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.     

Semiclassical asymptotic spectrum of a Hartree-type operator near the upper boundary of spectral clusters

We consider the problem for eigenvalues of a perturbed two-dimensional oscillator in the case of a resonance frequency. The exciting potential is given by a Hartree-type integral operator with a smooth self-action potential. We find asymptotic eigenvalues and asymptotic eigenfunctions near the upper boundary of spectral clusters, which form around energy levels of the nonperturbed operator. To calculate them, we use asymptotic formulas for quantum means.     

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.     

On Hill's operator with a matrix potential

In this article we obtain asymptotic formulas for eigenvalues and eigenfunctions of the self‐adjoint operator generated by a system of Sturm–Liouville equations with summable coefficients and quasiperiodic boundary conditions. Then using these asymptotic formulas, we find conditions on the potential for which the number of gaps in the spectrum of the Hill's operator with matrix potential is finite. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the asymptotics of eigenfunctions of Sturm-Liouville operators with distribution potentials

We consider a Sturm-Liouville operator in the space L ₂[0, π] and derive asymptotic formulas for the eigenvalues and eigenfunctions of this operator for the case of Dirichlet-Neumann boundary conditions. The leading and second terms of the asymptotics are found in closed form.     

On the riesz basisness of the root functions of the nonself-adjoint sturm-liouville operator

In this article we obtain the asymptotic formulas for eigenfunctions and eigenvalues of the nonself-adjoint Sturm-Liouville operators with periodic and antiperiodic boundary conditions, when the potential is an arbitrary summable complex-valued function. Then using these asymptotic formulas, we find the conditions on Fourier coefficients of the potential for which the eigenfunctions and associated functions of these operators form a Riesz basis inL ₂(0, 1).     

On non-self-adjoint Sturm-Liouville operators in the space of vector functions

In this article we obtain asymptotic formulas for the eigenvalues and eigenfunctions of the non-self-adjoint operator generated in L ₂ ^m [0, 1] by the Sturm-Liouville equation with m × m matrix potential and the boundary conditions which, in the scalar case (m = 1), are strongly regular. Using these asymptotic formulas, we find a condition on the potential for which the root functions of this operator form a Riesz basis.     

Discontinuous Boundary Value Problems with Retarded Argument and Eigenparameter-Dependent Boundary Conditions

In this paper we study the asymptotic formulas for the eigenvalues and corresponding eigenfunctions of discontinuous boundary value problems with retarded argument and eigenparameter-dependent boundary conditions.     

Uniform asymptotics of the eigenvalues and eigenfunctions of the Dirac system with an integrable potential

We consider the Dirac operator on a finite interval with a potential belonging to some set X completely bounded in the space L₁[0, π] and with strongly regular boundary conditions. We derive asymptotic formulas for the eigenvalues and eigenfunctions of the operator; moreover, the constants occurring in the estimates for the remainders depend on the boundary conditions and the set X alone.     

Spectral Properties of Discontinuous Sturm–Liouville Problems with a Finite Number of Transmission Conditions

In this paper we investigate discontinuous two-point boundary value problems with eigenparameter in the boundary conditions and with transmission conditions at the finitely many points of discontinuity. A self-adjoint linear operator A is defined in a suitable Hilbert space H such that the eigenvalues of the considered problem coincide with those of A. We obtain asymptotic formulas for the eigenvalues and eigenfunctions. Also we show that the eigenelements of A are complete in H.     

Asymptotic property and convergence estimation for the eigenelements of the Laplace operator

In this article, we provide a rigorous derivation of asymptotic expansions for eigenfunctions and we establish convergence estimation for both eigenvalues and eigenfunctions of the Laplacian. We address the integral equation method to investigate the interplay between the geometry, boundary conditions and spectral properties of the eigenelements of the Laplace operator under deformation of the domain. The asymptotic formula and convergence estimation are tested by numerical examples.     

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.     

Inverse boundary spectral problem for non–selfadjoint maxwells–s equations with incomplete data

The inverse boundary spectral problem for selfadjoint Maxwell–s equations is to reconstruct unknown coefficient functions in Maxwell– equations from the knowledge of the boundary spectral data, i.e. fromt eh eigenvalues and the boudnary value of the eigenfunctions. Since the spectrum of non–selfadjoint Maxwell–s operator consists of normal eigenvalues and an interval, the complete boundary spectral data can be defind only in a very complicated way. In this article we show that the coefficients can be reconstructed from incomplete data, that is, from the large eigenvalues and the boundary values of the generalized eigenfunctions. Particularly, we do not need the nfinit–dimensional data corresponding to the non–discrete spectrum.     

Spectral properties of the family of even order differential operators with a summable potential

A boundary value problem for a higher order differential operator with separated boundary conditions is considered. The asymptotics of solutions of the corresponding differential equation for large values of the spectral parameter is studied. The indicator diagram of the equation for the eigenvalues is studied. The asymptotic behavior of eigenvalues and the formula for calculation of eigenfunctions of the studied operator is obtained in different sectors of the indicator diagram.     

Convergence of solutions and eigenelements of Steklov type boundary value problems with boundary conditions of rapidly varying type

We consider boundary value problems for the Laplace operator in a domain with boundary conditions of rapidly varying type: the Dirichlet homogeneous condition and the third (Fourier) boundary condition or a Steklov type condition. We construct the limit (homogenized) problem and prove that solutions, eigenvalues, and eigenfunctions of the original problem converge respectively to solutions, eigenvalues, and eigenfunctions of the limit problem. Bibliography: 47 titles. Illustrations: 2 figures.     

Spectral analysis of the diffusion operator with random jumps from the boundary

Using an operator-theoretic framework in a Hilbert-space setting, we perform a detailed spectral analysis of the one-dimensional Laplacian in a bounded interval, subject to specific non-self-adjoint connected boundary conditions modelling a random jump from the boundary to a point inside the interval. In accordance with previous works, we find that all the eigenvalues are real. As the new results, we derive and analyse the adjoint operator, determine the geometric and algebraic multiplicities of the eigenvalues, write down formulae for the eigenfunctions together with the generalised eigenfunctions and study their basis properties. It turns out that the latter heavily depend on whether the distance of the interior point to the centre of the interval divided by the length of the interval is rational or irrational. Finally, we find a closed formula for the metric operator that provides a similarity transform of the problem to a self-adjoint operator.     

Spectral analysis and stabilization of one dimensional wave equation with singular potential

In this paper, we are interested in a boundary damped wave problem with a singular potential. Using a careful spectral analysis, asymptotic expressions of the eigenvalues and eigenvectors of the system operator are derived in terms of the dissipative coefficient and the potential. The Riesz basis property of eigenfunctions and generalized eigenfunctions is also studied. As a consequence, we obtained the exponential stability.     

On the Asymptotics of Eigenvalues of a Fourth-Order Differential Operator with Matrix Coefficients

We study a fourth-order differential operator with matrix coefficients whose domain is determined by the Dirichlet boundary conditions. An asymptotics of the weighted average of the eigenvalues of this operator is obtained in the general case. As a consequence of this result, we present the asymptotics of the eigenvalues in several special cases. The obtained results significantly improve the already known asymptotic formulas for the eigenvalues of a one-dimensional fourth-order differential operator.     

Eigenvalues of Elliptic Systems for the Mixed Problem in Perturbations of Lipschitz Domains with Nonhomogeneous Neumann Boundary Conditions

We study eigenvalues of an elliptic operator with mixed boundary conditions on very general decompositions of the boundary. We impose nonhomogeneous conditions on the part of the boundary where the Neumann term lies in a certain Sobolev or L^p space. Our work compares the behavior of and gives a relationship between the eigenvalues and eigenfunctions on the unperturbed and perturbed domains, respectively.     

Inverse spectral problems for Dirac operator with eigenvalue dependent boundary and jump conditions

We deal with the Dirac operator with eigenvalue dependent boundary and jump conditions. Properties of eigenvalues, eigenfunctions and the resolvent operator are studied. Moreover, uniqueness theorems of the inverse problem according to the Weyl functions and the spectral data (the sets of eigenvalues and norming constants; two different eigenvalues sets) are proved.