Green function methods time-ordered

For the special case of non relativistic Hydrogen, the multiphoton transition rate can be obtained exactly using methods based on Green function techniques, which avoid summations over intermediate states. This approach was introduced in order to treat time independent problems, and later extended to time dependent ones [2]. In the Green function method, the evaluation of the infinite sums over intermediate states is reduced to the solution of a linear differential equation. For systems other than Hydrogen, this method can also be used, but the associated differential equation has to be integrated numerically. The two-photon transition rate can also be evaluated exactly by performing explicitly the summation over the intermediate states. 

The wavefunctions in Eq. (2.34) are different from the wavefunctions of the free tip and free sample. The effect of the distortion potential (V = Us — Uso and V = Us - Uso), can be evaluated through time-independent perturbation. In the following, we present an approximate method based on the Green s function of the vacuum (see Appendix B). To first order, the distorted wavefunction i)i is related to the undistorted one, i]jo, by... 

The present study shows that It is possible to evaluate the variability of statically determinate and statically indeterminate structures due to spatial variation of elastic properties without resort to finite element analysis. If a Green s function formulation is used, the mean square statistics of the indeterminate forces are obtained in a simple Integral form which is evaluated by numerical methods in negligible computer time. It was shown that the response variability problem becomes a problem Involving only few random variables, even if the material property is considered to constitute stochastic fields. The response variability was estimated using two methods, the First-Order Second Moment method, and the Monte Carlo simulation technique. 

EOMCC = equation-of-motion coupled-clu.ster FOD = fourth-order differencing MBGF = many-body Green s function RR = resonance Raman RRGM = recursive-residue-generation method SIL = short-time iterative Lanczos SOD = second-order differencing. 

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