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Let $a(x) \in W^{1, \infty}(\mathbb{R})$ be real-valued such that $a(x) \ge a_0 > 0$. Let $A^2$ denote the second order differential operator $A^2 : = -\partial_x (a(x) \partial_x) + 1 : L^2(\...

We fix a Hilbert space isomorphism $\phi:H^{1}\to H^{2}$. Here by $H^{s},\;s=1,2,\;$ we mean the sobolev space on $\mathbb{R}^{2}$ or $S^{2}$. We consider the following polynomial vector field on ...