コロキウム

We study semiclassical perturbations of single-degree-of-freedom Hamiltonian systems possessing hyperbolic saddles with homoclinic orbits and one-parameter families of periodic orbits. The semiclassical systems give approximations of the expectation values of the positions and momenta to the semiclassical Schrodinger equations with Gaussian wave packets as the initial conditions. We provide sufficient conditions for the separatrices to split and for meromorphic nonintegrabilty such that the first integrals depend on the small parameter. A Melnikov-type approach and the Morales-Ramis theory are used to obtain the former and latter results, respectively. In particular, the occurrence of separatrix splitting explains a mechanism for the existence of trajectories to cross the separatrices on the classical phase plane in the expectation value dynamics. We illustrate our theory for the potential of a simple pendulum and give numerical computations for the stable and unstable manifolds in the semiclassical system as well as solutions crossing the separatrices. If we have time, a recent result on nonintegrability of planar systems will be presented shortly. The first part is joint work with Tomoki Ohsawa at University of Texas at Dallas.