Green’s operator

Ediund A and Peskin U 1998 A parallel Green s operator for multidimensional quantum scattering calculations Int. J. Quantum Chem. 69 167... 

The RWP method also has features in common with several other accurate, iterative approaches to quantum dynamics, most notably Mandelshtam and Taylor s damped Chebyshev expansion of the time-independent Green s operator [4], Kouri and co-workers time-independent wave packet method [5], and Chen and Guo s Chebyshev propagator [6]. Kroes and Neuhauser also implemented damped Chebyshev iterations in the time-independent wave packet context for a challenging surface scattering calculation [7]. The main strength of the RWP method is that it is derived explicitly within the framework of time-dependent quantum mechanics and allows one to make connections or interpretations that might not be as evident with the other approaches. For example, as will be shown in Section IIB, it is possible to relate the basic iteration step to an actual physical time step. 

From Eq. (5.23) or appropriate manipulations of it, the matrix elements of the Green s operator G (E) between any pair of atomic-like basis orbitals can be obtained. 

Note that in the original definition of the Green s operators (8.128), the argument of Ge was a vector. In expression (9.93) we extend this definition to include a tensor argument T ... 

It is well known that the conventional Born approximation can be applied iteratively, generating A -th order Born approximations. This approximation can be treated as the sum of N terms of the Born (or Neumann) series. However, the convergence of the Born series is questionable and depends on the norm of the integral equation (Green s) operator. It seems to be very attractive to construct similar series on the basis of the QL and QA approximations. 

Green s operator with norm less than 1 Gg < 1. 

The Born scries would be a powerful tool for EM modeling if they were convergent. However, in practice the condition (9.130) does not hold, because in a general case the L2 norm of the Green s operator is bigger than 1. That is why the Born series has not found a wide application in EM modeling. 

Quasi-linear approximation of the modified Green s operator We will obtain a more accurate approximation even on the first step if we assume that the anomalous field E inside the inhomogeneous domain is not equal to zero, as it was supposed in the previous section, but is linearly related to the background field E by some tensor A ... 

Taking into account the definition of the modified Green s operator (9.135) and formula (9.170), we obtain... 

Using the original Green s operator given by expression (9.37) and taking into account formula (9.205), one can rewrite equation (9.208) as follows ... 

The critical problem of both the Born iterative and the distorted-Born iterative methods is the convergence of the algorithms. To ensure convergence we can use a modified Green s operator G ", which has the contraction property... 

The modified Green s operator is introduced by formula (9.135), which we reproduce here ... 

The iterative Born inversion based on the modified Green s operator involves subsequently determining A from equation (10.42) for specified E, and then updating E from equation (10.47) for predetermined A5, etc. 

Expressing the time evolution in terms of the Green s operator... 

This is a Fourier transform of the diagonal 1,1 matrix element of the Green s operator G E - - zs) where... 

Covariant Evolution Operator and the Green s Operator 4.1 Definitions... 

The vacuum expectation (39) contains singularities, which are eliminated by the denominator in the definition of the GF (38). For the CEO, which is an operator, the situation is more complex. We shall refer to the regular part of the CEO as the Green s operator (GO), which we separate into open and closed parts... 

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