Calculation of the vortex pinning energy in a long periodically modulated Josephson contact

An analytical method for evaluating the Josephson and magnetic energies of a vortex, as well as the vortex pinning energy and its components, in a long periodically modulated Josephson contact is suggested. The method allows one to take into account the variation of the vortex shape with the position of the vortex. The results obtained with this method are much closer to those of exact computer analysis compared with results of the conventional techniques. The discrepancy between the exact shape of the vortex and a function obtained by solving a differential equation approximating the exact difference equation is studied.     

A new method of calculation in the Fractional Quantum Hall Effect regime

The electron-electron and electron-background interaction energies are calculated analytically for systems with up to N = 6 electrons. The method consists of describing the position vectors of electrons using complex coordinates and all the interaction energies with complex notation, whereby simplifications become possible. As is known, in this type of calculation, complicated expressions involving integrals over many variables are encountered and the trick of using complex coordinates greatly facilitates the exact calculation of various quantities. Contrary to previous analytical calculations, using complex coordinates avoids complicated trigonometric functions from appearing in the integrand, simplifying the exact evaluation of the integrals. The method we have used can be straightforwardly extended to larger systems with N > 6 electrons.     

Dirac方程的数值解法及其在原子总能量计算中的应用

中心势近似下径向Dirac方程的求解是相对论性原子(离子)结构计算的基础.本文通过相对论性方程中径向波函数大分量与非相对论方程径向波函数的类比,提出了径向Dirac方程的一种数值解法.为了验证数值解法的精度和可靠性,首先将数值结果与类氢势作用下的解析解进行比较.然后,将这种算法扩展到基于解析势的相对论性原子结构计算中,并将计算出的总能量与实验结果和其他方法得到的结果进行对比.     

Exact hole-bound states for semiconductor quantum wells with arbitrary potential profiles

An efficient numerical-analytical method for finding confined and continuum states in quantum-well systems with arbitrary potential profiles, described by coupled Schrödinger equations, is presented. The method is based on the analytical properties of the wave functions, in particular, the power series representation of solutions of the corresponding coupled differential equations. Using only the general properties of the coefficients of a system of an arbitrary number of coupled Schrödinger equations, and imposing for definiteness the simplest boundary conditions, we derive exact expressions for the wave functions and present methods for accurate calculations of the energies and wave functions of confined states and of the wave functions of continuum states in quantum wells. The method is applied to the calculation of the dispersion of hole bound states in a single GaAs quantum well with truncated parabolic confining potentials of different strengths. The results are compared with data available from previous calculations.     

EXACT BUCKLING AND VIBRATION SOLUTIONS FOR STEPPED RECTANGULAR PLATES

This paper is concerned with the determination of exact buckling loads and vibration frequencies of multi-stepped rectangular plates based on the classical thin (Kirchhoff) plate theory. The plate is assumed to have two opposite edges simply supported while the other two edges can take any combination of free, simply supported and clamped conditions. The proposed analytical method for solution involves the Levy method and the state-space technique. By using this analytical method, exact buckling and vibration solutions are obtained for rectangular plates having one- and two-step thickness variations. These exact solutions are extremely useful as benchmark values for researchers developing numerical techniques and software for analyzing non-uniform thickness plates.     

Exact eigenstates for a class of models describing multiphoton processes in the presence of seven bosonic modes

With recent advances in Bose-Einstein condensation (BEC)[1—3], there has beenmuch interest in nonlinear processes in the various disciplines of physics, such asnonlinear multi-wave mixing processes[4]. Now, great efforts are devoted to constructingthe nonlinear quantized models to numerate the energy spectra and eigenstates of thesenonlinear processes[5—8]. But there still remain some problems with regard to interactionsamong several bosonic modes, for example, how to explicitly obtain the…     

The laser-dressed potential binding energy of a hydrogenic impurity in a spherical quantum dot by the analytical transfer matrix method

The analytical transfer matrix method (ATMM) is efficient and accurate for understanding the nature of bound states. In this paper, it is applied to obtain the binding energy of a hydrogenic impurity placed at the center of the spherical quantum dots (QDs) in an intense laser field. Our results agree with the exact energies in Varshni [Superlattices Microstructures 30 (2001) 45]. Therefore, ATMM gives us an alternative approach to tackle the problem of impurities placed in nano-structures under intense laser fields.     

Eigenstates and eigenenergies of four-wave-mixing models

We present explicit analytical expressions of all the eigenenergies and eigenstates for two different kinds of four-wave-mixing model in terms of a parameter lambda without the so-called Bethe ansatz assumption. The parameter lambda is shown to be determined by the roots of a simple polynomial. We also obtain the exact explicit analytical expressions of infinite eigenstates and energies without any unknown parameter.     

Analytic presentation of a solution of the Schrödinger equation

High-precision approximate analytic expressions for energies and wave functions are found for arbitrary physical potentials. The Schrödinger equation is cast into the nonlinear Riccati equation, which is solved analytically in first iteration of the quasi-linearization method (QLM). The zeroth iteration is based on general features of the exact solution near the boundaries. The approach is illustrated on the Yukawa potential. The results enable accurate analytical estimates of effects of parameter variations on physical systems.     

Variational approach to spin fluctuations in the Hubbard model

We study a one parameter variational wave function to improve the spin density wave ground state of the Hubbard model by inclusion of quantum spin fluctuations. Using a perturbative approach and novel lattice summation techniques we present analytical as well as numerical results for the correlation energies and the staggered magnetizations in one and two dimensions. We find ground state energies which are satisfyingly close to known exact results and are significantly lower than those of existing Gutzwiller and numerical treatments.     

Accurate analytic presentation of solution for the spiked harmonic oscillator problem

High precision approximate analytic expressions of the ground state energies and wave functions for the spiked harmonic oscillator are found by first casting the correspondent Schrödinger equation into the nonlinear Riccati form and then solving that nonlinear equation analytically in the first iteration of the quasilinearization method (QLM). In the QLM the nonlinear differential equation is treated by approximating the nonlinear terms with a sequence of linear expressions. The QLM is iterative but not perturbative and gives stable solutions to nonlinear problems without depending on the existence of a smallness parameter. The choice of zero iteration is based on general features of exact solutions near the boundaries. Comparison of our approximate analytic expressions for binding energies and wave functions with the exact numerical solutions demonstrates their high accuracy in the wide range of parameters. The accuracy ranging between 10⁻³ and 10⁻⁷ for the energies and, correspondingly, 10⁻² and 10⁻⁷ for the wave functions in the regions, where they are not extremely small is reached. The derived formulas enable one to make accurate analytical estimates of how variation of different interactions parameters affects the correspondent physical systems.     

Optimized Lower Bound for Four-Body Hamiltonians

Generalizing a method elaborated for three-body systems, we derive a new lower bound on four-body ground-state energies in terms of two-body binding energies in the unequal-mass case. For simple power-law potentials, this bound is compared to variational calculations and is shown to be very close to the exact result. In particular, it gives the exact answer for harmonic interactions. Received November 6, 1997; accepted in final form February 6, 1998     

Sound transmission prediction by 3-D elasticity theory

The problem of sound transmission through layered panel structures is studied with the exact theory of three-dimensional (3-D) elasticity. The exact solution to the 3-D elasticity equations is obtained by the use of the Fourier spectral method. Based on this analytical solution, a transfer matrix is derived that relates the spectral displacements and stresses on the one surface of the panel to those on the opposite panel surface. The transfer matrix is then used to develop the analytical solutions for sound reflection and transmission coefficients. Explicit, concise expressions are obtained for the analytical solutions of the acoustic transmission and reflection coefficients under the general conditions of layered anisotropic panels. Examples are given for both single-layer and sandwich panels. Predictions on sound transmission from the 3-D elasticity theory are compared with available data from other methods, and the results are discussed.     

Torsional vibration of multi-step non-uniform rods with various concentrated elements

An analytical approach and exact solutions for the torsional vibration of a multi-step non-uniform rod carrying an arbitrary number of concentrated elements such as rigid disks and with classical or non-classical boundary conditions is presented. The exact solutions for the free torsional vibration of non-uniform rods whose variations of cross-section are described by exponential functions and power functions are obtained. Then, the exact solutions for more general cases, non-uniform rods with arbitrary cross-section, are derived for the first time. In order to simplify the analysis for the title problem, the fundamental solutions and recurrence formulas are developed. The advantage of the proposed method is that the resulting frequency equation for torsional vibration of multi-step non-uniform rods with arbitrary number of concentrated elements can be conveniently determined from a homogeneous algebraic equation. As a consequence, the computational time required by the proposed method can be reduced significantly as compared with previously developed analytical procedures. A numerical example shows that the results obtained from the proposed method are in good agreement with those determined from the finite element method (FEM), but the proposed method takes less computational time than FEM, illustrating the present methods are efficient, convenient and accurate.     

Triply coupled vibrations of thin-walled open cross-section beams including rotary inertia effects

An exact analytical method is presented for predicting the undamped natural frequencies of beams with thin-walled open cross-sections having no axis of symmetry. The governing differential equations give a characteristic equation of the 12th order with real coefficients. The roots are found numerically and the exact boundary conditions are considered especially for free ends to obtain natural frequencies. The simpler cases of neglecting cross-sectional warping and/or rotary inertia are also dealt with. It is seen that when the effect of rotary inertia is neglected significant errors incur for some boundary conditions, cross-section thicknesses and mode numbers. This is more profound when the warping effect is taken into account.     

用新公式精确研究N_2分子部分电子态的离解能

本文使用代数方法(AM),研究了N_2分子的X~1∑~+_g、A~3∑~+_u、B~(′3)∑~-_u、a′~1∑~-_u、b′~1∑~+_u、B~3∏_g、c′_4~1∑~+_u等七个电子态的离解能;然后使用新公式计算了这些电子态的离解能,并分别与离解能的实验值进行了分析对比.结果表明:使用新公式得到的分子离解能与离解能的实验值更为接近.对于那些用实验方法还难以获得其离解能的电子态,该式提供了一种获得其离解能的新途径.     

An analytical solution to Li/Li+ insertion into a porous electrode

The dynamic faradic properties of the lithium ion batteries are primarily determined by the process of lithium ion insertion into a porous electrode. In this paper, we present an analytical result of the intercalating process of Li/Li⁺ into a spherical particle of graphite or cobalt oxide immersed in a conductive electrolyte. Using the finite integral transform method, an exact solution to the concentration profile was obtained for arbitrary linear initial and boundary conditions. To avoid analytical difficulties with respect to the boundary conditions of second kind, the method of pseudo-steady-state is applied. The final solution uniformly converges, and can be used for accurate and fast dynamic modelling and simulation.     

Single parameter scaling in one-dimensional localization revisited

The variance of the Lyapunov exponent is calculated exactly in the one-dimensional Anderson model with random site energies distributed according to the Cauchy distribution. We find a new significant scaling parameter in the system, and derive an exact analytical criterion for single parameter scaling which differs from the commonly used condition of phase randomization. The results obtained are applied to the Kronig-Penney model with the potential in the form of periodically positioned delta functions with random strength.     

Exact calculations and a fermion pair description of the s-bosons used in the interacting boson approximation

A convenient method is proposed for exact calculations using wave functions that are products of correlated fermion pairs coupled to zero angular momentum. The method is valid for arbitrary forces and arbitrary single-particle energies. The exact results are compared to various approximations and are used to generate an equivalent s-boson Hamiltonian.     

Excitons in a homogeneous magnetic field: A modified perturbation approach

We propose a specially adapted perturbational scheme to calculate the energies and weve functions of excitons in a homogeneous magnetic fieldB. The strength ofB is arbitrary. In contrast to involved variational calculations, our final results are entirely analytical and may serve as a starting point for further applications. As for the energies, we find good agreement with previous work. Moreover we show that the well-known small-B and large-B asymptotics for the exact eigenvalues areboth contained in our unifying formulas as limiting cases.