Nonrelativistic Limit of Scattering Theory for Nonlinear Klein-Gordon Equations

We study the nonrelativistic limit of the nonlinear Klein-Gordon equation and prove that the wave operators, the inverses of them, and the scattering operator for the naturally modulated equation converge to those for the singular limit, the nonlinear Schrödinger equation.     

Nonrelativistic limit in the energy space for nonlinear Klein-Gordon equations

We study the nonrelativistic limit of the Cauchy problem for the nonlinear Klein-Gordon equation and prove that any finite energy solution converges to the corresponding solution of the nonlinear Schr?dinger equation in the energy space, after the infinite oscillation in time is removed. We also derive the optimal rate of convergence in . Received: 13 July 2000 / Published online: 1 February 2002     

Transverse electrical conductivity of a quantum collisional plasma in the Mermin approach

We derive formulas for the transverse electrical conductivity and the permittivity in a quantum collisional plasma using the kinetic equation for the density matrix in the relaxation approximation in the momentum space. We show that the derived formula becomes the classical formula when the Planck constant tends to zero and that when the electron collision rate tends to zero (i.e., the plasma becomes collisionless), the derived formulas become the previously obtained Lindhard formulas. We also show that when the wave number tends to zero, the quantum conductivity becomes classical. We compare the obtained conductivity with the conductivity obtained by Lindhard and with the classical conductivity     

Long-time self-similarity of classical solutions to the bipolar quantum hydrodynamic models

In this paper, a bipolar transient quantum hydrodynamic model (BQHD) for charge density, current density and electric field is considered on the one-dimensional real line. This model takes the form of the classical Euler-Poisson system with additional dispersion caused by the quantum (Bohn) potential. We investigate the long-time behavior of the BQHD model and show the asymptotical self-similarity property of the global smooth solution. Namely, both of the charge densities tend to a nonlinear diffusion wave in large time, which is not a solution to the BQHD equation, but to the combined quasi-neutral, relaxation and semiclassical limiting model. Next, as a by-product, we can compare the large-time behavior of the bipolar quantum hydrodynamic models and of the corresponding classical bipolar hydrodynamic models. As far as we know, the nonlinear diffusion phenomena about the 1D BQHD is new.     

Self-Dual Vortices in Chern–Simons Hydrodynamics

The classical theory of a nonrelativistic charged particle interacting with a U(1) gauge field is reformulated as the Schrödinger wave equation modified by the de Broglie–Bohm nonlinear quantum potential. The model is gauge equivalent to the standard Schrödinger equation with the Planck constant for the deformed strength of the quantum potential and to the pair of diffusion–antidiffusion equations for the strength . Specifying the gauge field as the Abelian Chern–Simons (CS) one in 2+1 dimensions interacting with the nonlinear Schrödinger (NLS) field (the Jackiw–Pi model), we represent the theory as a planar Madelung fluid, where the CS Gauss law has the simple physical meaning of creation of the local vorticity for the fluid flow. For the static flow when the velocity of the center-of-mass motion (the classical velocity) is equal to the quantum velocity (generated by the quantum potential velocity of the internal motion), the fluid admits an N-vortex solution. Applying a gauge transformation of the Auberson–Sabatier type to the phase of the vortex wave function, we show that deformation parameter , the CS coupling constant, and the quantum potential strength are quantized. We discuss reductions of the model to 1+1 dimensions leading to modified NLS and DNLS equations with resonance soliton interactions.     

View of bunching and antibunching from the standpoint of classical signals

The similarity between classical wave mechanics and quantum mechanics was noted in the works of De Broglie, Schr?dinger, ??late?? Einstein, Lamb, Lande, Mandel, Marshall, Santos, Boyer, and many others. We present a new wave model of quantum mechanics, the so-called prequantum classical statistical field theory, in which an analogy between some quantum phenomena and the classical theory of random fields is investigated. Quantum systems are interpreted as symbolic representations of such fields (not only for photons, cf. Lande and Lamb, but even for massive particles). All quantum averages and correlations (including composite systems in entangled states) can be represented as averages and correlations for classical random fields. We use the prequantum classical statistical field theory to obtain bunching and antibunching in the framework of classical signal theory. We note that antibunching at least is typically considered an essentially quantum (nonclassical) phenomenon.     

Transition to the condensate state for classical gases and clusterization

In this paper, using of the rigorous statement and rigorous proof the Maxwell distribution as an example, we establish estimates of the distribution depending on the parameter N, the number of particles. Further, we consider the problem of the occurrence of dimers in a classical gas as an analog of Bose condensation and establish estimates of the lower level of the analog of Bose condensation. We find the relationship of this level to “capture” theory in the scattering problem corresponding to an interaction of the form of the Lennard-Jones potential. This also solves the problem of the Gibbs paradox. We derive the equation of state for a nonideal gas as a result of pair interactions of particles in Lennard-Jones models and, for classical gases, discuss the λ-transition to the condensed state (the state in which V _spec does not vary with increasing pressure; for heat capacity, this is the λ-point). We also present new quantum equations of the flow of a neutral gas consisting of particles with an odd number of neutrons in the capillaries in the Sutherland model.     

Space-time fractional Schrödinger equation with time-independent potentials

We develop a space-time fractional Schrödinger equation containing Caputo fractional derivative and the quantum Riesz fractional operator from a space fractional Schrödinger equation in this paper. By use of the new equation we study the time evolution behaviors of the space-time fractional quantum system in the time-independent potential fields and two cases that the order of the time fractional derivative is between zero and one and between one and two are discussed respectively. The space-time fractional Schrödinger equation with time-independent potentials is divided into a space equation and a time one. A general solution, which is composed of oscillatory terms and decay ones, is obtained. We investigate the time limits of the total probability and the energy levels of particles when time goes to infinity and find that the limit values not only depend on the order of the time derivative, but also on the sign (positive or negative) of the eigenvalues of the space equation. We also find that the limit value of the total probability can be greater or less than one, which means the space-time fractional Schrödinger equation describes the quantum system where the probability is not conservative and particles may be extracted from or absorbed by the potentials. Additionally, the non-Markovian time evolution laws of the space-time fractional quantum system are discussed. The formula of the time evolution of the mechanical quantities is derived and we prove that there is no conservative quantities in the space-time fractional quantum system. We also get a Mittag-Leffler type of time evolution operator of wave functions and then establish a Heisenberg equation containing fractional operators.     

Wave function and the probability current distribution for a bound electron moving in a uniform magnetic field

We study the effects of electromagnetic fields on nonrelativistic charged spinning particles bound by a short-range potential. We analyze the exact solution of the Pauli equation for an electron moving in the potential field determined by the three-dimensional δ-well in the presence of a strong magnetic field. We obtain asymptotic expressions for this solution for different values of the problem parameters. In addition, we consider electron probability currents and their dependence on the magnetic field. We show that including the spin in the framework of the nonrelativistic approach allows correctly taking the effect of the magnetic field on the electric current into account. The obtained dependences of the current distribution, which is an experimentally observable quantity, can be manifested directly in scattering processes, for example.     

Initial boundary value problems for a quantum hydrodynamic model of semiconductors: Asymptotic behaviors and classical limits

The present paper proves the existence and the asymptotic stability of a stationary solution to the initial boundary value problem for a quantum hydrodynamic model of semiconductors over a one-dimensional bounded domain. We also discuss on a singular limit from this model to a classical hydrodynamic model without quantum effects. Precisely, we prove that a solution for the quantum model converges to that for the hydrodynamic model as the Planck constant tends to zero. Here we adopt a non-linear boundary condition which means quantum effect vanishes on the boundary. In the previous researches, the existence and the asymptotic stability of a stationary solution are proved under the assumption that a doping profile is flat, which makes the stationary solution also flat. However, the typical doping profile in actual devices does not satisfy this assumption. Thus, we prove the above theorems without this flatness assumption. Firstly, the existence of the stationary solution is proved by the Leray-Schauder fixed-point theorem. Secondly, we show the asymptotic stability theorem by using an elementary energy method, where the equation for an energy form plays an essential role. Finally, the classical limit is considered by using the energy method again.     

Using the Evolution Operator Method to Describe a Particle in a Homogeneous Alternating Field

Using the evolution operator method, we construct coherent states of a nonrelativistic free particle with a variable mass M(t) and a nonrelativistic particle with a variable mass M(t) in a homogeneous alternating field. Under certain physical conditions, they can be regarded as semiclassical states of particles. We discuss the properties (in particular, the completeness relation, the minimization of the uncertainty relations, and the time evolution of the corresponding probability density) of the found coherent states in detail. We also construct exact wave functions of the oscillator type and of the plane-wave type for the considered systems and compute the quantum Wigner distribution functions for the wave functions of coherent and oscillator states. We establish the unitary equivalence of the problems of a free particle and a particle in a homogeneous alternating field.     

Local controllability of a 1-D Schrödinger equation

We consider a nonrelativistic charged particle in a 1-D box of potential. This quantum system is subject to a control, which is a uniform electric field. It is represented by a complex probability amplitude solution of a Schrödinger equation. We prove the local controllability of this nonlinear system around the ground state. Our proof uses the return method, a Nash–Moser implicit function theorem and moment theory.     

Analysis of the projected coupled cluster method in electronic structure calculation

The electronic Schrödinger equation plays a fundamental role in molcular physics. It describes the stationary nonrelativistic behaviour of an quantum mechanical N electron system in the electric field generated by the nuclei. The (Projected) Coupled Cluster Method has been developed for the numerical computation of the ground state energy and wave function. It provides a powerful tool for high accuracy electronic structure calculations. The present paper aims to provide a rigorous analytical treatment and convergence analysis of this method. If the discrete Hartree Fock solution is sufficiently good, the quasi-optimal convergence of the projected coupled cluster solution to the full CI solution is shown. Under reasonable assumptions also the convergence to the exact wave function can be shown in the Sobolev H ¹-norm. The error of the ground state energy computation is estimated by an Aubin Nitsche type approach. Although the Projected Coupled Cluster method is nonvariational it shares advantages with the Galerkin or CI method. In addition it provides size consistency, which is considered as a fundamental property in many particle quantum mechanics.     

GL method for solving equations in math-physics and engineering

In this paper, we propose a GL method for solving the ordinary and the partial differential equation in mathematical physics and chemics and engineering. These equations govern the acustic, heat, electromagnetic, elastic, plastic, flow, and quantum etc. macro and micro wave field in time domain and frequency domain. The space domain of the differential equation is infinite domain which includes a finite inhomogeneous domain. The inhomogeneous domain is divided into finite sub domains. We present the solution of the differential equation as an explicit recursive sum of the integrals in the inhomogeneous sub domains. Actualy, we propose an explicit representation of the inhomogeneous parameter nonlinear inversion. The analytical solution of the equation in the infinite homogeneous domain is called as an initial global field. The global field is updated by local scattering field successively subdomaln by subdomain. Once all subdomains are scattered and the updating process is finished in all the sub domains, the solution of the equation is obtained. We call our method as Global and Local field method, in short , GL method. It is different from FEM method, the GL method directly assemble inverse matrix and gets solution. There is no big matrix equation needs to solve in the GL method. There is no needed artificial boundary and no absorption boundary condition for infinite domain in the GL method. We proved several theorems on relationships between the field solution and Green＇s function that is the theoretical base of our GL method. The numerical discretization of the GL method is presented. We proved that the numerical solution of the GL method convergence to the exact solution when the size of the sub domain is going to zero. The error estimation of the GL method for solving wave equation is presented. The simulations show that the GL method is accurate, fast, and stable for solving elliptic, parabolic, and hyperbolic equations. The GL method has advantages and wide applications in the 3D electromagnetic （EM）     

Singular soliton solution and bifurcation analysis of Klein-Gordon equation with power law nonlinearity

In this paper, the Klein-Gordon equation (KGE) with power law nonlinearity will be considered. The bifurcation analysis as well as the ansatz method of integration will be applied to extract soliton and other wave solutions. In particular, the second approach to integration will lead to a singular soliton solution. However, the bifurcation analysis will reveal several other solutions that are of prime importance in relativistic quantum mechanics where this equation appears.     

Taking parastatistical corrections to the Bose-Einstein distribution into account in the quantum and classical cases

We use number-theoretical methods to study the problem of particle Bose-condensation to zero energy. The parastatistical correction to the Bose-Einstein distribution establishes a relation between the quantum mechanical and statistical definitions of the Bose gas and permits correctly defining the condensation point as a gap in the spectrum in the one-dimensional case, proving the existence of the Bose condensate in the two-dimensional case, and treating the negative pressure in the classical theory of liquids as the pressure of nanopores (holes).     

Nonsingularity in the no-boundary Universe

In the no-boundary Universe of Hartle and Hawking, the path integral for the quantum state of the Universe must be summed only over nonsingular histories. If the quantum corrections to the Hamilton-lacobi equation in the interpretation of the wave packet is taken into account, then all classical trajectories should be nonsingular. The quantum behaviour of the classical singularity in theS ¹×S ^m model (m⩾2) is also clarified. It is argued that the Universe should evolve from the zero momentum state, instead from a zero volume state, to a 3-geometry state.     

A mathematical analysis of quantum transport in three-dimensional crystals

Summary We present an existence and uniqueness result for a quantum transport model in three dimensional crystals. The model consists of a quantum transport (Wigner) equation posed on the phase space consisting of a discrete position variable and a «continuous» wave vector, which is restricted to a bounded domain inR ³ (first Brillouin zone of the crystal). The potential is modeled self-consistently by a discrete Poisson equation (Coulomb interaction). Also we investigate the limits of solutions of this model as the grid spacing tends to zero and show that they converge to the solution of a quantum transport model posed on the «fully continuous» phase space. The transport model derived by this limiting procedure treats the band diagram of the crystal in a semi-classical way and the potential energy term quantum mechanically.     

New explicit solutions for the Klein-Gordon equation with cubic nonlinearity

In this paper, we investigate Klein-Gordon equation with cubic nonlinearity. All explicit expressions of the bounded travelling wave solutions for the equation are obtained by using the bifurcation method and qualitative theory of dynamical systems. These solutions contain bell-shaped solitary wave solutions, kink-shaped solitary wave solutions and Jacobi elliptic function periodic solutions. Moreover, we point out the region which these periodic wave solutions lie in. We present the relation between the bounded travelling wave solution and the energy level h. We find that these periodic wave solutions tend to the corresponding solitary wave solutions as h increases or decreases. Finally, for some special selections of the energy level h, it is shown that the exact periodic solutions evolute into solitary wave solution.     

Thermodynamic Equilibrium in the System of Chaotic Quantized Vortices in a Weakly Imperfect Bose Gas

In the example of a weakly imperfect Bose gas, we discuss the mechanism of establishing thermodynamic equilibrium for a chaotic set of quantum vortex filaments. We assume that the dynamics of the Bose condensate is described by the Gross–Pitaevsky equation with an additional noise satisfying the fluctuation–dissipation theorem. In considering a vortex filament as the intersection line of surfaces on which the real and imaginary parts of the order parameter (x,t) vanish, we obtain an equation of the Langevin type for elements of the vortex filament with an appropriately transformed random force. The Fokker–Planck equation for the probability density has a solution given by the Gibbs distribution at the temperature of the Bose condensate. In other words, when the Bose condensate is in thermal equilibrium and no other random actions exist, the system of vortices is also in thermal equilibrium.