Exact series expansions, recurrence relations, properties and integrals of the generalized exponential integral functions

Generalized exponential integral functions (GEIF) are encountered in multi-dimensional thermal radiative transfer problems in the integral equation kernels. Several series expansions for the first-order generalized exponential integral function, along with a series expansion for the general nth order GEIF, are derived. The convergence issues of these series expansions are investigated numerically as well as theoretically, and a recurrence relation which does not require derivatives of the GEIF is developed. The exact series expansions of the two dimensional cylindrical and/or two-dimensional planar integral kernels as well as their spatial moments have been explicitly derived and compared with numerical values.     

Symmetry groups and Gauss kernels of Schrödinger equations

The relationship between symmetries and Gauss kernels for the Schrödinger equation iu_t=u_xx+f(x)u is established. It is shown that if the Lie point symmetries of the equation are nontrivial, a classical integral transformations of the Gauss kernels can be obtained. Then the Gauss kernels of Schrödinger equations are derived by inverting the integral transformations. Furthermore, the relationship between Gauss kernels for two equations related by an equivalence transformation is identified.     

Coupled channel T-operator equations with connected kernels: (I). Three-body problem with pairwise interactions

Previous work on T-operator coupled equations for two-channel systems is generalized and applied to the problem of three bodies interacting via pair potentials. Sets of coupled, integral equations for the two-body arrangement channel T-operators are derived using a channel coupling array W, and the connectedness properties of the kernels of these equations are discussed. It is shown that either disconnected or connected (iterated) kernels can be obtained by various choices of W. One particular realization of the coupled equations is seen to be similar but not identical to the Lovelace form of the Faddeev equations. Since the matrix form of the coupled equations is similar to the one-body Lippmann-Schwinger equation, the introduction of Møller wave operators is straightforward, and these are used to derive coupled integral equations for the channel state vectors.     

Application of Synthetic Kernel (SKN) method to participating linearly anisotropically scattering solid spherical medium

The Synthetic Kernel (SK_N) method is applied to a solid spherical absorbing, emitting and linearly anisotropically scattering homogeneous and inhomogeneous medium. The SK_N method relies on approximating the integral transfer kernels by Synthetic Kernels. The radiative integral transfer equation is then reducible to a set of coupled second-order differential equations. The SK_N method, which uses Gauss quadratures, is tested against integral equation and the discrete-ordinates S₈ solutions for various optical radius and scattering albedo variations.     

Some general properties of the nonlinear collision integral in the Boltzmann equation

A new method for computing matrix elements of the collision integral in the Boltzmann equation makes it possible to consider many problems of the kinetic theory of gases in a new way. Nonlinear kernels of the collision integral are studied and similarity relations, which simplify significantly the problem of constructing of such kernels, are proved.     

Anisotropic scattering in inhomogeneous media—1. A new representation formula for the solution of the auxiliary integral equation for the source function

A new representation formula for the solution of the auxiliary integral equation for the source function in inhomogeneous, anisotropically scattering media is presented. It involves two new functions Φ and ψ of two variables instead of the original five variables. This generalizes earlier results of Kagiwada et al. (1969) and Sobolev (1972) applicable to homogeneous atmospheres. The corresponding Bellman-Krein formula for the resolvent kernel is also derived. The present representation for the solution of Fredholm integral equations of second kind with unsymmetric kernels provides a new approach to radiative transfer in anisotropic inhomogeneous media.     

On the calculation of memory functions for nonlinear irreversible processes

A linear inhomogeneous Volterra integral equation for the memory functions of the nonlinear theory of irreversible processes is derived. No projection operator is involved. Recursion relations for the solution of the integral equation are given. If the kernels are expanded about equilibrium, the exact linear and nonlinear Onsager coefficients are obtained.     

Calculation of the linear kernel of the collision integral for the hard-sphere potential

The expansion of a distribution function in spherical harmonics transforms the Boltzmann equation into a system of integro-differential equations with kernels depending only of the magnitudes of velocities. The kernels can be expressed in terms of the sums including the matrix elements (MEs) of the collision integral. The kernels are constructed using new results of ME calculations; analysis of errors is carried out with the help of analytic expressions for kernels, which were derived by Hilbert and Hecke for the hard-sphere model. The concept of generalized matrix elements is introduced and their asymptotic representation is constructed for large values of indices. Analytic expressions for the contribution from MEs with large indices to the kernels are constructed. The high accuracy of the construction of a kernel using MEs is demonstrated.     

Singular Values of Products of Ginibre Random Matrices,Multiple Orthogonal Polynomials and Hard Edge Scaling Limits

Akemann, Ipsen and Kieburg recently showed that the squared singular values of products of M rectangular random matrices with independent complex Gaussian entries are distributed according to a determinantal point process with a correlation kernel that can be expressed in terms of Meijer G-functions. We show that this point process can be interpreted as a multiple orthogonal polynomial ensemble. We give integral representations for the relevant multiple orthogonal polynomials and a new double contour integral for the correlation kernel, which allows us to find its scaling limits at the origin (hard edge). The limiting kernels generalize the classical Bessel kernels. For M = 2 they coincide with the scaling limits found by Bertola, Gekhtman, and Szmigielski in the Cauchy–Laguerre two-matrix model, which indicates that these kernels represent a new universality class in random matrix theory.     

Recurrence procedure for calculating kernels of the nonlinear collision integral of the Boltzmann equation

A recurrence procedure for a sequential construction of kernels \(G_{{l_1},{l_2}}^l\) (c, c₁, c₂) appearing upon the expansion of a nonlinear collision integral of the Boltzmann equation in spherical harmonics is developed. The starting kernel for this procedure is kernel G_0,0⁰ (c, c₁, c₂) of the collision integral for the distribution function isotropic with respect to the velocities. Using the recurrence procedure, a set of kernels \(G_{{l_1},{l_2}}^{ + l}\) (c, c₁, c₂) for a gas consisting of hard spheres and Maxwellian molecules is constructed. It is shown that the resultant kernels exhibit similarity and symmetry properties and satisfy the relations following from the conservation laws.     

Path integral solution of linear second order partial differential equations I: the general construction

A path integral is presented that solves a general class of linear second order partial differential equations with Dirichlet/Neumann boundary conditions. Elementary kernels are constructed for both Dirichlet and Neumann boundary conditions. The general solution can be specialized to solve elliptic, parabolic, and hyperbolic partial differential equations with boundary conditions. This extends the well-known path integral solution of the Schrödinger/diffusion equation in unbounded space. The construction is based on a framework for functional integration introduced by Cartier/DeWitt-Morette.     

A variational principle for the Landau equation

The Landau equation describing the collective modes of an infinite many-fermion system is cast into a form which displays explicitly its structure as a homogeneous Fredholm equation of the second kind with non-symmetric kernel. A variational principle appropriate to this equation is constructed by noting the analogy to the integral form of the Schrödinger equation. The Landau equation is also re-expressed in terms of two coupled equations with symmetric kernels.     

Non-Commutative Painlevé Equations and Hermite-Type Matrix Orthogonal Polynomials

We study double integral representations of Christoffel–Darboux kernels associated with two examples of Hermite-type matrix orthogonal polynomials. We show that the Fredholm determinants connected with these kernels are related through the Its–Izergin–Korepin–Slavnov (IIKS) theory with a certain Riemann-Hilbert problem. Using this Riemann-Hilbert problem we obtain a Lax pair whose compatibility conditions lead to a non-commutative version of the Painlevé IV differential equation for each family.     

Exact four-body calculation of the 4ΛHe-4ΛH binding energy difference

The coupled, two-variable integral equations that determine the ⁴_ΛHe and ⁴_ΛH bound states, when the NN and ΛN interactions are represented by separable potentials, are derived from the Schrödinger equation. The integral equations are solved numerically for simple s-wave potentials and for tensor potentials in the truncated t-matrix approximation without resort to separable expansion of the kernels. The Λ-separation energy difference ΔB_Λ resulting from the genuine four-body model is shown to be approximately twice as large as that coming from an “effective two-body” model calculation, when identical central potentials are used. The four-body model estimate of ΔB_Λ made with tensor forces is consistent with the experimental value, indicating that charge symmetry breaking implied by the low energy Λ N scattering parameters is compatible with that suggested by the known binding energy difference in the A = 4 hypernuclear isodoublet.     

Bethe-Salpeter equation and current conservation

Current conservation in the form of the Ward identity between the electromagnetic vertex and the propagator implies that the energy dependence of the BS kernel is restricted and that the propagator cannot be chosen independently from the kernel. It is rather determined from the BS kernel in terms of an integral equation. Convolution type, energy-independent kernels are compatible with current conservation. We study the propagator and form factor resulting from smooth kernels.     

Partial wave relations from fixed-u dispersion relations

Partial wave relations are derived form fixed-u disperation relations for meson-baryon and meson-meson scattering. General expressions for the integral kernels are given and some problems pertaining to the application of the relations are discussed.     

连续谱Faddeev积分方程的新解法

在动量空间中具有定域势的Faddeev方程是二维积分方程,在破裂过程和三体散射一类的连续谱情况下,方程的积分核是奇异的。本文根据奇异积分方程一般理论提出一种求解二维方程的数值方法。实践证明数值解是收敛的,全运动学微分截面的计算值与实验数据十分符合。     

An improved technique on the stochastic functional approach for randomly rough surface scattering –Numerical-analytical Wiener analysis

This paper proposes an improved technique on the stochastic functional approach for randomly rough surface scattering. Its first application is made on a TE plane wave scattering from a Gaussian random surface having perfect conductivity with infinite extent. The random wavefield becomes a ‘stochastic Floquet form’ represented by a Wiener–Hermite expansion with unknown expansion coefficients called Wiener kernels. From the effective boundary condition as a model of the random surface, a series of integral equations determining the Wiener kernels are obtained. By applying a quadrature method to the first three order hierarchical equations, a matrix equation is derived. By solving that matrix equation, the exact Wiener kernels up to second order are numerically obtained. Then the incoherent scattering cross-section and the optical theorem are calculated. A prediction is that the optical theorem always holds, which is derived from previous work is confirmed in a numerical sense. It is then concluded that the improved technique is useful.     

Functional integral equation for the complete effective action in quantum field theory

Based on a methodological analysis of the effective action approach, certain conceptual foundations of quantum field theory are reconsidered to establish a quest for an equation for the effective action. Relying on the functional integral formulation of Lagrangian quantum field theory, we propose a functional integral equation for the complete effective action which can be understood as a certain fixed-point condition. This is motivated by a critical attitude toward the distinction, artificial from an experimental point of view, between classical and effective action. While for free field theories nothing new is accomplished, for interacting theories the concept differs from the established paradigm. The analysis of this new concept concentrates on gauge field theories, treating QED as the prototype model. An approximative approach to the functional integral equation for the complete effective action of QED is exploited to obtain certain nonperturbative information about the quadratic kernels of the action. As a particular application the approximate calculation of the QED coupling constant α is explicitly studied. It is understood as one of the characteristics of a fixed point given as a solution of the functional integral equation proposed. Finally, within the present approach the vacuum energy problem is considered, as are possible implications for the concept of induced gravity.