Spectrum of the Liouville Operator

In summary, the Liouville differential operator, defined using Poisson brackets, is Hermitian in the Hilbert space and easy to compute for integrable systems. However, for chaotic systems, there is limited information available. One potential resource is the paper "Koopman-von-Neumann Mechanics and the Chaos Hierarchy" by J. S. Hwang and Y. Subasi. The paper discusses the use of the imaginary i in the operator for hermiticity in KvN mechanics.
  • #1
andresB
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TL;DR Summary
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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  • #3
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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What is the spectrum of the Liouville operator?

The spectrum of the Liouville operator is the set of all possible eigenvalues of the operator. In other words, it is the set of values for which the operator has a non-trivial solution.

How is the spectrum of the Liouville operator related to the dynamics of a system?

The spectrum of the Liouville operator is closely related to the dynamics of a system. The eigenvalues of the operator correspond to the frequencies at which the system can oscillate, while the corresponding eigenvectors represent the different modes of oscillation.

What is the physical significance of the eigenvalues in the spectrum of the Liouville operator?

The eigenvalues in the spectrum of the Liouville operator have physical significance as they determine the stability and behavior of a system. For example, if the largest eigenvalue is positive, the system is unstable and will exhibit exponential growth, while a negative largest eigenvalue indicates stability.

Can the spectrum of the Liouville operator change over time?

Yes, the spectrum of the Liouville operator can change over time as the system evolves. This is because the eigenvalues and eigenvectors are dependent on the initial conditions and any external forces acting on the system.

How is the spectrum of the Liouville operator used in practical applications?

The spectrum of the Liouville operator is used in various fields such as quantum mechanics, statistical mechanics, and fluid dynamics. It is used to analyze the behavior and stability of complex systems and to make predictions about their future evolution.

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