Dirac operators in Boson-Fermion Fock spaces and supersymmetric quantum field theory

Infinite dimensional analysis is developed on an abstract Boson-Fermion Fock space. A general class of Dirac operators acting there is introduced and properties of them are investigated. An index theorem for the Dirac operators is established in terms of a path integral on a loop space. It is shown that the abstract formalism presented here gives a mathematical unification for some models of supersymmetric quantum field theory.     

The quantum stochastic equation is unitarily equivalent to a symmetric boundary value problem for the Schrödinger equation

We prove that the solution of the Hudson-Parthasarathy quantum stochastic differential equation in the Fock space coincides with the solution of a symmetric boundary value problem for the Schrödinger equation in the interaction representation generated by the energy operator of the environment. The boundary conditions describe the jumps in the phase and the amplitude of the Fourier transforms of the Fock vector components as any of its arguments changes the sign. The corresponding Markov evolution equation (the Lindblad equation or the “master equation”) is derived from the boundary value problem for the Schrödinger equation.     

Method for construction of operators in Fock space

It is shown, by providing a general method for the construction that any Fock space linear operator defined on the dense linear manifold spanned by the particle number representation basis can be represented in terms of the annihilation and creation operators. The normal form of the representation is unique.     

利用|x〉导出λ■+μ■的本征矢

利用坐标本征矢|x〉在Fock空间的展开式,导出了算符λ■+μ■在Fock空间的本征矢|ψ〉,并证明了其正交完备性.结果表明,得到的|ψ〉既是完备的,又是正交的,完全可以作为一个表象使用.当λ和μ分别取1和0时,|ψ〉约化为坐标本征态|x〉,而当λ和μ分别取0和1时,|ψ〉约化为动量本征态|p〉.因此,|ψ〉所构成的表象是介于坐标表象和动量表象之间的中介表象.     

Fock空间上的对偶Toeplitz代数

通过符号映射研究Fock空间之正交补空间上对偶Toeplitz代数的结构,得到了Fock空间上对偶Toeplitz代数的一个短正合序列.并研究了对偶Toeplitz算子谱的性质.     

Cross approximation in tensor electron density computations

We propose new tensor approximation algorithms for certain discrete functions related with Hartree–Fock/Kohn–Sham equations. Given a canonical tensor representation for the electron density function (for example, produced by quantum chemistry packages such as MOLPRO), we obtain its Tucker approximation with much fewer parameters than the input data and the Tucker approximation for the cubic root of this function, which is part of the Kohn–Sham exchange operator. The key idea is in the fast and accurate prefiltering of possibly large‐scale factors of the canonical tensor input. The new algorithms are based on the incomplete cross approximation method applied to matrices and tensors of order 3 and outperform other tools for the same purpose. First, we show that the cross approximation method is robust and much faster than the singular value decomposition‐based approach. As a consequence, it becomes possible to increase the resolution of grid and the complexity of molecules that can be handled by the Hartree–Fock chemical models. Second, we propose a new fast approximation method for f^1/3(x, y, z), based on the factor prefiltering method for f(x, y, z) and certain mimic approximation hypothesis. Third, we conclude that the Tucker format has advantages in the storage and computation time compared with the ubiquitous canonical format. Copyright © 2009 John Wiley & Sons, Ltd.     

Convergence of Discrete Green's Functions for Finite Difference Schemes

AMS(MOS): 65L10

The convergence of the discrete Green's function g_h is studied for finite difference schemes approximating m-th order linear two-point boundary value problems. Schemes of noncompact form and in part of the paper also nonuniform grids are admitted. Sharp convergence results are obtained for the difference quotients of g_h up to order m-1.     

Semiempirical fine-tuning for Hartree–Fock ionization potentials of atomic ions with non-integral atomic number

Amovilli and March (2006) [8] used diffusion quantum Monte Carlo techniques to calculate the non-relativistic ionization potential I(Z)$I(Z)$ in He-like atomic ions for the range of (fractional) nuclear charges Z   lying between the known critical value _Zc=0.911$Z_{\mathrm{c}}=0.911$ at which I(Z)$I(Z)$ tends to zero and Z=2$Z=2$. They showed that it is possible to fit I(Z)$I(Z)$ to a simple quadratic expression. Following that idea, we present here a semiempirical fine-tuning of Hartree–Fock ionization potentials for the isoelectronic series of He, Be, Ne, Mg and Ar-like atomic ions that leads to excellent estimations of _Zc$Z_{\mathrm{c}}$ for these series. The empirical information involved is experimental ionization and electron affinity data. It is clearly demonstrated that Hartree–Fock theory provides an excellent starting point for determining I(Z)$I(Z)$ for these series.