# American Institute of Mathematical Sciences

doi: 10.3934/mcrf.2020042

## Stable determination of a vector field in a non-Self-Adjoint dynamical Schrödinger equation on Riemannian manifolds

 1 Université Tunis El Manar, Ecole Nationale d'ingénieurs de Tunis, ENIT-LAMSIN, B.P. 37, 1002 Tunis, Tunisia 2 Beijing Computational Science Research Center, Beijing 100193, China, and, Université Tunis El Manar, Faculté des Sciences de Tunis & ENIT-LAMSIN, B.P. 37, 1002 Tunis, Tunisia

Received  January 2020 Revised  July 2020 Published  October 2020

This paper deals with an inverse problem for a non-self-adjoint Schrödinger equation on a compact Riemannian manifold. Our goal is to stably determine a real vector field from the dynamical Dirichlet-to-Neumann map. We establish in dimension $n\geq2$, an Hölder type stability estimate for the inverse problem under study. The proof is mainly based on the reduction to an equivalent problem for an electro-magnetic Schrödinger equation and the use of a Carleman estimate designed for elliptic operators.

Citation: Mourad Bellassoued, Ibtissem Ben Aïcha, Zouhour Rezig. Stable determination of a vector field in a non-Self-Adjoint dynamical Schrödinger equation on Riemannian manifolds. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2020042
##### References:
 [1] M. Bellassoued, Stable determination of coefficients in the dynamical Schrödinger equation in a magnetic field, Inverse Problems, 33 (2017), 055009, 36 pp. doi: 10.1088/1361-6420/aa5fc5.  Google Scholar [2] M. Bellassoued and I. Ben Aïcha, Optimal stability for a first order coefficient in a non-self-adjoint wave equation from Dirichlet-to-Neumann map, Inverse Problems, 33 (2017), 105006, 23 pp. doi: 10.1088/1361-6420/aa8415.  Google Scholar [3] M. Bellassoued and H. Benjoud, Stability estimate for an inverse problem for the wave equation in a magnetic field, Appl. Anal, 87 (2008), 277-292.  doi: 10.1080/00036810801911264.  Google Scholar [4] M. Bellassoued and M. Choulli, Stability estimate for an inverse problem for the magnetic Schroödinger equation from the Dirichlet-to-Neumann map, J. Funct. Anal., 258 (2010), 161-195.  doi: 10.1016/j.jfa.2009.06.010.  Google Scholar [5] M. Bellassoued and Z. Rezig, Simultaneous determination of two coefficients in the Riemannian hyperbolic equation from boundary measurements, Ann. Global Anal. Geom., 56 (2019), 291-325.  doi: 10.1007/s10455-019-09668-7.  Google Scholar [6] M. Bellassoued and M. Yamamoto, Carleman Estimates and Applications to Inverse Problems for Hyperbolic Systems, Springer, Tokyo, 2017. doi: 10.1007/978-4-431-56600-7.  Google Scholar [7] H. Ben Joud, A stability estimate for an inverse problem for the Schrödinger equation in a magnetic field from partial boundary measurements, Inverse Problems, 25 (2009), 045012, 23 pp. doi: 10.1088/0266-5611/25/4/045012.  Google Scholar [8] J. Cheng, G. Nakamura and E. Somersalo, Uniqueness of identifying the convection term, Communications of the Korean Mathematical Society, 16 (2001), 405-413.   Google Scholar [9] N. S. Dairbekov, Integral geometry problem for nontrapping manifolds, Inverse Problems, 22 (2006), 431-445.  doi: 10.1088/0266-5611/22/2/003.  Google Scholar [10] 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 [11] Y. Kian and A. Tetlow, Hölder stable recovery of time-dependent electromagnetic potentials appearing in a dynamical anisotropic Schrödinger equation, Inverse Probl. Imaging, 14 (2020), 819-839.  doi: 10.3934/ipi.2020038.  Google Scholar [12] K. Krupchyk and G. Uhlmann, Inverse problems for advection diffusion equations in admissible geometries, Comm. Partial Differential Equations, 43 (2018), 585-615.  doi: 10.1080/03605302.2018.1446163.  Google Scholar [13] Y. V. Kurylev and M. Lassas, The multidimensional Gel'fand inverse problem for non-self-adjoint operators, Inverse Problems, 13 (1997), 1495-1501.  doi: 10.1088/0266-5611/13/6/006.  Google Scholar [14] R. G. Muhometov, The reconstruction problem of a two-dimensional Riemannian metric, and integral geometry, Dokl. Akad. Nauk SSSR, 232 (1977), 32-35.   Google Scholar [15] V. Pohjola, A uniqueness result for an inverse problem of the steady state convection diffusion equation, SIAM J. Math. Anal., 47 (2015), 2084-2103.  doi: 10.1137/140970926.  Google Scholar [16] L. Pestov and G. Uhlmann, On characterization of the range and inversion of formulas for the geodesic $X$-ray transform, Int. Math. Res. Not., 2004 (2004) 4331–4347. doi: 10.1155/S1073792804142116.  Google Scholar [17] M. Salo, Inverse problems for nonsmooth first order perturbation of the Laplacian, Ann. Acad. Sci. Fenn. Math. Diss., 139 (2004), 67 pp.  Google Scholar [18] V. Sharafutdinov, Integral Geometry of Tensor Fields, VSP, Utrecht, 1994. doi: 10.1515/9783110900095.  Google Scholar [19] P. Stefanov and G. Uhlmann, Stable determination of generic simple metrics from the hyperbolic Dirichlet-to-Neumann map, Int. Math. Res. Not., 17 (2005), 1047-1061.  doi: 10.1155/IMRN.2005.1047.  Google Scholar [20] 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 [21] Z. Q. Sun, An inverse boundary value problem for Schrödinger operators with vector potentials, Trans. Amer. Math. Soc., 388 (1993), 953-969.  doi: 10.2307/2154438.  Google Scholar

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##### References:
 [1] M. Bellassoued, Stable determination of coefficients in the dynamical Schrödinger equation in a magnetic field, Inverse Problems, 33 (2017), 055009, 36 pp. doi: 10.1088/1361-6420/aa5fc5.  Google Scholar [2] M. Bellassoued and I. Ben Aïcha, Optimal stability for a first order coefficient in a non-self-adjoint wave equation from Dirichlet-to-Neumann map, Inverse Problems, 33 (2017), 105006, 23 pp. doi: 10.1088/1361-6420/aa8415.  Google Scholar [3] M. Bellassoued and H. Benjoud, Stability estimate for an inverse problem for the wave equation in a magnetic field, Appl. Anal, 87 (2008), 277-292.  doi: 10.1080/00036810801911264.  Google Scholar [4] M. Bellassoued and M. Choulli, Stability estimate for an inverse problem for the magnetic Schroödinger equation from the Dirichlet-to-Neumann map, J. Funct. Anal., 258 (2010), 161-195.  doi: 10.1016/j.jfa.2009.06.010.  Google Scholar [5] M. Bellassoued and Z. Rezig, Simultaneous determination of two coefficients in the Riemannian hyperbolic equation from boundary measurements, Ann. Global Anal. Geom., 56 (2019), 291-325.  doi: 10.1007/s10455-019-09668-7.  Google Scholar [6] M. Bellassoued and M. Yamamoto, Carleman Estimates and Applications to Inverse Problems for Hyperbolic Systems, Springer, Tokyo, 2017. doi: 10.1007/978-4-431-56600-7.  Google Scholar [7] H. Ben Joud, A stability estimate for an inverse problem for the Schrödinger equation in a magnetic field from partial boundary measurements, Inverse Problems, 25 (2009), 045012, 23 pp. doi: 10.1088/0266-5611/25/4/045012.  Google Scholar [8] J. Cheng, G. Nakamura and E. Somersalo, Uniqueness of identifying the convection term, Communications of the Korean Mathematical Society, 16 (2001), 405-413.   Google Scholar [9] N. S. Dairbekov, Integral geometry problem for nontrapping manifolds, Inverse Problems, 22 (2006), 431-445.  doi: 10.1088/0266-5611/22/2/003.  Google Scholar [10] 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 [11] Y. Kian and A. Tetlow, Hölder stable recovery of time-dependent electromagnetic potentials appearing in a dynamical anisotropic Schrödinger equation, Inverse Probl. Imaging, 14 (2020), 819-839.  doi: 10.3934/ipi.2020038.  Google Scholar [12] K. Krupchyk and G. Uhlmann, Inverse problems for advection diffusion equations in admissible geometries, Comm. Partial Differential Equations, 43 (2018), 585-615.  doi: 10.1080/03605302.2018.1446163.  Google Scholar [13] Y. V. Kurylev and M. Lassas, The multidimensional Gel'fand inverse problem for non-self-adjoint operators, Inverse Problems, 13 (1997), 1495-1501.  doi: 10.1088/0266-5611/13/6/006.  Google Scholar [14] R. G. Muhometov, The reconstruction problem of a two-dimensional Riemannian metric, and integral geometry, Dokl. Akad. Nauk SSSR, 232 (1977), 32-35.   Google Scholar [15] V. Pohjola, A uniqueness result for an inverse problem of the steady state convection diffusion equation, SIAM J. Math. Anal., 47 (2015), 2084-2103.  doi: 10.1137/140970926.  Google Scholar [16] L. Pestov and G. Uhlmann, On characterization of the range and inversion of formulas for the geodesic $X$-ray transform, Int. Math. Res. Not., 2004 (2004) 4331–4347. doi: 10.1155/S1073792804142116.  Google Scholar [17] M. Salo, Inverse problems for nonsmooth first order perturbation of the Laplacian, Ann. Acad. Sci. Fenn. Math. Diss., 139 (2004), 67 pp.  Google Scholar [18] V. Sharafutdinov, Integral Geometry of Tensor Fields, VSP, Utrecht, 1994. doi: 10.1515/9783110900095.  Google Scholar [19] P. Stefanov and G. Uhlmann, Stable determination of generic simple metrics from the hyperbolic Dirichlet-to-Neumann map, Int. Math. Res. Not., 17 (2005), 1047-1061.  doi: 10.1155/IMRN.2005.1047.  Google Scholar [20] 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 [21] Z. Q. Sun, An inverse boundary value problem for Schrödinger operators with vector potentials, Trans. Amer. Math. Soc., 388 (1993), 953-969.  doi: 10.2307/2154438.  Google Scholar
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