# The semi-classical limit with a delta-prime potential

December 23, 2020

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.

Keywords:

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