Equations For Green Functions Research Articles

Fast Time Domain Integral Equation Solvers for Analyzing Two-Dimensional Scattering Phenomena; Part I: Temporal Acceleration

The primary impediment to analyzing two-dimensional transient scattering phenomena using classical marching-on-in-time–based integral equation solvers is these schemes' high computational complexity that scales as O (Ns 2 Nt 2, where Ns and Nt denote the number of spatial and temporal degrees of freedom of the current on the scatterer. Here, three schemes that reduce this cost by permitting the rapid evaluation of the temporal convolution of a bandlimited transient source signature with the two-dimensional wave equation Green function are studied; these three methods rely on (i) blocked fast Fourier transforms, (ii) truncated singular value decompositions, and (iii) multiresolution concepts. The computational cost of all three proposed algorithms scales as O(Ns 2Nt logα Nt) with α ≤ 2. The three schemes are compared on their respective multiplicative constants inherent in this cost estimate, their memory requirements, and their ease of implementation.

Resonant transmission through two impurities in a narrow quantum wire

We study electron transmission through two impurities in a narrow quantum wire by solving Dyson’s equations for single electron Green functions. We have verified that, for the delta-function potential of two impurities, the Green function can be factorized into a product of the ‘free’ Green function and current transmission amplitude. Meanwhile Green function and current transmission amplitude obey Fisher-Lee’s relation. An analytical expression of the electron transmission amplitude for intrasubband and intersubband is obtained as a function of Fermi energy and the distance between two impurities. The resonant behavior of the current transmission amplitude are detail discussed.

Langevin equation with scale-dependent noise

A new wavelet based technique for the perturbative solution of the Langevin equation is proposed. It is shown that for the random force acting in a limited band of scales the proposed method directly leads to a finite result with no renormalization required. The one-loop contribution to the Kardar-Parisi-Zhang equation Green function for the interface growth is calculated as an example.

Nonlinearσmodel for long-range disorder and quantum chaos

We suggest a new scheme of derivation of a non-linear ballistic $\sigma$-model for a long range disorder and quantum billiards. The derivation is based on writing equations for quasiclassical Green functions for a fixed long range potential and exact representation of their solutions in terms of functional integrals over supermatrices $Q$ with the constraint $Q^2=1$. Averaging over the long range disorder or energy we are able to write a ballistic $\sigma$-model for all distances exceeding the electron wavelength. Neither singling out slow modes nor a saddle-point approximation are used in the derivation. Carrying out a course graining procedure that allows us to get rid off scales in the Lapunov region we come to a reduced $\sigma$-model containing a conventional collision term. For quantum billiards, we demonstrate that, at not very low frequencies, one can reduce the $\sigma$-model to a one-dimensional $\sigma$-model on periodic orbits. Solving the latter model, first approximately and then exactly, we resolve the problem of repetitions.

Nonequilibrium Green's function formulation of quantum transport theory for multi-band semiconductors

We present a nonequilibrium Green's function formulation of many-body quantum transport theory for multi-band semiconductor systems with a phonon bath. The equations are expressed exactly in terms of single particle nonequilibrium Green's functions and self-energies, treating the open electron–hole system in weak interaction with the bath. A decoupling technique is employed to separate the individual band Green's function equations of motion from one another, with the band–band interaction effects embedded in “cross-band” self-energies. This nonequilibrium Green's function formulation of quantum transport theory is amenable to solution by parallel computing because of its formal decoupling with respect to inter-band interactions. Moreover, this formulation also permits coding the simulator of an n-band semiconductor in terms of that for an ( n−1)-band system, in step with the current tendency and development of programming technology. Finally, the focus of these equations on individual bands provides a relatively direct route for the determination of carrier motion in energy bands, and to delineate the influence of intra- and inter-band interactions. A detailed description is provided for three-band semiconductor systems.

Theory of strongly correlated electron systems: Hubbard–Anderson models from an exact Hamiltonian, and perturbation theory near the atomic limit within a nonorthogonal basis set

AbstractThe theory of correlated electron systems is formulated in a form that allows the use as a reference point a density functional theory in the local density approximation (LDA DFT) for solids or molecules. The theory is constructed in two steps. As a first step, the total Hamiltonian is transformed into a correlated form. To elucidate the microscopic origin of the parameters of the periodic Hubbard–Anderson model (PHAM) the terms of the full Hamiltonian that have the operator structure of PHAM are separated. It is found that the matrix element of mixing interaction includes ion configuration‐ and number of particles‐dependent contributions from the Coulomb interaction. In a second step the diagram technique (DT) is developed by means of generalization of the Baym–Kadanoff method for correlated systems. The advantages of the method are that: (1) A nonorthogonal basis can be used, in particular the one generated by LDA DFT; (2) the equations for Green's functions (GFs) for the Fermi and Bose types of quasiparticles can be formulated in the form of a closed system of functional equations. The latter allows us to avoid the question of the nonunique decoupling procedure existing in previous versions of the DT and perform the expansion in terms of dressed GFs. Although the expressions for all interactions depend on the overlap matrix, it is shown that the theory is formally equivalent to one with orthogonal states with redefined interactions. When the PHAM is treated from the atomic‐limit side the vertexes are generated by kinematic interactions. The latter arise due to nontrivial commutation relations between X‐operators and come from the mixing, hopping, and overlap of states. The equations for GFs are derived within the nonorthogonal basis set in Hubbard‐I, one‐loop and random‐phase approximations with respect to kinematic interactions. The self‐consistent equation for “Hubbard Us” is derived. The technique developed is general, in particular its “bosonic” part can be used for description of spin systems with arbitrary anisotropy, systems with orbital ordering, or ordering of multipoles. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem 94: 113–143, 2003

A new technique for the derivation of closed-form electromagnetic Green's functions for unbounded planar layered media

A closed-form electromagnetic Green's function for unbounded, planar, layered media is derived in terms of a finite sum of Hankel functions. The derivation is based on the direct inverse Hankel transform of a pole-residue representation of the spectral-domain form of the Green's function. Such a pole-residue form is obtained through the solution of the spectral-domain form of the governing Green's function equation numerically, through a finite-difference approximation, rather than analytically. The proposed methodology can handle any number of layers, including the general case where the planar media exhibit arbitrary variation in their electrical properties in the vertical direction. The numerical implementation of the proposed methodology is straightforward and robust, and does not require any preprocessing of the spectrum of the Green's function for the extraction of surface-wave poles or its quasi-static part. The number of terms in the derived closed-form expression is chosen adaptively with the distance between source and observation point as parameter. The development of the closed-form Green's function is presented for both vertical and horizontal dipoles. Its accuracy is verified through a series of numerical examples and comparisons with results from other established methods.

Ab initio calculation of optical absorption in semiconductors: A density-matrix description

We show how to describe Coulomb renormalization effects and dielectric screening in semiconductors and semiconductor nanostructures within a first-principles density-matrix description. Those dynamic variables and approximation schemes which are required for a proper description of dielectric screening are identified. It is shown that within the random-phase approximation the direct Coulomb interactions become screened, with static screening being a good approximation, whereas the electron-hole exchange interactions remain unscreened. Differences and similarities of our results with those obtained from a corresponding GW approximation and Bethe-Salpeter equation Green's function analysis are discussed.

Gravitational Potentials of Triaxial Ellipsoids in Weyl Gravity

We present analytic expressions for the gravitational potentials associated with triaxial ellipsoids, spheroids, spheres and disks in Weyl gravity. The gravitational potentials of these configurations in Newtonian gravity, i.e. the potentials derived by integration of the Poisson equation Green's function 1/|r − r′| over the volume of the configuration, are well known in the literature. Herein we present the results of the integration of |r − r′|, the Green's function associated with the fourth order Laplacian ∇4 of Weyl gravity, over the volume of the configuration to obtain the resulting gravitational potentials within this specific theory. As an application of our calculations, we solve analytically Euler's equations pertaining to incompressible rotating fluids to show that, as in the case of Newtonian gravity, homogeneous prolate configurations are not allowed within Weyl gravity either.

Integration of Poisson Equation for the Diamond-Structure Semiconductors by the Green Function Cellular Method (GFCM)

Solving the Poisson equation for the electrostatic potential in a solid is an integral part of a modern electronic structure calculation. In this work, the Poisson equation for the diamond-structure semiconductors is solved using the Green Function Cellular Method. The charge density was obtained from a first principle consideration of the atomic wave functions for the electrons. The real solid harmonics JL, and HL are used in expanding the Poisson equation Green function, therefore, most of the calculations were done in the spherical coordinates system. The maximum value of the index L considered is six. The Poisson equation is solved only for the state k = 0 in this work. The results show that the total electron-electron Coulomb potential, VCE(rn) is given mainly by the central cell contribution and thus varies in a similar fashion. From a certain value at the origin, the total potential initially increases with the distance up to a certain point and then decreases as the distance is further advanced. The peak potential is roughly at about 0.05d in carbon, 0.03d in silicon and germanium, and 0.02d in gray tin. The rate of convergence of the external cells potential contribution over the l summation is faster than the central cell contribution, while the total potential rate is the slowest. In general the rate of convergence at a given point depends on the distance and direction of the point. The mathematical formulation and the results of the calculations are presented and discussed

Two-Time Temperature Green's Functions in Kinetic Theory and Molecular Hydrodynamics: III. Taking the Interaction of Hydrodynamic Fluctuations into Account

We describe a systematic method for constructing equations for Green's functions in molecular hydrodynamics and kinetics problems. The method allows consecutively accounting for the contribution to the generalized kinetic coefficients due to the interaction of two, three, and more hydrodynamic modes. In contrast to the standard perturbation theory in the coupling constant, the consecutive approximations are taken with respect to the degree of higher correlations described by irreducible functions.

Remarks on vacuum fluctuations around a spinning cosmic string

We apply the point-splitting method to investigate vacuum fluctuations in the nonglobally hyperbolic background of a spinning cosmic string. Implementing renormalization by removing the Minkowski contribution it is shown that although the Green function in the literature satisfies the usual Green function equation, it leads to pathological physical results.

The connection between factorization properties and closed-form solutions of certain linear dyadic differential operators

Representation of the electromagnetic field in terms of dyadic Green functions leads to the requirement to solve dyadic partial differential equations for the dyadic Green functions. While a formal solution for the infinite-medium problem can be given in a straightforward manner for even the most general, linear medium, the extraction of closed-form expressions is a complicated issue. The existence of such expressions for the dyadic Green functions is closely linked to the factorization properties of the determinant operator, which is associated with the dyadic differential operator of the dyadic Green functions. This connection is investigated for a special type of homogeneous, anisotropic dielectric medium.

Transport parameters for an ultrasonic pulsed wave propagating in a multiple scattering medium

A set of ultrasonic experimental methods was developed to characterize a multiple scattering medium in terms of l(s), l*, l(a), respectively, the elastic, transport, and absorption mean free paths and D the diffusion constant. Actually, these quantities are the key parameters for a wave propagating in a disordered medium. Although they are widely used in optics, they are less common in acoustics. The underlying model is based on the expansion of the average solution for the heterogeneous Green's function equation. To validate this theoretical approach, a sample made of randomly located steel rods was used as a prototype. Through time-resolved measurements of the transmitted amplitude, the difference between the ballistic and the coherent wave is highlighted. In varying the sample thickness, l(s) is determined, the coherent and diffusive regime are distinguished, and the transition from one to the other is followed. Furthermore, as a limit to a description of the average intensity based on the diffusion approximation, the existence of a coherent backscattering effect is shown. This latter gives a method to estimate D and l*. These quantities being determined, it becomes possible to infer l(a) using average time-resolved intensity measurements. Finally, some applications to coarse-grain stainless steels are discussed.

Equation of motion approach to the two-dimensional Hubbard model

Using a Green's-function equation of motion approach the two-dimensional Hubbard model is studied. The decoupling of the hierarchy is performed according to the conserving of the first four spectral moments. Beyond the Roth's two-pole approximation, the damping effect occurs naturally. The energy dispersions at various fillings, the spectral weight, and the density of states are calculated. Comparisons with the quantum Monte Carlo results and the cluster perturbation theory show that our approach can capture some essential features of the Hubbard model.

New Developments in D-dimensional conformal quantum field theory

A review of recent developments in conformal quantum field theory in D-dimensional space is presented. The conformally invariant solution of the Ward identities is studied. We demonstrate the existence of D-dimensional analogues of primary and secondary fields, the central charge, and the null vectors. The Hilbert space is shown to possess a specific model-independent structure defined by the 1 2 (D+1)(D+2) -dimensional symmetry and the Ward identities. In particular, there exists a sector H of the Hilbert space related to an infinite family of “secondary” fields which are generated by the currents and the energy-momentum tensor. The general solution of the Ward identities in D>2 defining the sector H necessarily includes the contribution of the gauge fields. We derive the conditions which single out the conformal theories of a direct (non-gauge) interaction. We examine the class of models satisfying these conditions. It is shown that the Green functions of the current and the energy-momentum tensor in these models are uniquely determined by the Ward identities for any D≥2. The anomalous Ward identities containing contributions of c-number and operator analogues of the central charge, are discussed. Closed sets of expressions for the Green functions of secondary fields are obtained in D-dimensional space. A family of exactly solvable conformal models in D≥2 is constructed. Each model is defined by the requirement of vanishing of a certain field Q s , s=1,2…. The fields Q s are constructed as definite superpositions of secondary fields. After that, one requires each field Q s to be primary. The latter is possible for specific values of scale dimensions of fundamental fields (a D-dimensional analogue of the Kac formula). The states Q s |0〉 are analogous to null vectors. One can derive closed sets of differential equations for higher Green functions in each of the models. These results are demonstrated on examples of several exactly solvable models in D>2. The approach developed here is based on the finite-dimensional conformal symmetry for any D≥2. However the family of models under consideration does have the structure identical to that of two-dimensional conformal theories. This analogy is discussed in detail. It is shown that when D=2, the above family coincides with the well-known family of models based on infinite-dimensional conformal symmetry. The analysis of this phenomenon indicates the possibility of existence of D-dimensional analogue of the Virasoro algebra.

Wiener–Hopf Method Applied to the X-ray Edge Problem

We apply the Wiener–Hopf method of solving convolutive integral equations on a semi-infinite interval to the X-ray edge problem. Dyson equations for basic Green functions from the X-ray problem are rewritten as convolutive integral equations on a time-interval [0,t] with t→∞. The long-time asymptotics of solutions to these equations is derived with the aid of the Wiener–Hopf method. Although the Wiener–Hopf long-time exponents differ by a factor of two from the solution of Nozières and De Dominicis we demonstrate how the latter and the critical exponents of measurable amplitudes from the X-ray problem can be derived from the former. We explain that the difference in the exponents arises due to different ways of performing the long-time limit in the two solutions. To enable the infinite-time limit in the defining equations a new infinite-time scale τ→∞, interpreted as an effective lifetime of the core-hole, must be introduced. The ratio t/τ decides about the resulting critical exponent. The physical relevance of the Nozières and De Dominicis as well as of the Wiener–Hopf exponents is discussed.

Time domain direct and inverse problems for a nonuniform LCRG line with internal sources

In this paper, direct and inverse problems for a nonuniform LCRG line with internal transient voltage and current sources are considered. In the inverse source problem, the compact Green functions approach based on wave splitting is used to reconstruct the transient voltage and current sources from the transient signals that are received at the two endpoints of the nonuniform line. The location of the sources is assumed to be known or possible to estimate from the arrival times of the signals. The compact Green functions are convolution kernels in the map from the received signals (at the two endpoints) to the internal split voltages. The partial differential equations for the compact Green functions are given together with the initial and boundary conditions. The present approach to the inverse source problem is exact and noniterative. To obtain independent synthetic input data for the inverse problem, the direct problem is solved by a finite difference scheme. Numerical results for reconstructions using both clean and noisy data are presented.

Tight-Binding Linear Muffin-Tin Orbital Implementation of the Difference Equation Green's Function Approach for 2D-Periodic Systems

ABSTRACTThe central ideas and the advantages of the difference equation Green's function approach for layered systems with 2D-periodicity are presented. The tight-binding linear muffin-tin orbital and other practical aspects of the implementation are discussed. Preliminary test results are presented.

Reconstruction of Source and Medium Parameters via Wave-Splitting and Green Function Equations

This paper applies the time-domain wave-splitting and the Green function concepts to study two inverse problems involving an internal transient source embedded in a nondispersive inhomogeneous medium. These two inverse problems are (1) given the physical one-way travel time Green operator kernel $\bar G^ - ( {0,t} )$ and the source-space distribution function, reconstruct the wave-speed profile $c( z )$ of the medium; (2) given the same kernel $\bar G^ - ( {0,t} )$ and the medium wave-speed function, recover the source-space distribution profile $p( z )$. Both numerical inversion algorithms and approximate-reconstruction formulas are presented. Several computational examples are given.