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Stability estimates for time-dependent coefficients appearing in the magnetic Schrödinger equation from arbitrary boundary measurements
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October  2020, 14(5): 819-839. doi: 10.3934/ipi.2020038

## Hölder-stable recovery of time-dependent electromagnetic potentials appearing in a dynamical anisotropic Schrödinger equation

 1 Aix-Marseille Univ, Université de Toulon, CNRS, CPT, Marseille, France 2 Department of Mathematics, University College London, London, UK

Y. Kian was partially supported by the Agence Nationale de la Recherche under grant ANR-17-CE40-0029.

Received  November 2019 Revised  March 2020 Published  July 2020

Fund Project: A. Tetlow was supported by EPSRC DTP studentship EP/N509577/1

We consider the inverse problem of Hölder-stably determining the time- and space-dependent coefficients of the Schrödinger equation on a simple Riemannian manifold with boundary of dimension $n\geq2$ from the 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 Hölder-stably recovered from these data. Here we also remove the smallness assumption for the solenoidal part of the magnetic potential present in previous results.

Citation: Yavar Kian, Alexander Tetlow. Hölder-stable recovery of time-dependent electromagnetic potentials appearing in a dynamical anisotropic Schrödinger equation. Inverse Problems & Imaging, 2020, 14 (5) : 819-839. doi: 10.3934/ipi.2020038
##### References:
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##### References:
 [1] Y. E. Anikonov and V. G. Romanov, On uniqueness of determination of a form of first degree by its integrals along geodesics, J. Inv. Ill-Posed Problems, 5 (1997), 487-480.  doi: 10.1515/jiip.1997.5.6.487.  Google Scholar [2] M. Bellassoued, Stable determination of coefficients in the dynamical Schrödinger equation in a magnetic field, Inverse Problems, 33 (2017), 36pp. doi: 10.1088/1361-6420/aa5fc5.  Google Scholar [3] M. Bellassoued and M. Choulli, Stability estimate for an inverse problem for the magnetic Schrödinger equation from the Dirichlet-to-Neumann map, J. Funct. Anal., 258 (2010), 161-195.  doi: 10.1016/j.jfa.2009.06.010.  Google Scholar [4] M. Bellassoued and D. D. S. Ferreira, Stable determination of coefficients in the dynamical anisotropic Schrödinger equation from the Dirichlet-to-Neumann map, Inverse Problems, 26 (2010), 125010. doi: 10.1088/0266-5611/26/12/125010.  Google Scholar [5] M. Bellassoued, Y. Kian and E. Soccorsi, An inverse problem for the magnetic Schrödinger equation in infinite cylindrical domains, Publications of the Research Institute for Mathematical Sciences, 54 (2018), 679-728.  doi: 10.4171/PRIMS/54-4-1.  Google Scholar [6] M. Bellassoued and Z. Rezig, Simultaneous determination of two coefficients in the Riemannian hyperbolic equation from boundary measurements, Ann. Glob. Anal. Geom., 56 (2019), 291-325.  doi: 10.1007/s10455-019-09668-7.  Google Scholar [7] I. B. Aicha, Stability estimate for an inverse problem for the Schrödinger equation in a magnetic field with time-dependent coefficient, Journal of Mathematical Physics, 58 (2017), 071508. doi: 10.1063/1.4995606.  Google Scholar [8] M. Choulli, Y. Kian and E. Soccorsi, Stable determination of time-dependent scalar potential from boundary measurements in a periodic quantum waveguide, SIAM J. Math. Anal., 47 (2015), 4536-4558.  doi: 10.1137/140986268.  Google Scholar [9] G. Eskin, Inverse problems for the Schrödinger equations with time-dependent electromagnetic potentials and the Aharonov-Bohm effect, J. Math. Phys., 49 (2008), 022105. doi: 10.1063/1.2841329.  Google Scholar [10] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, London, 1985.  Google Scholar [11] Y. Kian and E. Soccorsi, Hölder stably determining the time-dependent electromagnetic potential of the Schrödinger equation, SIAM J. Math. Anal., 51 (2019), 627-647.  doi: 10.1137/18M1197308.  Google Scholar [12] J. M. Lee, Riemannian Manifolds: An Introduction to Curvature, Springer, 1997. doi: 10.1007/b98852.  Google Scholar [13] J-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Dunod, Paris, 1968.  Google Scholar [14] C. Montalto, Stable determination of a simple metric, a covector field and a potential from the hyperbolic Dirichlet-to-Neumann map, Communications in Partial Differential Equations, 39 (2012), 120-145.  doi: 10.1080/03605302.2013.843429.  Google Scholar [15] M. Salo, Inverse problems for nonsmooth first order perturbations of the Laplacian, Ann. Acad. Scient. Fenn. Math. Dissertations, 139 (2004).  Google Scholar [16] V. Sharafutdinov, Integral Geometry of Tensor Fields, VSP, Utrecht, the Netherlands, 1994. doi: 10.1515/9783110900095.  Google Scholar [17] M. Spivak, A Comprehensive Introduction to Differential Geometry, Publish or Perish Inc., USA, 1970.  Google Scholar [18] P. Stefanov and G. Uhlmann, Stability estimates for the X-ray transform of tensor fields and boundary rigidity, Duke Math. J., 123 (2004), 445-467.  doi: 10.1215/S0012-7094-04-12332-2.  Google Scholar [19] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970.   Google Scholar
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