Focal decomposition, renormalization and semiclassical physics

Abstract

We review some recent results concerning a connection between focal decomposition, renormalization and semiclassical physics. The dynamical behaviour of a family of mechanical systems which includes the pendulum at small neighbourhoods of the equilibrium but after long intervals of time can be characterized through a renormalization scheme acting on the dynamics of this family. We have proved that the asymptotic limit of this renormalization scheme is universal: it is the same for all the elements in the considered class of mechanical systems. As a consequence, we have obtained an asymptotic universal focal decomposition for this family of mechanical systems which can now be used to compute estimates for propagators in semiclassical physics.

Keywords:

• mechanical systems
• renormalization
• focal decomposition
• semiclassical physics

AMS Subject Classification::

• 70H03
• 70H09
• 37E20
• 34B15

Acknowledgements

We thank Robert MacKay and Michael Berry for helpful discussions. We thank the Calouste Gulbenkian Foundation, PRODYN-ESF, POCTI, and POSI by FCT and Ministério da Ciência, Tecnologia e Ensino Superior. We thank CEMAPRE, LIAAD-INESC Porto LA and Centro de Matemática da Universidade do Minho and their FCT Pluriannual Funding Program for their financial support. We also thank CNPq, FAPERJ and FUJB for financial support. Part of this research was done during visits by the authors to IMPA (Brazil), The University of Warwick (UK), IHES (France), CUNY (USA), SUNY (USA) and MSRI (USA), who thank them for their hospitality. D. Pinheiro's research was supported by FCT – Fundação para a Ciência e Tecnologia grant with reference SFRH/BPD/27151/2006 and the program “Ciência 2007”.