American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Stochastic quasi-subgradient method for stochastic quasi-convex feasibility problems
  • DCDS-S Home
  • This Issue
  • Next Article
    Switching mechanism-based event-triggered fuzzy adaptive control with prescribed performance for MIMO nonlinear systems
doi: 10.3934/dcdss.2021158
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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 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 & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021158
References:
[1]

A. Amirov, Boundary rigidity for Riemannian manifolds, Selcuk J. Appl. Math., 4 (2003), 5-12.   Google Scholar

[2]

A. Amirov, Integral Geometry and Inverse Problems for Kinetic Equations, Inverse and Ill-posed Problems Series. VSP, Utrecht, 2001. doi: 10.1515/9783110940947.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

G. Eskin, A new approach to hyperbolic inverse problems, Inverse Problems, 22 (2006), 815-831.  doi: 10.1088/0266-5611/22/3/005.  Google Scholar

[19]

G. Eskin, Inverse hyperbolic problems with time-dependent coefficients, Commun. Partial Differential Equations, 32 (2007), 1737-1758.  doi: 10.1080/03605300701382340.  Google Scholar

[20]

V. Isakov, An inverse hyperbolic problem with many boundary measurements, Comm. Partial Differential Equations, 16 (1991), 1183-1195.  doi: 10.1080/03605309108820794.  Google Scholar

[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.  Google Scholar

[22]

A. Katchalov, Y. Kurylev and M. Lassas, Inverse Boundary Spectral Problems, Chapman and Hall/CRC, Boca Raton, 2001. doi: 10.1201/9781420036220.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

A. Ramm and J. Sjöstrand, An inverse problem of the wave equation, Math. Z., 206 (1991), 119-130.  doi: 10.1007/BF02571330.  Google Scholar

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

A. Amirov, Boundary rigidity for Riemannian manifolds, Selcuk J. Appl. Math., 4 (2003), 5-12.   Google Scholar

[2]

A. Amirov, Integral Geometry and Inverse Problems for Kinetic Equations, Inverse and Ill-posed Problems Series. VSP, Utrecht, 2001. doi: 10.1515/9783110940947.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

G. Eskin, A new approach to hyperbolic inverse problems, Inverse Problems, 22 (2006), 815-831.  doi: 10.1088/0266-5611/22/3/005.  Google Scholar

[19]

G. Eskin, Inverse hyperbolic problems with time-dependent coefficients, Commun. Partial Differential Equations, 32 (2007), 1737-1758.  doi: 10.1080/03605300701382340.  Google Scholar

[20]

V. Isakov, An inverse hyperbolic problem with many boundary measurements, Comm. Partial Differential Equations, 16 (1991), 1183-1195.  doi: 10.1080/03605309108820794.  Google Scholar

[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.  Google Scholar

[22]

A. Katchalov, Y. Kurylev and M. Lassas, Inverse Boundary Spectral Problems, Chapman and Hall/CRC, Boca Raton, 2001. doi: 10.1201/9781420036220.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

A. Ramm and J. Sjöstrand, An inverse problem of the wave equation, Math. Z., 206 (1991), 119-130.  doi: 10.1007/BF02571330.  Google Scholar

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Oleg Yu. Imanuvilov, Masahiro Yamamoto. Stability for determination of Riemannian metrics by spectral data and Dirichlet-to-Neumann map limited on arbitrary subboundary. Inverse Problems & Imaging, 2019, 13 (6) : 1213-1258. doi: 10.3934/ipi.2019054
[2]

Mourad Bellassoued, David Dos Santos Ferreira. Stability estimates for the anisotropic wave equation from the Dirichlet-to-Neumann map. Inverse Problems & Imaging, 2011, 5 (4) : 745-773. doi: 10.3934/ipi.2011.5.745
[3]

Victor Isakov, Jenn-Nan Wang. Increasing stability for determining the potential in the Schrödinger equation with attenuation from the Dirichlet-to-Neumann map. Inverse Problems & Imaging, 2014, 8 (4) : 1139-1150. doi: 10.3934/ipi.2014.8.1139
[4]

Jussi Behrndt, A. F. M. ter Elst. The Dirichlet-to-Neumann map for Schrödinger operators with complex potentials. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 661-671. doi: 10.3934/dcdss.2017033
[5]

Victor Isakov. Increasing stability for the Schrödinger potential from the Dirichlet-to Neumann map. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 631-640. doi: 10.3934/dcdss.2011.4.631
[6]

Ihsane Bikri, Ronald B. Guenther, Enrique A. Thomann. The Dirichlet to Neumann map - An application to the Stokes problem in half space. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 221-230. doi: 10.3934/dcdss.2010.3.221
[7]

Aymen Jbalia. On a logarithmic stability estimate for an inverse heat conduction problem. Mathematical Control & Related Fields, 2019, 9 (2) : 277-287. doi: 10.3934/mcrf.2019014
[8]

Francis J. Chung. Partial data for the Neumann-Dirichlet magnetic Schrödinger inverse problem. Inverse Problems & Imaging, 2014, 8 (4) : 959-989. doi: 10.3934/ipi.2014.8.959
[9]

Mahamadi Warma. A fractional Dirichlet-to-Neumann operator on bounded Lipschitz domains. Communications on Pure & Applied Analysis, 2015, 14 (5) : 2043-2067. doi: 10.3934/cpaa.2015.14.2043
[10]

Wolfgang Arendt, Rafe Mazzeo. Friedlander's eigenvalue inequalities and the Dirichlet-to-Neumann semigroup. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2201-2212. doi: 10.3934/cpaa.2012.11.2201
[11]

Soumen Senapati, Manmohan Vashisth. Stability estimate for a partial data inverse problem for the convection-diffusion equation. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021060
[12]

Kevin Arfi, Anna Rozanova-Pierrat. Dirichlet-to-Neumann or Poincaré-Steklov operator on fractals described by d-sets. Discrete & Continuous Dynamical Systems - S, 2019, 12 (1) : 1-26. doi: 10.3934/dcdss.2019001
[13]

Atsushi Kawamoto. Hölder stability estimate in an inverse source problem for a first and half order time fractional diffusion equation. Inverse Problems & Imaging, 2018, 12 (2) : 315-330. doi: 10.3934/ipi.2018014
[14]

Shengbing Deng, Zied Khemiri, Fethi Mahmoudi. On spike solutions for a singularly perturbed problem in a compact riemannian manifold. Communications on Pure & Applied Analysis, 2018, 17 (5) : 2063-2084. doi: 10.3934/cpaa.2018098
[15]

Anna Maria Candela, J.L. Flores, M. Sánchez. A quadratic Bolza-type problem in a non-complete Riemannian manifold. Conference Publications, 2003, 2003 (Special) : 173-181. doi: 10.3934/proc.2003.2003.173
[16]

Bo Guan, Heming Jiao. The Dirichlet problem for Hessian type elliptic equations on Riemannian manifolds. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 701-714. doi: 10.3934/dcds.2016.36.701
[17]

Gareth Ainsworth. The attenuated magnetic ray transform on surfaces. Inverse Problems & Imaging, 2013, 7 (1) : 27-46. doi: 10.3934/ipi.2013.7.27
[18]

Gareth Ainsworth. The magnetic ray transform on Anosov surfaces. Discrete & Continuous Dynamical Systems, 2015, 35 (5) : 1801-1816. doi: 10.3934/dcds.2015.35.1801
[19]

Daniel Fusca. The Madelung transform as a momentum map. Journal of Geometric Mechanics, 2017, 9 (2) : 157-165. doi: 10.3934/jgm.2017006
[20]

Shumin Li, Masahiro Yamamoto, Bernadette Miara. A Carleman estimate for the linear shallow shell equation and an inverse source problem. Discrete & Continuous Dynamical Systems, 2009, 23 (1&2) : 367-380. doi: 10.3934/dcds.2009.23.367

2020 Impact Factor: 2.425

Tools

Metrics

  • PDF downloads (54)
  • HTML views (48)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy