence of solids for manganese complexes of benzyl-trimethylammonium and 2-chloropyridinium

A formula at low temperature for the function G($\text{1}\text{λ}$) = I_λλ⁶ has been derived in terms of configuration curve theory and checked on luminescence spectra near 77 K. The vibrational quantum in the excited state has been calculated from the decrease of G($\text{1}\text{λ}{}_{\mathrm{M}}$), where λ_M is the wavelength of the maxima of G at low temperature, as a function of temperature.     

Correlation length bounds for disordered Ising ferromagnets

Thed-dimensional, nearest-neighbor disordered Ising ferromagnet: $$H = - \sum {J_{ij} \sigma _i \sigma _j }$$ is studied as a function of both temperature,T, and a disorder parameter,λ, which measures the size of fluctuations of couplingsJ _ij ≧0. A finite-size scaling correlation length,ζ _f (T, λ), is defined in terms of the magnetic response of finite samples. This correlation length is shown to be equivalent, in the scaling sense, to the quenched average correlation lengthζ(T, λ), defined as the asymptotic decay rate of the quenched average two-point function. Furthermore, the magnetic response criterion which definesζ _f is shown to have a scale-invariant property at the critical point. The above results enable us to prove that the quenched correlation length satisfies: $$C\left| {\log \xi (T)} \right|\xi (T) \geqq \left| {T - T_c } \right|^{ - {2 \mathord{\left/ {\vphantom {2 d}} \right. \kern-\nulldelimiterspace} d}}$$ which implies the boundv≧2/d for the quenched correlation length exponent.     

Infrared contribution to the double-logarithmic small-x asymptotics of structure functions

The role playing by the “soft” (κ _T < 1 GeV/c) region in the small-x behaviour ofg ₁(x,Q ²) and the non-singlet structure functionf _{1, NS}(x, Q²) has been studied with the help of the effective QCD Lagrangian which takes into account the lightest degrees of freedom — the constituent quarks and the π-mesons (Goldstone bosons). It has been shown that the quark-quark interaction due to the pion exchange has a negative couplingg for the isovector component (I = 1 in thet-channel) off ₁(x,Q ²) and isosinglet component ofg ₁(x, Q²). Here the pion induced interaction changes mainly the normalization of the quark distribution (it decreasesf _1,NS ^{I = 1} (x,Q ²) two times atx < 3·10^?3) and changes slightly the effective exponents λ (?_1,NS ,g ₁ ～x ^?λ atx → 0). On the other hand due to a positive value of couplingg the value of λ increases by 15% for the isovector part ofg ₁ (x,Q ²) and up to λ ≈ 0.5 (instead of λ ≈ 0.2 without the pion contribution) for the isoscalar non-singlet structure functionf _1,NS ^{I = 0} (x,Q ²).     

The spectral class of the quantum-mechanical harmonic oscillator

The purpose of this paper is to study the so-calledspectral class Q of anharmonic oscillatorsQ=?D ²+q having the same spectrum λ _n =2n (n≧0) as the harmonic oscillatorQ ⁰=?D ²+x ²?1. Thenorming constants \(t_n = \mathop {\lim }\limits_{x \uparrow \infty } \ell g[( - 1)^n {{e_n (x)} \mathord{\left/ {\vphantom {{e_n (x)} {e_n }}} \right. \kern-0em} {e_n }}( - x)]\) of the eigenfunctions ofQ form a complete set of coordinates inQ in terms of which the potential may be expressed asq=x ²?1?2D ² ?g? with $$\theta = \det \left[ {\delta _{ij} + (e^{ti} - 1)\int\limits_x^\infty {e_i^0 e_j^0 :0 \leqq i,j,< \infty } } \right],$$ e _n ⁰ being then ^th eigenfunctionQ ⁰. The spectrum and norming constants are canonically conjugate relative to the bracket [F, G]=∫ΔFDΔGdx,to wit: [λ _i , λj=0, [t _i, 2λ _j ]=1 or 0 according to whetheri=j or not, and [t _i,t _j]=0. This prompts an investigation of the symplectic geometry ofQ. The function ? is related to the theta function of a singular algebraic curve. Numerical results are also presented.     

E.S.R. evidence for the planarity of the t-butyl radical

Monte Carlo calculations are reported for the radial distribution function g ₂(r; λ) of a fluid in which the intermolecular pair potential is [u ^ref(r) + λu ^p(r)], u ^ref(r) being the Weeks-Chandler-Andersen (WCA) reference fluid, and [u ^ref(r) + u ^p(r)] being the Lennard-Jones (6, 12) fluid. The calculations are performed for λ values in the range 0 to 1, at the state condition ρσ³ = 0·80, kT/ε = 0·719. It is shown that at high densities the perturbation expansion of g ₂(r; λ = 1) about g ₂(r; λ = 0) is rapidly convergent, but that the corresponding expansion for y ₂(r; λ) = exp [βu(r; λ)] × g ₂(r; λ) is not. In addition Monte Carlo estimates of the individual terms that contribute to the first-order perturbation term, (?g ₂/?λ)_λ=0, are presented. It is shown that these terms are individually large, but that (?g ₂/?λ)_λ=0 is small because there is strong cancellation between the various terms. Consequently, the calculation of (?g ₂/?λ)_λ=0 is highly sensitive to the approximation used to evaluate the individual terms.     

On the theory of the low-temperature electrical resistivity of potassium

Expressing the residual resistivity ?₀ as a force-force correlation, a first-principles basis is afforded for the low temperature expansion of the electrical resistivity ? as

(I)$\text{? = ?}{}_0\text{[1 +}\text{λ1}\text{λ}\text{+ … ]}$

where λ = const T^-2 is the non-resistive mean free path arising from electron-electron scattering. λ¹ is found to reflect directly long-range correlations in the electronic motions via the off-diagonal behaviour of the local density of states at the Fermi energy.     

Asymptotic perturbation expansion in theP(φ)2 quantum field theory

In theP(φ)₂ model it is proved that the perturbation series for the infinite volume Schwinger functionsS(λ) are asymptotic in the limit as the coupling constant λ goes to zero. We also give conditions which imply smoothness ofS(λ) at arbitrary λ.     

New soft porous frameworks based on lambda-zirconium phosphate and aliphatic dicarboxylates: Synthesis and structural characterization

New dicarboxylate-functionalized pillared materials with a general formula of λ-ZrPO₄(OH)_1-x(OOC(CH₂)_nCOO)_x/2(dmso) (n=6, 8 and 10) have been prepared by post-synthesis modification of the inorganic layers of λ-zirconium phosphate (λ-ZrP), where the superficial Chloride monovalent anionic ligands of λ-layer are partially exchanged with the divalent anionic ligands of a series of long-chain aliphatic dicarboxylic acids, namely octanedioic acid, decanedioic acid and dodecanedioic acid. The synthesized materials are characterized by X-ray diffractometry, FT-IR spectrophotometry, elemental and thermogravimetric analyses. The X-ray diffraction patterns show that the obtained solid phases are pure. Furthermore, the interlayer distance of λ-ZrP systematically increases from 1.02 to 1.59 nm as a result of the incorporation of the mentioned acids inside the interlayer gallery.     

The correlation function for a quantum oscillator in a low-temperature heat bath

The momentum autocorrelation function c(t) for a quantum oscillator coupled with harmonic forces to a heat bath of oscillators is calculated at low temperatures. It is found that c(t) contains two distinct terms: one, the zero-point contribution c₀(t), is temperature independent, and the other, c₁(t), does depend on temperature. We concentrate our attention on the low-temperature case. An expression for c₁(t) is obtained, which is valid for arbitrary strenghts of the coupling and for arbitrary times. It is shown that c₁(t) is governed by the low-frequency behaviour of F(λ) = A²(λ)ϱ(λ), where ϱ(λ) is the density of normal modes and A(λ) is the central-oscillator component of the λth normal mode; other details of the problem are irrelevant. It is found that c₁(t) decays in time as an inverse-power law, with a relaxation time t_q ≈ ħ/kT.     

Inelastic 2-prong events in 147 GeV/cπ−p collisions

An SU(3) gauge-field non-topological soliton in three space-dimensions is discussed. Let N be the largest among the magnitudes of eigenvalues of the matrix Σ₁⁸λ_aQ_a, where the Q_a's are the eight conserved charges in the classical SU(3) gauge theory, and the λ_a's the usual 3 × 3 Gell-Mann matrices. We show that stable solitons exist if N > a critical value N_S, and if the ratio between the mass of the SU(3) gauge field and that of a relevant spin-0 field is >√6. Our trial function analysis suggests that, for large N, states with dets (Σ₁⁸λ_aQ_a) ≠ 0 are unstable against fission.     

The N-mode squeezed state with enhanced squeezing

It is known that exp [iλ (Q1P1i/2)] is a unitary single-mode squeezing operator,where Q1,P1 are the coordinate and momentum operators,respectively.In this paper we employ Dirac’s coordinate representation to prove that the exponential operator S n ≡ exp [iλ sum((QiPi+1+Qi+1Pi))) from i=1 to n ],(Qn+1=Q1,Pn+1=P1),is an n-mode squeezing operator which enhances the standard squeezing.By virtue of the technique of integration within an ordered product of operators we derive S n ’s normally ordered expansion and obtain new n-mode squeezed vacuum states,its Wigner function is calculated by using the Weyl ordering invariance under similar transformations.     

Eigenvalue comparisons for quantum systems in a box

Comparison theorems are obtained for the first even and odd solutions of Schrödinger's equation −v″ + Q(t)v = λv, −l ≤ t ≤ l with boundary conditions v(−l) = v(l) = 0. The comparison functions Q_i(t), i = 1, 2, may intersect at a finite number of points within [−l,l]. Immediate extensions are possible for a more general class of Sturm-Liouville problems, and for problems in unbounded regions.     

Transition from antiferromagnetic to ferromagnetic state of systems LaMnO3+λ and La1−x Srx(Mn1−x/2Nb x/2)O3

The crystal structure and magnetic and elastic properties of the system LaMnO_3+λ are investigated for various concentrations of oxygen. Upon an increase in the oxygen concentration, the orbital-ordered phase is transformed into an orbital-disordered phase via a two-phase crystal-structure state in the interval 0.04<λ<0.06. The transition is accompanied by a jumplike increase in the Curie temperature and spontaneous magnetization. An analysis of the magnetic properties in weak fields and of the temperature dependence of the Young modulus reveals the properties typical of the orbital-ordered antiferromagnetic phase up to λ=0.08. It is proposed that the two-phase state is associated with the martensite type of the orbital order-disorder phase transformation. The system La_1?x Sr_x(Mn_{1 ?x/2}Nb _x/2)O₃ in which all manganese ions are in the trivalent state exhibits a sequence of antiferromagnetic-ferromagnetic (x>0.2) and ferromagnetic-spin glass (x>0.4) transitions. In both systems, the orbital-disordered phases are ferromagnetic, indicating the crucial role of orbital ordering in the formation of magnetic properties.