## 3 A Model for the Measured Spectral Shape
### The Memory Function Formalism
The last term accounts for the coupling of density fluctuations with the ultra-fast vibrational dynamics here accounted by a \(\delta(t)\) profile. It was often found that the amplitude \(2T_{\mu}\) has a quadratic \(Q\)-dependence and is essentially insensitive to thermodynamic changes [20]. It is immediate to recognize that real and imaginary parts of Fourier transform of the memory function in Equation (9) read as:  
\[\frac{dC(t)}{dt}=i\Omega\cdot C(t)-\int_{0}^{t}K(\tau)\cdot C(t-\tau)d\tau \tag{5}\]1 \[\frac{dC(t)}{dt}=i\Omega\cdot C(t)-\int_{0}^{t}K(\tau)\cdot C(t-\tau)d\tau \tag{5}\]2  
respectively. These two equations can be inserted in Equation (8) to obtain a model for the classical part of the line-shape. It should be emphasized that at mesoscopic scales all transport and thermodynamic coefficients introduced above and in the following, unless otherwise specified, are \(Q\)-dependent generalizations of their macroscopic counterparts, although such a dependence in not made explicit in the notation. The model used for the memory function often results from suitable approximation of Equation (9) as appropriate to the sample, thermodynamic conditions and dynamic range probed. For simplest systems as noble gases, the single exponential decay or viscoelastic model has shown to provide a reasonable approximation [42]. Regardless on the approximation adopted, the exponential ansatz for the viscoelastic term, lends itself two at least these two fundamental objections:  
1. In principle, structural relaxation phenomena in highly viscous systems can be better described by assuming a stretched exponential rather than a simple exponential time-decay of the memory function [74; 75; 17]. However, the use of this model would introduce an unwanted additional parameter (the stretching coefficient) and, perhaps more importantly, its Fourier transform cannot be cast on an analytical form. Fortunately, for simplest systems as noble gases, of interest in this paper, it is reasonable to assume that the simple exponential decay is a rather accurate approximation. 2. A second remark concerns the possible presence in the spectrum of a second inelastic excitation as observed in systems as diverse as liquid water [73; 76; 77; 78], tetrahedrally arranged glasses [79], glass formers [80] liquid metals [81; 82; 83; 84; 85], complex biophysical samples [86; 87; 88] and mixtures [89]. The presence of this low frequency mode in the \(S(Q,\omega)\) is customarily ascribed to the onset of shear mode propagation, although this assignment could seem suspicious as \(S(Q,\omega)\) couples primarily with longitudinal movements only. In a liquid, the onset of shear waves in the spectrum of density fluctuations can only occur via the so-called longitudinal-transverse coupling [73; 76; 90], that is a mixing between acoustic modes having orthogonal polarization. The presence of this mode-mixing is not properly accounted for by any known analytic model for the memory function. However, such a coupling has never been observed in noble gases, likely due to the highly isotropic character of their interatomic interaction, and will not be discussed in the remainder of this paper.