## 3 A Model for the Measured Spectral Shape  
### The Memory Function Formalism  
Let \(\overset{\rightarrow}{A}(t)=[A_{1}(t).........A_{v}(t)]\) be a vector whose components are stochastic variables describing the properties of a system of \(N\) interacting particles (with \(N>v\)). It can be shown that under very general conditions, the equation of motion of \(\overset{\rightarrow}{A}(t)\) has the following form:  
\[\frac{d\overset{\rightarrow}{A}(t)}{dt}=i\Omega\cdot\overset{\rightarrow}{A }(t)-\int_{0}^{t}K(\tau)\cdot\overset{\rightarrow}{A}(t-\tau)d\tau+\overset{ \rightarrow}{f}(t) \tag{4}\]  
In the formula above the following quantities have been introduced:  
* The antisymmetric matrix \(i\Omega=\left\langle\overset{\rightarrow}{A}(0),i\overset{\rightarrow}{ LA}(0)\right\rangle\cdot\left\langle\overset{\rightarrow}{A}(0),\overset{ \rightarrow}{A}(0)\right\rangle^{-1}\) is the proper frequency of the system, which describes the oscillatory behavior of \(\overset{\rightarrow}{A}(t)\). This matrix depends on equilibrium (static) properties of the system. Here the symbol \(\left\langle......\right\rangle\) indicates a statistical (thermal) average. * The variable \(\overset{\rightarrow}{f}(t)=e^{i(1-\widetilde{p})Lt}i(1-\widetilde{P})L \overset{\rightarrow}{A}(0)\) is the random fluctuating force. Noticeably, the presence of the \((1-\widetilde{P})\) term implies that \(\left\langle\overset{\rightarrow}{A}(0),\overset{\rightarrow}{f}(t)\right\rangle =\ 0\), namely that the fluctuating force is orthogonal (statistically uncorrelated) to \(\overset{\rightarrow}{A}(0)\). * The matrix \(K(t)=\left\langle\overset{\rightarrow}{f},\overset{\rightarrow}{f}(t) \right\rangle\cdot\left\langle\overset{\rightarrow}{A}(0),\overset{ \rightarrow}{A}(0)\right\rangle^{-1}\) is memory matrix, or memory function if \(\upsilon=1\), as in the cases of interest here. It is worth noticing that the integrand in Equation (4) includes all \(\overset{\rightarrow}{A}(t)\) values between \(0\) and \(t\), \"weighted\" by the memory, which thus defines the ability of the system to keep memory of the past. The equation of motion for the correlation matrix \(C(t)=\left\langle\overset{\rightarrow}{A}(t)\overset{\rightarrow}{A}(0)\right\rangle\) can be readily derived from Equation (4) through a scalar product, while using the orthogonality between \(\overset{\rightarrow}{A}(0)\) and \(\overset{\rightarrow}{f}(t)\). Explicitly:  
\[\frac{dC(t)}{dt}=i\Omega\cdot C(t)-\int_{0}^{t}K(\tau)\cdot C(t-\tau)d\tau \tag{5}\]  
Equations (4) and (5) are referred to as memory equation or generalized Langevin equation either for dynamic variables or for their correlations, respectively. When dealing with a single dynamic variable, time reversal requirements impose \(\Omega=0\). Using a procedure described in various monographies [63; 22], the Langevin equation can be solved following an iterative approach, which eventually leads to the following recursive formula:  
\[\frac{\widetilde{C}(s)}{C(0)}=[s+\frac{K_{1}\left(0\right)}{s+\frac{K_{2} \left(0\right)}{s+\frac{K_{3}\left(0\right)}{s+\dots}}}]^{-1} \tag{6}\]  
customarily referred to as the continued fraction expansion. Here \(K_{i}(t)\) is the \(i\)-th (iteration) order memory function associated to the variable of interest,\[\widetilde{C}(s)=\int\limits_{0}^{\infty}dte^{-zt}C\left(t\right)\]  
while is the Laplace transform of the correlation function.