Linear Boltzmann equation as the weak coupling limit of a random Schrödinger equation

We study the time evolution of a quantum particle in a Gaussian random environment. We show that in the weak coupling limit the Wigner distribution of the wave function converges to a solution of a linear Boltzmann equation globally in time. The Boltzmann collision kernel is given by the Born approximation of the quantum differential scattering cross section. © 2000 John Wiley & Sons, Inc.     

On dissipative stochastic equations in a Hilbert space

Summary A general existence and uniqueness theorem for solutions of linear dissipative stochastic differential equation in a Hilbert space is proved. The dual equation is introduced and the duality relation is established. Proofs take inspirations from quantum stochastic calculus, however without using it. Solutions of both equations provide classical stochastic representation for a quantum dynamical semigroup, describing quantum Markovian evolution. The problem of the mean-square norm conservation, closely related to the unitality (non-explosion) of the quantum dynamical semigroup, is considered and a hyperdissipativity condition, ensuring such conservation, is discussed. Comments are given on the existence of solutions of a nonlinear stochastic differential equation, introduced and discussed recently in physical literature in connection with continuous quantum measurement processes.     

The Stratonovich Interpretation of Quantum Stochastic Approximations

The Stratonovich version of non-commutative stochastic calculus is introduced and shown to be equivalent to the Itô version developed by Hudson and Parthasarathy [1]. The conversion from Stratonovich to Itô version is shown to be implemented by a stochastic form of Wick's theorem: that is, involving the normal ordering of time-dependent noise fields. It is shown for a model of a quantum mechanical system coupled to a Bosonic field in a Gaussian state that under suitable scaling limits, in particular the weak coupling limit (for linear interactions) and low density limit (for scattering interactions), the limit form of the dynamical equation of motion is most naturally described as a quantum stochastic differential equation of Stratonovich form. We then convert the limit dynamical equations from Stratonovich to Itô form. Thermal Stratonovich noises are also presented.     

A probabilistic framework for the continuous observation of a quantum process

This paper treats quantum measurement within von Neumann's abstract framework. Specifically, observation is defined as a fixed self-adjoint operator with countable spectrum and nondegenerate eigenstates. Suppose scenarios for the observation of a quantum process over time are expanded by adding extra observations at time points interspersed among those of a previous scenario. If each observation leads to a mixture of eigenstates rather than a pure state, then the naturally defined joint probability measures on observed results are not consistent as scenarios vary. Nevertheless, we characterize the limiting subprobability measure when the times of observation become infinitely dense in any finite interval. This limiting measure corresponds to a continuous-time sub-stochastic process which decays with exponential rate out of any initial state and never reappears in any other state. Thus the process loses probability exponentially over time, and this loss occurs equally fast in the case of nonselective observation as for selective observation.Previous treatments of this problem have concentrated on the special case when Zeno's Paradox is in force, i.e. the rate of decay out of any state is zero and the process is immobilized by continuous observation. This situation exists, for instance, when the initial state is in the domain of the generator for the unitary group underlying the quantum process.     

广义Liénard方程极限环的惟一性问题

本文证明了广义Lienard方程极限环的一个惟一性定理,并用它证明了具有 稀疏效应的捕食-食饵系统在其正奇点外围至多有一个极限环.     

Quantum and non-causal stochastic calculus

Summary The quantum stochastic calculus initiated by Hudson and Parthasarathy, and the non-causal stochastic calculus originating with the papers of Hitsuda and Skorohod, are two potent extensions of the Itô calculus, currently enjoying intensive development. The former provides a quantum probabilistic extension of Schrödinger's equation, enabling the construction of a Markov process for a quantum dynamical semigroup. The latter allows the treatment of stochastic differential equations which involve terms which anticipate the future. In this paper the close relationship between these theories is displayed, and a noncausal quantum stochastic calculus, already in demand from physics, is described.     

具有细链双曲无穷远鞍点的二次系统

本文得到：具有细链双曲无穷远鞍点和一个细焦点的二次系统至多存在一个极限环,若有细无穷远分界线环S,则其内部不存在极限环,其稳定性与它包围的奇点的稳定性相反．     

Uniqueness of Limit Cycle for the Quadratic Systems with Weak Saddle and Focus

It is proved that the quadratic system with a weak saddle has at most one limit cycle, and that if this system has a separatrix cycle passing through the weak saddle, then the stability of the separatrix cycle is contrary to that of the singular point surrounded by it.     

Normal forms of quadratic vector fields and the Shi Songling equation

We suggest a method that permits one to locate limit cycles of planar vector fields with a weak focus. We use this method to analyze a large domain on the phase plane for the Shi Songling equation. The limit cycles in a neighborhood of the singular point (0, 0) are located in narrow annuli, in which their uniqueness is proved.     

Uniqueness of Limit Cycle for the Quadratic Systems with Weak Saddle and Focus

It is proved that the quadratic system with a weak saddle has at most one limit cycle,andthat if this system has a separatrix cycle passing through the weak saddle,then the stability of theseparatrix cycle is contrary to that of the singular point surrounded by it.     

Phase space Feynman path integrals via piecewise bicharacteristic paths and their semiclassical approximations

We give a fairly general class of functionals for which the phase space Feynman path integrals have a mathematically rigorous meaning. More precisely, for any functional belonging to our class, the time slicing approximation of the phase space path integral converges uniformly on compact subsets of the phase space. Our class of functionals is rich because it is closed under addition and multiplication. The interchange of the order with the Riemann integrals, the interchange of the order with a limit and the perturbation expansion formula hold in the phase space path integrals. The use of piecewise bicharacteristic paths naturally leads us to the semiclassical approximation on the phase space.     

On the Navier-Stokes equations: Existence theorems and energy equalities

Currently available results on the solvability of the Navier-Stokes equations for incompressible non-Newtonian fluids are presented. The order of nonlinearity in the equations may be variable; the only requirement is that it must be a measurable function. Unsteady and steady equations are considered. A lot of attention is paid to the recovery of energy balance, whose violation is theoretically admissible, in particular, in the three-dimensional classical unsteady Navier-Stokes equation. When constructing a weak solution by a limit procedure, a measure arises as a limit of viscous energy densities. Generally speaking, the limit measure contains a nonnegative singular (with respect to the Lebesgue measure) component. It is this singular component that maintains energy balance. Sufficient conditions for the absence of a singular component are studied: in this case, the standard energy equality holds. In many respects, only the regular component of the limit measure is important: in the natural form it is equal to the product of the viscous stress tensor and the gradient of a solution; if this natural form is retained, then the problem is solvable. Conditions are found for the validity of the indicated fundamental representation of the absolutely continuous component of the limit measure.     

On the stability of some groups of formal diffeomorphisms by the Birkhoff decomposition

Let G_∞ be the group of one parameter identity-tangent diffeomorphisms on the line whose coefficients are formal Laurent series in the parameter ε with a pole of finite order at 0. It is well known that the Birkhoff decomposition can be defined in such a group. We investigate the stability of the Birkhoff decomposition in subgroups of G_∞ and give a formula for this decomposition.These results are strongly related to renormalization in quantum field theory, since it was proved by A. Connes and D. Kreimer that, after dimensional regularization, the unrenormalized effective coupling constants are the image by a formal identity-tangent diffeomorphism of the coupling constants of the theory. In the massless theory, this diffeomorphism is in G_∞ and its Birkhoff decomposition gives directly the bare coupling constants and the renormalized coupling constants.     

Existence of a nontrivial weak solution to quasilinear elliptic equations with singular weights and multiple critical exponents

In this paper, we consider the existence of a non-trivial weak solution to a quasilinear elliptic equation with singular weights and multiple critical exponents in the whole space. Firstly, we get the existence of a local Palais–Smale sequence by verifying the geometric conditions of the Mountain Pass Lemma. Secondly, we study the concentration properties of the Palais–Smale sequence of a zero weak limit. Thirdly, we deduce by contradiction to eliminate the possibility of a zero weak limit case. Lastly, applying a monotonic inequality, we shall prove that the nontrivial weak limit of the Palais–Smale sequence is indeed a weak solution.     

Lie algebras and recurrence relations IV: Representations of the Euclidean group and Bessel functions

Representations of a complex form of the Lie algebra of the Euclidean group of the plane are found. Bases for the corresponding Hilbert spaces are given in terms of Lommel polynomials and Bessel functions. An element of the Lie algebra is interpreted as a quantum random variable leading to the Lommel distributions. The tails of these distributions and limit theorems are studied. Applications to electronic music are noted, as well as connection with quantum mechanics on a one-dimensional lattice.     

Derivation of Hartreeʼs theory for generic mean-field Bose systems

In this paper we provide a novel strategy to prove the validity of Hartree?s theory for the ground state energy of bosonic quantum systems in the mean-field regime. For the known case of trapped Bose gases, this can be shown using the strong quantum de Finetti theorem, which gives the structure of infinite hierarchies of k-particles density matrices. Here we deal with the case where some particles are allowed to escape to infinity, leading to a lack of compactness. Our approach is based on two ingredients: (1) a weak version of the quantum de Finetti theorem, and (2) geometric techniques for many-body systems. Our strategy does not rely on any special property of the interaction between the particles. In particular, our results cover those of Benguria–Lieb and Lieb–Yau for, respectively, bosonic atoms and boson stars.     

On a model of quantum field theory

A model of quantum field theory in an accelerated frame of reference is considered. It was suggested by Unruh that a uniformly accelerated detector in vacuum would perceive a noise with a thermal Gibbsian distribution. However, in justifying the assertion a singular transformation was implicitly performed, and doubts were expressed by some researches. We discuss a model of quantum field theory in an accelerated frame of reference in the two-dimensional spacetime for the wave equation. By using the Mellin transform, we obtain a representation of solutions of the wave equation. The representation includes a dependence on a parameter. The Unruh field corresponds to a singular limit of the representation.     

Existence and blow-up of the solutions to the viscous quantum magnetohydrodynamic nematic liquid crystal model

In this paper, we investigate the coupled viscous quantum magnetohydrodynamic equations and nematic liquid crystal equations which describe the motion of the nematic liquid crystals under the magnetic field and the quantum effects in the two-dimensional case. We prove the existence of the global finite energy weak solutions by use of a singular pressure close to vacuum. Then we obtain the local-in-time existence of the smooth solution. In the final, the blow-up of the smooth solutions is studied. The main techniques are Faedo-Galerkin method, compactness theory, Arzela-Ascoli theorem and construction of the functional differential inequality.