Introduction Current in quasi free systems

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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


Publié le : lundi 18 juin 2012
Lecture(s) : 14
Source : www-fourier.ujf-grenoble.fr
Nombre de pages : 54
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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 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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N
R (βR νR)
Consider the case2(Z). We couple a finite sample to 2 reservoirs.
L (βL νL)
0
1
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R N (βR νR)
Consider the case2(Z couple a finite sample to 2 reservoirs.). We
To make things simple letβL=βRandνLνR are interested in. We the current (charge flux) in the system:
L 0 1 (βL νL)
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smet
L 0 1 (βL νL)
To make things simple letβL=βRandνLνR are interested in. We the current (charge flux) in the system: 1We let the system relax to the NESSω+.
Consider the case2(Z). We couple a finite sample to 2 reservoirs.
R N (βR νR)
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To make things simple letβL=βRandνLνR. We are interested in the current (charge flux) in the system: 1We let the system relax to the NESSω+. 2IfJLis the observable “current out ofL”, we calculate ω+(JL) =:hJLi+N(the sample has sizeN).
Consider the case2(Z). We couple a finite sample to 2 reservoirs.
s
R N (βR νR)
L 0 1 (βL νL)
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R (βR νR)
N
ems
To make things simple letβL=βRandνLνR. We are interested in the current (charge flux) in the system: 1We let the system relax to the NESSω+. 2IfJLis the observable “current out ofL”, we calculate ω+(JL) =:hJLiN+(the sample has sizeN). 3We study the behaviour ofhJLiN+asN→ ∞according to the properties ofV(or ofΔ +V it go to 0? at which rate? is): does there a non trivial (positive) limit?
Consider the case2(Z). We couple a finite sample to 2 reservoirs.
L 0 (βL νL)
1
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Independent electrons approximation: free fermi gas with a1 particle spaceof the form
h=hL2([0N])hR
temsitnoqeauauisivqa
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