Note on the Norm Convergence of the Unitary Trotter Product Formula

We discuss when the unitary Trotter product formula converges in operator norm and apply the results obtained to the unitary group generated by the Dirac operator and the relativistic Schrödinger operator with suitable potentials.     

On Error Estimates for the Trotter–Kato Product Formula

We study the error bound in the operator-norm topology for the Trotter exponential product formula as well as for its generalization à la Kato. Within the framework of an abstract setting, we give a simple proof of error estimates which improve some recent results in this direction.     

Optimal Error Estimates in Operator-Norm Approximations of Semigroups

We demonstrate that an idea related to the Central Limit Theorem and approximations by accompanying laws in probability theory is useful to get optimal convergence rates in some approximation formulas for operators. As examples we provide a bound for Euler approximations of bounded holomorphic semigroups; a bound for error in approximation of a power of operators by accompanying exponents, which is a useful tool in analysis of the Trotter–Kato formula, and can be considered as an extended version of Chernoff's lemma.     

A New Differential Formula About Product of Polynomials and Its Application in Multi-electron State Physics

We derive a new differential formula about a kind of product of polynomials and then apply it to analyze some properties of multi-electron state ＂related to Laughlin wave function＂. The entangled state representation for describing electrons in uniform magnetic field is used.     

A New Differential Formula About Product of Polynomials and Its Application in Multi-electron State Physics

We derive a new differential formula about a kind of product of polynomials and then apply it to analyze some properties of multi-electron state “related to Laughlin wave function“. The entangled state representation for describing electrons in uniform magnetic field is used.     

Note on the Paper “The Norm Convergence¶ of the Trotter–Kato Product Formula with Error Bound” by Ichinose and Tamura

The norm convergence of the Trotter–Kato product formula is established with ultimate optimal error bound for the selfadjoint semigroup generated by the operator sum of two selfadjoint operators. A generalization is also given to the operator sum of several selfadjoint operators. Received: 5 October 2000 / Accepted: 12 March 2001     

Weyl-Ordered Operator Product Formula Derived by Technique of Integration Within Weyl-Ordered Product of Operators

Based on the explicit Weyl-ordered form of Wigner operator and the technique of integration within Weylordered product of operators we derive the Weyl-ordered operator product formula. The formula is then generalized to the entangled form with the help of entangled state representations.     

Sharp Spectral Asymptotics and Weyl Formula for Elliptic Operators with Non-smooth Coefficients

The aim of this paper is to give the Weyl formula for eigenvalues of self-adjoint elliptic operators, assuming that first-order derivatives of the coefficients are Lipschitz continuous. The approach is based on the asymptotic formula of Hörmander"s type for the spectral function of pseudodifferential operators having Lipschitz continuous Hamiltonian flow and obtained via a regularization procedure of nonsmooth coefficients.     

关于结合能公式的一种改进方法与普适的修正公式

将核子间的相互作用简化为核力键,从而根据核力键假设推导出结合能公式中^[1]的体积能项与表面能项;并利用21H的结合能的实验值给出二者的系数值,从而使得结合能公式的参数个数减少了两个.     

重核自发裂变寿命的新公式

利用Swiatecki寿命公式及其推广形式系统计算了重核的自发裂变寿命，并提出一个新的四参数自发裂变寿命公式，公式计算结果和实验数据能够很好地符合. 对于Z≥90的重核存在一条N=Z+52的长寿命线. 新公式可以预言长寿命线附近的重核自发裂变寿命.     

Convergence acceleration for the multilevel Hartree–Fock model

In the previously introduced multilevel Hartree–Fock (HF) model, the electronic density is optimised in a given region of the molecular system. The approach is based on generating an active occupied and active virtual space by decomposing a start guess density for the entire system. In this work, a diagonalisation based implementation for Roothaan–Hall (RH) with direct inversion in iterative subspace (DIIS) and a quasi-Newton minimisation procedure using the augmented RH (ARH) approach are described for accelerating convergence for the multilevel HF model. The equations are derived to be consistent with convergence acceleration for traditional atomic orbital based HF calculations. The main idea is to formulate all quantities in the molecular orbital basis to exploit that the active molecular orbital basis is significantly smaller than the atomic orbital basis, and thus enable the application of wave function approaches that are well-studied for small molecular systems to large molecular systems. Thus, all equations are formulated such that no atomic orbital density or Fock matrices are needed for the DIIS and ARH algorithms. Results show that the acceleration schemes yield efficient optimisation of the multilevel HF wave function.     

Creation Operators for the Macdonald and Jack Polynomials

Formulas of Rodrigues-type for the Macdonald polynomials are presented. They involve creation operators, certain properties of which are proved and other conjectured. The limiting case of the Jack polynomials is discussed.     

The Entropy Formula for SRB-Measures of Lattice Dynamical Systems

We give a detailed proof of the entropy formula for SRB-measures of coupled hyperbolic attractors over integer lattices. We show that the topological pressure for the potential function of the SRB-measure is zero.     

用推广的Viola-Seaborg公式计算α衰变寿命

根据Audi等编评的最新核数据表, 对Z≥84, N≥128的核的α衰变实验寿命用Viola-Seaborg公式进行最小方差拟合, 得到一套新参数. 用新拟合参数计算的偶偶核的衰变寿命与实验寿命很好地符合, 平均偏差1.3倍.而对奇A和奇奇核理论寿命与实验寿命相差较远. 考虑到处于基态时奇A和奇奇核与偶偶核的主要区别在于自旋和宇称, 对Viola-Seaborg公式做了推广. 在公式中引入了离心势垒(即α粒子轨道角动量)对衰变寿命的影响. 由推广公式计算的奇A和奇奇核的寿命与实验寿命平均偏差2.5倍,最大偏差5—6倍.     

Convergence of a Distributional Monte Carlo Method for the Boltzmann Equation

Direct Simulation Monte Carlo (DSMC) methods for the Boltzmann equation employ a point measure approximation to the distribution function, as simulated particles may possess only a single velocity. This representation limits the method to converge only weakly to the solution of the Boltzmann equation. Utilizing kernel density estimation we have developed a stochastic Boltzmann solver which possesses strong convergence for bounded and $L^\infty$ solutions of the Boltzmann equation. This is facilitated by distributing the velocity of each simulated particle instead of using the point measure approximation inherent to DSMC. We propose that the development of a distributional method which incorporates distributed velocities in collision selection and modeling should improve convergence and potentially result in a substantial reduction of the variance in comparison to DSMC methods. Toward this end, we also report initial findings of modeling collisions distributionally using the Bhatnagar-Gross-Krook collision operator.     

Unifying the theory of integration within normal-, Weyl- and antinormal-ordering of operators and the s-ordered operator expansion formula of density operators

By introducing the s-parameterized generalized Wigner operator into phase-space quantum mechanics we invent the technique of integration within s-ordered product of operators(which considers normally ordered,antinormally ordered and Weyl ordered product of operators as its special cases).The s-ordered operator expansion(denoted by...) formula of density operators is derived,which is ρ = 2 1 s ∫ d2βπβ|ρ |β exp { 2 s 1(s|β|2 β a + βa a a) }.The s-parameterized quantization scheme is thus completely established.     

A Construction of Berezin–Toeplitz Operators via Schrödinger Operators and the Probabilistic Representation of Berezin–Toeplitz Semigroups Based on Planar Brownian Motion

First we discuss the construction of self-adjoint Berezin–Toeplitz operators on weighted Bergman spaces via semibounded quadratic forms. To ensure semiboundedness, regularity conditions on the real-valued functions serving as symbols of these Berezin–Toeplitz operators are imposed. Then a probabilistic expression of the sesqui-analytic integral kernel for the associated semigroups is derived. All results are the consequence of a relation of Berezin–Toeplitz operators to Schrödinger operators defined via certain quadratic forms. The probabilistic expression is derived in conjunction with the Feynman–Kac–Itô formula.     

A Partial Derivative Approach to the Change of Scale Formula for the Function Space Integral

We investigate the partial derivative approach to the change of scale formula for the functon space integral and we investigate the vector calculus approach to the directional derivative on the function space and prove relationships among the Wiener integral and the Feynman integral about the directional derivative of a Fourier transform.     

The First Principle Calculation of Green—Kubo Formula with the Two—Time Ensemble Technique

For weak-coupling electronic plasmas in the presence of small fluctuations,we calculate Green-Kuboformula for transport coefficients employing the ab initio theory,i.e.,the two-time ensemble technique.The microscopic interpretations of Onsager‘s hypothesis on fluctuations are presented in the framework.