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IntroductionCurertnniuqsa-irfeestsysTemnsrartpopSsvrtcefomuV+ΔurenLB.orthortfanspauTreregnido¨rhcSD1ei-asquianvioatqueefrstsysem
L. Bruneau
Grenoble, December 1st, 2010
Transportforthe1DSchr¨odingerequationvia quasi-free systems (Collaboration with V. Jaksic)
Univ. Cergy-Pontoise
transport/localization
pour
H
=Δ +V.
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2
litterature
the
In
Dynamical vs spectral
of
notions
temsesysf-erauisivqaitno
tnIsTemnsrartpoSpvsrtcefomuV+ΔanyDroductionCurrentniuqsa-irfeeystsarleptcvlssimacLB.urenuarTnaps
In the litterature 2 notions of transport/localization pourH=Δ +V.
Dynamical: behaviour ofhψthXinψtiast→ ∞and where ψt=eitHψandhXi(1 +X2)12. = Localization if supthψthXinψti ≤Cnand transport if hψthXinψti ≃Cntnβ(n)withβ(n)>0 (transport exponent).
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Dynamical: behaviour ofhψthXinψtiast→ ∞and where ψt=eitHψandhXi= (1 +X2)12. Localization if supthψthXinψti ≤Cnand transport if hψthXinψti ≃Cntnβ(n)withβ(n)>0 (transport exponent). Spectral:sppp(H) is associated to the notion of localization and spac(H) to the one of transport.
In the litterature 2 notions of transport/localization pourH=Δ +V.
vsspectrals
In the litterature 2 notions of transport/localization pourH=Δ +V. Dynamical: behaviour ofhψthXinψtiast→ ∞and where ψt=eitHψandhXi= (1 +X2)12 . Localization if supthψthXinψti ≤Cnand transport if hψthXinψti ≃Cntnβ(n)withβ(n)>0 (transport exponent). Spectral:sppp(H) is associated to the notion of localization and spac(H) to the one of transport. Between these 2 notions there are links butno equivalence: Esppp(H) andψEan eigenfunction, thenhψtEhXinψtEi=C: dynamical loc. dynamical loc.pp spectrum (RAGE theorem). ψ∈ Hac:1TR0ThψthXinψtidtCnTnd[Guarneri ’93]. pp spectrum6⇒dynamical loc., see e.g. [GKT,JSS,DJLS]. Huge amount of litterature on the subject.
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