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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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[10]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, London, 1985.

[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.

[12]

J. M. Lee, Riemannian Manifolds: An Introduction to Curvature, Springer, 1997. doi: 10.1007/b98852.

[13]

J-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Dunod, Paris, 1968.

[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.

[15]

M. Salo, Inverse problems for nonsmooth first order perturbations of the Laplacian, Ann. Acad. Scient. Fenn. Math. Dissertations, 139 (2004).

[16]

V. Sharafutdinov, Integral Geometry of Tensor Fields, VSP, Utrecht, the Netherlands, 1994. doi: 10.1515/9783110900095.

[17]

M. Spivak, A Comprehensive Introduction to Differential Geometry, Publish or Perish Inc., USA, 1970.

[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.

[19] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970. 

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[10]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, London, 1985.

[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.

[12]

J. M. Lee, Riemannian Manifolds: An Introduction to Curvature, Springer, 1997. doi: 10.1007/b98852.

[13]

J-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Dunod, Paris, 1968.

[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.

[15]

M. Salo, Inverse problems for nonsmooth first order perturbations of the Laplacian, Ann. Acad. Scient. Fenn. Math. Dissertations, 139 (2004).

[16]

V. Sharafutdinov, Integral Geometry of Tensor Fields, VSP, Utrecht, the Netherlands, 1994. doi: 10.1515/9783110900095.

[17]

M. Spivak, A Comprehensive Introduction to Differential Geometry, Publish or Perish Inc., USA, 1970.

[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.

[19] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970. 
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