Quantum mechanical Hamiltonians with large ground-state degeneracy

Nonrelativistic Hamiltonians with large, even infinite, ground-state degeneracy are studied by connecting the degeneracy to the property of a Dirac operator. We then identify a special class of Hamiltonians, for which the full space of degenerate ground states in any spatial dimension can be exhibited explicitly. The two-dimensional version of the latter coincides with the Pauli Hamiltonian, and recently-discussed models leading to higher-dimensional Landau levels are obtained as special cases of the higher-dimensional version of this Hamiltonian. But, in our framework, it is only the asymptotic behavior of the background ‘potential’ that matters for the ground-state degeneracy. We work out in detail the ground states of the three-dimensional model in the presence of a uniform magnetic field and such potential. In the latter case one can see degenerate stacking of all 2d Landau levels along the magnetic field axis.     

耦合量子点中空穴基态反键态特性研究

本文采用六带K·P理论计算了耦合量子点在不同耦合距离下空穴基态特性, 探讨了轻重空穴及轨道自旋相互作用对耦合量子点空穴基态反成键态特性的影响. 在考虑多带耦合的情况下, 耦合量子点随着耦合强度的变化, 价带基态能级和激发态能级发生反交叉现象. 同时, 随着耦合距离的增加, 量子点基态轻重空穴波函数的比重发生变化,导致量子点空穴基态波函数从成键态反转成为反成键态. 同时研究发现, 因空穴基态及激发态波函数特性的转变, 电子、空穴的基态及激发态波函数的叠加强度发生的明显变化. 关键词： 耦合量子点 反键态 多带理论 自旋轨道耦合     

Asymptotic formulations of the eigenvalues and eigenfunctions for a boundary value problem

In this work, a discontinuous boundary‐value problem with retarded argument that contains a spectral parameter in the transmission conditions at the point of discontinuity is investigated. We obtained asymptotic formulas for the eigenvalues and eigenfunctions. Copyright © 2012 John Wiley & Sons, Ltd.     

Exact Solutions of Klein-Gordon Equation with Exponential Scalar and Vector Potentials

We obtain the exact analytical solution of the Klein-Gordon equation for the exponential vector and scalar potentials by using the asymptotic iteration method. For the scalar potential greater than the vector potential case, the exact bound state energy eigenvalues and corresponding eigenfunctions are presented. The bound state eigenfunction solutions are obtained in terms of the confluent hypergeometric functions.     

Periodic solutions to nonlinear one dimensional wave equation with -dependent coefficients

This paper deals with -periodicity and regularity of solutions to the one dimensional nonlinear wave equation with -dependent coefficients

     

Eigenvalues and eigenfunctions of the Laplace operator on an equilateral triangle

A boundary value problem for the Laplace equation with Dirichlet and Neumann boundary conditions on an equilateral triangle is transformed to a problem of the same type on a rectangle. This enables us to use, e.g., the cyclic reduction method for computing the numerical solution of the problem. By the same transformation, explicit formulae for all eigenvalues and all eigenfunctions of the corresponding operator are obtained.     

Fine Properties of the Integrated Density of States and a Quantitative Separation Property of the Dirichlet Eigenvalues

We consider one-dimensional difference Schr?dinger equations with real analytic function V(x). Suppose V(x) is a small perturbation of a trigonometric polynomial V ₀(x) of degree k ₀, and assume positive Lyapunov exponents and Diophantine ω. We prove that the integrated density of states is H?lder continuous for any k > 0. Moreover, we show that is absolutely continuous for a.e. ω. Our approach is via finite volume bounds. I.e., we study the eigenvalues of the problem on a finite interval [1, N] with Dirichlet boundary conditions. Then the averaged number of these Dirichlet eigenvalues which fall into an interval , does not exceed , k > 0. Moreover, for , this averaged number does not exceed exp , for any . For the integrated density of states of the problem this implies that for any . To investigate the distribution of the Dirichlet eigenvalues of on a finite interval [1, N] we study the distribution of the zeros of the characteristic determinants with complexified phase x, and frozen ω, E. We prove equidistribution of these zeros in some annulus and show also that no more than 2k ₀ of them fall into any disk of radius exp. In addition, we obtain the lower bound (with δ > 0 arbitrary) for the separation of the eigenvalues of the Dirichlet eigenvalues over the interval [0, N]. This necessarily requires the removal of a small set of energies. Received: February 2006, Accepted: December 2007     

声子色散对极性晶体中磁极化子基态能量的影响

利用线性组合算符和幺正变换相结合的方法,研究了声子色散对极性晶体中磁极化子基态能量的影响.计及纵光学(LO)声子色散,在抛物近似下导出了极性晶体中磁极化子基态能量随电子-纵光学声子耦合常数、回旋共振频率和声子色散系数的变化关系.数值计算结果表明磁极化子基态能量随声子色散系数和电子-纵光学声子耦合常数的增大而减小,随回旋共振频率增大而增大.     

Non-linear vibration of rectangular reticulated shallow shell structures

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正三角形上本征函数的对称性

由D3群一维表示,得到了正三角形上满足Dirichlet边界条件拉普拉斯算子的本征值和本征函数,并讨论了本征函数的对称性.