Interacting bosons in an optical lattice

A strongly interacting Bose gas in an optical lattice is studied using a hard‐core interaction. Two different approaches are introduced, one is based on a spin‐1/2 Fermi gas with attractive interaction, the other one on a functional integral with an additional constraint (slave‐boson approach). The relation between fermions and hard‐core bosons is briefly discussed for the case of a one‐dimensional Bose gas. For a three‐dimensional gas we identify the order parameter of the Bose‐Einstein condensate through a Hubbard‐Stratonovich transformation and treat the corresponding theories within a mean‐field approximation and with Gaussian fluctuations. This allows us to evaluate the phase diagram, including the Bose‐Einstein condensate and the Mott insulator, the density‐density correlation function, the static structure factor, and the quasiparticle excitation spectrum. The role of quantum and thermal fluctuations are studied in detail for both approaches, where we find good agreement with the Gross‐Pitaevskii equation and with the Bogoliubov approach in the dilute regime. In the dense regime, which is characterized by the phase transition between the Bose‐Einstein condensate and the Mott insulator, we discuss a renormalized Gross‐Pitaevskii equation. This equation can describe the macroscopic wave function of the Bose‐Einstein condensate in the dilute regime as well as close to the transition to the Mott insulator. Finally, we compare the results of the attractive spin‐1/2 Fermi gas and those of the slave‐boson approach and find good agreement for all physical quantities.     

Loop Observables for BF Theories in Any Dimension and the Cohomology of Knots

A generalization of Wilson loop observables for BF theories in any dimension is introduced within the Batalin–Vilkovisky framework. The expectation values of these observables are cohomology classes of the space of imbeddings of a circle. One of the resulting theories discussed in the Letter has only trivalent interactions and, irrespective of the actual dimension, looks like a three-dimensional Chern–Simons theory.     

On the computation of molecular auxiliary functions A n and B n

Molecular auxiliary functions A _n (p) and B _n (pt), arising in the Hartree-Fock-Roothaan (HFR) approximation for molecules, Ewald’s crystal lattice theory, electromagnetic stopping theory, and other approximate methods, are evaluated and analysed in the range of 17≤n≤60 and 25≤pt≤60.     

Functional Integrals and Free Energy in sine-Gordon-Thirring Model with Impurity Coupling

The free energy at low temperature in 1D sine-Gordon-Thirring model with impurity coupling is studied by means of functional integrals method. For massive free sine-Gordon-Thirring model, free energy is obtained from perturbation expansion of functional determinant. Moreover, the free energy of massive model is calculated by use of an auxiliary Bose field method.     

On Evaluation of Overlap Integrals with Noninteger Principal Quantum Numbers

By use of complete orthonormal sets of ψ^α exponential-type orbitals (ψ^α-ETOs,α=1,0,-1,-2,...) the series expansion formulas for the noninteger n Slater-type orbitals (NISTOs) in terms of integer n Slater-type orbitals (ISTOs) are derived. These formulas enable us to express the overlap integrals with NISTOs through the overlap integrals over ISTOs with the same and different screening constants. By calculating concrete cases the convergence of the series for arbitrary values of noninteger principal quantum numbers and screening constants of NISTOs and internuclear distances is tested. The accuracy of the results is quite high for quantum numbers, screening constants and location of STOs.     

关于积分中值定理的一个注记

研究当积分区间长度趋于无穷时 ,积分中值定理中间点的渐近性质     

一类平方函数的加幂权有界性

本文讨论平方函数 gJ (f ) 和幂权 Lp 有界性 ,其中 J(x ) = K(x′ )g(| x| ) ,K∈ H1 (Sn- 1 ),且 g(| x| ) 满足一定的条件.作为推论 ,本文得到了 Marcinkiewicz积分 _K( f ) 加幂权的 Lp 有界性.     

On the Asymptotic Behavior of Weighted U-Statistics

A sufficient condition for the convergence of degenerated weighted U-statistics of order k is proved. From this, the previous results by O'Neil and Redner⁽¹⁰⁾ for k = 2 and Major,⁽⁷⁾ which appeared to be very different, are related.     

Generalized Killing Tensors

Generalized Killing tensors are defined and the integrability conditions discussed to show that the familiar result that a space of constant curvature admits the maximum number of Killing vectors and second order Killing tensors does not necessarily generalize. The existence of second order Generalized Killing Yano tensors in spherically symmetric static space-times is investigated and a non-redundant example is given. It is proved that the natural vector analogue of the Lenz-Runge vector does not exist.     

Integral Representations of the Wilson Polynomials and the Continuous Dual Hahn Polynomials

We give representations of the Wilson polynomials and the continuous dual Hahn polynomials in terms of multidimensional generalizations of Barnes type integrals. Motivation is to study the Barnes type integrals from the viewpoint of the de Rham theory and holonomic systems.