Explicit forms of the off-shell Jost solutions for general central potentials

We derive explicit forms of the regular solutions and the Jost solutions off the energy shell, which satisfy the inhomogeneous Schrödinger equation. The used forms of the Yukawa-like and Gauss-like potentials are related to the two known integral representations of the Hankel functions. The explicit form of the introduced fully off-shell Jost functions enables us to write it in the alternative integral forms, which contain the Jost solutions or the regular solution.     

Relations between the wave solutions and the jost solutions off the energy shell for local central potentials

Using the Green function techniques we express the wave solutions of the radial inhomogeneous Schrödinger equation by means of the on-shell Jost and regular solutions. Making use of their boundary behaviour atr = andr = 0 we reexpress them alternatively in terms of the off-shell Jost and regular solutions. Relations among the different generalized (fully off the energy shell) Jost functions are derived and the radial matrix elements of the transition and reaction (reactance) operators are given in terms of these Jost functions. The relations reflect the principle of detailed balance.     

On- and off-shell Jost functions and their integral representations

By judicious exploitation of the transpose operator relation in conjunction with the differential equations of special functions of mathematical physics, integral representations of the on- and off-shell Jost functions are derived from the particular integrals of the inhomogeneous Schrödinger equation. Using the particular integral of the inhomogeneous Schrödinger equation, exact analytical expressions for the Coulomb and Coulomb plus Yamaguchi off-shell Jost solutions are constructed in the maximal reduced form. As a case study, the limiting behaviours and the on-shell discontinuities of the Coulomb plus Yamaguchi Jost solutions are verified numerically.     

Jost functions in the two-potential formalism

For two inhomogeneous Schrödinger equations playing an important role within the framework of the Gell-Mann — Goldberger two-potential formalism we derive the integral equations for the off-shell solutions and give the relations between the regular and Jost solutions. We define the Jost functions fully off the energy shell. The obtained formulae give the possibility to extend the validity of various useful relations derived within the one-potential theory.     

Inverse scattering equation with a coulombic reference potential

The relation of Marchenko's inverse scattering integral equation for charged and non charged particles is derived. Analytic properties of scattering functions and Jost solutions for both situations are of central importance. Limiting situations fork → 0, and implications for practical applications are discussed.     

A DIRECT PERTURBATION THEORY OF THE NONLINEAR SCHR?DINGER EQUATION WITH CORRECTIONS

An exact direct perturbation theory of nonlinear Schrodinger equation with corrections is developed under the condition that the initial value of the perturbed solution is equal to the value of an exact multisoliton solution at a particular time. After showing the squared Jost functions are the eigenfunctions of the linearized operator with a vanishing eigenvalue,suitable definitions of adjoint functions and inner product are introduced. Orthogonal relations are derived and the expansion of the unity in terms of the squared Jost functions is naturally implied. The completeness of the squared Jost functions is shown by the generalized Marchenko equation. As an example,the evolution of a Raman loss compensated soliton in an optical fiber is treated.     

Multiple Pole Solutions of the Modified Nonlinear Schrödinger Equation

A general formula for the N-tuple polesoliton solutions of the modified nonlinear Schrödinger equation, which corresponds to a nonzero pole of order N of the Jost solution to the corresponding Lax-pair equations, is derived.     

Analytical Solutions of the Gravitational Field Equations in de Sitter and Anti-de Sitter Spacetimes

The generalized Laplace partial differential equation, describing gravitational fields, is investigated in de Sitter spacetime from several metric approaches—such as the Riemann, Beltrami, Börner-Dürr, and Prasad metrics—and analytical solutions of the derived Riccati radial differential equations are explicitly obtained. All angular differential equations trivially have solutions given by the spherical harmonics and all radial differential equations can be written as Riccati ordinary differential equations, which analytical solutions involve hypergeometric and Bessel functions. In particular, the radial differential equations predict the behavior of the gravitational field in de Sitter and anti-de Sitter spacetimes, and can shed new light on the investigations of quasinormal modes of perturbations of electromagnetic and gravitational fields in black hole neighborhood. The discussion concerning the geometry of de Sitter and anti-de Sitter spacetimes is not complete without mentioning how the wave equation behaves on such a background. It will prove convenient to begin with a discussion of the Laplace equation on hyperbolic space, partly since this is of interest in itself and also because the wave equation can be investigated by means of an analytic continuation from the hyperbolic space. We also solve the Laplace equation associated to the Prasad metric. After introducing the so called internal and external spaces—corresponding to the symmetry groups SO(3,2) and SO(4,1) respectively—we show that both radial differential equations can be led to Riccati ordinary differential equations, which solutions are given in terms of associated Legendre functions. For the Prasad metric with the radius of the universe independent of the parametrization, the internal and external metrics are shown to be of AdS-Schwarzschild-like type, and also the radial field equations arising are shown to be equivalent to Riccati equations whose solutions can be written in terms of generalized Laguerre polynomials and hypergeometric confluent functions.     

Riemann-Hilbert approach and N double-pole solutions for a nonlinear Schrödinger-type equation

We investigate the inverse scattering transform for the Schrödinger-type equation under zero boundary conditions with the Riemann-Hilbert (RH) approach. In the direct scattering process, the properties are given, such as Jost solutions, asymptotic behaviors, analyticity, the symmetries of the Jost solutions and the corresponding spectral matrix. In the inverse scattering process, the matrix RH problem is constructed for this integrable equation base on analyzing the spectral problem. Then, the reconstruction formula of potential and trace formula are also derived correspondingly. Thus, N double-pole solutions of the nonlinear Schrödinger-type equation are obtained by solving the RH problems corresponding to the reflectionless cases. Furthermore, we present a single double-pole solution by taking some parameters, and it is analyzed in detail.     

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.     

Relativistic jost functions and potentials

An extension of the solution of the inversion problem reproducing the potentials and wave functions from the given singularities of the Jost functions on a static limit relativistic case of theS-wave Klein-Gordon equation is investigated by way of the method elaborated by Petrá in Czech. J. Phys.B12 (1962), 67. It is shown that similarly as in the non-relativistic case, a transparent representation of the relativistic Jost solution permits to determine uniquely the potentials decreasing exponentially at large distances and leads to the dispersive Fredholm integral equations for the Jost solution components. The proposed method might be considered as an alternative to that used by Cornille in J. Math. Phys.11 (1970), 79.     

An Integral Transform of Coulomb Green’s Function via Sturmian Representation and Off-Shell Scattering

By exploiting the term by term separability of the Sturmian function representation of the Coulomb Green’s function an integral transform of the corresponding outgoing wave Green’s function, that plays a crucial role in the studies of off-shell properties of the Coulomb and Coulomb-like interactions, is derived. Working in the representation space the off-shell Jost solutions and T-matrices for motion in Coulomb and Coulomb-modified nuclear potentials are expressed in terms of simple expressions involving hypergeometric functions. The effectiveness of our constructed expressions for the T-matrices is examined through a model calculation.     

Makarov势的Dirac方程的束缚态解

给出了Makarov型标量势与矢量势相等条件下的Dirac方程的束缚态解. Dirac方程的角向方程用因子分解方法求解，在得出角向波函数的过程中，自然地得到了属于同一本征值的不同角向波函数间的递推操作. 径向束缚态波函数用合流超几何函数表示，束缚态的能量方程可由径向波函数满足的边界条件得到. 关键词： Makarov势 Dirac方程 束缚态 因子分解方法     

NLS+方程的平方Jost解的完备性

对于NLS⁺方程相应的平方Jost函数的完备性,通过求得的广义Marchenko方程给出了证明,从而建立了含修正项的直接微扰理论． 关键词：     

The classical field limit of scattering theory for non-relativistic many-boson systems. I

We study the classical field limit of non-relativistic many-boson theories in space dimensionn≧3. When ?→0, the correlation functions, which are the averages of products of bounded functions of field operators at different times taken in suitable states, converge to the corresponding functions of the appropriate solutions of the classical field equation, and the quantum fluctuations are described by the equation obtained by linearizing the field equation around the classical solution. These properties were proved by Hepp [6] for suitably regular potentials and in finite time intervals. Using a general theory of existence of global solutions and a general scattering theory for the classical equation, we extend these results in two directions: (1) we consider more singular potentials, (2) more important, we prove that for dispersive classical solutions, the ?→0 limit is uniform in time in an appropriate representation of the field operators. As a consequence we obtain the convergence of suitable matrix elements of the wave operators and, if asymptotic completeness holds, of theS-matrix.     

The singularity of Jost functions and potentials

In this paper definite analytical properties of the Jost functions are postulated and their appropriate potentials are sought. It is found that for the case of the simplest singularities of the Jost functions (poles on the positive side of the imaginary axis of the complex planek and poles at the pointk=0 to the orderl), this inversion problem of the potential scattering leads to the solution of a definite system of non-linear differential equations of the second order.Finally, the generalized Noyes-Wong equation is derived and it is proved that this equation (i) is not homogeneous when the potential forr very large decreases exponentially and (ii) is homogeneous when for larger the potential decreases rationally (proportionally tor ^–3.The author expresses his thanks to Dr M. Petrá and M. Blaek M.A. for valuable advice and remarks by which they helped to solve the given problem.     

Poisson finite difference boundary equations for the magnetic potentials

Magnetostatic solutions may be obtained by using Laplace's equation in finite difference form. The boundary equations are of primary importance and even these have been put forward as special forms of Laplace's equation. They can be derived in what appears to be a simpler and more flexible manner by assuming that Poisson's equation applies at the interface for both scalar and vector potentials.     

Complete set of radial Schrödinger equation solutions with a linear λ-dependent potential

The completeness relation is found for the set of Jost solutions of the radial Schrödinger equation with a linear λ-dependent potential in the space of twice continuously differentiable functions defined on the half-axis and satisfying some conditions.     

KdV方程的直接微扰方法

系统地讨论了含修正项的KdV方程的直接微扰方法.从反散射变换所得的不含修正项方程的严格多孤子解出发,导出了线性化算子的零本征值的所有本征函数——平方Jost函数.引入了它们所对应的伴随函数和定义了内积.计算了应有的正交关系,并自然得到单位元的平方Jost函数的展开式.利用广义的Marchenko方程,证明了平方Jost函数的完备性.同时得到展开式中的积分是沿实轴从－∞到∞,但在原点附近将从上方绕过.这不同于过去所得的Cauchy主值积分.为最明确显示这一差别,在单孤子情况下又用平方Jost函数的显式,直接作了验证．同时指出,以前由于取Cauchy主值积分导出的KdV方程所特有的孤子尾,在采用从上方绕过原点的积分时,则事实上并不存在． 关键词：     

Semianalytic solution of the Kramers exit problem for a small ferromagnetic particle

The distribution Q(t) of magnetization reversal times in a small uniaxial particle is computed here directly from Brown's Fokker-Planck equation. Constant applied field and axial symmetry are assumed. The Laplace transform of Q(t) has the form Q(z)=F(1)(z)/F(2)(z) where the regular functions F(i)(z) are defined by a solution of a Volterra integral equation. A separate integral equation is derived for the function dF(2)(z)/dz, and the poles and residues of Q(z) may then be found numerically with arbitary precision.