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=h(2)/2m d(2)/d(x2)+alpha delta 0 , with alpha is an element of 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 (h) over bar (3/2-lambda), 0<3/2, by the quasi-classical evolution generated by a self-adjoint extension of the restriction to C-c(infinity)(M-0)M-0:={(q,p) is an element of R-2 vertical bar q not equal 0 times}, 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.