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Annales Henri Poincaré

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

The Weyl Symbol of Schrödinger Semigroups

  • Laurent Amour
  • Lisette Jager
  • Jean Nourrigat
Article

Abstract

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.

Keywords

Large Dimension Wigner Function Neighbor Interaction Selfadjoint Extension Wiener Measure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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