  We prove that in 1-D the growth of Sobolev norms for time-dependent linear
Schr\"odinger equations is at most logarithmic in time for any (fixed)
potential which is analytic (or Gevrey). Recently it was proven in [N] that
almost surely the Sobolev norms are unbounded, which indicates that the log is
almost surely necessary. In [W], the author showed that the Sobolev norms
remain bounded for an explicit time periodic potential. This is in the
exceptional set in the sense of [N]. The present paper together with [N, W]
give a rather complete picture of time dependent linear Schr\"odinger equations
on the circle.
