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?