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Need your suggestions as to how I can arrive at the expression for the Dyadic Green's function.

The scalar case is simple:

Consider the standard equation of motion in Fourier space: ## \omega^2 \hat{x}(\omega) = \omega_0^2 \hat{x}(\omega) - i \delta \omega \hat{x}(\omega)+ F(\omega) ##, where ##F## is the forcing function, and ##\delta## the damping parameter. Defining the scalar Green's function from the equation: ## \hat{x}(\omega) = G(\omega) F(\omega) ##, and substituting this in the above equation of motion, we get the expression for the scalar green's function as:

$$ G(\omega) = \frac{1}{\omega^2 - \omega_0^2 + i \delta \omega };$$

Now consider going to the vector form for the displacement, and consequently the dyadic form for the Green's function - as seen for coupled harmonic oscillator problems. The equations of motion can be reduced to the form:

$$ \hat{\bf{x}}(\omega) = \bar{\bar{\bf{A}}} \hat{\bf{x}}(\omega) + \bf{F}(\omega) $$

where ##\bar{\bar{\bf{A}}}## is a matrix.

Now defining the Green's function as before: ## \hat{\bf{x}}(\omega) = \bar{\bar{\bf{G}}}(\omega) \bf{F}(\omega) ##, is it possible to arrive at an expression for the Green's function? The problem is complicated (as compared to the scalar problem) due to the inherent summations that exist in the matrix multiplication.

Can anybody suggest how I can proceed ?

Thanks!