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A Spectral Study of the Linearized Boltzmann Equation for Diffusively Excited Granular Media

In this work, we are interested in the spectrum of the diffusively excited granular gases equation, in a space inhomogeneous setting, linearized around an homogeneous equilibrium. We perform a study which generalizes to a non-hilbertian setting and to the inelastic case the seminal work of Ellis and Pinsky about the spectrum of the linearized Boltzmann operator. We first give a precise localization of the spectrum, which consists in an essential part lying on the left of the imaginary axis and a discrete spectrum, which is also of nonnegative real part for small values of the inelasticity parameter. We then give the so-called inelastic "dispersion relations", and compute an expansion of the branches of eigenvalues of the linear operator, for small Fourier (in space) frequencies and small inelasticity. One of the main novelty in this work, apart from the study of the inelastic case, is that we consider an exponentially weighted $L^1(m^{-1})$ Banach setting instead of the classical $L^2(\mathcal M_{1,0,1}^{-1})$ Hilbertian case, endorsed with Gaussian weights. We prove in particular that the results of Ellis and Pinsky holds also in this space.