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.

每当将哈密顿量H作为二次形式的和通过\（H =-\ frac {{\ hbar ^ {2}} {2m} \给出时，我们考虑高斯相干态量子演化的半经典极限。，\ frac {d ^ {2} \，} {dx ^ {2}} \，\ dot {+} \，\ alpha \ delta _ {0} \），带有\（\ alpha \ in \ mathbb R \ ）和\（\ delta _ {0} \） Dirac delta-distribution at \（x = 0 \）。我们证明，量子演化可以在碰撞时间之外的任何时间均匀地近似，并且具有\（{\ hbar ^ {3 / 2- \ lambda}} \）阶次的误差，\（0 \！<\ ！\ lambda \！<\！3/2 \），通过限制的自伴随扩展生成的准经典演化\（\ mathcal C ^ {\ infty} _ {c}（{\ mathscr {M}} _ {0}）\），\（{\ mathscr {M}} _ {0}：= \ {（q， p）\！\ in \！\ mathbb R ^ {2} \，||，q \！\ not = \！0 \} \）（）（\（-i \）次）中免费经典的生成器动力学; 由于无穷大的障碍，这种自伴的扩展不对应于描述完全反射的经典动力学。还为波和散射算子提供了类似的近似结果。