NONINTEGRABILITY OF SOME HAMILTONIAN-SYSTEMS, SCATTERING AND ANALYTIC CONTINUATION

We consider canonical two degrees of freedom analytic Hamiltonian systems with Hamiltonian function H = 1/2[p1(2) + p2(2)] + U(q1, q2), where U(q1, q2) = 1/2[ - v2q1(2) + omega2 q2(2)] + O(q1(2) + q2(2))3/2) and partial-derivative(q2) U(q1, 0) = 0. Under some additional, not so restrictive hypothesis, we present explicit conditions for the existence of transversal homoclinic orbits to some periodic orbits of these systems. We use a theorem of Lerman (1991) and an analogy between one of its conditions with the usual one dimensional quantum scattering problem. The study of the scattering equation leads us to an analytic continuation problem for the solutions of a linear second order differential equation. We apply our results to some classical problems.