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Recovery of transversal metric tensor in the Schrödinger equation from the Dirichlet-to-Neumann map
1. | University of Tunis El Manar, National Engineering School of Tunis, ENIT-LAMSIN, B.P. 37, 1002 Tunis, Tunisia |
2. | University of Tunis El Manar, Faculty of Sciences of Tunis, ENIT-LAMSIN, B.P. 37, 1002 Tunis, Tunisia |
In this paper, we deal with the inverse problem of determining simple metrics on a compact Riemannian manifold from boundary measurements. We take this information in the dynamical Dirichlet-to-Neumann map associated to the Schrödinger equation. We prove in dimension $ n\geq 2 $ that the knowledge of the Dirichlet-to-Neumann map for the Schrödinger equation uniquely determines the simple metric (up to an admissible set). We also prove a Hölder-type stability estimate by the construction of geometrical optics solutions of the Schrödinger equation and the direct use of the invertibility of the geodesical X-ray transform.
References:
[1] |
A. Amirov,
Boundary rigidity for Riemannian manifolds, Selcuk J. Appl. Math., 4 (2003), 5-12.
|
[2] |
A. Amirov, Integral Geometry and Inverse Problems for Kinetic Equations, Inverse and Ill-posed Problems Series. VSP, Utrecht, 2001.
doi: 10.1515/9783110940947. |
[3] |
M. Anderson, A. Katsuda, Y. Kurylev, M. Lassas and M. Taylor,
Boundary regularity for the Ricci equation, geometric convergence, and Gel'fand's inverse boundary problem, Invent. Math., 158 (2004), 261-321.
doi: 10.1007/s00222-004-0371-6. |
[4] |
S. A. Avdonin and M. Belishev, Dynamical inverse problem for the Schrödinger equation, Proc. St. Petersburg Math. Soc., 10 (2005), 3–18 (Russian); Amer. Math. Soc. Translations: Series 2, 214 (2005), 1–14.
doi: 10.1090/trans2/214/01. |
[5] |
S. A. Avdonin, S. Lenhart and V. Protopopescu,
Determining the potential in the Schrödinger equation from the Dirichlet to Neumann map by the boundary control method, J. Inverse Ill-Posed Probl., 13 (2005), 317-330.
doi: 10.1515/156939405775201718. |
[6] |
M. Belishev,
Boundary control in reconstruction of manifolds and metrics (BC method), Inverse Problems, 13 (1997), R1-R45.
doi: 10.1088/0266-5611/13/5/002. |
[7] |
M. Belishev and Y. Kurylev,
To the reconstruction of a Riemannian manifold via its spectral data (BC-method), Comm. Partial Differential Equations, 17 (1992), 767-804.
doi: 10.1080/03605309208820863. |
[8] |
M. Bellassoued,
Stable determination of coefficients in the dynamical Schrödinger equation in a magnetic field, Inverse Problems, 33 (2017), 055009.
doi: 10.1088/1361-6420/aa5fc5. |
[9] |
M. Bellassoued, I. Ben Aïcha and Z. Rezig,
A stable determination of a vector field in a non-self-adjoint dynamical Schrödinger equation on Riemannian manifolds, Math. Control Relat. Fields, 11 (2021), 403-431.
doi: 10.3934/mcrf.2020042. |
[10] |
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. |
[11] |
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. |
[12] |
M. Bellassoued and M. Choulli,
Logarithmic stability in the dynamical inverse problem for the Schrödinger equation by an arbitrary boundary observation, J. Math. Pures Appl., 91 (2009), 233-255.
doi: 10.1016/j.matpur.2008.06.002. |
[13] |
M. Bellassoued and D. Dos Santos Ferreira,
Stability estimates for the anisotropic wave equation from the Dirichlet-to-Neumann map, Inverse Probl. Imaging, 5 (2011), 745-773.
doi: 10.3934/ipi.2011.5.745. |
[14] |
M. Bellassoued and D. Dos Santos 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. |
[15] |
M. Bellassoued, D. Jellali and M. Yamamoto,
Stability estimate for the hyperbolic inverse boundary value problem by local Dirichlet-to-Neumann map, J. Math. Anal. Appl., 343 (2008), 1036-1046.
doi: 10.1016/j.jmaa.2008.01.098. |
[16] |
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. |
[17] |
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. |
[18] |
G. Eskin,
A new approach to hyperbolic inverse problems, Inverse Problems, 22 (2006), 815-831.
doi: 10.1088/0266-5611/22/3/005. |
[19] |
G. Eskin,
Inverse hyperbolic problems with time-dependent coefficients, Commun. Partial Differential Equations, 32 (2007), 1737-1758.
doi: 10.1080/03605300701382340. |
[20] |
V. Isakov,
An inverse hyperbolic problem with many boundary measurements, Comm. Partial Differential Equations, 16 (1991), 1183-1195.
doi: 10.1080/03605309108820794. |
[21] |
V. Isakov and Z. Sun,
Stability estimates for hyperbolic inverse problems with local boundary data, Inverse Problems, 8 (1992), 193-206.
doi: 10.1088/0266-5611/8/2/003. |
[22] |
A. Katchalov, Y. Kurylev and M. Lassas, Inverse Boundary Spectral Problems, Chapman and Hall/CRC, Boca Raton, 2001.
doi: 10.1201/9781420036220. |
[23] |
Y. V. Kurylev and M. Lassas, Hyperbolic inverse problem with data on a part of the boundary, Differential Equations and Mathematical Physics, AMS/IP Stud. Adv. Math., 16 (2000); Amer. Math. Soc., Providence, (2000), 259–272. |
[24] |
M. Lassas, V. Sharafutdinov and G. Uhlmann,
Semiglobal boundary rigidity for Riemannian metrics, Math. Ann., 325 (2003), 767-793.
doi: 10.1007/s00208-002-0407-4. |
[25] |
Ra kesh,
Reconstruction for an inverse problem for the wave equation with constant velocity, Inverse Problems, 6 (1990), 91-98.
doi: 10.1088/0266-5611/6/1/009. |
[26] |
Ra kesh and W. Symes,
Uniqueness for an inverse problem for the wave equation, Comm. Partial Differential Equations, 13 (1988), 87-96.
doi: 10.1080/03605308808820539. |
[27] |
A. Ramm and J. Sjöstrand,
An inverse problem of the wave equation, Math. Z., 206 (1991), 119-130.
doi: 10.1007/BF02571330. |
[28] |
V. Sharafutdinov, Integral Geometry of Tensor Fields, VSP, Utrecht, 1994.
doi: 10.1515/9783110900095. |
[29] |
P. Stefanov and G. Uhlmann,
Stability estimates for the hyperbolic Dirichlet to Neumann map in anisotropic media, J. Funct. Anal., 154 (1998), 330-358.
doi: 10.1006/jfan.1997.3188. |
[30] |
P. Stefanov and G. Uhlmann,
Stable determination of generic simple metrics from the hyperbolic Dirichlet-to-Neumann map, Int. Math. Res. Not., 2005 (2005), 1047-1061.
doi: 10.1155/IMRN.2005.1047. |
[31] |
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. |
[32] |
P. Stefanov and G. Uhlmann,
Boundary rigidity and stability for generic simple metrics, J. Amer. Math. Soc., 18 (2005), 975-1003.
doi: 10.1090/S0894-0347-05-00494-7. |
[33] |
Z. Sun,
On continuous dependence for an inverse initial boundary value problem for the wave equation, J. Math. Anal. App., 150 (1990), 188-204.
doi: 10.1016/0022-247X(90)90207-V. |
show all references
References:
[1] |
A. Amirov,
Boundary rigidity for Riemannian manifolds, Selcuk J. Appl. Math., 4 (2003), 5-12.
|
[2] |
A. Amirov, Integral Geometry and Inverse Problems for Kinetic Equations, Inverse and Ill-posed Problems Series. VSP, Utrecht, 2001.
doi: 10.1515/9783110940947. |
[3] |
M. Anderson, A. Katsuda, Y. Kurylev, M. Lassas and M. Taylor,
Boundary regularity for the Ricci equation, geometric convergence, and Gel'fand's inverse boundary problem, Invent. Math., 158 (2004), 261-321.
doi: 10.1007/s00222-004-0371-6. |
[4] |
S. A. Avdonin and M. Belishev, Dynamical inverse problem for the Schrödinger equation, Proc. St. Petersburg Math. Soc., 10 (2005), 3–18 (Russian); Amer. Math. Soc. Translations: Series 2, 214 (2005), 1–14.
doi: 10.1090/trans2/214/01. |
[5] |
S. A. Avdonin, S. Lenhart and V. Protopopescu,
Determining the potential in the Schrödinger equation from the Dirichlet to Neumann map by the boundary control method, J. Inverse Ill-Posed Probl., 13 (2005), 317-330.
doi: 10.1515/156939405775201718. |
[6] |
M. Belishev,
Boundary control in reconstruction of manifolds and metrics (BC method), Inverse Problems, 13 (1997), R1-R45.
doi: 10.1088/0266-5611/13/5/002. |
[7] |
M. Belishev and Y. Kurylev,
To the reconstruction of a Riemannian manifold via its spectral data (BC-method), Comm. Partial Differential Equations, 17 (1992), 767-804.
doi: 10.1080/03605309208820863. |
[8] |
M. Bellassoued,
Stable determination of coefficients in the dynamical Schrödinger equation in a magnetic field, Inverse Problems, 33 (2017), 055009.
doi: 10.1088/1361-6420/aa5fc5. |
[9] |
M. Bellassoued, I. Ben Aïcha and Z. Rezig,
A stable determination of a vector field in a non-self-adjoint dynamical Schrödinger equation on Riemannian manifolds, Math. Control Relat. Fields, 11 (2021), 403-431.
doi: 10.3934/mcrf.2020042. |
[10] |
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. |
[11] |
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. |
[12] |
M. Bellassoued and M. Choulli,
Logarithmic stability in the dynamical inverse problem for the Schrödinger equation by an arbitrary boundary observation, J. Math. Pures Appl., 91 (2009), 233-255.
doi: 10.1016/j.matpur.2008.06.002. |
[13] |
M. Bellassoued and D. Dos Santos Ferreira,
Stability estimates for the anisotropic wave equation from the Dirichlet-to-Neumann map, Inverse Probl. Imaging, 5 (2011), 745-773.
doi: 10.3934/ipi.2011.5.745. |
[14] |
M. Bellassoued and D. Dos Santos 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. |
[15] |
M. Bellassoued, D. Jellali and M. Yamamoto,
Stability estimate for the hyperbolic inverse boundary value problem by local Dirichlet-to-Neumann map, J. Math. Anal. Appl., 343 (2008), 1036-1046.
doi: 10.1016/j.jmaa.2008.01.098. |
[16] |
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. |
[17] |
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. |
[18] |
G. Eskin,
A new approach to hyperbolic inverse problems, Inverse Problems, 22 (2006), 815-831.
doi: 10.1088/0266-5611/22/3/005. |
[19] |
G. Eskin,
Inverse hyperbolic problems with time-dependent coefficients, Commun. Partial Differential Equations, 32 (2007), 1737-1758.
doi: 10.1080/03605300701382340. |
[20] |
V. Isakov,
An inverse hyperbolic problem with many boundary measurements, Comm. Partial Differential Equations, 16 (1991), 1183-1195.
doi: 10.1080/03605309108820794. |
[21] |
V. Isakov and Z. Sun,
Stability estimates for hyperbolic inverse problems with local boundary data, Inverse Problems, 8 (1992), 193-206.
doi: 10.1088/0266-5611/8/2/003. |
[22] |
A. Katchalov, Y. Kurylev and M. Lassas, Inverse Boundary Spectral Problems, Chapman and Hall/CRC, Boca Raton, 2001.
doi: 10.1201/9781420036220. |
[23] |
Y. V. Kurylev and M. Lassas, Hyperbolic inverse problem with data on a part of the boundary, Differential Equations and Mathematical Physics, AMS/IP Stud. Adv. Math., 16 (2000); Amer. Math. Soc., Providence, (2000), 259–272. |
[24] |
M. Lassas, V. Sharafutdinov and G. Uhlmann,
Semiglobal boundary rigidity for Riemannian metrics, Math. Ann., 325 (2003), 767-793.
doi: 10.1007/s00208-002-0407-4. |
[25] |
Ra kesh,
Reconstruction for an inverse problem for the wave equation with constant velocity, Inverse Problems, 6 (1990), 91-98.
doi: 10.1088/0266-5611/6/1/009. |
[26] |
Ra kesh and W. Symes,
Uniqueness for an inverse problem for the wave equation, Comm. Partial Differential Equations, 13 (1988), 87-96.
doi: 10.1080/03605308808820539. |
[27] |
A. Ramm and J. Sjöstrand,
An inverse problem of the wave equation, Math. Z., 206 (1991), 119-130.
doi: 10.1007/BF02571330. |
[28] |
V. Sharafutdinov, Integral Geometry of Tensor Fields, VSP, Utrecht, 1994.
doi: 10.1515/9783110900095. |
[29] |
P. Stefanov and G. Uhlmann,
Stability estimates for the hyperbolic Dirichlet to Neumann map in anisotropic media, J. Funct. Anal., 154 (1998), 330-358.
doi: 10.1006/jfan.1997.3188. |
[30] |
P. Stefanov and G. Uhlmann,
Stable determination of generic simple metrics from the hyperbolic Dirichlet-to-Neumann map, Int. Math. Res. Not., 2005 (2005), 1047-1061.
doi: 10.1155/IMRN.2005.1047. |
[31] |
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. |
[32] |
P. Stefanov and G. Uhlmann,
Boundary rigidity and stability for generic simple metrics, J. Amer. Math. Soc., 18 (2005), 975-1003.
doi: 10.1090/S0894-0347-05-00494-7. |
[33] |
Z. Sun,
On continuous dependence for an inverse initial boundary value problem for the wave equation, J. Math. Anal. App., 150 (1990), 188-204.
doi: 10.1016/0022-247X(90)90207-V. |
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