Extrapolation algorithm for the terminator problem in the recursion method

The problem of finding the Green's function of spin and electromagnetic waves in the sinusoidal superlattice is considered. An analytical expression for the spectral representation of the Green's function has been found in the form of ascending continued fractions, the particular denominators of which are ordinary continued fractions. The Green’s function in the-space has been found by the numerical Fourier transformation of the Greens’s function found in the spectral representation.

An efficient and accurate contour integration method is presented for large-scale electronic structure calculations based on the Green function. By introducing a continued fraction representation of the Fermi-Dirac function derived from a hypergeometric function, the Matsubara summation is generalized with respect to distribution of poles so that the integration of the Green function can converge rapidly. Numerical illustrations, evaluation of the density matrix for a simple model Green function and a total energy calculation for aluminum bulk within density functional theory, clearly show that the method provides remarkable convergence with a small number of poles, indicating that the method can be applied to not only the electronic structure calculations, but also a wide variety of problems.

We consider the application of the recursion method to the calculation of one-particle Green's functions for strongly correlated systems and propose a new way how to extract the information about the infinite system from the exact diagonalisation of small clusters. Comparing the results for several cluster sizes allows us to establish those Lanczos coefficients that are not affected by the finite size effects and provide the information about the Green's function of the macroscopic system. The analysis of this 'bulk-related' subset of coefficients supplemented by alternative analytic approaches allows to infer their asymptotic behaviour and to propose an approximate analytical form for the 'terminator' of the Green's function continued fraction expansion for the infinite system. As a result, the Green's function acquires the branch cut singularity corresponding to the incoherent part of the spectrum. The method is applied to the spectral function of one-hole in the Majumdar-Ghosh model (the one-dimensional $ t-J-J^{\prime}$ model at $J^{\prime}/J=1/2$). For this model, the branch cut starts at finite energy $\omega_0$, but there is no upper bound of the spectrum, corresponding to a linear increase of the recursion coefficients. Further characteristics of the spectral function are band gaps in the middle of the band and bound states below $\omega_0$ or within the gaps. The band gaps arise due to the period doubling of the unit cell and show up as characteristic oscillations of the recursion coefficients on top of the linear increase.

We applied special graphical technique to investigate the influence of side/end groups of macromolecules upon long range electron transfer. We show that in case of acyclic side/end groups the Green function may be cast in the form of continued fraction, similarly to the result of electron transfer in random media. The structure of continued fraction correlates exactly to the side/end graph's structure. The rules of graphical construction of Green functions (which are solutions of Schwinger–Dyson equations) are presented in concise form.

A procedure for calculating many‐particle Green's functions is suggested, which is based on the information supplied by the recursion method. The advantages of the procedure are the possibility of introducing a renormalized energy spectrum of the considered system, exactly definable within the recursion method, into calculations and their relative simplicity. As an illustration of the general method a calculation of a two‐particle Green's function, describing a resonant secondary radiation in the region of a three‐dimensional isolated band of excitons interacting mainly with optical phonons, is carried out. The range of parameters covers the cases of a weak exciton‐phonon coupling and a vibronic exciton. The main attention is paid to the relation of luminescence and scattering components. The obtained spectra are compared with experimental results.

Double-sided open periodic structures are analyzed using inhomogeneous plane wave scattering.The leaky and surface wave modes of several unit cells of different structures are computed using the poles of generalized reflection and transmission coefficients of inhomogeneous plane waves in the spectral domain.It is shown that the reflection and transmission coefficients of the zeroth order Floquet mode contain the poles of the Green's function of the complex stratified periodic structure.The properties of evanescent mode amplification as well as super resolution near field imaging in a wire medium are addressed.A balanced leaky wave antenna unit cell with double-sided radiation feature is introduced and it is shown that, in contrast to grounded structures, total absorption in lossless nonchiral double-sided open unit cells is not feasible as long as the behavior of the unit cell is well described by its fundamental mode.

In this chapter, we discuss the general theory of quadratic forms and the notion of congruence and its invariants, as well as the applications of the theory to the analysis of special forms. We will focus primarily on quadratic forms induced by rational functions, most notably the Hankel and Bezout forms, because of their connection to system-theoretic problems such as stability and signature symmetric realizations. These forms use as their data different representations of rational functions, namely power series and coprime factorizations respectively. But we will discuss also the partial fraction representation in relation to the computation of the Cauchy index of a rational function and the proof of the Hermite–Hurwitz theorem and the continued fraction representation as a tool in the computation of signatures of Hankel matrices as well as in the problem of Hankel matrix inversion. Thus, different representations of rational functions, i.e., different encodings of the information carried by a rational function, provide efficient starting points for different methods. The results obtained for rational functions will be applied to root-location problems for polynomials in the next chapter.

In this study, the authors present an electromagnetic model of a nanodipole array above a double-layer Graphene (DLG) operating at Terahertz. The graphene layers deposited between the dielectric mediums are modelled as surface impedances with conductivities described by the Kubo model. The electromagnetic fields are defined by the spectral representation of the periodic Green's function with impedance condition between the mediums. The main objective of this work is to propose an equivalent and analytical electromagnetic model of a nanodipole array above a DLG, which can be used to describe a surface plasmon resonance sensor that operates at the terahertz frequency. As a way of validating the method, the authors compared the proposed model with a software based on the finite element method. In addition, still, as a form of validation, the authors verified the convergence of the method from the spectral analysis of the proposed Green's Function. As results, spatial and spectral parametric analyses are shown as a function of the concentration of nanodipoles in the array, and the chemical potential applied to the graphene layers. In these results, the authors verify the appearance of plasmonic, optical and acoustic modes, and their contributions in the total field.

It was shown recently that the anomalous scaling of simultaneous correlation functions in turbulence is intimately related to the breaking of temporal scale invariance, which is equivalent to the appearance of infinitely many times scales in the time dependence of time-correlation functions. In this paper we derive a continued fraction representation of turbulent time correlation functions which is exact and in which the multiplicity of time scales is explicit. We demonstrate that this form yields precisely the same scaling laws for time derivatives and time integrals as the "multi-fractal" representation that was used before. Truncating the continued fraction representation yields the "best" estimates of time correlation functions if the given information is limited to the scaling exponents of the simultaneous correlation functions up to a certain, finite order. It is worth noting that the derivation of a continued fraction representation obtained here for a time evolution operator which is not Hermitian or anti-Hermitian may be of independent interest.

We investigate thermodynamic and mechanical properties of silicon nanostructures at finite temperature. Thermodynamic properties for finite-temperature solid systems under isothermal conditions are characterized by the Helmholtz free energy density. The static part of the Helmholtz free energy is obtained directly from the interatomic potential, while the vibrational part is calculated by using the theory of local phonon density of states (LPDOS). The LPDOS is calculated efficiently from the on-site phonon Green's function by using a recursion technique based on a continued fraction representation. The Cauchy-Born hypothesis is employed to compute the mechanical properties. By considering ideal Si{001}, $(2\ifmmode\times\else\texttimes\fi{}1)$ reconstructed Si{001}, and monolayer-hydrogen-passivated $(2\ifmmode\times\else\texttimes\fi{}1)$ reconstructed Si{001} surfaces of a silicon nanowire, we calculate the local phonon structure and local thermodynamic and mechanical properties at finite temperature and observe that the surface effects on the local thermal and mechanical properties are localized to within one or two atomic layers of the silicon nanowire.

A hybrid ray-mode representation of the Green's function in a rectangular cavity is developed using the finite Poisson summation formula. To obtain a numerically efficient scheme for computing the field generated by a point source in a large rectangular cavity, the conventional modal representation of the Green's function is modified in such a way that all the modes near resonance are retained while the truncated remainder of the mode series is expressed in terms of a weighted contribution of rays. For an electrically large cavity, the contribution of rays from distant images becomes small; therefore, the ray sum can be approximated by one or two dominant terms without a loss of numerical accuracy. To illustrate the accuracy and the computational simplification of this ray-mode representation, numerical examples are included with the conventional mode series (summed at the expense of long computation time) serving as a reference.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

This paper illustrates a TI N-Spire .tns file created by the author for generating continued fraction representations of real numbers and doing arithmetic with them. The continued fraction representation provides an alternative to the decimal representation. The .tns file can be used as tool for studying continued fractions and their properties as well as introducing the ideas of a best rational approximation to a real number and exploring some elementary ideas in error analysis. While the paper is a review of some of the classical works on continued factions, it does not present the material in a theorem/proof style. It motivates the results by citing examples and noting some interesting properties that will need to be verified. Notations are given to the references where exact and complete proofs of these facts can be found. The type of investigations done in this paper are motivational and the style of presentation is designed to pique students and teachers curiosity about the subject. Another purpose of this paper is to show that by looking at alternative representations of something that seems to be well known, such as the real numbers, can lead to exciting new and useful outcomes and applications.

In this note a representation of the discrete Green's function of a compact discretization of a two point boundary value problem of order n 2 is given which among other things allows a direct proff of the convergence (and divergence) properties.

The imaginary-time Green's function is a building block of various numerical methods for correlated electron systems. Recently, it was shown that a model-independent compact orthogonal representation of the Green's function can be constructed by decomposing its spectral representation. We investigate the performance of this so-called intermediate representation (IR) from several points of view. First, we develop an efficient algorithm for computing the IR basis functions of arbitrary high degree. Second, for two simple models, we study the number of coefficients required to represent the Green's function within a given tolerance. We show that the number of coefficients grows only as $O(ln\ensuremath{\beta})$ for fermions and converges to a constant for bosons as temperature $T=1/\ensuremath{\beta}$ decreases. Third, we show that this remarkable feature is ascribed to the properties of the physically constructed basis functions. The fermionic basis functions on the real-frequency axis have features whose width is scaled as $O(T)$, which is consistent with the low-$T$ properties of quasiparticles in a Fermi liquid state. On the other hand, the properties of the bosonic basis functions are consistent with those of spin/orbital susceptibilities at low $T$. These results demonstrate the potential wide applications of the IR to calculations of correlated systems.

The details of the electronic structure of the perfect crystal provide a critically important foundation for understanding the various defect states in uranium dioxide. In order to understand the local defect and impurity mechanism, we carry out the calculation of electronic structure of UO2 in the one-electron approximation, using a semi-empirical tight-binding formalism. The local density of states for both spin are calculated from the spectral representation of Green's function. The method of tight-binding within the virtual crystal approximation (VCA) is applied to calculate the electronic structure of UO2.12 (2:2:2 cluster). We construct the Green's function of the defect states and present quantities which are closely related to the Green's function. We calculate the local and total densities of states for uranium and three different types of defect oxygen; O, O′, O″.