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

De
26 pages
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


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On
the
for
role of geometry in scattering theory
nonlinearSchro¨dingerequations
R´emi
CarlesRN&SC(reisnUvitponeMt´erliel Orle´ans,April9,2008
2)
rFeecShro¨dingerequationon
Rn:
i∂tu2Δ+1u ;= 0u|t=0=u0L2(Rn). Solution known explicitly, and long time asymptotics available: eit2Δu)ei|x2|t21 0(xt→±∞(2iπt)n/2F(u0)xt. DefocusingnonlinearSchr¨odingerequationonRn: i∂tu1+Δ2u=|u|2σu;u|t=0=u0H1∩ F(H1)(Rn). If 0< σ <n22,uC(R;H1∩ F(H1)).