We fix a Hilbert space isomorphism $\phi:H^{1}\to H^{2}$. Here we use $H^{s}$ for the sobolev space on $\mathbb{R}^{2}$ or $S^{2}$.

We consider the following polynomial vector field on $\mathbb{R}^{2}$ $$ \begin{cases} \dot x=ax+by+cx^{2}+dxy+ey^{2}\\\dot y=a'x+b'y+c'x^{2}+d'xy+e'y^{2} \end{cases}$$

Using Poincare compactification, the above vector field can be considered as an analytic vector field on $S^{2}$. These vector fields define derivation on the plane and sphere, epectively. In both case this operator is denoted by $D:H^{2} \to H^{1}$.

My question is about the spectrum of $\phi \circ D$ as a bounded operator on $H^{2}$:

Can one compute this spectrum in terms of the coefficient of the above vector field? Is it true to say that the number of connected components of the spectrum is finite? Are there any relation between the dynamics of the vector field and the topology-geometric properties of the spectrum? More generally, are there some results about some elations between the dynamical behaviour of an arbitrary vector field $X$ and the spectrum of the above operator on Sobolev space?