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    Knauf’s Degree and Monodromy in Planar Potential Scattering

    2016, Volume 21, Number 6, pp.  697-706

    Author(s): Martynchuk N. N., Waalkens H.

    We consider Hamiltonian systems on (TR2,dqdp) defined by a Hamiltonian function H of the “classical” form H=p2/2+V(q). A reasonable decay assumption V(q)0,, 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 = 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.
    Keywords: Hamiltonian system, Liouville integrability, nontrapping degree of scattering, scattering monodromy
    Citation: Martynchuk N. N., Waalkens H., Knauf’s Degree and Monodromy in Planar Potential Scattering, Regular and Chaotic Dynamics, 2016, Volume 21, Number 6, pp. 697-706



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