Spectral study of the linearized Boltzmann operator in L^2 spaces with polynomial and Gaussian weights

TL;DR

This paper extends the spectral analysis of the linearized Boltzmann operator to L^2 spaces with polynomial and Gaussian weights, providing detailed spectral and semigroup insights for small and large frequencies.

Contribution

It introduces a spectral study of the linearized Boltzmann operator in weighted L^2 spaces beyond exponential weights, using perturbation theory and enlargement techniques.

Findings

Spectral properties are characterized for small frequencies as perturbations of the homogeneous case.

Large frequency analysis reveals detailed spectral and semigroup behavior.

The approach combines perturbation theory with enlargement arguments for comprehensive spectral analysis.

Abstract

The aim of this paper is to extend to the spaces L^2(R^d , (1+|v|)^2k dv) the spectral study led in L^2(R^d , exp(|v|^2/2)dv) by R. Ellis and M. Pinsky on the space inhomogeneous linearized Boltzmann operator for hard spheres. More precisely, we look at the Fourier transform in the space variable of the inhomogeneous operator and consider the dual Fourier variable as a fixed parameter. We then perform a precise study of this operator for small frequencies (by seeing it as a perturbation of the homogeneous one) and also for large frequencies from spectral and semigroup point of views. Our approach is based on perturbation theory for linear operators as well as enlargement arguments from M.P. Gualdani, S. Mischler and C. Mouhot.