The semi-classical limit with a delta-prime potential

We consider the quantum evolution $e^{-i\frac{t}{\hbar}H_{\beta}} \psi_{\xi}^{\hbar}$ of a Gaussian coherent state $\psi_{\xi}^{\hbar}\in L^{2}(\mathbb{R})$ localized close to the classical state $\xi \equiv (q,p) \in \mathbb{R}^{2}$, where $H_{\beta}$ denotes a self-adjoint realization of the formal Hamiltonian $-\frac{\hbar^{2}}{2m}\,\frac{d^{2}\,}{dx^{2}} + \beta\,\delta'_{0}$, 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 (w.r.t. the $L^{2}(\mathbb{R})$-norm, uniformly for any $t \in \mathbb{R}$ away from the collision time) by $e^{\frac{i}{\hbar} A_{t}} e^{it L_{B}} \phi^{\hbar}_{x}$, where $A_{t} = \frac{p^{2}t}{2m}$, $\phi_{x}^{\hbar}(\xi) := \psi^{\hbar}_{\xi}(x)$ and $L_{B}$ is a suitable self-adjoint extension of the restriction to $\mathcal{C}^{\infty}_{c}({\mathscr M}_{0})$, ${\mathscr M}_{0} := \{(q,p) \in \mathbb{R}^{2}\,|\,q \neq 0\}$, 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, A. Posilicano, The semi-classical limit with a delta potential, Annali di Matematica Pura e Applicata (2020)] regarding the semi-classical limit with a delta potential, in the present case the approximation gives a smaller error: it is of order $\hbar^{7/2-\lambda}$, $0 < \lambda < 1/2$, whereas it turns out to be of order $\hbar^{3/2-\lambda}$, $0 < \lambda < 3/2$, for the delta potential. We also provide similar approximation results for both the wave and scattering operators.