Semi-classical observation suffices for observability: wave and Schr\"odinger equations

For the wave and the Schr\"odinger equations we show how observability can be deduced from the observability of solutions localized in frequency according to a dyadic scale.


Waves and observability
On a bounded smooth open set Ω of R d , consider the operator A = −∆ = − 1≤j≤d ∂ 2 j .The associated wave equation in the case of homogeneous Dirichlet boundary conditions is (1.1) 1.1.Strong and weak solutions.For u 0 ∈ H 2 (Ω) ∩ H 1 0 (Ω) and u 1 ∈ H 1 0 (Ω), there exists a unique solution to (1.1).Such a solution is called a strong solution as (∂ 2 t + A) u = 0 holds in L 2 loc R; L 2 (Ω) .One denotes by the (strong) energy of u at time t.Since the equation (1.1) is homogeneous this energy is independent of time t that is, H 1 0 (Ω) .One thus simply writes E 2 (u).In particular this conservation of the energy states the continuity of the map For less regular intitial data one uses a notion of weak solution.For instance, if u 0 ∈ H 1 0 (Ω) and u 1 ∈ L 2 (Ω), there exists a unique that is a weak solution of (1.1), meaning u |t=0 = u 0 and ∂ t u |t=0 = u 1 and (∂ 2 t + A)u = 0 holds in D ′ R × Ω .For such a solution one considers the following energy independent of time t as above, that is, With the density of H 2 (Ω) ∩ H 1 0 (Ω) in H 1 0 (Ω) and of H 1 0 (Ω) in L 2 (Ω), one can approach u 0 and u 1 by smoother data, and thus approach u(t) by strong solutions.
1.2.Observation operator, admissiblity and observability.An observation operator is an operator L on L 2 (Ω), possibly unbounded, with values in a Hilbert space K. Basis examples in the framework of the present introduction are the following ones.The observation operator is said to satisfy an admissibility condition if an estimate of the following form holds for some S > 0, C > 0 and an energy level j = 1 or 2 (other energy levels are considered in the abstract development in what follows).For example, let us assume here that j = 2, that is, admissibility is given at the level of strong solutions.One says that observability holds with the operator L in time T > 0 if one has with ℓ = 1 or 2, for some C obs > 0 for any strong solution to wave equation.If ℓ = 1 one says that observability holds with some loss of derivative, or some loss of energy, here, a loss of one energy level.
Observability estimates are important in applications such as inverse problems or controllability issues.In particular, for waves, observability is equivalent to exact controllability; see e.g.[8].For more aspects on admissibility, observability and their connections with controllability, we refer the reader to the book of M. Tucsnak et G. Weiss [21].1.3.Derivation of an observability estimate.There are various methods to derive observability estimates for the wave equation.Some rely on a multiplier approach going back to the seminal work of J.-L.Lions [18].Others rely on microlocal methods following the celebrated article of C. Bardos, G. Lebeau, and J. Rauch [2].
The purpose of the present article is not the derivation of observability per se.We are rather interested in showing that observability, be it with energy loss or not, can be deduced from the observation of very particular types of waves.The waves we shall consider are localized in a frequency band making them easier to handle than general waves (in particular when applying microlocal techniques).The frequency band is indexed by an integer k and ranges from αρ |k| to ρ |k| /α for 0 < α < 1 and some ρ > 1.This framework is given a semi-classical aspect by using the small parameter h k = ρ −|k| .
If u k denotes a wave localized in frequency as described above, a very pleasant property is that u k fufills the half-wave equation (∂ t − sgn(k)iA 1/2 )u = 0. (1.3)This can greatly simplify the analysis necessary for the derivation of an observation inequality as compared to treating all solutions to the wave equation.Also, the frequency localization of u k allows one to use powerful tools from semi-classical analysis that are often easier to handle that the analogous tools from microlocal analysis.The use of such tools can allow one to treat the case of coefficients with limited regularity; see for instance [5] for this last point.Having in mind the analysis of the HUM control operator carried out in [9] the introduction of waves with frequencies limited to a narrow band is very natural.In [9], the authors show that the control operator acts microlocally with a highly separated treatment of frequency bands similar to those considered here.
The starting point of the present article is to assume that a uniform observability estimate holds for frequency localized waves like u k (t), that is, for some C obs > 0 one has for all k sufficiently large.Our main result, under a unique continuation property to be described below, is the derivation from (1.4) of the observability inequality for general waves u(t) in the considered energy level for any T ′ > T and some C ′ obs > 0. We shall also show that an admissibility condition can be used to give the proper energy level where this inequality holds.
To allow for a general use of this result, we present it in a general abstract framework.1.4.Schrödinger equation.In the same geometrical setting as above, the Schrödinger equation, in the case of Dirichlet boundary conditions reads in Ω. (1.5) is independant of t.As for the wave equation, other levels of regularity are possible.If and the norm u(t) L 2 (Ω) remains constant.
For an observation opertator as above, observability takes the form here at the regularity given by D(A ℓ ), for ℓ = 0, 1/2 or 1 in the above levels of solutions.As for the wave equation, under a unique continuation property, we shall derive such an observability inequality from a similar inequality holding for solutions localized in frequency.
The Schrödinger equation can be seen sometimes as a half-wave equation; compare (1.5) and (1.3).With respect to the analysis we carry out in the present paper, this comparison is very relevant and the analysis is more involved for the wave equation.In what follows, we shall thus cover the wave equation first and cover the case of the Schrödinger on a second pass, yet with all necessary details. 1.5.Other settings.In this introductory section we have concentrated our attention on the case of the wave and the Schrödinger equations stated on a bounded smooth open set Ω of R d , along with homogeneous Dirichlet boundary conditions, that is, Bu = 0 with Bu = u |∂Ω .This is done for the purpose of motivation.However, the abstract framework we present in what follows allows one to consider more general settings.We give a nonexhaustive list of such settings.
(1) One can consider the elliptic operator A to be the Lapace-Beltrami (up to principal part with the requirement that A be selfadjoint and nonnegative) on a smooth Riemannian manifold M without boundary.If viewed as an unbounded operator on L 2 (M), one sees that 0 is an eigenvalue associated with constant functions.Considering the operator acting on L

Abstract equations and semi-classical reduction
Let E be a Hilbert space.Consider a positive unbounded selfadjoint operator A on E with dense domain D(A).Assume that there exists a real Hibert basis (e ν ) ν∈N of E, associated with a nondecreasing sequence of eigenvalues, (λ ν ) ν∈N , with λ ν → +∞ as ν → +∞, for instance if A has a compact resolvent map.In the example of the introduction, one has E = L 2 (Ω) and For s ≥ 0 one has For s < 0, D(A s ) denotes the dual of D(A |s| ) using E as a pivot space, and if u ∈ D(A s ) then u = ν∈N u ν e ν with convergence for the natural dual norm on D(A s ) and A s u = ν∈N λ s ν u ν e ν ∈ E. In all cases, a norm on D(A s ) is given by . One has the continuous and dense injection 2.1.Abstract wave equation and energy levels.The wave equation reads With the initial conditions u 0 ∈ E = D(A 0 ) and u 1 ∈ D(A −1/2 ), the unique solution to (2.1) in C 0 (R; E) ∩ C 1 R; D(A −1/2 ) is given by ).Note that (u ± ν ) ν∈N ∈ ℓ 2 (C).In turn the r.h.s. of (2.2) is solution to the wave equation (2.1) with u 0 and u 1 given by ) .The energy of the solution is given by It is constant with respect to t, that is, We thus simply write E 0 (u) and one has for any time inteval [t 1 , t 2 ], leading to a well defined energy if only considering the solution u in L 2 loc (R; ) .More generally, if s ∈ R and u 0 ∈ D(A s/2 ) and u 1 ∈ D(A (s−1)/2 ), the unique solution to (2.1) ) is given by (2.2), and one can define the energy that is also constant with respect to t.Note that if u(t) is such a solution then ) as above, with We shall say that such a solution to the wave equation lies in the s-energy level.Similarly to (2.5) one has If u 0 , u 1 ∈ D(A ∞ ) the associated solution u(t) is such that u ∈ C k (R; A s ) for any k ∈ N and s ∈ R. One has E s (u) < ∞ and one says that u(t) lies in all energy levels.If u 0 ∈ D(A ℓ/2 ) and u 1 ∈ D(A (ℓ−1)/2 ) and if one denotes by u(t) the unique solution to the wave equation (2.1) that lies in the ℓ-energy level, there exists a sequence u n (t) of solutions that lie in all energy levels and such that ) → 0 and let u n (t) be the associated solution to the wave equation.

2.2.
Dyadic decomposition for waves.Let 0 < α < 1, ̺ ∈]1, 1/α[ and set Note that #J k < ∞ from the assumed properties of the eigenvalues.Set also At this stage it is important to note that J −k = J k implying E −k = E k .However, we shall identify u ∈ E k with the following solution of the wave equation (2.9) The sign of k here becomes important.Yet, note that u ∈ E k if and only if ū ∈ E −k through this identification since the eigenfunctions e ν are assumed real.
Following up, we identify as h 2 k λ ν 1 for ν ∈ J k .For u ∈ E k , the identified solution to the wave equation given in (2.9) lies in all energy level.One has In particular, note that for u ∈ E k both terms in the energy coincide; this is not the case in general for a solution of the wave equation for fixed time (while it is true in time average).
The reason is that u ∈ E k is in fact solution to the following half-wave equation We introduce the following sets of sequences of functions

Abstract Schrödinger equation and dyadic decomposition.
The Schrödinger equation associated with the operator A reads With the initial conditions u 0 ∈ D(A p ), for some p ∈ R, the unique solution to (2.11) As above let 0 Note that #J S k < ∞ from the assumed properties of the eigenvalues.Introduce We shall identify u ∈ E k with the following solution to the Schrödinger equation The counterpart to Lemma 2.1 is the following lemma.Lemma 2.2.For u ∈ E S k , and r ∈ N and s ∈ R the norm We introduce the following set of sequences of functions

3.1.
Observation operator and unique continuation assumption.For some Hilbert space K consider an observation operator L : E → K, possibly unbounded, with domain given by D(L) = D(A m 0 ) for some m 0 ∈ R, with We introduce the following assumption.
Observe that an eigenvector of A lies in D(A ∞ ) and thus lies in D(L).

3.2.
From semi-classical observation to observability for waves.Our starting point will be the following property.

Semi-classical observability property (wave equation).
For some ℓ 1 ∈ R, C > 0, k 0 ∈ N and some T > 0 one has Our main result in the case of the wave equation is the following theorem.
Assume that there exists C > 0, k 0 > 0, and T > 0 such that (3.2) holds for any U = (u k ) k∈Z ∈ B and any |k| ≥ k 0 .Under the unique continuation Assumption 3.1, for any T ′ > T there exists C ′ > 0 such that for any ) the solution to (2.1) given by (2.2) satisfies Hence, semi-classical observation on a interval of length T implies classical observation on any interval of greater length.
Note that the r.h.s. in (3.3) makes sense because of (3.1) and u(t) ∈ L 2 loc R; D(A m 0 ) .Note that the requirement ℓ 1 ≤ 2m 0 is natural since u(t) lies in the (2m 0 )-energy level.
Remark 3.3.Let ū denote the complex conjugate.In many cases one has . Consequently, if (3.4) does hold, then assuming (3.5) suffices to reach the conclusion of Theorem 3.2.
In the case ℓ 1 = 2m 0 , the argument we develop leading to a proof of Theorem 3.2 is based on [17] (see also [4]), yet with more details provided here.The proof of Theorem 3.2 in this first case is given in Section 5.1.The argument is further refined to treat the case ℓ 1 < 2m 0 , that is, the case of an observability estimate with some energy loss.The proof of Theorem 3.2 in this second case is carried out in Section 5.3.Even though the second case contains the first one, we chose to provide a simpler proof in the first case for the benefit of the reader.

3.3.
Admissibility condition for waves.In the introduction we also considered admissibility conditions.Such conditions are usefull in cases where L u makes sense in energy levels lower than 2m 0 .Note that the (2m 0 )-level is given by the boundedness of L on D(A m 0 ); see (3.1).Yet, since L u(t) K appears in a time-integrated form in the sought observability estimates, in some cases, one can expect some improvement as formulated with the following additional assumption.Assumption 3.4 (admissibility condition for waves at the ℓ 0 -energy level).For some ℓ 0 ≤ 2m 0 , the operator L extends as an unbounded operator from the subspace of L 2 loc (R; D(A ℓ 0 /2 )) into L 2 loc (R; K), also denoted by L, and for some S > 0 and C S > 0 one has ).In other words, Assumption 3.4 states that L is bounded from the space of solutions that lie in the ℓ 0 -energy level into L 2 (0, S; K).Considering only ℓ 0 ≤ 2m 0 is natural since (3.6) holds for ℓ 0 = 2m 0 by (3.1).
. One thus has ∇u ∈ C 0 R; L 2 (Ω) , a regularity too low to allow one to apply the trace theorem to define ∂ n u |∂Ω = (n • ∇u) |∂Ω .However, because of the so-called hidden regularity for such a solution to the wave equation, one finds that the trace ∂ n u |∂Ω makes sense and lies in L 2 loc (R; L 2 (∂Ω)); see for example [14].A weak solution lies in the 1-energy level we have defined and moreover one has, for any S > 0, In this case, one has 1 = ℓ 0 < 2m 0 = 3/2 + ε.
From the time invariance of the energy with (3.6) one finds for any interval J of length |J| = S.Moreover, for any bounded interval I one has for some C |I| > 0 only function of |I|.
With Assumption 3.4 one obtains the following corollary to Theorem 3.2.
Corollary 3.6.Let ℓ 1 ≤ ℓ 0 ≤ 2m 0 .Assume that there exists C > 0, k 0 > 0, and T > 0 such that (3.2) holds for any U = (u k ) k∈N ∈ B + and any k ≥ k 0 .Assume also that (3.4) holds.Under the unique continuation Assumption 3.1 and the admissibility Assumption 3.4 , for any T ′ > T there exists C ′ > 0 such that for any (u 0 , u 1 ) ∈ D(A ℓ 0 /2 ) × D(A (ℓ 0 −1)/2 ) the solution to (2.1) given by (2.2) satisfies The proof simply uses the density of solutions in the (2m 0 )-energy level in the space of solution in the ℓ 0 -energy level and that both sides of the inequality (3.8) are continuous with respect to the ℓ 0 -energy; continuity of the r.h.s. is precisely (3.7) that follows from Assumption 3.4.
A remark similar to Remark 3.3 can be made for the result of Corollary 3.6.

3.4.
Main result for the Schrödinger equation.We first state what is meant by semiclassical observability in the case of the Schrödinger equation.
Semi-classical observability property (Schrödinger equation).For some p 1 ∈ R, C > 0, k 0 ∈ N and some T > 0 one has Our main result in the case of a the Schrödinger equation is the following theorem.
Theorem 3.7.Let p 1 ≤ m 0 .Assume that there exists C > 0, k 0 > 0, and T > 0 such that (3.9) holds for any U = (u k ) k∈N ∈ B S and any k ≥ k 0 .Under the unique continuation Assumption 3.1, for any T ′ > T there exists C ′ > 0 such that for any u 0 ∈ D(A m 0 ) the solution to (2.11) given by (2.12) satisfies Recall that m 0 is as given by the continuity property (3.1) for L.
Similarly to waves an admissibility assumption reads as follows.
Assumption 3.8 (admissibility condition the Schrödinger equation in D(A p 0 )).For some p 0 ≤ m 0 , the operator L extends as an unbounded operator from the subspace of L 2 loc (R; D(A p 0 )) into L 2 loc (R; K), also denoted by L, and for some S > 0 and C S > 0 one has

11). Then, for any bounded interval I one has
for some C |I| > 0 only function of |I|.
With Assumption 3.8 one obtains the following corollary to Theorem 3.7.
Corollary 3.9.Let p 1 ≤ p 0 ≤ m 0 .Assume that there exists C > 0, k 0 > 0, and T > 0 such that (3.9) holds for any U = (u k ) k∈N ∈ B S and any k ≥ k 0 .Under the unique continuation Assumption 3.1 and the admissibility Assumption 3.8, for any T ′ > T there exists C ′ > 0 such that for any u 0 ∈ D(A p 0 ) the solution to (2.11) given by (2.12) satisfies 3.5.Existing and potential applications.In the introduction, we considered the wave equation on an open set of R d .This can be generalized to the manifold setting.Consider a compact connected Riemannian manifold M of dimension d with boundary endowed with a metric g = (g ij ).Introduce the elliptic operator where κ is a positive function on M. The metric g and the function κ can be assumed C k with k ≥ 1 or Lipschitz.The operator A is unbounded on E = L 2 (M).With the domain D(A) = H 2 (M)∩H 1 0 (M) one finds that A is selfadjoint, with respect to the L 2 -inner product, and A is negative.With the elliptic operator A one also defines the wave operator (3.14) P = P κ,g = ∂ 2 t − A κ,g , and one can consider the associated homogeneous wave equation For an open set ω ⊂ M one can consider the observation operator with K = L 2 (Ω) and the action on a solution to the wave equation given by For an open set Γ ⊂ ∂M one can consider the observation operator with K = L 2 (Γ) and the action on a solution to the wave equation given by where ∂ n is normal derivative at the boundary.In both cases the admissibility Assumption 3.4 holds as one has for a weak solution and for some S > 0. The second property is in fact the so-called hidden regularity property of waves; see e.g.[14].In both cases Assumption 3.1 holds with classical unique continuation results for elliptic operators; see for instance [11,Theorem 2.4] and [15,Theorems 5.11 and 5.13].The result of Corollary 3.6 thus applies.It is used without loss of energy, that is, in the case ℓ 1 = 1, in [4] for a boundary observation in the case of C 2coefficients and in [5] for both types of observations in the case of C 1 -coefficients with also result for Lipschitz coefficients by a perturbation argument.In these references, powerful tools of semi-classical analysis and semi-classical measures are key to prove a semi-classical observability estimate as in (3.2).
Here, we also treat the case of the loss of derivatives, that is, if ℓ 1 < ℓ 0 in the assumed semiclassical observability estimate (3.2) and in the resulting observability estimate in Theorem 3.2 and Corollary 3.6.Estimates with such losses can be found in the literature.We refer for instance to the work of F. Fanelli and E. Zuazua [10].Their result is in the case of very rough coefficients (log-Lipschitz) and only concerns the wave equation in one space dimension.Results in higher dimensions are open to our knowledge and the study of such cases could benefit from the use of simpler localized-in-frequency waves and their semi-classical setting.Though observation estimates with loss of derivatives are not so common for waves, they appear quite naturally for Schrödinger equations.See [3,6] for such results in the presence of weak (hyperbolic) trapping, or [7, Sections 6.4 & 6.5] and [19,Section 4] for the observability of Schrödinger on the square with an observation in (say) the vertical boundary.

Time microlocalization
Let H be a Hilbert space, It also maps S (R; H) (resp.S ′ (R; H)) into itself.We shall choose F according to the following lemma.The difference between the two bounds is 2 ln(a −1 )/ ln(ρ) > 2 and one has ln(|τ |)+ln(a −1 ) ln(ρ) Let T > 0. For j ∈ Z set Define H H as the space of functions w ∈ L 2 loc (R; H) such that w H H := sup that is, the space of uniformly locally L 2 -bounded functions with values in H. One Proof.Let w ∈ H H and set w j = 1 I j w.One has For the first point of the lemma we treat the case k > 0. The case k < 0 can be treated similarly.Consider φ ∈ C ∞ c (R) and j ∈ Z such that dist(supp(φ), I j ) ≥ γ > 0. Using that −i t − s ∂ τ e i(t−s)τ = e i(t−s)τ , for t = s, with N integrations by parts one writes One finds and that (4.3) holds.One also concludes that the estimate (4.4) holds using also (4.6) for a finite numbers of terms.
Proof.Observe that the series in (2.2) that defines u(t) converges in the space H D(A ℓ/2 ) .Hence, with the first point in Lemma 4.2 one finds One has F k (h k D t )e irt = F k (h k r)e irt , r ∈ R; see for instance (18.1.27)in [12].This gives This gives the result using the dependency of the support of F k upon the sign of k.

Because of the form of u
is also solution to the wave equation.Yet, as the sum is finite in (4.8) one has that is, the wave u k (t) lies in all energy levels.
We now consider the particular case of a solution u(t) that lies in the (2m 0 )-energy level with m 0 as appearing in the continuity property (3.1) of L. Lemma 4.4.Let u(t) be a solution to (2.1) that lies in the m 0 -energy level.One has One has yielding in turn by Lemma 4.2.As the sum defining U n is finite one has using the support property of F k for k > 0 and that F k (h k D t )e irt = F k (h k r)e irt , r ∈ R; see for instance (18.1.27)in [12].One observes that F K k (h k D t ) L U n = L u k for n chosen sufficiently large.The limit in (4.10) hence gives the result.
Lemma 4.6.Let u(t) be a solution to (2.11) The proof of Lemmata 4.3 and 4.4 can be adapted mutatis mutandis.
One sees that u k (t) is also solution to the Schrödinger equation and as the sum is finite in (4.11).

Proof of the main result for waves
As explained below Theorem 3.2, for the benefit of the reader, we have chosen to provide a proof for the case ℓ 1 = 2m 0 and a proof for the case ℓ 1 ≤ 2m 0 .Even though the second case contains the first one, the proof is the first case is less technical.
and we aim to prove that, for any δ ∈]0, δ 0 ], holds for any solution u to the wave equation (2.1) writen in (2.2) that lies in the ℓ 1 -energy level, that is, u 0 ∈ D(A m 0 ) and u 1 ∈ D(A m 0 −1/2 ) here.The simultaneous treatment of 0 < δ ≤ δ 0 is used for a technical argument in the proof of Lemma 5.1 below.For such a solution u, one notes that L u ∈ H K by (3.7) and one has (5.3) One has u k ∈ E k .With the semi-classical observation property (5.1) one has (5.4) One thus obtains with (5.4) for k 1 ≥ k 0 to be chosen below.
With Lemma 4.4, one can write With the third point of Lemma 4.2 and (5.2) one has using that h k = ρ −|k| with ρ > 1.With (5.6) one finds For k 1 ≥ k 0 chosen sufficiently large one obtains To remove the first term on the r.h.s. of (5.7) we shall use the following lemma that states that only the trivial solution is invisible for the observation operator L. Lemma 5.1 (absence of invisible waves).Let u ∈ ∩ k C k R; D(A m 0 −k/2 ) be solution to (2.1) and such that ψ δ L u = 0. Then u = 0.
Proof.For 0 < δ ≤ δ 0 as above, set that is, the space of invisible solutions in the sense of the observation operator ψ δ L. We equip N δ with the norm associated with the energy E ℓ 1 .With (5.7) one has As the maps u → u k have a finite rank, they are compact.With (5.8) it follows that N δ has a compact unit ball and is thus finite dimensional by the Riesz theorem.
We claim that The finite dimensional space N δ is thus stable under the action of the operator ∂ t .Consequently this operator has an eigenvector v ∈ N δ with associated eigenvalue µ.One finds Av = ∂ 2 t v = µ 2 v meaning that v(t) is an eigenfunction for A for all t ∈ R. As L v(t) = 0 if t ∈] − δ, T + δ[, with the unique continuation Assumption 3.1 one obtains v(t) = 0 for all t ∈] − δ, T + δ[.Hence, v = 0 since the energy of this solution is zero and one concludes that N δ = {0}.
We now prove our claim (5.9).Let 0 < δ ′ < δ and note that On the other hand, if one applies the operator F ) , for any r, m ∈ N and any bounded interval J. Hence, recalling that w k ε and ∂ t u k are solutions to the wave equation, one finds w k ε → ∂ t u k in the norm associated with the ℓ 1 -energy.With (5.8) one finds that (w ε ) ε is of Cauchy type in N δ ′ for this latter norm, as ε → 0. It thus converges to some w ∈ N δ ′ , as N δ ′ is complete since finite dimensional.Then, one has uniformly for t ∈ any bounded interval of R, Iterating the argument, one obtains We now conclude the proof of Theorem 3.2 by a classical argument by contradiction, assuming that the observation inequality does not hold.Then, there exists a sequence of initial conditions (u 0,n , u Some subsequence, that we also write (u 0,n , u 1,n ) for simplicity, weakly converges to some (u 0 , u 1 ) ∈ D(A m 0 ) × D(A m 0 −1/2 ).Associated with (u 0 , u 1 ) is a solution u, also in the ℓ 1 -energy level, and u n converges weakly to u in L 2 −δ, T+δ; D(A m 0 ) ∩ H 1 − δ, T + δ; D(A m 0 −1/2 ) .Moreover one has ψ δ L u = 0.In fact, one considers L : where v is the linear wave with initial conditions v 0 and v 1 as given by (2.2).With (5.2) the map L is continuous.It is thus also continuous for the weak topologies; see for instance [20,Proposition 35.8].Since L(u 0,n , u 1,n ) converges strongly to 0, and thus also weakly, this gives L(u 0 , u 1 ) = 0, that is, ψ δ L u = 0.With lemma 5.1 one concludes that u = 0, and thus u 0 = u 1 = 0.As above, for a linear wave v with initial conditions (v is compact since with a finite dimensional range; see the expression in Lemma 4.3.As one has (u 0 n , u 1 n ) ⇀ (0, 0), one obtains that u k n converges strongly to 0 in the norm given by the E ℓ 1 -energy, for 0 ≤ |k| < k 1 .Here, one thus obtains Estimate (5.7) applied to u n thus leads to a contradiction since both terms on the r.h.s.converge to zero and the l.h.s. is equal to 1.This concludes the contradiction argument and the proof of Theorem 3.2 in the case ℓ 1 = 2m 0 .

Refined time-microlocalization estimates.
Here, we consider a solution u(t) to the abstract wave equation (2.1) with for some m ∈ R.Then, u(t) lies in the (2m)-energy level and ) such that F = 1 in a neighborhood of supp(F ).A first result we shall use is the following one. (5.12) The definition of the Fourier multiplier Combined with the third item of Lemma 4.2 one has the following corollary.
Proof of Lemma 5.2.We consider the case k > 0. The case k < 0 is treated similarly.With Lemma 4.3 one has By Lemma 2.1 one has using that F k is a bounded function since compactly supported.This gives ũk (t) The result follows from the definition of .H D(A m ) in (4.2).
A second important result is given by the following lemma.
Note that here one assumes the function ψ to be smooth as opposed to the results in Lemma 4.2 and Corollary 5.3.In fact, the proof of Lemma 5.4 is based on a kernel regularization argument that requires smoothness of the function ψ.
Proof.As in other proofs we treat the case k > 0. The case k < 0 can be treated similarly.
With Lemma 4.2 the proof is clear in the case ℓ ≥ 2m.We shall thus only consider the case ℓ < 2m.Let r ∈ N be such that r ≥ m − ℓ/2 > 0.
With u solution to the wave equation one has It is also a solution to the wave equation.One has One thus considers the action of the operator t , on w.Note that P maps S ′ R; D(A m ) into itself.Thus the action of P on j∈Z w j yields j∈Z P w j with convergence in S ′ R; D(A m ) .Recall that I j = [jT, (j + 1)T[ and w j = 1 I j w.The kernel of this operator is given by the following oscillatory integral References on the subject of oscillatory integrals are [13,1,15].In particular, usual operations such as integrations by parts are licit.Since supp(F k ) ∩ supp(1 − Fk ) = ∅ one has τ ′ = τ in the integrand.In fact one has the following estimation.A proof of Lemma 5.5 is given below.
, N integrations by parts give This is the step of the proof where smoothness of the function ψ is used.
Using N ≥ 2r + 2 and Lemma 5.5 with this form of the kernel of P a first estimate one can write is the following (5.17) For j such that dist(supp(ψ), I j ) > 0, we can proceed as in the proof of Lemma 4.2.Set G k (σ) = σ 2r (1 − Fk )(σ).One has Set γ = dist(supp(ψ), I j ).One has −i t ′ −s ∂ τ e i(t ′ −s)τ = e i(t−t ′ )τ ′ +i(t ′ −s)τ .Thus, N ′′ integration by parts yield . Using Lemma 5.5 one obtains the following estimate P w j (t) D(A m ) (5.18) with the γ −N ′′ factor the sum with respect to j converges.Since j w j converges to w in S ′ R; D(A m ) one concludes that the action of P on w is equal to j P w j in S ′ R; D(A m ) and thus in L ∞ loc R; D(A m ) by estimate (5.18) for |j| sufficiently large and estimate (5.17) for the remaining finite number of terms.Moreover, one has First, consider the case h k τ ≤ 2α −1 .Then, h k τ −1 .With (5.19) one obtains the result.
Second, consider the case h k τ ≥ 2α −1 .Then, one has , yielding the result in this second case.(5.20) and we aim to prove that (5.21) holds for a solution u to the wave equation (2.1) writen in (2.2) that lies in the (2m 0 )-energy level.Thus, we consider u 0 ∈ D(A m 0 ) and The begining of the proof is similar to that given in Section 5.1 and one reaches the following estimate that is the counterpart to (5.6) yielding, with the second point of lemma 4.2 We now concentrate our attention on the terms in the last sum on the r.h.s.. First one writes using that L is bounded on D(A m 0 ); see (3.1).This gives Second, as in Section 5.2 consider F ∈ C ∞ c (R * + ) such that F = 1 in a neighborhood of supp(F ).With Corollary 5.3 and Lemma 5.4 one has ϕF for any M ∈ N. From (5.23) using that h k = ρ −|k| with ρ > 1 one obtains For k 1 ≥ k 0 chosen sufficiently large one obtains The definition of the Fourier multiplier F D(A m ) k (h k D t ) is as in the beginning of Section 4.
Proof.With Lemma 4.5 one has Combined with the third item of Lemma 4.2 one has the following corollary.
R) be such that ψ = 1 in a neighborhood of I 0 , then for any M ≥ 1 and p ∈ R there exists As ϕF (h k D t ) u(t) = P w(t), one concludes the proof with (6.4).One writes using that L is bounded on D(A m 0 ); see (3.1).This gives implying N S = span{e ν ; ν ∈ Υ} with #Υ < ∞.Moreover, if u ∈ N S then u ∈ C m R, D(A r ) for any m ≥ 0 and r ≥ 0, similarly to what one has in (4.9).On this finite dimensional space one has ψ L ∂ t u = ψ∂ t L u = 0. Thus ∂ t maps N S into itself and consequently it has an eigenvector v with associated eigenvalue µ.One finds Av = D t v = −iµv meaning that v(t) is an eigenfunction for A for all t.With the unique continuation Assumption 3.1 one obtains v(t) = 0 for all t.Hence N S = {0}.
We conclude the proof of Theorem 3.7 by an argument by contradiction similar to that in the proof of Theorem 3.2.Adaptation is left to the reader.

Example 1. 1 . ( 1 )
If ω is an open subset of Ω one can define L : v → 1 ω v, yielding a bounded operator on L 2 (Ω).(2) If Γ is an open set of ∂Ω one can define L : v → 1 Γ ∂ n v |∂Ω ,where n is the outgoing normal vector at ∂M, yielding an unbounded operator on L 2 (Ω).

4. 2 .Lemma 4 . 5 .
Action on solutions to the Schrödinger equation.The counterpart results of Lemmata 4.3 and 4.4 for the Schrödinger equation are the following ones.Let u(t) be a solution to the Schrödinger equation (2.11) with u 0

) for k 1 ≥
k 0 to be chosen below.The treatment of the terms in the second sum is different from what is done in Section 5.1.Consider ψ ∈ C ∞ c (] − δ, T + δ[) such that ψ = 1 in a neighborhood of I 0 = [0, T].One writes

6 . 1 .Lemma 6 . 1 .
.24) With (5.24) the result of Lemma 5.1 holds here too.Arguing as in the proof given in Section 5.1 one obtains the sought observability estimate.Proof of the main result for the Schrödinger equation 6.Refined time-microlocalization estimates.The results of Section 5.2 can be adapted to a solution u(t) of a Schrödinger equation (2.11) with u 0 ∈ D(A m ), for some m ∈ R.Then, u ∈ ∩ k C k R; D(A m−k ) .Let F ∈ C ∞ c (]α, α −1 [) be as given by Lemma 4.1.Consider F ∈ C ∞ c (]α, α −1 [) such that F = 1 in a neighborhood of supp(F ).Let p ∈ R.There exists C = C m,p > 0 such that

2 L 2
(R;D(A m )) ≤ Ch M k u 0 D(A p ) .(6.3) Proof.With Lemma 4.2 the proof is clear in the case p ≥ m.We shall thus only consider the case p < m.Let r ∈ N be such that r ≥ m − p.With u solution to the Schrodinger equation one has u = D r t A −r u.Set w = A −r u.It is also a solution of the Schrodinger equation that lies in D(A ∞ ).One hasw(t) D(A m ) = u(t) D(A m−r ) u(t) D(A p ) , t ∈ R. (6.4)One thus considers the action of the operatorP = ϕF D(A m ) k (h k D t )(1 − ψ) Id − F D(A m ) k (h k D t ) D r t , on w.As in the proof of Lemma 5.4 one obtainsP w(t) D(A m ) C M h M k sup j∈Z w j L 2 (R;D(A m )) = C M h M k w H D(A m ) C M h M k w(0) D(A m ) .

u 2 D(A p 1 ) 0≤|k|<k 1 u k 2 D(A p 1 ) + ψ L u 2 L 2 ( 2 L 2 ( 2 L 2 (h k D t )u 2 L 2 ((h k D t ) u 2 L 2 ( 2 D(A p 1 ) 0≤|k|<k 1 u k 2 D(A p 1 ) 2 D(A p 1 ) 0≤|k|<k 1 u k 2 D(A p 1 )D(A m 0 ) 0≤|k|<k 1 u k 2 D(A p 1 )
m 0 ) k (h k D t )(1 − ψ)u R;D(A m 0 )) .Second, as in Section 5.2 consider F ∈ C ∞ c (R * + ) such that F = 1 in a neighborhood of supp(F ).With Corollary 6.2 and Lemma 6.3 one has ϕFD(A m 0 ) k (h k D t )(1 − ψ)u (R;D(A m 0 )) ϕF D(A m 0 ) k (h k D t )(1 − ψ) F D(A m 0 ) k R;D(A m 0 ) + ϕF D(A m 0 ) k (h k D t )(1 − ψ) Id − F D(A m 0 ) k R;D(A m 0 )) h M k u 2 D(A p 1 ), for any M ∈ N. From (6.11) using that h k = ρ −|k| with ρ > 1 one obtainsu + ψ L u 2 L 2 (R;K) + h M k 1 u 2 D(A p 1 ) .For k 1 ≥ k 0 chosen sufficiently large one obtainsu + ψ L u 2 L 2 (R;K) .(6.12)The following lemma is the counterpart of Lemma 5.1.Lemma 6.4 (absence of invisible solutions to the Schrödinger equation).Let u ∈ ∩ k C k R; D(A m 0 −k ) be a solution to (2.11) such that ψ L u = 0. Then u = 0.The proof is very similar to that of Lemma 5.1.Proof.Set N S as the space of such invisible solutions (in the sense of the observation operator ψ L) equipped with the norm u 0 D(A m 0 ) .With (6.12) one has u 0 [16,ro boundary condition that encompasses both Dirichlet and Neumann conditions, with the requirement that the considered elliptic operator be selfadjoint and nonnegative; we refer for instance to[16, Chapters 2 and 4].Then, one has to consider a quotient with respect to the kernel of the resulting unbounded operator if this kernel is not trivial.(3)Above, the coefficients of the elliptic operator are considered smooth.This can be relaxed, down to Lipschitz regularity, yet preserving the properties needed in what follows.Similarly, the regularity of the open set Ω or the manifold M (and its boundary ∂M) can be chosen as low as W 2,∞ .
2 (M)/C one then obtains the setting developped in what follows.(2) On a bounded smooth open set or on a smooth Riemannian manifold M with boundary, one can consider Neumann boundary conditions, that is, Bu = 0 with Bu = ∂ n u |∂M , with n the outgoing normal vector at ∂M.The operator A can be the Laplace(-Beltrami) operator.Similarly to the case without boundary, 0 is an eigenvalue of the elliptic operator A associated with constant functions.The same quotient procedure yields a setting compatible with the analysis developped in what follows.More generally, one can consider a boundary operator B that fulfills the more general Lopatinskiȋ- and consider the operator F k (h k D t ) that simply acts as a Fourier multipliers on function of time t with values in H.Most often we shall write F H k (h k D t ) to keep explicit on which space the operator acts.Since F k has H H ⊂ S ′ (R; H) and thus F H k (h k D t )w is a well defined tempered distribution in time t with values in H.The following lemma improves upon this result.