一维有限超晶格的电子态与透射问题的转移矩阵方法研究

采用转移矩阵方法,研究了一维有限超晶格的电子态与透射问题.计算了一维有限超晶格含单个缺陷层或少量缺陷层的透射谱和波函数,以及当电子被束缚在一维有限超晶格中电子的本征值和相应的定态本征函数.给出的方法对于研究电子通过任意排列的一维有限超晶格的输运具有普适性.     

Linear differential equations for the one‐dimensional scattering problem

The transmission and reflection amplitudes of an electron moving in a one dimensional potential of arbitrary form are obtained using the transfer matrix method. It is shown that the one‐dimensional scattering problem, in its most general form, can be reduced to Cauchy problem for a set of two linear differential equations.     

Superfluid to Mott‐insulator transition in an anisotropic two‐dimensional optical lattice

We study the superfluid to Mott‐insulator transition of bosons in an optical anisotropic lattice by employing the Bose‐Hubbard model living on a two‐dimensional lattice with anisotropy parameter κ. The compressible superfluid state and incompressible Mott‐insulator (MI) lobes are efficiently described analytically, using the quantum U(1) rotor approach. The ground state phase diagram showing the evolution of the MI lobes is quantified for arbitrary values of κ, corresponding to various kind of lattices: from square, through rectangular to almost one‐dimensional.     

Gödel type metrics in three dimensions

We show that the Gödel type metrics in three dimensions with arbitrary two dimensional background space satisfy the Einstein-perfect fluid field equations. We also show that there exists only one first order partial differential equation satisfied by the components of fluid’s velocity vector field. We then show that the same metrics solve the field equations of the topologically massive gravity where the two dimensional background geometry is a space of constant negative Gaussian curvature. We discuss the possibility that the Gödel type metrics to solve the Ricci and Cotton flow equations. When the vector field u ^μ is a Killing vector field, we came to the conclusion that the stationary Gödel type metrics solve the field equations of the most possible gravitational field equations where the interaction lagrangian is an arbitrary function of the electromagnetic field and the curvature tensors.     

Exact sum rules for inhomogeneous drums

We derive general expressions for the sum rules of the eigenvalues of drums of arbitrary shape and arbitrary density, obeying different boundary conditions. The formulas that we present are a generalization of the analogous formulas for one dimensional inhomogeneous systems that we have obtained in a previous paper. We also discuss the extension of these formulas to higher dimensions. We show that in the special case of a density depending only on one variable the sum rules of any integer order can be expressed in terms of a single series. As an application of our result we derive exact sum rules for the homogeneous circular annulus with different boundary conditions, for a homogeneous circular sector and for a radially inhomogeneous circular annulus with Dirichlet boundary conditions.     

Wu's Equations and Quasi-Hypergeometric Functions

We investigate analytic solutions to Wu's equations and their symmetry property, singularities and monodromy, relating them to quasi hypergeometric functions. We finally give the monodromy theorem for them in one, two and arbitrary dimensional cases (see Theorem 1, 2 and 3). Received: 22 October 1999 / Accepted: 28 June 2001     

High order numerical methods to two dimensional delta function integrals in level set methods

In this paper we design and analyze a class of high order numerical methods to delta function integrals appearing in level set methods in two dimensional case. The methods comprise approximating the mesh cell restrictions of the delta function integral. In each mesh cell the two dimensional delta function integral can be rewritten as a one dimensional ordinary integral with the smooth integrand being a one dimensional delta function integral, and thus is approximated by applying standard one dimensional high order numerical quadratures and high order numerical methods to one dimensional delta function integrals proposed in [X. Wen, High order numerical methods to a type of delta function integrals, J. Comput. Phys. 226 (2007) 1952–1967]. We establish error estimates for the method which show that the method can achieve any desired accuracy by assigning the corresponding accuracy to the sub-algorithms and has better accuracy under an assumption on the zero level set of the level set function which holds generally. Numerical examples are presented showing that the second to fourth order methods implemented in this paper achieve or exceed the expected accuracy and demonstrating the advantage of using our high order numerical methods.     

Primary Branch Solutions of First Order Autonomous Scalar Partial Differential Equations via Lie Symmetry Approach

A primary branch solution (PBS) is defined as a solution with m independent n ? 1 dimensional arbitrary functions for an m order n dimensional partial differential equation (PDE). PBSs of arbitrary first order scalar PDEs can be determined by using Lie symmetry group approach companying with the introduction of auxiliary fields. Because of the intrusion of the arbitrary function, the PBSs have abundant and complicated structure. Usually, PBSs are implicit solutions. In some special cases, explicit solutions such as the instanton (rogue wave like) solutions may be obtained by suitably fixing the arbitrary function of the PBS.     

Two-loop anomalous-dimension matrix for soft-gluon exchange

The resummation of soft-gluon exchange for QCD hard scattering requires a matrix of anomalous dimensions. We compute this matrix directly for arbitrary 2-->n massless processes for the first time at two loops. Using color-generator notation, we show that it is proportional to the one-loop matrix. This result reproduces all pole terms in dimensional regularization of the explicit calculations of massless 2-->2 amplitudes in the literature, and it predicts all poles at next-to-next-to-leading order in any 2-->n process that has been computed at next-to-leading order. The proportionality of the one- and two-loop matrices makes possible the resummation in closed form of the next-to-next-to-leading logarithms and poles in dimensional regularization for the 2-->n processes.     

Local unitary classification of arbitrary dimensional multipartite pure states

We propose a practical entanglement classification scheme for general multipartite pure states in arbitrary dimensions under local unitary equivalence by exploiting the high order singular value decomposition technique and local symmetries of the states. By virtue of this scheme, the method of determining the local unitary equivalence of n-qubit states proposed by Kraus is extended to the case for arbitrary dimensional multipartite states.     

2＋1维双线性Sawada—Kotera方程的对称结构

对一类２＋１维双线性方程从两个不同程度建立了形式级数对称理论。从一已知的时间无关对称出发或从与一维空间坐标有关的任意函数发出，均可得到了包含时间任意函数的形式级数对称。对于２＋１维双线性Ｓａｗａｄａ－Ｋｏｔｅｒａ方程，存在６个载断对称。这些截断对称构成一无究维李代数。一些有意义的子代数（如Ｖｉｒａｓｏｒｏ代数等）也被给定。     

2+1维双线性Sawada-Kotera方程的对称结构

对一类２＋１维双线性方程从两个不同角度建立了形式级数对称理论。从一已知的时间无关对称出发或从与一维空间坐标有关的任意函数出发，均可得到一包含时间任意函数的形式级数对称。对于２＋１维双线性Ｓａｗａｄａ－Ｋｏｔｅｒａ方程，存在６个截断对称。这些截断对称构成一无穷维李代数。一些有意义的子代数（如Ｖｉｒａｓｏｒｏ代数等）也被给定。     

Localization or tunneling in asymmetric double-well potentials

An asymmetric double-well potential is considered, assuming that the wells are parabolic around the minima. The WKB wave function of a given energy is constructed inside the barrier between the wells. By matching the WKB function to the exact wave functions of the parabolic wells on both sides of the barrier, for two almost degenerate states, we find a quantization condition for the energy levels which reproduces the known energy splitting formula between the two states. For the other low-lying non-degenerate states, we show that the eigenfunction should be primarily localized in one of the wells with negligible magnitude in the other. Using Dekker’s method (Dekker, 1987), the present analysis generalizes earlier results for weakly biased double-well potentials to systems with arbitrary asymmetry.     

势阱中粒子能级与波函数微扰计算的代数递推公式

利用超位力定理（HVT）和Hellmann-Feynman定理（HFT）,导出了由有精确解的势阱的能级值用微扰法直接计算一维势阱的各级近似能级的普遍代数公式,并导出由能级近似值计算定态波函数近似表达式的代数公式,给出了代数公式具体应用的几个典型一维势阱实例,此法可推广到二维势阱与三维势阱的情形。     

Quantum well excitons in an electric field: two versus three dimensional behavior

Previous calculations have shown a transition between two dimensional and three dimensional behavior of excitons confined in a semiconducting quantum well structure as a function of electric field. We here present calculations of the exciton binding energy as a function of electric field using a two parameter variational wave function of the form used in the absence of the electric field by Matsuura and Shinozuka. Our calculations were performed using a finite potential barrier model for the confinement of the exciton in the quantum well. The results of our calculations confirm the validity of the conclusion that the variational exciton wave function goes from being of a purely 2D hydrogenic type at small well widths and/or low electric fields to a 3D hydrogenic type in wide wells and/or high electric fields.     

Semi-classical approximation to the single particle and spectral density

Approximating an arbitrary single particle potential by its expansion up to quadratic terms around the classical path may yield valuable approximations for the single particle and spectral density. We show this for a one dimensional model.     

Exact solution to a class of generalized Kitaev spin-1/2 models in arbitrary dimensions

We construct a class of exactly solvable generalized Kitaev spin-1/2 models in arbitrary dimensions, which is beyond the category of quantum compass models. The Jordan-Wigner transformation is employed to prove the exact solvability. An exactly solvable quantum spin-1/2 model can be mapped to a gas of free Majorana fermions coupled to static Z2 gauge fields. We classify these exactly solvable models according to their parent models. Any model belonging to this class can be generated by one of the parent models. For illustration, a two dimensional(2D) tetragon-octagon model and a three dimensional(3D) xy bond model are studied.     

Resonances in the current-voltage characteristics of a dissipative Josephson junction

The I-V characteristics of a Josephson junction shunted by an Ohmic resistor shows sharp peaks when levels in neighbouring wells are crossing. We consider the shape and size of these peaks using a double-well model where the wells are given in parabolic approximation. The friction of arbitrary strength is included by the help of a bath of infinitely many degrees of freedom. 74.50.+z.     

The quantum spectra analysis of the circular billiards in wells

We use a recently defined quantum spectral function and apply the method of closed-orbit theory to the 2D circular billiard system. The quantum spectra contain rich information of all classical orbits connecting two arbitrary points in the well. We study the correspondence between quantum spectra and classical orbits in the circular, 1/2 circular and 1/4 circular wells using the analytic and numerical methods. We find that the peak positions in the Fourier-transformed quantum spectra match accurately with the lengths of the classical orbits. These examples show evidently that semi-classical method provides a bridge between quantum and classical mechanics.     

有任意带满的紧束缚哈密顿量的实空间重整化群研究

本文运用实空间重整化群方法研究了有任意带满的无自旋的Fermi系统的行为。对于一维,求得了系统的基态能、相关函数和相关的临界分量,并与各向同性的s=1/2的X-Y模型进行了比较。对于D>1维,定性地讨论了系统的相关函数及临界分量。 关键词：