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

TL;DR

This paper establishes that Sobolev norms for 1-D time-dependent linear Schrödinger equations grow at most logarithmically over time for analytic or Gevrey potentials, providing a comprehensive understanding of their long-term behavior.

Contribution

It proves a universal logarithmic upper bound on Sobolev norm growth for such equations, extending previous results and completing the picture on their long-term dynamics.

Findings

Sobolev norms grow at most logarithmically in time

Almost sure unboundedness of Sobolev norms is necessary

Boundedness holds for explicit time periodic potentials

Abstract

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.