Green function definition

The longitudinal diffusion coefficient D has been formulated by the hole theory in Sect. 6.3.2. If the similarity ratio X in this theory is chosen to be 0.025 for the rod with the axial ratio 50, Eq. (58) with Eq. (56) gives the solid curve in Fig. 16a. Though it fits closely the simulation data, the chosen X is not definitive because the change in D(l is small and the definition of the effective axial ratio is ambiguous. Though not shown here, Eq. (53) for D, by the Green function method describes the simulation data equally well if P and C, are chosen to be 1000 and 1, respectively. 

The function /i(r) being the equal-time part of F1(t1,t2) is completely determined by the definition of the Green function (38). But instead of using Eq. (38) it is more convenient to derive equations that define /i(r) entirely in terms of the equal time Green functions... 

Now we turn to the function /2(r) Using the definition of the Green function (38) one can take the variation derivative in (42)) obtaining... 

To proceed, we use the relation between the Green function and the eigenfunctions Z a of a system, which are solutions of the Schrodinger equation (6). Let us define Exio) = c in the eigenstate A) in the sense of definition (5), then... 

Now we are able to define contour or contour-ordered Green function - the useful tool of Keldysh diagrammatic technique. The definition is similar to the previous one... 

We start from the general definition of a Green function as the average of two Heisenberg operators A(t) and B(t), denoted as... 

The general formalism developed in Ref. 43 provides the following definition of the matrix kernel (3.104) through the Green function of the pair evolution during encounter ... 

This problem can be avoided by expressing the MST equations in terms of the square matrix S C = CVS, Hermitian in consequence of the surface-matching theorem. This matrix has full rank because it is contracted over the larger index lg. From the definitions of the C and S matrices, the matrix product S C is a specific integral involving the Helmholtz Green function [281],... 

The definition of irregular functions depends on boundary conditions imposed on the Green function. Any variation of the set of functions defined by adding linear combinations of the regular functions f with coefficients that constitute an Hermitian matrix simply moves the corresponding term between the two summations in Eq. (7.37), leaving the net sum invariant, and preserving the Kronecker-delta... 

The definition of the Schrodinger Green function can be extended to a complexvalued energy parameter z. Then Eq. (7.23) for the Schrodinger equation can be written as... 

To construct the elastic Green function for an isotropic linear elastic solid, we make a special choice of the body force field, namely, f(r) = fo5(r), where fo is a constant vector. In this case, we will denote the displacement field as Gik r) with the interpretation that this is the component of displacement in the case in which there is a unit force in the k direction, fo = e. To be more precise about the problem of interest, we now seek the solution to this problem in an infinite body in which it is presumed that the elastic constants are uniform. In light of these definitions and comments, the equilibrium equation for the Green function may be written as... 

The free particle Green function operator, Go, and the full Green function, G, are given by definition respectively by ... 

Unfortunately, there is a price to be paid for this universal definition of the energy scale. NHiile the expectation values of (175) are automatically finite, the same is not true for (179). In order to understand the mechanism which leads to divergencies let us consider the energy of the perturbed vacuum with respect to the homogeneous vacuum (often called Casimir energy) within perturbation theory. The basic elements of the perturbation expansion are the Greens function of the perturbed vacuum. 

For the images calculation the Green function technique described in Section 7 was utilized. Two main configuration of illumination were considered s-polarization (incident polarization is perpendicular to the plain of incidence) and p-polarization (incident polarization is parallel to the plain of incidence). The electromagnetic field was calculated at a definite distance from the sample surface ( constant height of scanning). 

In this definition, the response function is given by Green function theory as... 

Another meaningful observation is that the Green function convolution (integration) with (source) wave-function is done upon the coordinate only in the basic definition, while the entirely time evolution is contained within the Green function. In relation with this aspect one may establish... 

At different spatial points, the conjugate products of the fluctuating fields produce the Green function (at the same time/temperature), according to its basic definition ... 

In conclusion, we observe that many writers in the modern literature seem to agree about the convenience of the definition (Eq. 11.67), but that there has also been a great deal of confusion. For comparison we would like to refer to Slater, and Arai (1957). Almost the only exception seems to be Green et al. (1953, 1954), where the exact wave function is expanded as a superposition of orthogonal contributions with the HF determinant as its first term ... 

The very definition implies that Green s function is nonnegative and symmetric ... 

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