The semi-classical limit with a delta potential

We consider the semi-classical limit of the quantum evolution of Gaussian coherent states whenever the Hamiltonian H is given, as sum of quadratic forms, by , with and the Dirac delta-distribution at . We show that the quantum evolution can be approximated, uniformly for any time away from the collision time and with an error of order , , by the quasi-classical evolution generated by a self-adjoint extension of the restriction to , , of ( times) the generator of the free classical dynamics; such a self-adjoint extension does not correspond to the classical dynamics describing the complete reflection due to the infinite barrier. Similar approximation results are also provided for the wave and scattering operators.