A Spectrum of the Liouville Operator

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The discussion centers on the Liouville differential operator defined using Poisson brackets in classical systems with Hamiltonians. It highlights that while the spectrum of this operator is easily computed for integrable systems, challenges arise for chaotic systems. Participants seek references for understanding the behavior of the Liouville operator in chaotic contexts, noting a lack of relevant literature beyond quantum mechanics. The conversation also touches on the Koopman-von Neumann mechanics, clarifying that the imaginary unit is necessary for the operator's Hermitian property. One contributor reveals authorship of a referenced paper that may provide further insights.
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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.
 
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vanhees71 said:
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.
 
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