Examples of non-Kähler Hamiltonian circle manifolds with the strong Lefschetz property

In this paper we construct six-dimensional compact non-Kähler Hamiltonian circle manifolds which satisfy the strong Lefschetz property themselves but nevertheless have a non-Lefschetz symplectic quotient. This provides the first known counterexamples to the question whether the strong Lefschetz property descends to the symplectic quotient. We also give examples of Hamiltonian strong Lefschetz circle manifolds which have a non-Lefschetz fixed point submanifold. In addition, we establish a sufficient and necessary condition for a finitely presentable group to be the fundamental group of a strong Lefschetz manifold. We then use it to show the existence of Lefschetz four-manifolds with non-Lefschetz finite covering spaces.     

Generalized 3G theorem and application to relativistic stable process on non-smooth open sets

Let G(x,y) and GD(x,y) be the Green functions of rotationally invariant symmetric α-stable process in Rd and in an open set D, respectively, where 0<α<2. The inequality GD(x,y)GD(y,z)/GD(x,z)?c(G(x,y)+G(y,z)) is a very useful tool in studying (local) Schrödinger operators. When the above inequality is true with c=c(D)∈(0,∞), then we say that the 3G theorem holds in D. In this paper, we establish a generalized version of 3G theorem when D is a bounded κ-fat open set, which includes a bounded John domain. The 3G we consider is of the form GD(x,y)GD(z,w)/GD(x,w), where y may be different from z. When y=z, we recover the usual 3G. The 3G form GD(x,y)GD(z,w)/GD(x,w) appears in non-local Schrödinger operator theory. Using our generalized 3G theorem, we give a concrete class of functions belonging to the non-local Kato class, introduced by Chen and Song, on κ-fat open sets. As an application, we discuss relativistic α-stable processes (relativistic Hamiltonian when α=1) in κ-fat open sets. We identify the Martin boundary and the minimal Martin boundary with the Euclidean boundary for relativistic α-stable processes in κ-fat open sets. Furthermore, we show that relative Fatou type theorem is true for relativistic stable processes in κ-fat open sets. The main results of this paper hold for a large class of symmetric Markov processes, as are illustrated in the last section of this paper. We also discuss the generalized 3G theorem for a large class of symmetric stable Lévy processes.     

哈密顿算符的对角化与能量本征值问题的代数解法

提出了将哈密顿算符对角化的一种方法.围绕一维谐振子讨论了如何将哈密顿算符对角化、引进的是玻色算符还是费米算符的问题,并分析了能量量子化的原因及能量子与声子的区别.     

Velocity Quantization Approach of the One-Dimensional Dissipative Harmonic Oscillator

Given a constant of motion for the one-dimensional harmonic oscillator with linear dissipation in the velocity, the problem to get the Hamiltonian for this system is pointed out, and the quantization up to second order in the perturbation approach is used to determine the modification on the eigenvalues when dissipation is taken into consideration. This quantization is realized using the constant of motion instead of the Hamiltonian. PACS: 03.20.+i, 03.30.+p, 03.65.−w,03.65.Ca     

DISSIPATIVE SCHEMES WITH PROPERTY OF INHERITING HOMOCLINIC ORBITS FOR 2N-DIMENSIONAL HAMILTONIAN SYSTEMS

1　ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｉｓｐａｐｅｒｉｓｄｅｖｏｔｅｄｔｏｓｔｕｄｙｗｈａｔｋｉｎｄｏｆｄｉｓｃｒｅｔｅｓｃｈｅｍｅｓｏｆｔｈｅｆｏｌｌｏｗｉｎｇ 2ｎ ｄｉｍｅｎ ｓｉｏｎａｌＨａｍｉｌｔｏｎｉａｎｓｙｓｔｅｍｓｗｉｔｈｐａｒａｍｅｔｅｒｉｎｎｏｒｍａｌｆｏｒｍ ｕ=Ｊ2ｎ Ｈ ｕＴ,　　Ｈ =Ｈ(ｕ ,λ) ,(1 )ｗｈｅｒｅｕ∈Ｒ2ｎ,λ∈Ｒ ,Ｈ∈Ｃｋ+1(Ｒ2ｎ×Ｒ ,Ｒ) ,ｋ≥ 6,ａｎｄＪ2ｎ =0Ｉｎ-Ｉｎ 0 ,Ｉｎ:ｕｎｉｔｍａｔｒｉｘｏｆｏｒｄｅｒｎｈａｓｔｈｅｐｒｏｐｅｒｔｙｏｆｉｎｈｅｒｉｔｉｎｇｈｏｍ…     

蒙特卡罗哈密顿新方案及其检验

蒙特卡罗哈密顿方法(MCH)是研究量子理论的数值模拟新方法, 其优点是可求出超出基态的能谱和波函数. 旧MCH方案需要自由粒子的信息, 较难推广应用于格点规范理论. 本文提出克服这个困难的新方案. 首先介绍这一方案的思想, 并以1维量子力学模型V(x)=μ²x²+λx⁴(其中μ²< 0,λ>0)为例说明实现这一新方案的具体计算步骤和方法.     

The semiclassical limit of a quantum Fermi accelerator

A classical Fermi accelerator model (FAM) is known to show chaotic behavior. The FAM is defined by a free particle bouncing elastically from two rigid walls, one fixed and the other oscillating periodically in time. The central aim of this paper is to connect the quantum and the classical solutions to the FAM in the semiclassical limit. This goal is accomplished using a finite inverted parametric oscillator (FIPO), confined to a box withfixed walls, as an alternative representation of the FAM. In the FIPO representation, an explicit correspondence between classical and quantum limits is accomplished using a Husimi representation of the quasienergy eigenfunctions.     

The numerical approximation of center manifolds in Hamiltonian systems

In this paper we develop a numerical method for computing higher order local approximations of center manifolds near steady states in Hamiltonian systems. The underlying system is assumed to be large in the sense that a large sparse Jacobian at the equilibrium occurs, for which only a linear solver and a low-dimensional invariant subspace is available. Our method combines this restriction from linear algebra with the requirement that the center manifold is parametrized by a symplectic mapping and that the reduced equation preserves the Hamiltonian form. Our approach can be considered as a special adaptation of a general method from Numer. Math. 80 (1998) 1-38 to the Hamiltonian case such that approximations of the reduced Hamiltonian are obtained simultaneously. As an application we treat a finite difference system for an elliptic problem on an infinite strip.     

高阶微商系统Dirac猜想的一个反例

由扩展正则作用量导出了高阶微商奇异Lagrange量系统的扩展正则Noether恒等式.从广义约束Hamilton系统相空间中对称性分析，给出高阶微商系统Dirac猜想的一个反例. 用正则Noether定理、 正则Noether恒等式和扩展正则Noether恒等式说明在此反例中Dirac猜想失效, 讨论中没有将约束线性化. 关键词： 高阶微商系统 约束Hamilton系统 正则对称性 Dirac猜想