Logarithmic bounds on Sobolev norms for time-dependent linear Schr\"odinger equations

Understand

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].