Approximate controllability of the bilinear Schrdinger equation - PowerPoint PPT Presentation

Schrödinger Equation Some known results Perturbation A new result Approximate controllability of the bilinear Schrödinger equation Thomas Chambrion Toulon, 24-28 October 2010

Schrödinger Equation Some known results Perturbation A new result 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 Perturbation A new result 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 Perturbation A new result Discrete spectrum Theorem If Ω is a bounded domain with smooth boundary and if V ∈ L ∞ (Ω) , then there exists a family of vectors H 2 (Ω) ∩ H 1 0 (Ω) made from eigenvectors of − ∆ + V that is an orthonormal basis of L 2 (Ω) .

Schrödinger Equation Some known results Perturbation A new result Discrete spectrum Theorem If Ω = R d and V ∈ L 1 loc is bounded from below such that | x |→∞ V ( x ) = + ∞ , lim then there exists a family of vectors H 2 ( R d ) made from eigenvectors of − ∆ + V that is an orthonormal basis of L 2 ( R d ) . If V ≥ 0 , then all eigenfunction have exponential decay at infinity.

Schrödinger Equation Some known results Perturbation A new result 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 Perturbation A new result 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 Perturbation A new result 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) ; D ( A ) , D ( B ) domains of A , B ;

Schrödinger Equation Some known results Perturbation A new result 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) ; D ( A ) , D ( B ) domains of A , B ; ( φ n ) n ∈ N orthonormal basis of H made from eigenvectors of A ;

Schrödinger Equation Some known results Perturbation A new result 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) ; D ( A ) , D ( B ) domains of A , B ; ( φ n ) n ∈ N orthonormal basis of H made from eigenvectors of A ; � B ψ � ≤ α � A ψ � + β � ψ � (hence φ n ∈ D ( B ) for every n ∈ N ).

Schrödinger Equation Some known results Perturbation A new result 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) ; D ( A ) , D ( B ) domains of A , B ; ( φ n ) n ∈ N orthonormal basis of H made from eigenvectors of A ; � B ψ � ≤ α � A ψ � + β � ψ � (hence φ n ∈ D ( B ) for every n ∈ N ). We do not assume that B is bounded.

Schrödinger Equation Some known results Perturbation A new result Definition of solutions d dt ψ = ( A + u ( t ) B ) ψ The well-posedness is far from obvious.

Schrödinger Equation Some known results Perturbation A new result Definition of solutions d dt ψ = ( A + u ( t ) B ) ψ The well-posedness is far from obvious. Under the above assumptions on A and B ∀ u ∈ U , ∃ e t ( A + uB ) : H → H group of unitary transformations

Schrödinger Equation Some known results Perturbation A new result Definition of solutions d dt ψ = ( A + u ( t ) B ) ψ The well-posedness is far from obvious. Under the above assumptions on A and B ∀ u ∈ U , ∃ e t ( A + uB ) : H → H group of unitary transformations The easy way, using piecewise constant controls Definition We call e t k ( A + u k B ) ◦ · · · ◦ e t 1 ( A + u 1 B ) ( ψ 0 ) the solution of the control system ( A , B , U ) 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 ] + · · · .

Schrödinger Equation Some known results Perturbation A new result Definition of solutions (II) d dt ψ = ( A + u ( t ) B ) ψ

Schrödinger Equation Some known results Perturbation A new result Definition of solutions (II) d dt ψ = ( A + u ( t ) B ) ψ If B is bounded, definition of solution for controls u that are only measurable bounded or locally integrable by standard fixed point theory, (see Beauchard 2005).

Schrödinger Equation Some known results Perturbation A new result Definition of solutions (II) d dt ψ = ( A + u ( t ) B ) ψ If B is bounded, definition of solution for controls u that are only measurable bounded or locally integrable by standard fixed point theory, (see Beauchard 2005). If B is unbounded, well-posedness for u of class C 1 (Reed and Simon, 1973) can be extended to L 1 loc and locally finite measure (Boussaid-Chambrion, 2010).

Schrödinger Equation Some known results Perturbation A new result Finite dimensional case A T = − A , ¯ B T = − B with ¯ x = ( A + uB ) x ˙ Ask Yuri : A T � = − A , ¯ B T � = − B ) is much harder (ask The non-compact case ( ¯ Jean-Paul).

Schrödinger Equation Some known results Perturbation A new result Finite dimensional case A T = − A , ¯ B T = − B with ¯ x = ( A + uB ) x ˙ Ask Yuri : The system is exactly controllable in SU ( n ) ⇔ Lie ( A , B ) = su ( n ) . (Jurdjevic-Sussmann) A T � = − A , ¯ B T � = − B ) is much harder (ask The non-compact case ( ¯ Jean-Paul).

Schrödinger Equation Some known results Perturbation A new result Finite dimensional case A T = − A , ¯ B T = − B with ¯ x = ( A + uB ) x ˙ Ask Yuri : The system is exactly controllable in SU ( n ) ⇔ Lie ( A , B ) = su ( n ) . (Jurdjevic-Sussmann) The system is exactly controllable on the complex sphere ⇔ Lie ( A , B ) is isomorphic to some su ( p ) or so ( q ) or ... (Brockett) A T � = − A , ¯ B T � = − B ) is much harder (ask The non-compact case ( ¯ Jean-Paul).

Schrödinger Equation Some known results Perturbation A new result 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 r controls, r > 1 ) 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 Perturbation A new result Density matrices A density matrix is a trace-class operator representing a mix of states (proportion P j of the system in the state ψ j ) � � P j ψ j ψ ∗ ρ = P j � ψ j , ·� ψ j = j j ∈ N j ∈ N ( P j ) j ∈ N sequence in [ 0 , 1 ] with � j P j = 1.

Schrödinger Equation Some known results Perturbation A new result Density matrices A density matrix is a trace-class operator representing a mix of states (proportion P j of the system in the state ψ j ) � � P j ψ j ψ ∗ ρ = P j � ψ j , ·� ψ j = j j ∈ N j ∈ N ( P j ) j ∈ N sequence in [ 0 , 1 ] with � j P j = 1. � aa ∗ − bb ∗ � < ǫ ⇒ ∃ θ s. t. � a − e i θ b � < ǫ

Schrödinger Equation Some known results Perturbation A new result Notions of controllability Exact point-wise controllability : x a , x b given. Is is possible to steer the system from x a to x b ? Approximate controllability : x a , x b in H , ǫ > 0 given.Is is possible to steer the system from x a to an ǫ neighborhood of x b ? b , . . . , x n Collective controllability : x 1 a , x 2 a , . . . , x a , x 1 b , x 2 b given, ǫ > 0 given. Is is possible to steer the system from x i a to an ǫ neighborhood of x i b ? Density matrix controllability : x 1 a , x 2 a , . . . , x a , x 1 b , x 2 b , . . . , x n b given, ǫ > 0 given. Is is possible to steer the system from x j a to an ǫ neighborhood of e i θ j x j b (for some θ j in R ) ?