I am writing an essay on the Berry Keating article proposing to use the $\mathcal{H}=xp$ hamiltonian to get a correspondence between the nontrivial riemann zeros and the eigenvalues of an Hermitian operator. Now classically, the time evolution of the system as follows from the Hamilton equations is:

$$ x(t)=x(0)e^t $$ $$p(t) = p(0)e^{-t} $$

It seems as if the particle is accelerating (since $\frac{dx}{dt}$ clearly increases), while $p$ obviously decreases. Moreover I do not immediately see how this system is chaotic, although clearly the distance as a result of different initial conditions between particles increases over time. Could you please shed some light on this?

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    $\begingroup$ Where is the contradiction? $p$ is not the classical momentum $p=m \dot{x}$ here. Indeed Hamilton equations just say that $\dot{x} = x$, differently from the case ${\cal H}= p^2/2m + U(x)$ where $\dot{x}= p/m$. The Hamiltonian decides what is the relation between $x$ and $p$... $\endgroup$ – Valter Moretti Dec 9 '14 at 10:58
  • $\begingroup$ ok thank you, i already thought something like that. But that also means that the momentum quantum operator is also different (it is different from $\frac{d}{dx}$ at least)? $\endgroup$ – user1043065 Dec 9 '14 at 12:12
  • $\begingroup$ also, do you know in which sense the system is chaotic then (equivalently what definition of chaos should be applied here?) $\endgroup$ – user1043065 Dec 9 '14 at 12:13
  • $\begingroup$ No it does not mean that. Regarding the second question, I do not know, since I do not know the definition of chaotic. However the Hamiltonian flow does not seem very ``chaotic'' to me in this case, as the maximal solutions are the trivial ones you wrote! $\endgroup$ – Valter Moretti Dec 9 '14 at 12:49
  • $\begingroup$ Related: physics.stackexchange.com/q/384151/2451 $\endgroup$ – Qmechanic Feb 4 '18 at 12:13

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