Wigner Measures and Effective Mass Theorems Victor Chabu 1 , Clotilde Fermanian Kammerer 1 , Fabricio Macia 2 1 Universit´ e Paris Est - Cr´ eteil & CNRS 2 Universidad Polit´ ecnica de Madrid, ETSI Navales Cergy-Pontoise, June 22nd. 2016 C. Fermanian Kammerer (U.P.E.) Wigner Measures and Eff. Mass Theo. Cergy-Pontoise, 22.6.2016 1 / 21
Content: 1 Schr¨ odinger Equation in a Lattice 2 Floquet-Bloch theory 3 Quantifying the lack of dispersion 4 Strategy of the proof 5 Back to Effective Mass Theory C. Fermanian Kammerer (U.P.E.) Wigner Measures and Eff. Mass Theo. Cergy-Pontoise, 22.6.2016 2 / 21
Schr¨ odinger Equation in a Lattice : The equation The equation: i � ∂ t Ψ + � 2 2 m ∆ x Ψ − Q per ( x ) Ψ − Q ext ( x )Ψ = 0 , ( t , x ) ∈ R × R d , Ψ | t =0 = Ψ 0 ∈ L 2 ( R d ) , where Q per is a potential periodic with respect to a lattice Γ = Z d . Let ε be the ratio between the mean spacing of the lattice and the characteristic length scale of variation of Q ext . ε ≪ 1 . ⇒ Change of units and rescaling the external potential and the wave function ! = (see [Poupaud & Ringhofer 96]). C. Fermanian Kammerer (U.P.E.) Wigner Measures and Eff. Mass Theo. Cergy-Pontoise, 22.6.2016 3 / 21
Schr¨ odinger Equation in a Lattice : The equation The equation: i � ∂ t Ψ + � 2 2 m ∆ x Ψ − Q per ( x ) Ψ − Q ext ( x )Ψ = 0 , ( t , x ) ∈ R × R d , Ψ | t =0 = Ψ 0 ∈ L 2 ( R d ) , where Q per is a potential periodic with respect to a lattice Γ = Z d . Let ε be the ratio between the mean spacing of the lattice and the characteristic length scale of variation of Q ext . ε ≪ 1 . ⇒ Change of units and rescaling the external potential and the wave function ! = (see [Poupaud & Ringhofer 96]). C. Fermanian Kammerer (U.P.E.) Wigner Measures and Eff. Mass Theo. Cergy-Pontoise, 22.6.2016 3 / 21
Schr¨ odinger Equation in a Lattice : The scaled equation The equation: � x � i ∂ t ψ ε + 1 2∆ x ψ ε − 1 ψ ε − V ( x ) ψ ε = 0 , ( t , x ) ∈ R × R d , ε 2 V per ε ψ ε | t =0 = ψ ε 0 ∈ L 2 ( R d ) , where V per is a potential periodic with respect to Z d . Question: Effective Mass Theory consists in showing situations where ψ ε ( t ) can be approximated by the solution of a Effective Mass Equation: i ∂ t φ ( t , x ) + 1 2 � M D x , D x � φ ( t , x ) − V ( x ) φ ( t , x ) = 0 . M is a d × d matrix called the effective mass tensor. C. Fermanian Kammerer (U.P.E.) Wigner Measures and Eff. Mass Theo. Cergy-Pontoise, 22.6.2016 4 / 21
Schr¨ odinger Equation in a Lattice : The scaled equation The equation: � x � i ∂ t ψ ε + 1 2∆ x ψ ε − 1 ψ ε − V ( x ) ψ ε = 0 , ( t , x ) ∈ R × R d , ε 2 V per ε ψ ε | t =0 = ψ ε 0 ∈ L 2 ( R d ) , where V per is a potential periodic with respect to Z d . Question: Effective Mass Theory consists in showing situations where ψ ε ( t ) can be approximated by the solution of a Effective Mass Equation: i ∂ t φ ( t , x ) + 1 2 � M D x , D x � φ ( t , x ) − V ( x ) φ ( t , x ) = 0 . M is a d × d matrix called the effective mass tensor. C. Fermanian Kammerer (U.P.E.) Wigner Measures and Eff. Mass Theo. Cergy-Pontoise, 22.6.2016 4 / 21
Schr¨ odinger Equation in a Lattice : Our purpose The two main questions of the literature : Finding initial conditions for which the previous analysis holds, Finding the corresponding M . [Bensoussan, Lions & Papanicolaou 78], [Poupaud & Ringhofer 96], [Allaire & Piatniski 05], [Hoefer &Weinstein 11], [Barletti & Ben Abdallah 11]. ⇒ Our purpose : = Getting rid of assumptions on the initial conditions, Clarifying the dependence of M on the parameter of the equation. C. Fermanian Kammerer (U.P.E.) Wigner Measures and Eff. Mass Theo. Cergy-Pontoise, 22.6.2016 5 / 21
Schr¨ odinger Equation in a Lattice : Our purpose The two main questions of the literature : Finding initial conditions for which the previous analysis holds, Finding the corresponding M . [Bensoussan, Lions & Papanicolaou 78], [Poupaud & Ringhofer 96], [Allaire & Piatniski 05], [Hoefer &Weinstein 11], [Barletti & Ben Abdallah 11]. ⇒ Our purpose : = Getting rid of assumptions on the initial conditions, Clarifying the dependence of M on the parameter of the equation. C. Fermanian Kammerer (U.P.E.) Wigner Measures and Eff. Mass Theo. Cergy-Pontoise, 22.6.2016 5 / 21
Schr¨ odinger Equation in a Lattice : our strategy Remark If v ε solves the semiclassical Schr¨ odinger equation � x � i ε∂ t v ε + ε 2 2 ∆ x v ε − V per v ε − ε 2 V ( x ) v ε = 0 , v ε | t =0 = ψ ε 0 . ε Then, ψ ε ( t , x ) = v ε � t � ε, x . [G´ erard], [GMMP] [Poupaud & Ringhofer], [Bechouche, Mauser & Poupaud], [Spohn & Teufel] [Panati, Spohn &Teufel], [Dimassi, Guillot & Ralston] [Allaire &Palombaro] [Carles &Sparber] = ⇒ Perform simultaneously the s.c. limit ε → 0 with the limit t /ε → + ∞ . [Macia and his collaborators Anantharaman, L´ eautaud, Rivi` ere & C.F.K.] (without periodic potential). Our goal : apply this viewpoint to effective mass theory. ⇒ a generalized effective mass equation of Heisenberg type. = C. Fermanian Kammerer (U.P.E.) Wigner Measures and Eff. Mass Theo. Cergy-Pontoise, 22.6.2016 6 / 21
Schr¨ odinger Equation in a Lattice : our strategy Remark If v ε solves the semiclassical Schr¨ odinger equation � x � i ε∂ t v ε + ε 2 2 ∆ x v ε − V per v ε − ε 2 V ( x ) v ε = 0 , v ε | t =0 = ψ ε 0 . ε Then, ψ ε ( t , x ) = v ε � t � ε, x . [G´ erard], [GMMP] [Poupaud & Ringhofer], [Bechouche, Mauser & Poupaud], [Spohn & Teufel] [Panati, Spohn &Teufel], [Dimassi, Guillot & Ralston] [Allaire &Palombaro] [Carles &Sparber] = ⇒ Perform simultaneously the s.c. limit ε → 0 with the limit t /ε → + ∞ . [Macia and his collaborators Anantharaman, L´ eautaud, Rivi` ere & C.F.K.] (without periodic potential). Our goal : apply this viewpoint to effective mass theory. ⇒ a generalized effective mass equation of Heisenberg type. = C. Fermanian Kammerer (U.P.E.) Wigner Measures and Eff. Mass Theo. Cergy-Pontoise, 22.6.2016 6 / 21
Schr¨ odinger Equation in a Lattice : our strategy Remark If v ε solves the semiclassical Schr¨ odinger equation � x � i ε∂ t v ε + ε 2 2 ∆ x v ε − V per v ε − ε 2 V ( x ) v ε = 0 , v ε | t =0 = ψ ε 0 . ε Then, ψ ε ( t , x ) = v ε � t � ε, x . [G´ erard], [GMMP] [Poupaud & Ringhofer], [Bechouche, Mauser & Poupaud], [Spohn & Teufel] [Panati, Spohn &Teufel], [Dimassi, Guillot & Ralston] [Allaire &Palombaro] [Carles &Sparber] = ⇒ Perform simultaneously the s.c. limit ε → 0 with the limit t /ε → + ∞ . [Macia and his collaborators Anantharaman, L´ eautaud, Rivi` ere & C.F.K.] (without periodic potential). Our goal : apply this viewpoint to effective mass theory. ⇒ a generalized effective mass equation of Heisenberg type. = C. Fermanian Kammerer (U.P.E.) Wigner Measures and Eff. Mass Theo. Cergy-Pontoise, 22.6.2016 6 / 21
Floquet-Bloch theory : Bloch waves and energies We use the Ansatz : ψ ε ( t , x ) = U ε � � t , x , x , where U ε ( t , x , y ) is assumed ε to be Z d -periodic in y and solves i ε 2 ∂ t U ε ( t , x , y ) = P ( ε D ) U ε ( t , x , y ) + ε 2 V ( x ) U ε ( t , x , y ) , U ε | t =0 = ψ ε 0 where P ( ξ ) = 1 2 ( ξ + D y ) 2 + V Γ ( y ) , y ∈ T d := R d \ Z d . The Bloch energies are the eigenvalues of the self-adjoint operator on the torus P ( ξ ) : λ 1 ( ξ ) ≤ λ 2 ( ξ ) ≤ · · · ≤ λ n ( ξ ) → + ∞ . They are 2 π Z d periodic and smooth in domain where they are of constant multiplicity. The Bloch waves are the orthonormal eigenfunctions of P ( ξ ) P ( ξ ) ϕ n ( ξ, y ) = λ n ( ξ ) ϕ n ( ξ, y ) , n ∈ N , y ∈ T 2 , ∀ ξ ∈ R d . C. Fermanian Kammerer (U.P.E.) Wigner Measures and Eff. Mass Theo. Cergy-Pontoise, 22.6.2016 7 / 21
Floquet-Bloch theory : Bloch waves and energies We use the Ansatz : ψ ε ( t , x ) = U ε � � t , x , x , where U ε ( t , x , y ) is assumed ε to be Z d -periodic in y and solves i ε 2 ∂ t U ε ( t , x , y ) = P ( ε D ) U ε ( t , x , y ) + ε 2 V ( x ) U ε ( t , x , y ) , U ε | t =0 = ψ ε 0 where P ( ξ ) = 1 2 ( ξ + D y ) 2 + V Γ ( y ) , y ∈ T d := R d \ Z d . The Bloch energies are the eigenvalues of the self-adjoint operator on the torus P ( ξ ) : λ 1 ( ξ ) ≤ λ 2 ( ξ ) ≤ · · · ≤ λ n ( ξ ) → + ∞ . They are 2 π Z d periodic and smooth in domain where they are of constant multiplicity. The Bloch waves are the orthonormal eigenfunctions of P ( ξ ) P ( ξ ) ϕ n ( ξ, y ) = λ n ( ξ ) ϕ n ( ξ, y ) , n ∈ N , y ∈ T 2 , ∀ ξ ∈ R d . C. Fermanian Kammerer (U.P.E.) Wigner Measures and Eff. Mass Theo. Cergy-Pontoise, 22.6.2016 7 / 21
Recommend
More recommend
