The Calderón problem for the fractional Schrödinger equation with drift
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This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ The Calderón problem for the fractional Schrödinger equation with drift © The Authors 2020 Published version Cekić, Mihajlo; Lin, Yi-Hsuan; Rüland, Angkana Cekić, M., Lin, Y.-H., & Rüland, A. (2020). The Calderón problem for the fractional Schrödinger equation with drift. Calculus of Variations and Partial Differential Equations, 59(3), Article 91. https://doi.org/10.1007/s00526-020-01740-6 2020
Calc. Var. (2020) 59:91 https://doi.org/10.1007/s00526-020-01740-6 Calculus of Variations The Calderón problem for the fractional Schrödinger equation with drift Mihajlo Ceki´c1,2 ·Yi-Hsuan Lin3,4 ·Angkana Rüland5,6 Received: 1 January 2019 / Accepted: 28 February 2020 © The Author(s) 2020 Abstract We investigate the Calderón problem for the fractional Schrödinger equation with drift, proving that the unknown drift and potential in a bounded domain can be determined simultaneously and uniquely by an infinite number of exterior measurements. In particular, in contrast to its local analogue, this nonlocal problem does not enjoy a gauge invariance. The uniqueness result is complemented by an associated logarithmic stability estimate under suitable apriori assumptions. Also uniqueness under finitely many generic measurements is discussed. Here the genericity is obtained through singularity theory which might also be interesting in the context of hybrid inverse problems. Combined with the results from Ghosh et al. (Uniqueness and reconstruction for the fractional Calderón problem with a single easurement, 2018. arXiv:1801.04449), this yields a finite measurements constructive reconstruction algorithm for the fractional Calderón problem with drift. The inverse problem is formulated as a partial data type nonlocal problem and it is considered in any dimension n≥1. Mathematics Subject Classification 35R30 ·26A33 ·35J10 ·35J70 Communicated by X. Cabre. BAngkana Rüland [email protected] Mihajlo Ceki´c [email protected] Yi-Hsuan Lin [email protected] 1Max-Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany 2Present Address: Laboratoire de Mathématiques d’Orsay, Université Paris-Saclay, CNRS, 91405 Orsay, France 3Department of Mathematics and Statistics, University of Jyväskylä, Jyvaskyla, Finland 4Present Address: Department of Applied Mathematics, National Chiao Tung University, Hsinchu 30050, Taiwan 5Max-Planck Institute for Mathematics in the Sciences, Inselstrasse 22, 04103 Leipzig, Germany 6Present Address: Ruprecht-Karls-Universität Heidelberg, Institut für Angewandte Mathematik, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany 0123456789().: V,-vol 123
91 Page 2 of 46 M. Ceki´ cetal. Contents 1 Introduction ................................................ 1.1 The main results ............................................ 1.2 Outline of the remaining article .................................... 2 The fractional Schrödinger equation with drift .............................. 2.1 Preliminaries ............................................. 2.2 Well-posedness ............................................ 3 Approximation property .......................................... 4 Global uniqueness ............................................. 5 Stability .................................................. 5.1 Auxiliary results ............................................ 5.2 A quantitative approximation result .................................. 5.3 Proof of Theorem 1.3 ......................................... 6 Reconstruction and finite measurements uniqueness ........................... 6.1 Higher order approximation ...................................... 6.2 Finite measurements reconstruction without openness ........................ 6.3 Variations and extensions of the reconstruction result ........................ 6.4 Proof of Theorem 1.4 and generic properties of determinants via singularity theory ........ Appendix A. Genericity of determinants ................................... A.1 Corank one case ............................................ A.2 Corank two case ............................................ A.3 Corank at least three .......................................... References ................................................... 1 Introduction In this article, we consider an inverse problem for a nonlocal Schrödinger equation with drift. Here we seek to study the uniqueness, stability and reconstruction properties in analogy to its localcounterpart,whichisatypeofmagnetic Schrödinger equation and had been investigated in Nakamura et al. [34]. As one of our main results, we prove that: In contrast to its local counterpart,forthefractionalCalderónproblemwithdriftthereisno gaugeinvariance present (see Theorem 1.1). In particular, this poses an obstruction in possibly extracting information from the nonlocal inverse problem for its local analogue (as s→1). As our second main result,weproveageneric, finite measurements reconstruction, which might alsobeofinterest in the context of hybrid inverse problems (see Theorem 1.4 and the following discussions). As a key tool for this, we rely on singularity theory from Whitney [53]. The classical Calderón problem with drift. Before turning to the nonlocal problem, let us recall its local analogue and the known results on this: In the classical Calderón problem for the magnetic Schrödinger equation, the objective is to determine the drift and potential coefficients simultaneously. More precisely, for n≥3, let ⊂Rnbe a bounded Lipschitz domain, then consider the following Dirichlet boundary value problem (−+b·∇+c)u=0in, u=fon ∂, (1.1) where band care sufficiently smooth functions which vanish on ∂. Assuming the wellposedness of the boundary value problem (1.1), one can define boundary measurements given by the Dirichlet-to-Neumann map (abbreviated as the DN map in the rest of this paper) b,c:H1/2(∂) →H−1/2(∂) with b,c:f→ ∂u ∂ν, 123
The Calderón problem for the fractional Schrödinger equation… Page 3 of 46 91 where u∈H1() is the unique solution to (1.1)andνis the unit outer normal on ∂. The Calderón problem for (1.1) consists of trying to recover the unknown coefficients b, c(which are assumed to be in appropriate function spaces) by the information encoded in the boundary measurement operator b,con ∂. In the case of the local magnetic Schrödinger equation, there is however an intrinsic obstruction to the unique identification of these coefficients. To observe this, consider the substitution v=eφuwhich results in an equation for vwhich is of a similar form as (1.1): −v +(b+2∇φ) ·∇v+(c+φ −b·∇φ−|∇φ|2)v =0in, v=eφfon ∂. (1.2) If now φ=0, ∂νφ=0on∂, then the DN maps for the old and the new equations (1.1) and (1.2) coincide; there is a hidden gauge invariance. As a consequence, one cannot expect to be able to recover the full information on the coefficients band cfrom the knowledge of the Dirichlet-to-Neumann map b,con ∂. As a matter of fact, one can at most hope to recover band cup to the described gauge invariance for the inverse boundary value problem with respect to (1.1). This is indeed the case (c.f. [28, Theorem 5.4.1]): If bjand cjare compactly supported in a simply connected domain for j=1,2, and if their DN maps coincide on the boundary ∂, then one can obtain uniqueness up to the described gauge invariance: curlb1=curlb2and 4c1+b1·b1−2divb1=4c2+b2·b2−2divb2in . (1.3) We again emphasize that this does not allow us to recover the full fields b,cbut only allows one to obtain information up to the above gauge invariance. Variants of this inverse boundary value problem have also been studied in [43,48,49]for the symmetric magnetic Schrödinger operator, in [25,30] for the low regularity setting and in [16] for the setting in which only partial data are available. The setting of flexible geometries was studied in [8,31]. The case of systems was considered in [13] and Yang–Mills potentials with arbitrary geometry in [9]. Stability results can be found in [29,50] and reconstruction results are given in [44]. For more detailed discussions of inverse problems for the magnetic Schrödinger equation, we refer to the book [28, Chapter 5] as well as to the articles [45,51] and their bibliographies. The fractional Calderón problem with drift. Keeping the situation of the local problem with s=1 in the back of our minds, we turn to the Calderón problem for the analogous fractional Schrödinger equation with drift, which is a nonlocal inverse problem. This inverse problem should be regarded as a generalization of the fractional Calderón problem, which had first been introduced and investigated in Ghosh et al. [20]. In the sequel, we describe this problem more precisely. Assume that ⊂Rnis a bounded Lipschitz domain for n≥1andlet1 2<s<1(sothat the fractional nonlocal operator is dominant). Given a drift b∈W1−s,∞()nand a potential c∈L∞(), we consider the following fractional exterior value problem ((−)s+b·∇+c)u=0in, u=fin e:= Rn\, (1.4) with some suitable exterior datum f(we will present a rigorous mathematical formulation of this in Sect. 2, c.f. also Remark 2.10). Here the fractional Laplacian (−)sis given by (−)su:= F−1|ξ|2su(ξ),for u∈Hs(Rn), 123
91 Page 4 of 46 M. Ceki´ cetal. where u=Fudenotes the Fourier transform of u. We assume that this problem and its “adjoint problem” are well-posed, which is guaranteed by imposing the eigenvalue condition that If w∈Hs(Rn)is a solution of (−)sw+b·∇w+cw=0inwith w=0ine, then we have w≡0. (1.5) In the sequel, we will always suppose that this condition is satisfied. As in the classical Calderón problem with drift, we are interested in recovering the drift coefficient b∈W1−s,∞()nand the potential c∈L∞() simultaneously from the associated DN map. With slight abuse of notation (for a precise definition we refer to Sect. 2), the DN map associated with the nonlocal problem can be thought of as the mapping b,c: Hs(e)→( Hs(e))∗,f→ (−)su|e,(1.6) where uis the solution to (1.4) with exterior data f.Here Hs(e)denotes the completion of C∞ c(e)with respect to a fractional Sobolev norm. We refer to (2.1) for the precise definition. The aim of this work is to prove the global uniqueness and stability of the drift coefficient and the potential for this nonlocal inverse problem. This is in strong contrast to the local case, i.e., the case s=1, as in the nonlocal setting the gauge invariance (1.3) which presented an obstruction to global uniqueness in the local case disappears. 1.1 The main results Let us formulate our main results. As a first property we obtain the global uniqueness of the drift coefficient and the potential for the nonlocal fractional Calderón problem with drift: Theorem 1.1 (Global uniqueness) For n ≥1,let⊂Rnbe a bounded, open, non-empty Lipschitz domain, and let 1 2<s<1.Letb j∈W1−s,∞()nbe two drift fields, and cj∈L∞() be potentials for j =1,2. Given arbitrary open, non-empty sets W1,W2⊂e, suppose that the DN maps for the equations ((−)s+bj·∇+cj)uj=0in satisfy b1,c1f|W2=b2,c2f|W2,for any f ∈C∞ c(W1). Then b1=b2and c1=c2in . Theorem 1.1 can be regarded as a partial data result for our nonlocal inverse problem. Different from the local case, i.e., s=1, there is no gauge invariance and thus no intrinsic obstruction to uniqueness in this nonlocal Calderón problem. We expect that it is possible to improve the regularity assumptions on the drift coefficient and the potential. As our main focus in the present article is however on the striking differences between the local and the nonlocal problems in terms of the existence/absence of a gauge, we do not elaborate on this here but postpone this to a future work. AsinpreviousresultsonthefractionalCalderónproblem(c.f.[20]),inthiswork,wearenot using complex geometrical optics solutions. Instead, we rely on the following approximation property. Theorem 1.2 (Runge approximation) For n ≥1and 1 2<s<1,let⊂Rnbe a bounded open, non-empty Lipschitz set and 1⊂Rnbe an arbitrary open set such that 1\=∅. 123
The Calderón problem for the fractional Schrödinger equation… Page 5 of 46 91 (a) Let b ∈W1−s,∞()nand c ∈L∞(). Then, for any g ∈L2() and >0, one can find a solution u∈Hs(Rn)of (−)s+b·∇+cu=0in , with supp(u)⊂1 such that u−gL2() <. (b) If we further assume that has a C∞-smooth boundary, b ∈C∞ c()nand c ∈C∞ c() with supp(b), supp(c), given any g ∈C∞(),>0and k ∈N, then there exists a solution u∈Hs(Rn)of (−)s+b·∇+cu=0in , with supp(u)⊂1 such that d−s(x)u−gCk() ≤. The function d(x)is any C∞-smooth function defined in such that d >0in and d(x)=dist(x,∂)whenever x is near ∂. A more quantitative Runge approximation for the fractional Schrödinger equation with drift will be discussed in Sect. 5in the context of stability estimates. Remark 1.1 The qualitative Runge approximation property as a key tool for studying fractional Schrödinger type inverse problems had been introduced in [20]. In order to infer such a result, the authors of [20] built on the unique continuation property for fractional Schrödinger equations in the form of Carleman estimates which had been derived in [35], c.f.also[14,15,19,36,47,54]forrelated unique continuation results for fractional Schrödinger equations. For variable coefficient fractional Schrödinger operators, the authors of [18] utilized Almgren’s frequency function to derive such a property. In the context of nonlocal elliptic equations, this had earlier been employed by [15,54], c.f. also Section 7 in [35]for the derivation of unique continuation properties with variable coefficients. Letusputtheseresultsintothecontextof the literature on the fractional Calderón problem: The problem was first introduced by [20], where the authors treated the case with c∈L∞(), b=0ands∈(0,1), and proved a global uniqueness result for c. For more general nonlocal variable coefficient Schrödinger operators, the fractional Calderón problem was studied in [18]. The techniques based on Runge approximation are strong enough to deal with the case of semilinear equations [32] and low regularity, almost critical function spaces for the potential [38]. Even single measurement results are possible [19] (c.f. the discussion below). Moreover, these techniques have been extended to other nonlocal problems [7,39]in a slightly different context. Also, for positive and general potentials, monotonicity inversion formulas have been successfully discovered in [26,27]. Very recently and independently from our work, uniqueness results have been obtained for equations with nonlocal lower order contributions [4]. In addition to uniqueness, stability is of central importance in inverse problems. Stability results for the fractional Calderón problem were first obtained in [38,40], where optimal logarithmic stability estimates had been derived (c.f. also [41] for improvements of this if structural apriori conditions like the finiteness of the underlying function space are satisfied). It is possible to extend the logarithmic estimates to the setting of the fractional Calderón problem with drift. Here we obtain the following result: 123
91 Page 6 of 46 M. Ceki´ cetal. Theorem 1.3 (Logarithmic stability) Let s ∈(1 2,1),⊂Rn,n≥1be a bounded open, non-empty smooth domain. Let W1,W2be open, non-empty sets with W 1,W2⊂e. Assume that for some constants M >0,δ>0 bjW1−s+δ,∞() +cjHs() +cjW1,n+δ() ≤M, and that supp(bj), supp(cj)for j =1,2. Then for some constants μ>0and C >0 which depend on , W1,W2,n,s,M,δ, we have c1−c2H−s() +b1−b2H−s() ≤Clog(b1,c1−b2,c2∗)−μ, if b1,c1−b2,c2∗≤1,whereA∗:= sup{(Af1,f2)W2:f1∈ Hs(W1), f2∈ Hs(W2)}. As in [38] this relies on quantitative Runge approximation arguments, which are derived from quantitative unique continuation properties. We adapt the arguments from [38]toinfer these results for the fractional Schrödinger equation with drift. Last but not least, based on the higher order Runge approximation property (Theorem 1.2 (b)), we can deduce finite measurements uniqueness results for the fractional Calderón problem with drift. Theorem 1.4 (Finite measurements uniqueness) Let ⊂Rnbe a bounded, non-empty domain with a C∞-smooth boundary. Let W ⊂ebe an open, non-empty smooth set such that W∩=∅.Lets ∈(1 2,1)and assume that bj∈C∞ c()n,c j∈C∞ c() satisfy (1.5)with supp(bj), supp(cj)for j =1,2.Thereexistn+1exterior data f1,..., fn+1∈C∞ c(W)such that if b1,c1(fl)=b2,c2(fl)for l ∈{1,...,n+1}, then b1=b2and c1=c2. Moreover, the set of exterior data f1,..., fn+1, which satisfies this property forms an open and dense subset in C∞ c(W). This is analogous to the single measurement results in [19] for the fractional Schrödinger equation, c.f. also [6] for a single measurement result on the detection of an embedded obstacle. However, compared to [19] a word of caution is needed here: In contrast to the result from [19]itisnot possible to work with an arbitrary nontrivial set of measurements f1,..., fn+1. The data f1,..., fn+1have to be chosen appropriately from a set which depends on the unknowns b,c. This is similar to results on hybrid inverse problems, c.f. [1,2]. While the dependence of the admissible exterior data on the unknown drift field and potential seems like a serious restriction at first sight, we emphasise that by proving that the data f1,..., fn+1can be chosen in an open and dense set in C∞ c(W), we show that the set of admissible exterior data is very large: Given a (random) exterior measurement f1,..., fn+1∈C∞ c(W), our result states that an arbitrarily small perturbation of this yields an admissible exterior datum from which we can reconstruct the drift field band potential c. This might also be of interest in the setting of hybrid inverse problems for which we could not find a statement on an open and dense set of admissible measurements. We plan to address this in future research. For an overview about the fractional Calderón problem, we refer to the surveys [37,46]. 1.2 Outline of the remaining article The paper is organized as follows. In Sect. 2, we review the notion of a weak solutions of the fractional Schrödinger equation with drift. With this at hand, we define the DN map 123
The Calderón problem for the fractional Schrödinger equation… Page 7 of 46 91 rigorously. Section 3demonstrates the L2-Runge approximation property, which proves Theorem 1.2(a). We will prove the global uniqueness result of Theorem 1.1 in Sect. 4,which shows that the nonlocal Calderón problem does not enjoy a gauge invariance in contrast to its local analogue. In Sect. 5, we also prove the stability result of Theorem 1.3 for the fractional Calderón problem with drift and potential with respect to the associated DN maps. In Sect. 6, in the end of this work, we present the proof of Theorem 1.4, where we prove several points on the reconstruction from finitely many exterior measurements. In addition, we study generic unique determination results via singularity theory in the “Appendix”, which is useful to construct the open and dense subset for the exterior data stated in Theorem 1.4. 2 The fractional Schrödinger equation with drift In this section, we recall the relevant function spaces, prove the well-posedness of the fractionalSchrödingerequationwithdriftandintroduceandderivepropertiesoftheDirichlet- to-Neumann map associated with (1.4). 2.1 Preliminaries We begin by recalling the relevant fractional Sobolev spaces on (bounded) domains. We define the L2-based fractional Sobolev spaces as follows: for 0 <s<1, we consider the fractional Sobolev spaces Hs(Rn)=Ws,2(Rn)with the norm uHs(Rn):= F−1ξsuL2(Rn), where ξ=(1+|ξ|2)1 2.LetO⊂Rnbe an arbitrary non-empty open set and 0 <s<1, then we define: Hs(O):= {u|O;u∈Hs(Rn)}, Hs(O):= closure of C∞ c(O)in Hs(Rn), Hs 0(O):= closure of C∞ c(O)in Hs(O), (2.1) and Hs O:= {u∈Hs(Rn);with supp(u)⊂O}. The norm of Hs(O)is denoted by uHs(O):= inf vHs(Rn);v∈Hs(Rn)and v|O=u. It is known that Hs(O)⊆Hs 0(O),andthatHs Ois a closed subspace of Hs(Rn). Further we have for arbitrary non-empty open sets O Hs(O)∗= H−s(O)and Hs(O)∗=H−s(O). Remark 2.1 When O⊂Rnis a bounded Lipschitz domain, we have that for any s∈R, Hs(O)=Hs OHs 0(O). If s>−1 2and s/∈{1 2,3 2,...}the last inclusion also becomes an equality. 123
91 Page 8 of 46 M. Ceki´ cetal. We further denote the homogeneous fractional Sobolev spaces as ˙ Hs(Rn),where ˙ Hs(Rn):= {u:Rn→R;F−1{|ξ|sˆu}L2(Rn)<∞}. We define the associated semi-norm as u˙ Hs(Rn):= F−1{|ξ|sˆu}L2(Rn). Note that the norm ·Hs(Rn)is equivalent to the norm ·L2(Rn)+·˙ Hs(Rn). For a more detailed introduction to fractional Sobolev spaces and related results, we refer the readers to [12,33]. Since we will use this for our drift fields, we also recall the Lpbased fractional Sobolev spaces: we set uWs,p(Rn):= DsuLp(Rn),whereξ=(1+|ξ|2)1/2and m(D)u=F−1{m(ξ)u(ξ)}for m∈C∞(Rn)such that mand all its derivatives are polynomially bounded, and uis a tempered distribution. For a non-empty open set O⊂Rnand p>1, we then define the space Ws,p(O)by Ws,p(O)={u|O;u∈Ws,p(Rn)}. This is equipped with the associated norm uWs,p(O)=inf{wWs,p(Rn);w∈Ws,p(Rn), w|O=u}. We also define Ws,p 0(O):= closure of C∞ c(O)in Ws,p(O). In the sequel, we will only use these more general Ws,pfunction spaces to quantify the size of the drift field b. We conclude this section by recalling a fractional Poincaré type inequality: Lemma 2.2 Let n ≥1and let s ∈(0,1). Assume that ⊂Rnis non-empty, open and bounded. Then, there exists a constant C >0such that vL2() ≤C(diam())s(−)s/2vL2(Rn)for v∈ Hs(), where diam() denotes the diameter of . We present the proof for self-containedness but follow the idea from the appendix in [42]. Proof of Lemma 2.2 We first assume that v∈C∞ c(). The result will then follow by density of C∞ c() in Hs().Letx∈be arbitrary and let x=x+2diam() x |x|. Let furthervbe the Caffarelli-Silvestre [5] extension of v, i.e. let vbe the solution of ∇·x1−2s n+1∇v=0inRn+1 +, v=von Rn×{0}. Then the fundamental theorem of calculus and the support condition for vimply that for any r∈(diam(), ∞) 123
The Calderón problem for the fractional Schrödinger equation… Page 15 of 46 91 Remark 2.10 We point out that: (a) Relying on our assumption (1.5), and combining this with the results of Proposition 2.6 and the Fredholm alternative, then implies the well-posedness of our problem (1.4). In particular, the Fredholm alternative also yields that (1.5) is equivalent to the following condition (for example, see [33, Theorem 2.27] or [21, Chapter 5.3]) w∗∈Hs(Rn)is a solution of (−)sw∗−∇·(bw∗)+cw∗=0inwith w∗=0ine, then we have w∗≡0. (b) The proofs of Lemma 2.4 and Proposition 2.6 relied on the fact that the fractional Laplacian was the leading order operator of our equations which was ensured by the condition 1 2<s<1. In particular, the above arguments and results do not persist in the regime 0 <s≤1 2. Now, let ⊂Rnbe a bounded, non-empty open set and 1 2<s<1. We consider the Dirichlet problem (2.2) with a zero source function, i.e., ((−)s+b·∇+c)u=0in, u−f∈ Hs(), (2.15) for some f∈Hs(e). Recall that the eigenvalue condition (1.5) in combination with Proposition 2.6 shows that there exists a unique solution u∈Hs(Rn)of (2.15). We would like to point out that the solution uhere depends only on fmodulo Hs(). In effect, we emphasize that the solution of the Dirichlet problem (2.15) is only affected by the exterior datum. Therefore, following [18,20], we introduce the quotient space X=Hs(Rn)/ Hs(). We will denote the equivalence class of f∈Hs(Rn)by [f]. We also recall that for a bounded, non-empty open Lipschitz set, we have X=Hs(e)by Remark 2.1. Based on the well-posedness of (1.4), one can define the DN map rigorously as follows. Definition 2.11 (DN map)Let⊂Rn,n≥1, be a bounded Lipschitz domain, let s∈(1 2,1) and let b∈W1−s,∞()n,c∈L∞().LetBb,c(·,·)be given as (2.3). Then we define the Dirichlet-to-Neumann operator associated with the Eq. (1.4)as b,c:X→X∗,( b,c[f],[g])=Bb,c(uf,g), where f,g∈Hs(Rn)and ufis the weak solution to (1.4) with exterior datum f. We first remark that this is well-defined, since by the definition of a solution to (1.4)we have Bb,c(uf,g)=Bb,c(uf,g+ψ)for any ψ∈ Hs(). Also Bb,c(uf+ϕ,g)=Bb,c(uf,g) for ϕ∈ Hs(),sinceuf+ϕ=uf(by the uniqueness of solutions). Finally, the boundedness of the DN map b,c:X→X∗follows easily, once we have the multiplier estimate (2.8) and part (a) of Proposition 2.6. Remark 2.12 Furthermore, we remark that a formal calculation using Definition 2.11 yields (b,c[f],[g])=Bb,c(uf,g) =Rn (−)s/2uf(−)s/2gdx + b·∇ufgdx + cu fgdx =e g(−)sufdx,(2.16) 123
91 Page 16 of 46 M. Ceki´ cetal. where ufis the weak solution to (1.4) with exterior datum f. Then from (2.16), one can formally obtain that b,c[f]=(−)sufe, which gives evidence of (1.6). We remark that in order to make this calculation rigorous, higher regularity assumptions have to be imposed. We refer to Section 3, Lemma 3.1, in [20] for details on this. We may also consider the adjoint problem with a zero source term: (−)su∗−∇·(bu∗)+cu∗=0in, u∗−f∈ Hs(), (2.17) Then by using the eigenvalue condition (1.5) together with Proposition 2.6, this problem has a unique solution u∗. With slight abuse of notation, one can analogously define the DN map ∗ b,cof the adjoint bilinear form (2.4), i.e., ∗ b,c:X→X∗,( ∗ b,c[f],[g])=B∗ b,c(u∗ f,g). Here u∗ f∈Hs(R)nis a weak solution of (2.17) with exterior data f. Next, let b,c:X→X∗be the adjoint operator with respect to the DN map b,c, i.e., the adjoint DN map b,cis defined via (b,cf,g)=(f, b,cg), for f,g∈Hs(Rn). Then we further observe the following relation between the DN map associated with (1.4) and the DN map of the adjoint bilinear form, which states that the adjoint DN map b,cand the DN map ∗ b,cwith respect to the adjoint equation coincide. In order to simplify notation, here and in the sequel, we drop the brackets [·]for the elements of X. Lemma 2.13 Let ⊂Rn,n≥1, be a bounded, non-empty open Lipschitz set, s ∈(1 2,1), b∈W1−s,∞()n,c∈L∞() and assume that (1.5)holds. Let Bb,c(·,·),B ∗ b,c(·,·)and b,c, ∗ b,cbe given as above. Then, for all f ,g∈X (b,cf,g)=(f, ∗ b,cg). (2.18) Proof.The claim follows from the independence of the DN map and the adjoint DN map of the extension of ginto in the duality pairing from Definition 2.11.Letufbe a solution to (2.2) with exterior data f∈Xand let u∗ gbe a solution of the adjoint equation (2.17) with exterior data g∈X. Then, by (2.11) and the definition of b,c,∗ b,c (b,cf,g)=Bb,c(uf,u∗ g)=B∗ b,c(u∗ g,uf)=(∗ b,cg,f)=(f, ∗ b,cg). With this at hand, we seek to derive a corresponding Alessandrini type identity. It will play a key role in our uniqueness and stability arguments. Lemma 2.14 (Alessandrini identity) Let ⊂Rn,n≥1, be a bounded Lipschitz domain and 1 2<s<1.Letb j∈W1−s,∞()nand cj∈L∞() be such that (1.5)holds for j =1,2. For any f1,f2∈X,wehave (b1,c1−b2,c2)f1,f2X∗×X=((b1−b2)·∇u1,u∗ 2)+((c1−c2)u1,u∗ 2),(2.19) where u1∈Hs(Rn)is the solution to ((−)s+b1·∇+c1)u1=0in and u∗ 2∈Hs(Rn) is the solution to (−)su∗ 2−∇·(b2u∗ 2)+c2u∗ 2=0in with u1−f1∈ Hs() and u2−f2∈ Hs(). 123
The Calderón problem for the fractional Schrödinger equation… Page 17 of 46 91 Proof.By (2.18), one has ((b1,c1−b2,c2)f1,f2)=(b1,c1f1,f2)−(f1, ∗ b2,c2f2) =Bb1,c1(u1,u∗ 2)−B∗ b2,c2(u∗ 2,u1) =((b1−b2)·∇u1,u∗ 2)+((c1−c2)u1,u∗ 2). Last but not least, we define the Poisson operator associated with the Eq. (1.4): Definition 2.15 Let ⊂Rn,n≥1, be a bounded, non-empty open Lipschitz set. Let s∈(1 2,1)and assume that (1.5) holds. Then, the Poisson operator Pb,cassociated with (1.4) is defined as Pb,c:X→Hs(Rn), f→ uf,(2.20) where uf∈Hs(Rn)with uf−f∈ Hs() denotes the unique solution to (1.4). 3 Approximation property In this section we discuss the Runge approximation property for solutions to (1.4). To this end, we first recall the strong uniqueness property for the fractional Laplacian, which was provedin[20, Theorem 1.2]. Proposition 3.1 (Global weak unique continuation) Let n ≥1,s∈(0,1)and u ∈H−r(Rn) for some r ∈R. Assume that for some non-empty open set W ⊂Rnwe have u=(−)su=0in W. Then u ≡0in Rn. By using the above strong uniqueness property and a duality argument, one can derive the following Runge approximation [(which then immediately entails Theorem 1.2 (a)]. Lemma 3.2 For n ≥1and 1 2<s<1,let⊂Rnbe a bounded, non-empty open Lipschitz set, b ∈W1−s,∞()nand c ∈L∞() and let Pb,cdenote the Poisson operator from Definition 2.15. Further let W ⊂ebe an arbitrary non-empty open set. Let D:= Pb,cf−f:f∈C∞ c(W). Then we have the following results: (a) The set Dis dense in L2(). (b) The set Dis dense in Hs(). Proof.The proofs of (a) and (b) are similar and follow from a duality argument. Hence, we only need to prove the case (b). First, it is easy to see that D⊂ Hs(). By the Hahn-Banach theorem, it suffices to show that for any F∈( Hs())∗such that F(v) =0foranyv∈D, we must have F≡0. Since F(v) =0foranyv∈D,wehave F(Pb,cf−f)=0for f∈C∞ c(W). (3.1) Next, we claim that F(Pb,cf−f)=−Bb,c(f,ϕ), for f∈C∞ c(W), (3.2) 123
91 Page 18 of 46 M. Ceki´ cetal. where ϕ∈ Hs() is the unique solution of (−)sϕ−∇·(bϕ) +cϕ=Fin with ϕ=0ine. We remark that by Proposition 2.6 and the assumption (1.5) this problem is well-posed. In its weak form, it becomes B∗ b,c(ϕ, w) =F(w) for w∈ Hs(). We next address the proof of (3.2). Let f∈C∞ c(W)and uf:= Pb,cf∈Hs(Rn),then uf−f∈ Hs() and F(Pb,cf−f)=B∗ b,c(ϕ, uf−f)=Bb,c(uf−f,ϕ)=−Bb,c(f,ϕ), where we have utilized (2.11) and the fact that ufis a solution to (1.4)andϕ∈ Hs().By means of the relations (3.1)and(3.2), we hence conclude that Bb,c(f,ϕ)=0for f∈C∞ c(W). Since f=0inRn\Wand ϕ∈ Hs(),Bb,c(f,ϕ)=0 implies that 0=((−)s/2ϕ,(−)s/2f)Rn=((−)sϕ, f)Rn,for f∈C∞ c(W). In particular, ϕ∈Hs(Rn)satisfies ϕ=(−)sϕ=0inW. By invoking Proposition 3.1, this implies ϕ≡0inRnand therefore F≡0 as well. The above lemma proves Theorem 1.2 (a). The proof of Theorem 1.2 (b) is postponed to the last section of this paper. Without major modifications of the above argument, we also obtain the Runge approximation for the adjoint equation. Corollary 3.3 For n ≥1and 1 2<s<1,let⊂Rnbe a bounded, non-empty open Lipschitz set, b ∈W1−s,∞()nand c ∈L∞() and let P∗ b,cdenote the Poisson operator for the Eq. (2.5),i.e.let P∗ b,c:X→Hs(Rn), f∗→ u∗ f∗, where u∗ f∗∈Hs(Rn)is the solution of the adjoint equation (−)su∗ f∗−∇·(bu∗ f∗)+cu∗ f∗=0in with u∗ f∗=f∗in e. Let W ⊂ebe an arbitrary non-empty open set. Then the sets D∗:= P∗ b,cf∗−f∗:f∗∈ C∞ c(W)is dense in L2() and Hs(). 4 Global uniqueness In this section, we prove the global uniqueness result from Theorem 1.1 which follows from the knowledge of the DN map b,cand its adjoint. 123
The Calderón problem for the fractional Schrödinger equation… Page 19 of 46 91 Proof of Theorem 1.1 Since b1,c1f|W2=b2,c2f|W2for any f∈C∞ c(W1),whereW1,W2 are arbitrary, but fixed non-empty open sets in e, by using the Alessandrini identity (2.19), we have (b1−b2)·∇u1u∗ 2+(c1−c2)u1u∗ 2dx =0,(4.1) whereu1∈Hs(Rn)isthesolutionto((−)s+b1·∇+c1)u1=0inandu∗ 2∈Hs(Rn)isthe solution to (−)su∗ 2−∇·(b2u∗ 2)+c2u∗ 2=0inwith exterior data u1|e=f1∈C∞ c(W1) and u∗ 2|e=f∗ 2∈C∞ c(W2). In the sequel, we seek to recover the potential and the drift coefficients separately. Step 1: Recovery of the potential c.Letψ2∈C∞ c() be arbitrary. Then choose ψ1∈ C∞ c() such that ψ1=1 on the set supp(ψ2). By the Runge approximation of (−)s+b·∇+cand its adjoint (see Lemma 3.2 and Corollary 3.3), there exist sequences of solutions {u1 j}∞ j=1and {u2,∗ j}∞ j=1in Hs(Rn)such that ((−)s+b1·∇+c1)u1 j=(−)su2,∗ j−∇·(b2u2,∗ j)+c2u2,∗ j=0in, supp(u1 j)⊂1and supp(u2,∗ j)⊂2, u1 j|=ψ1+r1 jand u2,∗ j|=ψ2+r2,∗ j, (4.2) where 1, 2are non-empty open sets in Rncontaining ,andr1 j,r2,∗ j→0 strongly in Hs() as j→∞. With these solutions at hand, we observe that as r1 j,r2,∗ j∈ Hs() (b1−b2)·∇u1 ju2,∗ jdx≤ (b1−b2)·∇r1 jr2,∗ jdx+ (b1−b2)·∇r1 jψ2dx + (b1−b2)·∇ψ1ψ2dx+ (b1−b2)·∇ψ1r2,∗ jdx ≤∇r1 jHs−1()(b1−b2)r2,∗ j H1−s() +∇r1 jHs−1()(b1−b2)ψ2 H1−s() +∇ψ1Hs−1()(b1−b2)r2,∗ j H1−s() ≤Cb1−b2W1−s,∞()∇r1 jHs−1()r2,∗ j H1−s() +Cb1−b2W1−s,∞()∇r1 jHs−1()ψ2 H1−s() +Cb1−b2W1−s,∞()r2,∗ j H1−s()∇ψ1Hs−1() ≤Cb1−b2W1−s,∞()r1 j Hs()r2,∗ j Hs() +r1 j Hs()ψ2 H1−s() +r2,∗ j Hs()ψ1 Hs() →0,as j→∞. (4.3) Here we have used that b1−b2∈W1−s,∞() is a bounded multiplier from H1−s() into itself (which follows from the same argument as in the proof of Proposition 2.6)and the estimate u H1−s() ≤Cu Hs() for s∈(1 2,1).Further,wemadestronguseofthe 123
91 Page 20 of 46 M. Ceki´ cetal. support assumptions for ψ1and ψ2, which in particular allow us to drop the third term in the first estimate in (4.3). Inserting these solutions {u1 j},{u2,∗ j}and the estimate (4.3)into(4.1) and taking j→∞, together with the assumptions on ψ1and ψ2,wederive (c1−c2)ψ2dx =0. Since ψ2∈C∞ c() was arbitrary, by density of C∞ c() in L2() and the previous identity, we obtain that c1=c2in . Step 2: Recovery of the drift b. Since we have c1=c2in ,(4.1) becomes (b1−b2)·∇u1u∗ 2dx =0.(4.4) Fixanarbitraryψ2∈C∞ c(). Then choose ψxk∈C∞ c() equal xkon supp(ψ2), where for k=1,2,...,nthe function xkdenotes the restriction to the k-th component of x. By using the Runge approximation again as in (4.2), we can find sequences of solutions u1 j=ψxk+r1 jand u2,∗ j=ψ2+r2,∗ j, with r1 j,r2,∗ j→0 strongly in Hs() as j→∞. Plugging the Runge approximations of the functions ψxkand ψ2into (4.4), we obtain (b1−b2)kψ2dx + (b1−b2)·∇r1 jψ2dx + (b1−b2)·∇r1 jr2,∗ jdx + (b1−b2)·∇ψxkr2,∗ jdx =0, for k=1,2,...,n,where(b1−b2)kdenotes the k-th component of the vector valued function b1−b2. By arguing similarly as in (4.3), in the limit j→∞we arrive at (b1−b2)kψ2dx =0fork=1,2,...,n. Since ψ2∈C∞ c() is arbitrary and as C∞ c() is dense in L2(), we also conclude that (b1−b2)k=0forallk=1,2,...,n. Therefore, b1=b2, which completes the proof. 5 Stability In this section, we study the stability result for the fractional Schrödinger equation with drift. 5.1 Auxiliary results We begin by proving several auxiliary results, which will be used in deducing a quantitative Runge approximation result in the next section. To this end, we will mainly be studying the dual equation to (1.4) (−)sw−∇·(bw) +cw=vin , w=0ine.(5.1) Throughout this section, we assume that the drift field band the potential csatisfy (1.5). 123
The Calderón problem for the fractional Schrödinger equation… Page 21 of 46 91 Lemma 5.1 Let ⊂Rnbe an open, non-empty bounded Lipschitz domain. Let s ∈(1 2,1), b∈W1−s,∞()n,c∈L∞(),v∈H−s() and assume that w∈Hs is the solution to (5.1). Then there exists a constant C >1independent of v, w such that C−1vH−s() ≤wHs ≤CvH−s(). Proof.The upper bound follows from the well-posedness result of Proposition 2.6 and the assumption (1.5). It hence remains to discuss the lower bound. To this end, we use the triangle inequality and the equation (5.1), which lead to vH−s() ≤(−)swH−s() +∇·(bw)H−s() +cwH−s() ≤(−)swH−s(Rn)+∇·(bw)H−s() +cwH−s() ≤wHs +∇·(bw)H−s() +cwH−s(). (5.2) We estimate the terms with the drift and the potential separately. On the one hand, by integration by parts, we observe that for the drift, we can find a constant C>0 independent of band wsuch that ∇ ·(bw)H−s() =sup φ Hs()=1|(∇·(bw), φ)| ≤sup φ Hs()=1bw H1−s()∇φHs−1() ≤Cw H1−s()bW1−s,∞() sup φ Hs()=1φ Hs() ≤CwHs bW1−s,∞(), whereweusedthatfors∈(1/2,1)it holds that w H1−s() ≤Cw Hs . On the other hand, in order to estimate the potential c, we use Hölder’s and Poincaré’s inequalities to observe cwH−s() =sup φHs =1|(cw, φ)| =sup φHs =1cL∞()wL2()φL2() ≤CcL∞()wHs . Inserting the estimates for the contributions involving b,cinto (5.2) then concludes the proof of Lemma 5.1. Next, we prove Vishik–Eskin type estimates for the fractional Schrödinger equation with drift, c.f. [52]andalso[23, Section 3]. Lemma 5.2 (Vishik–Eskin) Let δ∈(−1 2,1 2).Let⊂Rnbe an open, non-empty bounded Lipschitz domain. Let s ∈(1 2,1),b∈W1−s+δ,∞()n,c∈L∞() satisfy (1.5),v∈H−s() and assume that w∈Hs is the solution to (5.1). Then there exists a constant C >1such that wHs+δ ≤CvH−s+δ(). The argument for this follows from a perturbation of the original estimates due to Vishik and Eskin [52]. 123
91 Page 22 of 46 M. Ceki´ cetal. Proof.We rewrite the Eq. (5.1)as (−)sw=Gin , w=0ine, where G=∇·(bw) −cw+v. Then the estimates of Vishik and Eskin (see [24, Theorem 3.1]) yield that wHs+δ ≤CGH−s+δ() ≤C(cwH−s+δ() +∇·(bw)H−s+δ() +vH−s+δ()). (5.3) We estimate the drift and the potential contributions separately: for δ∈(0,min{2s−1,1 2}) by integration by parts and duality ∇ ·(bw)H−s+δ() =sup φ Hs−δ()=1(∇·(bw), φ)L2() =sup φ Hs−δ()=1(bw, ∇φ)L2() ≤sup φ Hs−δ()=1bw H1−s+δ()∇φHs−1−δ() ≤sup φ Hs−δ()=1bW1−s+δ,∞()w H1−s+δ()φ Hs−δ() ≤CbW1−s+δ,∞()wHs . The potential term is estimated by Hölder’s and Poincaré’s inequalities cwH−s+δ() =sup φHs =1|(cw, φ)| =sup φHs−δ =1cL∞()wL2()φL2() ≤CcL∞()wHs . Combining these bounds with the estimate wHs ≤CvH−s(), which follows from the well-posedness result of Proposition 2.6 and returning to (5.3) concludes the argument. 5.2 A quantitative approximation result As a final preparation for the stability proof, in this section we deduce a quantitative Runge approximation result for fractional Schrödinger equations with drift terms: Proposition 5.3 For n ≥1,let⊂Rnbe a bounded, non-empty open set with a C∞-smooth boundary and s ∈(1 2,1).LetW ebe non-empty and open such that ∩W=∅.Further suppose that δ∈(0,2s−1 2)and that b ∈W1−s+δ,∞()n,c∈L∞(). Then, for each >0 and for each v∈Hs there exists f∈Hs Wsuch that the following approximation estimates hold true Pb,cf−f−vHs−δ ≤vHs ,fHs W≤CeC−μ(δ) vHs−δ . As in [38] the approximation property follows from a quantitative unique continuation result: 123
The Calderón problem for the fractional Schrödinger equation… Page 23 of 46 91 Proposition 5.4 For n ≥1,let⊂Rnbe a bounded, non-empty open set with a C∞-smooth boundary and s ∈(1 2,1).LetW ebe non-empty and open such that ∩W=∅.Further suppose that δ∈(0,2s−1 2)and that b ∈W1−s+δ,∞()n,c∈L∞(). Assume that for each v∈Hs−δ it holds that vHs−2δ ≤C log CvHs−δ (−)swH−s() σ(δ)vHs−δ ,(5.4) where w∈Hs is the solution of (5.1). Then, for each >0and for each v∈Hs there exists f∈Hs Wsuch that the following approximation estimate holds true Pb,cf−f−vHs−δ ≤vHs ,fHs W≤CeC−μ(δ) vHs−δ . This statement is the exact analogue of Lemma 8.2 in [38] with the argument for Proposition 5.4 following verbatim as in the proof of Lemma 8.2 in [38]: Indeed, the only property of the Eq. (5.1) which is used, is its mapping property. By virtue of the regularity assumptions on b,cand the well-posedness results of Proposition 2.6, solutions to (5.4) also enjoy exactly the same regularity and compactness estimates as the ones from [38]. Proof of Proposition 5.4 We consider the operator A:Hs W→Hs →Hs−δ ,f→ j(Pb,c(f)−f), where j:Hs →Hs−δ is a compact embedding. Thus, Ais a compact, injective operator. Hereinjectivityfollows fromthestrong uniquenessresultofProposition3.1. Inaddition,by(a slightadaptationof)Lemma3.2,ithasadenserange.Thus,wemayapplythespectraltheorem for compact operators and obtain sequences {μj}∞ j=1⊂R+decreasing, and {wj}∞ j=1⊂Hs W such that A∗Awj=μjwj. The set {wj}∞ j=1forms an orthonormal basis with respect to the Hs Wscalar product. By the density of the range of A, it also follows that the set {ϕj}∞ j=1:= 1 σjAwj∞ j=1with σj:= μ 1 2 j is an orthonormal basis of Hs−δ . As a consequence, (σj,wj,ϕj)∞ j=1⊂R+×Hs W×Hs−δ is the singular value decomposition of A. By the characterization of A∗, the assumption (5.4) can be rephrased as the estimate vHs−2δ ≤C log CvHs−δ A∗vHs W σ(δ)vHs−δ .(5.5) Using this and the singular value decomposition from above, we deduce the approximation property along the same lines as in [38, Lemma 8.2]: Let ¯v∈Hs .Forα∈(0,1),let rα:= σj≤α(¯v,wj)wj∈Hs−δ and Rα¯v:= σj>α σ−1 j(¯v,wj)ϕj∈Hs W. Therefore, on the one hand, we have Rα(¯v)Hs W≤C α¯vHs−δ .(5.6) 123
91 Page 24 of 46 M. Ceki´ cetal. On the other hand, ¯v−ARα(¯v)2 Hs−δ = σj≤α|(¯v,wj)Hs−δ |2≤(¯v,rα)Hs−δ =(¯v,rα)Hs−δ(Rn)≤¯vHs(Rn)rαHs−2δ(Rn)=¯vHs rαHs−2δ ≤¯vHs C log CrαHs−δ A∗rαHs W σ(δ)rαHs−δ ≤¯vHs C log C1 ασ(δ)rαHs−δ . (5.7) Optimizing (5.6), (5.7)inαthen implies the claim. As a second main ingredient in the proof of Proposition 5.3, we rely on Theorem 5.1 from [38]. This is a propagation of smallness estimate from the boundary into the bulk for the Caffarelli–Silvestre extension of a general function. It does not use the specific equation at hand. For completeness, we recall the statement: Proposition 5.5 ([38], Proposition 5.1) For n ≥1,let⊂Rnbe an open, non-empty bounded and smooth domain. Let W ebe open, non-empty bounded and Lipschitz with ∩W=∅. Suppose that s ∈(0,1)and that wis a solution to ∇·x1−2s n+1∇w=0in Rn+1 +, w=won Rn×{0},(5.8) where w∈Hs(Rn)is a function which vanishes in an open neighbourhood of W. Assume further that for some constants C1>0and δ>0one has the a priori bounds x1−2s n+1∂n+1wH−s(W)≤η, x 1−2s 2 n+1wL2(Rn×[0,C1])+x 1−2s 2 n+1∇wL2(Rn+1 +)+x 1−2s 2−δ n+1∇wL2(Rn+1 +)≤E, for some constants η, E with E η>1. Then, there exist constants C >1,μ>0which depend on n,s,C1,δ,,W such that x 1−2s 2 n+1wL2(2×[0,1])+x 1−2s 2 n+1∇wL2(2×[0,1])≤CE log CE (−)swH−s(W)μ. Here 2:= {x∈Rn:dist(x,)≤min{1 2dist(W,), 2}}. We remark that although Proposition 5.1 in [38] is formulated for the Caffarelli–Silvestre extension of the solution to the fractional Schrödinger equation which is studied there (in [38] the situation b=0 is considered), only the vanishing of win a neighbourhood of Wis used in the argument which leads to Proposition 5.1. We thus do not present the proof of this result, but refer the reader to Section 5 in [38]. We will apply it to solutions to (5.1)inthe sequel. With Propositions 5.4 and 5.5 at hand, we address the proof of the approximation result of Proposition 5.3: 123
The Calderón problem for the fractional Schrödinger equation… Page 31 of 46 91 6.2 Finite measurements reconstruction without openness In this section, we discuss a first result towards the proof of Theorem 1.4 by using higher order Runge approximation [Theorem 1.2 (b)]. However, before proving the full result of Theorem 1.4, we prove a weaker (but technically considerably easier) result, which still proves finite measurement reconstruction but only asserts that the set of measurement data contains a non-empty open set (a priori this argument does not prove the density of the set of good data). The technically more involved statement on the openness and density of the set of good data will be proved in the subsequent sections. Proof of Theorem 1.4 without the density result We show that for any drift b∈C∞ c()n and any potential c∈C∞ c() with supp(b), supp(c), there exist exterior Dirichlet data f1,..., fn+1such that the band ccan be uniquely reconstructed from the knowledge of f1,..., fn+1and b,c(f1),..., b,c(fn+1). By Runge approximation in Ckspaces [see Theorem 1.2 (b)], we have that for any g∈ C∞() there exists a sequence of solutions {uj}j∈Nto the fractional Schrödinger equation (1.4) with drift (and compactly supported coefficients) such that for any k∈N g−d−sujCk() →0asj→∞, where d(x)=dist(x,∂) if x∈is sufficiently close to the boundary of and d(x)is extended to a positive function smoothly into the interior of . Next, we choose n+1 smooth functions g1,...,gn+1defined in with the property that h(g1,g2,...,gn+1)(x):= det ⎛ ⎜ ⎜ ⎜ ⎝ ∂1g1... ∂ ng1g1 . . .. . .. . .. . . ∂1gn... ∂ ngngn ∂1gn+1... ∂ ngn+1gn+1 ⎞ ⎟ ⎟ ⎟ ⎠(x)= 0.(6.2) An example for this would be the functions gj=xjfor j∈{1,...,n}and gn+1=1. We further set gl=d−s(χgl),whereχ∈C∞ c() and χ=1onK,forl∈{1,...,n+1}and apply Theorem 1.2 (b). As a consequence, for each l∈{1,...,n+1}, and in any compact subset Kthere exists a sequence of solutions {ul j,K}j∈Nsuch that for any k∈N d−s(gl−ul j,K)Ck(K)→0asj→∞. As K⊂and as d(x)>0inK,wethenalsohave gl−ul j,KCk(K)→0asj→∞. Hence, choosing j≥j0large enough and Ksuch that supp(b)∪supp(c)K, we obtain that h(u1 j,K,...,un+1 j,K)(x)=det ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ∂1u1 j,K... ∂ nu1 j,Ku1 j,K . . .. . .. . .. . . ∂1un j,K... ∂ nun j,Kun j,K ∂1un+1 j,K... ∂ nun+1 j,Kun+1 j,K ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ (x)= 0.(6.3) 123
91 Page 32 of 46 M. Ceki´ cetal. As a consequence, for these values of ul j,Kthe (linear) system for b1,...,bnand c ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ ∂1u1 j,K... ∂ nu1 j,Ku1 j,K . . .. . .. . .. . . ∂1un j,K... ∂ nun j,Kun j,K ∂1un+1 j,K... ∂ nun+1 j,Kun+1 j,K ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎜ ⎝ b1 . . . bn c ⎞ ⎟ ⎟ ⎟ ⎠=−⎛ ⎜ ⎜ ⎜ ⎜ ⎝ (−)su1 j,K . . . (−)sun j,K (−)sun+1 j,K ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ (6.4) is solvable (since the (n+1)×(n+1)matrix in the left hand side of (6.4) is invertible). Thus, from the knowledge of ul j,K,l∈{1,...,n+1}for some j≥j0, it is possible to uniquely recover the drift and the potential simultaneously. As by the global (nonlocal) unique continuation arguments in [19] (c.f. also Proposition 3.1 from above) it is possible to recover ul j,Kgiven the measurements fl j,Kand b,c(fl j,K) we infer the finite measurement recovery statement. Finally, in order to infer the openness of the set of possible exterior data, we note that for any >0 there exists δ>0 such that for any f=(f1,..., fn+1)∈C∞ c(W)n+1with f−fj,KCk c(W)<δ, k∈N, we have by boundedness of the mapping Ck c(W)n+1f→ u∈Ck(K) u−uj,KCk(K)<. Here fj,K:= (f1 j,K,..., fn+1 j,K)∈C∞ c(W)n+1are the exterior data from above, u=(u1,...,un+1)are the solutions to (1.4) corresponding to the data fand uj,K:= (u1 j,K,...,un+1 j,K)are the solutions to (1.4) corresponding to the data fj,K= (f1 j,K,..., fn+1 j,K). In particular, assuming that det ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ∂1u1 j,K... ∂ nu1 j,Ku1 j,K . . .. . .. . .. . . ∂1un j,K... ∂ nun j,Kun j,K ∂1un+1 j,K... ∂ nun+1 j,Kun+1 j,K ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠≥c>0inK, the triangle inequality implies that if >0 (and correspondingly δ) is chosen sufficiently small, also det ⎛ ⎜ ⎜ ⎜ ⎝ ∂1u1... ∂ nu1u1 . . .. . .. . .. . . ∂1un... ∂ nunun ∂1un+1... ∂ nun+1un+1 ⎞ ⎟ ⎟ ⎟ ⎠>0inK. This concludes the argument. Remark 6.3 We conclude this section by some comments on the assumptions of the theorem: (a) The compact support condition for the functions bj,cj,j=1,2, is assumed here in order to be able to apply the theory of Grubb [24]. (b) In order to obtain our result we in principle do not need the full strength of the Ck, k∈N, approximation result from [20]. It would for instance be sufficient to use a C1 approximation result for which only lower regularity on the coefficients is needed. Since 123
The Calderón problem for the fractional Schrödinger equation… Page 33 of 46 91 the theory of Grubb is however formulated in the smooth set-up, we do not optimize the regularity dependences here. (c) We again emphasize that although the variant of Theorem 1.4 whichisprovedinthis section is interesting from a theoretical point of view, a word of caution is needed as follows: In contrast to the single measurement result from [19], the exterior data f1,..., fn+1are not arbitrary. In general the specific choice of these data depends on b1,b2,c1,c2and hence the explicit choice of the functions f1,..., fn+1is not known in general. This has a similar background (see the following proof) as many results on hybrid inverse problems, where it is important that constraints are satisfied, see [1,2]. The result from this section will be improved considerably in the argument leading to the proof of Theorem 1.4. (d) In addition to the previous point, there are examples of matrices with entries satisfying elliptic equations, for which the determinant vanishes on an open set, see [10]. This indicates that the zero set of the determinant (6.3) can indeed be large. 6.3 Variations and extensions of the reconstruction result We discuss a slight variation of the reconstruction result from Sect. 6.2 by relaxing the condition that the fields b,care compactly supported in . Recall that a C∞-smooth function fis vanishing to infinite order at a point x0provided that ∂αf(x0)=0 holds for any multiindex α=(α1,...,α n)∈(N∪{0})n. Proposition 6.4 Let ⊂Rnbe a bounded domain with a C∞-smooth boundary. Let W ⊂ ebe a non-empty open, smooth domain containing an open neighbourhood of ∂.Let 1 2<s<1and assume that bj∈C∞()n,cj∈C∞() satisfy (1.5)and b1−b2,c1−c2vanish to infinite order on ∂. Then, there exist n +1exterior Dirichlet data f1,..., fn+1∈C∞ c(W)such that if b1,c1(fl)=b2,c2(fl)for l ∈{1,...,n+1}, then b1=b2and c1=c2in . Moreover, for any k ∈Nthe set of exterior data f1,..., fn+1, which satisfies this property forms an open subset in Ck c(W). This follows from an auxiliary result, which states that under geometric restrictions on the set where we measure the Dirichlet data, we may enlarge our domain and that the DN map on the larger domain is determined by the DN map on the smaller set. This is well-known in the study of local inverse problems – see e.g. [43, Lemma 4.2]. In what follows, we denote by rX Hs(Y)the set of all restrictions f|Xof functions f∈ Hs(Y)for open X,Y⊂Rn. Lemma 6.5 Assume that and with ⊂are bounded domains with Lipschitz boundaries and W1,W2⊂Rnare two non-empty open sets, such that \W1(so, in particular, W1∩ eis non-empty). Then let b1,b2∈W1−s,∞()and c1,c2∈L∞(), and 1 2<s<1, and assume c1=c2and b1=b2in \. Additionally, assume that the coefficients satisfy the eigenvalue condition (1.5)both on and on . Assume the equality of DN maps bj,cjwith respect to b1,c1f|W2=b2,c2f|W2for all f ∈rW1∩e Hs(W1), 123
91 Page 34 of 46 M. Ceki´ cetal. Then we have the equality of DN maps bj,cjwith respect to : b1,c1f|W2= b2,c2f|W2for all f ∈rW1∩ e Hs(W1). Proof.Assume u 1∈Hs(Rn)solves the Dirichlet problem for f∈ Hs(W1∩ e) ((−)s+b1·∇+c1)u 1=0in, u 1| e=f| e. Then solve the analogous Dirichlet problem with respect to the smaller domain : ((−)s+b2·∇+c2)u2=0in, u2|e=u 1|e. By using the computation from Remark 2.12 for φ∈C∞ c(e∩W2)(we need the support condition here) and the hypothesis of equality of the DN maps, we get: b1,c1f|W2∩e=b2,c2f|W2∩e, ⇒ B1(u 1,φ)=B2(u2,φ), ⇒ Rn φ(−)su 1dx =Rn φ(−)su2dx, ⇒ (−)su 1≡(−)su2on W2∩e. Here, for convenience, we have used the abbreviation Bj(·,·)=Bbj,cj(·,·)for j∈{1,2}. Furthermore, note that here we also use that u 1|e∈re∩W1 Hs(W1),since\W1. Therefore, we have the following relations: (−)s(u 1−u2)=0onW2∩eand u 1−u2=0onW2∩e.(6.5) By the strong uniqueness Proposition from 3.1, we conclude u 1≡u2on the whole of Rn. Now we proceed to compare the DN maps on the domain . We have the following chain of equalities for w∈C∞ c(W2∪) and u 1,u2as above: B 2(u2,w)=Rn (−) s 2u2·(−) s 2wdx + b2·∇u2wdx + c2u2wdx =B2(u2,w)+\ b2·∇u2wdx +\ c2u2w =B1(u 1,w)+\ b1·∇u 1wdx +\ c1u 1w =B 1(u 1,w). Here we split the integrals over and \, used the notation B 1,B 2for the bilinear form associated to the equation on , the definition of the DN map (see Definition 2.11), the fact that u2=u 1,andb1=b2and c1=c2on \. We conclude that u2solves (−)su2+b2·∇u2+c2u2=0in, with u2| e=f| e. Also, we conclude that for all was above ( b2,c2f,w)=B 2(u2,w)=B 1(u 1,w)=( b1,c1f,w). Here we use notation for the DN map on . The main claim follows by observing that we may pick arbitrary w∈C∞ c(W2). Note that we need to assume that the operators satisfy condition (1.5) on the bigger set to assume well-definedness of the DN maps. 123
The Calderón problem for the fractional Schrödinger equation… Page 35 of 46 91 Sketch of the argument for Proposition 6.4 The proof of Proposition 6.4 is similar to Section 6.2, so we only sketch the arguments here. With the auxiliary result of Lemma 6.5 at hand, we deduce the claim of Proposition 6.4 by extending to suitably such that the geometric conditions on the domains are satisfied. As our operator is not self-adjoint, in the extended domain we might possibly work with Cauchy data, as we could catch a finite number of Dirichlet eigenvalues (c.f. Remark 6.6 below on how to avoid this in some cases). However, using arguments as in [41], it is possible to deduce analogous results. Remark 6.6 (Domain monotonicity) We remark that when the drift term b=0, then it is possible to avoid dealing with Cauchy data by using perturbation of domain arguments. Indeed, for the fractional Laplacian (and self-adjoint fractional Schrödinger operators), it is possible to characterize the Dirichlet spectrum through min-max formulations [17](seealso [11] for similar settings for the classical Laplacian). Relying on the weak unique continuation property, it is possible to show the monotonicity of eigenvalues λk(1)>λ k(2)for 12and k∈N, where λkis the k-th Dirichlet eigenvalue of the fractional Laplacian. Thus, perturbing the domain suitably and only considering a finite number of eigenvalues in the case of self-adjoint fractional Schrödinger operators, one might work with the DN map instead of having to resort to Cauchy data. 6.4 Proof of Theorem 1.4 and generic properties of determinants via singularity theory In this section, we prove the full result of Theorem 1.4, i.e. we show that the set of exterior data from which we can choose n+1 measurements in order to recover the coefficients on a compactset Kasinprevioussections is open and dense. Thissignificantlyimprovestheresult from Sect. 6.2 in that the data still depend on the unknown potentials b,c,butinaprecise sense they form a large set, i.e. given (random) exterior data, we know that an arbitrarily small perturbation of them will render them admissible in our reconstruction scheme. The main idea of our argument is to relax the condition that for admissible exterior data f1,..., fn+1∈C∞ c(W)we require h(Pb,c(f1),...,Pb,c(fn+1)) = 0inK⊂, where hwas the function from (6.2). Instead, we consider data f1,..., fn+1∈C∞ c(W)such that h(Pb,c(f1),...,Pb,c(fn+1)) is only allowed to vanish to a finite (dimension-dependent) order. Then, by known results from [3, Lemma 3], it follows that the set x∈K;h(Pb,c(f1),...,Pb,c(fn+1))(x)=0 is of measure zero in K.1As a consequence, due to the continuity of b,c,itisthen possible to reconstruct both coefficients (see Lemma 6.8). Simultaneously, the set of exterior data f1,..., fn+1∈C∞ c(W)for which h(Pb,c(f1),...,Pb,c(fn+1)) vanishes only of finite (dimension-dependent) order immediately by definition is open in C∞ c(W)n+1.Thedensity of such data will be obtained via small perturbations, relying on ideas of Whitney’s work [53], which had been developed in the context of singularity theory. Technically, this is the most involved part of our arguments. Let us introduce the set of our admissible exterior conditions. 1More specifically, it is countably-C∞-rectifiable, i.e. covered by a countable union of hypersurfaces. 123
91 Page 36 of 46 M. Ceki´ cetal. Definition 6.7 Let b,csatisfy the conditions in Theorem 1.4.Letn∈Nand k(n)= √n+1∈N(i.e. k(n)is the smallest positive integer greater or equal to √n+1). Let ⊂Rnbe as in Theorem 1.4 and Kbe a compact set as in Sect. 6.2. Then, we define the set F⊂C∞ c(W)n+1to be the set of exterior data, such that (f1,... , fn+1)∈F,if h(Pb,cf1,... ,Pb,cfn+1)has at most order of vanishing k(n)−2 at each point of K. We claim that the set Fyields the desired set of exterior data for the proof of Theorem 1.4. To this end, we first show that given exterior data (f1,... , fn+1)∈F, it is possible to recover band c: Lemma 6.8 Assume that the conditions of Theorem 1.4 hold. Let (f1,... , fn+1)∈F.Then it is possible reconstruct b,c from the exterior measurements of fland b,c(fl),forl ∈ {1,... ,n+1}. Proof.We first recall that the zero set of a smooth function not vanishing to infinite order is contained in a countable union of codimension one submanifolds (see e.g. [3, Lemma 3]). In particular, by definition of the set F, we thus infer that h(Pb,c(f1),...,Pb,c(fn+1)) vanishes only on a set of Lebesgue measure zero. Let us denote this measure zero set by B⊂K. Therefore, the argument in (6.4) goes through on the set K\B.Butb,csatisfy supp(b)∪supp(c)Kand are smooth, so have unique continuous extensions to K,which can be determined from b|K\Band c|K\B. This concludes the argument for the reconstruction of b,cfrom fland b,c(fl)for l∈{1,... ,n+1}. Hence, it remains to prove the openness and density of the set F⊂C∞ 0(W)n+1. While the openness is a direct consequence of the definition of the set F,thedensity of the set F requires careful arguments. This will be the content of the remaining subsections. 6.4.1 Generic properties of determinants via singularity theory In order to deduce the density of the set F, we seek to argue by perturbation: The main idea is that for an m-tuple of functions f=(f1,... , fm)on Rn, and some differential relation P(x,D)( f)(x)=0onRn, we may generate a parametric family fα,insuchawaythaton a compact set, near any α0there is an αarbitrarily close, such that P(x,D)( fα)= 0onK. Here α∈RNfor some (large) N∈Nand fαis given by adding a polynomial of degree r (related to N) to each of the entries fiof f, with coefficients given by reading off indices of αin a suitable order. A famous example, due to Morse, of this fact is just a C2function f:Rn→R.Thenby looking at fα(x)=f(x)+α, x,where·,· is the inner product and α∈Rn,bySard’s theorem the set of αfor which fαhas a degenerate critical point is of measure zero. These ideas were generalised by Whitney [53] and others in the area of singularity theory to study generic maps Rn→Rm. In our case, the idea is that by adding generic polynomials of degree k(n)∈Nwith small coefficients to Pb,cf1,... ,Pb,cfn+1, we may obtain perturbations such that the determinant function hfrom (6.2) only vanishes of order at most k(n)−2. Here k(n)∈Nis the constant from Definition 6.7. Since the perturbation is just by polynomials of order k(n), i.e. by a linear combination of one of N0= k(n) j=0n+j−1 j=n+k(n) k(n)(6.6) 123
The Calderón problem for the fractional Schrödinger equation… Page 37 of 46 91 linearly independent polynomials of the form 1,xi,xixj, ..., by Runge approximation we may approximate these by Pb,c(fi,m)for i∈{1,... ,N0}arbitrarily close, for some suitable exterior data fi,m’s. By adding a linear combination of fi,mwith coefficients αto the exterior data, one can obtain an arbitrary close measurement for which the zero set of his “good”, i.e. is just given by a stratification of smooth hypersurfaces (see Propositions 6.10 and 6.14 below). In the sequel, we present the details of this argument. 6.4.2 Preliminaries We need to import some (old) technology from [53, Parts A and B], which allows us to modify functions in a favorable way by simply applying dimension-counting arguments. We consider a mapping f=(f1,... , fm):Rn→Rm. We consider derivatives of order up to r∈Nof fand a map ¯ f:Rn→RNgiven by arranging the partial derivatives of fin some fixed order. Then there is a “bad” set S⊂RNthat we would like to avoid. In general, the space Sis stratified, i.e. there is a splitting S=∪ i≤μSi,whereSiare smooth manifolds of dimension dim Sifor i∈{1,... ,μ}.Moreprecisely,wesaySis a manifold collection of defect δ,if∪i≤jSiis closed for all j∈{1,... ,μ}and codim Si≤δfor all i. We alter fby adding to it a polynomial of degree ≤rwhose coefficients form a set αof very small numbers. If fαis the resulting mapping Rn→RN, we may prove that for compact subsets K⊂Rnand T⊂Sthere exists αarbitrarily small such that fα(K)∩T=∅.Ifwe fix K,bytaking ¯ f(K)⊂BL⊂RNfor some large L, we prove that for any K,thereisan αsuch that fα(K)∩S=∅. Let N∈Nbe the number given as above. Assume that we have a smooth map for (p,α)∈×R1⊂Rn×RN F(p,α)=fα(p)=f∗ p(α) into RN. The family fαis called an N-parameter family of mappings of into RNif the matrix ∇αf∗ p(α) has full rank for all p∈. Next, we say a subset Qof a Euclidean space is of finite μ-extent if there is a number A with the following property. For any integer κ, there are sets Q1,... ,Qasuch that Q=Q1∪...∪Qa,diam(Qj)<1/2κ(for all j), a≤2μκ A.(6.7) Lemma 6.9 (Lemma 9a in [53]) Let the fαform an N-parameter family of mappings of ⊂Rninto RNand let 1⊂and Q be compact subsets of Rnand of RNof finite ω-extent and of finite q-extent, respectively, and suppose ω+q<N. Then for any α0∈RN and for any >0there exists αwith |α−α0|<such that fα(1)∩Q=0. By [53, Section 10], the family of functions fαgiven by adding polynomials of order ≤r is an N-parameter family of mapping of Rninto RN.Wecallδ:= N−qthe defect of Sin RN, so the condition in Lemma 6.9 can be restated as simply δ>ω. It can be easily seen that the conclusion of the above lemma holds if Qis a manifold collection of defect δ>ω. 6.4.3 Density argument Let F⊂C∞ c(W)n+1be the set, which contains exterior measurements from Definition 6.7. By Lemma 6.8, we may reconstruct the drift band potential cfor such exterior measurements. 123
91 Page 38 of 46 M. Ceki´ cetal. Notice that Fis an open set, since the set of all functions gwith finite order of vanishing at each point is open and the operator Pb,cis continuous in given topologies. In the sequel, we seek to prove that Fis also dense. We first illustrate the argument for the case n=1 in which case it is rather transparent. We will then present the proof for the general case below. Proposition 6.10 Assume n =1.ThenF⊂C∞ c(W)2is open and dense. Proof.Take f=(f1,f2)∈C∞ c(W)2to be any exterior data and consider gi:= Pb,cfi∈ C∞(K)for i=1,2. Write g=(g1,g2). We will construct an approximation of (f1,f2)in the topology of C∞ c(W)2lying in F. In one dimension, we have h(g1,g2)=g 1g1 g 2g2=g 1g2−g1g 2.(6.8) In other words, his the Wronskian of g1and g2. Now, in this case, we have ¯g(x)=g1(x), g2(x), g 1(x), g 2(x), g 1(x), g 2(x) and N=2N0=6[whereN0was defined in (6.6)]. We consider perturbations of the form gα(x)=g(x)+(α1+α 1x+α 1x2)e1+(α2+α 2x+α 2x2)e2∈R2, where e1,e2are the canonical coordinate vectors in R2. We compute the defect of the bad set given by h=0and∇h=0, i.e. S:= {J1=α 1α2−α1α 2=0}∩{J2=α 1α2−α1α 2=0}⊂R6. We need to show that the defect δ=6−q,whereqis the extent of S, is bigger that n=1 to apply Lemma 6.9. To this end, we compute the gradients ∇¯ J1=(−α 2,α 1,α 2,−α1,0,0), ∇¯ J2=(−α 2,α 1,0,0,α 2,−α1). These are clearly linearly independent for α1= 0orα2= 0ordetα 1α 2 α 1α 2= 0, so Sis a manifold collection of defect δ=2. So we apply Lemma 6.9 to obtain arbitrarily small values of α=(α1,α 1,α 1,α 2,α 2,α 2)such that gαsatisfies the property that h(gα)has an empty critical zero set on K⊂. Next, by Runge approximation (see Lemma 6.2), there exists f0,m,f1,m,f2,m∈C∞ c(W) with Pb,cfi,m→xiin C∞(K)for i=0,1,2andasm→∞. Therefore, we define fα,m=f+(α1f0,m+α 1f1,m+α 1f2,m)e1+(α2f0,m+α 2f1,m+α 2f2,m)e2∈R2. Now fix mlarge enough, so Pb,cfi,mis close to xi, such that the perturbation by elements fi,mof fmakes a 6-parameter family of mappings →R6,forα∈B1(0)⊂R6in the unit ball (say), where K⊂, by compactness. Then we again apply Lemma 6.9 and get that gα,m:= Pb,cfα,m→gin C∞(K)on a sequence of αconverging to zero. By construction fα,m∈F, so this finishes the proof. Remark 6.11 Toprovethedesiredgenericityproperty,weneedtoapproximate Pb,cfbyeither polynomials or other nice functions (analytic, generic etc.), by use of a linear approximation operator Tmf, but such that TmPb,cf=Pb,cTmfand lim m→∞Tmf=f. 123
The Calderón problem for the fractional Schrödinger equation… Page 39 of 46 91 This is the reason why the usual approximation operators, such is the Bernstein polynomials operator, or the general Weierstrass approximation theorem approach are not good for this purpose. The above approximation argument however proves the existence of such Tm,which is obtained by adding a finite linear combination of suitable functions with coefficients going to zero as m→∞. Remark 6.12 There is an alternative proof by hand of the above statement for n=1, not using the Whitney machinery, but only the genericity of Morse functions. We seek to extend the previous argument to dimension n=2 and higher. Unfortunately, in this context, it does not suffice to consider the critical zero set of h, i.e. the set h=0and ∇h=0. The computations below show that we have to include higher order derivatives of hto obtain genericity in the sense of the previous proposition. Let N=(n+1)×n+k(n) k(n)be the number of polynomials of degree ≤k(n)with which we perturb (we multiply by n+1 as this is our number of functions). Then we consider the map ¯g:Rn→RN(see previous subsection) of evaluating the derivatives of order ≤k(n) at each point. We define the bad set to be S=Sk(n)⊂RNconsisting of points given by the condition that hand its derivatives up to order k(n)−1 vanish. For a given function g=(g1,... ,gn+1), we denote this set by Z0(g)=h−1(0),Z1(g)=Z0∩(∇h)−1(0),and inductively we define Zj(g)=Zj−1(g)∩(∇jh)−1(0). Similarly as in one dimension, we then also have the following lemma, which we prove in the “Appendix”: Lemma 6.13 (Determinant genericity) The bad set Sk(n)⊂RNis a manifold collection of defect n +1. In particular, there are arbitrarily small perturbations gαwith α∈RN,such that Zk(n)−1(gα)=∅on an arbitrary compact set. As a corollary, we deduce the main result. Proposition 6.14 For any n ∈N,thesetF⊂C∞ c(W)n+1is open and dense. Proof.The proof is an immediate corollary of Lemma 6.13 and the method of proof of Proposition 6.10. Indeed, openness again follows from the definition of the set F.Inorderto inferthe density ofF,we arguealongthelines ofProposition6.10:Let f=(f1,..., fn+1)∈ C∞ c(W)n+1be arbitrary but fixed. By Runge approximation (see Theorem 1.2 (b)), for each β∈Nnwith |β|≤ √n+1there exists fβ,m∈C∞ c(W)such that Pb,c(fβ,m)→xβin C∞(K). As the set of polynomials up to degree |β|≤√n+1forms a ν-parameter family with ν=N0and N0as in (6.6), for m≥m0sufficiently large, also the set Pb,c(fβ,m)with |β|≤√n+1forms a N0-parameter family. As a result, Lemma 6.9 can be applied to Pb,c(fα):= Pb,c(f)+ n+1 j=1 β∈Nn,|β|≤√n+1αβjPb,c(fβ,m)ej, where {e1,...,en+1}denotes the canonical basis of Rn+1,αβj∈Rand xβ:= n =1xβ . Thus, for any >0 there exists α∈RN0(n+1)with |α|≤such that Zk(n)−1(Pb,c(fα)) = ∅on K⊂. By construction, we have fα∈F. This concludes the density proof. 123
91 Page 40 of 46 M. Ceki´ cetal. Combining Lemma 6.8 and Propositions 6.10,6.14 then implies the result of Theorem 1.4. Remark 6.15 We remark that Theorem 1.4 together with the results from [19] also yields a constructive reconstruction algorithm for the fractional Calderón problem with drift. Remark 6.16 Last but not least, we point out that similar openness and density results can also be obtained for the Jacobian by arguing along the same lines as in the “Appendix”. More specifically, genericity results in Lemma 6.13 can be shown to hold with the same critical index k(n)−1, if instead of the determinant in Eq. (6.2) we consider the Jacobian determinant of nfunctions, which might be of independent interest. Acknowledgements Open access funding provided by Projekt DEAL. This project strongly profited from many discussions among the authors during the HIM summer school “Unique continuation and inverse problems” and the MPI MIS summer school “Inverse and Spectral Problems for (Non)-Local Operators” at which the authors participated. The authors would like to thank that Hausdorff Center for Mathematics and the Max- Planck Institute for Mathematics in the Sciences for their support during these two weeks. YHL was supported by the Academy of Finland, under the Project Number 309963, 2018–2019. YHL is now supported by the Ministry of Science and Technology Taiwan, under the Columbus Program: MOST-109-2636-M-009-006, 2020–2025. In the course of writing of this work, MC was supported by the Max-Planck Institute for Mathematics in Bonn. MC is currently supported by the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (Grant Agreement No. 725967). Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. Appendix A. Genericity of determinants The aim of this appendix is to prove Lemma 6.13. In order to determine the conditions which are imposed by the sets Zifrom above, we first rewrite the condition ∇h=0inclosed form. Using that ∂det(M) ∂Mij =Cof(M)ij for a matrix M,whereCof(M)denotes the matrix of cofactors, we obtain ∂xjh(g1,...,gn+1)=∂h(M) ∂Mkl ∂Mkl ∂xj=Cof(M)kl ∂Mkl ∂xj , where we use summation convention of repeated indices, and M=M(g1,...,gn+1)=⎛ ⎜ ⎜ ⎝ ∂g1 ∂x1... ∂g1 ∂xng1 . . . ... . . .. . . ∂gn+1 ∂x1... ∂gn+1 ∂xngn+1 ⎞ ⎟ ⎟ ⎠ abbreviates the entries of h. Hence, by the column-wise expansion of the determinant the condition ∂jh=0 can be reformulated as n+1 l=1 det( Mj l)=0forj∈{1,...,n},(A.1) 123