The semi-classical limit with a delta-prime potential

We consider the quantum evolution e(-it/)((h) over barH beta)psi((h) over bar )(xi)of a Gaussian coherent state psi((h) over bar)(xi) is an element of L-2(R) localized close to the classical state xi (q, p) is an element of R-2, where H-beta denotes a self-adjoint realization of the formal Hamiltonian -(h) over bar (2)/2m d(2)/dx(2) + beta delta(0)' with delta(0)' the derivative of Dirac's delta distribution at x = 0 and beta a real parameter. We show that in the semi-classical , limit such a quantum evolution can be approximated (with respect to the L-2(R)-norm, uniformly for any t is an element of R away from the collision time) by e(i/)((h) over bar At) e(itLB )phi((h) over bar)(x), where A(t) = p(2)t/2m, phi((h) over bar)(x)(xi) := psi((h) over bar)(xi)(x) and L-B is a suitable 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}, of (-i times) the generator of the free classical dynamics. While the operator L-B here utilized is similar to the one appearing in our previous work [C. Cacciapuoti, D. Fermi and A. Posilicano, The semi-classical limit with a delta potential, Ann. Mat. Pura Appl. 200 (2021) 453-489], in the present case the approximation gives a smaller error: it is of order (h) over bar (7/2)(-lambda), 0 < lambda < 1/2, whereas it turns out to be of order (h) over bar (3/2-lambda), 0 < lambda < 3/2, for the delta potential. We also provide similar approximation results for both the wave and scattering operators.