Non-self-adjoint sturm-liouville operators with matrix potentials

We obtain asymptotic formulas for non-self-adjoint operators generated by the Sturm-Liouville system and quasiperiodic boundary conditions. Using these asymptotic formulas, we obtain conditions on the potential for which the system of root vectors of the operator under consideration forms a Riesz basis.     

Asymptotic behavior of the solutions and nodal points of Sturm-Liouville differential expressions

We obtain asymptotic formulas for the values of second-order differential operators with the coefficient of bounded variation depending on the spectral parameter. We give a counterexample showing that the requirement of boundedness for the variation is essential for the preservation of the error of the so-obtained asymptotic formulas.     

On the basis property of the root functions of differential operators with matrix coefficients

We obtain asymptotic formulas for eigenvalues and eigenfunctions of the operator generated by a system of ordinary differential equations with summable coefficients and periodic or antiperiodic boundary conditions. Then using these asymptotic formulas, we find necessary and sufficient conditions on the coefficients for which the system of eigenfunctions and associated functions of the operator under consideration forms a Riesz basis.     

On mean values of some arithmetic functions in number fields

We obtain, for quadratic and cyclotimic fields, asymptotic formulas for two arithmetic functions, which are similar to divisor function.     

Asymptotic formulas for solutions of nonlocal elliptic problems

We consider nonlocal elliptic problems in plane domains and obtain asymptotic formulas for solutions in weighted spaces near junction points.     

Shift formulas for root functions of odd-order differential operators with nonsmooth coefficients

We obtain shift formulas for the root functions of odd-order differential operators with nonsmooth coefficients (an exact formula for first-order operators and an asymptotic formula for operators of higher odd order); these formulas are needed when studying the convergence of spectral expansions in root functions.     

Approximation by Bernstein polynomials at the discontinuity points of derivatives

We obtain asymptotic formulas for the deviation of Bernstein polynomials from functions at the first-kind discontinuity points of the highest derivatives of odd order.     

On the nonself-adjoint ordinary differential operators with periodic boundary conditions

In this article we obtain asymptotic formulas for eigenvalues and eigenfunctions of the nonself-adjoint ordinary differential operator with periodic and antiperiodic boundary conditions, when coefficients are arbitrary summable complex-valued functions. Then using these asymptotic formulas, we obtain necessary and sufficient conditions on the coefficient for which the root functions of these operators form a Riesz basis.     

Spectral properties of a fourth-order differential operator with integrable coefficients

The aim of this paper is to study spectral properties of differential operators with integrable coefficients and a constant weight function. We analyze the asymptotic behavior of solutions to a differential equation with integrable coefficients for large values of the spectral parameter. To find the asymptotic behavior of solutions, we reduce the differential equation to a Volterra integral equation. We also obtain asymptotic formulas for the eigenvalues of some boundary value problems related to the differential operator under consideration.     

Tunnel splitting of the spectrum and bilocalization of eigenfunctions in an asymmetric double well

We consider the one-dimensional stationary Schrödinger equation with a smooth double-well potential. We obtain a criterion for the double localization of wave functions, exponential splitting of energy levels, and the tunneling transport of a particle in an asymmetric potential and also obtain asymptotic formulas for the energy splitting that generalize the formulas known in the case of a mirror-symmetric potential. We consider the case of higher energy levels and the case of energies close to the potential minimums. We present an example of tunneling transport in an asymmetric double well and also consider the problem of tunnel perturbation of the discrete spectrum of the Schrödinger operator with a single-well potential. Exponentially small perturbations of the energies occur in the case of local potential deformations concentrated only in the classically forbidden region. We also calculate the leading term of the asymptotic expansion of the tunnel perturbation of the spectrum.     

Electron in a multilayered magnetic structure: resonance asymptotics

We consider the Neumann Laplacian for two-dimensional weakly coupled strips and obtain asymptotic formulas for a quasieigenvalue close to the threshold. We study the scattering problem in the framework of the asymptotic approach and discuss possible applications of the results. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 3, pp. 429–442, March, 2006.     

Infinite number of eigenvalues of $$2{\times}2$$ operator matrices: Asymptotic discrete spectrum

Theoretical and Mathematical Physics - We study an unbounded $$2{\times}2$$ operator matrix $$ \mathcal{A} $$ in the direct product of two Hilbert spaces. We obtain asymptotic formulas for the...     

Number of irreducible polynomials and pairs of relatively prime polynomials in several variables over finite fields

We discuss several enumerative results for irreducible polynomials of a given degree and pairs of relatively prime polynomials of given degrees in several variables over finite fields. Two notions of degree, the total degree and the vector degree, are considered. We show that the number of irreducibles can be computed recursively by degree and that the number of relatively prime pairs can be expressed in terms of the number of irreducibles. We also obtain asymptotic formulas for the number of irreducibles and the number of relatively prime pairs. The asymptotic formulas for the number of irreducibles generalize and improve several previous results by Carlitz, Cohen and Bodin.     

Integral operators with the generalized sine kernel on the real axis

We study the asymptotic properties of integral operators with the generalized sine kernel acting on the real axis. We obtain the formulas for the Fredholm determinant and the resolvent in the large-x limit and consider some applications of the obtained results to the theory of integrable models.     

On some hybrid power moments of products of generalized quadratic Gauss sums and Kloosterman sums*

In this paper, we investigate hybrid power moments of generalized quadratic Gauss sums weighted with powers of Kloosterman sums and with powers of values of Dirichlet L-functions at 1. We obtain several exact formulas for prime and prime power modulus and some asymptotic formulas.     

Billiards in L-Shaped Tables with Barriers

We compute the volumes of the eigenform loci in the moduli space of genus-two Abelian differentials. From this, we obtain asymptotic formulas for counting closed billiards paths in certain L-shaped polygons with barriers.     

关于一些数值求积公式的渐近性

该文给出了一些数值求积公式的渐近性质,这些公式包括求积分的矩形法则、梯形法则和抛物线法则,包含于余项中的中介点的位置当积分区间的长度趋于零时被确定,对应于该法则的校正公式被得到,它们具有较高的代数精度,我们也进行了一些数值试验,得到较满意的数值结果。     

Asymptotics of the eigenvalues of a boundary value problem for a vector singular differential equation

By analyzing a system of integro-differential equations of the Volterra-Stieltjes type, for large parameter values, we obtain asymptotic formulas for a linearly independent system of solutions of ordinary differential equations with generalized functions in coefficients. These solutions permit one to construct asymptotic formulas for the eigenvalues of a boundary value problem in the case of regular boundary conditions.     

Comparison of numerical integration formulas

In this paper we study the mean square error of numerical integration, when the integrand is a random stationary process. We obtain exact asymptotic errors of classical quadrature formulas and give lower and upper bounds for the least mean square error.     

On sharp asymptotic formulas for the Sturm–Liouville operator with a matrix potential

In this article we obtain the sharp asymptotic formulas for the eigenvalues and eigenfunctions of the non-self-adjoint operators generated by a system of the Sturm–Liouville equations with Dirichlet and Neumann boundary conditions. Using these asymptotic formulas, we find a condition on the potential for which the root functions of these operators form a Riesz basis.