Propagation of singularities of solutions to semilinear Schrödinger equations (Q1083611)

The author proves a result about the propagation of singularities of a semilinear Schrödinger equation with a holomorphic nonlinearity along bicharacteristic strips which are in that case straight lines in the hyperplane \(t=const\). The result implies that a solution of \[ (1)\quad iu_ t=-\Delta u+f(u,\bar u) \] which lies in a certain micro-localized Sobolev space and which also is locally in a weighted Sobolev space of a certain order at some point \(z_ 0\) of \(p^{-1}(0)\) (where p is the symbol \(p=\tau +| \xi |^ 2\) of (1)) is in the same Sobolev space at all points of the bicharacteristic line through \(z_ 0.\) The paper contains an outline of the proof of that result which uses some lemmata on pseudo-differential operators by Lascar and Hörmander, and a lemma on the paraproduct of tempered distributions due to Yamazaki.