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 H=-ħ22md2dx2+˙αδ0, with α∈ R and δ the Dirac delta-distribution at x= 0. We show that the quantum evolution can be approximated,uniformly for any time away from the collision time and with an error of order ħ3/2-λ, 0<3/2, by the quasi-classical evolution generated by a self-adjoint extension of the restriction to Cc∞(M0), M0:={(q,p)∈R2|q≠0}, of (- i 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.