One-dimensional random field Ising model and discrete stochastic mappings

Previous results relating the one-dimensional random field Ising model to a discrete stochastic mapping are generalized to a two-valued correlated random (Markovian) field and to the case of zero temperature. The fractal dimension of the support of the invariant measure is calculated in a simple approximation and its dependence on the physical parameters is discussed.Contribution to the symposium Statistical Mechanics of Phase Transitions—Mathematical and Physical Aspects, Trebo, CSSR, September 1–6, 1986.     

One-dimensional three-body problem via symbolic dynamics

Symbolic dynamics is applied to the one-dimensional three-body problem with equal masses. The sequence of binary collisions along an orbit is expressed as a symbol sequence of two symbols. Based on the time reversibility of the problem and numerical data, inadmissible (i.e., unrealizable) sequences of collisions are systematically found. A graph for the transitions among various regions in the Poincare section is constructed. This graph is used to find an infinite number of periodic sequences, which implies an infinity of periodic orbits other than those accompanying a simple periodic orbit called the Schubart orbit. Finally, under reasonable assumptions on inadmissible sequences, we prove that the set of admissible symbol sequences forms a Cantor set. (c) 2000 American Institute of Physics.     

Spectral problem for the Schrodinger equation with singular potential polynomial of even degree

For the stationary Schrodinger equation with singular potential U(x)=x^2r, where r∈Z₊, that describes situations associated with phase transition in quantum systems, discovery of symmetries can be used to reduce problems to algebraic problems for finding spectra. We construct analytic procedures for the convergent theory of perturbations to find eigenvalues for potentials of the indicated type. We discuss the fine structure of Bohr—Sommerfeld quantum conditions. Moscow State Institute of Radio Engineering, Electronics, and Automation. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 55–70, May, 1996.     

Orthogonal polynomials on a family of Cantor sets and the problem of iterations of quadratic mappings

We first study a family of invariant transformations for the integer moment problem. The fixed point of these transformations generates a positive measure with support on a Cantor set depending on a parameter q. We analyze the structure and properties of the set of orthogonal polynomials with respect to this measure. Among these polynomials, we find the iterates of the canonical quadratic mapping: F(x)=(x–q) ², q2. It appears that the measure is invariant with respect to this mapping. Algebraic relations among these polynomials are shown to be analytically continuable below q=2, where bifurcation doubling among stable cycles occurs. As the simplest possible consequence we analyze the neighborhood of q=2 (transition region) for q<2.     

Boundary-free propagation with the time-dependent Schrodinger equation

We present two methods that allow for the efficient numerical propagation of continuum wave packets to large times. Time-dependent solutions of the Schrodinger equation that include continuum components are numerically challenging to solve because the wave packet travels, spreads, and acquires a spatial phase gradient. The methods we propose account for these kinematic effects analytically in general and numerically tractable schemes.     

用松弛法解薛定谔方程

求解定态薛定谔方程常常会涉及到常微分方程的本征值问题.目前解常微分方程本征值用的比较多的方法是以龙格-库塔方法为基础的打靶方法.打靶方法常用,但是计算时间长.当边界条件比较复杂或比较敏感的时候,用松弛法会有更好的效果.本文用松弛法解薛定谔方程,并和理论解进行比较.发现松弛法得到的数值解和理论解符合度很高,而且使用松弛法...     

Hamiltonian Formalism of the Derivative Nonlinear Schrodinger Equation

A particular form of poisson bracket is introduced for the derivative nonlinear Schrodinger (DNLS) equation.And its Hamiltonian formalism is developed by a linear combination method. Action-angle variables are found.     

Dressed Dark Solitons of the Defocusing Nonlinear Schrodinger Equation

The nonlinear Schrodinger equation is proved to be consistent-tanh-expansion （CTE） solvable. Some types of dark soliton solutions dressed by cnoidal periodic waves are obtained by means of the CTE method.     

Solving the Schrodinger Equation for the Feynman Quantum Computer

Feynman showed that a closed, locallyinteracting quantum system is capable of performingdeterministic computation. For a finite-size version ofsuch a computer, the Schrodinger equation is solvedanalytically. The probability that the computer yields afinal result upon measurement is derived.     

An Integrable Decomposition of the Derivative Nonlinear SchrSdinger Equation

The nonlinearization method of spectral problem is developed and applied to the derivative nonlinear Schr6dinger equation （DNLS）. As a result, an integrable decomposition of the DNLS equation is obtained.     

Reversible mappings of the plane

The critical exponents for period-doubling and for the breaking of KAM-tori in areapreserving reversible systems are known. We show that symmetric orbits in non-areapreserving reversible systems have the same exponents.     

Novel soliton solutions of the nonlinear Schrodinger equation model

The methodology developed provides for a systematic way to find an infinite number of the novel stable bright and dark "soliton islands" in a "sea of solitary waves" of the nonlinear Schrodinger equation model with varying dispersion, nonlinearity, and gain or absorption. It is shown that solitons exist only under certain conditions and the parameter functions describing dispersion, nonlinearity, and gain or absorption inhomogeneities cannot be chosen independently. Fundamental soliton management regimes are discovered.     

Marchenko Equation for the Derivative Nonlinear SchrSdinger Equation

A simple derivation of the Marchenko equation is given for the derivative nonlinear Schrodinger equation. The kernel of the Marchenko equation is demanded to satisfy the conditions given by the compatibility equations. The soliton solutions to the Marchenko equation are verified. The derivation is not concerned with the revisions of Kaup and Newell.     

One-dimensional quantum theory of the pomeron

The scattering amplitude, originated by the sum over all multi-pomeron exchanges, is evaluated in the zero-slope limit by solving a one-dimensional quantum field theory. In the classical limit, the theory gives back the tree-diagram results obtained in a preceding paper. The constant cross sections obtained in that case for a pomeron with intercept larger than one, are modified only at ultra-asymptotic energies by quantum effects. In the one-dimensional theory, this modification gives rise to cross sections behaving as s^?ΔE where ΔE is a small quantity generated by a tunnel effect.     

Inverse Scattering Transform for the Derivative Nonlinear Schrodinger Equation

Since the Jost solutions of the derivative nonlinear Schrodinger equation do not tend to the free Jost solutions, when the spectral parameter tends to infinity（｜A｜→∞）, the usual inverse scattering transform （IST） must be revised. If we take the parameter κ = λ^-1 as the basic parameter, the Jost solutions in the limit of ｜κ｜→∞ do tend to the free Jost solutions, hence the usual procedure to construct the equations of IST in κ-plane remains effective. After we derive the equation of IST in terms of κ, we can obtain the equation of IST in λ-plane by the simple change of parameters λ = κ^-1. The procedure to derive the equation of IST is reasonable, and attention is never paid to the function W（x） introduced by the revisions of Kaup and Newell. Therefore, the revision of Kaup and Newell can be avoided.     

One-dimensional ordering on the interface

Submonolayer deposition of Mo on Si( 100) below 650°C leads to two surface phases which exhibit short range ( 25 Å) order in one dimension only. The resulting surface was previously thought to be amorphous, but tunneling microscopy reveals the locally ordered character of the system.     

One-dimensional model of the quasicrystalline alloy

A one-dimensional chain of atoms of two types is investigated. It is proven exactly for the model of attracting hard spheres that if the ratio of the numbers of atoms of the two types is irrational, then the state of absolutely minimal energy is quasicrystalline. The quasicrystalline state is also investigated in the case of the Lennard-Jones interatomic potential. All the microscopic values (interatomic spacing, electronic density, etc.) are shown to be quasiperiodic functions varying on Cantor sets. Diffraction patterns, electronic properties, and vibrational spectra are also discussed.