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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

Yavar Kian 1, and  Alexander Tetlow 2,

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.

Keywords: Dynamic Schrödinger equation, Hölder stability, ray-transform, geometric optics, time-dependent potentials.
Mathematics Subject Classification: Primary: 35R30; Secondary: 35R01.
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:
[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

show all references

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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