Energy levels of the Schrödinger equation for some rational potentials in two-dimensional space using the inner product technique

The energy levels of a two-dimensional system are calculated for the rational potential,V(x, y; λ, g)=x ²+y ²+λ[x ²/(1+gx ²)+y ²/(1+gy ²)+a _xxx⁴+a _xyx² y ²+a _yyy⁴] using the inner product technique over a wide range of values of the perturbation parameters (g, λ) and for various eigenstates.     

傅里叶自函数的维格纳分布及应用

傅里叶变换是现代光学发展的重要理论工具。自1991年Caola首次定义傅里叶自函数以来1,它在光学领域的应用研究日趋活跃。本文首先对傅里叶自函数定义进行扩展,再讨论其维格纳分布函数及其矩,研究它们在光学中的应用。最后推导出傅里叶自函数应用于光学变换器成象时的变换矩阵。     

Properties of Perturbative Solutions of Unilateral Matrix Equations

A left-unilateral matrix equation is an algebraic equation of the form a ₀+a ₁ x+a ₂ x ²+·+a _n x ⁿ =0 where the coefficients a _r and the unknown x are square matrices of the same order and all coefficients are on the left (similarly for a right-unilateral equation). Recently certain perturbative solutions of unilateral equations and their properties have been discussed. We present a unified approach based on the generalized Bezout theorem for matrix polynomials. Two equations discussed in the literature, their perturbative solutions and the relation between them are described. More abstractly, the coefficients and the unknown can be taken as elements of an associative, but possibly noncommutative, algebra.     

The Fixed Point of a Generalization of the Renormalization Group Maps for Self-Avoiding Paths on Gaskets

Let W(x,y) = ax ³+ bx ⁴+ f ₅ x ⁵+ f ₆ x ⁶+ (3 ax ²)² y+ g ₅ x ⁵ y + h ₃ x ³ y ² + h ₄ x ⁴ y ² + n ₃ x ³ y ³+a ₂₄ x ² y ⁴+a ₀₅ y ⁵+a ₁₅ xy ⁵+a ₀₆ y ⁶, and X = , , where the coefficients are non-negative constants, with a > 0, such that X ²(x,x ²)−Y(x,x ²) is a polynomial of x with non-negative coefficients. Examples of the 2 dimensional map Φ: (x,y)↦ (X(x,y),Y(x,y)) satisfying the conditions are the renormalization group (RG) maps (modulo change of variables) for the restricted self-avoiding paths on the 3 and 4 dimensional pre-gaskets. We prove that there exists a unique fixed point (x _f ,y _f ) of Φ in the invariant set . 2000 Mathematics Subject Classification Numbers: 82B28; 60G99; 81T17; 82C41.     

Influence of Sr doping on the free carrier absorption in high-T c superconductors La2−x SrxCuO4

The reflectance spectra of polycrystalline La_2–x Sr _x CuO₄ samples were investigated in the energy range between 50 meV and 4 eV in dependence of the Sr content. The spectra are attributed to free carrier absorption of the Drude type, superimposed by optical phonon excitations below 0.1 eV and intrinsic absorption above 1 eV. From the influence of Sr doping onto the plasma energy it is deduced that La_2–x Sr _x CuO₄ is ap-type conductor with a maximum carrier concentration of 2.0×10²¹ cm^–3 forx=0.15. The results are interpreted in terms of a Hubbard model with an empty upper and ap-doped lower Hubbard band with a width of 1.9 eV.     

快离子导体AgxTiS2中Ag+离子－空位的二维基态结构与能量性质研究

根据Ag⁺离子-空位的二维有序结构建立了三维晶胞模型.采用局域密度近似下的平面波赝势方法,对有序Ag_xTiS₂(x=0,1/4,1/3,1/2,2/3,3/4,1)系列进行了几何结构优化和总能量计算,并与Li_xTiS₂系列进行了对比研究.有序Ag_xTiS₂系统的晶格参量增量Δa_{关键词： x}TiS₂')" href="#">Ag_xTiS₂ 有序—无序相变 离子扩散 第一性原理计算     

Small-x behaviour of the polarized photon structure function F_3^\gamma(x,Q^2)

We study the small-x behaviour of the polarized photon structure function F₃^gF_3^{\gamma}, measuring the gluon transversity distribution, in the leading logarithmic approximation of perturbative QCD. There are two contributions, both arising from two-gluon exchange. The leading contribution to small-x is related to the BFKL pomeron and behaves like x^-1-w₂x^{-1-\omega_2}, w₂ = O(a_S)\omega_2 ={\cal O}(\alpha_S). The other contribution includes in particular the ones summed by the DGLAP equation and behaves like x^1-w₀(+)x^{1-\omega_0^{(+)}}, w₀⁽⁺⁾ = O(?{a_S})\omega_0^{(+)} = {\cal O}(\sqrt{\alpha_S}).     

A model with persistent vacuum

It is shown that there exists a selfadjoint Hamilton operator corresponding to the interactionH ₀ ^(a) +H ₀ ^(b) + _b ⁺ (x) _a (x) _b ^– (x)d ³ x, wherea andb denote two types of scalar particles. We discuss the scattering theory of this operator.Work partially supported by the Swiss National Science Foundation.This paper contains results from the author's Ph. D. Thesis [3].     

The isotonic oscillator

The motion of a particle in the potential V(x) = V₀ (a/x - x/a)² is exactly soluble both classically and quantum mechanically. We have studied the features of the dynamics and the energies of a coherent state of this nonlinear system.     

Spectral Properties of the Periodic Magnetic Schrödinger Operator in the High-Energy Region. Two-Dimensional Case

The goal is to investigate spectral properties of the operator H=(–i +a(x))²+a₀(x) in the two-dimensional situation, a(x), a₀(x)) being periodic. We construct asymptotic formulae for Bloch eigenvalues and eigenfunctions in the high-energy region, describe properties of isoenergetic curves in the space of quasimomenta and give a new proof of the Bethe-Sommerfeld conjecture.Research partially supported by USNSF Grant DMS-0201383.Acknowledgements The author is thankful to Konstantin Makarov for very useful discussions and to Young-Ran Lee for her great help with pictures.     

Connection Considerations of Gravitational Field in Finsler Spaces

Some alternative connection structures of the Finslerian gravitational field are considered by modifying the independent variables (x,y) (x: point and y: vector) in various ways. For example, (x ^k ,y ⁱ ) (k,i = 1,2,3,4) are changed to (x ^k ,y ⁰) (y ⁰: scalar) or (x ⁰,y ⁱ ) (x ⁰: time axis); (x ^k ,y ⁱ ) are generalized to (x ^k ,y ⁱ ,p _i ) (p _i : covector dual to y ⁱ ) or (x ^k ,y ⁱ ,q _a ) (q _a : covector different from p _i ); (x ^k ,y ⁱ ) are further generalized to (x ^k ,y ^(a)i ) (a = 1,2,…,m), (y ^(a): (a)th vector), etc.     

The transverse correlation length for randomly rough surfaces

It is shown by numerical simulations for a random, one-dimensional surface defined by the equationx ₃=(x ₁), where the surface profile function (x ₁) is a stationary, stochastic, Gaussian process, that the transverse correlation lengtha of the surface roughness is a good measure of the mean distance d between consecutive peaks and valleys on the surface. In the case that the surface height correlation function (x ₁)(x ₁)/²(x ₁)=W (|x ₁–x ₁|) has the Lorentzian formW(|x ₁|)=a ²/(x ₁ ² +a ²) we find that d=0.9080a; when it has the Gaussian formW(|x ₁|)=exp(–x ₁ ² /a ²), we find that d=1.2837a; and when it has the nonmonotonic formW(|x ₁|)=sin(x ₁/a)/(x ₁/a), we find that d=1.2883a. These results suggest that d is larger, the faster the surface structure factorg(|Q|) [the Fourier transform ofW(|x ₁|)] decays to zero with increasing |Q|. We also obtain the functionP(itx ₁), which is defined in such a way that, ifx ₁=0 is a zero of (x ₁),P(x ₁)dx ₁ is the probability that the nearest zero of (x ₁) for positivex ₁ lies betweenx ₁ andx ₁+dx ₁.     

Dielectronic recombination and resonant transfer excitation processes for helium-like krypton

The relativistic configuration interaction method is employed to calculate the dielectronic recombination(DR) cross sections of helium-like krypton via the 1s2lnl ’(n = 2,3,...,15) resonances.Then,the resonant transfer excitation(RTE) processes of Kr 34+ colliding with H,He,H 2,and CH x(x = 0-4) targets are investigated under the impulse approximation.The needed Compton profiles of targets are obtained from the Hartree-Fock wave functions.The RTE cross sections are strongly dependent on DR resonant energies and strengths,and the electron momentum distributions of the target.For H 2 and H targets,the ratio of their RTE cross sections changes from 1.85 for the 1s2l2l ’ to 1.88 for other resonances,which demonstrates the weak molecular effects on the Compton profiles of H 2.For CH x(x = 0-4) targets,the main contribution to the RTE cross section comes from the carbon atom since carbon carries 6 electrons;as the number of hydrogen increases in CH x,the RTE cross section almost increases by the same value,displaying the strong separate atom character for the hydrogen.However,further comparison of the individual orbital contributions of C(2p,2s,1s) and CH 4(1t 2,2a 1,1a 1) to the RTE cross sections shows that the molecular effects induce differences of about 25.1%,19.9%,and 0.2% between 2p-1t 2,2s-2a 1,and 1s-1a 1 orbitals,respectively.     

Universality properties of the stationary states in the one-dimensional coagulation-diffusion model with external particle input

We investigate with the help of analytical and numerical methods the reactionA+AA on a one-dimensional lattice opened at one end and with an input of particles at the other end. We show that if the diffusion rates to the left and to the right are equal, for largex, the particle concentrationc(x) behaves likeA _s x ^–1 (x measures the distance to the input end). If the diffusion rate in the direction pointing away from the source is larger than the one corresponding to the opposite direction, the particle concentration behaves likeA _a x ^–1/2. The constantsA _s andA _a are independent of the input and the two coagulation rates. The universality ofA _a comes as a surprise, since in the asymmetric case the system has a massive spectrum.     

Null Surfaces and Their Singularities in Three-Dimensional Minkowski Space-Time

In this work we integrate the null geodesic equations in three-dimensional Minkowski space-time in order to obtain the light-cone cut function; that is, the function that describes the intersection, C_x ^a, of the light cone from each space-time point, _x ^a, with future null infinity I ⁺. Furthermore, using this result, we locate the singularities of the null surface obtained as the envelope of the past light cones from points on a deformed light-cone cut of I ⁺.     

高次正幂与逆幂势函数的叠加的径向薛定谔方程的解析解

当薛定谔方程中出现高次非谐振子势，电偶极矩势，分子晶体势，极化等效势等高次正幂与逆幂势函数以及它们的叠加时，薛定谔方程的求解变得非常复杂，采用奇点邻域附近的级数解法与求解渐近解相结合，在多种相互作用幂函数紧密耦合的条件下，得到势函数为V(r)=a₁r⁶+a₂r²+a₃r^{-4关键词： 级数解法 幂势函数 径向波函数 渐近解}     

More on Zeta-Function Regularization of High-Temperature Expansions

A recent paper using the Riemann ζ-function to regularize the (divergent) coefficients occurring in the high-temperature expansions of one-loop thermodynamic potentials is extended. This method proves to be a powerful tool for converting Dirichlet-type series Σ_ma_m(x_i)/m^s into powerseries in the dimensionless parameters x_i. The coefficients occurring in the power series are (proportional to) ζ-functions evaluated away from their poles - this is where the regularization occurs. High-temperature expansions are just one example of this highly-nontrivial rearrangement of Dirichlet series into power series form. We discuss in considerable detail series in which a_m(x_i) is a product of trigonometrie, algebraic and Bessel function factors. The ζ-function method is carefully explained, and a large number of new formulae are provided. The means to generalize these formulae are also provided. Previous results on thermodynamic potentials are generalized to include a nonzero constant term in the gauge potential (time component) which can be used to probe the electric sector of temperature gauge theories.     

Energy levels of a two-dimensional anharmonic oscillator: Hill determinant approach

The energy levels of the Schrödinger equation for the potentialx ²+y ²+λ[a _xxx⁴+2a _xyx² y ²+a _yyy⁴] are calculated using Hill determinant approach for several eigenstates and over a wide range of values of perturbation parameters. The obtained numerical results agree with those previously reported by other methods.     

Geometry and Growth Rate of Frobenius Numbers of Additive Semigroups

Linear combinations x ₁ a ₁ ⋯ x _n a _n of n given natural numbers a _s (with nonnegative integral coefficients x _s ) attain all the integral values, starting from some integer N(a), called the Frobenius number of vector a (provided that the integers a _s have no common divisor, greater than 1). The growth rate of N(a) with the large value of σ = ta ₁ ⋯ a _n depends peculiarly from the direction α of the vector a = σα. The article proves the lower bound of order and the upper bound of order σ ². Both orders are reached from some directions α. The averaging of N(a) along all directions, performed for σ = 7, 19, 41 and 97, provides the values, confirming the rate σ ^p for some p between 3/2 and 2 (for n = 3), excluding neither 3/2 nor 2, for the asymptotic behaviour at large σ. One gets check p ≈ 1, 66 for σ between 100 and 200. These unexpected results, based on some strange relations of the Frobenius numbers to the higher-dimensional continued fractions geometry, lead to many facts of this arithmetic trubulence theory, discussed in this article both as theorems and as conjectures. ^★ Partially supported by RFBR grant 05-01-00104.     

Central Limit Theorems for Nonlinear Hierarchical Sequences of Random Variables

Let a random variable x ₀ and a function f:[a, b] ^k [a, b] be given. A hierarchical sequence {x _n :n=0, 1, 2,...} of random variables is defined inductively by the relation x _n =f(x _{n–1, 1}, x _{n–1, 2}..., x _{n–1, k} ), where {x _{n–1, i} :i=1, 2,..., k} is a family of independent random variables with the same distribution as x _n–1. We prove a central limit theorem for this hierarchical sequence of random variables when a function f satisfies a certain averaging condition. As a corollary under a natural assumption we prove a central limit theorem for a suitably normalized sequence of conductivities of a random resistor network on a hierarchical lattice.