Introduction Current in quasi free systems

L. Bruneau

Description

Introduction Current in quasi-free systems Transport vs Spectrum of ?∆ + V Transport for the 1D Schrodinger equation via quasi-free systems (Collaboration with V. Jaksic) L. Bruneau Univ. Cergy-Pontoise Grenoble, December 1st, 2010 L. Bruneau Transport for the 1D Schrodinger equation via quasi-free systems

• transport exponent


• pp spectrum


• ?x ?


• transport


• cergy pontoise


• huge amount


• free systems


• bruneau transport

Sujets

• Transport
• Cergy Pontoise

Informations

• Publié par: pefav
• Nombre de lectures: 15
• Langue: Français
• Poids de l'ouvrage: 1 Mo

Extrait

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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

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In the litterature 2 notions of transport/localization pourH=−Δ +V.

Dynamical: behaviour ofhψ t hXi n ψ t iast→ ∞and where ψ t =e −itH ψandhXi(1 +X 2 ) 12 . = Localization if sup t hψ t hXi n ψ t i ≤C n and transport if hψ t hXi n ψ t i ≃C n t nβ(n) withβ(n)>0 (transport exponent).

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Dynamical: behaviour ofhψ t hXi n ψ t iast→ ∞and where ψ t =e −itH ψandhXi= (1 +X 2 ) 12 . Localization if sup t hψ t hXi n ψ t i ≤C n and transport if hψ t hXi n ψ t i ≃C n t nβ(n) withβ(n)>0 (transport exponent). Spectral:sp pp (H) is associated to the notion of localization and sp ac (H) to the one of transport.

In the litterature 2 notions of transport/localization pourH=−Δ +V.

vsspectral s

In the litterature 2 notions of transport/localization pourH=−Δ +V. Dynamical: behaviour ofhψ t hXi n ψ t iast→ ∞and where ψ t =e −itH ψandhXi= (1 +X 2 ) 12 . Localization if sup t hψ t hXi n ψ t i ≤C n and transport if hψ t hXi n ψ t i ≃C n t nβ(n) withβ(n)>0 (transport exponent). Spectral:sp pp (H) is associated to the notion of localization and sp ac (H) to the one of transport. Between these 2 notions there are links butno equivalence: E∈sp pp (H) andψ E an eigenfunction, thenhψ tE hXi n ψ tE i=C: dynamical loc. dynamical loc.⇒pp spectrum (RAGE theorem). ψ∈ H ac : 1 T R 0 T hψ t hXi n ψ t idt≥C n T nd [Guarneri ’93]. pp spectrum6⇒dynamical loc., see e.g. [GKT,JSS,DJLS]. Huge amount of litterature on the subject.

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