WKB Wave Function of the General Time-Dependent Quadratic Hamiltonian System

We derived the WKB wave function for the general time-dependent quadratic Hamiltonian system using a unitary transformation method. We applied our research to sinusodially drived Caldirola–Kanai oscillator and confirmed that the time evolution of our approximated WKB wave function is similar to that of the exact one. This wave function can be used to analyze the interference between the probability amplitudes contributed by the area of overlap in phase space of quantum states.     

Line graphs of bipartite graphs with Hamiltonian paths

It is shown that, if t is an integer ≥3 and not equal to 7 or 8, then there is a unique maximal graph having the path P_t as a star complement for the eigenvalue ?2. The maximal graph is the line graph of K_m,m if t = 2m?1, and of K_m,m+1 if t = 2m. This result yields a characterization of L(G ) when G is a (t + 1)‐vertex bipartite graph with a Hamiltonian path. The graphs with star complement P_r ∪ P_s or P_r ∪ C_s for ?2 are also determined. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 137–149, 2003     

Effective stability for generalized Hamiltonian systems

An effective stability result for generalized Hamiltonian systems is obtained by applying the simultaneous approximation technique due to Lochak. Among these systems, dimensions of action variables and angle variables might be distinct.     

A test problem for molecular dynamics integrators

We derive a test problem for evaluating the ability of time-steppingmethods to preserve the statistical properties of systems inmolecular dynamics. We consider a family of deterministic systemsconsisting of a finite number of particles interacting on acompact interval. The particles are given random initial conditionsand interact through instantaneous energy- and momentum-conservingcollisions. As the number of particles, the particle density,and the mean particle speed go to infinity, the trajectory ofa tracer particle is shown to converge to a stationary Gaussianstochastic process. We approximate this system by one describedby a system of ordinary differential equations and provide numericalevidence that it converges to the same stochastic process. Wesimulate the latter system with a variety of numerical integrators,including the symplectic Euler method, a fourth-order Runge-Kuttamethod, and an energyconserving step-and-project method. Weassess the methods' ability to recapture the system's limitingstatistics and observe that symplectic Euler performs significantlybetter than the others for comparable computational expense.     

Critical Point Theorems and Applications to Differential Equations

This paper contains a generalization of the well-known Palais-Smale and Cerami compactness conditions. The compactness condition introduced is used to prove some general existence theorems for critical points. Some applications are given to differential equations.     

A COMPLETE METRIC OF POSITIVE CURVATURE ON R~n AND EXISTENCE OF CLOSED GEODESICS

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The Hamiltonian Canonical Form for Euler-Lagrange Equations

Based on the theory of calculus of variation, some suffcient conditions are given for some Euler-Lagrangcequations to be equivalently represented by finite or even infinite many Hamiltonian canonical equations. Meanwhile,some further applications for equations such as the KdV equation, MKdV equation, the general linear Euler Lagrangeequation and the cylindric shell equations are given.     

Finite-Dimensional PT-Symmetric Hamiltonians

This paper investigates finite-dimensional PT-symmetric Hamiltonians. It is shown here that there are two ways to extend real symmetric Hamiltonians into the complex domain: (i) The usual approach is to generalize such Hamiltonians to include complex Hermitian Hamiltonians. (ii) Alternatively, one can generalize real symmetric Hamiltonians to include complex PT-symmetric Hamiltonians. In the first approach the spectrum remains real, while in the second approach the spectrum remains real if the PT symmetry is not broken. Both generalizations give a consistent theory of quantum mechanics, but if D>2, a D-dimensional Hermitian matrix Hamiltonian has more arbitrary parameters than a D-dimensional PT-symmetric matrix Hamiltonian.     

PT Symmetry in Quantum Field Theory

The Hamiltonian H specifies the energy levels and the time evolution of a quantum theory. It is an axiom of quantum mechanics that H be Hermitian. The Hermiticity of H guarantees that the energy spectrum is real and that the time evolution is unitary (probability preserving). In this talk we investigate an alternative formulation of quantum mechanics in which the mathematical requirement of Hermiticity is replaced by the more physically transparent condition of space-time reflection (PT) symmetry. We show that if the PT symmetry of a Hamiltonian H is not broken, then the spectrum of H is real. Examples of PT-symmetric non-Hermitian Hamiltonians are H=p ²+ix ³ and H=p ²-x ⁴. The crucial question is whether PT-symmetric Hamiltonians specify physically acceptable quantum theories in which the norms of states are positive and the time evolution is unitary. The answer is that a Hamiltonian that has an unbroken PT symmetry also possesses a physical symmetry that we call C. Using C, we show how to construct an inner product whose associated norm is positive definite. The result is a new class of fully consistent complex quantum theories. Observables exhibit CPT symmetry, probabilities are positive, and the dynamics is governed by unitary time evolution.     

电子储存环中插入元件的非线性束流动力学的Lie代数方法

介绍了用能保证哈密顿系统“辛”性质的Lie映射方法来研究电子储存环中的插入元件对束流动力学的影响.并以此方法对合肥光源中将要安装的波荡器UD?–?1对储存环的影响作了初步的研究