On the spectrum of the free streaming operator in the three‐dimensional case

We study the spectrum of the free streaming operator in the three‐dimensional case in an L ¹ ‐contest. We prove that the spectrum is contained in the imaginary axis of the complex plane. We also perform a study of the semigroup generation properties of the free streaming operator and prove that the streaming operator generates a group of unitary norm in the whole space. In the half‐space case, the free streaming operator generates a group is of unitary norm only if the boundary conditions are conservative.

Keywords:

• Spectrum
• Free streaming operator
• Semigroup theory
• Boundary conditions

Acknowledgments

This work was partially supported by Par 2004—Research Project “Nuovi approcci matematici per lo studio di problemi in biologia, medicina, trasporto di fotoni e finanza” of the University of Siena funds, as well as by MURST research funds.