H\"older Stable Recovery of Time-Dependent Electromagnetic Potentials Appearing in a Dynamical Anisotropic Schr\"odinger Equation

We consider the inverse problem of H\"oldder-stably determining the time- and space-dependent coefficients of the Schr\"odinger equation on a simple Riemannian manifold with boundary of dimension $n\geq2$ from knowledge of the Dirichlet-to-Neumann map. Assuming the divergence of the magnetic potential is known, we show that the electric and magnetic potentials can be H\"older-stably recovered from these data. Here we also remove the smallness assumption for the solenoidal part of the magnetic potential present in previous results.

1.2.History of the Problem.In the case of the dynamic Schrödinger equation with time-independent potentials, Hölder-stable recovery of the magnetic field from knowledge of the Dirichlet-to-Neumann map was shown in [3], and stable recovery of the electric potential of the Schrödinger equation on a Riemannian manifold was proved in [4].This latter result is extended to stable determination of the electromagnetic potentials on a Riemannian manifold from the D-to-N map in [2].We mention also the recent work of [5], where such results have been extended to unbounded cylindrical domain.
Literature dealing with the inverse problem of recovering time-dependent potentials of the Schrödinger equation is rather sparse.To the best of the authors knowledge, the only results establishing recovery of time-dependent potentials of the Schrödinger equation from the D-to-N map deal with Euclidean domains.In particular, it was proved in [8] that the time-dependent electric and magnetic potentials are uniquely determined by the D-to-N map.Logarithmic-stable determination was shown for the electric potential in [7].This result was extended to the full electromagnetic potential in [6], provided that the time-independent part of the magnetic potential is sufficiently small.Indeed, it was only recently shown in [10] that the electromagnetic potential in a Euclidean domain can be Hölder-stably recovered from knowledge of the D-to-N map.
In the current work, we show that it is possible to Hölder-stably recover the time-and-space-dependent coefficients of the dynamic Schrödinger equation on a simple Riemannian manifold.

Main Results.
Here and in the rest of this paper we write • for the norm of an operator in In this paper we aim to prove the following: Theorem 1. (Uniqueness):For j = 1, 2, let A j ∈ W 6,∞ ((0, T ) × M; T * M) and q j ∈ W 4,∞ ((0, T ) × M).Assume also that Then the condition Λ A1,q1 = Λ A2,q2 implies that (A 1 , q 1 ) and (A 2 , q 2 ) are gauge equivalent.
Theorem 2. (Stable Recovery of the Magnetic Potential): Let the condition of Theorem 1 be fulfilled and, Assume also that there exists a constant B such that

Then we have
where s 1 > 0 is a general constant, C > 0 a constant depending only on B, T , M and A sol j is the solenoidal part of the Hodge decomposition of A j , given in Lemma 1.
Theorem 3. (Stable Recovery of the Electric Potential): Let the condition of Theorem 2 be fulfilled with Fix also q j ∈ W 4,∞ ((0, T ) × M) ∩ H 5 ((0, T ) × M) and assume that the condition , is fulfilled.We also assume that there exists a constant B 1 > 0 such that Then we have (1.9) where C depends only on B, B 1 T , and M, and s 2 is a general constant.
As far as the authors are aware, the present work is the first dealing with recovery of time-dependent potentials appearing in a Schrödinger equation with variable coefficients of order two.In fact, the above estimates are the first showing Hölder-stable recovery of a coefficient dependent on all variables of a second order partial differential equation with variable coefficients of order two.The only other work where similar results have been obtained is [10], where the authors consider the case of a bounded subset of R n with the Euclidean metric.
Furthermore, stable recovery of a magnetic potential appearing in a Schrödinger equation on a manifold with non-Euclidean metric has, thus far, relied upon the a priori assumption that the magnetic potential is small in some appropriate norm, even in the time-independent case (see, for example, [2]).This smallness assumption is also utilized when recovering the magnetic potential of the wave equation (as seen in [12]).In fact, it happens that this assumption is not necessary when dealing with the Schrödinger equation, even when the magnetic potential is allowed to depend on time, as we shall demonstrate herein.
In Section 2, we introduce the geodesic ray-transforms for 1-forms and for functions.In Section 3 we construct geometric optics solutions to the equation (1.1).We devote Section 4 to the proof of Theorem 1, using the geometric optics solutions as the main tool.The estimate of Theorem 2 is proved in Section 5, whereas the estimate of Theorem 3 is proved in Section 6.

Notations
In this section, we list some notation used in the rest of the paper.We denote by •, • g the inner product with respect to g on T M, that is for Similarly, we denote by •, • g the inner product with respect to g on T * M, that is for U, V ∈ T * x M given by U = n j=1 u j dx j , V = n j=1 v j dx j we have We denote by dV g the Riemannian volume on M, which is given in local coordinates by dV g = |g| We further define on ∂M the surface measure σ g such that for X ∈ H 1 (M; T M) we have Additionally, we recall the Riemannian gradient operator given by We recall the coderivative operator δ is the operator sending the 1-form ω We recall also the definition of a simple manifold.Let D be the Levi-Civita connection on (M, g).For x ∈ ∂M we consider the second quadratic form of the boundary We say that ∂M is strictly convex if the form Π is positive-definite for every x ∈ ∂M.Definition 1.We say that (M, g) is simple if ∂M is strictly convex, M is simply connected, and for any x ∈ M the exponential map exp x : exp −1  x (M) → M is a diffeomorphism.
We write γ x,θ for the unique geodesic in M with initial point x ∈ M and initial direction θ ∈ T x M. We define the sphere bundle of M by and likewise the submanifold of inner vectors ∂ + SM by Given that M is assumed to be simple, we can also define τ + (x, θ) to be the maximal time of existence in M of the geodesic γ x,θ for x ∈ ∂M, that is We also introduce here the geodesic ray transforms on a simple Riemannian manifold M. Definition 2. The geodesic ray transform for 1-forms is the linear operator Definition 3. The geodesic ray transform for functions is the linear operator which is given by

Geometric Optics Solutions
We now seek to construct GO solutions of the magnetic Schrödinger equation in (0, T ) × M. We fix We consider the equations i∂ t u j + ∆ g,Aj (t) u j + q j u j = 0 in (0, T ) × M, We seek to find, for λ > 1, j = 1, 2, solutions u j ∈ H 1,2 ((0, T ) × M) of (3.2) of the form In (3.3) above, ψ, a j , b j satisfy the following eikonal and transport equations: Taken together, equations (3.4) -(3.6) yield We also assume that there exists τ ∈ 0, T 4 ) such that a j , b j are supported in [τ, T − τ ] × M and further assume that a j , b j ∈ H 3 ((0, T ) × M), whence (i∂ t + ∆ g,Aj (t) + q j )b j ∈ H 1 (0, T ; L 2 (M)).Thus we can choose R j,λ solving Since (M, g) is simple, the eikonal equation (3.4) can be solved globally on M. To see this, we first extend the simple manifold (M, g) to a simple, compact manifold (M 1 , g) with M contained in the interior of M 1 .We pick y ∈ ∂M 1 and consider polar normal coordinates (r, θ) on M 1 given by x = exp y (rθ) for r > 0 and θ ∈ S y M 1 = {v ∈ T y M 1 : |v| g(y) = 1}.Letting ν(y) denote the outward unit normal to ∂M 1 with respect to the metric g, we define ∂ + S y M 1 = {θ ∈ S y M 1 : θ, ν(y) g(y) < 0}.According to the Gauss Lemma (see e.g.[15,Chapter 9,Lemma 15]), in these coordinates the metric takes the form g(r, θ) = dr 2 + g 0 (r, θ) with g 0 (r, θ) a metric on {θ ∈ S y M 1 : ν(y), θ g(y) ≤ 0} depending smoothly on r.In polar normal coordinates drdθ, where µ = det g 0 and dθ is the usual spherical volume form on ∂ + S y M. For a function f ∈ L 1 (M) extended by zero to M 1 , we can extend dV g to a volume form on T y (M 1 ) and get We choose where dist g denotes the Riemannian distance function.Since ψ(r, θ) = r, we can easily check that ψ solves the eikonal equation (3.4).
For any h ∈ H 5 ((0, T ) × ∂ + S y M 1 ), the functions are solutions to the transport equations (3.10).In the same way, for βj = (i∂ t + ∆ g, Ãj (t) + qj )a j /2, we fix which is a solution of (3.11).Here we fix χ Let us now consider the remainder terms R j,λ , j = 1, 2. In view of (3.12)-(3.14),we deduce the following bounds: where C depends only on M, T and Then applying [10, Lemma 2.1], we see that problem (3.7) admits unique solutions R j,λ for j = 1, 2 with R j,λ ∈ C([0, T ]; ).On the other hand, from the a priori estimate [11, (10.10), page 324], we deduce that Moreover, applying [10, Lemma 2.1] we find that and by interpolation between this estimate and (3.18) we deduce Combining this with (3.18) we obtain In a similar manner, we derive the estimate This completes our construction of the geometric optics solutions of (3.2).
We start by considering the implication Λ A1,q1 = Λ A2,q2 ⇒ A sol = 0, where A sol is the solenoidal part of the Hodge decomposition (4.1) of A. For this purpose, we establish the following intermediate result.

3), and fix
In particular, for Ãj the extension of A j to (0, T ) × M 1 introduced in the previous section, we have A = Ã1 − Ã2 .Assuming these conditions are fulfilled, we find that Proof.We fix u j , j = 1, 2 the solutions for j = 1, 2 respectively of (3.2) taking the form (3.3).We write also and consider w = v − u 1 which solves Multiplying this equation by u 2 and integrating by parts yields and (3.15)-(3.16)imply 2 ((0,T )×M) Here C is a generic constant which depends only on M, T and A 1 W 5,∞ ((0,T )×M) + A 2 W 5,∞ ((0,T )×M) .
On the other hand, we have that We then divide (4.5) by λ and apply (3.19)-(3.20) to obtain Using polar normal coordinates in the left hand side of the above gives us Using now the fact that µ(r, θ) − 1 2 dV g (r, θ) = drdθ, we conclude that We use this last estimate together with (4.3) and (4.4) to obtain (4.2).
Armed with the above, we are now in a position to complete the proof of the uniqueness result.
Proof of Theorem 1.Let us assume that Λ A1,q1 = Λ A2,q2 , and begin by proving that this condition implies that A sol = 0. We recall also Definition 2 of I 1 , the geodesic ray transform for 1-forms given by (2.2).According to s-injectivity of the transform I 1 (consult e.g.[1] or [16,Theorem 4]), it is enough to show that On the other hand, notice that, due to (3.9), for A = n j=1 a j dx j we have Thus we deduce that Using this identity in (4.6) and applying Fubini's theorem, we get But since τ ∈ (0, T 4 ) is arbitrary and and hence deduce that for all t ∈ [0, T ], •)](y, θ) is constant.On the other hand, note that A = 0 on M 1 \ M, so that for any y ∈ ∂M 1 there exists θ ∈ ∂ + S y M 1 such that for all t ∈ [0, T ] we have I 1 [A(t, •)](y, θ) = 0. Therefore we conclude that A sol = 0.

Stable Determination of the Magnetic Potential
In this section we establish the stability estimate in the recovery of the magnetic potential stated in Theorem 2. For j = 1, 2, we assume that ).We will also assume for the moment that for some small ε > 0 it holds that (5.1) A sol L 2 ((0,T )×M1) ≤ ε.Before proving Theorem 2, let us recall some facts about the geodesic ray transform I 1 .Firstly, according to [14, Theorem 4.2.1], the ray transform for 1-forms extends to a bounded linear operator . Fixing w(x, θ) = θ, ν(x) g , we can also extend I 1 to a bounded linear operator , where L 2 w (∂ + SM 1 ) is the L 2 space with respect to the weighted measure w(y, θ)dθdσ g (y), and thus define as the adjoint of I 1 .By condition (1.3) we have A ∈ H 5 ((0, T ) × M 1 ; T * M 1 ) with supp A(t, •) ⊂ M for t ∈ (0, T ).Moreover, according to [16,Section 8], the operator I * 1 I 1 , is an elliptic pseudodifferential operator of order −1.Together with condition (1.5), we have for 0 ≤ k ≤ 5 (5.2) [16,Section 8], we can find constants C 1 , C 2 > 0 such that for 0 ≤ k ≤ 5 Proof of Theorem 2 subject to (5.1).Following the work of the previous section, we allow h(t, θ) to depend on y ∈ ∂M 1 .We can rewrite inequality (4.2) in the form We can use the Taylor expansion e t = 1 + t + t 2 1 0 e st (1 − s)ds to see that and using this identity in (5.4) yields
Thus for small ε we deduce that A sol L 2 ((0,T )×M) ≤ Cγ Thus the proof of Theorem 2 is complete, subject to the smallness assumption (5.1).
We will now show that the assumption that (5.1) holds a priori is unnecessary.Define η ∈ C ∞ (R n ) by where C > 0 is chosen so that R n η(x)dx = 1.We further define the function Note that η ρ approximates the Dirac delta distribution on R n as ρ → 0. Arguing as we did in (5.8), we use the estimate (5.4) to deduce that Since A is extended by 0 to (0, T ) × (M 1 \ M), it follows that e iI1[A(t,•)](y,θ) − 1 is compactly supported in [0, T ] × ∂ + S y M 1 .We can find a finite open cover {U i } N i=1 of ∂M 1 so that for all y ∈ U i we can choose the same spherical coordinates θ := R n−1 ∋ α → θ(α) on S y M 1 in such a way that θ(α) gives coordinates in a neighborhood of supp(e iI1[A(t,•)](y,θ) − 1) ⊂ ∂ + S y M 1 .