Maximal Lyapunov exponent of Schrödinger-Newton equation

Question

I am trying to determine the sign of the maximal Lyapunov exponent of the Schrödinger-Newton equation $$ \partial_t \psi(t,\vec{x}) = i\left(a\nabla^2 + \int_{\mathbb{R}^3} \frac{|\psi(t,\vec{y})|^2}{|\vec{x} - \vec{y}|} d^3\vec{y} \right) \psi(t, \vec{x}), $$ where $\psi : \mathbb{R}^4 \rightarrow L^2(\mathbb{R}^3)$ and $a > 0$. So far I've tried applying the definition of the maximal LE, and also tried studying how a perturbation evolves in time, but both got complicated fast. I'm therefore wondering if there is a more tractable technique to do this.