# Extension of classical Liouville operator

Let us consider a classical Hamiltonian system described by the Hamiltonian $$\begin{equation} H(q,p) =\frac{p^2}{2m}+V(q) \end{equation}$$ where we stick to the case of single particle for simplicity. I am interesting in the dynamics generated by the extended Liouville operator \begin{align} i\mathcal L = \frac p m\frac{\partial}{\partial q}-V'(q)\frac{\partial}{\partial p}-V'''(q)\frac{\partial^3}{\partial p^3} \end{align} The equations of motion generated by $$e^{i\mathcal L t}$$ can no longer be written in a linear symplectic form, because by interpreting $$\frac{\mathrm d}{\mathrm d t}\leftrightarrow i\mathcal L$$ \begin{align}\label{eom} \dot q = \frac{p}{m} \hspace{10mm}\dot p = -\frac{\partial V}{\partial q} \end{align} we would identically neglect the cubic derivative in the momentum. Is there a way to extend the above dynamical scheme in compliance with the modified Liouville operator? I am ultimately interested in a numerical method aimed at integrating $$(q(t),p(t))$$.

## 1 Answer

I am not experienced with cubic PDEs... only Infinite dimensional such. Your system looks like the first quantum correction in Wigner flows in phase space, but I cannot get much help from that magnificent formulation, off the cuff...

I could only formulate your problem in 19th century language.

You are seeking the Green's function for $$\left (\partial_t - \frac{p}{m} \partial_q + V(q)'\partial_p +V(q)'''\partial_p^3 \right ) f(q,p)=0 ,$$

One could Fourier-transform to $$f(q,p)=\int dy ~e^{iyp} F(q,y)$$ , $$\left (\partial_t +\frac{i}{m} \partial_y \partial_q + iV(q)' y -iV(q)''' y^3 \right ) F(q,y)=0 ,$$ but it may not be clear this is friendlier to you purposes.

• Indeed, I am looking for a first order nonlinear correction in the propagation of Wigner functions for a nonlinear quantum system. It seems like the problem is not straightforward, but I can look up whether the reformulation in terms of Green function offers some computational advantage. – Graz Mar 11 '20 at 6:49