Estimates of solutions of linear Boltzmann equation at large time and spectral singularities

TL;DR

This paper analyzes the spectral properties of the linear Boltzmann operator at large times, providing estimates for solution asymptotics and identifying spectral singularities and eigenvalues.

Contribution

It applies the Szokefalvi-Nagy - Foias functional model to the dissipative Boltzmann operator, offering new asymptotic estimates and spectral analysis results.

Findings

Exact asymptotic estimate in the isotropic case

Finite eigenvalues and spectral singularities in the general case

Upper bounds for the remainder in solution estimates

Abstract

The spectral analysis of the dissipative linear transport (Boltzmann) operator with polynomial collision integral by the Szokefalvi-Nagy - Foias functional model is given. An exact estimate for the reminder in the asymptotic of the corresponding evolution semigroup is proved in the isotropic case. In the general case, it is shown that the operator has finitely many eigenvalues and spectral singularities and an absolutely continuous essential spectrum, and an upper estimate for the reminder is established.