The semi-classical limit with a delta-prime potential

We consider the quantum evolution e-i t/h Hβψζh of a Gaussian coherent state ψζh ∈ L2(ℝ) localized close to the classical state ζ ≡ (q,p) ℝ2, where Hβ denotes a self-adjoint realization of the formal Hamiltonian - h2/2m d2/dx2 + βδ0′, with δ0′ the derivative of Dirac's delta distribution at x = 0 and β a real parameter. We show that in the semi-classical limit such a quantum evolution can be approximated (with respect to the L2(ℝ)-norm, uniformly for any t ℝ away from the collision time) by ei/h AteitLB φxh;, where At = p2t/2m, φxh(ζ): = ψ ζh(x) and LB is a suitable self-adjoint extension of the restriction to c∞c), (ℳ0) ℳ0: = {(q,p) ∈ ℝ2|q ≠ 0}, of (- i times) the generator of the free classical dynamics. While the operator LB 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 h7/2-λ, 0 < λ < 1/2, whereas it turns out to be of order h3/2-λ, 0 < λ < 3/2, for the delta potential. We also provide similar approximation results for both the wave and scattering operators.