Annales Henri Poincaré

, Volume 16, Issue 6, pp 1479–1488 | Cite as

The Weyl Symbol of Schrödinger Semigroups

  • Laurent AmourEmail author
  • Lisette Jager
  • Jean Nourrigat


In this paper, we study the Weyl symbol of the Schrödinger semigroup etH , H = −Δ + V, t > 0, on \({L^2(\mathbb{R}^n)}\) , with nonnegative potentials V in \({L^1_{\rm loc}}\) . Some general estimates like the L norm concerning the symbol u are derived. In the case of large dimension, typically for nearest neighbor or mean field interaction potentials, we prove estimates with parameters independent of the dimension for the derivatives \({\partial_x^\alpha\partial_\xi^\beta u}\) . In particular, this implies that the symbol of the Schrödinger semigroups belongs to the class of symbols introduced in Amour et al. (To appear in Proceedings of the AMS) in a high-dimensional setting. In addition, a commutator estimate concerning the semigroup is proved.


Large Dimension Wigner Function Neighbor Interaction Selfadjoint Extension Wiener Measure 
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Authors and Affiliations

  1. 1.LMR EA 4535 and FR CNRS 3399Université de Reims Champagne-ArdenneReims Cedex 2France

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