Global approximate controllability for Schrödinger equation in higher Sobolev norms and applications

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www.elsevier.com/locate/anihpc

Global approximate controllability for Schrödinger equation in higher Sobolev norms and applications

Vahagn Nersesyan

Laboratoire de Mathématiques, Université de Paris-Sud XI, Bâtiment 425, 91405 Orsay Cedex, France Received 18 May 2009; received in revised form 14 December 2009; accepted 14 December 2009

Available online 7 January 2010

Abstract

We prove that the Schrödinger equation is approximately controllable in Sobolev spacesH^s,s >0, generically with respect to the potential. We give two applications of this result. First, in the case of one space dimension, combining our result with a local exact controllability property, we get the global exact controllability of the system in higher Sobolev spaces. Then we prove that the Schrödinger equation with a potential which has a random time-dependent amplitude admits at most one stationary measure on the unit sphereSinL².

1. Introduction

In this paper, we study the problem

iz˙= −z+V (x)z+u(t )Q(x)z, x∈D, (1.1)

z|∂D=0, (1.2)

z(0, x)=z₀(x), (1.3)

whereD⊂R^mis a bounded domain with smooth boundary,V ∈C^∞(D,R)is an arbitrary given function,uis the control, andzis the state. We prove that this system is approximately controllable to the eigenfunctions of−+V in Sobolev spaces H^s, s >0 generically with respect to the dipolar moment Q. In the case m=1 and V =0, combination of our result with the local exact controllability property obtained by Beauchard [7,8] gives the global exact controllability of the system in the spacesH⁵⁺^ε. Approximate controllability property implies also that the random Schrödinger equation admits at most one stationary measure on the unit sphereSinL².

The problem of controllability of the Schrödinger equation has been largely studied in the literature. Let us mention some previous results closely related to the present paper. Ball, Marsden and Slemrod [5] and Turinici [35] show that the set of attainable points from any initial data inS∩H² by system (1.1), (1.2) admits a dense complement in S∩H². In [7], Beauchard proves an exact controllability result for the system withm=1, D=(−1,1), V (x)=0 andQ(x)=xinH⁷-neighborhoods of the eigenstates. Beauchard and Coron [9] established later a partial global exact

E-mail address:[email protected].

doi:10.1016/j.anihpc.2010.01.004

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controllability result, showing that the system in question is also controlled between some neighborhoods of any two eigenstates. Chambrion et al. [14] and Privat and Sigalotti [31] prove that (1.1), (1.2) is approximately controllable in L² generically with respect to the functions V , Q and the domainD. See also the papers [32,36,4,3,1,10] for controllability of finite-dimensional systems and the papers [24,25,6,38,16,26,11,17] for controllability properties of various Schrödinger systems.

Let us recall that, in the case of the spaceH², we established a stabilization property for system (1.1), (1.2) in [28].

Namely, we introduce a Lyapunov function V(z)0 that controls theH²-norm ofzand possesses the following properties:

• V(z)=0 if and only ifz=ce1,V, wheree1,V is the first eigenfunction of the operator−+V,

• _dt^dV(z(t ))=uG(z(t )), wherez(t )=z(t, u)is the solution of (1.1)–(1.3) andG(z)is a function given explicitly.

Choosing the feedback lowu(z)= −G(z), we see that _dt^dV(z(t ))= −u² 0, i.e., the functionV decreases on the trajectories of (1.1) corresponding to the feedbacku. Thus, to conclude, it suffices to prove thatV(z(t ))→0. To this end, we use an iteration argument and show that theH²-weakω-limit set of any trajectoryz(t )contains a minimum point of the functionV, i.e., the eigenfunctionce1,V, wherec∈C,|c| =1. Thus we construct explicitly a feedback lowu(z)which forces the trajectories of the system to converge to the eigenstate{ce1,V: c∈C,|c| =1}.

The aim of this paper is to generalize these ideas to the case of the spacesH^k, k >2. The main difficulty is that we are not able to construct a Lyapunov functionV(z)such that _dt^dV(z(t ))=uG(z(t ))for some functionG. However, notice that for anyw∈C^∞([0, T],R)we can calculate explicitly the derivative _dσ^dV(z(t, σ w))|σ=0. We show that there is a timeT >0 and a controlwsuch that _dσ^dV(z(t, σ w))|σ=0=0. Hence we can chooseσ0close to zero such that

V

z(T , σ₀w)

<V z(T ,0)

=V(z₀).

Thus for any pointz0we find a timeT >0 and a controlusuch that V

z(T , u)

<V(z₀).

Using an iteration argument close to that of [28], we conclude that there are sequencesT_n>0 andu_n∈C^∞([0, Tn],R) such thatz(t_n, u_n)→e1,V. Thus any pointz0can be approximately controlled toe1,V.

Then, in the casem=1 and V =0, combining this controllability property with the local exact controllability result obtained by Beauchard [8], we see that the system is globally exactly controllable inS∩H⁵⁺^εgenerically with respect toQ∈C^∞(D,R), i.e., for anyz₀, z₁∈S∩H⁵⁺^εthere is a timeT >0 and a controlu∈H₀¹([0, T],R)such thatz(0, u)=z₀andz(T , u)=z₁.

Next we apply approximate controllability property to prove that the random Schrödinger equation has at most one stationary measure onS. This follows from uniform Feller property and irreducibility of the transition functions of the Markov chain associated to the system in question. Existence of a stationary probability different from the Dirac measure concentrated at zero is an open problem. There are several results on the existence of stationary measures for deterministic Schrödinger equations. Bourgain [12] and Tzvetkov [37] prove the existence of stationary Gibbs measures for different nonlinear Schrödinger systems. Kuksin and Shirikyan [23] construct a stationary measure as a limit of the unique stationary measure of the randomly forced complex Ginzburg–Landau equation when the viscosity goes to zero. In [15], Debussche and Odasso prove existence and uniqueness of stationary measure and a polynomial mixing property for a damped 1D Schrödinger equation. For finite-dimensional approximations of the Schrödinger equation, existence and uniqueness of stationary measure and an exponential mixing property is obtained in [27].

1.1. Notation

In this paper we use the following notation. LetD⊂R^m, m1, be a bounded domain with smooth boundary. Let H^s:=H^s(D)be the Sobolev space of orders0 endowed with the norm · s. Consider the operators−z+V z, z∈D(−+V ):=H₀¹∩H²,whereV ∈C^∞(D,R). We denote by{λ_j,V}and{e_j,V}the sets of eigenvalues and normalized eigenfunctions of−+V. Define the spacesH_{(V )}^s :=D((−+V )²^s). Let ·,·and · be the scalar product and the norm in the spaceL². LetSbe the unit sphere inL². For a Polish spaceX, we shall use the following notation.

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B(X)is theσ-algebra of Borel subsets ofX.

C(X)is the space of real-valued continuous functions onX.

C_b(X)is the space of bounded functionsf ∈C(X).

L(X)is the space of functionsf∈Cb(X)such that f_L:= f_∞+sup

u=v

|f (u)−f (v)|

u−v <+∞. P(X)is the set of probability measures on(X,B(X)).

2. Main results

2.1. Approximate controllability toe1,V

The following lemma shows the well-posedness of system (1.1), (1.2) in Sobolev spacesH^s,s0.

Lemma 2.1.For any control u∈C^∞([0,∞),R)such that ^d_dt^k^uk(0)=0 for all k1 and for anyz₀∈H_(V^s ₊_u(0)Q) problem (1.1)–(1.3) has a unique solution z∈C([0,∞), H^s). Furthermore, if ^d_dt^k^uk(T )=0 for all k1, then z(T )∈ H_(V^s ₊_{u(T )Q)}.

See [7] for the proof. Denote byUt(·, u):L²→L²the resolving operator of (1.1), (1.2). As in [28], we assume that the functionsV andQsatisfy the following condition.

Condition 2.2.The functionsV , Q∈C^∞(D,R)are such that:

(i) Qe_1,V, e_j,V =0 for allj2,

(ii) λ_1,V−λ_j,V =λ_p,V −λ_q,V for allj, p, q1 such that{1, j} = {p, q}andj =1.

We say that problem (1.1), (1.2) is approximately controllable to e_1,V in H^s, s >0 if for any integer k1, any constantsε, δ, d >0 and for any pointz0∈S∩H_{(V )}^s there is a timeT >0 and a controlu∈C₀^∞((0, T ),R), uC^k([0,T])< dsuch that

UT(z₀, u)−e_1,V

s−δ< ε.

The theorem below is one of the main results of the present paper.

Theorem 2.3.Under Condition2.2, for anys >0, problem(1.1), (1.2)is approximately controllable toe1,V inH^s. In many relevant examples, the spectrum of the operator−+V is degenerate, hence property (ii) in Condition 2.2 is not verified. In fact, the proof of Theorem 2.3 can be adapted to show the approximate controllability of (1.1), (1.2) for any potentialV, under stronger assumptions on the functionQ. More precisely, we have the following result.

Theorem 2.4.LetV ∈C^∞(D,R)be arbitrary. The set of functionsQsuch that problem(1.1), (1.2)is approximately controllable toe1,V inH^sfor anys >0is residual inC^∞(D,R), i.e., contains a countable intersection of open dense sets inC^∞(D,R).

HereC^∞(D,R)is endowed with its usual topology given by the countable family of norms:

p_n(Q):=

|α|n

sup

x∈D

∂^αQ(x).

See Section 2.3 for the proof this theorem.

We say that problem (1.1), (1.2) is approximately controllable inL²if for any integerk1, any constantsε, d >0 and for any pointsz₀, z₁∈Sthere is a timeT >0 and a controlu∈C₀^∞((0, T ),R),uC^k([0,T])< dsuch that

Uk(z₀, u)−z₁< ε.

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Combination of Theorem 2.4 with the time reversibility property of the Schrödinger equation implies approximate controllability inL².

Theorem 2.5.LetV ∈C^∞(D,R)be arbitrary. The set of functionsQsuch that problem(1.1), (1.2)is approximately controllable inL²is residual inC^∞(D,R).

This result is proved exactly in the same way as Theorem 3.5 in [28].

Proof of Theorem 2.3. By Lemma 3.4 in [28], it suffices to prove the theorem for any initial dataz₀∈S∩H_{(V )}^s with z₀, e_1,V =0. Let us introduce the Lyapunov function

V(z):=α(−+V )^s²P_1,Vz²+1−z, e_1,V², z∈S∩H_{(V )}^s , (2.1) whereα >0 and P_1,Vz:=z− z, e_1,Ve_1,V is the orthogonal projection inL²onto the closure of the vector span of {e_k,V}k2. Notice that V(z)0 for all z∈S∩H_{(V )}^s and V(z)=0 if and only if z=ce_1,V,|c| =1. For any z∈S∩H_{(V )}^s we have

V(z)α(−+V )^s²P_1,Vz²C₁z²s−C₂. Thus

C

1+V(z)

zs (2.2)

for some constantC >0. We need the following result proved in Section 2.2.

Proposition 2.6.Fix any constantss >0andd >0and an integerk1. There is a finite or countable setJ⊂R^∗₊ such that for anyα /∈J and for anyz0∈S∩H_{(V )}^s with z0, e1,V =0and 0<V(z0)there is a timeT >0and a controlu∈C₀^∞((0, T ),R),uC^k([0,T])< dverifying

V

UT(z₀, u)

<V(z₀).

Let us take anyz₀∈S∩H_{(V )}^s with z₀, e_1,V =0 and 0<V(z₀), and chooseα >0 in (2.1) such thatV(z₀) <1.

Define the set K=

z∈H_{(V )}^s :UT_n(z0, un)→zinH^s⁻^δfor someTn0, un∈C₀^∞

(0, Tn),R

,unC^k([0,Tn])< dand for anyδ >0 . Let

m:= inf

z∈KV(z).

This infimum is attained, i.e., there ise∈Ksuch that V(e)=inf

z∈KV(z). (2.3)

Indeed, take any minimizing sequence z_n∈K,V(z_n)→m. By (2.2),z_n is bounded in H^s. Thus, without loss of generality, we can assume thatz_n einH^s for somee∈H_{(V )}^s . This implies thatV(e)lim infn→∞V(z_n)=m. Let us show thate∈K. Asz_n∈K, there are sequencesT_n>0 andu_n∈C₀^∞((0, Tn),R)such that

UT_n(z0, un)−zn

s−δ1

n. (2.4)

On the other hand,zn→einH^s⁻^δ, and (2.4) implies thatUT_n(z0, un)→einH^s⁻^δ. Thuse∈KandV(e)=m.

Let us show thatV(e)=0.Suppose, by contradiction, thatV(e) >0. It follows from (2.3) and from the choice ofα thatV(e)V(z₀) <1. This shows that e, e_1,V =0. Proposition 2.6 implies that there is a timeT >0 and a control u∈C₀^∞((0, T ),R)such that

V

UT(e, u)

<V(e). (2.5)

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Defineu˜_n(t )=u_n(t ),t∈ [0, Tn]andu˜_n(t )=u(t−T_n),t∈ [T_n, T_n+T]. Then u˜_n∈C₀^∞((0, Tn+T ),R)and, by continuity inH^s⁻^δof the resolving operator for (1.1), (1.2) (e.g., see [13]),

UTn+T(z₀,u˜_n)→UT(e, u) inH^s⁻^δ.

This implies thatUT(e, u)∈K. Clearly, (2.5) contradicts (2.3). It follows thatV(e)=0, hencee=ce_1,V for some c:=c(e)∈C,|c| =1. AsUτ(ce_1,V,0)=e^iτce_1,V =e_1,V forτ= −arg(c), we see thate_1,V∈K. 2

2.2. Proof of Proposition 2.6

Take anyz₀∈S∩H_{(V )}^s such that z₀, e_1,V =0 and 0<V(z₀). This implies that also z₀, e_p,V =0 for some p2. Let us consider the mapping

V UT

z₀, (·)w

:R→R, σ→V

UT(z0, σ w) ,

whereT >0 and w∈C₀^∞((0, T ),R). We are going to show that, for an appropriate choice of T andw, we have

dV(UT(z₀,σ w))

dσ |σ=0=0. Clearly, dV(UT(z0, σ w))

dσ

σ=0

=2αRe(−+V )^s²P_1,VUT(z₀,0), (−+V )^s²P_1,VRT(w)

−2 ReUT(z0,0), e1,V e1,V,RT(w)

, (2.6)

whereRt(·)is the resolving operator of problem

iz˙= −z+V (x)z+w(t )Q(x)Ut(z₀,0), x∈D, (2.7)

z|∂D=0, (2.8)

z(0)=0. (2.9)

System (2.7)–(2.9) is the linearization of (1.1), (1.2) around the solutionUt(z₀,0). Note that (2.7)–(2.9) is equivalent to

z(t )= −i t

0

e⁻ⁱ⁽⁻⁺^{V )(t}⁻^{τ )}w(τ )Q(x)Uτ(z0,0)dτ. (2.10)

Taking into account the fact that Ut(z₀,0)=

∞ k=1

e⁻^iλ^k,V^t z₀, e_k,Ve_k,V, (2.11)

we get from (2.10) RT(w), e_j,V

= −ie⁻^iλ^j,V^T ∞ k=1

z0, e_k,V Qe_k,V, e_j,V T

0

e⁻^i(λ^k,V⁻^λ^j,V^)τw(τ )dτ. (2.12) Replacing (2.11) and (2.12) into (2.6), we get

dV(UT(z0, σ w)) dσ

σ=0

= −2αIm ∞ j=2,k=1

λ^s_j,V z₀, e_j,V e_k,V, z₀ Qe_k,V, e_j,V × T

0

e^i(λ^k,V⁻^λ^j,V^)τw(τ )dτ

+2 Im ∞ k=1

z₀, e_1,V e_k,V, z₀ Qe_k,V, e_1,V T

0

e^i(λ^k,V⁻^λ^1,V^)τw(τ )dτ

= T

0

Φ(τ )w(τ )dτ,

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where

iΦ(τ ):= −α ∞ j=2,k=1

λ^s_j,V z0, ej,V ek,V, z0 Qek,V, ej,Ve^i(λ^k,V⁻^λ^j,V^)τ +α

∞ j=2,k=1

λ^s_j,V e_j,V, z₀ z₀, e_k,V Qe_k,V, e_j,Ve⁻^i(λ^k,V⁻^λ^j,V^)τ +

∞ k=2

z₀, e_1,V e_k,V, z₀ Qe_k,V, e_1,Ve^i(λ^k,V⁻^λ^1,V^)τ

− ∞ k=2

e1,V, z0 z0, ek,V Qek,V, e1,Ve⁻^i(λ^k,V⁻^λ^1,V^)τ

= ^∞

j=2,k=2

P (z₀, Q, j, k)e⁻^i(λ^j,V⁻^λ^k,V^)t

+ ∞ j=2

αλ^s_j,V+1

z₀, e_1,V e_j,V, z₀ Qe_j,V, e_1,Ve⁻^i(λ^1,V⁻^λ^j,V^)t

−^∞

j=2

αλ^s_j,V+1

e1,V, z0 z0, e_j,V Qe_j,V, e1,Ve^i(λ^1,V⁻^λ^j,V^)t. (2.13) HereP (z₀, Q, j, k)are constants. Define the set

J:=

α∈R: αλ^s_j,V = −1 for somej1 ,

and take anyα /∈J. As z₀, e_j,V =0 forj =1, p, Condition 2.2 and Lemma 3.10 in [28] imply that there isT >0 such thatΦ(T )=0. Thus there is a controlw∈C₀^∞((0, T ),R),w_C^k₍_[0,T])< dsuch that

dV(UT(z₀, σ w)) dσ

σ=0

= T

0

Φ(τ )w(τ )dτ=0.

This implies that there isσ₀∈Rclose to zero such that V

UT(z₀, σ₀w)

<V

UT(z₀,0)

=V(z₀), which completes the proof.

Remark 2.7.Modifying slightly Condition 2.2, Theorem 2.3 can be restated for the eigenfunctione_i,V,i1. Indeed, one should replaceλ_1,V ande_1,V byλ_i,V ande_i,V in Condition 2.2 and use the Lyapunov function

Vi(z):=α(−+V )²^sP_i,Vz²+1− z, e_i,V², z∈S∩H_{(V )}^s ,

whereP_i,V is the orthogonal projection inL²onto the closure of the vector span of{e_k,V}k=i. 2.3. Proof of Theorem 2.4

Let us look for a control in the formσ+u(t ), whereσ∈R\{0}is a constant. Then (1.1) takes the form iz˙= −z+

V (x)+σ Q(x)

z+u(t )Q(x)z,

and the idea of the proof is to show that the set of dipolar momentsQsuch that Condition 2.2 holds forV , Qreplaced byV +σ Q, Qis residual. The proof is divided into three steps.

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Step 1.First let us show that the setQof all functionsWverifying

λ_1,W−λ_j,W=λ_p,W−λ_q,W (2.14)

for all integersj, p, q1 such that{1, j} = {p, q}andj =1 is residual inC^∞(D,R). Fixj, p, q1 and denote by Qj,p,q the set of functionsW∈C^∞(D,R)satisfying (2.14). AsQ=

j,p,qQj,p,q, it suffices to show thatQj,p,qis open and dense. Continuity of the eigenvaluesλk,W fromC^∞(D,R)toR(e.g., see Theorem 8.1.2, [18]) implies that Qj,p,q is open. Let us show thatQj,p,qis dense inC^∞(D,R). Indeed, by [2], the setQof functionsW∈C^∞(D,R) such that the spectrum of the operator−+W is non-degenerate is residual inC^∞(D,R). Take anyW∈Qand P ∈C^∞(D,R). It is well known thatλ_k,W₊_{σ P} ande_k,W₊_{σ P}are analytic inσ in the neighborhood of 0 inR(e.g., see Theorem XII.8, [33]). Differentiating the identity

(−+W+σ P−λ_k,W₊_{σ P})e_k,W₊_{σ P} =0 with respect toσ atσ=0, we get

(−+W−λ_k,W)dek,W+σ P

dσ

σ=0

+

P−dλk,W+σ P

dσ

σ=0

e_k,W =0.

Taking the scalar product of this identity withek,W, we obtain dλk,W+σ P

dσ

σ=0

= P , e_k,W²

. (2.15)

Thus d

dσ(λ_1,W₊_{σ P}−λ_j,W₊_{σ P}−λ_p,W₊_{σ P}+λ_q,W₊_{σ P}) σ=0

= P , e_1,W² −e²_j,W−e_p,W² +e_q,W²

. (2.16)

By Theorem 4.1, the setAof all functionsWsuch that the system{e²_j,W}^∞_j₌₁is rationally independent is residual in C^∞(D,R). Hence the setQ∩Ais residual. Fix anyW∈Q∩Aand takeP ∈C^∞(D,R)such that

P , e²_1,W−e_j,W² −e²_p,W+e²_q,W

=0.

Then (2.16) implies thatW +σ P ∈Qj,p,q for any σ sufficiently close to 0. This proves thatQj,p,q is dense in C^∞(D,R).

Step 2.Here we reduce the proof of the controllability of system (1.1), (1.2) with functionsV andQto that of withV +σ QandQ.

First let us suppose thatV∈C^∞(D,R)is such that the spectrum of the operator−+V is non-degenerate. Take any sequenceσn→0,σn=0. Then the setP1of all functionsQ∈C^∞(D,R)such thatV +σnQ∈Qfor alln1 is residual as a countable intersection of residual sets. On the other hand, the set P2 of functionsQ∈C^∞(D,R) such that Qe1,V, ej,V =0 for allj2 is also residual (see Section 3.4 in [28]). DefineP:=P1∩P2. Let us show that problem (1.1), (1.2) is approximately controllable toe1,V for anyQ∈P. Fix an integern1 and consider the problem

iz˙= −z+V (x)z+σ_nQ(x)z+u(t )Q(x)z, x∈D, (2.17)

z|∂D=0, (2.18)

z(0, x)=z₀(x). (2.19)

LetUtⁿ(·, u)be the resolving operator for problem (2.17), (2.18). Then we haveUtⁿ(·, u)=Ut(·, u+σn). Notice that we cannot apply Theorem 2.3 with functionsV (x)+σnQ(x)andQ. Indeed, Condition 2.2, (i) is not necessarily satisfied. However, asσn→0,Q∈P2and the spectrum of−+V is non-degenerate, for anyN1 there isn^∗1 such that Qe1,V+σ_nQ, ej,V+σ_nQ =0, j =1, . . . , N andnn^∗. Modifying slightly the arguments of the proof of Theorem 2.3, we show that this property is enough to conclude the approximate controllability. We need the following result.

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Lemma 2.8.Fix any constantss >0,d >0 andδ >0, an integerk1and functionsQ∈P andV ∈C^∞(D,R) such that the spectrum of the operator−+V is non-degenerate. For anyM >0andε >0there is an integernˆ1 such that for anynnˆandz₀∈S∩H_(V^s ₊_σ

nQ)with z₀, e_1,V₊_σ_n_Q =0andz₀s< Mwe have U_Tⁿ(z0, u)−e1,V+σ_nQ

s−δ< ε

for some timeT >0and controlu∈C₀^∞((0, T ),R),uC^k([0,T])< d.

To prove Theorem 2.4, letz₀∈S∩H_{(V )}^s be such thatz₀s< M and z₀, e_1,V =0. Asz₀is not necessarily in H_(V^s ₊_σ

nQ), we cannot apply Lemma 2.8. Take anyv∈C^∞([0, η],R)such that ^d_dt^k^vk(0)= ^d_dt^k^vk(η)=0 for allk1 andv(0)=0. By Lemma 2.1, we havey:=Uη(z₀, v)∈H_(V^s ₊_v(η)Q). Ifk1 is sufficiently large andv_C^k₍_[_0,η_]₎ is sufficiently small, thenys < M and y, e_1,V₊_v(η)Q =0. We can choosev such thatv(η)=σ_n for somenn.ˆ Applying Lemma 2.8 for the initial datay, we see that there is a timeT >˜ 0 and a controlu˜ such thatu˜−v(η)∈ C₀^∞((0,T ),˜ R))and

U_T_˜(y,u)˜ −e_1,V₊_v(η)Q

s−δ<ε 2. For sufficiently smallvC^k([0,η])+ηwe have

Uη

U_T_˜(y,u), v(η˜ − ·)

−e_1,V

s−δ< ε.

This proves Theorem 2.4, when the spectrum of the operator−+V is non-degenerate.

To prove the theorem for any functionV ∈C^∞(D,R), notice that, Theorem 2.4 holds for system (1.1), (1.2) with functionsV +σ_nQandQfor any Q∈P andn1. Indeed, from the construction of the setP it follows that the spectrum of −+V +σ_nQ is non-degenerate. Repeating literally the arguments of the previous paragraph with Lemma 2.8 replaced by Theorem 2.4 for functionsV +σ_nQandQ, we complete the proof.

Step 3.To prove Lemma 2.8, letV be defined by (2.1) withV +σnQinstead ofV. As z0, e1,V+σ_nQ =0, we can chooseα >0 so small thatV(z0) <1. Notice that ifV(z) <1,z∈S then z, e1,V+σ_nQ =0.On the other hand, for anyε >0 there is an integerN1 such that ifzsM,z∈S∩H_(V^s ₊_σ

nQ)and z, ej,V+σ_nQ =0 for anyj∈ [2, N], thenz−ce1,V+σ_nQs−δ< εfor somec:=c(z)∈Cwith|c| =1.

We need the following lemma.

Lemma 2.9.There is a finite or countable set J⊂R^∗₊ and an integer nˆ1 such that for any α /∈J,nnˆ and z∈H_(V^s ₊_σ

nQ)with z, e1,V+σ_nQ =0and z, ej,V+σ_nQ =0for some integerj∈ [2, N], there is a timeT >0and a controlu∈C₀^∞((0, T ),R)verifying

V

UTⁿ(z, u)

<V(z).

Proof. AsQ∈P, for sufficiently largenˆwe have Qe1,V+σ_nQ, ej,V+σ_nQ =0 forj =2, . . . , Nandnn. Repeatingˆ the arguments of the proof of Proposition 2.6, one should just notice that if sum (2.13) equals zero for allτ0, then

z0, ej,V+σ_nQ =0 for allj∈ [2, N]. This contradicts the hypothesis of the lemma. 2 As in the proof of Theorem 2.3, we define the set

K=

z∈H_(V^s ₊_σ

nQ): U_Tⁿ(z₀, u)→zinH^s⁻^δfor someT0, u∈C₀^∞

(0, T),R

,uC^k([0,T])< dand for anyδ >0 . Let

m:= inf

z∈KV(z).

This infimum is attained at some pointe∈K. Let us show that e−ce_1,V₊_σ_n_Qs−δ< ε

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for some c:=c(z)∈ C with |c| =1. Indeed, it follows from the choice of α that V(e)V(z₀) <1. Hence e, e_1,V₊_σ_n_Q =0. Suppose that e, e_j,V₊_σ_n_Q =0 for some integerj∈ [2, N]. By Lemma 2.9, there is a timeT >0 and a controlu∈C₀^∞((0, T ),R)such that

V

U_Tⁿ(e, u)

<V(e). (2.20)

Defineu˜(t )=u(t ),t∈ [0, T]andu˜(t )=u(t ),t∈ [T, T+T]. Thenu˜∈C₀^∞((0, T+T ),R)and U_Tⁿ₊_T(z0,u˜)→U_Tⁿ(e, u) inH^s⁻^δ.

This implies thatU_Tⁿ(e, u)∈K. Clearly, (2.20) contradicts the definition ofe. It follows that e, e_j,V₊_σ_n_Q =0 for any j∈ [2, N], hencee−ce_1,V₊_σ_n_Qs−δ< εfor somec:=c(e)∈C,|c| =1. Without loss of generality, we can suppose thatc=1.

3. Applications

3.1. Global exact controllability

The following controllability result for system (1.1), (1.2) is obtained by Beauchard [8] in the case m=1 and V =0.

Theorem 3.1.There is a residual setQinW^3,^∞((0,1),R)such that for anyQ∈Qand some constantsT >0and ε >0the following exact controllability property holds:for anyz0, z1∈S∩H₍₀₎⁵ with

z_i−e_1,05< ε, i=1,2

there is a controlu∈H₀¹([0, T],R)such that UT(z₀, u)=z₁.

On the other hand, by Theorem 2.4, problem (1.1), (1.2) withm=1 andV =0 is approximately controllable to e_1,0inH⁵⁺^δ generically with respect toQinC^∞([0,1],R). Literally the same proof shows that in the case of the phase spaceH⁵⁺^δ the approximate controllability property holds generically with respect toQinW^3,^∞((0,1),R).

Thus, combining Theorems 2.4 and 3.1, we obtain.

Theorem 3.2.There is a residual setQinW^3,^∞((0,1),R)such that for anyQ∈Qand anyz₀, z₁∈S∩H₍₀₎⁵⁺^δthere is a timeT >0and a controlu∈H₀¹([0, T],R)verifying

UT(z₀, u)=z₁.

Remark 3.3.It is shown in [7,11] that if we takeQ(x)=x, then for anyN1 there is a constantσ^∗>0 such that for anyσ∈(0, σ^∗)we have

(i) xe1,σ x, e_{j,σ x} =0 for allj2,

(ii) λ_{1,σ x}−λ_{j,σ x}=λ_{p,σ x}−λ_{q,σ x} for allj, p, q1 such that{1, j} = {p, q}and 2j N.

These properties imply that the proof of Theorem 2.4 works and the system withQ(x)=xis approximately controllable. Indeed, properties (i) and (ii) are sufficient to conclude that in the proof of Lemma 2.9 sum (2.13) is not equal to zero for allτ0. This is the only place, where the hypotheses on the dipolar momentQare used.

On the other hand, by Beauchard [7], forQ(x)=x the problem is exactly controllable in a neighborhood ofe_1,0. Thus global exact controllability property holds forQ(x)=x.

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3.2. Uniqueness of stationary measure

Let us consider the Schrödinger equation with a potential which has a random time-dependent amplitude:

iz˙= −z+V (x)z+β(t )Q(x)z, x∈D, (3.1)

z|∂D=0, (3.2)

z(0)=z₀, (3.3)

whereV , Q∈C^∞(D,R)are given functions. We assume thatβ(t )is a random process of the form β(t )=

+∞

k=0

Ik(t )ηk(t−k), t0, (3.4)

whereI_k(·)is the indicator function of the interval[k, k+1)andη_k are independent identically distributed random variables inL²([0,1],R). ThenUk(·, β)is a homogeneous Markov chain with respect to the filtrationFk generated byη0, . . . , ηk−1 (e.g., see [29]). For anyz∈S andΓ ∈B(S), the transition functions corresponding to the process Uk(·, β)are defined byP_k(z, Γ )=P{Uk(z, β)∈Γ}and the Markov operators by

Pkf (z)=

S

Pk(z,dv)f (v), P^∗_kμ(Γ )=

S

Pk(v, Γ )μ(dv),

wheref∈C_b(S)andμ∈P(S).Let us recall that a measureμ∈P(S)is stationary for (3.1), (3.2), (3.4) ifP^∗₁μ=μ.

The question of existence of a stationary measure is an open problem. In this section, we derive the uniqueness from Theorem 2.3. We need the following condition.

Condition 3.4.The random variablesη_khave the form η_k(t )=^∞

j=1

b_jξ_{j k}g_j(t ), t∈ [0,1],

where{g_j}is an orthonormal basis inL²([0,1],R),b_j>0 are constants with ∞

j=1

b²_j<∞,

andξ_{j k} are independent real-valued random variables such thatEξ_{j k}² =1. Moreover, the distribution ofξ_{j k} possesses a continuous densityρ_j with respect to the Lebesgue measure andρ_j(r) >0 for allr∈R.

Theorem 3.5.Under Conditions2.2and3.4, problem(3.1), (3.2), (3.4)has at most one stationary measure onS.

This theorem is derived from the following general result (cf. [22]). Let X be a Polish space and let P_k(z, Γ ) be a Markov transition function satisfying the Feller property. We denote byP_k andP^∗_k the corresponding Markov semigroups. Recall that a stationary measureμforP^∗_k is said to be ergodic if

σn(f ):=1 n

n−1

i=0

Pif (z)→(f, μ) (3.5)

for anyf ∈C_b(X)and forμ-a.e.z∈X, where(f, μ)=

Xf (z)μ(dz).

Theorem 3.6.Suppose thatP_k satisfies the following two conditions.

(i) For anyf ∈L(X)there is a constantL_f >0such thatP_kf isL_f-Lipschitz for anyk0.

(ii) For any pointz∈Xand any ballB⊂Xthere isl1such thatP_l(z, B) >0.

ThenP^∗_khas at most one stationary distribution.

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Proof of Theorem 3.5. Let us show that properties (i) and (ii) are verified for system (3.1), (3.2), (3.4). Take any functionf∈L(S). Then

P_kf (z₁)−P_kf (z₂)=E f

Uk(z₁, β)

−f

Uk(z₂, β)

f_LEUk(z₁, β)−Uk(z₂, β)= f_Lz₁−z₂, which implies (i). To show (ii), notice that Condition 3.4 implies that

P

u−β_L²₍_[0,l])< ε

>0

for anyu∈L²([0, l])andε >0. Moreover, using the continuity of the mappingUl(z₀,·):L²([0, l])→L²(D), for anyδ >0 we can find a constantε >0 such that

PUl(z₀, β)−Ul(z₀, u)< δ

P

u−βL²([0,l])< ε

>0.

Hence, any point Ul(z₀, u), u∈ L²([0, l]) is in the support of the measure D(Ul(z₀, β))=P_l(z₀,·). By Theo- rem 2.3, problem (1.1), (1.2) is approximately controllable inS(cf. Theorem 3.5 in [28]), hence the set{Ul(z₀, u):

u∈L²([0, l]), l0}is dense inS. This implies (ii). Applying Theorem 3.6, we complete the proof. 2

Proof of Theorem 3.6. In view of ergodic decomposition of stationary distributions (e.g., see [21]), it suffices to prove that there is at most one ergodic stationary measure. Letμ1andμ2be two ergodic stationary measures. Suppose there is a functionf ∈L(X)such that(f, μ1)=(f, μ2). LetXi, i=1,2, be the set of convergence in (i) withμ=μi. Thenμi(Xi)=1 andX1∩X2= ∅. Furthermore, in view of condition (ii), for any ballB⊂Xthere isl1 such that

μ_i(B)=

X

P_l(z, B)μ_i(dz) >0.

Thus suppμi=X, and thereforeXi =X. Now letKn⊂Xbe an increasing sequence of compact subsets such that μi(Kn) >1−2⁻ⁿ. Then, by condition (i) and the Arzelà theorem, there is a subsequencekj → ∞such that for any n1 the sequenceσ_k_j(f )converges uniformly onK_nto anL_f-Lipschitz bounded functionf_n. Let us setY =

nK_n and define anL_f-Lipschitz functionf :Y →Rsuch thatf|Kn=f_n. Sinceμ_i(Y )=1, we see thatμ_i(Y∩X_i)=1 andY∩Xi =X, where we used again condition (ii). We conclude thatf must be a constant function onXequal to (f, μ_i). The contradiction obtained shows thatμ₁=μ₂. 2

4. Independence of squares of eigenfunctions

Recall that the functions{fj}^∞_j₌₁⊂C(D)are said to be rationally independent, if for anyN1 andαk∈Q, k=1, . . . , Nwith|α1| + · · · + |αN|>0 we have

N

k=1

α_kf_k≡0.

Theorem 4.1.The setAof all functionsQ∈C^∞(D,R)such that the system{e²_j,Q}^∞_j₌₁is rationally independent is residual inC^∞(D,R).

Proof. The proof of this theorem is inspired by the paper [31] by Privat and Sigalotti, where the linear independence of the squares of the eigenfunctions of the Dirichlet Laplacian is established to hold generically with respect to the domainD.

Take anyN1 andαk∈Q, k=1, . . . , N, with|α1| + · · · + |αN|>0. It suffices to show that the setAα,N of all functionsQ∈C^∞(D,R)such that the eigenvaluesλj,Q,j=1, . . . , N, are simple and

N

k=1

α_ke²_k,Q≡0