伪自伴量子系统的酉演化与绝热定理

经典量子系统的哈密尔顿是自伴算子.哈密尔顿算符的自伴性不仅确保了系统遵循酉演化,而且也保证了它自身具有实的能量本征值.但是,确实有一些物理系统,其哈密尔顿是非自伴的,但也具有实的能量本征值,这种具有非自伴哈密尔顿的系统就是非自伴量子系统.具有伪自伴哈密尔顿的系统是一类特殊的非自伴量子系统,其哈密尔顿相似于一个自伴算子.本文研究伪自伴量子系统的酉演化与绝热定理.首先,给出了伪自伴算子定义及其等价刻画;其次,对于伪自伴哈密尔顿系统,通过构造新内积,证明了伪自伴哈密尔顿在新内积下是自伴的,并给出了系统在新内积下为酉演化的充分必要条件.最后,建立了伪自伴量子系统的绝热演化定理及与绝热逼近定理.     

The Quantum Faedo-Galerkin Equation: Evolution Operator and Time Discretization

Time dependent quantum systems have become indispensable in science and nanotechnology. Disciplines including chemical physics and electrical engineering have used approximate evolution operators to solve these systems for targeted physical quantities. Here, we discuss the approximation of closed time dependent quantum systems on bounded domains via evolution operators. The work builds upon the use of weak solutions, which includes a framework for the evolution operator based upon dual spaces. We are able to derive the corresponding Faedo-Galerkin equation as well as its time discretization, yielding a fully discrete theory. We obtain corresponding approximation estimates. These estimates make no regularity assumptions on the weak solutions, other than their inherent properties. Of necessity, the estimates are in the dual norm, which is natural for weak solutions. This appears to be a novel aspect of this approach.     

Feynman operator calculus: The constructive theory

In this paper, we survey progress on the Feynman operator calculus and path integral. We first develop an operator version of the Henstock-Kurzweil integral, construct the operator calculus and extend the Hille-Yosida theory. This shows that our approach is a natural extension of operator theory to the time-ordered setting. As an application, we unify the theory of time-dependent parabolic and hyperbolic evolution equations. Our theory is then reformulated as a sum over paths, providing a completely rigorous foundation for the Feynman path integral. Using our disentanglement approach, we extend the Trotter-Kato theory.     

Approximation by Bézier variants of the BBHK operators

In this work a new type of approximation operator—the Bézier variant of the BBHK operator—is introduced. Its approximation properties are studied. A convergence theorem for such approximation operators for locally bounded functions is established by means of some techniques of probability theory and analysis methods. This convergence theorem subsumes the approximation of functions of bounded variation as a special case.     

Feynman's Operational Calculus and Evolution Equations

We give an exposition and an extension of the ideas of Feynman's time-ordered operational calculus for noncommuting operators. Various directions for 'disentangling' functions of such operators are provided by measures on the time intervals in question. We concentrate especially on exponentials of sums of noncommuting operators and prove that the unique solution of a broad class of evolution equations is given by the time-dependent operators arrived at by disentangling such exponential expressions.     

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.     

Sesquilinear quantum stochastic analysis in Banach space

A theory of quantum stochastic processes in Banach space is initiated. The processes considered here consist of Banach space valued sesquilinear maps. We establish an existence and uniqueness theorem for quantum stochastic differential equations in Banach modules, show that solutions in unital Banach algebras yield stochastic cocycles, give sufficient conditions for a stochastic cocycle to satisfy such an equation, and prove a stochastic Lie–Trotter product formula. The theory is used to extend, unify and refine standard quantum stochastic analysis through different choices of Banach space, of which there are three paradigm classes: spaces of bounded Hilbert space operators, operator mapping spaces and duals of operator space coalgebras. Our results provide the basis for a general theory of quantum stochastic processes in operator spaces, of which Lévy processes on compact quantum groups is a special case.     

Quantum systems and representation theorem

In this paper we investigate quantum systems which are locally convex versions of abstract operator systems. Our approach is based on the duality theory for unital quantum cones. We prove the unital bipolar theorem and provide a representation theorem for a quantum system being represented as a quantum $L^{\infty }$ -system.     

Semigroup-theoretic proofs of the central limit theorem and other theorems of analysis

The theory of one parameter semigroups of bounded linear operators on Banach spaces has deep and far reaching applications to partial differential equations and Markov processes. Here we present some known elementary applications of operator semigroups to approximation theory, a new proof of the central limit theorem, and we give E. Nelson's rigorous interpretation of Feynman integrals. Our main tools are (i) a special case of the Trotter-Neveu-Kato approximation theorem, of which we give a new elementary proof, and (ii) P. Chernoff's product formula.     

Multi-Product Expansion with Suzuki's Method:Generalization

In this paper we discuss the extension to exponential splitting methods with respect to time-dependent operators. We concentrate on the Suzuki's method, which incorporates ideas into the time-ordered exponential of [3, 11, 12, 34]. We formulate the methods with respect to higher order by using kernels for an extrapolation scheme. The advantages include more accurate and less computational intensive schemes for special time-dependent harmonic oscillator problems. The benefits of the higher order kernels are given on different numerical examples.     

Time Evolution of Quadratic Quantum Systems: Evolution Operators,Propagators, and Invariants

Theoretical and Mathematical Physics - We use the evolution operator method to describe time-dependent quadratic quantum systems in the framework of nonrelativistic quantum mechanics. For...     

Integrable Potentials by Darboux Transformations in Rings and Quantum and Classical Problems

We study a problem in associative rings of left and right factorization of a polynomial differential operator regarded as an evolution operator. In a direct sum of rings, the polynomial arising in the problem of dividing an operator by an operator for two commuting operators leads to a time-dependent left/right Darboux transformation based on an intertwining relation and either Miura maps or generalized Burgers equations. The intertwining relations lead to a differential equation including differentiations in a weak sense. In view of applications to operator problems in quantum and classical mechanics, we derive the direct quasideterminant or dressing chain formulas. We consider the transformation of creation and annihilation operators for specified matrix rings and study an example of the Dicke model.     

Adiabatic approximation for the evolution generated by an <Emphasis Type="Italic">A</Emphasis>-uniformly pseudo-Hermitian Hamiltonian

We discuss an adiabatic approximation for the evolution generated by an A-uniformly pseudo-Hermitian Hamiltonian H(t). Such a Hamiltonian is a time-dependent operator H(t) similar to a time-dependent Hermitian Hamiltonian G(t) under a time-independent invertible operator A. Using the relation between the solutions of the evolution equations H(t) and G(t), we prove that H(t) and H^? (t) have the same real eigenvalues and the corresponding eigenvectors form two biorthogonal Riesz bases for the state space. For the adiabatic approximate solution in case of the minimum eigenvalue and the ground state of the operator H(t), we prove that this solution coincides with the system state at every instant if and only if the ground eigenvector is time-independent. We also find two upper bounds for the adiabatic approximation error in terms of the norm distance and in terms of the generalized fidelity. We illustrate the obtained results with several examples.     

Existence results for evolution hemivariational inequalities of second order

In this paper, we consider evolution hemivariational inequalities of second order with a time-dependent pseudomonotone operator and nonmonotone multivalued perturbations. We present the existence of solutions for such inequality. The proof profits from a result on the surjectivity of operators of pseudomonotone type. We discuss some examples which indicate the practical importance of our theoretical findings.     

The best approximation theorems and variational inequalities for discontinuous mappings in Banach spaces

We discuss Ky Fan's theorem and the variational inequality problem for discontinuous mappings f in a Banach space X. The main tools of analysis are the variational characterizations of the metric projection operator and the order-theoretic fixed point theory. Moreover, we derive some properties of the metric projection operator in Banach spaces. As applications of our best approximation theorems, three fixed point theorems for non-self maps are established and proved under some conditions. Our results are generalizations and improvements of various recent results obtained by many authors.     

Statistical approximation properties of q-Bleimann,Butzer and Hahn operators

The main aim of this study is to introduce a new generalization of q-Bleimann, Butzer and Hahn operators and obtain statistical approximation properties of these operators with the help of the Korovkin type statistical approximation theorem. Rates of statistical convergence by means of the modulus of continuity and the Lipschitz type maximal function are also established. Our results show that rates of convergence of our operators are at least as fast as classical BBH operators. The second aim of this study is to construct a bivariate generalization of the operator and also obtain the statistical approximation properties.     

非线性 Lipschitz 连续算子的定量性质(III)------glb-Lipschitz数

本文引进非线性Lipschitz算子T的glb-Lipschitz数l(T),并证明l(T)定量刻画非线性Lipschitz连续算子全体所构成的赋半范算子空间中可逆算子T保持可逆的最大扰动半径,因而具有特别重要意义.所获结果被应用来建立``非线性扰动引理'、非线性算子条件数、推广线性算子逼近理论和建立与矩阵理论中Gerschgorin圆盘定理对应的非线性Lipschitz连续算子谱集的包含域.     

Explicit Green operators for quantum mechanical Hamiltonians. I. The hydrogen atom

We study a new approach to determine the asymptotic behaviour of quantum many-particle systems near coalescence points of particles which interact via singular Coulomb potentials. This problem is of fundamental interest in electronic structure theory in order to establish accurate and efficient models for numerical simulations. Within our approach, coalescence points of particles are treated as embedded geometric singularities in the configuration space of electrons. Based on a general singular pseudo-differential calculus, we provide a recursive scheme for the calculation of the parametrix and corresponding Green operator of a nonrelativistic Hamiltonian. In our singular calculus, the Green operator encodes all the asymptotic information of the eigenfunctions. Explicit calculations and an asymptotic representation for the Green operator of the hydrogen atom and isoelectronic ions are presented.     

Estimates for restrictions of monotone operators on the cone of decreasing functions in Orlicz space

We introduce a number of notions related to the Lyapunov transformation of linear differential operators with unbounded operator coefficients generated by a family of evolution operators. We prove statements about similar operators related to the Lyapunov transformation and describe their spectral properties. One of the main results of the paper is a similarity theorem for a perturbed differential operator with constant operator coefficient, an operator which is the generator of a bounded group of operators. For the perturbation, we consider the operator ofmultiplication by a summable operator function. The almost periodicity (at infinity) of the solutions of the corresponding homogeneous differential equation is established.     

Vector-valued Ruelle operators

The scalar Ruelle operators have been extensively studied both in dynamical systems and iterated function systems. And the Ruelle-Perron-Frobenius theorem for scalar Ruelle operator was well-known. We mainly set up a vector analogue theory in this paper.