Stochastic Schrodinger Equation for a Quantum Oscillator with Dissipation

In this paper, we construct an exact solution of the stochastic Schrodinger equation for a quantum oscillator with possible dissipation of energy taken into account. Using the explicit form of the solution, we calculate estimates for the characteristic damping time of free damped oscillations. In the case of forced oscillations, we obtain formulas for the Q-factor of the system and for the variance of the coordinate and momentum of a quantum oscillator with dissipation. We obtain the quantum analog of the classical diffusion equation and explicitly show that the equations of motion for the mean value of the momentum operator following from the solution of the stochastic Schrodinger equation play the role of the quantum Langevin equation describing Brownian motion under the action of a stochastic force.     

A Generalized Coordinate-Momentum Representation in Quantum Mechanics

We obtain a one-parameter family of (q, p)-representations of quantum mechanics; the Wigner distribution function and the distribution function we previously derived are particular cases in this family. We find the solutions o the evolution equations or the microscopic classical and quantum distribution functions in the form of integrals over paths in a phase space. We show that when varying canonical variables in the Green’s function of the quantum Liouville equation, we must use the total increment o the action functional in its path-integral representation, whereas in the Green’s function of the classical Liouville equation, the linear part o the increment is sufficient. A correspondence between the classical and quantum schemes holds only under a certain choice of the value of the distribution family parameter. This value corresponds to the distribution unction previously found.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 143, No. 3, pp. 401–416, June, 2005.     

Quantum Laplacian and Applications

In this paper, we give an explicit solution of some linear quantum white noise differential equations by applying the convolution calculus on a suitable distribution space. In particular, we give an integral representation for the solution of the quantum heat equation.     

What Is a Quantum Stochastic Differential Equation from the Point of View of Functional Analysis?

We prove that a quantum stochastic differential equation is the interaction representation of the Cauchy problem for the Schrödinger equation with Hamiltonian given by a certain operator restricted by a boundary condition. If the deficiency index of the boundary-value problem is trivial, then the corresponding quantum stochastic differential equation has a unique unitary solution. Therefore, by the deficiency index of a quantum stochastic differential equation we mean the deficiency index of the related symmetric boundary-value problem.In this paper, conditions sufficient for the essential self-adjointness of the symmetric boundary-value problem are obtained. These conditions are closely related to nonexplosion conditions for the pair of master Markov equations that we canonically assign to the quantum stochastic differential equation.     

A competition between Fujita and Strauss type exponents for blow-up of semi-linear wave equations with scale-invariant damping and mass

We obtain a blow-up result for solutions to a semi-linear wave equation with scale-invariant dissipation and mass and power non-linearity, in the case in which the model has a “wave like” behavior. We perform a change of variables that transforms our starting equation in a strictly hyperbolic semi-linear wave equation with time-dependent speed of propagation. Applying Kato's lemma we prove a blow-up result for solutions to the transformed equation under some assumptions on the initial data. The limit case, that is, when the exponent p is exactly equal to the upper bound of the range of admissible values of p yielding blow-up needs special considerations. In this critical case an explicit integral representation formula for solutions of the corresponding linear Cauchy problem in 1d is derived. Finally, carrying out the inverse change of variables we get a non-existence result for global (in time) solutions to the original model.     

Equilibration Rate for the Linear Inhomogeneous Relaxation-Time Boltzmann Equation for Charged Particles

Abstract

We study the long-time behavior of a linear inhomogeneous Boltzmann equation. The collision operator is modeled by a simple relaxation towards the Maxwellian distribution with zero mean and fixed lattice temperature. Particles are moving under the action of an external potential that confines particles, i.e., there exists a unique stationary probability density. Convergence rate towards global equilibrium is explicitly measured based on the entropy dissipation method and apriori time independent estimates on the solutions. We are able to prove that this convergence is faster than any algebraic time function, but we cannot achieve exponential convergence.     

Vacuum and Thermal Energies for Two Oscillators Interacting Through A Field

We consider a simple (1+1)-dimensional model for the Casimir–Polder interaction consisting of two oscillators coupled to a scalar field. We include dissipation in a first-principles approach by allowing the oscillators to interact with heat baths. For this system, we derive an expression for the free energy in terms of real frequencies. From this representation, we derive the Matsubara representation for the case with dissipation. We consider the case of vanishing intrinsic frequencies of the oscillators and show that the contribution from the zeroth Matsubara frequency is modified in this case and no problem with the laws of thermodynamics appears.     

Is nature quantum non-local,complex holistic,or what?: II — Applications

In Part I of this paper [A. Sengupta, Is nature quantum non-local, complex holistic, or what? I–Theory & analysis. Nonlinear Anal.: RWA (2009) (in press)] to be referenced “I-”, we examined the linear–nonlinear divide of the natural world in an attempt to seek a rationale for the question “Is nature interactively nonlinear and holistic, or is it additively linear and reductionist?”: Is Nature governed by entanglements of linear superposition or does it represent the nonlinear holism of emergence, self-organization, and complexity? This second part carries the debate forward to propose that Quantum Mechanics is an effective linear representation of a fully chaotic, maximally illposed, multifunctional negworld that obviously is not just a mirror image of the functional real world we inhabit: in fact we argue that nonlinear complex holism represents a stronger form of entanglement than linear quantum non-locality. The bi-directionality of a self-organized, emergent, engine-pump system is analyzed with reference to the role of gravity as the compressive agent responsible for generation and maintenance of structures and life in Nature; we also explore the applicability of chanoxity to the metaphorical resolution of some of the long-standing paradoxes and puzzles in quantum measurement and non-locality, in Prigoginian intrinsic irreversibility, and in some core issues in cosmology and gravitational black holes.Holism is to be seen as complementing mainstream reductionism–linear science has after all stood the test of the last 400 years as quantum mechanics is acknowledgedly one of the most successful yet possibly one of the most mysterious of scientific theories: the success lies in its capacity to classify and predict the physical world, the mystery in what this physical world must be like to behave quantum mechanically–providing a unified picture of the dialectics of the evolutionary dynamics of Nature.     

一类线性积分方程解的存在唯一性

在这篇文章内我们证明了一类线性积分方程存在唯一解．并给出解的表示式．它以Ｆｒｅｄｈｏｌｍ方程和Ｖｏｌｔｅｒｒａ方程为特殊情形．     

Nonelement boundary representation with Bézier surface patches for 3D linear elasticity problems in parametric integral equation system (PIES) and its solving using Lagrange polynomials

In this article, we present a strategy of using rectangular and triangular Bézier surface patches for nonelement representation of 3D boundary geometries for problems of linear elasticity. The boundary generated in this way is directly incorporated in the parametric integral equation system (PIES), which has been developed by the authors. The boundary values on each surface patch are approximated by Lagrange polynomials. Three illustrative examples are presented to confirm the effectiveness of the proposed boundary representation in connection with PIES and to show good accuracy of numerical results.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 34: 51–79, 2018     

Global nonlinear stability of longitudinal wave for the planar motion of elastic string with linear Hooke's law

In this paper, we consider the global nonlinear stability of longitudinal wave for the planar motion of elastic string with linear Hooke's law. First, we will give the representation of the traveling wave solution to the recast hyperbolic system. Then, by using the weighted function method, we obtain the global stability of longitudinal wave solution for nonlinear elastic string equation.     

Numbers of subnormal solutions for higher order periodic differential equations

In this paper, we estimate the number of subnormal solutions for higher order linear periodic differential equations, and estimate the growth of subnormal solutions and all other solutions. We also give a representation of subnormal solutions of a class of higher order linear periodic differential equations.     

非线性Schrdinger方程的高精度守恒差分格式

本文首先分析线性Schrodinger方程一种高阶差分格式的构造方法,得到方程的耗散项.在此基础上对三次非线性Schrodinger方程,提出了一种精度为O(r2 h2)的差分格式,证明了该格式保持了连续方程的两个守恒量,且是收敛的与稳定的.并通过数值例子与已有隐格式进行了比较,结果表明,本文格式在计算量类似的情况下,提高了数值精度.     

“Quantizations” of the second Painlevé equation and the problem of the equivalence of its <Emphasis Type="Italic">L</Emphasis>-<Emphasis Type="Italic">A</Emphasis> pairs

We show that the known Flaschka-Newell L-A pair for the second Painlevé equation gives solutions to linear evolutionary equations similar to the quantum Schrödinger equations. Using the Fourier transform for distributions, we derive this pair from the classical Garnier pair.     

Aspects of compact quantum group theory

We show that if a compact quantum semigroup satisfies certain weak cancellation laws, then it admits a Haar measure, and using this we show that it is a compact quantum group. Thus, we obtain a new characterization of a compact quantum group. We also give a necessary and sufficient algebraic condition for the Haar measure of a compact quantum group to be faithful, in the case that its coordinate -algebra is exact. A representation is given for the linear dual of the Hopf -algebra of a compact quantum group, and a functional calculus for unbounded linear functionals is derived.

     

Stability of discrete-time positive evolution operators on ordered Banach spaces and applications

In this paper we discuss stability problems for a class of discrete-time evolution operators generated by linear positive operators acting on certain ordered Banach spaces. Our approach is based upon a new representation result that links a positive operator with the adjoint operator of its restriction to a Hilbert subspace formed by sequences of Hilbert–Schmidt operators. This class includes the evolution operators involved in stability and optimal control problems for linear discrete-time stochastic systems. The inclusion is strict because, following the results of Choi, we have proved that there are positive operators on spaces of linear, bounded and self-adjoint operators which have not the representation that characterize the completely positive operators. As applications, we introduce a new concept of weak-detectability for pairs of positive operators, which we use to derive sufficient conditions for the existence of global and stabilizing solutions for a class of generalized discrete-time Riccati equations. Finally, assuming weak-detectability conditions and using the method of Lyapunov equations we derive a new stability criterion for positive evolution operators.     

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In this paper, we establish a local representation theorem for generators of reflected backward stochastic differential equations (RBSDEs), whose generators are continuous with linear growth. It generalizes some known representation theorems for generators of backward stochastic differential equations (BSDEs). As some applications, a general converse comparison theorem for RBSDEs is obtained and some properties of RBSDEs are discussed.     

Nonautonomous Hamiltonian quantum systems,operator equations,and representations of the Bender–Dunne Weyl-ordered basis under time-dependent canonical transformations

We solve the problem of integrating operator equations for the dynamics of nonautonomous quantum systems by using time-dependent canonical transformations. The studied operator equations essentially reproduce the classical integrability conditions at the quantum level in the basic cases of one-dimensional nonautonomous dynamical systems. We seek solutions in the form of operator series in the Bender–Dunne basis of pseudodifferential operators. Together with this problem, we consider quantum canonical transformations. The minimal solution of the operator equation in the representation of the basis at a fixed time corresponds to the lowest-order contribution of the solution obtained as a result of applying a canonical linear transformation to the basis elements.     

耗散力学和非线性耗散型方程的精确解——耗散力学原理和应用(Ⅰ)

本文是文[1～4]的继续和升华.(1)在本文中,我们根据互补性原理,建立了耗散力学.它是与量子力学相对应的一种耗散理论.可以用这种理论来统一地处理非平衡态热力学和粘滞流体动力学问题,并可以用它来处理量子力学中各种耗散和不可逆的问题.耗散力学的基本方程是与Schr?dinger方程或Dirac方程相对应的一类本征值方程;(2)在本文中,我们将一些基本的非线性耗散型方程,特别是作为宏观非平衡态热力学和粘滞流体动力学基本方程的Navier-Stokes方程,统一地归结为耗散力学基本方程的可积性条件,从而为利用散射反演方法求它们的精确解扫平了道路.     

Quantum volterra model and the universalR-matrix

We prove that an integrable system solved by the quantum inverse scattering method can be described by a purely algebraic object (universal R-matrix) and a proper algebraic representation. For the quantum Volterra model, we construct the L-operator and the fundamental R-matrix from the universal R-matrix for the quantum affine algebra U_q(ŝl₂) and the q-oscillator representation for it. Thus, there is an equivalence between an integrable system with the symmetry algebra A and the representation of this algebra. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 113, No. 3, pp. 384–396, December, 1997.