Effective Approximation for the semiclassical Schrödinger equation.

The computation of the semiclassical Schrödinger equation presents a
number of difficult challenges because of the presence of high oscillation
and the need to respect unitarity. Given periodic boundary conditions, the
typical approach consists, basically, of two steps: Semi-discretisation
with spectral method in space and Strang splitting in time, however this
strategy occurs to be of low accuracy and sensitive to high oscillation.
In this talk we sketch an alternative approach. Our analysis commences not with semi-discretisation, but with the investigation of the free Lie
algebra generated by differentiation and by multiplication with the
interaction potential: it turns out that this algebra possesses a
structure which renders it amenable to a very effective form of asymptotic splitting: exponential splitting where consecutive terms are scaled by increasing powers of the small parameter.
The semi-discretisation in deferred to the very end of computations.