On the Long Time Behavior of a Particle in an Infinitely Extended System in One Dimension

We study the long time behavior of a non-equilibrium infinite particle system in one dimension. First, we show that the velocity of a particle increases at most linearly in time. Then we discuss at a heuristic level the displacement of a particle when the mutual interaction is singular. Finally we study the motion of a fast particle interacting with a background of slow particles and we prove that the velocity of the fast particle remains almost unchanged for a long time (at least proportional to the velocity itself).     

Behavior for Large Time of a Two-Component Chain of Harmonic Oscillators

We consider a model consisting of two semi-infinite chains of harmonic oscillators with a special particle subject to a harmonic external potential.We study the Cauchy problem for this model. The main goal is to derive the dispersive bounds for the solutions in the energy weighted norms.     

Central Limit Theorem for Partial Linear Eigenvalue Statistics of Wigner Matrices

In this paper, we study the complex Wigner matrices $M_{n}=\frac{1}{\sqrt{n}}W_{n}$ whose eigenvalues are typically in the interval [?2,2]. Let λ ₁≤λ ₂?≤λ _n be the ordered eigenvalues of M _n . Under the assumption of four matching moments with the Gaussian Unitary Ensemble (GUE), for test function f 4-times continuously differentiable on an open interval including [?2,2], we establish central limit theorems for two types of partial linear statistics of the eigenvalues. The first type is defined with a threshold u in the bulk of the Wigner semicircle law as $\mathcal{A}_{n}[f; u]=\sum_{l=1}^{n}f(\lambda_{l})\mathbf{1}_{\{\lambda_{l}\leq u\}}$ . And the second one is $\mathcal{B}_{n}[f; k]=\sum_{l=1}^{k}f(\lambda_{l})$ with positive integer k=k _n such that k/n→y∈(0,1) as n tends to infinity. Moreover, we derive a weak convergence result for a partial sum process constructed from $\mathcal{B}_{n}[f; \lfloor nt\rfloor]$ . The main difficulty is to deal with the linear eigenvalue statistics for the test functions with several non-differentiable points. And our main strategy is to combine the Helffer-Sjöstrand formula and a comparison procedure on the resolvents to extend the results from GUE case to general Wigner matrices case. Moreover, the results on $\mathcal{A}_{n}[f;u]$ for the real Wigner matrices will also be briefly discussed.     

Test of a Jastrow-type Wavefunction for a Trapped Few-body System in One Dimension

For a system with interacting quantum mechanical particles in a one-dimensional harmonic oscillator, a trial wavefunction with simple structure based on the solution of the corresponding two-particle system is suggested and tested numerically.With the inclusion of a scaling parameter for the distance between particles, at least for the very small systems tested here the ansatz gives a very good estimate of the ground state energy, with the error being of the order of ~1 % of the gap to the first excited state.     

The NLS Equation in Dimension One with Spatially Concentrated Nonlinearities: the Pointlike Limit

In the present paper, we study the following scaled nonlinear Schrödinger equation (NLS) in one space dimension: $$ i\frac{\rm d}{{\rm d}t}\psi^{\varepsilon}(t)=-\Delta\psi^{\varepsilon}(t) +\frac{1}{\varepsilon}V\left(\frac{x}{\varepsilon} \right)|\psi^{\varepsilon}(t)|^{2\mu}\psi^{\varepsilon}(t)\quad \varepsilon > 0\,\quad V\in L^1(\mathbb{R},(1+|x|){\rm d}x) \cap L^\infty(\mathbb{R}).$$ This equation represents a nonlinear Schrödinger equation with a spatially concentrated nonlinearity. We show that in the limit \({\varepsilon\to 0}\) the weak (integral) dynamics converges in \({H^1(\mathbb{R})}\) to the weak dynamics of the NLS with point-concentrated nonlinearity: $$ i\frac{{\rm d}}{{\rm d}t} \psi(t) =H_{\alpha} \psi(t) .$$ where H _α is the Laplacian with the nonlinear boundary condition at the origin \({\psi'(t,0+)-\psi'(t,0-)=\alpha|\psi(t,0)|^{2\mu}\psi(t,0)}\) and \({\alpha=\int_{\mathbb{R}}V{\rm d}x}\) . The convergence occurs for every \({\mu\in \mathbb{R}^+}\) if V ≥  0 and for every  \({\mu\in (0,1)}\) otherwise. The same result holds true for a nonlinearity with an arbitrary number N of concentration points.     

五角棱镜在建立大尺寸平面基准中的应用

从五角棱镜的原理和特性出发，叙述了利用五角棱镜扫描建立大平面基准过程中，由于加工角差和当五角棱镜相对入射光束有相对运动时其出射光束的横移对平面基准的影响，提出了用衍射成像技术消除光束横移，数据计算修正棱镜的角度差的影响。     

Exponentially Long Stability Times for a Nonlinear Lattice in the Thermodynamic Limit

In this paper, we construct an adiabatic invariant for a large 1–d lattice of particles, which is the so called Klein Gordon lattice. The time evolution of such a quantity is bounded by a stretched exponential as the perturbation parameters tend to zero. At variance with the results available in the literature, our result holds uniformly in the thermodynamic limit. The proof consists of two steps: first, one uses techniques of Hamiltonian perturbation theory to construct a formal adiabatic invariant; second, one uses probabilistic methods to show that, with large probability, the adiabatic invariant is approximately constant. As a corollary, we can give a bound from below to the relaxation time for the considered system, through estimates on the autocorrelation of the adiabatic invariant.     

Levinson's Theorem for the Nonlocal Interaction in One Dimension

The Levinson theorem for the one-dimensional Schrödinger equation with both local and the nonlocal symmetric potentials is established by the Sturm–Liouville theorem. The critical case where the Schro;audinger equation has a finite zero-energy solution is also analyzed. It is shown that the number n ₊ (n _–) of bound states with even (odd) parity is related to the phase shift ₊(0)[_–(0)] of the scattering states with the same parity at zero momentum as

The Hierarchy Principle and the Large Mass Limit of the Linear Sigma Model

In perturbation theory we study the matching in four dimensions between the linear sigma model in the large mass limit and the renormalized nonlinear sigma model in the recently proposed flat connection formalism. We consider both the chiral limit and the strong coupling limit of the linear sigma model. Our formalism extends to Green functions with an arbitrary number of pion legs, at one loop level, on the basis of the hierarchy as an efficient unifying principle that governs both limits. While the chiral limit is straightforward, the matching in the strong coupling limit requires careful use of the normalization conditions of the linear theory, in order to exploit the functional equation and the complete set of local solutions of its linearized form.     

Associated Productions of HZZ and HHZ at Linear Colliders in Large Extra Dimension Model

In this paper we investigate the effects of the large extra dimensions on the two processes e⁺e^-→ H⁰Z⁰Z⁰ and e⁺e^-→ H⁰Z⁰Z⁰ at linear colliders in both unpolarized and polarized collision modes. We find that the virtual Kaluza-Klein graviton exchange can significantly enhance the cross section from their standard model expectations for these two processes. The results show that the LED effect on the process e⁺e^-→ H⁰Z⁰Z⁰ allows the observation limits on the effective scale M_S to be probed up to 9.75 TeV and 10.1 TeV in the unpolarized and +-(λ_e⁺=1/2, λ_e^-=-1/2) polarized beam collision modes (with P_e⁺=0.6, P_e^-=0.8), respectively. For the process e⁺e^-→ H⁰Z⁰Z⁰, these limits on M_S can be probed up to 6.06 TeV and 6.38 TeV in the unpolarized and polarized collision modes separately. We find that the λ_e⁺=1/2, λ_e^-=-1/2 polarization collision mode in both process e⁺e^-→ H⁰Z⁰Z⁰ and e⁺e^-→ H⁰Z⁰Z⁰ may provide a possibility to improve the sensitivity in probing the LED effects.     

Long Wave Limit of the Complex Dielectric Function for Coulomb System with Bound States

For plasmas consisting of charged particles an improvement of the RPA dielectric function is considered which accounts especially for the possibility of the formation of bound states. The polarizability contains contributions of bound states as well as of scattering states. The imaginary part of the dielectric function describes optical transitions, and expressions for the electrical conductivity are obtained.     

Long Time Behavior of a Modified Becker-Döring System

A modification, based on asymptotic behavior, of the Becker-Döring system is introduced in which the concentration of monomers is slaved to the concentrations of the other clusters. This modified system has the same continuum limit as the usual Becker-Döring system. For one member of the modified systems it is proved, for compact initial data, that all solutions will converge to the same self-similar form as time tends to infinity.     

Limit Cycles Arising in a Chain of Inhibitorily Coupled Identical Relaxation Oscillators Near the Self-Oscillation Threshold

We consider the dynamic regimes arising in a linear chain of four identical stiff FitzHugh- Nagumo oscillators existing in the vicinity of the bifurcation of limit cycle emergence. It is shown that in a broad range of coupling forces, slow-variable exchange between the oscillators gives rise to multiple limit cycles with different periods and different phase relations. In addition to the expected antiphase solutions, three families of stable limit cycles that differ in the number of bursts of the fast variable in the neighboring elements and in the number of bursts per period are detected. The boundaries of attractor stability are calculated and the parameter regions of their coexistence are found.__________Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 48, No. 3, pp. 238–248, March 2005.     

Linear Boltzmann Equation as the Long Time Dynamics of an Electron Weakly Coupled to a Phonon Field

We consider the long time evolution of a quantum particle weakly interacting with a phonon field. We show that in the weak coupling limit the Wigner distribution of the electron density matrix converges to the solution of the linear Boltzmann equation globally in time. The collision kernel is identified as the sum of an emission and an absorption term that depend on the equilibrium distribution of the free phonon modes.     

一种强健的守恒型间断跟踪法在一维Euler方程组上的实现

文[8,9]研究设计了一种守恒型的间断跟踪法,其主要特点是以解的守恒性作为跟踪的机制而不是像传统的间断跟踪法利用Rankine-Hugoniot条件,将该跟踪法对一维Euler方程组进行实现.对一维单个守恒律凸流的实现见文[7].由于Euler方程组有3类特征线,信息沿每类特征线传播并相互穿透.如果仍然象单个守恒律那样间断两侧的数值解的计算只用到来自间断同侧的信息,则间断一侧的信息就不能传到间断的另一侧,而会堆积在间断附近.设计了一种几乎是二阶的按特征方式分离信息的方法,将堆积在间断附近的信息分离.而且按此思想可做成高阶方法.该方法中不仅用到了精确Riemann分解,还用到了Roe^[12]的近似Riemann分解及各种插值和重构技巧.数值实验表明这样的处理是有效的.     

The a Posteriori Error Estimates for Chebyshev-Galerkin Spectral Methods in One Dimension

In this paper, the Chebyshev-Galerkin spectral approximations are employed to investigate Poisson equations and the fourth order equations in one dimension. Meanwhile, $p$-version finite element methods with Chebyshev polynomials are utilized to solve Poisson equations. The efficient and reliable a posteriori error estimators are given for different models. Furthermore, the a priori error estimators are derived independently. Some numerical experiments are performed to verify the theoretical analysis for the a posteriori error indicators and a priori error estimations.     

Proof of a Conjecture on the Infinite Dimension Limit of a Unifying Model for Random Matrix Theory

Journal of Statistical Physics - We study the large N limit of a sparse random block matrix ensemble. It depends on two parameters: the average connectivity Z and the size of the blocks d, which is...     

Dualities for a Class of Finite Range Probabilistic Cellular Automata in One Dimension

In this paper we study dualities for a class of one-dimensional probabilistic cellular automata with finite range interactions by using a sequence of extended cellular automata.     

Three-Body Dynamics in One Dimension: A Test Model for the Three-Nucleon System with Irreducible Pionic Diagrams

 We formulate the three-body problem in one dimension in terms of the (Faddeev-type) integral equation approach. As an application, we develop a spinless, one-dimensional (1-D) model that mimics three-nucleon dynamics in one dimension. Using simple two-body potentials that reproduce the deuteron binding, we obtain that the three-body system binds at about 7.5 MeV. We then consider two types of residual pionic corrections in the dynamical equation; one related to the 2π-exchange three-body diagram, the other to the 1π-exchange three-body diagram. We find that the first contribution can produce an additional binding effect of about 0.9 MeV. The second term produces smaller binding effects, which are, however, dependent on the uncertainty in the off-shell extrapolation of the two-body t-matrix. This presents interesting analogies with what occurs in three dimensions. The paper also discusses the general three-particle quantum scattering problem, for motion restricted to the full line. Received March 5, 2002; accepted July 19, 2002