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The semi-classical limit with a delta-prime potential

    We consider the quantum evolution eitHβψξ of a Gaussian coherent state ψξL2() localized close to the classical state ξ(q,p)2, where Hβ denotes a self-adjoint realization of the formal Hamiltonian 22md2dx2+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()-norm, uniformly for any t away from the collision time) by eiAteitLBϕx, where At=p2t2m, ϕx(ξ):=ψξ(x) and LB is a suitable self-adjoint extension of the restriction to 𝒞c(0), 0:={(q,p)2|q0}, 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 7/2λ, 0<λ<1/2, whereas it turns out to be of order 3/2λ, 0<λ<3/2, for the delta potential. We also provide similar approximation results for both the wave and scattering operators.

    AMSC: 81Q20, 81Q10, 47A40


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