New approach to the solution of the scattering of spinless particles by central potentials

The explicit forms of the regular solutions, of the Jost solutions and functions for the radial Schrödinger equation, which describe the scattering of spinless particles by central potentials, are found. The regular solutions are derived from the iterative solution of the integral equation which their suitably modified Laplace transforms fulfil. Two general classes of potentials are used each of them being expressed by the corresponding inverse Laplace transform. As such forms of the regular solutions are related to those of the Jost solutions, the Jost solutions (along with the Jost functions) are written directly. The regions of the complex angular momenta and wave numbers, to which they can be analytically continued, are specified. Some testing relations are also derived.Dedicated to Academician Václav Votruba on the occasion of his seventieth birthday.     

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.     

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.     

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.     

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.     

An integral transform of Green’s function, off-shell Jost solution and T-matrix for Coulomb-Yamaguchi potential in coordinate representation

By exploiting the theory of ordinary differential equations together with certain properties of higher transcendental functions, a useful analytical expression for the integral transform of the Green’s function for motion in Coulomb-Yamaguchi potential is derived via the r-space approach. This integral transform is applied to construct an analytical expression for off-shell Jost solution in its ‘maximal reduced form’ involving confluent and Gaussian hypergeometric functions. Corresponding Jost functions automatically follow from this solution. Finally, as another application of the off-shell Jost solution, the off-shell T-matrix is calculated by using a modified relation between off-shell physical wave function and T-matrix which does not involve the potential explicitly, thereby avoiding certain difficult integrals, and expressed it in terms of rational functions and simple hypergeometric functions which is in exact agreement with the results given previously by other authors.      

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：     

Green's Function Method for Perturbed sine-Gordon Equation

The usual Green's function method is introduced to show the completeness of the squared Jost solutions of the multi-soliton case in this work. Since sine-Gordon equation contains second derivative in time, the known procedure with generalized Marchenko equation to show completeness relation of the eigenfunctions of linearized equation is unavailable. And the explicit expressions of Jost solutions are not necessary here. Thus a general method of direct perturbation method for the perturbed sine-Gordon equation is developed.     

A Characterisation of Locality in Momentum Space

It is proved that a Poincaré invariant Wightman function which fulfils the spectral property and can be defined at sharp times, is local if and only if the integration over both the energy variables of a commutator in momentum space is a polynomial in the momentum conjugated to the spacial difference variable of the commutator with distributional coefficients depending on the remaining energy and momentum variables. Using this characterisation of locality in momentum space, the locality of a sequence of Wightman functions with nontrivial scattering behavior (associated to some quantum field in indefinite metric) can be proved by explicit calculations. We compare the above characterisation of locality with the classical integral representation method of Jost, Lehmann and Dyson.     

A relationship between Gel'fand-Levitan and Marchenko kernels

A new integral equation which relates the output kernels of the Gel'fand-Levitan and Marchenko inverse scattering equations in a continuous range of their variables is specified. Structural details of this integral equation are studied when theS-matrix is a rational function, and the output kernels are separable in terms of Bessel, Hankel and Jost solutions.     

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.     

On the analytic properties of nonrelativistic two-body off-shell amplitudes II. Off-shell generalizations of the jost formalism

The connections between several studies of the off-shell amplitude, based on apparently different criteria, are first clarified and expressed through more concise operator notation. In all cases the resolution of the underlying dynamical relations is reduced to a two-step procedure. Under conditions which are discussed, the latter implies only Volterra-like integral equations followed by simple quadratures and algebraic operations. Then it is shown that the off-shell generalizations of the Jost formalism which are defined by such approaches can as well be introduced without reference to any explicit dynamical framework. Examples are given specifying in such model independent ways the main properties of the associated Jost-type functions. The relative interest of different representations built with these functions is also examined, and on this occasion new three-separable-term approximations are proposed. Finally, it is noticed that the present work, together with I and III, can provide a useful guideline for introducing similar formalisms in other potential-like frameworks, and further comments on such possible extensions are added.     

Axial Green’s function method for steady Stokes flow in geometrically complex domains

Axial Green’s function method (AGM) is developed for the simulation of Stokes flow in geometrically complex solution domains. The AGM formulation systematically decomposes the multidimensional steady-state Stokes equations into 1D forms. The representation formula for the solution variables can then be derived using the 1D Green’s functions only, from which a system of 1D integral equations is obtained. Furthermore, the explicit representation formula for the pressure variable enable the unique AGM approach to facilitating the stabilization issue caused by the saddle structure between velocity and pressure. The convergence of numerical solutions, the simple axial discretization of complex solution domains, and the nature of integral schemes are demonstrated through a variety of numerical examples.     

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.     

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.     

Solutions of singular integral equations from gas dynamics and plasma physics

In this paper we give the explicit form of the solutions of the singular integral equations associated with some models of gas dynamics and plasma physics which are extensively investigated in the existing literature. In particular, we deal with equations on infinite and semi-infinite contours, where the data are assumed to be meromorphic functions. In this context we rederive some published results and present some new results which show how our method can be successfully used to obtain the explicit form of the solutions in much more general cases than those found in the literature.     

Three-dimensional analytical solution for a transversely isotropic functionally graded piezoelectric circular plate subject to a uniform electric potential difference

This paper studies the problem of a functionally graded piezoelectric circular plate subjected to a uniform electric potential difference between the upper and lower surfaces. By assuming the generalized displacements in appropriate forms, five differential equations governing the generalized displacement functions are derived from the equilibrium equations. These displacement functions are then obtained in an explicit form, which still involve four undetermined integral constants, through a step-by-step integration which properly incorporates the boundary conditions at the upper and lower surfaces. The boundary conditions at the cylindrical surface are then used to determine the integral constants. Hence, three-dimensional analytical solutions for electrically loaded functionally graded piezoelectric circular plates with free or simply-supported edge are completely determined. These solutions can account for an arbitrary material variation along the thickness, and thus can be readily degenerated into those for a homogenous plate. A numerical example is finally given to show the validity of the analysis, and the effect of material inhomogeneity on the elastic and electric fields is discussed. Supported by the National Natural Science Foundation of China (Grant Nos. 10472102 and 10432030) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20060335107)     

A Reduced System of Linear Algebraic Equations for Giving Soliton Solutions of the Generalized ZS/AKNS Systems

A simple method for finding soliton solutions of the generaked ZS/AKNS systems whose Lax pairs are matrices with high orders is considered. An explicit expreesion of transformation between the Jost solution relating to the (n-1)-soliton solution and that relating to the n-soliton solution is found. A reduced system of N algebraic equations for giving N soliton solutions is deduced, it has an identical form no matter how high the order of matrices of the Lax pain is.     

Three Types of Expression in Dark-Soliton Perturbation Theory Based on Squared Jost Solutions

Three types of expression in the dark-soliton perturbation theory based on squared Jost solutions are invesgigaged in ghis paper. It is shown that there are three formally different results about the effects of perturbagion on a dark soliton, and it is proved by means of a transformation between two integral variables that they are essentially equivalent.     

Application of the Riemann method to the Bethe-Salpeter equation

The Bethe-Salpeter equation describing the interaction of two scalar particles via the exchange of a third scalar particle with mass 0 is in configuration space a hyperbolic partial differential equation of fourth order which will be studied with the help of the Riemann method. This method yields two Volterra equations the solutions of which are special solutions of the Bethe-Salpeter equation. The wave function is a superposition of the special solutions. For the coefficients one gets a system of two integral equations. The Fredholm determinant of the system is the generalization of the nonrelativistic Jost function.