Finite-Rank Multivariate-Basis Expansions of the Resolvent Operator as a Means of Solving the Multivariable Lippmann–Schwinger Equation for Two-Particle Scattering

Finite-rank expansions of the two-body resolvent operator are explored as a tool for calculating the full three-dimensional two-body T-matrix without invoking the partial-wave decomposition. The separable expansions of the full resolvent that follow from finite-rank approximations of the free resolvent are employed in the Low equation to calculate the T-matrix elements. The finite-rank expansions of the free resolvent are generated via projections onto certain finite-dimensional approximation subspaces. Types of operator approximations considered include one-sided projections (right or left projections), tensor-product (or outer) projection and inner projection. Boolean combination of projections is explored as a means of going beyond tensor-product projection. Two types of multivariate basis functions are employed to construct the finite-dimensional approximation spaces and their projectors: (i) Tensor-product bases built from univariate local piecewise polynomials, and (ii) multivariate radial functions. Various combinations of approximation schemes and expansion bases are applied to the nucleon-nucleon scattering employing a model two-nucleon potential. The inner-projection approximation to the free resolvent is found to exhibit the best convergence with respect to the basis size. Our calculations indicate that radial function bases are very promising in the context of multivariable integral equations.     

Higher Order Corrections to the Asymptotic Perturbative Solution of a Schwinger–Dyson Equation

Building on our previous works on perturbative solutions to a Schwinger–Dyson for the massless Wess–Zumino model, we show how to compute 1/n corrections to its asymptotic behavior. The coefficients are analytically determined through a sum on all the poles of the Mellin transform of the one-loop diagram. We present results up to the fourth order in 1/n as well as a comparison with numerical results. Unexpected cancellations of zetas are observed in the solution, so that no even zetas appear and the weight of the coefficients is lower than expected, which suggests the existence of more structure in the theory.     

An Efficient Method for the Solution of Schwinger–Dyson Equations for Propagators

Efficient computation methods are devised for the perturbative solution of Schwinger–Dyson equations for propagators. I show how a simple computation allows to obtain the dominant contribution in the sum of many parts of previous computations. This allows for an easy study of the asymptotic behavior of the perturbative series. In the cases of the four-dimensional supersymmetric Wess–Zumino model and the f₆³{\phi_6^3} complex scalar field, the singularities of the Borel transform for both positive and negative values of the parameter are obtained and compared.     

Spinodal Decomposition¶for the Cahn–Hilliard–Cook Equation

This paper gives theoretical results on spinodal decomposition for the stochastic Cahn–Hilliard–Cook equation, which is a Cahn–Hilliard equation perturbed by additive stochastic noise. We prove that most realizations of the solution which start at a homogeneous state in the spinodal interval exhibit phase separation, leading to the formation of complex patterns of a characteristic size. In more detail, our results can be summarized as follows. The Cahn–Hilliard–Cook equation depends on a small positive parameter ε which models atomic scale interaction length. We quantify the behavior of solutions as ε→ 0. Specifically, we show that for the solution starting at a homogeneous state the probability of staying near a finite-dimensional subspace ?_ε is high as long as the solution stays within distance r _ε=O(ε ^R ) of the homogeneous state. The subspace ?_ε is an affine space corresponding to the highly unstable directions for the linearized deterministic equation. The exponent R depends on both the strength and the regularity of the noise. Received: 2 May 2000 / Accepted: 8 July 2001     

Analytical Solution of the Klein–Fock–Gordon Equation for the Rosen–Morse Potential

Russian Physics Journal - With the help of a successful scheme for overcoming difficulties arising for l ≠ 0 in the centrifugal part of the Rosen–Morse potential with bound states, a...     

Solution of the Dirac Equation on the Bertotti–Robinson Metric

It is shown that each component of the Dirac field satisfies a decoupled equation, which admits separable solutions, when the background spacetime is the Bertotti–Robinson metric, which is a solution of the Einstein vacuum field equations with a cosmological constant. Furthermore, the seperated functions appearing in the solutions are shown to obey identities of the Teukolsky–Starobinsky type and the separable solutions are shown to be eigenfunctions of a certain differential operator.     

Analytical Solution of the Duffin–Kemmer–Petiau Equation for the Sum of Manning–Rosen and Yukawa Class Potentials

Russian Physics Journal - This paper presents an analytical bound-state solution to the Duffin–Kemmer–Petiau equation for the new putative combined Manning–Rosen and Yukawa class...     

Solution of the Logunov–Tavkhelidze Equation for the Three-Dimensional Oscillator Potential in the Relativistic Configuration Representation

Russian Physics Journal - Approximate analytical and numerical solutions of the three-dimensional Logunov–Tavkhelidze equation are found for the spherically symmetric case. Solutions are...     

Analytical Solution of the Klein–Fock–Gordon Equation for the Modified Klingberg Plus Ring Shaped Potentials

Russian Physics Journal - In this study, the bound state solution of the modified Klein–Fock–Gordon equation is found for new combined Klingberg and ring shaped potentials. Analytical...     

Low-T Asymptotic Expansion of the Solution to the Yang–Yang Equation

We prove that the unique solution to the Yang–Yang equation arising in the context of the thermodynamics of the so-called non-linear Schrödinger model admits a low-temperature expansion to all orders. Our approach provides a rigorous justification, for a certain class of non-linear integral equations, of the low-temperature asymptotic expansions that were argued previously in various works related to the low-temperature behaviour of integrable models.     

Adomian Decomposition Method for the One-dimensional Nonlocal Fisher–Kolmogorov–Petrovsky–Piskunov Equation

Russian Physics Journal - The Adomian decomposition method is applied to construct an approximate solution of the generalized one-dimensional Fisher–Kolmogorov–Petrovsky–Piskunov...     

Solution of Gauss-Codazzi Equation with Applications in the Tzitzeica Equation

The surface in R3 associated with the Tzitzeica equation ＆ considered. By curvature coordinate transformation and surface imbedding, the Gauss-Codazzi equation is presented. Resorting to the solutions of the Gauss-Codazzi equation, the solution of the Tzitzeica equation ＆ obtained under a restrictive condition.     

New Formulas for the Eigenfunctions of the Two-Particle Difference Calogero–Moser System

We present a new proof of the integrability of the DDPT-I equation. The DDPT-I equation represents a functional-difference deformation of the well-known Darboux–Pöschl–Teller equation. The proof is based on some formula for special Casorati determinants established in the paper. This formula provides some new representation for the DDPT-I potentials and for the general solution for the DDPT-I equation. It allows also a very easy computation of the action of the difference KdV flow on the DDPT-I initial data. In other words we obtain the new formulas for the eigenfunctions of the Hamiltonians of the two-particle difference BC ₁ Calogero–Moser system also known as quantum relativistic Calogero–Moser, (QRCM), system.     

Estimates for the KdV-Limit of the Camassa–Holm Equation

In a number of scaling limits, we prove estimates relating the solutions of the Camassa–Holm equation to the solutions of the associated KdV equation. As a consequence, suitable solutions of the water wave problem and solutions of the Camassa–Holm equation stay close together for long times.     

A Harmonic Solution for the Hyperbolic Heat Conduction Equation and Its Relationship to the Guyer–Krumhansl Equation

A particular solution of the hyperbolic heat-conduction equation was constructed using the method of operators. The evolution of a harmonic solution is studied, which simulates the propagation of electric signals in long wire transmission lines. The structures of the solutions of the telegraph equation and of the Guyer–Krumhansl equation are compared. The influence of the phonon heat-transfer mechanism in the environment is considered from the point of view of heat conductivity. The fulfillment of the maximum principle for the obtained solutions is considered. The frequency dependences of heat conductivity in the telegraph equation and in an equation of the Guyer–Krumhansl type are studied and compared with each other. The influence of the Knudsen number on heat conductivity in the model of thin films is studied.     

Any J-State Solution of the Duffin–Kemmer–Petiau Equation for a Vector Deformed Woods–Saxon Potential

By using the Pekeris approximation, the Duffin–Kemmer–Petiau (DKP) equation is investigated for a vector deformed Woods–Saxon (dWS) potential. The parametric Nikiforov–Uvarov (NU) method is used in calculations. The approximate energy eigenvalue equation and the corresponding wave function spinor components are calculated for any total angular momentum J in closed form. The exact energy equation and wave function spinor components are also given for the J?=?0 case. We use a set of parameter values to obtain the numerical values for the energy states with various values of quantum levels (n, J) and potential’s deformation constant q and width R.     

Solving the Dyson–Schwinger Equation at Zero and Finite Temperatures

The properties of the solutions of the truncated Dyson–Schwinger equation for the quark propagator at finite temperatures within the rainbow-ladder approximation are analysed in some detail.