Recent Results in Semiclassical Approximation with Rough Potentials

Abstract

Quantum Mechanics was invented for stability reasons. In fact it is striking to notice the difference of regularity that needs the potential of a Schrödinger operator to insure unitary of the quantum flow (e.g. $V \in {L}_{\mbox{ loc}}^{1}{,\ \ \lim {}_{\epsilon \rightarrow 0}\sup }_{x}\int\limits_{\vert x-y\vert \leq \epsilon }\vert x - {y\vert }^{2-N}\vert V (y)\vert dy\,=\,0$) compared to the classical Cauchy-Lipshitz condition for vector fields.