Quasi-Weyl asymptotics of the spectrum of the vector Dirichlet problem

In a space of vector functions, we consider the spectral problem

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, where

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, and the a _αjk and p _jk are constants, x ∈ Ω, and Ω is a bounded open set. The boundary conditions correspond to the Dirichlet problem. Let N _±(μ) be the positive and negative spectral counting functions. We establish the asymptotics N _±(μ) ～ (mes_mΩ)φ_±(μ) as μ → +0. The functions φ_±(μ) are independent of Ω. In the nonelliptic case, these asymptotics are in general different from the classical (Weyl) asymptotics.

     

Estimate of the second term in the spectral asymptotic of Cauchy transform

We give the best possible (respecting to order) estimate of the remainder term in spectral asymptotics of the Cauchy operator and the logarithmic potential type operator. Also, we state a conjecture on the exact value of the remainder term.     

On the asymptotics of the spectrum of a nonsemibounded vector Sturm-Liouville operator

On the half-line, we consider a vector Sturm-Liouville operator with a potential that is unbounded below. Asymptotic formulas for the spectrum are given. These formulas involve the eigenvalues of the matrix potential as well as the “rotational velocities” of the eigenvectors.     

Inverse spectral reconstruction problem for the convolution operator perturbed by a one-dimensional operator

We consider a one-dimensional perturbation of the convolution operator. We study the inverse reconstruction problem for the convolution component using the characteristic numbers under the assumption that the perturbation summand is known a priori. The problem is reduced to the solution of the so-called basic nonlinear integral equation with singularity. We prove the global solvability of this nonlinear equation. On the basis of these results, we prove a uniqueness theorem and obtain necessary and sufficient conditions for the solvability of the inverse problem.     

Inverse spectral problems for the Sturm-Liouville operator on a d-star graph

In this work, we consider inverse spectral problems for the Sturm-Liouville differential operator on a d-star-type graph with standard matching conditions in the internal vertex, where the integer d?2. By using the Yurko's method (Yurko (2008) [27], Yurko (2009) [28]) we show that

(1)

if the potential function qj(x) on a fixed edge ej is prescribed on the interval , then the reciprocal of d of the spectrum suffices to determine qj(x) on the whole interval [0,π];

(2)

the 2 over d of the spectrum suffices to determine qj(x) on a fixed edge ej.

     

Spectral asymptotics for an elliptic operator in a locally periodic perforated domain

We consider the homogenization of an elliptic spectral problem with a large potential stated in a thin cylinder with a locally periodic perforation. The size of the perforation gradually varies from point to point. We impose homogeneous Neumann boundary conditions on the boundary of perforation and on the lateral boundary of the cylinder. The presence of a large parameter 1/ε in front of the potential and the dependence of the perforation on the slow variable give rise to the effect of localization of the eigenfunctions. We show that the jth eigenfunction can be approximated by a scaled exponentially decaying function that is constructed in terms of the jth eigenfunction of a one-dimensional harmonic oscillator operator.     

A trace formula and high-energy spectral asymptotics for the perturbed Landau Hamiltonian

A two-dimensional Schrödinger operator with a constant magnetic field perturbed by a smooth compactly supported potential is considered. The spectrum of this operator consists of eigenvalues which accumulate to the Landau levels. We call the set of eigenvalues near the nth Landau level an nth eigenvalue cluster, and study the distribution of eigenvalues in the nth cluster as n→∞. A complete asymptotic expansion for the eigenvalue moments in the nth cluster is obtained and some coefficients of this expansion are computed. A trace formula involving the eigenvalue moments is obtained.     

Semiclassical spectral series of the Schrödinger operator on surfaces in magnetic fields

We consider the spectral problem for the Schrödinger operator describing a charged particle confined by a homogeneous magnetic field to a certain two-dimensional symmetric surface. Spectral asymptotic series are calculated for either strong or weak magnetic field.     

An inverse spectral problem for a nonsymmetric differential operator: Reconstruction of eigenvalue problem

In this paper, an algorithm is established to reconstruct an eigenvalue problem from the given data satisfying certain conditions. These conditions are proved to be not only necessary but also sufficient for the given data to coincide with the spectral characteristics corresponding to the reconstructed eigenvalue problem.     

Exact tail asymptotics for the Israeli queue with retrials and non-persistent customers

In this paper, we consider the Israeli queue which consists of a main queue with at most N groups and an infinite capacity retrial orbit. The retrial customers may become non-persistent before receiving service. This model was considered before and the decay rate function of the stationary distribution was obtained. To strengthen the result, we characterize the exact tail asymptotics by calculating the coefficient before the decay rate function.     

The Schrödinger operator: Perturbation determinants,the spectral shift function,trace identities,and all that

We discuss applications of the M. G. Krein theory of the spectral shift function to the multidimensional Schrödinger operator. Specific properties of this function, for example, its high-energy asymptotics are studied. Trace identities are derived.     

Inverse problems for Sturm–Liouville operators on a star-shaped graph with mixed spectral data

The partial inverse spectral problem for Sturm–Liouville operators on a star-shaped graph was studied. The authors showed that if the potentials but one were known a priori, then the unknown potential on the whole interval can be uniquely determined by part of information of the potential and part of eigenvalues. The methods employed rest on the Weyl's m-function and theory concerning densities of zeros of entire functions.     

The Dirichlet problem for second order parabolic operators in non‐cylindrical domains

In this paper we develope a perturbation theory for second order parabolic operators in non‐divergence form. In particular we study the solvability of the Dirichlet problem in non cylindrical domains with L^p ‐data on the parabolic boundary (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Estimates for eigenvalues of Laplacian operator with any order

Let D be a bounded domain in an n-dimensional Euclidean space Rn. Assume that 0 ＜ λ1 ≤λ2 ≤ … ≤ λκ ≤ … are the eigenvalues of the Dirichlet Laplacian operator with any order l{(-△)lu=λu, in D u=(δ)u/(δ)(→n)=…(δ)l-1u/(δ)(→n)l-1=0,on (δ)D.Then we obtain an upper bound of the (k 1)-th eigenvalue λκ 1 in terms of the first k eigenvalues.k∑i=1(λκ 1-λi) ≤ 1/n[4l(n 2l-2)]1/2{k∑i=1(λκ 1-λi)1/2λil-1/l k∑i=1(λκ 1-λi)1/2λ1/li}1/2.This ineguality is independent of the domain D. Furthermore, for any l ≥ 3 the above inequality is better than all the known results. Our rusults are the natural generalization of inequalities corresponding to the case l = 2 considered by Qing-Ming Cheng and Hong-Cang Yang. When l = 1, our inequalities imply a weaker form of Yang inequalities. We aslo reprove an implication claimed by Cheng and Yang.     

Sturm-LiouVille算子部分谱信息的逆问题

研究了Sturm-Liouville算子Aq,h,Hj,j=1,2,…中势函数q(x)的确定性问题,即已知部分区间[a,1](a∈(0,1))上的势函数q(x),则无限组部分谱信息可唯一确定整个区间[0,1]上的势函数.推广了Simon的方法,且将选择条件的范围从一组谱扩展到了无限组.     

分数次Orlicz极大算子在齐型空间中的局部加权端点估计

利用齐型空间中的覆盖引理及其有界区域的二进方体分解得到了分数次Orlicz极大算子在齐型空间（X,d,μ）中的有界区域Ω上的局部加权端点估计．该工作为分数次积分交换子[b,Iα】在欧式空间R^n中的有界区域上的加权端点弱型估计推广到齐型空间奠定了基础．