Scattering by a Cylindrical Trap in the Critical Case

We study a two-dimensional analog of the Helmholtz resonator with walls of finite thickness in the critical case, for which there exists a frequency which is simultaneously the limit of poles generated both by the bounded component of the resonator and by a narrow communication channel. Under the assumption that the limit frequency is a simple frequency for the bounded component, by using the method of matched asymptotic expansions, we construct asymptotics for the two sequences of poles converging to this frequency. We obtain explicit formulas for the leading terms of the asymptotics of poles and for the solution of the scattering problem.     

How poles of orthogonal rational functions affect their Christoffel functions

We show that even a relatively small number of poles of a sequence of orthogonal rational functions approaching the interval of orthogonality, can prevent their Christoffel functions from having the expected asymptotics. We also establish a sufficient condition on the rate for such asymptotics, provided the rate of approach of the poles is sufficiently slow. This provides a supplement to recent results of the authors where poles were assumed to stay away from the interval of orthogonality.     

On the spectral instability of the Sturm-Liouville operator with a complex potential

Using the Sturm-Liouville operator with a complex potential as an example, we analyze the spectral instability effect for operators that are far from being self-adjoint. We show that the addition of an arbitrarily small compactly supported function with an arbitrarily small support to the potential can substantially change the asymptotics of the spectrum. This fact justifies, in a sense, the necessity of well-known sufficient conditions for the potential under which the spectrum of the operator is localized around some ray. For an operator with a logarithmic growth, we construct a perturbation that preserves the asymptotics of the spectrum but has infinitely many poles inside the main sector.     

Correlation between pole location and asymptotic behavior for Painlevé I solutions

We extend the technique of asymptotic series matching to exponential asymptotics expansions (transseries) and show by using asymptotic information that the extension provides a method of finding singularities of solutions of nonlinear differential equations. This transasymptotic matching method is applied to Painlevé's first equation, P1. The solutions of P1 that are bounded in some direction towards infinity can be expressed as series of functions obtained by generalized Borel summation of formal transseries solutions; the series converge in a neighborhood of infinity. We prove (under certain restrictions) that the boundary of the region of convergence contains actual poles of the associated solution. As a consequence, the position of these exterior poles is derived from asymptotic data. In particular, we prove that the location of the outermost pole x_p(C) on ℝ⁺ of a solution is monotonic in a parameter C describing its asymptotics on anti‐Stokes lines and obtain rigorous bounds for x_p(C). We also derive the behavior of x_p(C) for large C ∈ ℂ. The appendix gives a detailed classical proof that the only singularities of solutions of P1 are poles. © 1999 John Wiley & Sons, Inc.     

Imbalance in Random Digital Trees

The imbalance factor of the nodes containing keys in a trie (a sort of digital trees) is investigated. Accurate asymptotics for the mean are derived for a randomly chosen key in the trie via poissonization and the Mellin transform, and the inverse of the two operations. It is also shown from an analysis of the moving poles of the Mellin transform of the poissonized moment generating function that the imbalance factor (under appropriate centering and scaling) follows a Gaussian limit law.      

Resonances for Dirac operators on the half-line

We consider the 1D Dirac operator on the half-line with compactly supported potentials. We study resonances as the poles of scattering matrix or equivalently as the zeros of modified Fredholm determinant. We obtain the following properties of the resonances: (1) asymptotics of counting function, (2) estimates on the resonances and the forbidden domain.     

The Cauchy problem for a singularly perturbed integro-differential Fredholm equation

An initial problem is considered for an ordinary singularly perturbed integro-differential equation with a nonlinear integral Fredholm operator. The case when the reduced equation has a smooth solution is investigated, and the solution to the reduced equation with a corner point is analyzed. The asymptotics of the solution to the Cauchy problem is constructed by the method of boundary functions. The asymptotics is validated by the asymptotic method of differential inequalities developed for a new class of problems.     

Equilibrium of vector potentials and uniformization of the algebraic curves of genus 0

We consider equilibrium problems for the logarithmic vector potential related to the asymptotics of the Hermite-Padé approximants. Solutions of such problems can be expressed by means of algebraic functions. The goal of this paper is to describe a procedure for determining the algebraic equation for this function in the case when the genus of this algebraic function is equal zero. Using the coefficients of the equation we compute the extremal cuts of the Riemann surfaces. These cuts are attractive sets for the poles of the Hermite-Padé approximants. We demonstrate the method by an example of the equilibrium problem related to a special system that is called the Angelesco system.     

Resonances for 1D massless Dirac operators

We consider the 1D massless Dirac operator on the real line with compactly supported potentials. We study resonances as the poles of scattering matrix or equivalently as the zeros of modified Fredholm determinant. We obtain the following properties of the resonances: (1) asymptotics of counting function, (2) estimates on the resonances and the forbidden domain, (3) the trace formula in terms of resonances.     

The inverse laplace transform of some analytic functions with an application to the eternal solutions of the Boltzmann equation

When the Laplace transform F(p) of a function f(x) has no poles but is singular only on the real negative semiaxis because of a cut required to make it single-valued, the inverse transform f(x) can easily be computed by means of the integral of a real-valued function. This result is applied to the calculation of a class of exact eternal solutions of the Boltzmann equation, recently found by the authors. The new approach makes it easier to prove that these solutions are positive, as well as to study their asymptotics.     

Poles of tritronquée solution to the Painlevé I equation and cubic anharmonic oscillator

The distribution of poles of zero-parameter solution to Painlevé I, specified by P. Boutroux as intégrale tritronquée, is studied in the complex plane. This solution has regular asymptotics ? $ \sqrt {{z \mathord{\left/ {\vphantom {z 6}} \right. \kern-0em} 6}} $ + O(1) as z → ∞, | arg z| < 4π/5. At the sector | arg z| > 4π/5 it is a meromorphic function with regular asymptotic distribution of poles at infinity. This fact together with numeric simulations for |z| < const allowed B. Dubrovin to make a conjecture that all poles of the intégrale tritronquée belong to this sector. As a step to prove this conjecture, we study the Riemann-Hilbert (RH) problem related to the specified solution of the Painlevé I equation. It is “undressed” to a similar RH problem for the Schrödinger equation with cubic potential. The latter determines all coordinates of poles for the intégrale tritronquée via a Bohr-Sommerfeld quantization conditions.     

Global Asymptotics of Krawtchouk Polynomials — a Riemann-Hilbert Approach

In this paper, we study the asymptotics of the Krawtchouk polynomials KnN(z;p,q) as the degree n becomes large. Asymptotic expansions are obtained when the ratio of the parameters n/N tends to a limit c ∈ (0, 1) as n →∞. The results are globally valid in one or two regions in the complex z-plane depending on the values of c and p;in particular, they are valid in regions containing the interval on which these polynomials are orthogonal. Our method is based on the Riemann-Hilbert approach introduced by Deift and Zhou.     

Spherical convolution operators in spaces of variable Hölder order

In this paper, we study the images of operators of the type of spherical potential of complex order and of spherical convolutions with kernels depending on the inner product and having a spherical harmonic multiplier with a given asymptotics at infinity. Using theorems on the action of these operators in Hölder-variable spaces, we construct isomorphisms of these spaces. In Hölder spaces of variable order, we study the action of spherical potentials with singularities at the poles of the sphere. Using stereographic projection, we obtain similar isomorphisms of Hö lder-variable spaces with respect to n-dimensional Euclidean space (in the case of its one-point compactification) with some power weights.     

The approximate spectral projection method

In this paper we develop a general method for investigating the spectral asymptotics for various differential and pseudo-differential operators and their boundary value problems, and consider many of the problems posed when this method is applied to mathematical physics and mechanics. Among these problems are the Schrödinger operator with growing, decreasing and degenerating potential, the Dirac operator with decreasing potential, the quasi-classical spectral asymptotics for Schrödinger and Dirac operators, the linearized Navier-Stokes equation, the Maxwell system, the system of reactor kinetics, the eigenfrequency problems of shell theory, and so on. The method allows us to compute the principal term of the spectral asymptotics (and, in the case of Douglis-Nirenberg elliptic operators, also their following terms) with the remainder estimate close to that for the sharp remainder.     

Asymptotics of Orthogonal Polynomials: Some Old, Some New, Some Identities

We briefly review some asymptotics of orthonormal polynomials. Then we derive the Bernstein–Szeg, the Riemann–Hilbert (or Fokas–Its–Kitaev), and Rakhmanov projection identities for orthogonal polynomials and attempt a comparison of their applications in asymptotics.     

Random fractal strings: Their zeta functions, complex dimensions and spectral asymptotics

In this paper a string is a sequence of positive non-increasing real numbers which sums to one. For our purposes a fractal string is a string formed from the lengths of removed sub-intervals created by a recursive decomposition of the unit interval. By using the so-called complex dimensions of the string, the poles of an associated zeta function, it is possible to obtain detailed information about the behaviour of the asymptotic properties of the string. We consider random versions of fractal strings. We show that by using a random recursive self-similar construction, it is possible to obtain similar results to those for deterministic self-similar strings. In the case of strings generated by the excursions of stable subordinators, we show that the complex dimensions can only lie on the real line. The results allow us to discuss the geometric and spectral asymptotics of one-dimensional domains with random fractal boundary.

     

Spectral properties of a Sturm-Liouville type differential operator with a retarding argument

Spectral properties of a differential operator of Sturm-Liouville type are studied in the case of retarding argument with different boundary conditions. The asymptotics of solutions to the corresponding differential equation is studied in the case of a summable potential. An asymptotics of eigenvalues and an asymptotics of eigenfunctions of the differential operator are calculated for each considered case.     

On the spectrum of a vector Schrödinger operator

We consider the asymptotics of the spectrum of a Sturm-Liouville operator acting on a space of vector functions and show that this asymptotics is affected by “rotation” of eigenvectors of the potential. A similar result is obtained for a vector Schrödinger operator.     

On existence of quadratic differentials with poles of high orders

Conditions of existence of a quadratic differential which has poles of high orders at the points c_k, k=1,...,p, and is the limit of a sequence of quadratic differentials of a special type are established. The quadratic differential mentioned has no poles of order greater than 2 and has poles of order 2 at the points situated in a suitable uniform way on the circles |z-c_k|=ɛ_k,ɛ_k→0, k=1, ..., p. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 212, 1994, pp. 129–138 Translated by G. V. Kuz'mina     

Asymptotics of solutions to the time-dependent Schrödinger equation with a small Planck constant

A regularized asymptotics of the solution to the time-dependent Schrödinger equation in which the spatial derivative is multiplied by a small Planck constant is constructed. It is shown that the asymptotics of the solution contains a rapidly oscillating boundary layer function.