Graduate Spectrum of the Liouville Operator

[andresB]

TL;DR

I would like some references that study the spectrum of the Liouville operator in the general case.

Context: Consider a classical system with Hamiltonian $H$. The Liouville differential operator can be defined using the Poisson brackets as $$L=-i\left \{ ,H \right \}.$$
$L$ is Hermitian in the Hilbert space of square integrable wavefunctions over phase space. The spectrum of $L$ is easy to compute for all systems that are integrable in the Arnold-Liouville sense. What about the chaotic systems? is there any good reference for this? Google is only giving me papers for the Liouville- von Neumann operator of QM.

 

[vanhees71]

Are you working within Koopman-von-Neumann mechanics? Otherwise I don't understand your notation. Maybe this paper helps:

https://arxiv.org/abs/2204.02955
https://doi.org/10.1088/1751-8121/ac8f75

 

[andresB]

Are you working within Koopman-von-Neumann mechanics? Otherwise I don't understand your notation. Maybe this paper helps:

https://arxiv.org/abs/2204.02955
https://doi.org/10.1088/1751-8121/ac8f75

Yes, the imaginary i is to make the operator hermitian in the KvN mechanics.

On the other hand, I have to mention that I wrote that paper.