The structure of semiclassical asymptotic expansions of antisymmetric solutions of the stationary Schrödinger equation

We consider the semiclassical asymptotics of eigenfunctions for the Hamiltonian of a quantum-mechanical system ofN identical fermions withd degrees of freedom without spin interaction. In the one-dimensional case (d=1), examples are known in which the ground antisymmetric state in the semiclassical limit is the product ofN(N−1)/2 two-particle wave functions. We construct a nontrivial generalization of this property ford>1. Translated fromMatematicheskie Zametki, Vol. 67, No. 2, pp. 257–269, February, 2000.     

Construction and implementation of asymptotic expansions for Jacobi–type orthogonal polynomials

We are interested in the asymptotic behavior of orthogonal polynomials of the generalized Jacobi type as their degree n goes to \(\infty \). These are defined on the interval [?1, 1] with weight function

$$w(x)=(1-x)^{\alpha}(1+x)^{\beta}h(x), \quad \alpha,\beta>-1 $$

and h(x) a real, analytic and strictly positive function on [?1, 1]. This information is available in the work of Kuijlaars et al. (Adv. Math. 188, 337–398 2004), where the authors use the Riemann–Hilbert formulation and the Deift–Zhou non-linear steepest descent method. We show that computing higher-order terms can be simplified, leading to their efficient construction. The resulting asymptotic expansions in every region of the complex plane are implemented both symbolically and numerically, and the code is made publicly available. The main advantage of these expansions is that they lead to increasing accuracy for increasing degree of the polynomials, at a computational cost that is actually independent of the degree. In contrast, the typical use of the recurrence relation for orthogonal polynomials in computations leads to a cost that is at least linear in the degree. Furthermore, the expansions may be used to compute Gaussian quadrature rules in \(\mathcal {O}(n)\) operations, rather than \(\mathcal {O}(n^{2})\) based on the recurrence relation.

     

Some asymptotic expansions on hyperfactorial functions and generalized Glaisher–Kinkelin constants

In this paper, by the Bernoulli numbers and the exponential complete Bell polynomials, we establish two general asymptotic expansions on the hyperfactorial functions \(\prod _{k=1}^nk^{k^q}\) and the generalized Glaisher–Kinkelin constants \(A_q\), where the coefficient sequences in the expansions can be determined by recurrences. Moreover, the explicit expressions of the coefficient sequences are presented and some special asymptotic expansions are discussed. It can be found that some well-known or recently published asymptotic expansions on the factorial function n!, the classical hyperfactorial function \(\prod _{k=1}^nk^k\), and the classical Glaisher–Kinkelin constant \(A_1\) are special cases of our results, so that we give a unified approach to dealing with such asymptotic expansions.     

Global asymptotic expansions of the Laguerre polynomials—a Riemann–Hilbert approach

By using the steepest descent method for Riemann–Hilbert problems introduced by Deift–Zhou (Ann Math 137:295–370, 1993), we derive two asymptotic expansions for the scaled Laguerre polynomial $L^{(\alpha)}_n(\nu z)$ as n→∞, where ν=4n+2α+2. One expansion holds uniformly in a right half-plane $\text{Re}\; z\geq \delta_1, 0<\delta_1<1$ , which contains the critical point z=1; the other expansion holds uniformly in a left half-plane $\text{Re}\; z\leq 1-\delta_2, 0<\delta_2<1-\delta_1$ , which contains the other critical point z=0. The two half-planes together cover the entire complex z-plane. The critical points z=1 and z=0 correspond, respectively, to the turning point and the singularity of the differential equation satisfied by $L^{(\alpha)}_n(\nu z)$ .     

New asymptotic expansions related to Somosʼ quadratic recurrence constant

We derive new asymptotic expansions related to Somos? quadratic recurrence constant, in terms of the ordered Bell numbers.     

On asymptotic expansions in nonlinear,singularly perturbed difference equations ∗

The present paper deals with the problem of constructing and proving asymptotic expansions for nonlinear, singularly perturbed difference equations. New methods for the construction of asymptotic expansions are presented and compared with well-known ones. For the proof of their validity, fundamental principles for the treatment of nonlinear singular perturbation problems are applied, based on the concepts of e-stability, formal asymptotic expansions, matching and asymptotic expansions. The results are derived from a general theory of asymptotic expansions of nonlinear operator equations that has been developed recently by the author.     

Superconvergence and asymptotic expansions for linear finite element approximations on criss-cross mesh

In this paper, we discuss the error estimation of the linear finite element solution on criss-cross mesh. Using space orthogonal decomposition techniques, we obtain an asymptotic expansion and superconvergence results of the finite element solution. We first prove that the asymptotic expansion has different forms on the two kinds of nodes and then derive a high accuracy combination formula of the approximate derivatives.     

Fourier–Jacobi expansions in Morrey spaces

In this paper we obtain a characterization of the convergence of the partial sum operator related to Fourier–Jacobi expansions in Morrey spaces.     

The global existence and asymptotic stability of solutions for a reaction–diffusion system

This paper studies the solutions of a reaction–diffusion system with nonlinearities that generalize the Lengyel–Epstein and FitzHugh–Nagumo nonlinearities. Sufficient conditions are derived for the global asymptotic stability of the solutions. Furthermore, we present some numerical examples.     

High order explicit Runge–Kutta Nyström pairs

Explicit Runge–Kutta Nyström pairs provide an efficient way to find numerical solutions to second-order initial value problems when the derivative is cheap to evaluate. We present new optimal pairs of orders ten and twelve from existing families of pairs that are intended for accurate integrations in double precision arithmetic. We also present a summary of numerical comparisons between the new pairs on a set of eight problems which includes realistic models of the Solar System. Our searching for new order twelve pairs shows that there is often not quantitative agreement between the size of the principal error coefficients and the efficiency of the pairs for the tolerances we are interested in. Our numerical comparisons, as well as establishing the efficiency of the new pairs, show that the order ten pairs are more efficient than the order twelve pairs on some problems, even at limiting precision in double precision.     

Post-Newtonian Approximation of the Vlasov–Nordström System

ABSTRACT

We study the Nordström–Vlasov system, which describes the dynamics of a self-gravitating ensemble of collisionless particles in the framework of the Nordström scalar theory of gravitation. If the speed of light c is considered as a parameter, it is known that in the Newtonian limit c → ∞ the Vlasov–Poisson system is obtained. In this paper we determine a higher approximation and establish a pointwise error estimate of order 𝒪(c ^?4). Such an approximation is usually called a 1.5 post-Newtonian approximation.