Investigation of Angular Dependence of Strong-Field Tunneling Ionization for Asymmetric Diatomic Molecule HeH~(2+)

We determine the structure parameters for the asymmetric heteronuclear diatomic molecule HeH 2+ at several internuclear distances with the molecular wavefunctions obtained by solving the time-independent Schr¨odinger equation with B-spline basis.Then the angular dependence of strong-field ionization rates of HeH 2+ are investigated with the molecular tunneling ionization theory.We show that the shape of several lowly excited states(i.e.2pσ,2pπ,3dδ) for HeH 2+ are reflected in the orientation dependent ionization rates very well,however,the angle-dependent ionization rate fails to follow the angular distribution of the asymptotic electron density for the ground state 1sσ.We also show that the internuclear distance dependent ionization probabilities are in a good agreement with the more accurate result obtained from the numerical solution of the time-dependent Schro¨dinger equation.     

Relativistic symmetries with the trigonometric Pschl-Teller potential plus Coulomb-like tensor interaction

The Dirac equation is solved to obtain its approximate bound states for a spin-1/2 particle in the presence of trigonometric Pschl-Teller(tPT) potential including a Coulomb-like tensor interaction with arbitrary spin-orbit quantum number κ using an approximation scheme to substitute the centrifugal terms κ(κ± 1)r-2.In view of spin and pseudo-spin(p-spin) symmetries,the relativistic energy eigenvalues and the corresponding two-component wave functions of a particle moving in the field of attractive and repulsive tPT potentials are obtained using the asymptotic iteration method(AIM).We present numerical results in the absence and presence of tensor coupling A and for various values of spin and p-spin constants and quantum numbers n and κ.The non-relativistic limit is also obtained.     

Spectrum for heavy quarkonia and mixture of the relevant wave functions within the framework of Bethe-Salpeter equation

Considering the fact that some excited states of the heavy quarkonia (charmonium and bottomonium) are still missing in experimental observations and potential applications of the relevant wave functions of the bound states,we re-analyze the spectrum and the relevant wave functions of the heavy quarkonia within the framework of Bethe-Salpeter (B.S.) equation with a proper QCDinspired kernel.Such a kernel for the heavy quarkonia,relating to potential of the non-relativistic quark model,is instantaneous,so we call the corresponding B.S.equation as BS-In equation throughout the paper.Particularly,a new way to solve the B.S.equation,which is different from the traditional ones,is proposed here,and with it not only the known spectrum for the heavy quarkonia is re-generated,but also an important issue is brought in,i.e.,the obtained solutions of the equation ‘automatically’ include the ‘fine’,‘hyperfine’ splittings and the wave function mixture,such as S-D wave mixing in J PC = 1-states,P-F wave mixing in J PC = 2 ++ states for charmonium,bottomonium etc.It is pointed out that the best place to test the wave mixture probably is at Z-factory (e + e-collider running at Z-boson pole with extremely high luminosity).     

A Corresponding Lie Algebra of a Reductive homogeneous Group and Its Applications

With the help of a Lie algebra of a reductive homogeneous space G/K, where G is a Lie group and K is a resulting isotropy group, we introduce a Lax pair for which an expanding(2+1)-dimensional integrable hierarchy is obtained by applying the binormial-residue representation(BRR) method, whose Hamiltonian structure is derived from the trace identity for deducing(2+1)-dimensional integrable hierarchies, which was proposed by Tu, et al. We further consider some reductions of the expanding integrable hierarchy obtained in the paper. The first reduction is just right the(2+1)-dimensional AKNS hierarchy, the second-type reduction reveals an integrable coupling of the(2+1)-dimensional AKNS equation(also called the Davey-Stewartson hierarchy), a kind of(2+1)-dimensional Schr¨odinger equation, which was once reobtained by Tu, Feng and Zhang. It is interesting that a new(2+1)-dimensional integrable nonlinear coupled equation is generated from the reduction of the part of the(2+1)-dimensional integrable coupling, which is further reduced to the standard(2+1)-dimensional diffusion equation along with a parameter. In addition, the well-known(1+1)-dimensional AKNS hierarchy, the(1+1)-dimensional nonlinear Schr¨odinger equation are all special cases of the(2+1)-dimensional expanding integrable hierarchy. Finally, we discuss a few discrete difference equations of the diffusion equation whose stabilities are analyzed by making use of the von Neumann condition and the Fourier method. Some numerical solutions of a special stationary initial value problem of the(2+1)-dimensional diffusion equation are obtained and the resulting convergence and estimation formula are investigated.     

A(2+1)-dimensional nonlinear model for Rossby waves in stratified fluids and its solitary solution

In this paper,we investigate a(2+1)-dimensional nonlinear equation model for Rossby waves in stratified fluids.We derive a forced Zakharov–Kuznetsov(ZK)–Burgers equation from the quasigeostrophic potential vorticity equation with dissipation and topography under the generalized beta effect,and by utilizing temporal and spatial multiple scale transform and the perturbation expansion method.Through the analysis of this model,it is found that the generalized beta effect and basic topography can induce nonlinear waves,and slowly varying topography is an external impact factor for Rossby waves.Additionally,the conservation laws for the mass and energy of solitary waves are analyzed.Eventually,the solitary wave solutions of the forced ZK–Burgers equation are obtained by the simplest equation method as well as the new modified ansatz method.Based on the solitary wave solutions obtained,we discuss the effects of dissipation and slowly varying topography on Rossby solitary waves.     

Relativistic symmetries with the trigonometric Poeschl-Teller potential plus Coulomb-like tensor interaction

The Dirac equation is solved to obtain its approximate bound states for a spin-1/2 particle in the presence of trigonometric Poeschl-Teller （tPT） potential including a Coulomb-like tensor interaction with arbitrary spin-orbit quantum number κ using an approximation scheme to substitute the centrifugal terms κ（κ± i 1）r^-2. In view of spin and pseudo-spin （p-spin） symmetries, the relativistic energy eigenvalues and the corresponding two-component wave functions of a particle moving in the field of attractive and repulsive tPT potentials are obtained using the asymptotic iteration method （AIM）. We present numerical results in the absence and presence of tensor coupling A and for various values of spin and p-spin constants and quantum numbers n and κ. The non-relativistic limit is also obtained.     

A comprehensive approach to an equation of state for hard spheres and Lennard-Jones fluids

We present a simple method of obtaining various equations of state for hard sphere fluid in a simple unifying way.We will guess equations of state by using suitable axiomatic functional forms (n=1,2,3,4,5) for surface tension S n m (r),r ≥ d/2 with intermolecular separation r as a variable,where m is an arbitrary real number (pole).Among the equations of state obtained in this way are Percus-Yevick,scaled particle theory and Carnahan-Starling equations of state.In addition,we have found a simple equation of state for the hard sphere fluid in the region that represents the simulation data accurately.It is found that for both hard sphere fluids as well as Lennard-Jones fluids,with m=3/4 the derived equation of state (EOS) gives results which are in good agreement with computer simulation results.Furthermore,this equation of state gives the Percus-Yevick (pressure) EOS for the m=0,the Carnahan-Starling EOS for m=4/5,while for the value of m=1 it corresponds to a scaled particle theory EOS.     

Approximate solution of the spin-one Duffin-Kemmer-Petiau （DKP） equation under a non-minimal vector Yukawa potential in （1-F 1）-dimensions

<正>We solve the Duffin-Kemmer-Petiau(DKP) equation with a non-minimal vector Yukawa potential in(1 + 1)- dimensional space-time for spin-1 particles.The Nikiforov-Uvarov method is used in the calculations,and the eigenfunctions as well as the energy eigenvalues are obtained in a proper Pekeris-type approximation.     

The ■ tetraquark states and the structure of X(2239) observed by the BESⅢ collaboration

We investigate the mass spectrum of the■tetraquark states in the relativized quark model.By solving the Schrodinger-like equation with the relativized potential,the masses of S-and P-wave■tetraquarks are obtained.The screening effects are also taken into account.It is found that the resonant structureX(2239)observed in thee+e-→K+K-process by the BESIII collaboration can be assigned as a P-wave 1--■tetraquark state.Furthermore,the radiative transition and the strong decay behavior of this structure are also estimated,which can provide helpful information for future experimental searches.     

Soliton structures in the (1+1)-dimensional Ginzburg–Landau equation with a parity-time-symmetric potential in ultrafast optics

In this paper, the(1+1)-dimensional variable-coefficient complex Ginzburg–Landau(CGL) equation with a paritytime(PT) symmetric potential U(x) is investigated. Although the CGL equations with a PT-symmetric potential are less reported analytically, the analytic solutions for the CGL equation are obtained with the bilinear method in this paper. Via the derived solutions, some soliton structures are presented with corresponding parameters, and the influences of them are analyzed and studied. The single-soliton structure is numerically verified, and its stability is analyzed against additive and multiplicative noises. In particular, we study the soliton dynamics under the impact of the PT-symmetric potential. Results show that the PT-symmetric potential plays an important role for obtaining soliton structures in ultrafast optics, and we can design fiber lasers and all-optical switches depending on the different amplitudes of soliton-like structures.     

Exact solutions of the Klein—Gordon equation with ring-shaped oscillator potential by using the Laplace integral transform

We present exact solutions for the Klein-Gordon equation with a ring-shaped oscillator potential. The energy eigenvalues and the normalized wave functions are obtained for a particle in the presence of non-central oscillator potential. The angular functions are expressed in terms of the hypergeometric functions. The radial eigenfunctions have been obtained by using the Laplace integral transform. By means of the Laplace transform method, which is efficient and simple, the radial Klein-Gordon equation is reduced to a first-order differential equation.     

Solvability of a class of PT-symmetric non-Hermitian Hamiltonians:Bethe ansatz method

We use the Bethe ansatz method to investigate the Schrdinger equation for a class of PT-symmetric non-Hermitian Hamiltonians. Elementary exact solutions for the eigenvalues and the corresponding wave functions are obtained in terms of the roots of a set of algebraic equations. Also, it is shown that the problems possess sl(2) hidden symmetry and then the exact solutions of the problems are obtained by employing the representation theory of sl(2) Lie algebra. It is found that the results of the two methods are the same.     

von Neumann measurement-related matrices and the nullity condition for quantum correlation

We study von Neumann measurement-related matrices, and the nullity condition of quantum correlation. We investigate the properties of these matrices that are related to a von Neumann measurement. It is shown that these(m~2-1)×(m~2-1) matrices are idempotent, and have rank m-1. These properties give rise to necessary conditions for the nullity of quantum correlations in bipartite systems. Finally, as an example we discuss quantum correlation in Bell diagonal states.     

Study on an extended Boussinesq equation

An extended Boussinesq equation that models weakly nonlinear and weakly dispersive waves on a uniform layer of water is studied in this paper. The results show that the equation is not Painlev\'e-integrable in general. Some particular exact travelling wave solutions are obtained by using a function expansion method. An approximate solitary wave solution with physical significance is obtained by using a perturbation method. We find that the extended Boussinesq equation with a depth parameter of $1/\sqrt 2$ is able to match the Laitone's (1960) second order solitary wave solution of the Euler equations.     

Relativistic Treatment of Spinless Particles Subject to a q-Deformed Morse Potential

The approximate analytical solutions of the Klein-Gordon equation with equal scalar and vector q-deformed Morse potential are presented for arbitrary e-states by using Laplace integral transform. The energy eigenvalues and corresponding wave functions are obtained for n and e values. In this study, in the non-relativistic limit c→∞, it has been also provided that the energy eigenfunctions for Klein-Gordon system turn into those for Schrdinger one.     

Four-level model of ion-ion laser-induced collisional energy transfer and numerical calculations for Ca~+-Sr~+ system

The four-level model of laser-induced collisional energy transfer(LICET) for the ion-ion collision system is established based on the time-dependent Schro¨dinger equation for the electron dynamics,through which the equations of motion of the probability amplitudes and cross section of the collision system are obtained.Numerical calculations are performed for the Ca+-Sr+ system,with the results showing that the peak of the LICET spectrum appears at a resonant frequency of the transfer laser.The magnitude of the obtained collision cross section is in the order of 10 16 cm2,and is comparable to that obtained in atomic systems,which indicates the validity of the established four-level model.     

Generalized Master Equation for Space-Time Coupled Continuous Time Random Walk

The generalized master equation for the space-time coupled continuous time random walk is derived analytically,in which the space-time coupling is considered through the correlated function g(t)~t~γ,0≤γ 2, and the probability density function ω(t) of a particle's waiting time t follows a power law form for large t: ω(t)~t~(-(1+α)),0 α 1. The results indicate that the expressions of the generalized master equation are determined by the correlation exponent γ and the long-tailed index α of the waiting time. Moreover, the diffusion results obtained from the generalized master equation are in accordance with the previous known results and the numerical simulation results.     

Ultraslow optical solitons via electromagnetically induced transparency: a density-matrix approach

We study the ultraslow optical solitons in a resonant three-level atomic system via electromagnetically induced transparency under a density-matrix (DM) approach. The results of linear and nonlinear optical properties are compared with those obtained by using an amplitude variable (AV) approach. It is found that the results for both approaches are the same in the linear regime if the corresponding relations between the population-coherence decay rates in the DM approach and the energy-level decay rates in the AV approach are appropriately imposed. However, in the nonlinear regime there is a small difference for the self-phase modulation coefficient of the nonlinear Schr\"{o}dinger equation that governs the time evolution of probe pulse envelope. All analytical predicts are checked by numerical simulations.     

Shannon information entropies for position-dependent mass Schrdinger problem with a hyperbolic well

The Shannon information entropy for the Schrodinger equation with a nonuniform solitonic mass is evaluated for a hyperbolic-type potential. The number of nodes of the wave functions in the transformed space z are broken when recovered to original space x. The position Sx and momentum S p information entropies for six low-lying states are calculated. We notice that the Sx decreases with the increasing mass barrier width a and becomes negative beyond a particular width a,while the Sp first increases with a and then decreases with it. The negative Sx exists for the probability densities that are highly localized. We find that the probability density ρ(x) for n = 1, 3, 5 are greater than 1 at position x = 0. Some interesting features of the information entropy densities ρs(x) and ρs(p) are demonstrated. The Bialynicki–Birula–Mycielski(BBM)inequality is also tested for these states and found to hold.