Finite energy solutions of self‐adjoint elliptic mixed boundary value problems

This paper describes existence, uniqueness and special eigenfunction representations of H¹‐solutions of second order, self‐adjoint, elliptic equations with both interior and boundary source terms. The equations are posed on bounded regions with Dirichlet conditions on part of the boundary and Neumann conditions on the complement. The system is decomposed into separate problems defined on orthogonal subspaces of H¹(Ω). One problem involves the equation with the interior source term and the Neumann data. The other problem just involves the homogeneous equation with Dirichlet data. Spectral representations of the solution operators for each of these problems are found. The solutions are described using bases that are, respectively, eigenfunctions of the differential operator with mixed null boundary conditions, and certain mixed Steklov eigenfunctions. These series converge strongly in H¹(Ω). Necessary and sufficient conditions for the Dirichlet part of the boundary data to have a finite energy extension are described. The solutions for a problem that models a cylindrical capacitor is found explicitly. Copyright © 2009 John Wiley & Sons, Ltd.     

非协调有限元与本征值下界

与协调有限元相比,非协调有限元常常给出本征值下界.这种现象已经由很多数值例子观察到.但是理论上的证明最近才做的更多.以Morley非协调元作为更简单的范例,来说明这种理论为何成功.     

Thermodynamic properties and the approximate solutions of the Schrödinger equation with the shifted Deng–Fan potential model

With the introduction of a new improved approximation scheme (Pekeris-type approximation) to deal with the centrifugal term, the energy eigenvalues and the wave functions of the Schrödinger equation of the shifted Deng–Fan molecular potential are obtained, via the asymptotic iteration method. Rotational–vibrational energy eigenvalues of some diatomic molecules are presented, these results are in good agreement with other results in the literature. For these selected diatomic molecules, energy eigenvalues obtained are in much better agreement with the results obtained from the rotating Morse potential model for moderate values of rotational and vibrational quantum numbers. Furthermore, thermodynamic properties such as the vibrational mean U, specific heat C, free energy F and entropy S for the pure vibrational state in the classical limit for these energy eigenvalues are studied.     

Bueckner's work conjugate integrals and weight functions for a crack in anisotropic solids

The Bueckner work conjugate integrals are studied for cracks in anisotropic elastic solids. The difficulties in separating Lekhnitskii's two complex arguments involved in the integrals are overcome and explicit functional representations of the integrals are given for several typical cases. It is found that the pseudoorthogonal property of the eigenfunction expansion forms presented previously for isotropic cases, isotropic bimaterials, and orthotropic cases, are proved to be also valid in the present case of anisotropic material. Finally, Some useful path-independent integrals and weight functions are proposed. The project supported by the National Natural Science Foundation of China (19891180) and Doctorate Foundation of Xi'an Jiaotong University     

Some mathematical aspects of Anderson localization: boundary effect,multimodality, and bifurcation

Anderson localization is a famous wave phenomenon that describes the absence of diffusion of waves in a disordered medium. Here we generalize the landscape theory of Anderson localization to general elliptic operators and complex boundary conditions using a probabilistic approach, and further investigate some mathematical aspects of Anderson localization that are rarely discussed before. First, we observe that under the Neumann boundary condition, the low energy quantum states are localized on the boundary of the domain with high probability. We provide a detailed explanation of this phenomenon using the concept of extended subregions and obtain an analytical expression of this probability in the one-dimensional case. Second, we find that the quantum states may be localized in multiple different subregions with high probability in the one-dimensional case and we derive an explicit expression of this probability for various boundary conditions. Finally, we examine a bifurcation phenomenon of the localization subregion as the strength of disorder varies. The critical threshold of bifurcation is analytically computed based on a toy model and the dependence of the critical threshold on model parameters is analyzed.     

空间轴对称变形界面裂纹尖端的应力奇异性分析

假定位移场,并将位移势函数展开成一系列的特征函数,代入空间轴对称变形的基本方程,推得界面裂纹的特征值理论。把这一理论应用到单向增强的连续纤维复合材料上,建立了纤维断裂、基体开裂等模型下应力奇异性的特征方程,分析了界面裂纹尖端的应力奇异性。     

一般非均匀凸介质的迁移算子广义本征函数系统的完整性

该文讨论一般非均匀凸介质所确定的迁移算子的广义本征函数系统的完整性问题，利用我们探索的相对收敛方法，完整地刻划了一般非均匀凸介质中迁移问题的广义本征函数系统的完整性问题，我们证明了，对迁移半群E（t），当0∈Pσ（E（t）），迁移算子广义本征函数系统不完整，当0∈Pσ（E（t））时，仅当满足相对收敛条件时，迁移算子的广义本征函数系统完整．     

三维各向同性谐振子的代数解法

用带角动量的玻色算子统一构成三维各向同性谐振子的能量本征函数，并讨论了它们和直接求解微分方程得到的本征解之间的关系．这种方法快捷优美，而且有统一的表达式和简单的李代数关系。     

A note on the harmonic plus inverse-harmonic potential in quantum mechanics

The ground state of a class of potentials described byV(x)=a ₂₀ x ²+a ₁/x ² is investigated. This potential possesses some distinct features like that of a symmetric double-well potential, and can be used in modelling the fusion process of two identical nuclei as well.