Darboux partners of pseudoscalar Dirac potentials associated with exceptional orthogonal polynomials

We introduce a method for constructing Darboux (or supersymmetric) pairs of pseudoscalar and scalar Dirac potentials that are associated with exceptional orthogonal polynomials. Properties of the transformed potentials and regularity conditions are discussed. As an application, we consider a pseudoscalar Dirac potential related to the Schrödinger model for the rationally extended radial oscillator. The pseudoscalar partner potentials are constructed under the first- and second-order Darboux transformations.     

Multiscaling for classical nanosystems: Derivation of Smoluchowski & Fokker–Planck equations

Using multiscale analysis and methods of statistical physics, we show that a solution to the N-atom Liouville equation can be decomposed via an expansion in terms of a smallness parameter , wherein the long scale time behavior depends upon a reduced probability density that is a function of slow-evolving order parameters. This reduced probability density is shown to satisfy the Smoluchowski equation up to O(²) for a given range of initial conditions. Furthermore, under the additional assumption that the nanoparticle momentum evolves on a slow time scale, we show that this reduced probability density satisfies a Fokker–Planck equation up to O(²). This approach has applications to a broad range of problems in the nanosciences.     

Quasi-Hermitian Hamiltonians associated with exceptional orthogonal polynomials

Using the method of point canonical transformation, we derive some exactly solvable rationally extended quantum Hamiltonians which are non-Hermitian in nature and whose bound state wave functions are associated with Laguerre- or Jacobi-type _X1$X_1$ exceptional orthogonal polynomials. These Hamiltonians are shown, with the help of imaginary shift of coordinate: e^−αpxe^αp=x+iα$e^{-\alpha p}x{e}^{\alpha p}=x+i\alpha$, to be both quasi- and pseudo-Hermitian. It turns out that the corresponding energy spectra is entirely real.     

Generalized Fokker–Planck equations derived from generalized linear nonequilibrium thermodynamics

Recently, Compte and Jou derived nonlinear diffusion equations by applying the principles of linear nonequilibrium thermodynamics to the generalized nonextensive entropy proposed by Tsallis. In line with this study, stochastic processes in isolated and closed systems characterized by arbitrary generalized entropies are considered and evolution equations for the process probability densities are derived. It is shown that linear nonequilibrium thermodynamics based on generalized entropies naturally leads to generalized Fokker–Planck equations.     

Dirac constraint analysis and symplectic structure of anti-self-dual Yang-Mills equations

We present the explicit form of the symplectic structure of anti-self-dual Yang-Mills (ASDYM) equations in Yang’s J- and K-gauges in order to establish the bi-Hamiltonian structure of this completely integrable system. Dirac’s theory of constraints is applied to the degenerate Lagrangians that yield the ASDYM equations. The constraints are second class as in the case of all completely integrable systems which stands in sharp contrast to the situation in full Yang-Mills theory. We construct the Dirac brackets and the symplectic 2-forms for both J- and K-gauges. The covariant symplectic structure of ASDYM equations is obtained using the Witten-Zuckerman formalism. We show that the appropriate component of the Witten-Zuckerman closed and conserved 2-form vector density reduces to the symplectic 2-form obtained from Dirac’s theory. Finally, we present the Bäcklund transformation between the J- and K-gauges in order to apply Magri’s theorem to the respective two Hamiltonian structures.     

涉及Hermite多项式的二项式定理和Laguerre多项式的负二项式定理

提出量子力学算符Hermite多项式方法,即将若干常用的特殊函数的宗量由普通数变为算符,并用它来发现涉及Hermite多项式（单变数和双变数）的二项式定理和涉及Laguerre多项式的负二项式定理,它们在计算若干量子光场的物理性质时有实质性的应用. 该方法不但具有简捷的优点,而且能导出很多新的算符恒等式,成为发展数学物理理论的一个重要分支. 关键词： 量子力学 Hermite多项式 二项式定理 Laguerre多项式     

Solution of Spin and Pseudo-Spin Symmetric Dirac Equation in (1+1) Space-Time Using Tridiagonal Representation Approach

The aim of this work is to find exact solutions of the Dirac equation in(1+1) space-time beyond the already known class.We consider exact spin(and pseudo-spin) symmetric Dirac equations where the scalar potential is equal to plus(and minus) the vector potential.We also include pseudo-scalar potentials in the interaction.The spinor wavefunction is written as a bounded sum in a complete set of square integrable basis,which is chosen such that the matrix representation of the Dirac wave operator is tridiagonal and symmetric.This makes the matrix wave equation a symmetric three-term recursion relation for the expansion coefficients of the wavefunction.We solve the recursion relation exactly in terms of orthogonal polynomials and obtain the state functions and corresponding relativistic energy spectrum and phase shift.     

Solvable potentials with exceptional orthogonal polynomials

Classes of solvable potentials are presented within an standard application of supersymmetric quantum mechanics. Sets of exceptional orthogonal polynomials generated by these solvable potentials are introduced and examined in detail. Several properties of these polynomials including orthogonality conditions, weight functions, differential equations, the Wronskains, possible recurrence relations are also investigated.

     

Analytical solution of the PNP equations at AC applied voltage

A symmetric binary polymer electrolyte subjected to an AC voltage is considered. The analytical solution of the Poisson–Nernst–Planck equations (PNP) is found and analyzed for small applied voltages. Three distinct time regimes offering different behavior can be discriminated. The experimentally realized stationary behavior is discussed in detail. An expression for the external current is derived. Based on the theoretical result a simple method is suggested of measuring the ion mobility and their concentration separately.     

Methodology for the solutions of some reduced Fokker‐Planck equations in high dimensions

In this paper, a new methodology is formulated for solving the reduced Fokker‐Planck (FP) equations in high dimensions based on the idea that the state space of large‐scale nonlinear stochastic dynamic system is split into two subspaces. The FP equation relevant to the nonlinear stochastic dynamic system is then integrated over one of the subspaces. The FP equation for the joint probability density function of the state variables in another subspace is formulated with some techniques. Therefore, the FP equation in high‐dimensional state space is reduced to some FP equations in low‐dimensional state spaces, which are solvable with exponential polynomial closure method. Numerical results are presented and compared with the results from Monte Carlo simulation and those from equivalent linearization to show the effectiveness of the presented solution procedure. It attempts to provide an analytical tool for the probabilistic solutions of the nonlinear stochastic dynamics systems arising from statistical mechanics and other areas of science and engineering.     

The probability representation of quantum states and integral relations for Hermite and Laguerre polynomials

Some integral relations for orthogonal polynomials are elucidated. We review the generic scheme of the star-product construction and study in detail the star-product scheme based on the tomographic map. The dual star-product operator symbols are also considered and studied. Some integral kernels related to the star-product are calculated and new integral formulas for special functions are derived.     

Exact solutions of Dirac equation with Pöschl–Teller double-ring-shaped Coulomb potential via Nikiforov–Uvarov method

Exact analytical solutions of the Dirac equation are reported for the Pöschl-Teller double-ring-shaped Coulomb potential. The radial, polar, and azimuthal parts of the Dirac equation are solved by using the Nikiforov-Uvarov method, and exact bound state energy eigenvalues and the corresponding two-component spinor wavefunctions are reported.     

Numerical algorithm for the time fractional Fokker–Planck equation

Anomalous diffusion is one of the most ubiquitous phenomena in nature, and it is present in a wide variety of physical situations, for instance, transport of fluid in porous media, diffusion of plasma, diffusion at liquid surfaces, etc. The fractional approach proved to be highly effective in a rich variety of scenarios such as continuous time random walk models, generalized Langevin equations, or the generalized master equation. To investigate the subdiffusion of anomalous diffusion, it would be useful to study a time fractional Fokker–Planck equation. In this paper, firstly the time fractional, the sense of Riemann–Liouville derivative, Fokker–Planck equation is transformed into a time fractional ordinary differential equation (FODE) in the sense of Caputo derivative by discretizing the spatial derivatives and using the properties of Riemann–Liouville derivative and Caputo derivative. Then combining the predictor–corrector approach with the method of lines, the algorithm is designed for numerically solving FODE with the numerical error O(k^min{1+2α,2})+O(h²), and the corresponding stability condition is got. The effectiveness of this numerical algorithm is evaluated by comparing its numerical results for α=1.0 with the ones of directly discretizing classical Fokker–Planck equation, some numerical results for time fractional Fokker–Planck equation with several different fractional orders are demonstrated and compared with each other, moreover for α=0.8 the convergent order in space is confirmed and the numerical results with different time step sizes are shown.     

Multisymplectic implicit and explicit methods for Klein–Gordon–Schrödinger equations

We propose multisymplectic implicit and explicit Fourier pseudospectral methods for the Klein-Gordon-Schrödinger equations. We prove that the implicit method satisfies the charge conservation law exactly. Both methods provide accurate solutions in long-time computations and simulate the soliton collision well. Numerical results show the abilities of the two methods in preserving charge, energy, and momentum conservation laws.     

Woods-Saxon势下的赝自旋对称性

通过求解具有Woods-Saxon径向标量势与矢量势的Dirac方程, 分别分析了原子核中赝自旋双重态能级劈裂和波函数下分量劈裂随着Woods-Saxon势参数的变化关系, 其中Wood-Saxon势中的参数a以及R对于赝自旋能级劈裂和波函数劈裂的影响最大, 而势阱深度V₀只产生微小的影响. 随着平均场的变化, 赝自旋双重态会出现能级反转现象. 具有较大n或l的赝自旋双重态能级劈裂对于参数的变化更敏感. 赝自旋波函数下分量劈裂随着参数a和R的增大向核表面扩散, 并且在核表面附近达到最大. 赝自旋波函数劈裂随着参数a的增大而增大, 但是随着参数R或V₀的增大它却在减小的. 由于在特定的同位素链中, 参数R和V₀都与核子数有关系, 而参数a又是描述核的表面性质的, 所以以这些参数为变量对于赝自旋劈裂的研究是有意义的, 研究的结果至少在定性上可以应用到大部分原子核中.     

Dirac Equation and Some Quasi-Exact Solvable Potentials in the Turbiner’s Classification

In this paper quasi-exact solvability (QES) of Dirac equation with some scalar potentials based on sl(2) Lie algebra is studied. According to the quasi-exact solvability theory, we construct the configuration of the classes II, IV, V, and X potentials in the Turbiner's classification such that the Dirac equation with scalar potential is quasi-exactly solved and the Bethe ansatz equations are derived in order to obtain the energy eigenvalues and eigenfunctions.