Knauf’s Degree and Monodromy in Planar Potential Scattering

We consider Hamiltonian systems on $(T^{*}\mathbb R^2, dq \wedge dp)$ defined by a Hamiltonian function $H$ of the “classical” form $H = p^2/2 + V(q)$. A reasonable decay assumption $V(q) \to 0, \, \|q\| \to \infty,$ allows one to compare a given distribution of initial conditions at $t = - \infty$ with their final distribution at $t = + \infty$. To describe this Knauf introduced a topological invariant $\text{deg}(E)$, which, for a nontrapping energy $E>0$, is given by the degree of the scattering map. For rotationally symmetric potentials $V(q) = W(\|q\|)$, scattering monodromy has been introduced independently as another topological invariant. In the present paper we demonstrate that, in the rotationally symmetric case, Knauf's degree $\text{deg}(E)$ and scattering monodromy are related to one another. Specifically, we show that scattering monodromy is given by the jump of the degree $\text{deg}(E)$, which appears when the nontrapping energy $E$ goes from low to high values.