Stability estimates for time-dependent coefficients appearing in the magnetic Schrödinger equation from arbitrary boundary measurements

In this work, we study the stable determination of time-dependent coefficients appearing in the Schrodinger equation from partial Dirichlet-to-Neumann map measured on an arbitrary part of the boundary. Specifically, we establish stability estimates up to the natural gauge for the magnetic potential.

In this paper, we focus on measuring arbitrarily, so that we consider an arbitrary non-empty relatively open subset Γ of Γ. Then we set Σ = (0, T ) × Γ and we introduce the partial Dirichlet-to-Neumann map (D-to-N map in short) associated to the IBVP (1) which is bounded and where ν is the outward unit normal vector to Γ. The goal of this work is to treat the inverse problem of stably recuperating the electromagnetic potential (A, q) from the knowledge of the partial operator Λ A,q .
Yet, we find an obstruction for the uniqueness of the present problem in terms that the D-to-N map Λ A,q is not injective. As a matter of fact, and as it was mentioned in [22], the D-to-N map is invariant under the gauge transformation of the magnetic potential. Precisely, for any Φ ∈ W 3,∞ (Q) such that Φ = 0 on Σ, one has e −iΦ H A,q e iΦ = H A+∇Φ,q−∂tΦ , e −iΦ Λ A,q e iΦ = Λ A+∇Φ,q−∂tΦ = Λ A,q .
Hence, the magnetic potential cannot be uniquely determined by the D-to-N map Λ A,q . In geometric terms, the vector field A defines the connection given by the oneform A = n j=1 a j dx j , and the non-uniqueness illustrated in (3), leads us to wish to restore the magnetic field curl(A) from the D-to-N map Λ A,q , which is defined by That's the case in [6], where Bellassoued and Choulli treated the dynamical Schrödinger equation in the presence of a time-independent magnetic potential, and stably recovered the magnetic field curl(A) induced by the magnetic potential A from the knowledge of the global Dirichlet-to-Neumann map on the whole boundary. Besides, and by imposing some geometric conditions on the domain, Chung showed in [15] that the knowledge of the Dirichlet-to-Neumann map on certain subsets of the boundary uniquely determines the magnetic Schrödinger operator.
Hence, and when working on a Riemannian manifold, Bellassoued-Dos Santos Ferreira [7] and Bellassoued [4] stably recovered the electric potential or the magnetic field from the knowledge of the dynamical D-to-N map associated to the Schrödinger equation. We mention also the recent work of Bellassoued, Kian and Soccorsi [9] where such results have been extended to unbounded cylindrical domain. Further, we can refer to several papers which similarly treated such inverse problems such as [23,26].
In the case of a finite number of boundary obervations of the solutions, Bellassoued and Choulli derived in [5] a logarithmic stability estimate for the inverse problem of determining the potential appearing in the dynamical Schrödinger equation which is assumed to be known in a neighborhood of the boundary, with a single measurement on an arbitrary given subboundary. In addition, we cite [16] where Cristofol and Soccorsi restored the magnetic potential in the Coulomb gauge class by finitely measuring the solution of the Schrödinger equation. Recently, and by a similar type of boundary data, Ben Aïcha and Mejri recovered simultaneously in [3] the electric potential and the divergence free magnetic potential.
For stationary Schrödinger equation and from local observations as shown by Dos Santos Ferreira, Sjöstrand, Kenig and Uhlmann in [17], the magnetic field curl(A) and the electric potential q are uniquely determined even if the measurements are taken only on a small part of the boundary. Further, in [34], Tzou established a log-log type stability estimate for curl(A) and q when measuring only on a chosen subset of ∂Ω which is slightly larger than half of the boundary, and if one has full data measurements, the result can be improved to a log-type estimate.
Clearly, all the above cited results are specific with time-independent coefficients and the publications concerned with the study of the inverse problem of determining the time-dependent potentials in a Schrödinger equation are very limited. This is not the case for hyperbolic and parabolic equations where similar inverse problems have been extensively treated such as [10,1,12,13,21,24,25,27,28,31,33]. Further, we cite [14] in which Choulli, Kian and Soccorsi proved a logarithmic stability estimate of determining a time-dependent electric potential from boundary measurements in a one-periodic quantum cylindrical waveguide. This result was extended by Ben Aïcha [2] to the full electromagnetic potential, where the knowledge of the D-to-N map stably determines the magnetic field and the time-dependent electric potential.
In addition, it was proved by Eskin [22] that the time-dependent electric and magnetic potentials are uniquely determined by the D-to-N map in domains with obstacles. This paper is an extension of the author's works [18,19,20] to the case of time-dependent potentials. Yet, Kian and Tetlow considered in [30] the inverse problem of Hölder stably determining the time-and space-dependent coefficients of the dynamical Schrödinger equation on a simple Riemannian manifold from the knowledge of the Dirichlet-to-Neumann map on the whole boundary. Similar results have been obtained by Kian and Soccorsi in [29] where the authors treated the case of a bounded subset of R n with the Euclidean metric. Furthermore, and as it was shown in [29], the time-dependent magnetic potential A itself and the electric potential q are Hölder stably determined from the global Dirichlet-to-Neumann map: Λ A,q : f → (∂ ν + iA · ν)u |Σ , provided that the divergence ∇ · A is known in Q. We prove in this paper an estimate, which shows that the electromagnetic potential (A, q) depends stably on the partial D-to-N map Λ A,q . More precisely, we generalize the previous result by observing on an arbitrary subboundary Σ = (0, T ) × Γ and assuming that the magnetic potential A and the electric potential q are known in a neighborhood of the boundary. Further, and by removing the condition on ∇ · A, we only recover the magnetic field curl(A) induced by the magnetic potential A, provided that the electromagnetic potential (A, q) vanishes near the boundary Γ.
First of all, we consider O ⊂ Ω an arbitrary neighborhood of Γ and we introduce the admissible sets of coefficients A and q : For M > 0, we define Then, we establish a stability estimate for the electromagnetic potential (A, q) appearing in the Schrödinger equation (1), from observations given by the partial D-to-N map Λ A,q .
Then, for any T * < T, there exist C > 0 depending on Ω, T, T * , M and β ∈ (0, 1) such that we obtain provided that Λ A1,q1 − Λ A2,q2 is small. Further, if we assume the following condition then, there exist C > 0 and β 1 , β 2 ∈ (0, 1) such that This article is organized as follows. In Section 2, we introduce some preliminaries and estimates for the Schrödinger equation. In Section 3, we construct geometric optics solutions used for the proof of the main results in Section 4 and 5. For the last Section, we establish a special unique continuation estimate by applying the specific Fourier-Bros-Iagolnitzer (F.B.I) transformation.

2.
Well-posedness and unique continuation. This section will be devoted to determine some results which are strongly needed to prove our Theorem. First, let's introduce the following notations which will be used in the whole coming parts. We consider three open subsets O j , j = 1, 2, 3 of O neighborhood of the boundary Γ such that 2.1. The magnetic Schrödinger equation. Let's start by the following Lemma for the existence of unique solutions proved in [29].

2.2.
A unique continuation estimate. With the following Lemma, we will establish a stability estimate in the unique continuation of the solutions for the magnetic Schrödinger equation on an arbitrary open subset Γ of Γ from lateral boundary data.
From Lemma 2.2, we can derive the following unique continuation property.
The proof of Lemma 2.2 is given in Section 6.
3. Geometric optics solutions. In this section, we construct special solutions for the electromagnetic Schrödinger equation. These solutions called geometric optics solutions are the main key for the proof of our main results. Let's first introduce some notations and definitions needed for all the following parts. The structure of our geometric optics solutions is similar to the one used in [29].
Let T * < T and let h ∈ (0, T * /4), we set the first function where the constant C depends only on M , and τ, ξ denotes for (1 + τ 2 + ξ 2 ) 1/2 . At this stage, we introduce the two following functions in W 5,∞ (Q) defined by Then, we get that these functions satisfy (12) ω · ∇ Aj a j,1 = 0, in Q, where C depends only on Ω, T and M . In addition, by decomposing x ∈ R n into x = y + sω where s = x · ω and y = x − sω ∈ ω ⊥ , for j = 1, 2, we set Then, a direct calculation shows that a j,2 , j = 1, 2, solves (14) 2iω · ∇ Aj a j,2 + (i∂ t + ∆ Aj + q j ) a j,1 = 0, in Q, and satisfies Thus, we get the following estimates We are now in position to present the following Lemma in which we construct special solutions to our electromagnetic Schrödinger equation and which are the main tool strongly needed for the proof of the basic results.
The following equation has a solution of the form where r 1,σ satisfies Moreover, there exists C > 0 such that Proof. We give an idea of the proof. By considering (17), we see that r 1,σ must be the solution of the following IBVP (20) Here a σ (t, . Now, an elementary calculation gives that which implies, by recalling (12) and (14), that Now, and by referring to Lemma 2.1, we show that r 1,σ is a well defined solution of the equation (20), and it satisfies (10) and (16) entail In addition, we get from (9) and (15) that Then, by interpolating with (21), we obtain So that we conclude (18) from (21)-(23) and we obtain (19) from (22).
By a similar way and strategy for the proof, we get a second result for the backward equation.
The following equation has a solution of the form where r 2,σ satisfies 4. Stable recovery of the magnetic potential. At the first stage, we establish stability estimates for the magnetic potential A. The key ingredients for the proof will be based on using the geometric optics solutions already constructed in Section 3 and on applying the special continuation estimate established in Lemma 2.2.

4.1.
Recovery of the magnetic potential without Coulomb gauge. In the beginning, we will deal with any magnetic potentials A j where ∇ · (A 1 − A 2 ) is not fixed.
Lemma 4.1. There exist C > 0, m 1 > 0 and µ < 1 such that the following estimate holds true uniformly in ξ ∈ ω ⊥ , where C depends only on Ω, T, T * and M.
At this stage, we define the Fourier transform of a function f ∈ L 1 (R n+1 ) by Further, we set Then, we get an estimate on the Fourier transform of ρ jk , j, k = 1, . . . , n, given by the following Lemma.
So, if |ξ m | ≤ |ξ |, we take j = in (44) to find by (43) that That's how In the other hand and if |ξ m | ≥ |ξ |, we take j = m in (44) and we obtain similarly that Collecting (45) and (46), we get the desired estimate. Now, we are arranged to establish the first magnetic stability estimate (4) for our problem. We start by estimating the H −1 (R n+1 ) norm of ρ jk , to get for fixed R > 0. Then, by combining (47) with (39), we obtain Further, by interpolating we get Then, we get On the other hand, we use the fact that , and we keep in mind that the function (1−θ) is valued in [0, 1] and verifies 1−θ(t) = 0 if t ∈ [2h, T * − 2h], so that which leads to Thus, by (48) and (49), we obtain where N 1 and N 2 are positive constants depending on n, and by a similar way, we pick R that is to satisfy R N2 γ −µ = R −1/2 in such a way that if Λ A1,q1 − Λ A2,q2 is sufficiently small, we end up getting that where N 3 < 1 and m > 0 depending on n and µ.
Thus, we pick γ such that to find a constant β < 1 such that for some constant p ∈ (0, 1).
That's how we complete the proof of the first stability estimate (4) dealing with the magnetic potential.

4.2.
Recovery of the magnetic potential with Coulomb gauge. This part will be devoted to treat the case where the magnetic potential A = A 1 − A 2 satisfies the Coulomb gauge (5), that is By using the following Lemma, the result will be based on the previous part as we will see.  for some constant C > 0.
We return now to prove the second stability result (6). We recall (50) and (4) to obtain Then, by interpolating, we get from this the desired estimate for some s ∈ (0, 1). We complete then the proof of the second stability estimate for the magnetic potential (6) .

5.
Stable recovery of the electric potential. At this stage, we are interested to prove a stability estimate for the electric potential q. The work will be based on the previous stability estimate proved for the magnetic field, besides, we keep the same previous notations and definitions, unless the function η which will be given by the new definition as follows Here we assume that div(A) = 0. Then, we recall the equality (33) but we focus now on the electric potential term to get using the estimates (13), (24), (36)-(38) that We pick γ such that γ µ1 = σ 7 , to get for some α > 0. Now, we recall the definition of G A,q to get This, (13), (24) and (52) imply which leads to by applying the mean value theorem. That's how we find, using (53) and (54), that Now, we need to estimate the L ∞ -norm of A by the norm of the associated partial D-to-N map. To do this, we start by choosing n 0 > n + 1 and applying the Sobolev embedding theorem (see e.g. [11]), getting
The next step will be similar to the one achieved in the previous section, and that's how it leads to where α 1 , β < 1 depending on n, n 0 , α and β 1 . Thus, we pick γ in such a way that to find a constant β 2 < 1 such that This completes the proof of the estimate (7).
6. Unique continuation estimate. We prove now the unique continuation estimate which presented an important ingredient in the proof of our main results.
6.1. A parabolic Carleman estimate. We begin our section with a special inequality for PDE solutions which is the Carleman estimate. To do this, we start, by introducing the set Γ 0 defined by and we may assume, without loss of generality, that Γ 0 is C 2 -smooth. Also, we choose a function ψ ∈ C 2 (O) such that Then, for β > 0, we define the functions ϑ 0 and ϑ by where k(s) = (1 − s)(1 + s) and ψ is defined by (57) and (58), and Further, in connection with the Schrödinger operator (i∂ t + H A2,q2 ), we define a parabolic operator, associated with some fixed parameter ∈ (0, 1), by Now, we are ready to express the following Carleman estimate for the operator L .
The proof of a similar result can be found in [8].
Here := T /T 0 . Since and ∆w γ,t (s, x) = F γ (∆w)(z, x), we get by applying the FBI transform F γ to (65) that We continue our proof by applying the parabolic Carleman estimate of Lemma 6.1 to the solution w γ,t of (70) in purpose to get the next result.