Knauf’s degree and monodromy in planar potential scattering

We consider Hamiltonian systems on ( T *ℝ 2 , dq ∧ dp ) defined by a Hamiltonian function H of the “classical” form H = p 2 /2 + V ( q ). A reasonable decay assumption V ( q ) → 0, ‖ q ‖ → ∞, allows one to compare a given distribution of initial conditions at t = −∞ with their final distribution at t = +∞. To describe this Knauf introduced a topological invariant 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 deg( E ) and scattering monodromy are related to one another. Specifically, we show that scattering monodromy is given by the jump of the degree deg( E ), which appears when the nontrapping energy E goes from low to high values.