Logarithmic Bounds on Sobolev Norms for Time Dependent Linear Schrödinger Equations
By W.-M. Wang. 2008
63 citations recorded by the source corpus.
We prove that in 1-D the growth of Sobolev norms for time dependent linear Schrödinger equations on the circle is at most logarithmic in time for any fixed potential which is analytic (or Gevrey). Recently it was proven in [5] in a probabilistic setting that almost surely the Sobolev norms are unbounded, which indicates that the log is almost surely necessary. In [6], the author proved that the Sobolev norms remain bounded for an explicit time periodic potential. This is in the exceptional set probabilistically. The present paper together with [5, 6] give a rather complete picture of time dependent linear Schrödinger equations on the circle.
Subfields: Atomic and Molecular Physics, and Optics, Mathematical Physics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/7de6c87c99176a92837ad6148c0f634ca023e391
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In Atomic and Molecular Physics, and Optics
In Mathematical Physics
In Statistical and Nonlinear Physics
Citations in the corpus
- Bounds on the growth of high Sobolev norms of solutions to nonlinear Schrödinger equations on $S^1$ (2010, methodology): “We will use the idea, used in [18, 17, 111], of estimating the high-frequency part of the 41”
- Growth of Sobolev Norms of Solutions of Linear Schrödinger Equations on Some Compact Manifolds (2009, methodology): “In dimension d = 1, Wei-Min Wang [13] showed recently that such a logarithmic bound holds for any real analytic potential, whose holomorphic extension to a complex strip around R × S 1 is bounded, adapting the method of [4] to this setting.”