On the role of the flux in scattering theory
In Holen et al., (Hrsg.), Stochastic Processes, Physics and Geometry: New Interplays I.
Canadian Mathematical Society, 2001. Conference Proceedings.
[FluxAsym.ps (223.7 kB)]
Zusammenfassung
The often ignored quantum probability flux is fundamental for a genuine understanding of scattering theory as, in particular, expressed in the flux-across-surfaces theorem. This work splits into two parts. First we show how the flux enters into scattering theory and we give an elementary proof of the free flux-across-surfaces theorem. At least heuristically, the free theorem together with completeness of the wave operators implies the full fluxacross-surfaces theorem. Therefore, in the second part, we discuss the proof of asymptotic completeness in potential scattering—the main focus of mathematical scattering theory so far. Of course this is well known, however, we found that the presentations of the proof (we looked at) showed no awareness of the crucial physical ingredient, namely the current positivity condition, a condition on the quantum flux. We wish to present here our understanding of the issues involved and we wish to emphasize that the arguments are all straightforward and natural: The proof uses Riemann-Lebesgue, compactness of operators and the current positivity condition.