Introduction Time averaging and main results Main Tools Global in time estimates Conclusion and perspectives Effective mass theorem for a bidimensional electron gas under a strong magnetic field Fanny Fendt, Florian M´ ehats IRMAR Universit´ e de Rennes Journ´ ee de l’ANR Quatrain 20 mai 2008
Introduction Time averaging and main results Main Tools Global in time estimates Conclusion and perspectives Motivation In this talk, we present an asymptotic model that describes the transport of 3D quantum gas confined in one direction ( z ∈ R ) and subject to a strong magnetic field whose direction is in the transport plane (the horizontal x plane).
Introduction Time averaging and main results Main Tools Global in time estimates Conclusion and perspectives Introduction 1 The physical problem and the associated model Mathematical background Rescaling The Poisson non-linearity Asymptotic model Time averaging and main results 2 Second order averaging Main theorem Main Tools 3 Adapted functional framework Asymptotics for the Poisson kernel A priori local in time estimates Global in time estimates 4 Conclusion and perspectives 5
Introduction Time averaging and main results Main Tools Global in time estimates Conclusion and perspectives The physical problem and the associated model Physical problem and associated model Schr¨ odinger-Poisson system i ∂ t Ψ ε = ( i ∇ ) 2 Ψ ε + V ε Ψ ε , − ∆ V ε = | Ψ ε | 2 .
Introduction Time averaging and main results Main Tools Global in time estimates Conclusion and perspectives The physical problem and the associated model Physical problem and associated model Schr¨ odinger-Poisson system perturbed by a confinement potential i ∂ t Ψ ε = ( i ∇ ) 2 Ψ ε + V ε c Ψ ε + V ε Ψ ε , 1 V ε = | Ψ ε | 2 � � | x | 2 + z 2 ∗ . � 4 π x = ( x 1 , x 2 ) ∈ R 2 et z ∈ R are the transport and confinement directions respectively.
Introduction Time averaging and main results Main Tools Global in time estimates Conclusion and perspectives The physical problem and the associated model Physical problem and associated model Schr¨ odinger-Poisson system perturbed by a confinement potential and a strong uniform magnetic potential : i ∂ t Ψ ε = ( i ∇ − A ε ) 2 Ψ ε + V ε c Ψ ε + V ε Ψ ε , 1 V ε = | Ψ ε | 2 � � | x | 2 + z 2 ∗ . � 4 π x = ( x 1 , x 2 ) ∈ R 2 et z ∈ R are the transport and confinement directions respectively. Strong and uniform magnetic field : B ε e 2 , where B ε > 0 is fixed. Associated magnetic potential A ε : we choose the gauge A ε ( z ) = B ε ze 1 .
Introduction Time averaging and main results Main Tools Global in time estimates Conclusion and perspectives The physical problem and the associated model Scale of the confinement and magnetic field Scale of the Magnetic field : B ε = B ε 2 with B > 0. c ( z ) = 1 ε 2 V c ( z Scale of the Confinement potential : V ε ε ). Confinement assumption : V c is assumed to be even, positive and smooth such that : V c ( z ) − | z |→∞ + ∞ . → + a gap assumption : If ( E n ) n ≥ 0 denote the eigenvalues of z + B 2 z 2 + V c ( z ), we assume that operator − ∂ 2 ∃ γ > 0 , ∀ n , m ≥ 0 , n � = m , | E n − E m | > γ > 0 .
Introduction Time averaging and main results Main Tools Global in time estimates Conclusion and perspectives The physical problem and the associated model Scale of the confinement and magnetic field Scale of the Magnetic field : B ε = B ε 2 with B > 0. c ( z ) = 1 ε 2 V c ( z Scale of the Confinement potential : V ε ε ). Confinement assumption : V c is assumed to be even, positive and smooth such that : V c ( z ) − | z |→∞ + ∞ . → + a gap assumption : If ( E n ) n ≥ 0 denote the eigenvalues of z + B 2 z 2 + V c ( z ), we assume that operator − ∂ 2 ∃ γ > 0 , ∀ n , m ≥ 0 , n � = m , | E n − E m | > γ > 0 .
Introduction Time averaging and main results Main Tools Global in time estimates Conclusion and perspectives The physical problem and the associated model Scale of the confinement and magnetic field Scale of the Magnetic field : B ε = B ε 2 with B > 0. c ( z ) = 1 ε 2 V c ( z Scale of the Confinement potential : V ε ε ). Confinement assumption : V c is assumed to be even, positive and smooth such that : V c ( z ) − | z |→∞ + ∞ . → + a gap assumption : If ( E n ) n ≥ 0 denote the eigenvalues of z + B 2 z 2 + V c ( z ), we assume that operator − ∂ 2 ∃ γ > 0 , ∀ n , m ≥ 0 , n � = m , | E n − E m | > γ > 0 .
Introduction Time averaging and main results Main Tools Global in time estimates Conclusion and perspectives The physical problem and the associated model Scale of the confinement and magnetic field Scale of the Magnetic field : B ε = B ε 2 with B > 0. c ( z ) = 1 ε 2 V c ( z Scale of the Confinement potential : V ε ε ). Confinement assumption : V c is assumed to be even, positive and smooth such that : V c ( z ) − | z |→∞ + ∞ . → + a gap assumption : If ( E n ) n ≥ 0 denote the eigenvalues of z + B 2 z 2 + V c ( z ), we assume that operator − ∂ 2 ∃ γ > 0 , ∀ n , m ≥ 0 , n � = m , | E n − E m | > γ > 0 .
Introduction Time averaging and main results Main Tools Global in time estimates Conclusion and perspectives Mathematical background State of Art 1 Schr¨ odinger-Poisson system : Brezzi, Markowich ’91, Illner, Zweifel, Lange ’94, Castella ’97 2 Stationary Schr¨ odinger-Poisson system : Ben Abdallah ’00, Nier ’90, ’93 3 Schr¨ odinger-Poisson system confined along a plane : Pinaud ’03, Ben-Abdallah, M´ ehats, Pinaud ’05, Stage de M2 de F.F 4 Time Averaging for the nonlinear Schr¨ odinger equation : Ben-Abdallah, Castella, M´ ehats ’08
Introduction Time averaging and main results Main Tools Global in time estimates Conclusion and perspectives Rescaling Rescaling : the initial system In order to observe the system at the gas scale, we rescale the variables : z = z ˜ t = t , ˜ x = x , ˜ ε. Which leads to the new initial system : i ∂ t ψ ε = − ∆ x ψ ε − 2 iB ε z ∂ 1 ψ ε + 1 ε 2 H z ψ ε + V ε ψ ε , where z + B 2 z 2 + V c ( z ) H z = − ∂ 2 and 1 V ε = | x | 2 + ε 2 z 2 ∗ | ψ ε | 2 . � 4 π
Introduction Time averaging and main results Main Tools Global in time estimates Conclusion and perspectives Rescaling Rescaling : the initial system In order to observe the system at the gas scale, we rescale the variables : z = z ˜ t = t , ˜ x = x , ˜ ε. Which leads to the new initial system : i ∂ t ψ ε = − ∆ x ψ ε − 2 iB ε z ∂ 1 ψ ε + 1 ε 2 H z ψ ε + V ε ψ ε , where z + B 2 z 2 + V c ( z ) H z = − ∂ 2 and 1 V ε = | x | 2 + ε 2 z 2 ∗ | ψ ε | 2 . � 4 π
Introduction Time averaging and main results Main Tools Global in time estimates Conclusion and perspectives The Poisson non-linearity Introduction 1 The physical problem and the associated model Mathematical background Rescaling The Poisson non-linearity Asymptotic model Time averaging and main results 2 Second order averaging Main theorem Main Tools 3 Adapted functional framework Asymptotics for the Poisson kernel A priori local in time estimates Global in time estimates 4 Conclusion and perspectives 5
Introduction Time averaging and main results Main Tools Global in time estimates Conclusion and perspectives The Poisson non-linearity The Poisson non-linearity The asymptotic obtained for the solution V ε of the Poisson equation as ε tends to 0 gives : 1 � | ψ ε ( t , · , z ′ ) | 2 dz ′ V ( t , x , z ) ∼ W ( t , x ) = 4 π | x | ∗ x R The B=0 case : The Schr¨ odinger-Poisson system confined on the plane : i ∂ t ψ ε = − ∆ x ψ ε + 1 ε 2 H z ψ ε + V ε ψ ε We filter out by e itH z /ε 2 : φ ε ( t , x , z ) = e itH z /ε 2 ψ ε ( t , x , z ) satisfies i ∂ t φ ε = − ∆ x φ ε + e − itH z /ε 2 V ε e itH z /ε 2 φ ε . It converges towards the limit model : i ∂ t φ = − ∆ x φ + W φ
Introduction Time averaging and main results Main Tools Global in time estimates Conclusion and perspectives The Poisson non-linearity The Poisson non-linearity The asymptotic obtained for the solution V ε of the Poisson equation as ε tends to 0 gives : 1 � | ψ ε ( t , · , z ′ ) | 2 dz ′ V ( t , x , z ) ∼ W ( t , x ) = 4 π | x | ∗ x R The B=0 case : The Schr¨ odinger-Poisson system confined on the plane : i ∂ t ψ ε = − ∆ x ψ ε + 1 ε 2 H z ψ ε + V ε ψ ε We filter out by e itH z /ε 2 : φ ε ( t , x , z ) = e itH z /ε 2 ψ ε ( t , x , z ) satisfies i ∂ t φ ε = − ∆ x φ ε + e − itH z /ε 2 V ε e itH z /ε 2 φ ε . It converges towards the limit model : i ∂ t φ = − ∆ x φ + W φ
Introduction Time averaging and main results Main Tools Global in time estimates Conclusion and perspectives The Poisson non-linearity Results for the B = 0 case ehats, Pinaud, ’05 : L 2 convergence result for 1 Ben Abdallah, M´ an initial datum polarized on an eigenmode of the operator − ∂ 2 ∂ z 2 + V ε c ( z ). ehats : L 2 convergence result 2 F.F under direction of F. M´ extended to any initial datum bounded uniformly in ε in the energy space.
Recommend
More recommend
