# The semi-classical limit with delta potentials

July 12, 2019

We consider the semi-classical limit of the quantum evolution of Gaussian
coherent states whenever the Hamiltonian $\mathsf H$ is given, as sum of
quadratic forms, by $\mathsf H=
-\frac{\hbar^{2}}{2m}\,\frac{d^{2}\,}{dx^{2}}\,\dot{+}\,\alpha\delta_{0}$, with
$\alpha\in\mathbb R$ and $\delta_{0}$ the Dirac delta-distribution at $x=0$. We
show that the quantum evolution can be approximated, uniformly for any time
away from the collision time and with an error of order $\hbar^{3/2-\lambda}$,
$0\!<\!\lambda\!<\!3/2$, by the quasi-classical evolution generated by a
self-adjoint extension of the restriction to $\mathcal C^{\infty}_{c}({\mathscr
M}_{0})$, ${\mathscr M}_{0}:=\{(q,p)\!\in\!\mathbb R^{2}\,|\,q\!\not=\!0\}$, of
($-i$ times) the generator of the free classical dynamics; such a self-adjoint
extension does not correspond to the classical dynamics describing the complete
reflection due to the infinite barrier. Similar approximation results are also
provided for the wave and scattering operators.

Keywords:

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