PT-对称算子的表示及非正交量子态的区分

经典量子系统中的哈密尔顿为自伴算子，这不仅保证了系统能量本征值全部为实数，而且相应的本征态（单位长度的特征向量）构成了状态空间的一组正规正交基.然而存在一类PT-对称的物理系统，哈密尔顿的自伴性（共轭转置）被物理的PT-对称性所代替.一个完整的PT-对称哈密尔顿，其谱全部为实数且能构造一个合理的CPT-内积.本文研究一类PT-对称算子.固定时间反演算子T，得到宇称算子P的矩阵表示，进而给出每一组PT-对称哈密尔顿的具体表示形式.作为应用，选择一组确定的{P，T}算子，及PT-对称的哈密尔顿，给出两个在传统量子力学中不正交的量子态区分的刻画.     

复完备随机内积模上的随机酉算子群的Stone表示定理

设[a,b]是有限实区间,(S,∥·∥)是完备的随机赋范模并赋予(ε,λ)-拓扑.在本文中,我们首先引进了从[a,b]到S的抽象值函数的Riemann积分并给出值域几乎处处有界的连续函数Riemann可积的一个充分条件.然后我们研究了随机谱测度和随机测度之间的关系.最后,在上述两个准备工作的基础之上,我们建立了复完备随机内积模上随机酉算子群的Stone表示定理.     

二项自伴向量微分算子的本质谱

首先利用算子比较的方法,研究了二项自伴向量微分算子的本质谱,得到了这类微分算子的本质谱分布范围;然后利用算子分解定理,得到了这类算子谱的离散性的一个充分条件;最后得到了Sturm-Liouville算子和Schr?dinger算子的本质谱范围,以及这两类算子谱的离散性的一个充分条件.     

Krein空间上算子的可定化性

从Hilbert空间(H,(·,·))上的一个有界自伴算子G可以导出不定内积[·,·]=(G·,·),本文给出了由G所导出的Krein空间上的G-自伴、G-酉以及G-正常算子的可定化、强可定化和一致可定化性质以及这三种不同的可定化性之间的关系.     

Krein空间上算子的可定化性

从Hilbert空间(H,(·,·))上的一个有界自伴算子G可以导出不定内积[·,·]:=(G·,·),本文给出了由G所导出的Krein空间上的G-自伴、G-酉以及G-正常算子的可定化、强可定化和一致可定化性质以及这三种不同的可定化性之间的关系.     

含内积倍数的两区间奇数阶微分算子自伴域的刻画

研究了含内积倍数的两端奇异两区间奇数阶自伴微分算子及其在直和空间上自伴域的刻画,并证明了在直和空间中运用内积倍数可以扩大自伴算子实现的范围.     

关于J－对称微分算子的J－自伴扩张的若干注记

本文给出了一条解析描述Ｊ－对称微分算子Ｊ－自伴扩张的新途径．我们借助方程Ｔ（ｙ）＝λｏｙ的解，而不是如文［３］利用方程＋（ｙ）＇＝－ｙ的解来描述Ｊ－对称微分算式的所有Ｊ－自伴域在奇异端点的边条件，不过我们假设生成的最小算子具非空正则域．我们对主要定理给出了若干有趣的注，得到了二阶极限圆情形的有趣结果．     

关于J－对称微分算子的J－自伴扩张的若干注记

本文给出了一条解析描述Ｊ－对称微分算子Ｊ－自伴扩张的新途径．我们借助方程Ｔ（ｙ）＝λｏｙ的解，而不是如文［３］利用方程＋（ｙ）＇＝－ｙ的解来描述Ｊ－对称微分算式的所有Ｊ－自伴域在奇异端点的边条件，不过我们假设生成的最小算子具非空正则域．我们对主要定理给出了若干有趣的注，得到了二阶极限圆情形的有趣结果．     

算子值酉随机矩阵及其渐近自由性

该文在算子值非交换概率空间上引入半标准酉随机矩阵的概念, 证明了它是算子值Haar酉元的矩阵模型,并给出了半标准酉随机矩阵的渐近自由判定定理.     

不定奇异微分算子的非实谱问题

考 虑具中间亏指数 的非负系数对 称微分算式 ，在 Ｋｒｅｉｎ 空间 Ｌ２ｒ ［ａ，∞） 内生 成的自伴 算子，本文证明此算 子是可定化算 子，从而得到其非 实谱仅由有限 个成共轭对的特 征值构成     

极限圆型Hamilton算子乘积的自伴性

本文讨论了极限圆型Hamilton算子乘积的自伴性,利用Calkin方法及奇异Hamilton系统自伴扩张的一般构造理论,给出了在极限圆型时判定Hamilton算子乘积自伴的一个充要条件.     

Pontryagin’s theorem and spectral stability analysis of solitons

The main result of the present paper is the use of Pontryagin’s theorem for proving a criterion, based on the difference in the number of negative eigenvalues between two self-adjoint operators L _? and L ₊, for the linear part of a Hamiltonian system to have eigenvalues with strictly positive real part (unstable eigenvalues).     

Rotating Singular Perturbations of the Laplacian

We study a system of a quantum particle interacting with a singular time-dependent uniformly rotating potential in 2 and 3 dimensions: in particular we consider an interaction with support on a point (rotating point interaction) and on a set of codimension 1 (rotating blade). We prove the existence of the Hamiltonians of such systems as suitable self-adjoint operators and we give an explicit expression for the unitary dynamics. Moreover we analyze the asymptotic limit of large angular velocity and we prove strong convergence of the time-dependent propagator to some one-parameter unitary group as Communicated by Gian Michele GrafSubmitted 07/10/03, accepted 09/12/03     

Level shift operators for open quantum systems

Level shift operators describe the second-order displacement of eigenvalues under perturbation. They play a central role in resonance theory and ergodic theory of open quantum systems at positive temperatures. We exhibit intrinsic properties of level shift operators, properties which stem from the structure of open quantum systems at positive temperatures and which are common to all such systems. They determine the geometry of resonances bifurcating from eigenvalues of positive temperature Hamiltonians and they relate the Gibbs state, the kernel of level shift operators, and zero energy resonances. We show that degeneracy of energy levels of the small part of the open quantum system causes the Fermi Golden Rule Condition to be violated and we analyze ergodic properties of such systems.     

N-body quantum scattering theory in two Hilbert spaces. III. Theory of approximations

A rigorous mathematical theory of approximations is developed for abstract nonrelativistic quantum scattering systems within the two-Hilbert-space framework. An approximate space of asymptotic states and an approximate asymptotic Hamiltonian must be specified initially. An approximate N-particle Hamiltonian is then constructed and proved to be self-adjoint. Approximate wave operators are shown to exist and, in certain interesting cases, to be asymptotically complete. Certain sequences of the approximate wave operators are proved to converge to the exact wave operators in an appropriate limit. Thus approximate scattering operators are unitary and converge to the exact scattering operator.     

On Dynamical Properties of a One-Parameter Family of Transformations Arising in Averaging of Semigroups

To describe the dynamics of quantum systems with degenerate symmetric but not self-adjoint Hamiltonian, we consider the Naimark extension of the Hamiltonian to a self-adjoint operator in an extended Hilbert space. We relate to the symmetric Hamiltonian a one-parameter family of averaged dynamical transformations of the set of quantum states obtained from a unitary group of transformations of the extended Hilbert space by using a conditional expected value to an algebra of bounded operators acting in the original space. We establish the absence of the semigroup property and injectivity of the family of averaged dynamical transformations. We obtain a representation of trajectories of the averaged family of dynamical transformations by maximum points of functionals on the space of mappings of the time interval into the set of quantum states.     

Discretization of time-dependent quantum systems: real-time propagation of the evolution operator

We discuss time-dependent quantum systems on bounded domains. Our work may be viewed as a framework for several models, including linear iterations involved in time-dependent density functional theory, the Hartree-Fock model, or other quantum models. A key aspect of the analysis of the algorithms is the use of time-ordered evolution operators, which allow for both a well-posed problem and its approximation. The approximation theorems obtained for the time-ordered evolution operators complement those in the current literature. We discuss the available theory at the outset, and proceed to apply the theory systematically in later sections via approximations and a global existence theorem for a nonlinear system, obtained via a fixed point theorem for the evolution operator. Our work is consistent with first-principle real-time propagation of electronic states, aimed at finding the electronic responses of quantum molecular systems and nanostructures. We present two full 3D quantum atomistic simulations using the finite element method for discretizing the real space, and the FEAST eigenvalue algorithm for solving the evolution operator at each time step. These numerical experiments are representative of the theoretical results.     

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.