Asymptotic Behavior Of Eigenvalues Research Articles

Inverse nodal problems for Sturm–Liouville operators on quantum star graphs

In this work, we study the nodal inverse problem for Sturm–Liouville operators on quantum star graphs. We show that the zeros of eigenfunctions characterize the potential, and we present a simple algorithm that enables us to approximate it. The method does not rely on a priori information about the asymptotic behavior of eigenvalues. We manage to reduce the problem to a family of constant coefficient eigenvalue problems that can be solved explicitly and contain enough information to recover the potential. We also consider the multiplicity of eigenvalues and their vanishing properties, which only hold generically.

Continuous dependence of eigenvalues on potential functions for nonlocal Sturm–Liouville equations

In this paper, we discuss the continuous dependence of eigenvalues on the potential function for a nonlocal Sturm–Liouville equation with truncated Marchaud fractional derivative term by two‐parameter method. To this end, we first study the properties of eigenvalues for a two‐parameter nonlocal Sturm–Liouville eigenvalue problem. We obtain the two‐parameter nonlocal Sturm–Liouville eigenvalue problem that has countable number of simple real eigenvalues. Meanwhile, the asymptotic behavior of eigenvalues is studied by aid of the analytic perturbation theory.

Inverse problem for Dirac operators with a constant delay less than half the length of the interval

We study inverse spectral problems for Dirac-type functional-differential operators with a constant delay a ? [?/3, ?/2).We consider the asymptotic behavior of eigenvalues and research the inverse problem of recovering operators from two spectra. The main result of the paper refers to the proof that the operator could be recovered uniquely from two spectra in the case a ? [2?/5, ?/2), as well as the proof that it is not possible in the case a ? [?/3, 2?/5).

Asymptotic behavior of Eigenvalues and Eigenfunctions of T.Regge Fractional Problem

The asymptotic behavior of eigenvalues and eigenfunctions of T.Regge fractional boundary value problem has been shown, and we state and prove some theorems for many results, also some necessary definitions and results. In this paper, we look into a group of fractional boundary value problem equations involving fractional derivative fractional orders and with two boundary value conditions. We will which are important to state and prove those theorems; our primary findings are illustrated using examples.

An improved subspace weighting method using random matrix theory

The weighting subspace fitting (WSF) algorithm performs better than the multi-signal classification (MUSIC) algorithm in the case of low signal-to-noise ratio (SNR) and when signals are correlated. In this study, we use the random matrix theory (RMT) to improve WSF. RMT focuses on the asymptotic behavior of eigenvalues and eigenvectors of random matrices with dimensions of matrices increasing at the same rate. The approximative first-order perturbation is applied in WSF when calculating statistics of the eigenvectors of sample covariance. Using the asymptotic results of the norm of the projection from the sample covariance matrix signal subspace onto the real signal in the random matrix theory, the method of calculating WSF is obtained. Numerical results are shown to prove the superiority of RMT in scenarios with few snapshots and a low SNR.

Asymptotic behaviour of resonance eigenvalues of the Schrödinger operator with a matrix potential

We will discuss the asymptotic behaviour of the eigenvalues of a Schrodinger operator with a matrix potential defined by the Neumann boundary condition in L₂^{m}(F), where F is a d-dimensional rectangle and the potential is an m×m matrix with m≥2, d≥2 , when the eigenvalues belong to the resonance domain, roughly speaking they lie near the planes of diffraction.

On Spectral Properties of the One-Dimensional Stark Operator on the Semiaxis

We consider a one-dimensional Stark operator on the semiaxis with Dirichlet boundary condition at the origin. The asymptotic behavior of eigenvalues at infinity is analyzed.

A Sturm-Liouville Type Problem with Retarded Argument Which Contains a Spectral Parameter in the Boundary Condition

The aim of this study is to investigate a discontinuous problem of the Sturm-Liouville type with a retarded argument containing a spectral parameter in the boundary condition and two additional conditions of transmission at the two discontinuity points. Also, eigenparameter dependent boundary conditions and obtains asymptotic formulas for the eigenvalues and eigenfunctions. And the spectral parameter is real. And the real valued function is continuous on that interval and it has a finite limit. The goal of this article is to obtain asymptotic formulas for eigenvalues of eigenfunctions for problem of the form : with boundary conditions and transmission conditions To this aim, first, the principal term of asymptotic distribution of eigenvalues and eigenfunctions of given problem was obtained up to but, afterwards under some additional conditions, we improve these formulas up to Thus, when the number of points of discontinuity is more than one, we see how the asymptotic behaviour of eigenvalues and eigenfunctions of a boundary value problem with retarded argument which contains a spectral parameter in the boundary conditions change.

On a New Characterization of Some Class Nonlinear Eigenvalue Problem

A normal mode analysis of a vibrating mechanical or electrical system gives rise to an eigenvalue problem. Faber made a fairly complete study of the existence and asymptotic behavior of eigenvalues and eigenfunctions, Green’s function, and expansion properties. We will investigate a new characterization of some class nonlinear eigenvalue problem.

On the solution of an inverse Sturm–Liouville problem with a delay andeigenparameter-dependent boundary conditions

In this paper, a boundary value problem consisting of a delay differential equation of the Sturm-Liouville type with eigenparameter-dependent boundary conditions is investigated. The asymptotic behavior of eigenvalues is studied and the parameter of delay is determined by eigenvalues. Then we obtain the connection between the potential function and the canonical form of the characteristic function.

Asymptotics of eigenvalues and eigenvectors of Toeplitz matrices

Analysis of the asymptotic behaviour of the spectral characteristics of Toeplitz matrices as the dimension of the matrix tends to infinity has a history of over 100 years. For instance, quite a number of versions of Szegő's theorem on the asymptotic behaviour of eigenvalues and of the so-called strong Szegő theorem on the asymptotic behaviour of the determinants of Toeplitz matrices are known. Starting in the 1950s, the asymptotics of the maximum and minimum eigenvalues were actively investigated. However, investigation of the individual asymptotics of all the eigenvalues and eigenvectors of Toeplitz matrices started only quite recently: the first papers on this subject were published in 2009–2010. A survey of this new field is presented here. Bibliography: 55 titles.

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.

Eigenvalue asymptotics for the damped wave equation on metric graphs

We consider the linear damped wave equation on finite metric graphs and analyse its spectral properties with an emphasis on the asymptotic behaviour of eigenvalues. In the case of equilateral graphs and standard coupling conditions we show that there is only a finite number of high-frequency abscissas, whose location is solely determined by the averages of the damping terms on each edge. We further describe some of the possible behaviour when the edge lengths are no longer necessarily equal but remain commensurate.

Asymptotic behavior of eigenvalues of the Laplacian on a thin domain under the mixed boundary condition

Asymptotic behavior of eigenvalues of the Laplacian on a thin domain under the mixed boundary condition

Asymptotic behavior of eigenvalues of hydrogen atom equation

We consider a spectral problem for a class of singular Sturm-Liouville operators on the unit interval with explicit singularity $2/x$ - $2/x^{2}$ , related to the Schrodinger operator with radially symmetric potential. In particular, we give the asymptotic behavior of the eigenvalues of the hydrogen atom equation.

Eigenvibrations of thick cascade junctions with `very heavy' concentrated masses

We study small-parameter asymptotics of eigenelements of a boundary-value problem for the Laplace operator in a thick cascade junction with concentrated masses. There are five qualitatively different cases in the asymptotic behaviour of eigenvalues and eigenfunctions as the small parameter tends to zero (`light', `intermediate', `slightly heavy', `intermediate heavy' and `very heavy' concentrated masses). We study the influence of concentrated masses on the asymptotics of eigenvibrations in the last two cases. We construct the leading terms of asymptotic expansions for eigenfunctions and eigenvalues and rigorously justify them by appropriate asymptotic estimates. We also find new types of high-frequency eigenvibrations.

Necessary and sufficient conditions for the solvability of inverse problem for a class of Dirac operators

In this paper we consider a problem for the first order Dirac differential equations system with spectral parameter dependent on boundary condition. The asymptotic behaviors of eigenvalues, eigenfunctions and normalizing numbers of this system are investigated. The expansion formula with respect to eigenfunctions is obtained and Parseval equality is given. The main theorem on necessary and sufficient conditions for the solvability of inverse problem is proved and the algorithm of reconstruction of potential from spectral data (the sets of eigenvalues and normalizing numbers) is given.

Asymptotic behavior of eigenvalues and eigenfunctions of Sturm-Liouville problems with coupled boundary conditions and transmission conditions

Asymptotic behavior of eigenvalues and eigenfunctions of Sturm-Liouville problems with coupled boundary conditions and transmission conditions

Eigenvalues of the Laplace-Beltrami operator on a large spherical cap under the Robin problem

Eigenvalues of the Laplace-Beltrami operator on a spherical cap is considered under the homogeneous Robin condition. The asymptotic behavior of eigenvalues and the influence of the eigenvalues by the boundary conditions are discussed as the cap becomes large so that the domain covers almost the whole sphere.

INVERSE PROBLEM FOR A CLASS OF DIRAC OPERATOR

In this paper, we consider a problem for the first order canonical Dirac differential equations system with piecewise continuous coefficient and spectral parameter dependent in boundary condition. The asymptotic behavior of eigenvalues, eigenfunctions and normalizing numbers of this system is investigated. The completeness theorem is proved. The spectral expansion formula with respect to eigenvector functions or equivalently Parseval equality is obtained. Weyl solution and Weyl function for the problem are constructed. Uniqueness theorem for inverse problem by the Weyl function and by the spectral data are proved.