Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Logarithmic bounds on Sobolev norms for time-dependent linear
  Schr\"odinger equations   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.

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The categories of this paper are: patt-sol
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The categories of this paper are: supr-con
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The categories of this paper are: econ
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The categories of this paper are: nucl-th
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The categories of this paper are: bayes-an
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The categories of this paper are: stat
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The categories of this paper are: cmp-lg
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The categories of this paper are: physics
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The categories of this paper are: dg-ga
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The categories of this paper are: acc-phys
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The categories of this paper are: ao-sci
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The categories of this paper are: hep-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: chem-ph
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The categories of this paper are: q-bio
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The categories of this paper are: solv-int