Introduction Current in quasi-free systems Transport vs Spectrum of − ∆ + V Transport for the 1D Schr¨ odinger equation via quasi-free systems (Collaboration with V. Jaksic) L. Bruneau Univ. Cergy-Pontoise Grenoble, December 1st, 2010 L. Bruneau Transport for the 1D Schr¨ odinger equation via quasi-free systems
Introduction Current in quasi-free systems Transport vs Spectrum of − ∆ + V Dynamical vs spectral In the litterature 2 notions of transport/localization pour H = − ∆ + V . L. Bruneau Transport for the 1D Schr¨ odinger equation via quasi-free systems
Introduction Current in quasi-free systems Transport vs Spectrum of − ∆ + V Dynamical vs spectral In the litterature 2 notions of transport/localization pour H = − ∆ + V . Dynamical: behaviour of � ψ t , � X � n ψ t � as t → ∞ and where ψ t = e − itH ψ and � X � = (1 + X 2 ) 1 / 2 . Localization if sup t � ψ t , � X � n ψ t � ≤ C n and transport if � ψ t , � X � n ψ t � ≃ C n t n β ( n ) with β ( n ) > 0 (transport exponent). L. Bruneau Transport for the 1D Schr¨ odinger equation via quasi-free systems
Introduction Current in quasi-free systems Transport vs Spectrum of − ∆ + V Dynamical vs spectral In the litterature 2 notions of transport/localization pour H = − ∆ + V . Dynamical: behaviour of � ψ t , � X � n ψ t � as t → ∞ and where ψ t = e − itH ψ and � X � = (1 + X 2 ) 1 / 2 . Localization if sup t � ψ t , � X � n ψ t � ≤ C n and transport if � ψ t , � X � n ψ t � ≃ C n t n β ( n ) with β ( n ) > 0 (transport exponent). Spectral: sp pp ( H ) is associated to the notion of localization and sp ac ( H ) to the one of transport. L. Bruneau Transport for the 1D Schr¨ odinger equation via quasi-free systems
Introduction Current in quasi-free systems Transport vs Spectrum of − ∆ + V Dynamical vs spectral In the litterature 2 notions of transport/localization pour H = − ∆ + V . Dynamical: behaviour of � ψ t , � X � n ψ t � as t → ∞ and where ψ t = e − itH ψ and � X � = (1 + X 2 ) 1 / 2 . Localization if sup t � ψ t , � X � n ψ t � ≤ C n and transport if � ψ t , � X � n ψ t � ≃ C n t n β ( n ) with β ( 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 but no equivalence: E ∈ sp pp ( H ) and ψ E an eigenfunction, then � ψ E t , � X � n ψ E t � = C : dynamical loc. dynamical loc. ⇒ pp spectrum (RAGE theorem). � T 0 � ψ t , � X � n ψ t � d t ≥ C n T n / d [Guarneri ’93]. 1 ψ ∈ H ac : T pp spectrum �⇒ dynamical loc., see e.g. [GKT,JSS,DJLS]. Huge amount of litterature on the subject. L. Bruneau Transport for the 1D Schr¨ odinger equation via quasi-free systems
Introduction Current in quasi-free systems Transport vs Spectrum of − ∆ + V Approach via quantum statistical mechanics Consider the case ℓ 2 ( Z ). We couple a finite sample to 2 reservoirs. L R 0 1 N ( β L , ν L ) ( β R , ν R ) L. Bruneau Transport for the 1D Schr¨ odinger equation via quasi-free systems
Introduction Current in quasi-free systems Transport vs Spectrum of − ∆ + V Approach via quantum statistical mechanics Consider the case ℓ 2 ( Z ). We couple a finite sample to 2 reservoirs. L R 0 1 N ( β L , ν L ) ( β R , ν R ) To make things simple let β L = β R and ν L ≥ ν R . We are interested in the current (charge flux) in the system: L. Bruneau Transport for the 1D Schr¨ odinger equation via quasi-free systems
Introduction Current in quasi-free systems Transport vs Spectrum of − ∆ + V Approach via quantum statistical mechanics Consider the case ℓ 2 ( Z ). We couple a finite sample to 2 reservoirs. L R 0 1 N ( β L , ν L ) ( β R , ν R ) To make things simple let β L = β R and ν L ≥ ν R . We are interested in the current (charge flux) in the system: We let the system relax to the NESS ω + . 1 L. Bruneau Transport for the 1D Schr¨ odinger equation via quasi-free systems
Introduction Current in quasi-free systems Transport vs Spectrum of − ∆ + V Approach via quantum statistical mechanics Consider the case ℓ 2 ( Z ). We couple a finite sample to 2 reservoirs. L R 0 1 N ( β L , ν L ) ( β R , ν R ) To make things simple let β L = β R and ν L ≥ ν R . We are interested in the current (charge flux) in the system: We let the system relax to the NESS ω + . 1 If J L is the observable “current out of L ”, we calculate 2 ω + ( J L ) =: � J L � N + (the sample has size N ). L. Bruneau Transport for the 1D Schr¨ odinger equation via quasi-free systems
Introduction Current in quasi-free systems Transport vs Spectrum of − ∆ + V Approach via quantum statistical mechanics Consider the case ℓ 2 ( Z ). We couple a finite sample to 2 reservoirs. L R 0 1 N ( β L , ν L ) ( β R , ν R ) To make things simple let β L = β R and ν L ≥ ν R . We are interested in the current (charge flux) in the system: We let the system relax to the NESS ω + . 1 If J L is the observable “current out of L ”, we calculate 2 ω + ( J L ) =: � J L � N + (the sample has size N ). We study the behaviour of � J L � N + as N → ∞ according to the 3 properties of V (or of − ∆ + V ): does it go to 0? at which rate? is there a non trivial (positive) limit? L. Bruneau Transport for the 1D Schr¨ odinger equation via quasi-free systems
Introduction Current in quasi-free systems Transport vs Spectrum of − ∆ + V Quasi-free systems Independent electrons approximation: free fermi gas with a 1 particle space of the form h = h L ⊕ ℓ 2 ([0 , N ]) ⊕ h R . L. Bruneau Transport for the 1D Schr¨ odinger equation via quasi-free systems
Introduction Current in quasi-free systems Transport vs Spectrum of − ∆ + V Quasi-free systems Independent electrons approximation: free fermi gas with a 1 particle space of the form h = h L ⊕ ℓ 2 ([0 , N ]) ⊕ h R . 1 particle hamiltonian: h = h 0 + w where h 0 = h L ⊕ ( − ∆+ V ) D ⊕ h R , w = | δ L �� δ 0 | + | δ 0 �� δ L | + | δ R �� δ N | + | δ N �� δ R | . ( h L / R , h L / R ): “free” reservoirs with good ergodic properties: we assume that the spectral measures µ L / R of h L / R for δ L / R are purely a.c. L. Bruneau Transport for the 1D Schr¨ odinger equation via quasi-free systems
Introduction Current in quasi-free systems Transport vs Spectrum of − ∆ + V Quasi-free systems Independent electrons approximation: free fermi gas with a 1 particle space of the form h = h L ⊕ ℓ 2 ([0 , N ]) ⊕ h R . 1 particle hamiltonian: h = h 0 + w where h 0 = h L ⊕ ( − ∆+ V ) D ⊕ h R , w = | δ L �� δ 0 | + | δ 0 �� δ L | + | δ R �� δ N | + | δ N �� δ R | . ( h L / R , h L / R ): “free” reservoirs with good ergodic properties: we assume that the spectral measures µ L / R of h L / R for δ L / R are purely a.c. Without loss of generality we now take h L / R = L 2 ( R , d µ L / R ( E )) , h L / R = mult par E , δ L / R = 1 . Examples: free Laplacian on half-line , full line , Bethe lattice , 1 / 2-space,... L. Bruneau Transport for the 1D Schr¨ odinger equation via quasi-free systems
Introduction Current in quasi-free systems Transport vs Spectrum of − ∆ + V Quasi-free systems The full Hilbert space is then H = Γ − ( h ), the algebra of observables is O = CAR ( h ). L. Bruneau Transport for the 1D Schr¨ odinger equation via quasi-free systems
Introduction Current in quasi-free systems Transport vs Spectrum of − ∆ + V Quasi-free systems The full Hilbert space is then H = Γ − ( h ), the algebra of observables is O = CAR ( h ). The uncoupled hamiltonian is H 0 = d Γ( h 0 ) , and the full one is H = d Γ( h ) = H 0 + a ∗ ( δ L ) a ( δ 0 )+ a ∗ ( δ 0 ) a ( δ L )+ a ∗ ( δ R ) a ( δ N )+ a ∗ ( δ N ) a ( δ R ) . For any A ∈ O , τ t ( A ) := e itH A e − itH . In particular, for f ∈ h one has τ t ( a # ( f )) = a # ( e ith f ). L. Bruneau Transport for the 1D Schr¨ odinger equation via quasi-free systems
Introduction Current in quasi-free systems Transport vs Spectrum of − ∆ + V Quasi-free systems The full Hilbert space is then H = Γ − ( h ), the algebra of observables is O = CAR ( h ). The uncoupled hamiltonian is H 0 = d Γ( h 0 ) , and the full one is H = d Γ( h ) = H 0 + a ∗ ( δ L ) a ( δ 0 )+ a ∗ ( δ 0 ) a ( δ L )+ a ∗ ( δ R ) a ( δ N )+ a ∗ ( δ N ) a ( δ R ) . For any A ∈ O , τ t ( A ) := e itH A e − itH . In particular, for f ∈ h one has τ t ( a # ( f )) = a # ( e ith f ). Initial state of the system: quasi-free state ω 0 associated to the density matrix T = (1 + e β ( h L − ν L ) ) − 1 ⊕ ρ S ⊕ (1 + e β ( h R − ν R ) ) − 1 , i.e. ω 0 is such that ω 0 ( a ∗ ( g n ) · · · a ∗ ( g 1 ) a ( f 1 ) · · · a ( f m )) = δ nm det ( � f i , Tg j � ) i , j . L. Bruneau Transport for the 1D Schr¨ odinger equation via quasi-free systems
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