Asymptotic Estimates Of Eigenvalues Research Articles

On spectral properties of one class difference operators

In this paper, we analyse basic facts of infinite matrix theory. We construct a similarity transform which allows one to represent matrices in a certain class of 5-th diagonal matrices of a difference operator in a diagonal or block-diagonal form. For such matrices, asymptotic estimates of eigenvalues and eigenvectors are obtained. Such matrices are considered in game theory. They are also used in the study fourth-order difference operators with growing potential. The problem of invariant subspaces is also considered.

Spectral analysis of one class of the integro-differential operators

Let ℋ = L2[0,ω] be the Hilbert space of equivalence classes of square integrable Lebesgue measurable complex functions on the closed interval [0,ω]. In this paper, we study the spectral properties of a second-order differential operator which perturbed by an integral operator. The integral operator belong to the two-sided ideal of Hilbert-Schmidt operator in ℋ. We use similar operators method for studying this operator. The similar operators method was developed by A. G. Baskakov and his collaborators. This method allows us to reduce the study of the operator to the one with a diagonal and a block-diagonal matrix. Asymptotic estimates of eigenvalues, eigenvectors, and spectral projections of an integro-differential operator are obtained.

Spectral analysis of a difference operator with a growing potential

In this paper, we study the spectral properties of a second order difference operator with a growing potential. The operator acts in the complex Hilbert space ℓ2(Z) of square summable complex sequences indexed by the integers. This operator is a discrete analogue of a second order differential operator with a growing complex potential. The study is based on a method of similar operators developed by A. G. Baskakov and his collaborators. This method allows us to reduce the study of the operator to one with a block-diagonal matrix. Asymptotic estimates of eigenvalues, eigenvectors, and spectral projections of a difference operator are obtained.

Spectral properties of unbounded $J$-self-adjoint block operator matrices

We study the spectrum of unbounded J -self-adjoint block operator matrices. In particular, we prove enclosures for the spectrum, provide a sucient condition for the spectrum being real and derive variational principles for certain real eigenvalues even in the presence of non-real spectrum. The latter lead to lower and upper bounds and asymptotic estimates for eigenvalues.

Inverse nodal problem for p-Laplacian energy-dependent Sturm-Liouville equation

In this study, the inverse nodal problem is solved for p-Laplacian Schrodinger equation with energy-dependent potential function with the Dirichlet conditions. Asymptotic estimates of eigenvalues, nodal points and nodal lengths are given by using Prufer substitution. Especially, an explicit formula for a potential function is given by using nodal lengths. Results are more general than the classical p-Laplacian Sturm-Liouville problem. For the proofs, methods previously developed by Law et al. and Wang et al., in 2009 and 2011, respectively, are used. In there, they solved an inverse nodal problem for the classical p-Laplacian Sturm-Liouville equation with eigenparameter boundary conditions. MSC:34A55, 34L20.

Boundary-layer eigen solutions for multi-field coupled equations in the contact interface

The dissipative equilibrium dynamics studies the law of fluid motion under constraints in the contact interface of the coupling system. It needs to examine how constraints act upon the fluid movement, while the fluid movement reacts to the constraint field. It also needs to examine the coupling fluid field and media within the contact interface, and to use the multi-scale analysis to solve the regular and singular perturbation problems in micro-phenomena of laboratories and macro-phenomena of nature. This paper describes the field affected by the gravity constraints. Applying the multi-scale analysis to the complex Fourier harmonic analysis, scale changes, and the introduction of new parameters, the complex three-dimensional coupling dynamic equations are transformed into a boundary layer problem in the one-dimensional complex space. Asymptotic analysis is carried out for inter and outer solutions to the perturbation characteristic function of the boundary layer equations in multi-field coupling. Examples are given for disturbance analysis in the flow field, showing the turning point from the index oscillation solution to the algebraic solution. With further analysis and calculation on nonlinear eigenfunctions of the contact interface dynamic problems by the eigenvalue relation, an asymptotic perturbation solution is obtained. Finally, a boundary layer solution to multi-field coupling problems in the contact interface is obtained by asymptotic estimates of eigenvalues for the G-N mode in the large flow limit. Characteristic parameters in the final form of the eigenvalue relation are key factors of the dissipative dynamics in the contact interface.

Spectral estimates of least energy solutions to the Brezis–Nirenberg problem with a variable coefficient

Let u ε be a least energy solution to the Brezis–Nirenberg problem: − Δ u = c 0 u p + ε k ( x ) u in Ω , u > 0 in Ω , u | ∂ Ω = 0 , where Ω ⊂ R N ( N ⩾ 6 ) is a smooth bounded domain, k ∈ C 2 ( Ω ¯ ) is a nonnegative function, c 0 = N ( N − 2 ) , p = ( N + 2 ) / ( N − 2 ) is the critical Sobolev exponent and ε > 0 is a small parameter. We prove several asymptotic estimates of eigenvalues λ i , ε and corresponding eigenfunctions v i , ε to the eigenvalue problem: { − Δ v i , ε = λ i , ε ( c 0 p u ε p − 1 + ε k ( x ) ) v i , ε in Ω , v i , ε = 0 on ∂ Ω , ‖ v i , ε ‖ L ∞ ( Ω ) = 1 as ε → 0 , for i = 1 , 2 , … , N + 1 , N + 2 .

Finite-dimensional perturbations of self-adjoint operators

We study finite-dimensional perturbationsA+γB of a self-adjoint operatorA acting in a Hilbert space\(\mathfrak{H}\). We obtain asymptotic estimates of eigenvalues of the operatorA+γB in a gap of the spectrum of the operatorA as γ → 0, and asymptotic estimates of their number in that gap. The results are formulated in terms of new notions of characteristic branches ofA with respect to a finite-dimensional subspace of\(\mathfrak{H}\) on a gap of the spectrum σ(A) and asymptotic multiplicities of endpoints of that gap with respect to this subspace. It turns out that ifA has simple spectrum then under some mild conditions these asymptotic multiplicities are not bigger than one. We apply our results to the operator(Af)(t)=tf(t) onL2([0, 1],ρc), whereρc is the Cantor measure, and obtain the precise description of the asymptotic behavior of the eigenvalues ofA+γB in the gaps of\(\sigma (A) = \mathfrak{C}\)(= the Cantor set).