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May  2022, 15(5): 1061-1084. doi: 10.3934/dcdss.2021158

Recovery of transversal metric tensor in the Schrödinger equation from the Dirichlet-to-Neumann map

Mourad Bellassoued 1,, and  Zouhour Rezig 2,

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

*Corresponding author: Mourad Bellassoued

Received  September 2021 Published  May 2022 Early access  December 2021

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.

Keywords: Inverse problem, Riemannian manifold, stability estimate, Dirichlet-to-Neumann map, geodesical ray transform.
Mathematics Subject Classification: Primary: 35R30; Secondary: 35J10.
Citation: Mourad Bellassoued, Zouhour Rezig. Recovery of transversal metric tensor in the Schrödinger equation from the Dirichlet-to-Neumann map. Discrete and Continuous Dynamical Systems - S, 2022, 15 (5) : 1061-1084. doi: 10.3934/dcdss.2021158
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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