The Green function

If there is an external field f4(r) acting on each segment, the equilibrium distribution of the Gaussian chain is modified by the Boltzmann factor 

To discuss the statistical properties of such a system it is convenient to consider the Green function defined by,  

In the general case of (4= 0, G R,R N) represents the statistical weight (or the partition function) of the chain which starts from R and ends at H in Af steps. The partition function for all possible conformations is given by 

The physical significance of this equation is clear. The factor G(R, R N -n)G(R , R n) represents the statistical weight of the chain which starts at R, passes through R in n steps, and ends at If in iV steps (see Fig. 2.5). The integration of this statistical weight over all R gives the statistical weight of the chain which starts at R and ends at R. 

Given G(R, R ,N), the average of an arbitrary physical quantity A is easily calculated. If A depends only on the position of the n-th segment, then 

---

If we know the Green function of the non-perturbed problem, we can found a solution by using the Lippman-Schwinger integral equation ... 

In a subsequent treatment from the time-dependent response point of view, connection with the Greens function... 

Without going into details we shall quote the final result of the orbital peeling method which expresses the EPI as a sum of terms involving the zeroes and the poles of peeled Green functions (G/j,, ) Gij k) denotes IJ block of the Green function corresponding to the Hamiltonian, where two atoms I and J are embedded at sites i and j and the orbitals from 1 to (k-1) are deleted at the site i.) ... 

For the electronic structure calculations in a disordered system, f is chosen to be the Green function (zI-H( n )) where H is the Hamiltonian of the system and n are the random site occupation variables. According to ASF configuration averaged density of states (DOS) is given by ... 

The LDOS have been calculated using 10 exact levels in the continued fraction expansion of the Green functions. For clean surfaces the quantities A Vi are the same for all atoms in the same plane they have been determined up to p = 2, 4, 6 for the (110), (100) and (111) surfaces, respectively, and neglected beyond. The cluster C includes the atoms located at the site occupied by the impurity and at al the neighbouring sites up to the fourth nearest neighbour. 

In some cases, one is not interested in the Green function but in the Hamiltonian. Grigore, Nenciu and Purice (1989) and Thaller (1992, p. 184) gave " ormula for the relativistic corrections to the non relativistic eigenstates of energy Eq. The following discussion is a bit abstract, but it will be illustrated by examples in the next two sections. Equation (2) is rewritten as... 

Using the usual expression for the Green function, we have... 

The first section is devoted to the derivation of the two-center expansion of the Green function in overlapping spheres. This expansion is a notoriously difficult problem (Sack, 1964) that was finally solved by Sun and Su. In the second section, we discuss the use of this overlapping Green function. 

We follow the same reasoning for the integration on the lower contour, and adding the two contributions, we obtain the following expression for the Green function in the overlapping region ... 

In giving the response of the system to an impulse change to the initial configuration, Gij t) may thus be considered as being the Green Function for CA, If j > i,... 

Hence, by virtue of the equation of motion (10-1), which >fi(x) obeys, and the equal time commutation rules (10-8), the Green function GA obeys the following equation... 

Other ideas are connected with a possibility of arranging the Green function such as... 

Lemma For the Green function G(x,() related to problem (6)-(7) the uniform estimates... 

Remember that the quantity of interest is njfi). It now becomes evident why it was worth introducing the Greens function as... 

The dynamical elastic and inelastic scattering ofhigh-energy electrons by solids may be described by three fundamental equations [5]. The first equation determines the wave amplitude G ( r, r, E), or the Green function, at point r due to a point source of electrons at r in the averaged potential (V (r)) ... 

For large P, (3/P is small and it is possible to find a good short-time approximation to the Green function p. This is usually done by employing the Trotter product formula for the exponentials of the noncommuting operators K and V... 

Rather than looking at the spectrum obtained from the secular determinant (5), we will here consider the spectrum SG for fixed wavenumber k and than average over k. One can write the spectrum in terms of a periodic orbit trace formula reminiscent to the celebrate Gutzwiller trace formula being a semiclassical approximation of the trace of the Green function (Gutzwiller 1990). We write the density of states in terms of the traces of SG, that is,... 

Such equation is termed the KMS (Kubo, Martin and Schwinger) relation and describes the conditions of periodicity to be obeyed by a correlation, in particular the Green functions. 

The Green function for the massless scalar field in the region defined by (1) is given by... 

The evolution of the wave functional may be found in terms of the Green function... 

We first find the Green function Go for H0 and then obtain pertur-batively the wave functional for the total Hamiltonian. In fact, each mode of the quadratic part Ho can be solved exactly in terms the time-dependent creation and annihilation operators (S.P. Kim et.al., 2000 2002 2001 S.P. Kim et.al., 2003)... 

We applied the Liouville-von Neumann (LvN) method, a canonical method, to nonequilibrium quantum phase transitions. The essential idea of the LvN method is first to solve the LvN equation and then to find exact wave functionals of time-dependent quantum systems. The LvN method has several advantages that it can easily incorporate thermal theory in terms of density operators and that it can also be extended to thermofield dynamics (TFD) by using the time-dependent creation and annihilation operators, invariant operators. Combined with the oscillator representation, the LvN method provides the Fock space of a Hartree-Fock type quadratic part of the Hamiltonian, and further allows to improve wave functionals systematically either by the Green function or perturbation technique. In this sense the LvN method goes beyond the Hartree-Fock approximation. 

---