Rules of correspondence in atomic physics

The Smirnov method of analytic continuation (B.M. Smirnov, Sov. Phys. JETP 20, 345 (1964)) has been justified and developed for atomic physics. It has been shown that the polarizability of alkali atoms α, their van der Waals interaction constant C ₆, and the oscillator strength of the transition to the first P state f ₀₁ are related to the parameter 〈r ²〉 and gap in the spectrum \(\frac{3}{2}\frac{f}{\Delta } \approx \frac{3}{2}\alpha \Delta \approx {\left( {3{C_6}\Delta } \right)^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$2$}}}} \approx \left\langle {{r^2}} \right\rangle \). The average square of the coordinate of the valence electron 〈r ²〉 in the first approximation has a hydrogen dependence \({J_1} = \frac{1}{{2{v^2}}}.\) on the filling factor ν, which is defined in terms of the first ionization potential: xxxxxxxxx     

The general linear theory of dynamic vibration absorbers

It is shown that vibrations of an elastic platform, induced by an external force \(f\left( t \right) = \sum\limits_{j = 0}^n {{A_j}} \sin \left( {{\omega _j}t + {\varphi _j}} \right)\), can be suppressed using n dynamic vibration absorbers with eigenfrequencies ω _j .     

A new method for calculating three-particle interaction in the theory of modulus elasticity

We propose a new method for calculating the potential of multiparticle interaction. Our method considers the energy symmetry for clusters that contain N identical particles with respect to permutation of the number of atoms and free rotation in three-dimensional space. As an example, we calculate moduli of third-order rigidity for copper considering only the three-particle interaction. We analyze nine models of energy dependence on the polynomials that form the integral rational basis of invariants (IRBI) for the group G ₃ = O(3) ? P ₃. In this work, we use only the simplest relation between energy and the invariants forming the IRBI: \(\varepsilon \left( {\left. {i,k,l} \right|j} \right) = \sum\nolimits_{i,k,l} {\left[ { - A_1 r_{ik}^{ - 6} + A_2 r_{ik}^{ - 12} + Q_j I_j^{ - n} } \right]}\), where I _j is the invariant number j (j = 1, 2,..., 9). The results are in good agreement with the experimental values. The best agreement is observed at n = 2, j = 4: \(I_4 = \left( {\vec r_{ik} \vec r_{kl} } \right)\left( {\vec r_{kl} \vec r_{li} } \right) + \left( {\vec r_{kl} \vec r_{li} } \right)\left( {\vec r_{li} \vec r_{ik} } \right) + \left( {\vec r_{li} \vec r_{ik} } \right)\left( {\vec r_{ik} \vec r_{kl} } \right)\).     

Gleichzeitige Messung von Hyperfeinstruktur,Stark-Effekt und Zeeman-Effekt des39K19F mit einer Molekülstrahlresonanzapparatur

An electric Molecular-Beam-Resonance-Spectrometer has been used to measure simultanously the Zeeman- and Stark-effect splitting of the hyperfine structure of³⁹K¹⁹ F. Electric four pole lenses served as focusing and refocusing fields of the spectrometer. A homogenous magnetic field (Zeeman field) was superimposed to the electric field (Stark field) in the transition region of the apparatus. The observed (Δm _J =±1)-transitions were induced electrically. Completely resolved spectra of KF in theJ=1 rotational state have been measured. The obtained quantities are: The electric dipolmomentμ _e l of the molecul forv=0,1 and 2; the rotational magnetic dipolmomentμ _J forv=0,1; the difference of the magnetic shielding (σ_⊥ ? σ_∥) by the electrons of both nuclei as well as the difference of the molecular susceptibility (ξ_⊥ ? ξ_∥). The numerical values are

$$\begin{array}{*{20}c} {\mu _{e1} = 8,585(4)deb,} \\ {\frac{{(\mu _{e1} )_{\upsilon = 1} }}{{(\mu _{e1} )_{\upsilon = 0} }} = 1,0080,} \\ {{{\mu _J } \mathord{\left/ {\vphantom {{\mu _J } J}} \right. \kern-\nulldelimiterspace} J} = ( - )2352(10) \cdot 10^{ - 6} \mu _B ,} \\ {(\sigma _ \bot - \sigma _\parallel )F = ( - )2,19(9) \cdot 10^{ - 4} ,} \\ {(\sigma _ \bot - \sigma _\parallel )K = ( - )12(9) \cdot 10^{ - 4} ,} \\ {(\xi _ \bot - \xi _\parallel ) = 3 (1) \cdot 10^{ - 30} {{erg} \mathord{\left/ {\vphantom {{erg} {Gau\beta ^2 }}} \right. \kern-\nulldelimiterspace} {Gau\beta ^2 }}} \\ \end{array} $$     

Entanglement Involved in Time Evolution of Two-Mode Squeezed State in Single-Mode Diffusion Channel

We derive the evolution law of an initial two-mode squeezed vacuum state \( \text {sech}^{2}\lambda e^{a^{\dag }b^{\dagger }\tanh \lambda }\left \vert 00\right \rangle \left \langle 00\right \vert e^{ab\tanh \lambda }\) (a pure state) passing through an a-mode diffusion channel described by the master equation

$$\frac{d\rho \left( t\right) }{dt}=-\kappa \left[ a^{\dagger}a\rho \left( t\right) -a^{\dagger}\rho \left( t\right) a-a\rho \left( t\right) a^{\dagger}+\rho \left( t\right) aa^{\dagger}\right] , $$

since the two-mode squeezed state is simultaneously an entangled state, the final state which emerges from this channel is a two-mode mixed state. Performing partial trace over the b-mode of ρ(t) yields a new chaotic field, \(\rho _{a}\left (t\right ) =\frac {\text {sech}^{2}\lambda }{1+\kappa t \text {sech}^{2}\lambda }:\exp \left [ \frac {- \text {sech}^{2}\lambda }{1+\kappa t\text {sech}^{2}\lambda }a^{\dagger }a \right ] :,\) which exhibits higher temperature and more photon numbers, showing the diffusion effect. Besides, measuring a-mode of ρ(t) to find n photons will result in the collapse of the two-mode system to a new Laguerre polynomial-weighted chaotic state in b-mode, which also exhibits entanglement.

     

Coherent pion production in neutrino and antineutrino interactions on nuclei of heavy freon molecules

Studying the coherent diffractive production of pions in neutrino and antineutrino scattering off the nuclei of freon molecules we have observed for the first time in one experiment all three states of the isospin triplet of the axial part of the weak charged and neutral currents. For the corresponding cross sections we derive $$\begin{array}{*{20}c} {\sigma _{coh}^v (\pi ^ + ) = (106 \pm 16) \cdot 10^{ - 40} {{cm^2 } \mathord{\left/ {\vphantom {{cm^2 } {\left\langle {nucl.} \right\rangle }}} \right. \kern-\nulldelimiterspace} {\left\langle {nucl.} \right\rangle }}} \\ {\sigma _{coh}^{\bar v} (\pi ^ - ) = (113 \pm 35) \cdot 10^{ - 40} {{cm^2 } \mathord{\left/ {\vphantom {{cm^2 } {\left\langle {nucl.} \right\rangle }}} \right. \kern-\nulldelimiterspace} {\left\langle {nucl.} \right\rangle }}and} \\ {\sigma _{coh}^v (\pi ^0 ) = (52 \pm 19) \cdot 10^{ - 40} {{cm^2 } \mathord{\left/ {\vphantom {{cm^2 } {\left\langle {nucl.} \right\rangle }}} \right. \kern-\nulldelimiterspace} {\left\langle {nucl.} \right\rangle }}} \\ \end{array} $$ . Comparing our data with theoretical predictions based on the standard model of weak interactions we find reasonable agreement. Independently from any model of coherent pion production we determine the isovector axial vector coupling constant to be |β|=0.99±0.20.     

Higher recurrences for Apostol-Bernoulli-Euler numbers

We present explicit formulas for sums of products of Apostol-Bernoulli and Apostol-Euler numbers of the form

$\sum\limits_{_{m_1 , \cdots ,m_N \geqslant n}^{m_1 + \cdots + m_N = n} } {\left( {_{m_1 , \cdots m_N }^n } \right)B_{m_1 } (q) \cdots B_{m_N } (q),} \sum\limits_{_{m_1 , \cdots ,m_N \geqslant n}^{m_1 + \cdots + m_N = n} } {\left( {_{m_1 , \cdots m_N }^n } \right)E_{m_1 } (q) \cdots E_{m_N } (q),}$

where N and n are positive integers, B _m (q) _n stand for the Apostol-Bernoulli numbers, E _m (q) for the Apostol-Euler numbers, and \(\left( {\begin{array}{*{20}c} n \\ {m_1 , \cdots ,m_N } \\ \end{array} } \right) = \frac{{n!}}{{m_1 ! \cdots m_N !}}.\) Our formulas involve Stirling numbers of the first kind. We also derive results for Apostol-Bernoulli and Apostol-Euler polynomials. As an application, for q = 1 we recover results of Dilcher, and our paper can be regarded as a q-extension of that of Dilcher.

     

Scaling exponents of forced polymer translocation through a nanopore

We investigate several properties of a translocating homopolymer through a thin pore driven by an external field present inside the pore only using Langevin Dynamics (LD) simulations in three dimensions (3D). Motivated by several recent theoretical and numerical studies that are apparently at odds with each other, we estimate the exponents describing the scaling with chain length (Nof the average translocation time \(\ensuremath \langle\tau\rangle\) , the average velocity of the center of mass \(\ensuremath \langle v_{{\rm CM}}\rangle\) , and the effective radius of gyration \(\ensuremath \langle {R}_g\rangle\) during the translocation process defined as \(\ensuremath \langle\tau\rangle \sim N^{\alpha}\) , \(\ensuremath \langle v_{{\rm CM}} \rangle \sim N^{-\delta}\) , and \(\ensuremath {R}_g \sim N^{\bar{\nu}}\) respectively, and the exponent of the translocation coordinate (s -coordinate) as a function of the translocation time \(\ensuremath \langle s^2(t)\rangle\sim t^{\beta}\) . We find \(\ensuremath \alpha=1.36 \pm 0.01\) , \(\ensuremath \beta=1.60 \pm 0.01\) for \(\ensuremath \langle s^2(t)\rangle\sim \tau^{\beta}\) and \(\ensuremath \bar{\beta}=1.44 \pm 0.02\) for \(\ensuremath \langle\Delta s^2(t)\rangle\sim\tau^{\bar{\beta}}\) , \(\ensuremath \delta=0.81 \pm 0.04\) , and \(\ensuremath \bar{\nu}\simeq\nu=0.59 \pm 0.01\) , where \( \nu\) is the equilibrium Flory exponent in 3D. Therefore, we find that \(\ensuremath \langle\tau\rangle\sim N^{1.36}\) is consistent with the estimate of \(\ensuremath \langle\tau\rangle\sim\langle R_g \rangle/\langle v_{{\rm CM}} \rangle\) . However, as observed previously in Monte Carlo (MC) calculations by Kantor and Kardar (Y. Kantor, M. Kardar, Phys. Rev. E 69, 021806 (2004)) we also find the exponent α = 1.36 ± 0.01 < 1 + ν. Further, we find that the parallel and perpendicular components of the gyration radii, where one considers the “cis” and “trans” parts of the chain separately, exhibit distinct out-of-equilibrium effects. We also discuss the dependence of the effective exponents on the pore geometry for the range of N studied here.     

The Andreev conductance in superconductor–insulator–normal metal structures

The Andreev subgap conductance at 0.08–0.2 K in thin-film superconductor (aluminum)–insulator–normal metal (copper, hafnium, or aluminum with iron-sublayer-suppressed superconductivity) structures is studied. The measurements are performed in a magnetic field oriented either along the normal or in the plane of the structure. The dc current–voltage (I–U) characteristics of samples are described using a sum of the Andreev subgap current dominating in the absence of the field at bias voltages U < (0.2–0.4)Δ_c/e (where Δ_c is the energy gap of the superconductor) and the single-carrier tunneling current that predominates at large voltages. To within the measurement accuracy of 1–2%, the Andreev current corresponds to the formula \({I_n} + {I_s} = {K_n}\tanh \left( {{{eU} \mathord{\left/ {\vphantom {{eU} {2k{T_{eff}}}}} \right. \kern-\nulldelimiterspace} {2k{T_{eff}}}}} \right) + {K_s}{{\left( {{{eU} \mathord{\left/ {\vphantom {{eU} {{\Delta _c}}}} \right. \kern-\nulldelimiterspace} {{\Delta _c}}}} \right)} \mathord{\left/ {\vphantom {{\left( {{{eU} \mathord{\left/ {\vphantom {{eU} {{\Delta _c}}}} \right. \kern-\nulldelimiterspace} {{\Delta _c}}}} \right)} {\sqrt {1 - {{eU} \mathord{\left/ {\vphantom {{eU} {{\Delta _c}}}} \right. \kern-\nulldelimiterspace} {{\Delta _c}}}} }}} \right. \kern-\nulldelimiterspace} {\sqrt {1 - {{eU} \mathord{\left/ {\vphantom {{eU} {{\Delta _c}}}} \right. \kern-\nulldelimiterspace} {{\Delta _c}}}} }}\) following from a theory that takes into account mesoscopic phenomena with properly selected effective temperature T _eff and the temperature- and fieldindependent parameters K _n and K _s (characterizing the diffusion of electrons in the normal metal and superconductor, respectively). The experimental value of K _n agrees in order of magnitude with the theoretical prediction, while K _s is several dozen times larger than the theoretical value. The values of T _eff in the absence of the field for the structures with copper and hafnium are close to the sample temperature, while the value for aluminum with an iron sublayer is several times greater than this temperature. For the structure with copper at T = 0.08–0.1 K in the magnetic field B_|| = 200–300 G oriented in the plane of the sample, the effective temperature T _eff increases to 0.4 K, while that in the perpendicular (normal) field B _⊥ ≈ 30 G increases to 0.17 K. In large fields, the Andreev conductance cannot be reliably recognized against the background of single- carrier tunneling current. In the structures with hafnium and in those with aluminum on an iron sublayer, the influence of the magnetic field is not observed.     

Classical Dynamics Based on the Minimal Length Uncertainty Principle

In this paper we consider the quadratic modification of the Heisenberg algebra and its classical limit version which we call the β-deformed Poisson bracket for corresponding classical variables. We use the β-deformed Poisson bracket to discuss some physical problems in the β-deformed classical dynamics. Finally, we consider the (α,β)- deformed classical dynamics in which minimal length uncertainty principle is given by \( [ \hat {x} , \hat {p}] = i \hbar (1 + \alpha \hat {x}^{2} + \beta \hat {p}^{2} ) \). For two small parameters α,β, we discuss the free fall of particle and a composite system in a uniform gravitational field.     

Krein formula with compensated singularities for the ND-mapping and the generalized Kirchhoff condition at the Neumann Schrödinger junction

The Neumann Schrödinger operator \(\mathcal{L}\) is considered on a thin 2D star-shaped junction, composed of a vertex domain Ω_int and a few semi-infinite straight leads ω _m , m = 1, 2, ..., M, of width δ, δ ? diam Ω_int, attached to Ω_int at Γ ? ?Ω_int. The potential of the Schrödinger operator l ^ω on the leads vanishes, hence there are only a finite number of eigenvalues of the Neumann Schrödinger operator L _int on Ω_int embedded into the open spectral branches of l ^ω with oscillating solutions χ _±(x, p) = \(e^{ \pm iK_ + x} e_m \) of l ^ω χ _± = p ² χ _±. The exponent of the open channels in the wires is

$K_ + (\lambda ) = p\sum\limits_{m = 1}^M {e^m } \rangle \langle e^m = \sqrt \lambda P_ + $

, with constant e _m , on a relatively small essential spectral interval Δ ? [0, π ² δ ^?2). The scattering matrix of the junction is represented on Δ in terms of the ND mapping

$\mathcal{N} = \frac{{\partial P_ + \Psi }}{{\partial x}}(0,\lambda )\left| {_\Gamma \to P_ + \Psi _ + (0,\lambda )} \right|_\Gamma $

as

$S(\lambda ) = (ip\mathcal{N} + I_ + )^{ - 1} (ip\mathcal{N} - I_ + ), I_ + = \sum\limits_{m = 1}^M {e^m } \rangle \langle e^m = P_ + $

. We derive an approximate formula for \(\mathcal{N}\) in terms of the Neumann-to-Dirichlet mapping \(\mathcal{N}_{\operatorname{int} } \) of L _int and the exponent K _? of the closed channels of l ^ω . If there is only one simple eigenvalue λ ₀ ∈ Δ, L _intφ0 = λ _0φ0 then, for a thin junction, \(\mathcal{N} \approx |\vec \phi _0 |^2 P_0 (\lambda _0 - \lambda )^{ - 1} \) with

$\vec \phi _0 = P_ + \phi _0 = (\delta ^{ - 1} \int_{\Gamma _1 } {\phi _0 (\gamma )} d\gamma ,\delta ^{ - 1} \int_{\Gamma _2 } {\phi _0 (\gamma )} d\gamma , \ldots \delta ^{ - 1} \int_{\Gamma _M } {\phi _0 (\gamma )} d\gamma )$

and \(P_0 = \vec \phi _0 \rangle |\vec \phi _0 |^{ - 2} \langle \vec \phi _0 \),

$S(\lambda ) \approx \frac{{ip|\vec \phi _0 |^2 P_0 (\lambda _0 - \lambda )^{ - 1} - I_ + }}{{ip|\vec \phi _0 |^2 P_0 (\lambda _0 - \lambda )^{ - 1} + I_ + }} = :S_{appr} (\lambda )$

. The related boundary condition for the components P ₊Ψ(0) and P ₊Ψ′(0) of the scattering Ansatz in the open channel \(P_ + \Psi (0) = (\bar \Psi _1 ,\bar \Psi _2 , \ldots ,\bar \Psi _M ), P_ + \Psi '(0) = (\bar \Psi '_1 , \bar \Psi '_2 , \ldots , \bar \Psi '_M )\) includes the weighted continuity (1) of the scattering Ansatz Ψ at the vertex and the weighted balance of the currents (2), where

$\frac{{\bar \Psi _m }}{{\bar \phi _0^m }} = \frac{{\delta \sum\nolimits_{t = 1}^M { \bar \Psi _t \bar \phi _0^t } }}{{|\vec \phi _0 |^2 }} = \frac{{\bar \Psi _r }}{{\bar \phi _0^r }} = :\bar \Psi (0)/\bar \phi (0), 1 \leqslant m,r \leqslant M$

(1)

,

$\sum\limits_{m = 1}^M {\bar \Psi '_m } \bar \phi _0^m + \delta ^{ - 1} (\lambda - \lambda _0 )\bar \Psi /\bar \phi (0) = 0$

(1)

. Conditions (1) and (2) constitute the generalized Kirchhoff boundary condition at the vertex for the Schrödinger operator on a thin junction and remain valid for the corresponding 1D model. We compare this with the previous result by Kuchment and Zeng obtained by the variational technique for the Neumann Laplacian on a shrinking quantum network.

     

Compositions,Random Sums and Continued Random Fractions of Poisson and Fractional Poisson Processes

In this paper we consider the relation between random sums and compositions of different processes. In particular, for independent Poisson processes N _α (t), N _β (t), t>0, we have that \(N_{\alpha}(N_{\beta}(t)) \stackrel{\mathrm{d}}{=} \sum_{j=1}^{N_{\beta}(t)} X_{j}\), where the X _j s are Poisson random variables. We present a series of similar cases, where the outer process is Poisson with different inner processes. We highlight generalisations of these results where the external process is infinitely divisible. A section of the paper concerns compositions of the form \(N_{\alpha}(\tau_{k}^{\nu})\), ν∈(0,1], where \(\tau_{k}^{\nu}\) is the inverse of the fractional Poisson process, and we show how these compositions can be represented as random sums. Furthermore we study compositions of the form Θ(N(t)), t>0, which can be represented as random products. The last section is devoted to studying continued fractions of Cauchy random variables with a Poisson number of levels. We evaluate the exact distribution and derive the scale parameter in terms of ratios of Fibonacci numbers.     

The Finite-Size Scaling Study of the Ising Model for the Fractals

The fractals are obtained by using the model of diffusion-limited aggregation (DLA) for 40 ≤ L ≤ 240. The two-dimensional Ising model is simulated on the Creutz cellular automaton for 40 ≤ L ≤ 240. The critical exponents and the fractal dimensions are computed to be β = 0.124(8), γ = 1.747(10), α = 0.081(21), δ = 14.994(11), η = 0.178(10), ν = 0.960(23) and \(d_{f}^{\beta } =1.876(8), \,d_{f}^{\gamma } =3.747(10), \,d_{f}^{\alpha } =2.081(68), \,d_{f}^{\delta } =1.940(22)\), \(d_{f}^{\eta } =2.178(10)\), \(d_{f}^{\nu } =2.960(22)\), which are consistent with the theoretical values of β = 0.125, γ = 1.75, α = 0, δ = 15, η = 0.25, ν = 1 and \(d_{f}^{\beta } =1.875, \,d_{f}^{\gamma } =3.75, \,d_{f}^{\alpha } =2, \,d_{f}^{\delta } =1.933, \,d_{f}^{\eta } =2.25, \,d_{f}^{\nu } =3\).     

Random Walk with Barycentric Self-interaction

We study the asymptotic behaviour of a d-dimensional self-interacting random walk (X _n ) _n∈? (?:={1,2,3,…}) which is repelled or attracted by the centre of mass \(G_{n} = n^{-1} \sum_{i=1}^{n} X_{i}\) of its previous trajectory. The walk’s trajectory (X ₁,…,X _n ) models a random polymer chain in either poor or good solvent. In addition to some natural regularity conditions, we assume that the walk has one-step mean drift

$\mathbb{E}[X_{n+1} - X_n \mid X_n - G_n = \mathbf{x}] \approx\rho\|\mathbf{x}\|^{-\beta}\hat{ \mathbf{x}}$

for ρ∈? and β≥0. When β<1 and ρ>0, we show that X _n is transient with a limiting (random) direction and satisfies a super-diffusive law of large numbers: n ^?1/(1+β) X _n converges almost surely to some random vector. When β∈(0,1) there is sub-ballistic rate of escape. When β≥0 and ρ∈? we give almost-sure bounds on the norms ‖X _n ‖, which in the context of the polymer model reveal extended and collapsed phases.

Analysis of the random walk, and in particular of X _n ?G _n , leads to the study of real-valued time-inhomogeneous non-Markov processes (Z _n ) _n∈? on [0,∞) with mean drifts of the form

$ \mathbb{E}[ Z_{n+1} - Z_n \mid Z_n = x ] \approx\rho x^{-\beta} - \frac {x}{n},$

(0.1)

where β≥0 and ρ∈?. The study of such processes is a time-dependent variation on a classical problem of Lamperti; moreover, they arise naturally in the context of the distance of simple random walk on ? ^d from its centre of mass, for which we also give an apparently new result. We give a recurrence classification and asymptotic theory for processes Z _n satisfying (0.1), which enables us to deduce the complete recurrence classification (for any β≥0) of X _n ?G _n for our self-interacting walk.

     

Lightest scalar resonances and the dynamics of the γγ → ππ reactions

The high-statistics Belle data on the γγ → π⁺π^? and γγ → π⁰π⁰ reactions have been jointly analyzed. The main dynamical mechanisms of these reactions for energies below 1.5 GeV have been revealed. It has been shown that the direct coupling constants of the σ(600) and f ₀(980) resonances with a γγ pair are small and that the σ(600) → γγ and f ₀(980) → γγ decays are four-quark transitions due primarily to π⁺π^? and K ⁺ K ^?-loop mechanisms, respectively. The role of the chiral shielding of the σ(600) resonance is emphasized. The widths of the f ₀(980) → γγ and σ(600) → γγ decays averaged over the resonance mass distributions, as well as the width of the f ₂(1270) → γγ decay, are estimated as \(\langle \Gamma _{f_0 \to \gamma \gamma } \rangle _{\pi \pi } \approx 0.19 keV, \langle \Gamma _{\sigma \to \gamma \gamma } \rangle _{\pi \pi } \approx 0.45 keV\), and \(\Gamma _{f_2 \to \gamma \gamma } (m_{f_2 }^2 ) \approx 3.8 keV\).     

Quantum Fisher Information for <Emphasis Type="Italic">s</Emphasis><Emphasis Type="Italic">u</Emphasis>(2) Atomic Coherent States and <Emphasis Type="Italic">s</Emphasis><Emphasis Type="Italic">u</Emphasis>(1, 1) Coherent States

We investigate quantum Fisher information (QFI) for s u(2) atomic coherent states and s u(1, 1) coherent states. In this work, we find that for s u(2) atomic coherent states, the QFI with respect to \(\vartheta ~(\mathcal {F}_{\vartheta })\) is independent of φ, the QFI with respect to \(\varphi (\mathcal {F}_{\varphi })\) is governed by ??. Analogously, for s u(1,1) coherent states, \(\mathcal {F}_{\tau }\) is independent of φ, and \(\mathcal {F}_{\varphi }\) is determined by τ. Particularly, our results show that \(\mathcal {F}_{\varphi }\) is symmetric with respect to ?? = π/2 for s u(2) atomic coherent states. And for s u(1,1) coherent states, \(\mathcal {F}_{\varphi }\) also possesses symmetry with respect to τ = 0.     

The convergence behaviour of expansions in the classical collision theory

In the classical collision theory the scattering angle? depends on the impact parameterb and on the kinetic energyE _r of the relative motion. This angle?(b, E _r ) is expanded for two limiting cases: 1. Expansion in powers of the potentialV(r)/E _r (momentum approximation). 2. Expansion in powers of the impact parameterb (central collision approximation). The radius of convergence of the series depends onb andE _r . It will be given for the following potentialsV(r):

$$A\left( {\frac{a}{r}} \right)^\mu ;Ae^{ - \frac{r}{a}} ;A\frac{a}{r}e^{ - \frac{r}{a}} ;A\left( {\frac{a}{r}} \right)^2 e^{ - \left( {\frac{r}{a}} \right)^2 } .$$     

A Note on Quantum Coherence

In this paper, we discuss the coherence of the reduced state in system H _A ?H _B under taking different quantum operations acting on subsystem H _B . Firstly, we show that for a pure bipartite state, the coherence of the final subsystem H _A under the sum of two orthonormal rank 1 projections acting on H _B is less than or equal to the sum of the coherence of the state after two orthonormal projections acting on H _B , respectively. Secondly, we obtain that the coherence of reduced state in subsystem H _A under random unitary channel \({\Phi }(\rho )={\sum }_{s}\lambda _{s}U_{s}\rho U_{s}^{\ast }\) acting on H _B , is equal to the coherence of the state after each operation \({\Phi }_{s}(\rho )=\lambda _{s}U_{s}\rho U_{s}^{\ast }\) acting on H _B for every s. In addition, for general quantum operation \({\Phi }(\rho )={\sum }_{s}F_{s}\rho F_{s}^{\ast }\) on H _B , we get the relation

$$ C\left (\left ((I\otimes {\Phi })\rho ^{AB}\right )^{A}\right )\leq \sum \limits _{s}C\left (\left ((I\otimes {\Phi }_{s})\rho ^{AB}\right )^{A}\right ). $$