The semi-classical limit with a delta potential
Tóm tắt
We consider the semi-classical limit of the quantum evolution of Gaussian coherent states whenever the Hamiltonian H is given, as sum of quadratic forms, by
$$ H= -\frac{{\hbar ^{2}}}{2m}\,\frac{d^{2}\,}{dx^{2}}\,\dot{+}\,\alpha \delta _{0}$$
, 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.
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