Logarithmic bounds on Sobolev norms for time-dependent linear Schrödinger equations

2008/05/24 by W.-M. Wang, Wang, W. -M.
Mathematics · #35xx #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.0805.3771

openalex publication_date 2008/05/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that in 1-D the growth of Sobolev norms for time-dependent linear Schrödinger 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 Schrödinger equations on the circle.

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