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 Rd, or Rd, or SO(3),... ψ ∈ L2(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 Rd, or Rd, or SO(3),... ψ ∈ L2(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 H2(Ω) ∩ H1

0(Ω) made from

eigenvectors of −∆ + V that is an orthonormal basis of L2(Ω).

Schrödinger Equation Some known results Perturbation A new result

Discrete spectrum

Theorem If Ω = Rd and V ∈ L1

loc is bounded from below such that

lim

|x|→∞ V (x) = +∞,

then there exists a family of vectors H2(Rd) made from eigenvectors of −∆ + V that is an orthonormal basis of L2(Rd). If V ≥ 0, then all eigenfunction have exponential decay at infinity.

Schrödinger Equation Some known results Perturbation A new result

Abstract form

dψ 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

dψ 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

dψ 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

dψ 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 ; (φn)n∈N orthonormal basis of H made from eigenvectors of A ;

Schrödinger Equation Some known results Perturbation A new result

Abstract form

dψ 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 ; (φn)n∈N orthonormal basis of H made from eigenvectors of A ; Bψ ≤ αAψ + βψ (hence φn ∈ D(B) for every n ∈ N).

Schrödinger Equation Some known results Perturbation A new result

Abstract form

dψ 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 ; (φn)n∈N orthonormal basis of H made from eigenvectors of A ; Bψ ≤ αAψ + βψ (hence φn ∈ 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, ∃ et(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, ∃ et(A+uB) : H → H group of unitary transformations The easy way, using piecewise constant controls Definition We call etk(A+ukB) ◦ · · · ◦ et1(A+u1B)(ψ0) the solution of the control system (A, B, U) starting from ψ0 associated to the piecewise constant control u1χ[0,t1] + u2χ[t1,t1+t2] + · · · .

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 L1

loc and locally finite measure

(Boussaid-Chambrion, 2010).

Schrödinger Equation Some known results Perturbation A new result

Finite dimensional case

˙ x = (A + uB)x with ¯ AT = −A, ¯ BT = −B Ask Yuri : The non-compact case (¯ AT = −A, ¯ BT = −B) is much harder (ask Jean-Paul).

Schrödinger Equation Some known results Perturbation A new result

Finite dimensional case

˙ x = (A + uB)x with ¯ AT = −A, ¯ BT = −B Ask Yuri : The system is exactly controllable in SU(n) ⇔ Lie(A, B) = su(n). (Jurdjevic-Sussmann) The non-compact case (¯ AT = −A, ¯ BT = −B) is much harder (ask Jean-Paul).

Schrödinger Equation Some known results Perturbation A new result

Finite dimensional case

˙ x = (A + uB)x with ¯ AT = −A, ¯ BT = −B 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) The non-compact case (¯ AT = −A, ¯ BT = −B) is much harder (ask 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 Lr 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 Pj of the system in the state ψj) ρ =

• j∈N

Pjψj, ·ψj =

• j∈N

Pjψjψ∗

j

(Pj)j∈N sequence in [0, 1] with

j Pj = 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 Pj of the system in the state ψj) ρ =

• j∈N

Pjψj, ·ψj =

• j∈N

Pjψjψ∗

j

(Pj)j∈N sequence in [0, 1] with

j Pj = 1.

aa∗ − bb∗ < ǫ ⇒ ∃θ s. t.a − eiθb < ǫ

Schrödinger Equation Some known results Perturbation A new result

Notions of controllability

Exact point-wise controllability : xa, xb given. Is is possible to steer the system from xa to xb ? Approximate controllability : xa, xb in H, ǫ > 0 given.Is is possible to steer the system from xa to an ǫ neighborhood of xb ? Collective controllability : x1

a , x2 a , . . . , xa,x1 b, x2 b, . . . , xn b given,

ǫ > 0 given. Is is possible to steer the system from xi

a to an ǫ

neighborhood of xi

b ?

Density matrix controllability : x1

a , x2 a , . . . , xa,x1 b, x2 b, . . . , xn b

given, ǫ > 0 given. Is is possible to steer the system from xj

a to

an ǫ neighborhood of eiθjxj

b (for some θj in R) ?

Schrödinger Equation Some known results Perturbation A new result

Approximate controllability

Two ways of thinking : We are not able to reach every point on the sphere. But we can reach some points. How could we characterize these points ? Which unitary operators can be written in the form etk(A+ukB) ◦ · · · ◦ et1(A+u1B) ? Common corollary : approximate controllability on the unit sphere. Question Can you give a necessary and sufficient condition for approximate controllability (on the group or on the unit sphere) ?

Schrödinger Equation Some known results Perturbation A new result

Non controllability of the harmonic oscillator (I)

i ∂ψ ∂t = −1 2 ∂2ψ ∂x2 + 1 2x2ψ − u(t)xψ with ψ ∈ L2(R, C). Theorem (Mirrahimi-Rouchon, 2004) The quantum harmonic oscillator is not controllable (in any reasonable sense).

Schrödinger Equation Some known results Perturbation A new result

Non controllability of the harmonic oscillator (II)

The Galerkyn approximation of order n is controllable (in U(n)) : A = − i 2       1 · · · 3 ... . . . . . . ... ... · · · 2n + 1       B = −i            1 · · · · · · 1 √ 2 ... . . . √ 2 √ 3 ... . . . . . . ... ... ... ... . . . ... ... √n + 1 · · · · · · √n + 1           

Schrödinger Equation Some known results Perturbation A new result

Exact controllability for the potential well

Ω = (−1/2, 1/2) i ∂ψ ∂t = −1 2 ∂2ψ ∂x2 − u(t)xψ Theorem (Beauchard, 2005) The system is exactly controllable in the intersection of the unit sphere of L2 with H7

(0).

Proof : Coron’s return method and sophisticated fixed point theory.

Schrödinger Equation Some known results Perturbation A new result

Generic controllability results via geometric methods

Theorem (Boscain-Chambrion-Mason-Sigalotti, 2009) If (λn+1 − λn)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 (λn+1 − λn)n∈N is Q-linearly independent if for every N ∈ N and (q1, . . . , qN) ∈ QN {0} one has N

n=1 qn(λn+1 − λn) = 0.

B is connected w.r.t. A if for every {j, k} in N2, ∃p ∈ N, ∃j = l1, l2, . . . , lp = k such that bli,li+1 = 0, for 1 ≤ i ≤ p.

Schrödinger Equation Some known results Perturbation A new result

Lyapounov techniques

i ∂ψ ∂t (x, t) = −∆ψ(x, t) + V (x)ψ(x, t)

• Aψ

+u(t) W (x)ψ(x, t)

• Bψ

Ω is a bounded domain of Rd, with smooth boundary. Theorem (Nersesyan, 2009) If b1,j = 0 for every j ≥ 1 and |λ1 − λj| = |λk − λl| for every j > 1, {1, j} = {k, l} then the control system is approximately controllable on the unit sphere.

Schrödinger Equation Some known results Perturbation A new result

Coron’s return method comes back

Ω = (0, 1) i ∂ψ ∂t (x, t) = −∆ψ(x, t)

• Aψ

+u(t) W (x)ψ(x, t)

• Bψ

Theorem (Beauchard-Laurent, 2009) If there exists C > 0 such that for every j ∈ N, |b1,j| > C j3 then the system is exactly controllable in the intersection of the unit sphere with H3

(0).

Schrödinger Equation Some known results Perturbation A new result

A tempting idea... (I)

All the previous positive controllability results have the same structure : Unfortunate misconception (Chambrion et al., 2007-2008) If the operators A and B satisfy : a spectral non-resonance condition (this condition only involves A) a coupling condition (this condition only involves B) then the system (A, B, (0, δ)) is approximately controllable on the unit sphere.

Schrödinger Equation Some known results Perturbation A new result

A tempting idea... (II)

Beauchard-Laurent 2009 (and Beauchard 2005) :

λk = k2π2/2 for every k |b1,j| > C/j3 for every j ≥ 2

Nersesyan 2009 :

|λ1 − λj| = |λk − λl| for every j, k, l such that {1, j} = {k, l} |b1,j| = 0 for every j ≥ 2

Boscain-Chambrion-Mason-Sigalotti 2009 :

Q-linearly independence of the spectrum of A there exists a connectedness chain for B : for every j, k, there exists i1 = j, i2, ..., ip = k such that bil ,il+1 = 0 for 1 ≤ l ≤ p − 1.

Schrödinger Equation Some known results Perturbation A new result

...yet too simplistic

When B is relatively bounded w.r.t. A, we can use classical perturbation results : A + uB = A + ¯ uB

A¯

u

+ (u − ¯ u) new control B By analyticity, the existence of one ¯ u for which (A¯

u, B) satisfies the

non-resonance and coupling hypotheses ensures that this is also the case for almost every ¯ u.

Schrödinger Equation Some known results Perturbation A new result

...yet too simplistic

When B is relatively bounded w.r.t. A, we can use classical perturbation results : A + uB = A + ¯ uB

A¯

u

+ (u − ¯ u) new control B By analyticity, the existence of one ¯ u for which (A¯

u, B) satisfies the

non-resonance and coupling hypotheses ensures that this is also the case for almost every ¯ u. Theorem (Boscain-Chambrion-Mason-Sigalotti, 2009) Let Ω = R, V : x → x2 and Wa,b : x → eax2+bx (A = (−∆+V )/2, B = Wa,b). If √1 − a and b are algebraically independent, then the system is approximately controllable on the unit sphere of L2(R, C).

Schrödinger Equation Some known results Perturbation A new result

Collective approximate controllability

Interesting byproducts of the geometric approach Theorem (Boscain-Chambrion-Mason-Sigalotti, 2009) Let Φ : H → H be a unitary operator. If the spectrum of A is Q-linearly independent and if B is connected with respect to A, then for every n, for every δ > 0, for every ǫ > 0, there exists a piecewise constant u : [0, T] → [0, δ] such that ˙ x = (A + uB)x steers φj to Φ(φj) for 1 ≤ j ≤ n, up to ǫ.

Schrödinger Equation Some known results Perturbation A new result

Output tracking

Theorem (Chambrion, 2009) Let a curve t ∈ [0, T] → ψref (t) be given. Under the usual non-resonance and coupling hypotheses, the system ˙ x = (A + uB)x can track in modulus the curve ψref : ∀ǫ, δ > 0, ∃u ∈ PC([0, Tu], [0, δ]), Φ : [0, T] → [0, Tu] increasing bijection, such that, for every t ∈ [0, T], for every k,

• |ψref (t), φk| − |ψ(Φ(t)), φk|
• < ǫ

and ψref (T) − ψ(Tu) < ǫ.

Schrödinger Equation Some known results Perturbation A new result

Output tracking

Theorem (Chambrion, 2009) Let a curve t ∈ [0, T] → ψref (t) be given. Under the usual non-resonance and coupling hypotheses, the system ˙ x = (A + uB)x can track in modulus the curve ψref : ∀ǫ, δ > 0, ∃u ∈ PC([0, Tu], [0, δ]), Φ : [0, T] → [0, Tu] increasing bijection, such that, for every t ∈ [0, T], for every k,

• |ψref (t), φk| − |ψ(Φ(t)), φk|
• < ǫ

and ψref (T) − ψ(Tu) < ǫ. This property extends to the tracking of density matrices. Recall that the question was about approximate controllability on the sphere !

Schrödinger Equation Some known results Perturbation A new result

Idea of the geometric proof

Up to a time reparametrization, et(A+uB) = etu( 1

u A+B) that

exchange time and L1 norm, the control system is ˙ x = uAx + Bx After the change of variable Y = e−

R uAX, one finds

˙ Y = e−

R uABe− R uAY

Galerkyn approximation : ˙ Y =

• ei(λj−λk)

R ubj,k

• j,k Y .

Schrödinger Equation Some known results Perturbation A new result

Tracking

If (λk+1 − λk)k is Q-linearly independent, for every ǫ > 0, there exists a piecewise constant control u such that the system can track (in projection), up to ǫ, the finite dimensional system : ˙ Y =         · · · · · · . . . bj,k . . . bk,j · · · . . . · · · · · · · · · · · ·         Y

Schrödinger Equation Some known results Perturbation A new result

Tracking

If |λj − λk| = |λl − λm| for every {j, k} = {l, m}, for every ǫ > 0, there exists a piecewise constant control u such that the system can track (in projection), up to ǫ the finite dimensional system : ˙ Y = ρ         · · · · · · . . . bj,k . . . bk,j · · · . . . · · · · · · · · · · · ·         Y with ρ ≥

∞

• k=2

cos π 2k

• ≈ 0.4298156...

Schrödinger Equation Some known results Perturbation A new result

A sufficient condition for approximate controllability

Theorem (Boscain-Caponigro-Chambrion-Sigalotti, not published yet) If there exists a subset S of N2 such that for every s in S, |bs| = 0 ; for every (j, k) in N2, there exists s1, s2, . . . , sp in S such that s1(1) = j, sp(2) = k and sl(2) = sl+1(1) ; for any s in S, for any (j, k) in N2 s.t. {j, k} = {s(1), s(2)}, |λs(1) − λs(2)| = |λj − λk| ; then the control system is approximately controllable on the sphere.

Schrödinger Equation Some known results Perturbation A new result

A sufficient condition for approximate controllability

Theorem (Boscain-Caponigro-Chambrion-Sigalotti, not published yet) If there exists a subset S of N2 such that for every s in S, |bs| = 0 ; for every (j, k) in N2, there exists s1, s2, . . . , sp in S such that s1(1) = j, sp(2) = k and sl(2) = sl+1(1) ; for any s in S, for any (j, k) in N2 s.t. {j, k} = {s(1), s(2)}, |λs(1) − λs(2)| = |λj − λk| ; then the control system is approximately controllable on the sphere. Same hypotheses as for the finite result of controllability on the sphere for finite dimensional systems, obtained in 2000 by Turinici.

Schrödinger Equation Some known results Perturbation A new result

Estimates of the L1-norm of the control

Theorem (Boscain-Caponigro-Chambrion-Sigalotti) Assume that φj, Bφj+1 = 0 for every j and |λj − λj+1| = |λp − λq| for every {j, j + 1} = {p, q}. Then, for every n ∈ N, for every σ ∈ Sn, for every ǫ, δ > 0, there exists a control u : [0, Tu] → (0, δ) steering up to the phase, up to ǫ in modulus, d

dt ψ = (A + uB)ψ from φj to φσ(j) for every j ≤ n and

uL1 ≤ 5π (2n−1 − 1) 4 minj≤n |φj, Bφj+1|

Schrödinger Equation Some known results Perturbation A new result

Estimates of the L1-norm of the control

Theorem (Boscain-Caponigro-Chambrion-Sigalotti) Assume that φj, Bφj+1 = 0 for every j and |λj − λj+1| = |λp − λq| for every {j, j + 1} = {p, q}. Then, for every n ∈ N, for every σ ∈ Sn, for every ǫ, δ > 0, there exists a control u : [0, Tu] → (0, δ) steering up to the phase, up to ǫ in modulus, d

dt ψ = (A + uB)ψ from φj to φσ(j) for every j ≤ n and

uL1 ≤ 5π (2n−1 − 1) 4 minj≤n |φj, Bφj+1| Method : use Agrachev time estimates for d

dt ψ = (uA + B)ψ.

Schrödinger Equation Some known results Perturbation A new result

Simultaneous control

(Σ)

• d

dt ψ1

= (A1 + u(t)B1)ψ1 (Σ1)

d dt ψ2

= (A2 + u(t)B2)ψ2 (Σ2) Theorem Assume that for j = 1, 2, (Σj) admits a non-resonant chain of connectedness Sj. If at least one gap of S1 is not a gap of A2, then (Σ1) and (Σ2) are approximately simultaneously controllable. The result extends to the tracking of density matrices.

Schrödinger Equation Some known results Perturbation A new result

Example on SO(2)

i ∂ψ ∂t = −∆ψ + u(t) cos(θ)ψ ψ ∈ L2(SO(2), C)

Schrödinger Equation Some known results Perturbation A new result

Example on SO(2)

i ∂ψ ∂t = −∆ψ + u(t) cos(θ)ψ ψ ∈ L2(SO(2), C) No global controllability : the set of odd (resp. even) functions is invariant.

Schrödinger Equation Some known results Perturbation A new result

Example on SO(2)

i ∂ψ ∂t = −∆ψ + u(t) cos(θ)ψ ψ ∈ L2(SO(2), C) No global controllability : the set of odd (resp. even) functions is invariant. Spectrum for the even part : {k2|k ≥ 0}. For the odd part, {k2|k ≥ 1} (all eigenvalues are simple)

Schrödinger Equation Some known results Perturbation A new result

Example on SO(2)

i ∂ψ ∂t = −∆ψ + u(t) cos(θ)ψ ψ ∈ L2(SO(2), C) No global controllability : the set of odd (resp. even) functions is invariant. Spectrum for the even part : {k2|k ≥ 0}. For the odd part, {k2|k ≥ 1} (all eigenvalues are simple) Connectedness chain for the even part : {(k, k + 1), k ≥ 0} Connectedness chain for the odd part : {(k, k + 1), k ≥ 1}

Schrödinger Equation Some known results Perturbation A new result

Example on SO(2)

i ∂ψ ∂t = −∆ψ + u(t) cos(θ)ψ ψ ∈ L2(SO(2), C) No global controllability : the set of odd (resp. even) functions is invariant. Spectrum for the even part : {k2|k ≥ 0}. For the odd part, {k2|k ≥ 1} (all eigenvalues are simple) Connectedness chain for the even part : {(k, k + 1), k ≥ 0} Connectedness chain for the odd part : {(k, k + 1), k ≥ 1} p2 − k2 = 1 ⇒ (p − k)(p + k) = 1 ⇒ p = 1, k = 0

Schrödinger Equation Some known results Perturbation A new result

Example on SO(2)

i ∂ψ ∂t = −∆ψ + u(t) cos(θ)ψ ψ ∈ L2(SO(2), C) No global controllability : the set of odd (resp. even) functions is invariant. Spectrum for the even part : {k2|k ≥ 0}. For the odd part, {k2|k ≥ 1} (all eigenvalues are simple) Connectedness chain for the even part : {(k, k + 1), k ≥ 0} Connectedness chain for the odd part : {(k, k + 1), k ≥ 1} p2 − k2 = 1 ⇒ (p − k)(p + k) = 1 ⇒ p = 1, k = 0 The gap 12 − 02 = 1 does not appear in the gaps of the odd part.

Schrödinger Equation Some known results Perturbation A new result

Example on SO(2)

i ∂ψ ∂t = −∆ψ + u(t) cos(θ)ψ ψ ∈ L2(SO(2), C) No global controllability : the set of odd (resp. even) functions is invariant. Spectrum for the even part : {k2|k ≥ 0}. For the odd part, {k2|k ≥ 1} (all eigenvalues are simple) Connectedness chain for the even part : {(k, k + 1), k ≥ 0} Connectedness chain for the odd part : {(k, k + 1), k ≥ 1} p2 − k2 = 1 ⇒ (p − k)(p + k) = 1 ⇒ p = 1, k = 0 The gap 12 − 02 = 1 does not appear in the gaps of the odd part. Approximate simultaneous controllability.

Schrödinger Equation Some known results Perturbation A new result

Example on SO(2)

i ∂ψ ∂t = −∆ψ + u(t) cos(θ)ψ ψ ∈ L2(SO(2), C) No global controllability : the set of odd (resp. even) functions is invariant. Spectrum for the even part : {k2|k ≥ 0}. For the odd part, {k2|k ≥ 1} (all eigenvalues are simple) Connectedness chain for the even part : {(k, k + 1), k ≥ 0} Connectedness chain for the odd part : {(k, k + 1), k ≥ 1} p2 − k2 = 1 ⇒ (p − k)(p + k) = 1 ⇒ p = 1, k = 0 The gap 12 − 02 = 1 does not appear in the gaps of the odd part. Approximate simultaneous controllability. And tracking.

Schrödinger Equation Some known results Perturbation A new result

Example on SO(2)

i ∂ψ ∂t = −∆ψ + u(t) cos(θ)ψ ψ ∈ L2(SO(2), C) No global controllability : the set of odd (resp. even) functions is invariant. Spectrum for the even part : {k2|k ≥ 0}. For the odd part, {k2|k ≥ 1} (all eigenvalues are simple) Connectedness chain for the even part : {(k, k + 1), k ≥ 0} Connectedness chain for the odd part : {(k, k + 1), k ≥ 1} p2 − k2 = 1 ⇒ (p − k)(p + k) = 1 ⇒ p = 1, k = 0 The gap 12 − 02 = 1 does not appear in the gaps of the odd part. Approximate simultaneous controllability. And tracking. And simultaneous approximate collective controllability.

Schrödinger Equation Some known results Perturbation A new result

Concluding remarks (that do not commit anyone but me)

Despite recent developments...

Schrödinger Equation Some known results Perturbation A new result

Concluding remarks (that do not commit anyone but me)

Despite recent developments... ...we are still far from a necessary and sufficient condition for the approximate controllability of density matrices,..

Schrödinger Equation Some known results Perturbation A new result

Concluding remarks (that do not commit anyone but me)

Despite recent developments... ...we are still far from a necessary and sufficient condition for the approximate controllability of density matrices,.. ...and we are even further from a necessary and sufficient condition for the approximate controllability on the sphere.