Abstract: A very successful paradigm of semiclassical analysis has been that of Wigner measures: the Wigner function for a quantum problem satisfies a quantum kinetic equation; in the semiclassical limit the Wigner function converges to a phase-space probability measure, satisfying a classical kinetic equation. Such convergence results often only ensure that the limit is a solution of the classical problem; i.e. the classical problem might be ill-posed.
In the last few years this possible loss of uniqueness in the limit classical equation has attracted systematic attention. In some cases it is possible to extract information from the family of quantum problems which allows the selection of the correct weak solution of the classical problem. This information is not contained in the Wigner measure itself; it is thus a sort of "quantum residual" datum.
I will talk about such a result, obtained jointly with T. Paul, about linear problems with non-regular potentials (less than C^{1,1}, or even less than C^1). Moreover, I will attempt to place it in perspective among other recent works on ill-posed semiclassical limits, and finally highlight some technical questions that arise. |