On the role of geometry in scattering theory for nonlinear Schrödinger equations

Rémi Carles

Description

On the role of geometry in scattering theory for nonlinear Schrodinger equations Remi Carles (CNRS & Universite Montpellier 2) Orleans, April 9, 2008

• l2 ??


• large time


• repulsive harmonic


• euclidean geometries


• linear potential


• free schrodinger


• schrodinger equation

Sujets

• Carles
• Schrödinger equation

Informations

• Publié par: pefav
• Nombre de lectures: 10
• Langue: English

Extrait

On

the

for

role of geometry in scattering theory

nonlinearSchro¨dingerequations

R´emi

Carles RN&SC(reisnUvitponeMt´erliel Orle´ans,April9,2008

2)

rFeecShro¨dingerequationon

R n :

i∂ t u 2 Δ+ 1 u ;= 0u |t=0 =u 0 ∈L 2 ( R n ).
Solution known explicitly, and long time asymptotics available: e i t 2 Δ u)∼e i |x 2 | t 2 1 0 (x t→±∞ (2iπt) n/2 F ( u 0 )  x t  . DefocusingnonlinearSchr¨odingerequationon R n : i∂ t u 1 +Δ 2 u=|u| 2σ u;u |t=0 =u 0 ∈H 1 ∩ F(H 1 )( R n ). If 0< σ < n 2 −2 ,u∈C( R ;H 1 ∩ F(H 1 )).