NuHAG :: TALKS

Talks given at NuHAG events

Maximal Covariance Properties for Pseudo-Differential Operators


  Maurice de Gosson

  given at  esi12 (..)
  id:  2357
  length:  min
  status: 
  type: 
  LINK-Presentation:  http://univie.ac.at/nuhag-php/dateien/talks/de Gosson_2012-09_Talk_M.deG.pdf
  ABSTRACT:
Let A be a Weyl operator with symbol a: A <-Weyl-> a. It is
well-known that Ŝ^(-1) AŜ <-Weyl-> a o S for every linear symplectic automorphism
S where Ŝ is anyone of the two elements of the metaplectic group covering
S. We ask the question whether this covariance property can be extended to
larger classes of pseudo-differential operators or to larger groups of automor-
phisms (symplectic, or not, linear, or not) of phase space (R^2d; sigma). We show
that the answer is negative in both cases: symplectic covariance of Weyl op-
erators cannot be extended to the non-linear case, moreover the symplectic
group is the largest group of linear operators having this property. More-
over this property characterizes the Weyl calculus, and cannot thus be fully
extended to other classes of pseudo-differntial operators. We brifly discuss
partial symplectic covariance for Shubin and Born–Jordan operators.


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