Asymptotic Behavior Of Eigenvalues Research Articles

Asymptotic Behaviors of the Eigenvalues of Schrödinger Operator with Critical Potential

We study the asymptotic behaviors of the discrete eigenvalue of Schrödinger operatorP(λ)=P0+λVwithP0=-Δ+qθ/r2.We obtain the leading terms of discrete eigenvalues ofP(λ)when the eigenvalues tend to 0. In particular, we obtain the asymptotic behaviors of eigenvalues when(P0−α)−1has singularity atα=0.

Asymptotic behavior of eigenvalues of the two-particle discrete Schrödinger operator

We consider two-particle Schrodinger operator H(k) on a three-dimensional lattice ℤ 3 (here k is the total quasimomentum of a two-particle system, $$k \in \mathbb{T}^3 : = \left( { - \pi ,\pi ]^3 } \right)$$ . We show that for any $$k \in S = \mathbb{T}^3 \backslash ( - \pi ,\pi )^3$$ , there is a potential $$\hat v$$ such that the two-particle operator H(k) has infinitely many eigenvalues zn(k) accumulating near the left boundary m(k) of the continuous spectrum. We describe classes of potentials W(j) and W(ij) and manifolds S(j) ⊂ S, i, j ∈ {1, 2, 3}, such that if k ∈ S(3), (k 2 , k 3 ) ∈ (−π,π) 2 , and $$\hat v \in W(3)$$ , then the operator H(k) has infinitely many eigenvalues zn(k) with an asymptotic exponential form as n → ∞ and if k ∈ S(i) ∩ S(j) and $$\hat v \in W(ij)$$ , then the eigenvalues znm(k) of H(k) can be calculated exactly. In both cases, we present the explicit form of the eigenfunctions.

Spatial‐skin effect for eigenvibrations of a thick cascade junction with ‘heavy’ concentrated masses

A spectral problem for the Laplace operator in a thick cascade junction with concentrated masses is considered. This cascade junction consists of the junction's body and a great number of ϵ‐alternating thin rods belonging to two classes. One class consists of rods of finite length, and the second one consists of rods of small length of order . The density of the junction is of order on the rods from the second class and outside of them. The asymptotic behavior of eigenvalues and eigenfunctions of this problem is studied as ϵ → 0. There exist five qualitatively different cases in the asymptotic behavior of eigenmagnitudes as ϵ → 0, namely the case of ‘light’ concentrated (α ∈ (0,1)), ‘middle’ concentrated (α = 1), and ‘heavy’ concentrated masses (α ∈ (1, + ∞ )) that we divide into ‘slightly heavy’ concentrated (α ∈ (1,2)), ‘intermediate heavy’ concentrated (α = 2), and ‘very heavy’ concentrated masses (α > 2). In the paper, we study in detail the influence of the concentrated masses on the asymptotic behavior if α ∈ (1,2). We construct the leading terms of asymptotic expansions both for the eigenvalues and eigenfunctions and prove the corresponding asymptotic estimates. Copyright © 2013 John Wiley & Sons, Ltd.

Система Дирака с недифференцируемым потенциалом и антипериодическими краевыми условиями

Saratov State University, Russia, 410012, Saratov, Astrahanskaya st., 83, KornevVV@info.sgu.ru, KhromovAP@info.sgu.ruTheobjectofthepaperisDiracsystemwithantiperiodicboundaryconditionsandcomplex-valuedconditionspotential.Anewmethodis suggested for investigating spectral properties of this boundary problem. The method is based on the formulas of the transformoperators type. It is rather elementary and simple. Using this method asymptotic behaviour of eigenvalues is specificated and it isproved that eigen and associated functions form Riesz basis with brackets in the space of quadratic summerable two-dimensionalvector-functions since eigenvalues may be multiple. The structure of Riesz projection operators is also studied. The results of thepaper can be used in spectral problems for equations with partial derivatives of the 1-st order containing involution.Key words: Dirac system, spectrum, asymptotics, Riesz basis.

Estimates for the eigenfunctions of the Regge problem

the problem is considered in the reduced form (1) (2). From now on, we assume that the functions q(x) and ρ(x) are integrable on the interval [0, a] and nonnegative. The asymptotic behavior of eigenvalues, completeness properties, and the basis property of eigenfunctions for the Regge problem were studied by many authors; see [2]–[8] and the references therein. This paper contains results on the character of growth of the norms of eigenfunctions for the Regge problem in the space C[0, 1] under the following normalization condition on functions from L2[0, a]: ˆ a

Eigenvalue problems of the model from nonlocal continuum mechanics

This article studies the eigenvalue problem of a fractional differential equation which is a foundation model of a bar of finite length with long-range interactions arising from non-local continuum mechanics. We show that this problem has countable simple real eigenvalues and the corresponding eigenfunctions form a complete orthogonal system in the Hilbert space L2. Furthermore, the asymptotic behavior of eigenvalues and the numbers of zeros of eigenfunctions are studied by using the analytic perturbation theory.

On the spectrum of Euler–Bernoulli beam equation with Kelvin–Voigt damping

The spectral property of an Euler–Bernoulli beam equation with clamped boundary conditions and internal Kelvin–Voigt damping is considered. The essential spectrum of the system operator is rigorously identified to be an interval on the left real axis. Under some assumptions on the coefficients, it is shown that the essential spectrum contains continuous spectrum only, and the point spectrum consists of isolated eigenvalues of finite algebraic multiplicity. The asymptotic behavior of eigenvalues is presented.

Spectral analysis of a wave equation with Kelvin‐Voigt damping

AbstractA vibrating system with some kind of internal damping represents a distributed or passive control. In this article, a wave equation with clamped boundary conditions and internal Kelvin‐Voigt damping is considered. It is shown that the spectrum of the system operator is composed of two parts: point spectrum and continuous spectrum. The point spectrum consists of isolated eigenvalues of finite algebraic multiplicity, and the continuous spectrum that is identical to the essential spectrum is an interval on the left real axis. The asymptotic behavior of eigenvalues is presented.

Two-Scale Convergence of Stekloff Eigenvalue Problems in Perforated Domains

By means of the two-scale convergence method, we investigate the asymptotic behavior of eigenvalues and eigenfunctions of Stekloff eigenvalue problems in perforated domains. We prove a concise and precise homogenization result including convergence of gradients of eigenfunctions which improves the understanding of the asymptotic behavior of eigenfunctions. It is also justified that the natural local problem is not an eigenvalue problem.

Asymptotic behavior of eigenvalues of the Laplace operator in thin infinite tubes

In this paper, we obtain an asymptotic expansion for the eigenvalues of the Laplace operator with zero Dirichlet conditions in tubes, i.e., in infinite bent cylinders with internal torsion under uniform contraction of their cross-sections, with respect to a small parameter characterizing the transverse dimensions of the tube. A method of reducing the problem of determining the eigenvalues to the solution of an implicit equation is proposed.

Homogenization of a boundary value problem with mixed type of boundary conditions in a thick junction

Numerous papers deal with asymptotic methods for boundary value problems in domains depending on a small parameter in a complicated way (perforated domains, partially perforated domains, framework structures, thin domains); e.g., see [1–9]. Boundary value problems in thick singularly degenerating junctions (the number of components of such junctions grows infinitely as the perturbation parameter tends to zero) have specific difficulties and require a separate consideration. It was shown in [10] that boundary value problems in thick junctions lose their coercivity under the passage to the limit, which substantially complicates asymptotic studies. The papers [11, 12] were the first in this direction. In [13–19], a classification of thick junctions was given and rigorous mathematical methods were developed for analyzing the main boundary value problems of mathematical physics in thick singularly degenerating junctions of various types. The study of boundary value problems in thick junctions is focused on the asymptotic behavior of their solutions as e → 0, i.e., as the number of thin attached domains grows infinitely and their thickness tends to zero. In the present paper, we consider a thick junction whose thin attached cylinders are divided into two levels depending on the boundary conditions posed on the lateral surface of these cylinders (inhomogeneous Neumann boundary conditions and homogeneous Dirichlet conditions). In addition, thin cylinders of each level e-periodically alternate along the junction zone. Such thick junctions will be called thick two-level junctions. A problem on a thick plane two-level junction was considered for the first time in [20], where the asymptotic behavior of eigenvalues and eigenfunctions of a spectral boundary value problem was analyzed. Other boundary value problems in plane two-level junctions were considered in [21–23]. Asymptotic analysis of boundary value problems with a periodic change of boundary conditions (Neumann and Dirichlet conditions) on the boundary of smooth unperturbed domains was carried out in [24–26], where it was shown that the first term of the asymptotics is mainly the solution of the corresponding boundary value problem with the Dirichlet conditions. The qualitatively new result of the present paper implies that the first term of the asymptotics is a vector function whose components are solutions of two independent boundary value problems (one in the junction body and another in a domain filled with thin cylinders in the limit) with homogeneous Dirichlet conditions in the junction zone. Note also that the second problem is a problem for a second-order ordinary differential equation with a new right-hand side that “remembers” the inhomogeneity in the Neumann boundary conditions of the original problem and the specific “packing density” of thin cylinders.

Asymptotic behavior of eigenvalues of the Laplace operator in infinite cylinders perturbed by transverse extensions

We present sufficient conditions for the existence of an eigenvalue of the Laplace operator with zero Dirichlet conditions in a weakly perturbed infinite cylinder in the case of localized perturbations which are extensions along the transverse coordinates with coefficients depending on the longitudinal coordinate. If such an eigenvalue exists, then, for this eigenvalue, we obtain an asymptotic formula with respect to a small parameter characterizing the values of extensions.

Optimal Weyl-type Inequalities for Operators in Banach Spaces

Let (s n ) be an s-number sequence. We show for each k = 1, 2, . . . and n ≥ k + 1 the inequality between the eigenvalues and s-numbers of a compact operator T in a Banach space. Furthermore, the constant (k + 1)1/2 is optimal for n = k + 1 and k = 1, 2, . . .. This inequality seems to be an appropriate tool for estimating the first single eigenvalues. On the other hand we prove that the Weyl numbers form a minimal multiplicative s-number sequence and by a well-known inequality between eigenvalues and Weyl numbers due to A. Pietsch they are very good quantities for investigating the optimal asymptotic behavior of eigenvalues.

Asymptotic behavior of eigenvalues and eigenfunctions of the Fourier problem in a thick multilevel junction

A spectral boundary-value problem is considered in a plane thick two-level junction Ωe formed as the union of a domain Ω0 and a large number 2N of thin rods with thickness of order e = O(N −1). The thin rods are split into two levels depending on their length. In addition, the thin rods from the indicated levels are e-periodically alternating. The Fourier conditions are given on the lateral boundaries of the thin rods. The asymptotic behavior of the eigenvalues and eigenfunctions is investigated as e → 0, i.e., when the number of thin rods infinitely increases and their thickness approaches zero. The Hausdorff convergence of the spectrum is proved as e → 0, the leading terms of asymptotics are constructed, and the corresponding asymptotic estimates are justified for the eigenvalues and eigenfunctions.

Analysis of Wave Propagation in 1D Inhomogeneous Media

In this paper, we consider the one-dimensional inhomogeneous wave equation with particular focus on its spectral asymptotic properties and its numerical resolution. In the first part of the paper, we analyze the asymptotic nodal point distribution of high-frequency eigenfunctions, which, in turn, gives further information about the asymptotic behavior of eigenvalues and eigenfunctions. We then turn to the behavior of eigenfunctions in the high- and low-frequency limit. In the latter case, we derive a homogenization limit, whereas in the first we show that a sort of self-homogenization occurs at high frequencies. We also remark on the structure of the solution operator and its relation to desired properties of any numerical approximation. We subsequently shift our focus to the latter and present a Galerkin scheme based on a spectral integral representation of the propagator in combination with Gaussian quadrature in the spectral variable with a frequency-dependent measure. The proposed scheme yields accurate resolution of both high- and low-frequency components of the solution and as a result proves to be more accurate than available schemes at large time steps for both smooth and nonsmooth speeds of propagation.

Functional-discrete Method for Eigenvalue Transmission Problem with Periodic Boundary Conditions

AbstractWe apply a functional-discrete method with convergence rate not worse than that of a geometric series for an eigenvalue transmission problem with periodic boundary conditions. It is shown that the convergence rate increases with an increase in the ordinal number of trial eigenvalue. Based on asymptotic behavior of eigenvalues of the basic problem and the functional-discrete method, qualitative results concerning the arrangement of the eigenvalues of the original eigenvalue transmission problem with periodic boundary conditions are proved. A number of numerical examples are given to support the theory.

Scheme of Investigation of the Spectrum of a Family of Perturbed Operators and Its Application to Spectral Problems in Thick Junctions

We develop a scheme for the investigation of the asymptotic behavior of eigenvalues and eigenvectors of a family of self-adjoint compact operators {Ae: e > 0} that act in different spaces \({\mathcal{H}}_\varepsilon\) and lose their compactness in the limit case e → 0. We prove the Hausdorff convergence of the spectrum of the operator Ae to the spectrum of the limit operator A0, obtain asymptotic estimates for this convergence both to points of the discrete spectrum and to points of the essential spectrum of the operator A0, and prove asymptotic estimates for eigenvectors of Ae. This scheme is applied to the investigation of the asymptotic behavior of eigenvalues and eigenfunctions of the Neumann problem in a thick singularly degenerate junction that consists of two domains connected by an e-periodic system of thin rods of fixed length.

Approximation of Eigenvalues of Elliptic Operators with Discontinuous Coefficients

The asymptotic behavior of eigenvalues of an elliptic operator with a divergence form is discussed. The coefficients of the operator are discontinuous through a boundary of a subdomain and degenerate to zero on the subdomain when a parameter tends to zero. We will prove that the eigenvalues approach eigenvalues of the Laplacian on the subdomain or on the complement. We will obtain precise asymptotic behavior of their convergence.

The Asymptotic Behavior of Eigenvalues of a Boundary Value Problem for a Second-Order Elliptic Operator-Differential Equation with a Discontinuous Coefficient

The Asymptotic Behavior of Eigenvalues of a Boundary Value Problem for a Second-Order Elliptic Operator-Differential Equation with a Discontinuous Coefficient

On the Asymptotics of Eigenvalues of an Analytical Pencil of Operators

We study the asymptotic behavior of eigenvalues of a perturbed pencil of operators in Banach spaces. We prove theorems on branching of an isolated eigenvalue into a finite number of simple eigenvalues. In addition, we deduce conditions for obtaining the asymptotics of a given form.