Green function Fourier transforms

Computational Algorithm for Green s Functions Fourier Transform of the Newton Polynomial Expansion. 

S. M. Auerbach and C. Leforestier, A new computational algorithm for Green s functions Fourier transform of the Newton polynomial expansion, Comp. Phys. Comm. 78 55 (1993). 

In the further manipulations the site representation will be used for convenience. The Fourier transform with respect to time of a single-exciton retarded Green s function (t) of a system under the Hamiltonian in the site representation for exciton coordinates and Fock s representation for phonon coordinates is written as... 

The Fourier transform of the propagator (3.1) gives the energy Green function,... 

For a pure state density operator, the Fourier transform of this double-time Green s function yields the spectral representation of the propagator (21)... 

Figure 5. The Fourier transformed signal AS[r, i] of I2/CCI4. The pump-probe delay times are I = 200 ps, 1 ns, and 1 ps. The green bars indicate the bond lengths of iodine in the X and A/A states. The blue bars show the positions of the first two intermolecular peaks in the pair distribution function gci-ci- (See color insert.)...

Different techniques are commonly used to solve the diffusion equation (Carslaw and Jaeger, 1959). Analytic solutions can be found by variable separation, Fourier transforms or more conveniently Laplace transforms and other special techniques such as point sources or Green functions. Numerical solutions are calculated for the cases which have no simple analytic solution by finite differences (Mitchell, 1969 Fletcher, 1991), which is the simplest technique to implement, but also finite elements, particularly useful for complicated geometry (Zienkiewicz, 1977), and collocation methods (Finlayson, 1972). 

The experimental evidence, then, suggests that quantum interference is absent in liquid metals. At first sight this might seem to contradict Ziman s (1961) theory of liquid metals, in which waves scattered by different atoms do interfere this, however, is not so. Following arguments of Baym (1964), Greene and Kohn (1965) and Faber (1972), one should not use in that theory the Fourier transform S(k) of the instantaneous pair-distribution function, but rather... 

In an unbounded medium, and with the condition that G -> 0 as x -> > permits the application of Fourier transformation of the Green s function. This is defined as... 

After Fourier transformation, the time domain of the Green s function is translated to frequency dependency. We start with the resulting spectral representation of the one-electron propagator... 

If there is no explicit external electromagnetic field, the covariant field equations determine a self-interaction energy that can be interpreted as a dynamical electron mass Sm. Since this turns out to be infinite, renormalization is necessary in order to have a viable physical theory. Field quantization is required for quantitative QED. The classical field equation for the electromagnetic field can be solved explicitly using the Green function or Feynman propagator GPV, whose Fourier transform is —gllv/K2, where k = kp — kq is the 4-momentum transfer. The product of y0 and the field-dependent term in the Dirac Hamiltonian, Eq. (10.3), is... 

The 7V-particle Green s function is a function of energy simply defined through a Fourier transform of Eq. (1) ... 

For arbitrary initial conditions, the solution of (4.27) may be derived from the knowledge of the response, at an arbitrary site m, to an excitation localized at t = 0 on an arbitrary site n (this is the Green s function of the system see Appendix A). As we are interested in spectral data, it is natural to use the Fourier transform of (4.27) corresponding to an initial condition of one excitation localized on site n> ... 

The nonequilibrium zeroth Green s functions are determined by the Dyson equations (62) and (63) on the Keldysh contour. The standard way to solve these equations is to perform a Fourier transform and then solve the algebraic matrix equations for the Green s functions. For the Keldysh functions, this procedure cannot be implemented in a straightforward way because of two time branches. Thus, we should find the Fourier transform for each Keldysh function after applying the Langreth s mapping procedure described in Section 2 [41, 45]. In particular for — t ), the Dyson... 

The one-exciton Green functions, the scattering matrix, and the irreducible part of the two-exciton Green function entering these equations are given by a Fourier transformation of Equations (18)—(21) to the frequency domain. 

Key features of both the non-interacting and the interacting Fermi liquid theory can be illustrated through the retarded one-particle Green function, which can be defined from the following Fourier transform... 

Let us now find the Green s function of the corresponding Helmholtz equation (13.56). We can solve this problem by applying the Fourier transform to expression... 

The calculation of the Fourier space contribution is the most time consuming part of the Ewald sum. The essential idea of P M is to replace the simple continuous Fourier transformations in (3) by discrete Fast Fourier Transformations, that are numerical faster to calculate. The charges are interpolated onto a regular mesh. Since this introduces additional errors, the simple Coulomb Green function as used in the second term in (3), is cleverly adjusted in... 

In this notation the inverse Fourier transform of the characteristic function in Eq. (167) is Green s function for Eq. (171), but in a notation where... 

Interesting strategies have been adopted for explicitly determining G(r) both in real space as well as Fourier space. Here we follow de Wit (1960) in adopting the Fourier space scheme, leaving the real space solution for the ambitious reader. To find the Green function, we exploit our Fourier transform convention ... 

In the previous sections, we have utilized Green s function techniques to eliminate some of the summations involved in the calculations of nonlinear susceptibilities. The general expression for R(t3,t2,t1) [Eq. (49) or (60)], involves four summations over molecular states a, b, c, d. In Eq. (80) we carried out two of these summations for harmonic molecules. It should be noted that for this particular model it is possible to carry out formally all the summations involved, resulting in a closed time-domain expression for R(t3,t2,t1). This expression, however, cannot be written in terms of simple products of functions of , r2 and t3. Therefore, calculating the frequency-domain response function / via Eq. (30) requires the performing of a triple Fourier transform (rather than three one-dimensional transforms). This formula is, therefore, useful for extremely short pulses when a time-domain expression is needed. Otherwise, it is more convenient to use the expressions of Section VI, whereby only two of the four summations were carried out, but the transformation to... 

To illustrate how classically forbidden processes are described with this type of approximation, consider tunnelling through a one-dimensional potential barrier as sketched in Fig. 9. Apart from some irrelevant constants, the amplitude for the particle going from the left to the right of the barrier at fixed energy is a matrix element of the Green s function, which in term is a Fourier transform of the propagator ... 

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