The semi-classical limit with delta potentials

We consider the semi-classical limit of the quantum evolution of Gaussian coherent states whenever the Hamiltonian $mathsf H$ is given, as sum of quadratic forms, by $mathsf H= -frac{hbar^{2}}{2m},frac{d^{2},}{dx^{2}},dot{+},alphadelta_{0}$, with $alphainmathbb R$ and $delta_{0}$ 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 $hbar^{3/2-lambda}$, $0!<!lambda!<!3/2$, by the quasi-classical evolution generated by a self-adjoint extension of the restriction to $mathcal C^{infty}_{c}({mathscr M}_{0})$, ${mathscr M}_{0}:={(q,p)!in!mathbb R^{2},|,q! ot=!0}$, 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.