Non-self-adjoint sturm-liouville operators with matrix potentials

We obtain asymptotic formulas for non-self-adjoint operators generated by the Sturm-Liouville system and quasiperiodic boundary conditions. Using these asymptotic formulas, we obtain conditions on the potential for which the system of root vectors of the operator under consideration forms a Riesz basis.     

On the inverse spectral theory in a non-homogeneous interior transmission problem

We study the inverse spectral problem in an interior transmission eigenvalue problem. The Cartwright’s theory in value distribution theory gives a connection between the distributional structure of the eigenvalues and the asymptotic behaviours of its defining functional determinants. Given a sufficient quantity of transmission eigenvalues, we prove a uniqueness of the refraction index in inhomogeneous medium as an uniqueness problem in entire function theory. The asymptotically periodical structure of the zero set of the solutions helps to locate infinitely many eigenvalues of infinite degree of freedom.     

The mutual variational principle of free wave propagation in periodic structures

By taking infinite periodic beams as examples, the mutual variational principle for analyzing the free wave propagation in periodic structures is established and demonstrated through the use of the propagation constant in the present paper, and the corresponding hierarchical finite element formulation is then derived. Thus, it provides the numerical analysis of that problem with a firm theoretical basis of variational principles, with which one may conveniently illustrate the mathematical and physical mechanisms of the wave propagation in periodic structures and the relationship with the natural vibration. The solution is discussed and examples are given. Supported by Doctorate Training Fund of National Education Commission of China     

Modified classification of non-self-adjoint second-order difference equations

This paper is concerned with the classification of non-self-adjoint second-order difference equations. The relationship between the number of summable solutions of non-self-adjoint difference equations and that of the discrete linear Hamiltonian system is discussed. A classification for non-self-adjoint second-order difference equations is established, which is independent of the choice of the rotated half plane and the fixed point.     

上三角无穷维Hamilton算子点谱的对称性

In this paper, by using characterization of the point spectrum of the upper triangular infinite dimensional Hamiltonian operator H, a necessary and sufficient condition is obtained on the symmetry of σP（A） and σ1/P（-A^＊） with respect to the imaginary axis. Then the symmetry of the point spectrum of H is given, and several examples are presented to illustrate the results.     

Exact controllability of damped coupled Euler-Bernoulli and Timoshenko beam model

^** Email: marianna.shubov{at}euclid.unh.edu The zero controllability problem for the system of two coupledhyperbolic equations which governs the vibrations of the coupledEuler–Bernoulli and Timoshenko beam model is studied inthe paper. The system is considered on a finite interval witha two-parameter family of physically meaningful boundary conditionscontaining damping terms. The controls are introduced as separableforcing terms g_i(x)f_i(t), i = 1, 2, on the right-hand sidesof both equations. The force profile functions g_i(x), i = 1,2, are assumed to be given. To construct the controls f_i(t),i = 1, 2, which bring a given initial state of the system tozero on the specific time interval [0, T], the spectral decompositionmethod has been applied. The approach, used in the present paper,is based on the results obtained in the recent works by theauthor and the collaborators. In these works, the detailed asymptoticand spectral analyses of the non-self-adjoint operators generatingthe dynamics of the coupled beam have been carried out. It hasbeen shown that for each set of the boundary parameters, theaforementioned operator is Riesz spectral, i.e. its generalizedeigenvectors form a Riesz basis in the energy space. Explicitasymptotic formulas for the two-branch spectrum have also beenderived. Based on these spectral results, the control problemhas been reduced to the corresponding moment problem. To solvethis moment problem, the asymptotical representation of thespectrum and the Riesz basis property of the generalized eigenvectorshave been used. The necessary and/or sufficient conditions forthe exact controllability are proven in the paper and the explicitformulas for the control laws are given. The case of the approximatecontrollability is discussed in the paper as well.     

Quantized Riemann surfaces and semiclassical spectral series for a non-self-adjoint Schrödinger operator with periodic coefficients

We consider a non-self-adjoint Schrödinger operator describing the motion of a particle in a one-dimensional space with an analytic potential iV (x) that is periodic with a real period T and is purely imaginary on the real axis. We study the spectrum of this operator in the semiclassical limit and show that the points of its spectrum asymptotically belong to the so-called spectral graph. We construct the spectral graph and evaluate the asymptotic form of the spectrum. A Riemann surface of the particle energy-conservation equation can be constructed in the phase space. We show that both the spectral graph and the asymptotic form of the spectrum can be evaluated in terms of integrals of the pdx form (where x ∈31 ?/T? and p ∈, ? are the particle coordinate and momentum) taken along basis cycles on this Riemann surface. We use the technique of Stokes lines to construct the asymptotic form of the spectrum.     

二阶非自伴两点边值问题Garlerkin有限元的超收敛算法

提出了基于改进位移模式的二阶非自伴两点边值问题Garlerkin有限元的超收敛算法. 用常规有限元解的位移模式与高阶有限元解的位移模式之和构造新的位移模式,基于Garlerkin 方法,采用积分形式推导了单元平衡方程. 对于线性单元,本文给出了有代表性的算例,结点和单元的位移、导数都达到了h⁴阶的超收敛精度.