Strong phase-space semiclassical asymptotics

Thierry Paul (1) , Agissilaos Athanassoulis (1, 2, 3)

1 CMLS - Centre de Mathématiques Laurent Schwartz
2 BANG - Nonlinear Analysis for Biology and Geophysical flows
3 DMA - Département de Mathématiques et Applications - ENS Paris

Wigner and Husimi transforms have long been used for the phase- space reformulation of SchrÄodinger-type equations, and the study of the corresponding semiclassical limits. Most of the existing results provide approximations in appropriate weak topologies. In this work we are con- cerned with semiclassical limits in the strong topology, i.e. approxima- tion of Wigner functions by solutions of the Liouville equation in L2 and Sobolev norms. The results obtained improve the state of the art, and highlight the role of potential regularity, especially through the regularity of the Wigner equation. It must be mentioned that the strong conver- gence can be shown up to logarithmic time-scales, which is well known to be, in general, the limit of validity of semiclassical asymptotics.

Domaines: Equations aux dérivées partielles [math.AP]

Thierry Paul, Agissilaos Athanassoulis. Strong phase-space semiclassical asymptotics. SIAM Journal on Mathematical Analysis, 2011, 43 (5), pp.2116-2149. ⟨10.1137/10078712X⟩. ⟨hal-00496375⟩