Asymptotic behavior of a differential operator with discontinuities at two points

In this paper, we study a Sturm–Liouville operator with eigenparameter‐dependent boundary conditions and transmission conditions at two interior points. By establishing a new operator A associated with the problem, we prove that the operator A is self‐adjoint in an appropriate space H, discuss completeness of its eigenfunctions in H, and obtain its Green function. Copyright © 2010 John Wiley & Sons, Ltd.     

The Rayleigh-Ritz method, refinement and Arnoldi process for periodic matrix pairs

We extend the Rayleigh-Ritz method to the eigen-problem of periodic matrix pairs. Assuming that the deviations of the desired periodic eigenvectors from the corresponding periodic subspaces tend to zero, we show that there exist periodic Ritz values that converge to the desired periodic eigenvalues unconditionally, yet the periodic Ritz vectors may fail to converge. To overcome this potential problem, we minimize residuals formed with periodic Ritz values to produce the refined periodic Ritz vectors, which converge under the same assumption. These results generalize the corresponding well-known ones for Rayleigh-Ritz approximations and their refinement for non-periodic eigen-problems. In addition, we consider a periodic Arnoldi process which is particularly efficient when coupled with the Rayleigh-Ritz method with refinement. The numerical results illustrate that the refinement procedure produces excellent approximations to the original periodic eigenvectors.     

一个Krasnoselski定理的推广

主要讨论了非线性方程F(λ,u)=λu-G(u)=θ的分歧问题,其中G:X→X为非线性可微映射,X为Banach空间.在G′(θ)为紧算子,N(λ~*I-G′(θ))\R(λ~*I-G′(θ))≠{θ}的条件下,利用Lyapunov-Schmidt约化过程和隐函数定理证得了方程F(λ,u)=θ在多重特征值处的分歧定理,推广了Krasnoselski的经典分歧定理.     

Bounding the sum of powers of the Laplacian eigenvalues of graphs

For a non-zero real number α, let s _α (G) denote the sum of the αth power of the non-zero Laplacian eigenvalues of a graph G. In this paper, we establish a connection between s _α (G) and the first Zagreb index in which the Hölder’s inequality plays a key role. By using this result, we present a lot of bounds of s _α (G) for a connected (molecular) graph G in terms of its number of vertices (atoms) and edges (bonds). We also present other two bounds for s _α (G) in terms of connectivity and chromatic number respectively, which generalize those results of Zhou and Trinajsti? for the Kirchhoff index [B Zhou, N Trinajsti?. A note on Kirchhoff index, Chem. Phys. Lett., 2008, 455: 120–123].     

Multiscale finite element discretizations based on local defect correction for the biharmonic eigenvalue problem of plate buckling

In this paper, we study the multiscale finite element discretizations about the biharmonic eigenvalue problem of plate buckling. On the basis of the work of Dai and Zhou (SIAM J. Numer. Anal. 46[1] [2008] 295‐324), we establish a three‐scale scheme, a multiscale discretization scheme, and the associated parallel version based on local defect correction. We first prove a local priori error estimate of finite element approximations, then give the error estimates of multiscale discretization schemes. Theoretical analysis and numerical experiments indicate that our schemes are suitable and efficient for eigenfunctions with local low smoothness.     

Solvability of an inverse problem for discontinuous Sturm–Liouville operators

In this paper, we consider the Sturm–Liouville equation with the jump conditions inside the interval (0,π). The inverse problem is studied, which consists in recovering operator coefficients from two spectra, corresponding to different boundary conditions. We prove the uniqueness theorem and provide necessary and sufficient conditions for solvability of the inverse problem. We also obtain the oscillation theorem for the eigenfunctions of the considered discontinuous boundary value problem.     

Spectral properties of the operator of Riesz potential type

For the operator of the Riesz potential type we estimate the second term in the asymptotic behavior of the spectrum and find its ``regularized' trace.

     

On spectral properties of periodic polyharmonic matrix operators

We consider a matrix operatorH = (-Δ)^l +V inR ⁿ, wheren ≥ 2,l ≥ 1, 4l > n + 1, andV is the operator of multiplication by a periodic inx matrixV(x). We study spectral properties ofH in the high energy region. Asymptotic formulae for Bloch eigenvalues and the corresponding spectral projections are constructed. The Bethe-Sommerfeld conjecture, stating that the spectrum ofH can have only a finite number of gaps, is proved.     

Persistence of Eigenvalues and Multiplicity in the Dirichlet Problem for the Laplace Operator on Nonsmooth Domains

We consider the Dirichlet eigenvalue problem for the Laplace operator on a variable nonsmooth domain. We extend a result of Lupo and Micheletti concerning the structure of the set of domain perturbations which leave the multiplicity of an eigenvalue unchanged, and we study the set of perturbations which leave a certain eigenvalue unchanged.