Breakdown of Stability of Motion in Superquadratic Potentials

Based on the KAM theory, investigation of the equation of motion of a classical particle in a one-dimensional superquadratic potential well, under the influence of an external time-periodic forcing, raised a hope that all the solutions are bounded. Indeed, due to the superquadraticity of the potential the frequency of oscillations of the solutions in the system tends to infinity as the amplitude increases. Therefore, because of this relationship between the frequency and the amplitude, intuitively one might expect that all resonances that could cause the accumulation of energy would be destroyed, and thus all solutions would stay bounded for all time. More formally, according to Moser's twist theorem, this could mean the existence of invariant tubes in the extended phase space and therefore would result in the boundedness of the solutions. Actually, the boundedness results have been established for a large class of superquadratic potentials, but in general, the above intuition turns out to be wrong. Littlewood showed it by creating a superquadratic potential in which an unbounded motion occurs in the presence of some particular piecewise constant forcing. Moreover, Littlewood's result holds for a larger class of forcings. Here it is proven for the continuous time-periodic forcing. For this purpose a new averaging technique for the forced motions in superquadratic potentials with rather weak assumptions on the differentiability of the potentials has been developed. Received: 1 February 1995 / Accepted: 15 March 1997     

Geodesic Study of a Charged Black Hole

The behavior of the timelike and null geodesics of charged E. Ayón-Beato and A. Garcia (ABG) black hole are investigated. For circular and radial geodesics, we investigate all the possible motions by plotting the effective potentials for different parameters. In conclusion, we have shown that there is no phenomenon of superradiance in this case.     

Particle interaction measurements in a Coulomb crystal using caged-particle motion

A technique for characterizing the particle interaction potential of a Coulomb crystal is developed. The mean-square displacement (MSD) is measured, showing both caged- and superdiffusive-particle motions. By subtracting the center of mass of neighboring particles in computing MSD, only short-wavelength particle motions are retained. This yields the lattice Einstein frequency, which contains information about the interparticle forces and potentials. Video measurements of particle motions in a complex (dusty) plasma are used to demonstrate the technique.     

Connecting cluster dynamics and protein folding

The relaxation dynamics of clusters can be interpreted in terms of the topographies of their potential surfaces. Systems with short-range potentials have sawtooth-like potential surfaces with small drops in energy from one local minimum to the next and few-body motions as the clusters move from one minimum to another; such systems readily take on amorphous structures. These are called “glass-formers". Systems with long-range forces have potentials whose topographies are like rough staircases, with some large drops in energy from one minimum to the next; their well-to-well passages involve very collective motions and such systems are excellent structure-seekers. They find their way to well-ordered, highly selective structures under almost all circumstances. These characteristics generalize to describe the potential surfaces and folding behavior of polypeptides and proteins. The forces are effective long-range forces due to the polymer chain. Staircase topographies emerge both from direct sampling of potential surfaces and from the inversion of the kinetics generated by a much more aaabstract topological model, from which folding pathways can be inferred. Received 4 December 2000     

Relationship Between Two Classes of Shape-Invariant Potentials

We show that two classes of shape-invariant potentials are interrelated to each other. For all one-dimensional shape-invariant potentials with parameters related by translation, i.e. the first class of shape-invariant potentials (SIP1), we can find their multi-parameter deformations with q acting as the deformation parameter, i.e. the second class of shape-invariant potentials (SIP2) with parameters related by scaling. In order to get closed solution of SIP2, we consider two infinitesimal intervals, one is close to q=0 another close to q=1, and show that in these intervals we can get separately two first-order approximate solutions in closed form, furthermore we prove that all SIP1 can be obtained by the limiting procedures for corresponding SIP2.     

Relationship Between Two Classes of Shape-Invariant Potentials

We show that two classes of shape-invariant potentials are interrelated to each other. For all one-dimensional shape-invariant potentials with parameters related by translation, i.e. the first class of shapc-invariant potentials (SIP1),we can find their multi-parameter deformations with q acting as the deformation parameter, i.e. the second class of shape-invariant potentials (SIP2) with parameters related by scaling. In order to get closed solution of SIP2, we consider two infinitesimal intervals, one is close to q= 0 another close to q = 1, and show that in these intervals we can get separately two first-order approximate solutions in closed form, furthermore we prove that all SIP1 can be obtained by the limiting procedures for corresponding SIP2.``     

Homothetic motions and vacuum Robinson-Trautman solutions

The approach to symmetries given by Kerr and Debney (1970) is applied to the class of vacuum Robinson-Trautman solutions. The coordinate freedom is extended and used to define the pull-backs of local diffeomorphisms which are then shown to be homothetic motions. The resulting homothetic Killing's equations are solved to give a form of all homothety groups for this class.     

Fractional extensions of the classical isotropic oscillator and the Kepler problem

The class of fractional Hamiltonian systems that generalize the classical problem of the two-dimensional (2D) isotropic harmonic oscillator and the Kepler problem is considered. It is shown that, in the 4D space of structural parameters, the 2D isotropic harmonic oscillator can be extended along a line in such a way that the orbits remain closed and oscillations remain isochronous. Likewise, the Kepler problem can be extended along a line in such a way that the orbits remain closed for all finite motions and the third Kepler law remains valid. These curves lie on the 2D surfaces where any dynamical system is characterized by the same rotation number for all finite motions.     

Contribution of quantum molecular flexibility to the second virial coefficient of water vapor

The contribution of essentially quantum internal molecular motions to the second virial coefficient B2 of water vapor is analyzed in the framework of the path integral approach. A general purpose ab initio polarizable force field QMPFF2 or a nonpolarizable three-site water model are used with oscillator and Morse valence potentials. It is demonstrated that the contribution may be significant but depends strongly on the form of the intramolecular potential. In the case of the more realistic stretching Morse potential, inclusion of quantum molecular flexibility into the simulation reduces the virial coefficient by 20%-40%. Also, the internal modes make a contribution to the difference in the virial coefficient for light and heavy water, which is opposite to that of the intermolecular motions, so that the net effect can even change the sign at higher temperatures.     

On the Reality of the Eigenvalues for a Class of -Symmetric Oscillators

 We study the eigenvalue problem with the boundary conditions that decays to zero as z tends to infinity along the rays , where is a real polynomial and . We prove that if for some we have for all , then the eigenvalues are all positive real. We then sharpen this to a larger class of polynomial potentials. In particular, this implies that the eigenvalues are all positive real for the potentials when with , and with the boundary conditions that decays to zero as z tends to infinity along the positive and negative real axes. This verifies a conjecture of Bessis and Zinn-Justin. Received: 16 January 2002 / Accepted: 1 May 2002 Published online: 6 August 2002     

NiAl中空位迁移机制的计算机模拟

运用分子动力学方法采用F-S多体势函数从原子尺度上模拟了NiAl金属间化合物中单空位的迁移运动行为，认为空位随成分的变化而采取不同的迁移方式：成分在理想化合比附近空位迁移主要以六步循环方式进行，其中V_Al主要以直型［100］六步循环方式迁移，V_Ni以［110］型六步循环方式占优势；当成分偏离时在富Ni一侧空位迁移则以ASB方式占很大的优势.计算所得NiAl金属间化合物中单空位迁移激活能与实验值相符，从微观上合理地解释了NiAl金属间化合物淬火实验中较高淬火温度对 关键词：     

Stability of metal vicinal surfaces revisited

The stability of metal vicinal surfaces with respect to faceting is investigated using empirical potentials as well as electronic structure calculations. It is proven that for a wide class of empirical potentials all vicinal surfaces between (100) and (111) are unstable at 0 K when the role of third and farther nearest neighbors is negligible. However, electronic structure calculations reveal that the answer concerning the stability of vicinal surfaces is not so clear-cut. Finally, it is shown that surface vibrations at finite temperatures have little effect on the stability of vicinal surfaces.     

Regular Spacings of Complex Eigenvalues in the One-Dimensional Non-Hermitian Anderson Model

We prove that in dimension one the non-real eigenvalues of the non-Hermitian Anderson (NHA) model with a selfaveraging potential are regularly spaced. The class of selfaveraging potentials which we introduce in this paper is very wide and in particular includes stationary potentials (with probability one) as well as all quasi-periodic potentials. It should be emphasized that our approach here is much simpler than the one we used before. It allows us a) to investigate the above mentioned spacings, b) to establish certain properties of the integrated density of states of the Hermitian Anderson models with selfaveraging potentials, and c) to obtain (as a by-product) much simpler proofs of our previous results concerned with non-real eigenvalues of the NHA model.     

Superintegrability and higher-order constants for classical and quantum systems

We extend recent work by Tremblay, Turbiner, and Winternitz which analyzes an infinite family of solvable and integrable quantum systems in the plane, indexed by the positive parameter k. Key components of their analysis were to demonstrate that there are closed orbits in the corresponding classical system if k is rational, and for a number of examples there are generating quantum symmetries that are higher order differential operators than two. Indeed they conjectured that for a general class of potentials of this type, quantum constants of higher order should exist. We give credence to this conjecture by showing that for an even more general class of potentials in classicalmechanics, there are higher-order constants of the motion as polynomials in the momenta. Thus these systems are all superintegrable.     

Ergodic and topological properties of coulombic periodic potentials

The motion of a classical pointlike particle in a two-dimensional periodic potential with negative coulombic singularities is examined. This motion is shown to be Bernoullian for many potentials and high enough energies. Then the motion on the plane is a diffusion process. All such motions are topologically conjugate and the periodic orbits can be analysed with the help of a group.This work is part of a thesis submitted to Freie Universität Berlin     

ON FINITE NORMAL SERIES SOLUTIONS OF THE SCHRÖDINGER EQUATION WITH IRREGULAR SINGULARITY

We compute all potentials with the following property: The one-dimensional nonrelativistic Schrödinger equation for these potentials has irregular singular points at infinity and/or zero and is solved by a finite normal series. We restrict to expansion order zero, discuss some properties of the potentials obtained and, as an application, calculate for some given potentials exact solutions and energies. The aim of this paper is to provide a tool for finding exact solutions of the Schrödinger equation for a large class of singular potentials.     

Competition between direct and concerted movements in surface diffusion with application to the Au(110) surface

Recent calculations of the energetic barriers for diffusion of Au atoms on the Au(110) surface have shown that for moves perpendicular to the close-packed atomic rows, concerted, or multi-atom movements are energetically favorable over single-atom movements for the same displacement. Entropic hindrances, however are expected to limit concerted moves. In the present paper this competition is investigated via simulation techniques based on model potentials that reproduce the relevant energtic and entropic features of the Au(110) situation. Apparently non-Arrhenius behavior is found for the concerted motions. A new scheme, which avoids the necessity of long simulation runs to determine the “hop” rate at several temperatures, for determining entropic barriers directly is proposed and tested. Though this method is applied in the present context to a simplified model for the interatomic potentials, it can be straightforwardly extended to realistic treatments of those interactions.     

A relativistic generalisation of rigid motions

A weaker substitute for the too restrictive class of Born-rigid motions is proposed, which we call radar-holonomic motions. The definition is expressed as a set of differential equations. Integrability conditions and Cauchy problem are studied. We finally obtain an example of a radar-holonomic congruence containing a given worldline with a given value of the rotation on this line.     

Identity for the Exponential-Type Molecule Potentials and the Supersymmetry Shape Invariance

The identity and the supersymmetry shape invariance for a class of exponential-type molecule potentials are studied by introducing a deformed five-parameter exponential-type potential (DFPEP) and via the multi-parameter deformations. It has been shown that the DFPEP is a shape-invariant potential with a translation of parameters. By making use of the shape invariance approach, the exact energy levels are determined for the bound states with zero angular momentum. A class of molecule potentials and their exact energy spectra for the zero angular momentum states are reduced from the DFPEP and a general energy spectrum formula, respectively. The interrelations for some molecule potentials are also discussed.