Vahagn Nersesyan
email: vahagn.nersesyan@math.uvsq.fr

Hayk Nersisyan
email: hayk.nersisyan@u-cergy.fr

Global exact controllability in infinite time of Schrödinger equation

Keywords: Schrödinger equation, controllability, multivalued mapping, Kolmogorov ε-entropy multifunctions . . . . . . . . . . . . 28

In this paper, we study the problem of controllability of Schrödinger equation. We prove that the system is exactly controllable in infinite time to any position. The proof is based on an inverse mapping theorem for multivalued functions. We show also that the system is not exactly controllable in finite time in lower Sobolev spaces.

Résumé. Dans cet article, nous étudions le problème de contrôlabilité pour l'équation de Schrödinger. Nous montrons que le système est exactement contrôlable en temps infini. La preuve est basée sur un théorème d'inversion locale pour des multifonctions. Nous montrons aussi que le système n'est pas exactement contrôlable en temps fini dans les espaces de Sobolev d'ordre inférieur.

Introduction

The paper is devoted to the study of the following controlled Schrödinger equation

i ż = -∆z + V (x)z + u(t)Q(x)z, (1.1)
z| ∂D = 0, (1.2) z(0, x) = z 0 (x).

(1.3)

We assume that space variable x belongs to a rectangle D ⊂ R d , V, Q ∈ C ∞ (D, R) are given functions, u is the control, and z is the state. We prove that the linearization of this system is exactly controllable in Sobolev spaces in infinite time. Application of this result gives global exact controllability in infinite time in H 3 for d = 1. We show also that the system is not exactly controllable in finite time in lower Sobolev spaces. Let us recall some previous results on the controllability problem of Schrödinger equation. In [START_REF] Beauchard | Local controllability of a 1D Schrödinger equation[END_REF], Beauchard proves an exact controllability result for the system with d = 1, D = (-1, 1) and Q(x) = x in H 7 -neighborhoods of the eigenfunctions. Beauchard and Coron [START_REF] Beauchard | Controllability of a quantum particle in a moving potential well[END_REF] established later a partial global exact controllability result, showing that the system in question is also controlled between neighborhoods of eigenfunctions. Recently, Beauchard and Laurent [START_REF] Beauchard | Local controllability of linear and nonlinear Schrödinger equations with bilinear control[END_REF] simplified the proof of [START_REF] Beauchard | Local controllability of a 1D Schrödinger equation[END_REF] and generalized it to the case of the nonlinear equation. The proofs of [START_REF] Beauchard | Local controllability of a 1D Schrödinger equation[END_REF][START_REF] Beauchard | Controllability of a quantum particle in a moving potential well[END_REF][START_REF] Beauchard | Local controllability of linear and nonlinear Schrödinger equations with bilinear control[END_REF] work also for the neighborhoods of finite linear combinations of eigenfunctions. In the case of infinite linear combinations, these arguments do not work, since the linearized system does not verify the property of spectral gap (even if the problem is 1-D), hence the Ingham inequality cannot be applied.

Chambrion et al. [START_REF] Chambrion | Controllability of the discrete-spectrum Schrödinger equation driven by an external field[END_REF], Privat, Sigalotti [START_REF] Privat | The squares of Laplacian-Dirichlet eigenfunctions are generically linearly independent[END_REF], and Mason, Sigalotti [START_REF] Mason | Generic controllability properties for the bilinear Schrödinger equation[END_REF] prove that (1.1), (1.2) is approximately controllable in L 2 generically with respect to function Q and domain D. In [START_REF] Nersesyan | Global approximate controllability for Schrödinger equation in higher Sobolev norms and applications[END_REF][START_REF] Nersesyan | Growth of Sobolev norms and controllability of the Schrödinger equation[END_REF], the first author of this paper proves a stabilization result and a property of global approximate controllability to eigenstates for Schrödinger equation. Combination of these results with the local exact controllability property obtained by Beauchard [START_REF] Beauchard | Local controllability of a 1D Schrödinger equation[END_REF] gives global exact controllability in finite time for d = 1 in the spaces H 3+ε , ε > 0. See also papers [START_REF] Ramakrishna | Controllability of molecular systems[END_REF][START_REF] Turinici | Quantum wavefunction controllability[END_REF][START_REF] Altafini | Controllability of quantum mechanical systems by root space decomposition of su(n)[END_REF][START_REF] Albertini | Notions of controllability for bilinear multilevel quantum systems[END_REF][START_REF] Agrachev | An estimation of the controllability time for single-input systems on compact Lie groups[END_REF][START_REF] Beauchard | Implicit Lyapunov control of finite dimensional Schrödinger equations[END_REF] for controllability of finite-dimensional systems and papers [START_REF] Lebeau | Contrôle de l'équation de Schrödinger[END_REF][START_REF] Machtyngier | Stabilization of the Schrödinger equation[END_REF][START_REF] Baudouin | Uniqueness and stability in an inverse problem for the Schrödinger equation[END_REF][START_REF] Zuazua | Remarks on the controllability of the Schrödinger equation[END_REF][START_REF] Dehman | Stabilization and control for the nonlinear Schrödinger equation on a compact surface[END_REF][START_REF] Mirrahimi | Lyapunov control of a particle in a finite quantum potential well[END_REF][START_REF] Ervedoza | Approximate controllability for a system of Schrödinger equations modeling a single trapped ion[END_REF] for controllability properties of various Schrödinger systems.

In this article, we study the properties of control system on the time half-line R + instead of a finite interval [0, T ], as in all above cited papers. We study the mapping, which associates to initial condition z 0 and control u the ω-limit set of the corresponding trajectory. We consider this mapping as a multivalued function in the phase space. We show that this multivalued function is differentiable with differential equal to the limit of the linearization of (1.1), (1.2), when time t goes to infinity. Observing that the linearized system is controllable in infinite time at almost any point, we conclude the controllability of the nonlinear system (in the case d = 1), using an inverse mapping theorem for multivalued functions [START_REF] Nachi | Inversion of multifunctions and differential inclusions[END_REF] by Nachi and Penot. Thus (1.1), (1.2) is exactly controllable near any point in the phase space, hence globally. The controllability of the linearized system is proved for any d ≥ 1, but this result is not directly applicable to the study of the nonlinear system with d ≥ 2. We have a loss of regularity: the solution of the nonlinear problem exists for more regular controls than the ones used to control the linear problem. The multidimensional case is treated in our forthcoming paper.

To our knowledge, the inverse mapping theorem for multivalued functions was never used before in the theory of control of PDEs. Our proof does not rely on the particular asymptotics of the eigenvalues of Dirichlet Laplacian, so it is likely to work in other settings. Considering the problem in infinite time enables us to prove the controllability of the linearized system in the case of any space dimension d ≥ 1, even when the gap condition is not verified for the eigenvalues (which is the case for d ≥ 3).

In the second part of the paper, we study the problem of non-controllability for (1.1), (1.2) in finite time. We prove that the system is not exactly controllable in finite time in the spaces H k with k ∈ (0, d). Let us recall that previously Ball, Marsden and Slemrod [START_REF] Ball | Controllability for distributed bilinear systems[END_REF] and Turinici [START_REF] Turinici | On the controllability of bilinear quantum systems[END_REF] have shown that the problem is not controllable in the space H 2 . Our result is inspired by the paper [START_REF] Shirikyan | Euler equations are not exactly controllable by a finitedimensional external force[END_REF] of Shirikyan, where the non-controllability of 2D Euler equation is established. More precisely, it is proved in [START_REF] Shirikyan | Euler equations are not exactly controllable by a finitedimensional external force[END_REF] that, if the Euler system is controlled by finite dimensional external force, then the set of all reachable points in a given time T > 0 cannot cover a ball in the phase space. Later this result was generalized by the second author of the present paper, in [START_REF] Nersisyan | Controllability of 3D incompressible Euler equations by a finite-dimensional external force[END_REF]: in the case of 3D Euler equation it is proved that the union of all sets of reachable points at all times T > 0 also does not cover a ball.

Using ideas of Shirikyan, we prove that the image by the resolving operator of a ball in the space of controls has a Kolmogorov ε-entropy strictly less than that of a ball in the phase space H k . This implies the non-controllability.

Notation

In this paper, we use the following notation. Let

ℓ 2 := {{a j } ∈ C ∞ : {a j } 2 ℓ 2 = +∞ j=1 |a j | 2 < +∞} ℓ 2 0 := {{a j } ∈ ℓ 2 : a 1 ∈ R}.
We denote by H s := H s (D) the Sobolev space of order s ≥ 0. Consider the Schrödinger operator

-∆ + V , V ∈ C ∞ (D, R) with D(-∆ + V ) := H 1 0 ∩ H 2 .
Let {λ j,V } and {e j,V } be the sets of eigenvalues and normalized eigenfunctions of this operator. Let •, • and • be the scalar product and the norm in the space L 2 . Define the space

H s (V ) := D((-∆ + V ) s 2 ) endowed with the norm • s,V = (λ j,V ) s 2 •, e j,V ℓ 2 . When D is the rectangle (0, 1) d and V (x 1 , . . . , x d ) = V 1 (x 1 ) + . . . + V d (x d ), V k ∈ C ∞ ([0, 1], R), the eigenvalues and eigenfunctions of -∆ + V on D are of the form λ j1,...,j d ,V = λ j1,V1 + . . . + λ j d ,V d , (1.4) e j1,...,j d ,V (x 1 , . . . , x d ) = e j1,V1 (x 1 ) • . . . • e j d ,V d (x d ), (x 1 , . . . , x d ) ∈ D, (1.5)
where {λ j,V k } and {e j,V k } are the eigenvalues and eigenfunctions of operator

-d 2 dx 2 + V k on (0, 1)
. Define the space

V := {z ∈ L 2 : z 2 V := +∞ j1,...,j d =1 |(j 3 1 • . . . • j 3 d z, e j1,...,j d ,V | 2 < +∞}. (1.6)
Notice that, in the case d = 1, the space V coincides with H 3 (V ) . The eigenvalues and eigenfunctions of Dirichlet Laplacian on the interval (0, 1) are λ k,0 = k 2 π 2 and e k,0 (x) = √ 2 sin(kπx), x ∈ (0, 1). It is well known that for any

V ∈ L 2 ([0, 1], R) λ k,V = k 2 π 2 + 1 0 V (x)dx + r k , (1.7)
e k,V -e k,0 L ∞ ≤ C k , (1.8) de k,V dx - de k,0 dx L ∞ ≤ C, (1.9)
where +∞ k=1 r 2 k < +∞ (e.g., see [START_REF] Pöschel | Inverse Spectral Theory[END_REF]). For a Banach space X, we shall denote by B X (a, r) the open ball of radius r > 0 centered at a ∈ X. For a set A, we write 2 A for the set consisting of all subsets of A. We denote by C a constant whose value may change from line to line.

2 Controllability of linearized system

Main result

In this section, we suppose that d = 1 and D = (0, 1). For any z ∈ H 3 (V ) , let U t (z, 0) be the solution of (1.1)-(1.3) with z 0 = z and u = 0. Clearly,

U t (z, 0) = +∞ j=1
e -iλj,V t z, e j,V e j,V .

(2.1)

Lemma 2.1. There is a sequence T n → + ∞ such that for any z ∈ H 3 (V ) we have U Tn (z, 0)→z in H 3 (V ) .
Proof. The proof uses the following well known result (e.g., see [START_REF] Katok | Introduction to the modern theory of dynamical systems[END_REF]).

Lemma 2.2. For any ε > 0, N ≥ 1 and

α j ∈ R, j = 1, . . . , N , there is k ∈ N such that N j=1 |e iαj k -1| < ε.
Applying this lemma, we see that for any ε > 0 and for sufficiently large N ≥ 1, we have

U k (z, 0) -z 2 3,V ≤ j≤N |e -iλj,V k -1| 2 |λ 3 2 j,V z, e j1,...,j d ,V | 2 + 2 j>N |λ 3 2 j,V z, e j1,...,j d ,V | 2 ≤ ε 2 + ε 2 = ε
for an appropriate choice of k ∈ N. This proves Lemma 2.1.

This subsection is devoted to the study of the linearization of (1.1), (1.2) around the trajectory U t (z, 0):

i ż = - ∂ 2 z ∂x 2 + V (x)z + u(t)Q(x)U t (z, 0), (2.2
)

z| ∂D = 0, (2.3)
z(0, x) = z 0 . (2.4)
Let S be the unit sphere in L 2 . For y ∈ S, let T y be the tangent space to S at y ∈ S:

T y = {z ∈ S : Re z, y = 0}. Lemma 2.3. For any z 0 ∈ T z ∩ H 2 (0) and u ∈ L 1 loc (R + , R), problem (2.2)-(2.4) has a unique solution z ∈ C(R + , H 2 (0) ). Furthermore, if R t (•, •) : T z ∩ H 2 (0) × L 1 ([0, t], R) → H 2 (0) , (z 0 , u) → z(t)
is the resolving operator of the problem, then (i) R t (z 0 , u) ∈ T Ut(z,0) for any t ≥ 0,

(ii) The operator R t is linear continuous from T z ∩ H 2 (0) × L 1 ([0, t], R) to H 2 (0) .
Proof. The proof of existence and (ii) is standard (e.g., see [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF]). To prove (i), notice that

d dt Re R t , U t = Re Ṙt , U t + Re R t , Ut = Re i( ∂ 2 ∂x 2 -V )R t -iu(t)Q(x)U t , U t + Re R t , i( ∂ 2 ∂x 2 -V )U t = Re i( ∂ 2 ∂x 2 -V )R t , U t + Re R t , i( ∂ 2 ∂x 2 -V )U t = 0.
Since Re R 0 , U 0 = Re z 0 , z = 0, we get (i).

As (2.2)-(2.4) is a linear control problem, the controllability of system with z 0 = 0 is equivalent to that with any z 0 ∈ T z . Henceforth, we take z 0 = 0 in (2.4). Let us rewrite this problem in the Duhamel form

z(t) = -i t 0 S(t -s)u(s)Q(x)U s (z, 0)ds, (2.5)
where S(t) = e it( ∂ 2 ∂x 2 -V ) is the free evolution. Using (2.1) and (2.5), we obtain

z(t), e m,V = -i +∞ k=1 e -iλm,V t z, e k,V Q mk t 0 e iω mk s u(s)ds, m ≥ 1, (2.6)
where ω mk = λ m -λ k and Q mk := Qe m,V , e k,V . Let T n → +∞ be the sequence in Lemma 2.1. Then e -iλm,V Tn → 1 as n → +∞. Let us take t = T n in (2.6) and pass to the limit as n → +∞. For any u ∈ L 1 (R + , R) the righthand side has a limit. Equality (2.6) implies that the following limit exists in the L 2 -weak sense

R ∞ (0, u) := lim n→+∞ z(T n ) = lim n→+∞ R Tn (0, u). (2.7)
The choice of the sequence T n implies that

R ∞ (0, u), e m,V = -i +∞ k=1 z, e k,V Q mk +∞ 0 e iω mk s u(s)ds. (2.8)
Moreover, R ∞ (0, u) ∈ T z . Indeed, using (2.7) and the convergence U Tn (z, 0)→z in H 3 (V ) , we get Re R ∞ (0, u), z = lim n→∞ Re R Tn (0, u), U Tn (z, 0) = 0, by property (i).

For any u ∈ L 1 (R + , R), denote by ǔ the inverse Fourier transform of the function obtained by extending u as zero to R

B := {u ∈ L 2 loc (R + , R) : u 2 B := +∞ p=1 p 2 u 2 L 2 ([p-1,p]) < +∞}, C := {u ∈ L 1 (R + , R) : {ǔ(ω mk )} ∈ l2 }.
The set of admissible controls is the Banach space

Θ := u ∈ B ∩ C ∩ H s (R + , R) endowed with the norm u Θ := u B + u L 1 + {ǔ(ω mk )} l2 + u H s ,
where s ≥ 1 is any fixed constant. Clearly, the space Θ is nontrivial. The presence of the space B in the definition of Θ is motivated by the application to the nonlinear control system that we give in Section 3 (this guarantees that the trajectories of the nonlinear system with controls from B are bounded in the phase space).

The space C in the definition of Θ ensures that the operator R ∞ (0, •) takes its values in H 3 (V ) .

Lemma 2.4. For any z ∈ S ∩H 3 (V ) , R ∞ (0, •) is linear continuous mapping from Θ to T z ∩ H 3 (V ) .

Proof.

Step 1. Let us admit that for any m, k ≥ 1 we have

m 3 k 3 Qe k,V , e m,V ≤ C. (2.10)
Then (1.7), (2.8), (2.10) and the Schwarz inequality imply that

R ∞ (0, u) 2 3,V ≤ C +∞ m=1 |m 3 R ∞ (0, u), e m,V | 2 ≤ C +∞ m=1 m 3 z, e m,V Qe m,V , e m,V +∞ 0 u(s)ds 2 + C z 2 3,V +∞ m,k=1,m =k m 3 k 3 Qe k,V , e m,V +∞ 0 e iω mk s u(s)ds 2 ≤ C z 2 3,V u 2 Θ < +∞.
Step 2. Let us prove (2.10). Integration by parts gives

Qe k,V , e m,V = 1 λ 2 m,V (- ∂ 2 ∂x 2 + V )(Qe k,V ), ( -
∂ 2 ∂x 2 + V )(e m,V ) = 1 λ 2 m,V (-2 ∂Q ∂x ∂e k,V ∂x ∂e m,V ∂x x=1 x=0 + ∂ ∂x (- ∂ 2 ∂x 2 + V )(Qe k,V ), ∂e m,V ∂x + (- ∂ 2 ∂x 2 + V )(Qe k,V ), V e m,V ).
In view of (1.4)-(1.9), this implies (2.10).

We prove the controllability of (2.2), (2.3) under below condition with d = 1.

Condition 2.5. Suppose that D is the rectangle (0, 1) d , d ≥ 1 and the functions

V, Q ∈ C ∞ (D, R) are such that (i) inf p1,j1,...,p d ,j d ≥1 |(p 1 j 1 • . . . • p d j d ) 3 Q pj | > 0,Q pj := Qe p1,...,p d ,V , e j1,...,j d ,V , (ii) λ i,V -λ j,V = λ p,V -λ q,V
for all i, j, p, q ≥ 1 such that {i, j} = {p, q} and i = j.

See Appendix for the proof of genericity of this condition. Let us introduce the set E := {z ∈ S : ∃p, q ≥ 1, p = q,z = c p e p,V + c q e q,V ,

|c p | 2 Qe p,V , e p,V -|c q | 2 Qe q,V , e q,V = 0}.
The following result is proved in next subsection.

Theorem 2.6. Under Condition 2.5 with

d = 1, for any z ∈ S ∩ H 3 (V ) \ E, the mapping R ∞ (0, •) : Θ → T z ∩ H 3 (V ) admits a continuous right inverse, where the space T z ∩ H 3 (V ) is endowed with the norm of H 3 (V ) . If z ∈ S ∩ H 3 (V ) ∩ E, then R ∞ (0, •) is not invertible.
Remark 2.7. The invertibility of the mapping R T (0, •) with finite T > 0 and z = e 1 is studied by Beauchard et al. [START_REF] Beauchard | Spectral controllability of 2D and 3D linear Schrödinger equations[END_REF]. They prove that for space dimension d ≥ 3 the mapping is not invertible. By Beauchard [START_REF] Beauchard | Local controllability of a 1D Schrödinger equation[END_REF], R T is invertible in the case d = 1 and z = e 1 . The case d = 2 is open to our knowledge.

Remark 2.8. Let us emphasize that the set {ω mk } does not verify the gap condition (even in the case

d = 1) inf (m,k) =(m ′ ,k ′ ) |ω mk -ω m ′ k ′ | > 0.
Thus one cannot prove exact controllability in finite time near points, which are not eigenfunctions, using arguments based on the Ingham inequality.

Proof of Theorem 2.6

The proof of the theorem is based on the following proposition, which is proved in next subsection. < +∞ and ω ij = ω pq for all i, j, p, q ≥ 1 such that {i, j} = {p, q} and i = j. Let us take any y ∈ T z ∩ H 3 (V ) . Define

d mk := i y, e m e k , z -i e k , y z, e m Q mk + C mk ,
where C mk ∈ C and e k = e k,V . The fact that z ∈ S implies

-i +∞ k=1 z, e k Q mk d mk = +∞ k=1 y, e m | z, e k | 2 - +∞ k=1 e k , y z, e m z, e k -i +∞ k=1 z, e k Q mk C mk = y, e m -z, e m z, y -i +∞ k=1 z, e k Q mk C mk .
By (2.8), we have y = R ∞ (0, u), when

i +∞ k=1 z, e k Q mk C mk = -z, e m z, y (2.11)
for all m ≥ 1. Thus if we show that there are C mk ∈ C such that (2.11) is verified and d = {d mk } ∈ l2 , then the proof of the theorem will be completed, in view of Proposition 2.9. Notice that, under Condition 2.5, we have

+∞ m,k=1,m =k y, e m e k , z Q mk 2 ≤ C y 2 3,V z 2 3,V < +∞. Thus {d mk } ∈ l2 , if C mk ∈ C are such that d mm = i y, e m e m , z -i e m , y z, e m Q mm + C mm = d 0 , (2.12
)

C mk = C km , (2.13) +∞ m,k=1,m =k |C mk | 2 < +∞, ( 2
d 0 = Im z,y
Qpp .

Case 2. Suppose z = c p e p + c q e q , where c p , c q ∈ C, |c p | 2 + |c q | 2 = 1 and p = q. For any m ≥ 1, define C mm by (2.16). If m = p, we set

C mp := -c m (Im z, y + Q mm C mm ) c p Q mp , (2.17)
where c m = 0 for m = q, and C mk = 0 for any k ≥ 1 such that k = m, p. Then all the equations in (2.15) are verified, excepted the case m = p. Let us show that, for an appropriate choice of d 0 ∈ R, this equation is also satisfied. Equation (2.15) for m = p is

c p Q pp C pp + c q Q pq C pq = -c p Im z, y .
Using (2.17) for m = q (taking C pq = C qp ) and (2.16) for m = p, we get

-c p Im z, y =c p Q pp d 0 + 2 Im( y, e p e p , z ) Q pp + c q Q pq -c q (Im z, y + Q qq C qq ) c p Q qp =c p Q pp d 0 + 2 Im( y, e p e p , z ) Q pp + c q Q pq -c q Im z, y c p Q qp + c q Q pq -c q Q qq C qq c p Q qp .
Now using (2.16) for m = q, we rewrite this equality in an equivalent form

(|c p | 2 Q pp -|c q | 2 Q qq )d 0 = A for some constant A ∈ R. Thus if z is such that |c p | 2 Q pp -|c q | 2 Q qq = 0, then we are able to find C mk satisfying (2.13)-(2.16). If |c p | 2 Q pp -|c q | 2 Q qq = 0, then linear system (2.
2), (2.3) is not controllable, since for any u ∈ Θ and t ≥ 0 we have d dt Im R t (0, u), c p e -iλpt e p -c q e -iλqt e q = Im i( ∂ 2 ∂x 2 -V )R t (0, u) -iuQ(c p e -iλpt e p + c q e -iλqt e q ), c p e -iλpt e p -c q e -iλq t e q + Im R t (0, u), i( ∂ 2 ∂x 2 -V )(c p e -iλpt e p -c q e -iλq t e q ) = Im -iuQ(c p e -iλpt e p + c q e -iλqt e q ), c p e -iλpt e p -c q e -iλqt e q

= -u(|c p | 2 Q pp -|c q | 2 Q qq ) = 0.
This non-controllability property is a remark of Beauchard and Coron [START_REF] Beauchard | Controllability of a quantum particle in a moving potential well[END_REF].

Case 3. Here we suppose that z = +∞ j=1 c j e j with c p c q c r = 0, and p, q, r are not equal to each other. If we define again C mp , m = p by (2.17) and C mk = 0 for any k ≥ 1 such that k = m, p, then the arguments of case 2 give the following equation for d 0

(|c p | 2 Q pp - m =p |c m | 2 Q mm )d 0 = Ã
for some constant à ∈ R. This implies that for any z such that

|c p | 2 Q pp - m =p |c m | 2 Q mm = 0, we can find C mk satisfying (2.13)-(2.16). Let us suppose that |c p | 2 Q pp - m =p |c m | 2 Q mm = 0. (2.18)
In this case, we define C mp by (2.17) only for integers m ≥ 1 such that m = p, q, r and C mk = 0 for any k ≥ 1 such that k = m, p, q, r. Then all the equations in (2.15) are verified, except for m = p, q, r. We take any C qp ∈ C and choose C qr and C rp such that

c p Q rp C rp + c q Q rq C rq + c p Q rr C rr = -c r Im z, y , (2.19)
c p Q qp C qp + c q Q qq C qq + c r Q qr C qr = -c q Im z,
(|c p | 2 Q pp + |c q | 2 Q qq - m =p,q |c m | 2 Q mm )d 0 = à for some constant à ∈ R. Equality (2.18) implies that |c p | 2 Q pp + |c q | 2 Q qq - m =p,q |c m | 2 Q mm = 0
if and only if |c q | 2 Q qq = 0, which is not the case: c q = 0, Q qq = 0. Thus solution d 0 ∈ R exists, and the sequence C mk is constructed for any z / ∈ E.

Multidimensional case

In this section, we suppose that D is the rectangle (0, 1)

d , d ≥ 1 and V (x 1 , . . . , x d ) = V 1 (x 1 ) + . . . + V d (x d ), V k ∈ C ∞ ([0, 1], R).
This subsection is devoted to the study of the linearization of (1.1), (1.2) around the trajectory U t (z, 0):

i ż = -∆z + V (x)z + u(t)Q(x)U t (z, 0), (2.21) z| ∂D = 0, (2.22) z(0, x) = z 0 . (2.23)
The proof of Theorem 2.6 does not work in the multidimensional case for a general z. Indeed, the well-known asymptotic formula for eigenvalues

λ k,V ∼ C d k 2 d
implies that the frequencies ω mk are dense in R for space dimension d ≥ 3. Thus the moment problem ǔ(ω mk ) = d mk cannot be solved in the space L 1 (R + , R) for a general d mk ∈ l2 . The asymptotic formula for eigenvalues implies that the moment problem cannot be solved also in this case d = 2. Clearly, this does not imply the non-controllability of linearized system. Let us prove the controllability of (2.21), (2.22) for z = e k,V . See our forthcoming publication for the case of a general z and for an application to the nonlinear control problem.

For z = e k,V the mapping R ∞ (0, u) is given by

R ∞ (0, u), e m,V = -iQ mk ǔ(ω mk ) (cf. 2.8). Lemma 2.10. The mapping R ∞ (0, •) is linear continuous from Θ to T e k,V ∩ V,
where V is defined by (1.6).

Proof.

Step 1. Let us admit that for any m j , k j ≥ 1, j = 1, . . . , d we have

(m 1 • . . . • m d ) 3 (k 1 • . . . • k d ) 3 Qe k1,...,k d ,V , e m1,...,m d ,V ≤ C. (2.24)
Then (2.8), (2.24) and the Schwarz inequality imply that

R ∞ (0, u) 2 V = +∞ m1,...,m d =1 |m 3 1 • . . . • m 3 d R ∞ (0, u), e m1,...,m d ,V | 2 ≤ C +∞ m=1 m 3 1 • . . . • m 3 d z, e m1,...,m d ,V Qe m,V , e m,V +∞ 0 u(s)ds 2 + C z 2 V +∞ m,k=1,m =k (m 1 • . . . • m d ) 3 (k 1 • . . . • k d ) 3 Qe k1,...,k d ,V , e m1,...,m d ,V +∞ 0 e iω mk s u(s)ds 2 ≤ C z 2 V u 2 Θ < +∞.
Step 2. Let us prove (2.24). To simplify notation, let us suppose that d = 2; the proof of the general case is similar. Let

V (x 1 , x 2 ) = V 1 (x 1 ) + V 2 (x 2 ).

Integration by parts gives

Qe k1,k2,V , e m1,m2,V = 1 λ 2 m1,V1 (- ∂ 2 ∂x 2 1 + V 1 )(Qe k1,k2,V ), (- ∂ 2 ∂x 2 1 + V 1 )(e m1,m2,V ) = 1 λ 2 m1,V1 ( 1 0 -2 ∂Q ∂x 1 ∂e k1,k2,V ∂x 1 ∂e m1,m2,V ∂x 1 x1=1 x1=0 dx 2 + ∂ ∂x 1 (- ∂ 2 ∂x 2 1 + V 1 )(Qe k1,k2,V ), ∂e m1,m2,V ∂x 1 + (- ∂ 2 ∂x 2 1 + V 1 )(Qe k1,k2,V ), V 1 e m1,m2,V ) =:I 1 + I 2 + I 3 .
Again integrating by parts, we get

I 1 = -2 λ 2 m1,V1 λ 2 m2,V2 1
0 (- ∂ 2 ∂x 2 2 + V 2 )( ∂Q ∂x 1 ∂e k1,k2,V ∂x 1 )(- ∂ 2 ∂x 2 2 + V 2 ) ∂e m1,m2,V ∂x 1 x1=1 x1=0 dx 2 = -2 λ 2 m1,V1 λ 2 m2,V2 (-2 ∂ 2 Q ∂x 1 ∂x 2 ∂ 2 e k1,k2,V ∂x 1 ∂x 2 ∂ 2 e m1,m2,V ∂x 1 ∂x 2 x1=1 x1=0 x2=1 x2=0 + 1 0 ∂ ∂x 2 (- ∂ 2 ∂x 2 2 + V 2 )( ∂Q ∂x 1 ∂e k1,k2,V ∂x 1 ) ∂ 2 e m1,m2,V ∂x 1 ∂x 2 x1=1 x1=0 dx 2 + 1 0 (- ∂ 2 ∂x 2 2 + V 2 )( ∂Q ∂x 1 ∂e k1,k2,V ∂x 1 )V 2 ∂e m1,m2,V ∂x 1 x1=1 x1=0 dx 2 ).
In view of (1.4)-(1.9), this implies that

(m 1 • . . . • m d ) 3 (k 1 • . . . • k d ) 3 I 1 ≤ C.
The terms I 2 , I 3 are treated in the same way. We omit the details.

We rewrite (2.8) in the form

ǔ(ω mk ) = d m , (2.25)
where

d m = R∞(0,u),em,V -iQ mk . We have ∞ m=1,m =k 1 |ω mk | d < +∞ for fixed k ≥ 1. Under Condition 2.5, (i), d m ∈ ℓ 2 0 .
Applying Proposition 2.9, we obtain the following theorem.

Theorem 2.11. Under Condition 2.5, the mapping R ∞ (0, •) : Θ → T e k,V ∩ V admits a continuous right inverse, where the space T e k,V ∩ V is endowed with the norm of V.

Proof of Proposition 2.9

The construction of the operator A is based on the following lemma.

Lemma 2.12. Under the conditions of Proposition 2.9, for any d ∈ ℓ 2 0 and ε > 0, there is u ∈ B Θ (0, ε) such that {ǔ(ω m )} = d.

Proof of Proposition 2.9. Let d n be any orthonormal basis in ℓ 2 0 . Applying Lemma 2.12, we find a sequence

u n ∈ B Θ (0, 1 n ) such that {ǔ n (ω m )} = d n . For any d ∈ ℓ 2 0 , there is c ∈ ℓ 2 such that d = +∞ n=1 c n d n .
Let us define A in the following way

A(d) = +∞ n=1 c n u n .
As u n ∈ B Θ (0, 1 n ), this sum converges in Θ:

A(d) Θ ≤ +∞ n=1 |c n | u n Θ ≤ +∞ n=1 |c n | 2 1 2 +∞ n=1 u n 2 Θ 1 2 ≤ C d ℓ 2 0 .
Thus A : ℓ 2 0 →Θ is linear continuous and { Ǎ (d)(ω m )} = d, by construction.

Proof of Lemma 2.12. Let us take any d ∈ ℓ 2 0 and ε > 0 and introduce the functional

H(u) := {ǔ(ω m )} -d 2 ℓ 2 0 = +∞ m=1 |ǔ(ω m ) -d m | 2
defined on the space Θ.

Step 1. First, let us show that there is u 0 ∈ B Θ (0, ε) such that

H(u 0 ) = inf u∈BΘ(0,ε)

H(u). (2.26)

To this end, let u n ∈ B Θ (0, ε) be an arbitrary minimizing sequence. Since B ∩ H s (R + , R) is reflexive, without loss of generality, we can assume that there is Again extracting a subsequence, if it is necessary, one gets {ǔ n (ω m )} ⇀ {ǔ 0 (ω m )} in ℓ 2 0 as n→ + ∞. Indeed, the tails on [T, +∞), T ≫ 1 of the integrals (2.9) are small uniformly in n (this comes from the boundedness of u n in B), and on the finite interval [0, T ] the convergence is uniform.) This implies that u 0 ∈ Θ and

u 0 ∈ B B∩H s (R+,R) (0, ε) such that u n ⇀ u 0 in B ∩ H s (R + , R).
H(u 0 ) ≤ inf u∈BΘ(0,ε) H(u).
The fact that u 0 ∈ B Θ (0, ε) follows from the Fatou lemma and lower weak semicontinuity of norms. Thus we have (2.26).

Step 2. To complete the proof, we need to show that H(u 0 ) = 0. Suppose, by contradiction, that H(u 0 ) > 0. As we shall see below, this implies that there

is v ∈ B Θ (0, ε) such that d dt H((1 -t)u 0 + tv) t=0 < 0. (2.27)
Since (1-t)u 0 +tv ∈ B Θ (0, ε) for all t ∈ [0, 1], (2.27) is a contradiction to (2.26).

To construct such a function v, notice that the derivative is given explicitly by

d dt H((1 -t)u 0 + tv) t=0 = 2 +∞ m=1 Re[(v(ω m ) -ǔ0 (ω m ))(ǔ 0 (ω m ) -d m )].
In view of this equality, the existence of v follows immediately from the following lemma.

Lemma 2.13. Under the conditions of Proposition 2.9, the set

U := {{ǔ(ω m )} : u ∈ B Θ (0, ε)} is dense in ℓ 2 0 .
Proof. Suppose that h ∈ ℓ 2 0 is orthogonal to U . Then for any u ∈ B Θ (0, ε) ∩ C ∞ 0 ((0, +∞)) we have e iωms (-iω m ) p h m ds = 0, where P p is a polynomial of degree p ≥ 1. Since this equality holds for any u ∈ B Θ (0, ε) ∩ C ∞ 0 ((0, +∞)), there is a polynomial Pp-1 (s) of degree p -1 such that for any s ≥ 0

P p (s)h 1 + +∞ m=2 e iωms (-iω m ) p h m = Pp-1 (s).
By Lemma 2.14, we have h m = 0 for any m ≥ 2. Equality (2.28) implies that h 1 = 0. This proves that U is dense.

The following lemma is a generalization of Lemma 3.10 in [START_REF] Nersesyan | Growth of Sobolev norms and controllability of the Schrödinger equation[END_REF].

Lemma 2.14. Suppose that r j ∈ R * and r k = r j for k = j and P p is a polynomial of degree p ≥ 1.

If ∞ j=1 c j e irj s = P p (s) (2.29)
for any s ≥ 0 and for some sequence c j ∈ C such that ∞ j=1 |c j | < ∞, then c j = 0 for all j ≥ 1 and P p ≡ 0.

Proof. Since the sum in the left hand side of (2.29) is bounded in s, the polynomial P p (s) is constant. The case of constant right hand side follows from Lemma 3.10 in [START_REF] Nersesyan | Growth of Sobolev norms and controllability of the Schrödinger equation[END_REF].

3 Controllability of nonlinear system

Well-posedness of Schrödinger equation

In this section, we suppose that d = 1, D = (0, 1). We consider the following Schrödinger equation

i ż = - ∂ 2 z ∂x 2 + V (x)z + u(t)Q(x)z + v(t)Q(x)y, (3.1
)

z| ∂D = 0, (3.2)
z(0, x) = z 0 (x). (3.3)
See Proposition 2 in [START_REF] Beauchard | Local controllability of linear and nonlinear Schrödinger equations with bilinear control[END_REF] for the proof of well-posedness of this system with V = 0. Here we prove well-posedness in the case of V = 0 and we give an estimate for the solution which is important for the study of the controllability property.

Proposition 3.1. For any

z 0 ∈ H 3 (V ) , u, v ∈ L 1 (R + , R)∩B and y ∈ C(R + , H 3 (V ) ), problem (3.1)-(3.3) has a unique solution z ∈ C(R + , H 3 (V ) ). Furthermore, there is a constant C > 0 such that sup t∈R+ z(t) 3,V ≤ C( z 0 3,V + sup t∈R+ y(t) 3,V ( v L 1 (R+) + v B )) × exp C( u L 1 (R+) + 1) exp( u 2 B ) . (3.4)
If v = 0, then for all t ≥ 0 we have

z(t) = z 0 . (3.5)
Proof. The proof follows the ideas of Proposition 2 in [START_REF] Beauchard | Local controllability of linear and nonlinear Schrödinger equations with bilinear control[END_REF]. We give all the details for the sake of completeness. Let us rewrite (3.1)-(3.3) in the Duhamel form

z(t) = S(t)z 0 -i t 0 S(t -s)[u(s)Qz(s) + v(s)Qy(s)]ds. (3.6) For any u ∈ L 1 (R + , R) ∩ B and z ∈ C(R + , H 3 (V )
), we estimate the function

G t (z) := t 0 S(-s) u(s)Qz(s) ds.
Integration by parts gives (we write λ j , e j instead of λ j,V , e j,V )

Qz(s), e j = 1 λ j (- ∂ 2 ∂x 2 + V )(Qz), e j = 1 λ 2 j (- ∂ 2 ∂x 2 + V )(Qz), ( -
∂ 2 ∂x 2 + V )e j = 1 λ 2 j ∂ 2 ∂x 2 (Qz) ∂ ∂x e j x=1 x=0 + 1 λ 2 j ( V (- ∂ 2 ∂x 2 + V )(Qz), e j + ∂ ∂x (- ∂ 2 ∂x 2 + V )(Qz), ∂ ∂x e j ) = : I j + J j . Thus G t (z) 2 3,V = +∞ j=1 j 3 t 0 e iλj s u(s) Qz(s), e j ds 2 = +∞ j=1 j 3 t 0 e iλj s u(s)(I j + J j )ds 2 . (3.7)
Using (1.9), we get

∂ ∂x (- ∂ 2 ∂x 2 + V )Qz, ∂ ∂x e j = jπ ∂ ∂x (- ∂ 2 ∂x 2 + V )Qz, √ 2 cos(jπx) + s j (z),
where |s j (z)| ≤ C z 3,V for all j ≥ 1. The definition of J j , the fact that { √ 2 cos(jπx)} is an orthonormal system in L 2 , (1.7) and the Minkowski inequality yield

+∞ j=1 j 3 t 0 e iλj s u(s)J j ds 2 ≤ C t 0 |u(s)| z(s) 3,V ds 2 .
(3.8)

On the other hand, (1.9) implies that

∂ 2 ∂x 2 (Qz) ∂ ∂x e j x=1 x=0 = jπ ∂ 2 ∂x 2 (Qz) √ 2 cos(jπx) x=1 x=0 + sj (z) =: jc j (z) + sj (z),
where |s j | ≤ C z 3,V for all j ≥ 1. Again applying the Minkowski inequality, we obtain

+∞ j=1 j 3 λ 2 j t 0 e iλj s u(s)s j (z)ds 2 ≤ C t 0 |u(s)| z(s) 3,V ds 2 .
(3.9)

Since c j (z) depends on the parity of j, without loss of generality, we can assume that c(z) := c j (z) does not depend on j. Thus we cannot conclude as in the case of J j . Here we use the fact that u ∈ B. Let P ≥ 1 be an integer such that P ≤ t < P + 1. Using the Cauchy-Schwarz and the Ingham inequalities, we obtain

≤ C u(s)c(z) 2 L 2 ([P,t]) + C P p=1 p 2 u(s)c(z) 2 L 2 ([p-1,p]) ≤ C t 0 w(s) z(s)
G t (z) 3,V ≤ C t 0 w(s) z(s) 2 3,V ds 1 2 + C t 0 |u(s)| z(s) 3,V ds. (3.11)
The quantity

Gt (f ) := t 0 S(-s) v(s)Qy(s) ds is estimated in a similar way Gt 3,V ≤ C t 0 w(s) y(s) 2 3,V ds 1 2 + C t 0 |v(s)| y(s) 3,V ds ≤ C sup s∈[0,T ] y(s) 3,V ( v L 1 (R+) + v B ), ( 3
z(t) 2 3,V ≤ C( z 0 2 3,V + Gt 2 3,V + G t 2 3,V ) ≤ C z 0 2 3,V + Gt 2 3,V + t 0 w(s) z(s) 2 3,V ds + t 0 |u(s)| z(s) 3,V ds 2 .
The Gronwall inequality implies

z(t) 2 3,V ≤ C z 0 2 3,V + Gt 2 3,V + t 0 |u(s)| z(s) 3,V ds 2 × exp C t 0 w(s)ds .
Taking the square root of this inequality, using (3.10) and the Gronwall inequality, we obtain

z(t) 3,V ≤ C( z 0 3,V + Gt 3,V ) × exp C( t 0 w(s)ds + t 0 |u(s)|ds exp( t 0 w(s)ds)) ≤ C( z 0 3,V + Gt 3,V ) exp C( u L 1 (R+) + 1) exp( u 2 B ) .
In view of (3.12), this completes the proof of the proposition.

Remark 3.2. Let us notice that, one should not expect to have a well-posedness property in any Sobolev space H k with controls in L 1 . Indeed, exact controllability property in H 3 , proved by Beauchard and Laurent [START_REF] Beauchard | Local controllability of linear and nonlinear Schrödinger equations with bilinear control[END_REF] in the case d = 1, implies that the problem is not well posed in spaces H 3+σ for any σ > 0 (a point z 1 ∈ H 3 \ H 3+σ would not be accessible from a point z 0 ∈ H 3+σ ). Schrödinger equation is well posed in higher Sobolev spaces, when control u is more regular.

Corollary 3.3. Denote by U t (•, •) : H 3 (V ) × L 1 (R + , R) ∩ B→H 3 (V )
the resolving operator of (1.1), (1.2). Then U t (•, •) is locally Lipschitz continuous, i.e., for any δ > 0 there is C > 0 such that

sup t∈R+ U t (z 0 , u) -U t (z ′ 0 , u ′ ) 3,V ≤ C (z 0 , u) -(z ′ 0 , u ′ ) H 3 (V ) ×L 1 (R+,R)∩B (3.13)
for all (z 0 , u),

(z ′ 0 , u ′ ) ∈ B H 3 (V ) ×L 1 (R+,R)∩B (0, δ), where L 1 (R + , R)∩B is endowed with the norm • L 1 (R+,R)∩B := • L 1 + • B . Proof. Notice that z(t) := U t (z 0 , u) -U t (z ′ 0 , u ′ ) is a solution of problem i ż = - ∂ 2 z ∂ 2 x + u(t)Q(x)z + (u(t) -u ′ (t))Q(x)U t (z ′ 0 , u ′ ), z| ∂D = 0, z(0, x) = z 0 (x) -z ′ 0 (x).
Applying Proposition 3.1, we get (3.13).

Exact controllability in infinite time

For any control u ∈ Θ, problem (

is well-posed in Sobolev space H 3 (V ) . Equality (3.5) implies that it suffices to consider the controllability properties of (3.1), (3.2) on the unit sphere S in L 2 . Let U ∞ (z 0 , u) be the H 3 (V ) -weak ωlimit set of the trajectory corresponding to control u ∈ Θ and initial condition

z 0 ∈ H 3 (V ) : U ∞ (z 0 , u) := {z ∈ H 3 (V ) : U tn (z 0 , u) ⇀ z in H 3 (V ) for some t n → +∞}. (3.14) By (3.4), U t (z 0 , u) is bounded in H 3 (V ) , thus U ∞ (z 0 , u) is non-empty.
Definition 3.4. We say that (3.1), (3.2) is exactly controllable in infinite time in subset H ⊂ S, if for any

z 0 , z 1 ∈ H there is a control u ∈ Θ such that z 1 ∈ U ∞ (z 0 , u).
Below theorem is one of the main results of this paper.

Theorem 3.5. Under Condition 2.5, for any z ∈ S ∩ H 3 (V ) there is δ > 0 such that problem (3.1), (3.2) is exactly controllable in infinite time in S∩B H 3 (V ) (z, δ). See Section 3.3 for the proof. Remark 3.6. Let us emphasize that the novelty of Theorem 3.5 with respect to the previous result proved for (3.1), (3.2) in [START_REF] Nersesyan | Global approximate controllability for Schrödinger equation in higher Sobolev norms and applications[END_REF] (see Theorem 3.1) is that the controllability here is realized with controls which have small norms.

Working in higher Sobolev spaces, one can prove similar exact controllability results with more regular controls. For example:

δ > 0 for which e( Ũ∞ (z 0 , u)-R ∞ (z 0 , u), Ũ∞ (z ′ 0 , u ′ )-R ∞ (z ′ 0 , u ′ )) ≤ ε (z 0 , u)-(z ′ 0 , u ′ ) Tz∩H 3 (V ) ×Θ , (3.16
) whenever (z 0 , u), (z ′ 0 , u ′ ) ∈ B Tz ∩H 3 (V ) ×Θ ((0, 0), δ). Here e(•, •) stands for the Hausdorff distance (see Appendix for the definition). It is clear from the definition of e(•, •), that (3.16) follows from the following stronger estimate sup ×Θ . To prove this estimate, notice that the function

t∈R+ U t (P -1 z 0 , u) -R t (z 0 , u) -U t (P -1 z ′ 0 , u ′ ) + R t (z ′ 0 , u ′ ) Tz∩H 3 (V ) ≤ ε (z 0 , u) -(z ′ 0 , u ′ ) Tz∩H 3 (V )
y(t) := U t (P -1 z 0 , u) -R t (z 0 , u) -U t (P -1 z ′ 0 , u ′ ) + R t (z ′ 0 , u ′ ) is a solution of the problem i ẏ = - d 2 y dx 2 + (u -u ′ )Q U t (P -1 z 0 , u) -U t (z, 0) + u ′ Q U t (P -1 z 0 , u) -U t (P -1 z ′ 0 , u) , y| ∂D = 0, y(0, x) = P -1 z 0 -z 0 -P -1 z ′ 0 + z ′ 0 . We have y(0) 3,V ≤ ε z 0 -z ′ 0 3,V (3.17)
for any z 0 , z ′ 0 ∈ B Tz ∩H 3 (V ) (0, δ) and for sufficiently small δ > 0. Using (3.4) (we use the version of the inequality with v 1 f 1 + v 2 f 2 instead of vf ), Corollary 3.3 and (3.17), we get sup t∈R+ y(t) 3,V ≤ C y(0) 3,V + sup

t∈R+ U t (P -1 z 0 , u) -U t (z, 0) 3,V u -u ′ Θ + sup t∈R+ U t (P -1 z 0 , u) -U t (P -1 z ′ 0 , u) 3,V u ′ Θ ≤ C y(0) 3,V +( z 0 3,V + u Θ ) u-u ′ Θ + z 0 -z ′ 0 3,V u ′ Θ ≤ ε (z 0 , u) -(z ′ 0 , u ′ ) Tz∩H 3 (V ) ×Θ
for sufficiently small δ. This proves the proposition.

4 Non-controllability result Lemma 4.1. For any z 0 ∈ L 2 and for any u ∈ L 1 loc (R + , R), problem (1.1)-(1.3) has a unique solution z ∈ C(R + , L 2 ). Furthermore, the resolving operator U t (•, u) : L 2 → L 2 taking z 0 to z(t) satisfies the relation

U t (z 0 , u) = z 0 , t ≥ 0.
See [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF] for the proof. Let us define the set of attainability of system (1.1), (1.2) from an initial point z 0 ∈ S:

A(z 0 ) := {U t (z 0 , u) : for all u ∈ W 1,1 loc (R + , R) and t ≥ 0 }. (4.1)
The following theorem is the main result of this section.

Theorem 4.2. For any constant k ∈ (0, d), any initial condition z 0 ∈ S and any ball B ⊂ H k (V ) , we have

A c (z 0 ) ∩ B ∩ S = ∅.
Let us emphasize that this theorem does not exclude exact controllability in H k (V ) with controls form a larger space than W 1,1 loc (R + , R). The proof of this theorem is an adaptation of ideas of Shirikyan [START_REF] Shirikyan | Euler equations are not exactly controllable by a finitedimensional external force[END_REF] to the case of Schrödinger equation. Using a Hölder type estimate for the solution of the equation, we show that the image by the resolving operator U of a ball in the space of controls has a Kolmogorov ε-entropy strictly less than that of a ball B in the phase space H k (V ) . As we show, this implies the non-controllability.

Some ε-entropy estimates

Let X be a Banach space. For any compact set K ⊂ X and ε > 0, we denote by N ε (K, X) the minimal number of sets of diameters ≤ 2ε that are needed to cover K. The Kolmogorov ε-entropy of K is defined as H ε (K, X) = ln N ε (K, X).

Let Y be another Banach space and let f : K → Y be a Hölder continuous function:

f (u 1 ) -f (u 2 ) Y ≤ L u 1 -u 2 θ X (4.2)
for any u 1 , u 2 ∈ K and for some constants L > 0 and θ ∈ (0, 1). The following lemma follows immediately from the definition of ε-entropy (cf. Lemma 2.1 in [START_REF] Shirikyan | Euler equations are not exactly controllable by a finitedimensional external force[END_REF]).

Lemma 4.3. For any compact set K ⊂ X and any function f : K → Y satisfying inequality (4.2), we have

H ε (f (K), Y ) ≤ H ( ε L ) 1 θ (K, X
) for all ε > 0. We also need the following two lemmas. Lemma 4.4. For any T > 0 and for any closed ball B ⊂ W 1,1 ([0, T ], R), there is a constant C > 0 such that

H ε (B, L 1 ([0, T ], R)) ≤ C ε ln 1 ε .
This is Proposition 2.3 in [START_REF] Shirikyan | Euler equations are not exactly controllable by a finitedimensional external force[END_REF].

Lemma 4.5. For any k > 0 and any closed ball

B := B H k (V ) (z 0 , r) such that B H k (V ) (z 0 , r) ∩ S = ∅ there is a constant C > 0 such that H ε (B ∩ S, H k-1 ) ≥ C 1 ε d . (4.3)
Proof. It is well known that

C 1 1 ε d ≤ H ε (B, H k-1 ) ≤ C 2 1 ε d (4.4)
for some constants C 1 , C 2 > 0 (e.g., see [START_REF] Edmunds | Function Spaces, Entropy Numbers, Differential Operators[END_REF]). Consider the mapping

f : [ 1 2 , 3 2 ] × B ∩ S→H k-1 , (s, z)→sz.
The set f ([ 1 2 , 3 2 ] × B ∩ S) has a non-empty interior, so there is a ball B in

H k such that B ⊂ f ([ 1 2 , 3 2 ] × B ∩ S). (4.5)
Clearly,

f (s 1 , z 1 ) -f (s 2 , z 2 ) k-1 ≤ C(|s 1 -s 2 | + z 1 -z 2 k-1 ).
Using (4.5) and Lemma 4.3, we get

H ε ( B, H k-1 ) ≤ H ε (f ([ 1 2 , 3 2 ] × B ∩ S), H k-1 ) ≤ H ε C ([ 1 2 , 3 2 ] × B ∩ S, R × H k-1 ) ≤ H ε C ([ 1 2 , 3 2 ], R) + H ε C (B ∩ S, H k-1 ) ≤ C ln 1 ε + H ε (B ∩ S, H k-1 ) .
Combining this with (4.4) for B, we obtain (4.3).

Proof of Theorem 4.2

Let us suppose, by contradiction, that there is k ∈ (0, d), an initial point z 0 ∈ S and a ball B ⊂ H k (V ) such that

B ∩ S ⊂ A(z 0 ), (4.6)
where A is the set of attainability of system (1.1), (1.2) from the initial point z 0 defined by (4.1). Let us set

B m : = [0, m] × B W 1,1 ([0,m],R) (0, m), U(B m ) : = {U t (z 0 , u) : for all (t, u) ∈ B m }.
We have

R × W 1,1 loc (R + , R) = ∞ m=1 B m , A(z 0 ) = ∞ m=1 U(B m ). ( 4.7)
Combining (4.6), (4.7) and the Baire lemma, we see that there is a ball

Q ⊂ H k (V )
and an integer m ≥ 1 such that U(B m ) is dense in Q ∩ S with respect to H knorm.

Step 1. Let us define the set

Bm = {(t, u) ∈ B m : such that U t (z 0 , u) ∈ Q}.
Here we prove that Bm is compact in

[0, m] × L 1 ([0, m], R). Indeed, take any sequence (t n , u n ) ∈ Bm . As (t n , u n ) ∈ B m and B m is compact in [0, m] × L 1 ([0, m], R), there is a sequence n k → ∞ and (t 0 , u 0 ) ∈ B m such that |t n k -t 0 | + u n k -u 0 L 1 ([0,m],R) → 0, k → ∞.
We need to show that (t 0 , u 0 ) ∈ Bm . As U tn k (z 0 , u n k ) ∈ Q, there is z ∈ Q such that U tn k (z 0 , u n k ) ⇀ z in H k (again extracting a subsequence, if necessary). On the other hand, Lemma 4.1 implies that U tn k (z 0 , u n k ) → U t0 (z 0 , u 0 ) in L 2 . Thus U t0 (z 0 , u 0 ) = z and (t 0 , u 0 ) ∈ Bm . Thus Bm is compact in [0, m] × L 1 ([0, m], R).

In particular, this implies that U( Bm ) is compact in L 2 , as an image of a compact set by a continuous mapping. On the other hand, U( Bm ) is dense in the compact set Q ∩ S in L 2 . Thus Q ∩ S = U( Bm ).

Step 2. Using standard arguments, one can show that we have

U t (z 0 , u) -U t ′ (z 0 , u ′ ) ≤ C(|t -t ′ | + u -u ′ L 1 ([0,m],R) )
for any (t, u), (t ′ , u ′ ) ∈ Bm , where C > 0 is a constant not depending on (t, u) and (t ′ , u ′ ). Combining this with the interpolation inequality

z k-1 ≤ C z 1 k z k-1 k k , we get U t (z 0 , u) -U t ′ (z 0 , u ′ ) k-1 ≤ C(|t -t ′ | 1 k + u -u ′ 1 k L 1 ([0,m],R) )
for any (t, u), (t ′ , u ′ ) ∈ Bm . Here we used the fact that U t (z 0 , u), U t ′ (z 0 , u ′ ) ∈ Q. Appying Lemmas 4.3 and 4.4 and of the fact that Q ∩ S ⊂ U( Bm ), we obtain

H ε (Q ∩ S, H k-1 ) ≤ H ε (U( Bm ), H k-1 ) ≤ CH ε k ( Bm , [0, m] × L 1 ([0, m], R)) ≤ CH ε k (B m , [0, m] × L 1 ([0, m], R)) ≤ C ε k ln 1 ε k .
This estimate contradicts Lemma 4.5 and proves the theorem.

Remark 4.6. The same proof works also in the case of Schrödinger equation with any finite number of controls:

i ż = -∆z + V (x)z + u 1 (t)Q 1 (x)z + . . . + u n (t)Q n (x)z,
where n ≥ 1 is any integer, Q j ∈ C ∞ (D, R) are arbitrary functions and u j are the controls j = 1, . . . , n.

Appendix

Genericity of Condition 2.5

Let us assume that D = (0, 1) d and introduce the space

G := {V ∈ C ∞ (D, R) : V (x 1 , . . . , x d ) = V 1 (x 1 ) + . . . + V d (x d ) for some V k ∈ C ∞ ([0, 1], R), k = 1, . . . , d}.
Then G, endowed with the metric of C ∞ (D, R), is a closed subspace in C ∞ (D, R). By Lemma 3.12 in [START_REF] Nersesyan | Growth of Sobolev norms and controllability of the Schrödinger equation[END_REF], the set A of all functions V ∈ G such that property (ii) in Condition 2.5 is verified is G δ set (i.e., countable intersection of dense open sets). First let us prove genericity of property (i) in the case d = 1.

Lemma 5.1.

For any V ∈ C ∞ ([0, 1], R), the set of functions Q ∈ C ∞ ([0, 1], R) such that inf p,j≥1 |p 3 j 3 Qe p,V , e j,V | > 0 (5.1) is dense in C ∞ ([0, 1], R).
Proof. If V = 0, then a straightforward calculation gives

x 2 e p,0 , e j,0 = (-1) p+j 8pj

π 2 (p 2 -j 2 ) 2 , if p = j, 2 3 -1 p 2 π 2 , if p = j,
which implies (5.1) for Q = x 2 and V = 0. In the general case, taking any p = j, we integrate by parts (we write λ j , e j and z ′′ , z ′ instead of λ j,V , e j,V and

d
+ V )(Q ′ e ′ p
), e j = V Q ′ e ′ p , e j + -Q ′′′ e ′ p , e j + -Q ′′ e ′′ p , e j + λ p Q ′ e ′ p , e j -Q ′ (V e p ) ′ , e j .

Replacing this into (5.3), we get Let Q be such that A := Q ′ (x) cos(pπx) cos(jπx) x=1 x=0 = 0. Clearly, this is verified for almost any Q, since A depends only on the parity of p and j. Let us choose Q such that Qe p , e j = 0 for all p, j ≥ 1; the set of such functions Q is G δ , by Section 3.4 in [START_REF] Nersesyan | Growth of Sobolev norms and controllability of the Schrödinger equation[END_REF]. Using the estimates (1.7)-(1.9), it is easy to see that inf p,j≥1,p =j |p 3 j 3 I 2 | > 0. Iterating the same arguments for I 1 , we see that inf p,j≥1,p =j |p 3 j 3 Qe p,V , e j,V | > 0 for almost any polynomial Q.

Q ′ e ′ p ,
If p = j, using (1.8), we get

Inverse mapping theorem for multifunctions

In this section, we recall the statement of the inverse mapping theorem for multivalued functions or multifunctions. We refer the reader to the paper [START_REF] Nachi | Inversion of multifunctions and differential inclusions[END_REF] by Nachi and Penot for details and for a review of the literature on this subject.

Let X and Y be Banach spaces. We call a multifunction from X to Y any mapping F from X to 2 Y . Definition 5.2. A multifunction F from an open set X 0 ⊂ X to Y is said to be strictly differentiable at (x 0 , y 0 ) if there exists some continuous linear map A : X→Y such that for any ε > 0 there exist β, δ > 0 for which e(F (x) ∩ B Y (y 0 , β) -A(x), F (x ′ ) -A(x ′ )) ≤ ε x -x ′ X , whenever x, x ′ ∈ B(x 0 , δ). The map A is called a derivative of F at (x 0 , y 0 ).

The following theorem is a generalization of the classical inverse function theorem to the case of multifunctions.

Theorem 5.3. Let F be a multifunction from an open set X 0 ⊂ X to Y with closed non-empty values. Suppose F is strictly differentiable at (x 0 , y 0 ) ∈ Gr(F ), and some derivative A of F at (x 0 , y 0 ) has a right inverse. Then for any neighborhood U of x 0 there exists a neighborhood V of y 0 such that V ⊂ F (U ). See Theorem 22 in [START_REF] Nachi | Inversion of multifunctions and differential inclusions[END_REF] for the proof.

  {d = {d mk } : d 2 l2 := |d 11 | 2 + +∞ m,k=1,m =k |d mk | 2 < +∞, d mm = d 11 and d mk = d km for all m, k ≥ 1},

Proposition 2 . 9 . 4 mk

 294 If the sequence ω m ∈ R, m ≥ 1 is such that ω 1 = 0 and ∞ m=2 1 |ωm| p < +∞ for some p ≥ 1 and ω i = ω j for i = j, then there is a linear continuous operator A from ℓ 2 0 to Θ such that { Ǎ (d)(ω m )} = d for any d ∈ ℓ 2 0 . The idea of the proof of Theorem 2.6 is to rewrite (2.8) in the form d mk = ǔ(ω mk ) with d = {d mk } ∈ l2 and to apply the proposition. Notice that ∞ m,k=1,m =k 1 ω

  m )h m = 0.(2.28) Replacing in this equality ǔ(ω m ) by its integral representation, we get integrat-m ) p u (p) (s)dsh m = +∞ 0 u (p) (s) P p (s)h 1 + +∞ m=2

1 e 2 ≤ 1 e

 121 iλj s u(s)c(z)ds C u(s)c(z) 2 L 2 ([P,t]) + C iλj s u(s)c(z)ds

2

 2

4. 1

 1 Main resultIn this section, we study the problem of non-controllability of Schrödinger system (1.1)-(1.3), where D ⊂ R d is a bounded domain with smooth boundary, V, Q ∈ C ∞ (D, R) are arbitrary given functions. The following lemma establishes the well-posedness of system (1.1)-(1.3) in the space L 2 .

+

  V Q ′ e ′ p , e j + -Q ′′′ e ′p , e j + -Q ′′ e ′′ p , e j -Q ′ (V e p ) ′ , e j ).(5.4) Using (5.2) and (5.4) and the fact that-Q ′′ e ′′ p , e j = -Q ′′ V e p , e j + λ p Q ′′ e p , e j ,we obtainQe p , e j =(-1 λ j -λ p Q ′′ e p , e j -λ p (λ j -λ p ) 2 Q ′′ e p , e j ) -1 (λ j -λ p ) 2 (-Q ′ e ′ p e ′ j x=1 x=0 + V Q ′ e ′ p , e j + -Q ′′′ e ′ p , e j -Q ′′ V e p , e j -Q ′ (V e p, )′ , e j ) =:I 1 + I 2 .

Qe p , e p = 2 Q 1 0Taking Q such that 1 0

 211 , sin 2 (pπx) + s p , where s p →0. Thus Qe p , e p = Q, 1 -cos 2pπx + s p = Qdx -Q, cos 2pπx + s p . Qdx = 0, we complete the proof of the lemma.Take any functionsQ k ∈ C ∞ ([0, 1], R), k = 1, . . . , d in the dense set of Lemma 5.1 corresponding to V k ∈ C ∞ ([0, 1], R), k = 1, . . . , d. Then Q(x 1 , . . . , x d ) := Q 1 (x 1 ) • . . . • Q d (x d ) satisfies (i) with V (x 1 , . . . , x d ) := V 1 (x 1 ) + . . . + V d (x d ).

  For any non-empty sets C, D ⊂ X, define the Hausdorff distance d(x, D) = inf y∈D x -y X , e(C, D) = sup x∈C d(x, D).

  .14) where d 0 ∈ R. Let us show that, for an appropriate choice of d 0 , there are C mk satisfying (2.11)-(2.14). Since y ∈ T z , we have z, y = i Im z, y . We can rewrite (2.11) and (2.12) in the following form Case 1. Let as suppose that z = ce p , where c ∈ C, |c| = 1 and p ≥ 1. Then (2.13)-(2.16) is verified for C mk = 0, if m = k and C mm defined by (2.16) with

	+∞
	k=1	z, e k Q mk C mk = -z, e m Im z, y ,	(2.15)
	d mm =	-2 Im( y, e m e m , z ) Q mm	+ C mm = d 0 .	(2.16)

  Existence of a solution is obtained easily from (3.11) and (3.12), by a fixed point theorem (cf. Proposition 2 in[START_REF] Beauchard | Local controllability of linear and nonlinear Schrödinger equations with bilinear control[END_REF]). Uniqueness follows from (3.4).

	Let us prove (3.4). From (3.6) and (3.11) we have

.12) where w(s) = |v(s)| 2 χ [P,t] (s) + P p=1 p 2 |v(s)| 2 χ [p-1,p] (s).

  2 z dx 2 , dz dx , respectively) ′′ e p , e j + -Q ′ e ′ p , e j + λ p Qe p , e j ).Qe p , e j = -1 λ j -λ p ( Q ′′ e p , e j + Q ′ e ′ p , e j ).

		Qe p , e j =	1 λ j	(-	d 2 dx 2 + V )(Qe p ), e j
	= ( -Q This implies that 1 λ j
										(5.2)
	Again integrating by parts, we get
		Q ′ e ′ p , e j =	1 λ j	Q ′ e ′ p , (-	d 2 dx 2 + V )e j
		= -	1 λ j	Q ′ e ′ p e ′ j	x=1 x=0	+	1 λ j	(-	d 2 dx 2 + V )(Q ′ e ′ p , e j .	(5.3)
	Notice that
	(-	d 2 dx 2

Acknowledgments. The authors would like to thank Armen Shirikyan for many fruitful conversations.

Theorem 3.7. Under Condition 2.5, for any z ∈ S ∩ H 3+σ (V ) , σ ∈ (0, 2] there is δ > 0 such that problem (3.1), (3.2) is exactly controllable in infinite time in S ∩ B H 3+σ (V ) (z, δ) with controls u ∈ W 1,1 (R + , R) ∩ H s (R + , R) for any s ≥ 1.

These local exact controllability properties imply the following global exact controllability result. Theorem 3.8. Under Condition 2.5, problem (3.1), (3.2) is exactly controllable in infinite time in S ∩ H 3 (V ) in the following sense: for any

for any s ∈ [0, 1). Using the compactness of the curve γ and Theorem 3.7, we prove that there is a control v and time

, where δ z1 > 0 is the constant in Theorem 3.5 corresponding to z 1 . This completes the proof. Remark 3.9. We do not know if problem (1.1)-(1.3) is well posed in the space V for d ≥ 2 with Θ-controls. Well-posedness in V with u ∈ Θ would imply the controllability of the multidimensional problem. The nonlinear problem's solution is in V for more regular controls.

Proof of Theorem 3.5

The proof is based on an inverse mapping theorem for multivalued functions. We apply the inverse mapping theorem established by Nachi and Penot [START_REF] Nachi | Inversion of multifunctions and differential inclusions[END_REF], which suits well to the setting of Schrödinger equation. For the reader's convenience, we recall the statement of their result in Appendix (see Theorem 5.3).

Let us first slightly modify the definition (3.14) of the set U ∞ (z 0 , u). Let T n → + ∞ be the sequence defined in Section 2.1. Define

Consider the multivalued function

Since the result of Nachi and Penot is stated in the case of Banach spaces, we cannot apply it directly to U ∞ . Following Beauchard and Laurent [START_REF] Beauchard | Local controllability of linear and nonlinear Schrödinger equations with bilinear control[END_REF], we project the system onto the tangent space T z . We apply Theorem 5.3 to the following multivalued function

where P is the orthogonal projection in L 2 onto T z , i.e., P z = z -Re z, z z, z ∈ L 2 . Notice that P -1 : B Tz (0, δ)→S is well defined for sufficiently small δ > 0.

By the definition of T n , we have lim n→+∞ U Tn (z, 0) = z. Hence (3.15) implies that U ∞ (z, 0) = z and Ũ∞ (0, 0) = {0}. If we show that Ũ∞ is strictly differentiable at (x 0 , y 0 ) with x 0 = (0, 0) ∈ T z ∩ H 3 (V ) × Θ and y 0 = 0 ∈ T z ∩ H 3 (V ) (see Definition 5.2), and the derivative admits a right inverse, then Theorem 3.5 will be proved as a consequence of Theorem 5.3.

Proposition 3.10. The multifunction Ũ∞ is strictly differentiable at (0, 0) ∈ T z ∩ H 3 (V ) × Θ in the sense of Definition 5.2. Moreover, the differential is the mapping

where R ∞ is defined in Section 2.1.

Proof of Theorem 3.5. Case 1. Let us suppose that z ∈ S ∩ H 3 (V ) \ E. For any (z 0 , u) ∈ B Tz ∩H 3 (V ) ×Θ (0, δ), the set Ũ∞ (z 0 , u) is closed and non-empty, if δ > 0 is sufficiently small. The mapping R ∞ is invertible in view of Theorem 2.6. Thus Theorem 5.3 completes the proof. Remark 3.11. Let us point out that in case 1 the controls u can be chosen such that u(0) = . . . = u (s-1) (0) = 0.

Case 2. In the case z ∈ S ∩ H 3 (V ) ∩ E, the linearized system (2.2), (2.3) is not controllable, and R ∞ is not invertible. Controllability in finite time near z is obtained combining the results of [START_REF] Beauchard | Controllability of a quantum particle in a moving potential well[END_REF] and [START_REF] Beauchard | Local controllability of linear and nonlinear Schrödinger equations with bilinear control[END_REF]: there is a constant δ > 0 and a time T > 0 such that for any z 0 , z 1 ∈ S ∩ B H 3 (V ) (z, δ) there is a control v ∈ L 2 ([0, T ], R) verifying U T (z 0 , v) = z 1 . Let us prove that the problem is exactly controllable in infinite time in S ∩ B H 3 (V ) (z, δ). Take any z 1 ∈ S ∩ B H 3 (V ) (z, δ) and let us show that there is a control u ∈ Θ such that z 1 ∈ U ∞ (z, u). Let us suppose first that z 1 / ∈ E. Then, by case 1, there is δ z1 > 0 such that exact controllability in infinite time holds in S ∩ B H 3 (V ) (z 1 , δ z1 ). By exact controllability property in finite time and by an approximation argument, one can find a control u 1 ∈ C ∞ 0 ((0, T ), R) such that U T (z, u 1 ) ∈ B H 3 (V ) (z 1 , δ z1 ). Thus the existence of u 1 follows from case 1 and Remark 3.11. Now let us suppose that z 1 ∈ E. Since E ⊂ ∩ ∞ k=1 H k (V ) , by [START_REF] Beauchard | Controllability of a quantum particle in a moving potential well[END_REF] and [START_REF] Beauchard | Local controllability of linear and nonlinear Schrödinger equations with bilinear control[END_REF], there is a control u 1 ∈ C s ([0, T ], R) such that U T (z, u 1 ) = z 1 and u(0) = . . . = u (s) (0) = u(T ) = . . . = u (s) (T ) = 0. Extending u 1 by 0 on [T, +∞), we obtain z 1 ∈ U ∞ (z, u 1 ).

Proof of Proposition 3.10. It suffices to show that for any ε > 0 there exists