**Abstract:**
It is well known that coherent states provide approximate
solutions
to the time dependent Schrodinger equation in the semiclassical limit if the
centre and the
shape of the state evolve according to the Hamiltonian dynamics of the
corresponding
classical system. We study the case that the Hamilton operator is no
longer self-adjoint and
find that coherent states still provide approximate solutions, but that the
underlying classical system
which describes the motion of the center and the shape of the state is no
longer Hamiltonian.
Instead the dynamics is governed by the combination of a Hamiltonian
vectorfield and a gradient vectorfield which are
coupled by a time dependent metric. We will concentrate in particular on the
quadratic case and show
how the above dynamics is related to a projection of the complexified
Hamiltonian dynamics
and a special complex structure on phase space. This is joint work with
Eva-Maria Graefe.