Re: IS quantum mechanics a limit cycle theory??


hello
let we have a classical hamiltonian H,and we quantize it.
I want to learn some relations between the behavior of solutions of X_H(the classic hamiltonian vector field) and some operetor theoretic invariants of the quantum operatores.In particular a nice interprewtation of the number of closed orbits of X_H in quantum language!
Thank you for your suggestions for some deep references
Ali Taghavi

> Hi
> I Am Intersted in the Hilbert 16th problem which main
> object is "Limit
> Cycle"!
> Last Year I Found in the web a paper by Cetto and De
> La Penna :with
> The Title "Is Quantum Mechanics A Limit Cycle
> Theory?"
> this paper is available in Mathscinet.
> I Invite you to review this paper and discuss on a
> possible relation
> between Hilbert 16th Problem And Mathematical Aspect
> Of QM,
> for begining:Let We Have A Planner Vector field
> L(Lienard Equation )
> as follow:
> x'=y-F(x) y'=-x where F is A non even_polynomial ,we
> are intersted in
> the number of Limit Cycles Of L,consider the
> following two
> questions:(Assume F'(0) is not zero)
> 1)Does there exist a correspondence between {closed
> orbites} of L and
> {closed orbits} of 4 dimensional (classical)
> Hamiltonian (y-F(x))z-xw?
> please see also the similar question in :
>
> http://www.arxiv.org/abs/math.CA/0409594
>
> 2)Let's Quantize the above 4 dimensional
> hamiltonian:x,y stand for
> operators multiplication by x and y,while z,w stand
> for partial
> derivative with respect to x ,y resp.
> What Is The Quantum interpretations for (the number
> of) closed orbits
> of classical Hamiltonian
> (y-F(x))z-xw?
> Thank you
> Ali Taghavi
> Iran
>

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