Hulthén势散射态的解析解

在任意l波的离心项1/r²用λ²e^-λr/(1-e^-λr)²近似表达的条件下,对Hulthén势的径向Schrdinger方程作自变量指数变换,使此转化为超几何微分方程,获得了Hulthén势s波散射态的精确解和非s波散射态的近似解析解.给出了相移的解析表达式和按“k/2π标度”归一化的用超几何函数表示的径向波函数.讨论了解析解的意义.     

Structure of quantum-mechanical relativistic two-body interactions for spinning particles

Relativistic constraint mechanics yields consistent systems of coupled Dirac equations for pairs of spinning particles. We explicitly connect these equations to the Bethe-Salpeter equation of quantum field theory and to the interactions of classical Fokker-Tetrode dynamics (and hence to classical relativistic field theory) to obtain versions of these equations governed by systems of (possibly noncoulombic) relativistic potentials whose detailed structures contain important relativistic effects like correct Darwin interactions. We recast the defining pair of Dirac equations in a number of equivalent but important forms—“external potential,” Sazdjian, hyperbolic, and Breit— and examine their interconnection. Since the potentials in these equations are no more singular than — 1/4r² we are able to solve appropriate versions of them nonperturbatively for the qˉq system to obtain a very good fit to the entire meson spectrum and for the e ⁺ e ⁻ system to calculate the positronium spectrum of QED correct through order α ⁴ .     

相对论性非球谐振子势场中的赝自旋对称性

求解了非球谐振子势场中1/2自旋粒子满足的Dirac方程,Dirac哈密顿量包含有标量非球谐振子势S(r)和矢量非球谐振子势V(r).在Σ(r)=S(r)+V(r)=0和Δ(r)=V(r)-S(r)=0的条件下,解析地得到了Dirac旋量波函数的束缚态解和能谱方程,结果表明非球谐振子势 关键词： 非球谐振子势 Dirac方程 赝自旋对称性 束缚态     

The Vortex-Wave Equation with a Single Vortex as the Limit of the Euler Equation

In this article we consider the physical justification of the Vortex-Wave equation introduced by Marchioro and Pulvirenti (Mechanics, analysis and geometry: 200 years after Lagrange, North-Holland Delta Ser., Amsterdam, North-Holland, pp. 79–95, 1991), in the case of a single point vortex moving in an ambient vorticity. We consider a sequence of solutions for the Euler equation in the plane corresponding to initial data consisting of an ambient vorticity in L ¹ ∩ L ^∞ and a sequence of concentrated blobs which approach the Dirac distribution. We introduce a notion of a weak solution of the Vortex-Wave equation in terms of velocity (or primitive variables) and then show, for a subsequence of the blobs, the solutions of the Euler equation converge in velocity to a weak solution of the Vortex-Wave equation.     

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}     

Klein paradox in the Breit equation

We demonstrate that in the Breit equation with a central potentialV(r) having the propertyV(r ₀)=E there appears a Klein paradox atr=r ₀. This phenomenon, besides the previously found Klein paradox arr→∞ appearing ifV(r)→∞ atr→∞, seems to indicate that in the Breit equation valid in the singleparticle theory the sea of particle-antiparticle pairs is not well separated from the considered two-body configuration. We conjecture that both phenomena should be absent from the Salpeter equation which is consistent with the hole theory. We prove this conjecture in the limit ofm ⁽¹⁾→∞ andm ⁽²⁾→∞, where we neglect the terms ～1/m ⁽¹⁾ and 1/m ⁽²⁾. In Appendix I we show that in the Breit equation the oscillations accumulating atr=r ₀ in the case ofm ⁽¹⁾≠m ⁽²⁾ are normalizable to the Dirac δ-function. In Appendix II the analogical statement is justified for the nonoscillating singular behaviour appearing atr=r ₀ in the case ofm ⁽¹⁾=m ⁽²⁾.     

Coulomb-Like Interactions in the Dirac Equation

The Dirac equation for the Coulomb-like problem is modified by incorporating minimal interactions into the Dirac Hamiltonian, that keep the 1/r potential dependence. We determine the general energy eigenvalues and the corresponding eigenfunctions.     

Relativistic calculations of electronic excitation probabilities in U<Superscript>92+</Superscript>-U<Superscript>91+</Superscript>(1<Emphasis Type="Italic">s</Emphasis>) collisions in the monopole approximation

Heavy ion collisions U⁹²⁺-U⁹¹⁺(1s) at a 6 MeV incident particle energy per nucleon are considered. The wave function of the electron was determined by solving the Dirac time-dependent equation in a central field, which is the sum of the potential of the target nucleus and the potential of the incident nucleus taken in the monopole approximation. The Dirac equation was solved using the basis set of Hermitian B-splines. The probabilities of remaining in the 1s target state were calculated as functions of the impact parameter.     

Approximate <Emphasis Type="Italic">κ</Emphasis>-state solutions to the Dirac-Yukawa problem based on the spin and pseudospin symmetry

Using an approximation scheme to deal with the centrifugal (pseudo-centrifugal) term, we solve the Dirac equation with the screened Coulomb (Yukawa) potential for any arbitrary spin-orbit quantum number κ. Based on the spin and pseudospin symmetry, analytic bound state energy spectrum formulas and their corresponding upper- and lower-spinor components of two Dirac particles are obtained using a shortcut of the Nikiforov-Uvarov method. We find a wide range of permissible values for the spin symmetry constant C _s from the valence energy spectrum of particle and also for pseudospin symmetry constant C _ps from the hole energy spectrum of antiparticle. Further, we show that the present potential interaction becomes less (more) attractive for a long (short) range screening parameter α. To remove the degeneracies in energy levels we consider the spin and pseudospin solution of Dirac equation for Yukawa potential plus a centrifugal-like term. A few special cases such as the exact spin (pseudospin) symmetry Dirac-Yukawa, the Yukawa plus centrifugal-like potentials, the limit when α becomes zero (Coulomb potential field) and the non-relativistic limit of our solution are studied. The nonrelativistic solutions are compared with those obtained by other methods.     

Exact asymptotic solution of the Della Sala-Görling integral equation for the exchange-only potential for Be-like atomic ions at large Z

Della Sala and Görling (DSG) have written an integral equation for the exchange-only potential Vx(r) in terms of the Dirac density matrix. Here, an exact asymptotic solution of this integral equation is presented, for the ground state of Be-like atomic ions, in terms of γ(r,r^′) plus the 2s HOMO orbital. In the large Z limit of such ions, the DSG integral equation corrects the asymptotic form −e²/r of Vx(r) by exponentially decaying terms. This amounts to setting the polarizability equal to zero.     

Can quarks always be confined by a linear potential?

It is demonstrated on the basis of the Dirac equation that quarks cannot be confined by a vector gluon potential of the form(r/r ₀)^a or[ln(r/r ₀]^a, a>0, if the quark-gluon interaction conserves parity. In order to confine quarks with the parity-conserving interaction, the effective gluon potential must be a pseudovector or a scalar. These are shown in a simple Yang-Mills field with theSU(2) group.     

类Quesne环状球谐振子势场中的赝自旋对称性

提出了一种新的类Quesne环状球谐振子势,应用二分量方法求解1/2-自旋粒子满足的Dirac方程, Dirac哈密顿量由标量和矢量类Quesne环状球谐振子势构成.在Σ=S(r)+V(r)=0的条件下,得到了Dirac旋量波函数下分量的束缚态解和能谱方程, 显示出类Quesne环状球谐振子势场中的赝自旋对称性.讨论了束缚态波函数和能谱方程的有关性质. 关键词： 类Quesne环状球谐振子势 Dirac方程 赝自旋对称性 束缚态     

Poisson-Lie Dynamical <Emphasis Type="Italic">r</Emphasis>-matrices from Dirac Reduction

The Dirac reduction technique used previously to obtain solutions of the classical dynamical Yang-Baxter equation on the dual of a Lie algebra is extended to the Poisson-Lie case and is shown to naturally yield certain dynamical r-matrices on the duals of Poisson-Lie groups found by Etingof, Enriquez and Marshall in math.QA/0403283.     

幂函数叠加势的径向薛定谔方程的解析解

研究多种正幂势函数与逆幂势函数紧密耦合条件下薛定谔径向方程解析解的求解方法.对势函数为V（r）=α₁r⁸＋α₂r³+α₃r²+β₃r^-1＋β₂r^-3＋β₁r^-4的径向薛定谔方程存在解析解的条件以及精确的解析解进行了研究. 根据量子系统波函数必须满足单值、有界和连续的标准条件,首先求出径向坐标r→∞以及r→0时的渐近解,然后采用非正则奇点邻域附近的波函数级数解法与求得的渐近解相结合,通过幂级数系数比较法得到径向薛定谔方程在势函数系数紧密耦合条件下的一系列定态波函数解析解以及相应的能级结构,并作适当讨论与结论. 关键词： 级数解法 幂势函数 径向波函数 渐近解     

Profiles and Quantization of the Blow Up Mass for Critical Nonlinear Schrödinger Equation

We consider finite time blow up solutions to the critical nonlinear Schrödinger equation For a suitable class of initial data in the energy space H¹, we prove that the solution splits in two parts: the first part corresponds to the singular part and accumulates a quantized amount of L² mass at the blow up point, the second part corresponds to the regular part and has a strong L² limit at blow up time.Part of this work has been supported by grant DMS-0111298.     

Radiative kinks and fermionic zero modes in a (λφ6)1+1 model

We consider bidimensional scalar models including kink solutions _k (x). Using the hidden supersymmetric properties of the Dirac equation, we describe a general method to find normalizable fermionic zero modes. In particular, we apply the technique to a ( ⁶)₁₊₁ model. Going to the one-loop order of the effective potential, the emergence of a radiative kink provides an interesting scalar background in order to discuss the Dirac equation.     

Poisson Homology of r-Matrix Type Orbits I: Example of Computation

Abstract

In this paper we consider the Poisson algebraic structure associated with a classical r-matrix, i.e. with a solution of the modified classical Yang–Baxter equation. In Section 1 we recall the concept and basic facts of the r-matrix type Poisson orbits. Then we describe the r-matrix Poisson pencil (i.e the pair of compatible Poisson structures) of rank 1 or CP ⁿ-type orbits of SL(n, C). Here we calculate symplectic leaves and the integrable foliation associated with the pencil. We also describe the algebra of functions on CP ⁿ-type orbits. In Section 2 we calculate the Poisson homology of Drinfeld–Sklyanin Poisson brackets which belong to the r-matrix Poisson family.     

Quantum Spectrum of Some Anharmonic Central Potentials: Wave Functions Ansatz

Based on the ansatz to the wave functions, the quasi-exact solutions of the 2D Schrödinger equation with some anharmonic potentials are reviewed and analyzed if admitting restrictions on the parameters of the potential and the angular momentum m. These potentials are taken as the screened Coulomb potential V(r)=a/r+b/(r+), the singular one-fraction power one V(r)=ar ^–1/2+br ^–3/2 and the singular two-fraction one V(r)=ar ^2/3+br ^–2/3+cr ^–4/3. The latter one is found that the hidden symmetry exists if substituting r–ir. It will reverse the signs of E and c of quantum system, leaving the remaining parameters invariant.     

Fast and Slow Convergence to Equilibrium for Maxwellian Molecules via Wild Sums

We consider the spatially homogeneous Boltzmann equation for Maxwellian molecules and general finite energy initial data: positive Borel measures with finite moments up to order 2. We show that the coefficients in the Wild sum converge strongly to the equilibrium, and quantitatively estimate the rate. We show that this depends on the initial data F essentially only through on the behavior near r=0 of the function J _F (r)=_|v|>1/r |v|² dF(v). These estimates on the terms in the Wild sum yield a quantitative estimate, in the strongest physical norm, on the rate at which the solution converges to equilibrium, as well as a global stability estimate. We show that our upper bounds are qualitatively sharp by producing examples of solutions for which the convergence is as slow as permitted by our bounds. These are the first examples of solutions of the Boltzmann equation that converge to equilibrium more slowly than exponentially.