Some estimates for the bilinear Schrdinger equation with discrete - PowerPoint PPT Presentation

Schrödinger Equation Some known results A new result Numerical simulations Some estimates for the bilinear Schrödinger equation with discrete spectrum Thomas Chambrion (joint work with U. Boscain, M. Caponigro and M. Sigalotti) IHP, 8-11 December 2010

Schrödinger Equation Some known results A new result Numerical simulations Quantum systems The state of a quantum system evolving in a space (Ω , µ ) can be represented by its wave function ψ . Under suitable hypotheses, the dynamics for ψ is given by the Schrödinger equation : i ∂ψ ∂ t ( x , t ) = − ∆ ψ ( x , t ) + V ( x ) ψ ( x , t ) Ω : finite dimensional manifold, for instance a bounded domain of R d , or R d , or SO ( 3 ) ,... ψ ∈ L 2 (Ω , C ) : wave function (state of the system) V : Ω → R : physical potential

Schrödinger Equation Some known results A new result Numerical simulations Quantum systems The state of a quantum system evolving in a space (Ω , µ ) can be represented by its wave function ψ . Under suitable hypotheses, the dynamics for ψ is given by the Schrödinger equation : i ∂ψ ∂ t ( x , t ) = − ∆ ψ ( x , t ) + V ( x ) ψ ( x , t ) + u ( t ) W ( x ) ψ ( x , t ) Ω : finite dimensional manifold, for instance a bounded domain of R d , or R d , or SO ( 3 ) ,... ψ ∈ L 2 (Ω , C ) : wave function (state of the system) V : Ω → R : physical potential W : Ω → R : control potential

Schrödinger Equation Some known results A new result Numerical simulations Quantum systems The state of a quantum system evolving in a space (Ω , µ ) can be represented by its wave function ψ . Under suitable hypotheses, the dynamics for ψ is given by the Schrödinger equation : i ∂ψ ∂ t ( x , t ) = − ∆ ψ ( x , t ) + V ( x ) ψ ( x , t ) + u ( t ) W ( x ) ψ ( x , t ) Ω : finite dimensional manifold, for instance a bounded domain of R d , or R d , or SO ( 3 ) ,... ψ ∈ L 2 (Ω , C ) : wave function (state of the system) V : Ω → R : physical potential W : Ω → R : control potential The well-posedness is far from obvious. It may require to add boundary conditions ( ψ | ∂ Ω = 0 if Ω is a bounded subspace of R d ) and hypotheses on V and W .

Schrödinger Equation Some known results A new result Numerical simulations Abstract form d ψ dt = A ( ψ ) + uB ( ψ ) , u ∈ U ( A , B , U ) with the assumptions H complex Hilbert space ; U ⊂ R ;

Schrödinger Equation Some known results A new result Numerical simulations Abstract form d ψ dt = A ( ψ ) + uB ( ψ ) , u ∈ U ( A , B , U ) with the assumptions H complex Hilbert space ; U ⊂ R ; A , B skew-adjoint operators on H (not necessarily bounded) ;

Schrödinger Equation Some known results A new result Numerical simulations Abstract form d ψ dt = A ( ψ ) + uB ( ψ ) , u ∈ U ( A , B , U ) with the assumptions H complex Hilbert space ; U ⊂ R ; A , B skew-adjoint operators on H (not necessarily bounded) ; ( φ n ) n ∈ N orthonormal basis of H made from eigenvectors of A ;

Schrödinger Equation Some known results A new result Numerical simulations Abstract form d ψ dt = A ( ψ ) + uB ( ψ ) , u ∈ U ( A , B , U ) with the assumptions H complex Hilbert space ; U ⊂ R ; A , B skew-adjoint operators on H (not necessarily bounded) ; ( φ n ) n ∈ N orthonormal basis of H made from eigenvectors of A ; every eigenspace of A is finite-dimensional ;

Schrödinger Equation Some known results A new result Numerical simulations Abstract form d ψ dt = A ( ψ ) + uB ( ψ ) , u ∈ U ( A , B , U ) with the assumptions H complex Hilbert space ; U ⊂ R ; A , B skew-adjoint operators on H (not necessarily bounded) ; ( φ n ) n ∈ N orthonormal basis of H made from eigenvectors of A ; every eigenspace of A is finite-dimensional ; φ n ∈ D ( B ) for every n ∈ N ;

Schrödinger Equation Some known results A new result Numerical simulations Abstract form d ψ dt = A ( ψ ) + uB ( ψ ) , u ∈ U ( A , B , U ) with the assumptions H complex Hilbert space ; U ⊂ R ; A , B skew-adjoint operators on H (not necessarily bounded) ; ( φ n ) n ∈ N orthonormal basis of H made from eigenvectors of A ; every eigenspace of A is finite-dimensional ; φ n ∈ D ( B ) for every n ∈ N ; for every u in U , A + uB has a unique self-adjoint extension.

Schrödinger Equation Some known results A new result Numerical simulations Abstract form d ψ dt = A ( ψ ) + uB ( ψ ) , u ∈ U ( A , B , U ) with the assumptions H complex Hilbert space ; U ⊂ R ; A , B skew-adjoint operators on H (not necessarily bounded) ; ( φ n ) n ∈ N orthonormal basis of H made from eigenvectors of A ; every eigenspace of A is finite-dimensional ; φ n ∈ D ( B ) for every n ∈ N ; for every u in U , A + uB has a unique self-adjoint extension. Under these assumptions ∀ u ∈ U , ∃ e t ( A + uB ) : H → H group of unitary transformations

Schrödinger Equation Some known results A new result Numerical simulations Definition of solutions i ∂ψ ∂ t ( x , t ) = − ∆ ψ ( x , t ) + V ( x ) ψ ( x , t )+ u ( t ) W ( x ) ψ ( x , t ) We choose piecewise constant controls Definition T ( ψ 0 ) = e t k ( A + u k B ) ◦ · · · ◦ e t 1 ( A + u 1 B ) ( ψ 0 ) the solution of We call Υ u the system starting from ψ 0 associated to the piecewise constant control u 1 χ [ 0 , t 1 ] + u 2 χ [ t 1 , t 1 + t 2 ] + · · · . If B is bounded, it is possible to extend this definition for controls u that are only measurable bounded or locally integrable.

Schrödinger Equation Some known results A new result Numerical simulations Controllability Exact controllability ψ a , ψ b given. Is it possible to find a control u : [ 0 , T ] → U such that Υ u T ( ψ a ) = ψ b ? Approximate controllability ǫ > 0, ψ a , ψ b given. Is it possible to find a control u : [ 0 , T ] → U such that � Υ u T ( ψ a ) − ψ b � < ǫ ? Simultaneous approximate controllability a , . . . , ψ p b , . . . , ψ p ǫ > 0, ψ 1 a , ψ 2 a , ψ 1 b given. Is it possible to find a T ( ψ j a ) − ψ j control u : [ 0 , T ] → U such that � Υ u b � < ǫ for every j ?

Schrödinger Equation Some known results A new result Numerical simulations A negative result Theorem (Ball-Marsden-Slemrod, 1982 and Turinici, 2000) If ψ �→ W ψ is bounded, then the reachable set from any point (with L 1 + r controls) of the control system : i ∂ψ ∂ t ( x , t ) = − ∆ ψ ( x , t ) + V ( x ) ψ ( x , t )+ u ( t ) W ( x ) ψ ( x , t ) has dense complement in the unit sphere.

Schrödinger Equation Some known results A new result Numerical simulations Non controllability of the harmonic oscillator (I) Ω = R ∂ 2 ψ i ∂ψ ∂ t = − 1 ∂ x 2 + 1 2 x 2 ψ − u ( t ) x ψ 2 Theorem (Mirrahimi-Rouchon, 2004) The quantum harmonic oscillator is not controllable. (see also Illner-Lange-Teismann 2005 and Bloch-Brockett-Rangan 2006)

Schrödinger Equation Some known results A new result Numerical simulations Non controllability of the harmonic oscillator (II) The Galerkin approximation of order n is controllable (in U ( n ) ) :   1 0 · · · 0 . ...   . A = − i 0 3 .     . ... ... 2  .  . 0   0 · · · 0 2 n + 1   0 1 0 · · · · · · 0 √ . ... .   1 0 2 .   √ √  .  ... .   0 2 0 3 .   B = − i .  ... ... ... ...  .   . 0   √ n + 1 .  ... ...  .  . 0  √ n + 1 0 · · · · · · 0 0

Schrödinger Equation Some known results A new result Numerical simulations Exact controllability for the potential well Ω = ( − 1 / 2 , 1 / 2 ) ∂ 2 ψ i ∂ψ ∂ t = − 1 ∂ x 2 − u ( t ) x ψ 2 Theorem (Beauchard, 2005) The system is exactly controllable in the intersection of the unit sphere of L 2 with H 7 ( 0 ) .

Schrödinger Equation Some known results A new result Numerical simulations Generic controllability results via geometric methods Theorem (Boscain-Chambrion-Mason-Sigalotti, 2009) If ( λ n + 1 − λ n ) n ∈ N is Q -linearly independent and if B is connected w.r.t. A, then for every δ > 0 ( A , B , ( 0 , δ )) is approximately controllable on the unit sphere. The family ( λ n + 1 − λ n ) n ∈ N is Q -linearly independent if for every N ∈ N and ( q 1 , . . . , q N ) ∈ Q N � { 0 } one has � N n = 1 q n ( λ n + 1 − λ n ) � = 0 . B is connected w.r.t. A if for every { j , k } in N 2 , ∃ p ∈ N , ∃ j = l 1 , l 2 , . . . , l p = k such that b l i , l i + 1 � = 0, for 1 ≤ i ≤ p .

Schrödinger Equation Some known results A new result Numerical simulations Lyapounov techniques i ∂ψ ∂ t ( x , t ) = − ∆ ψ ( x , t ) + V ( x ) ψ ( x , t ) + u ( t ) W ( x ) ψ ( x , t ) � �� � � �� � A ψ B ψ Ω is a bounded domain of R d , with smooth boundary. Theorem (Nersesyan, 2009) If b 1 , j � = 0 for every j ≥ 1 and | λ 1 − λ j | � = | λ k − λ l | for every j > 1 , { 1 , j } � = { k , l } then the control system is approximately controllable on the unit sphere of L 2 for H s norms.

Schrödinger Equation Some known results A new result Numerical simulations Fixed point theorem Ω = ( 0 , 1 ) i ∂ψ ∂ t ( x , t ) = − ∆ ψ ( x , t ) + u ( t ) W ( x ) ψ ( x , t ) � �� � � �� � A ψ B ψ Theorem (Beauchard-Laurent, 2009) If there exists C > 0 such that for every j ∈ N , | b 1 , j | > C j 3 then the system is exactly controllable in the intersection of the unit sphere with H 3 ( 0 ) .