American Institute of Mathematical Sciences

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October  2021, 15(5): 929-950. doi: 10.3934/ipi.2021022

Stable recovery of a non-compactly supported coefficient of a Schrödinger equation on an infinite waveguide

Yosra Soussi

Université de Tunis El Manar, Ecole Nationale d'Ingénieurs de Tunis, ENIT-LAMSIN, B.P. 37, 1002 Tunis, Tunisia, Aix Marseille Université, Université de Toulon, CNRS, CPT, Marseille, France

Received  February 2020 Revised  October 2020 Published  October 2021 Early access  March 2021

We study the stability issue for the inverse problem of determining a coefficient appearing in a Schrödinger equation defined on an infinite cylindrical waveguide. More precisely, we prove the stable recovery of some general class of non-compactly and non periodic coefficients appearing in an unbounded cylindrical domain. We consider both results of stability from full and partial boundary measurements associated with the so called Dirichlet-to-Neumann map.

Keywords: Inverse problem, stability estimate, Schrödinger equation, electric potential, Dirichlet-to-Neumann map, infinite cylindrical waveguide, partial data, Carleman's estimate.
Mathematics Subject Classification: Primary: 35R30; Secondary: 35J15.
Citation: Yosra Soussi. Stable recovery of a non-compactly supported coefficient of a Schrödinger equation on an infinite waveguide. Inverse Problems & Imaging, 2021, 15 (5) : 929-950. doi: 10.3934/ipi.2021022
References:
[1]

G. Alessandrini, Stable determination of conductivity by boundary measurements, Appl. Anal., 27 (1988), 153-172.  doi: 10.1080/00036818808839730.  Google Scholar

[2]

H. Ammari and G. Uhlmann, Reconstruction from partial Cauchy data for the Schrödinger equation, Indiana University Math J., 53 (2004), 169-183.  doi: 10.1512/iumj.2004.53.2299.  Google Scholar

[3]

J. Behrndt and J. Rohleder, Inverse problems with partial data for elliptic operators on unbounded Lipschitz domains, Inverse Problems, 36 (2020), 035009, 18 pp. doi: 10.1088/1361-6420/ab603d.  Google Scholar

[4]

M. Bellassoued, Y. Kian and E. Soccorsi, An inverse stability result for non compactly supported potentials by one arbitrary lateral Neumann observation, J. Differential Equations, 260 (2016), 7535-7562.  doi: 10.1016/j.jde.2016.01.033.  Google Scholar

[5]

M. Bellassoued, Y. Kian and E. Soccorsi, An inverse problem for the magnetic Schrödinger equation in infinite cylindrical domains, Publ. Research Institute Math. Sci., 54 (2018), 679-728.  doi: 10.4171/PRIMS/54-4-1.  Google Scholar

[6]

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

[7]

A. L. Bukhgeim and G. Uhlmann, Recovering a potential from partial Cauchy data, Comm. PDE, 27 (2002), 653-668.  doi: 10.1081/PDE-120002868.  Google Scholar

[8]

A. P. Calderón, On an inverse boundary value problem, Seminar on Numerical Analysis and its Applications to Continuum Physics, Rio de Janeiro, Sociedade Brasileira de Matematica, (1980), 65–73. doi: 10.1590/S0101-82052006000200002.  Google Scholar

[9]

P. Caro, D. Dos Santos Ferreira and A. Ruiz, Stability estimates for the Radon transform with restricted data and applications, Advances in Math., 267 (2014), 523-564.  doi: 10.1016/j.aim.2014.08.009.  Google Scholar

[10]

P. Caro, D. Dos Santos Ferreira and A. Ruiz, Stability estimates for the Calderón problem with partial data, J. Diff. Equa., 260 (2016), 2457-2489.  doi: 10.1016/j.jde.2015.10.007.  Google Scholar

[11]

P. Caro and K. Marinov, Stability of inverse problems in an infinite slab with partial data, Commun. Partial Diff. Equa., 41 (2016), 683-704.  doi: 10.1080/03605302.2015.1127967.  Google Scholar

[12]

P. Caro and V. Pohjola, Stability estimates for an inverse problem for the magnetic Schrödinger operator, IMRN, (2015), 11083–11116. doi: 10.1093/imrn/rnv020.  Google Scholar

[13]

M. Choulli, Une Introduction Aux Problèmes Inverses Elliptiques et Paraboliques, Mathematics & Applications, Vol. 65, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-642-02460-3.  Google Scholar

[14]

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.  Google Scholar

[15]

M. Choulli, Y. Kian and E. Soccorsi, Double logarithmic stability estimate in the identification of a scalar potential by a partial elliptic Dirichlet-to-Neumann map, Bulletin of the South Ural State University, Ser. Mathematical Modelling, Programming and Computer Software (SUSUMMCS), 8 (2015), 78-94.  doi: 10.14529/mmp150305.  Google Scholar

[16]

M. Choulli, Y. Kian and E. Soccorsi, On the Calderón problem in periodic cylindrical domain with partial Dirichlet and Neumann data, Mathematical Methods in the Applied Sciences, 40 (2017), 5959-5974.  doi: 10.1002/mma.4446.  Google Scholar

[17]

M. Choulli, Y. Kian and E. Soccorsi, Stability result for elliptic inverse periodic coefficient problem by partial Dirichlet-to-Neumann map, J. Spec. Theory, 8 (2018), 733-768.  doi: 10.4171/JST/212.  Google Scholar

[18]

M. Choulli and E. Soccorsi, An inverse anisotropic conductivity problem induced by twisting a homogeneous cylindrical domain, J. Spec. Theory, 5 (2015), 295-329.  doi: 10.4171/JST/99.  Google Scholar

[19]

O. Yu. Èmanuvilov, Controllability of evolution equations, Sb. Math., 186 (1995), 879-900.  doi: 10.1070/SM1995v186n06ABEH000047.  Google Scholar

[20]

O. Yu. Imanuvilov, G. Uhlmann and M. Yamamoto, The Calderón problem with partial data in two dimensions, Journal American Math. Society, 23 (2010), 655-691.  doi: 10.1090/S0894-0347-10-00656-9.  Google Scholar

[21]

O. Yu. Imanuvilov and M. Yamamoto, Lipschitz stability in inverse parabolic problems by the Carleman estimate, Inverse Problems, 14 (1998), 1229-1245.  doi: 10.1088/0266-5611/14/5/009.  Google Scholar

[22]

O. Kavian, Y. Kian and E. Soccorsi, Uniqueness and stability results for an inverse spectral problem in a periodic waveguide, J. Math. Pures Appl., 104 (2015), 1160-1189.  doi: 10.1016/j.matpur.2015.09.002.  Google Scholar

[23]

C. Kenig and M. Salo, The Calderón problem with partial data on manifolds and applications, Analysis & PDE, 6 (2013), 2003-2048.  doi: 10.2140/apde.2013.6.2003.  Google Scholar

[24]

C. E. Kenig, J. Sjöstrand and G. Uhlmann, The Calderón problem with partial data, Ann. of Math., 165 (2007), 567-591.  doi: 10.4007/annals.2007.165.567.  Google Scholar

[25]

Y. Kian, Recovery of non-compactly supported coefficients of elliptic equations on an infinite waveguide, Journal of the Institute of Mathematics of Jussieu, 19 (2020), 1573-1600.  doi: 10.1017/S1474748018000488.  Google Scholar

[26]

Y. Kian, Determination of non-compactly supported electromagnetic potentials in an unbounded closed waveguide, Revista Matemática Iberoamericana, 36 (2020), 671-710.  doi: 10.4171/rmi/1143.  Google Scholar

[27]

Y. Kian, Stability of the determination of a coefficient for wave equations in an infinite waveguide, Inverse Probl. Imaging, 8 (2014), 713-732.  doi: 10.3934/ipi.2014.8.713.  Google Scholar

[28]

Y. Kian, D. Sambou and E. Soccorsi, Logarithmic stability inequality in an inverse source problem for the heat equation on a waveguide, Applicable Analysis, 99 (2020), 2210-2228.  doi: 10.1080/00036811.2018.1557324.  Google Scholar

[29]

Y. Kian, Q. S. Phan and E. Soccorsi, A carleman estimate for infinite cylindrical quantum domains and application to inverse problems, Inverse Problems, 30 (2014), 055016, 16 pp. doi: 10.1088/0266-5611/30/5/055016.  Google Scholar

[30]

Y. Kian, Q. S. Phan and E. Soccorsi, Hölder stable determination of a quantum scalar potential in unbounded cylindrical domains, Jour. Math. Anal. Appl., 426 (2015), 194-210.  doi: 10.1016/j.jmaa.2015.01.028.  Google Scholar

[31]

K. Krupchyk, M. Lassas and G. Uhlmann, Inverse problems with Partial data for a magnetic Schrödinger operator in an infinite slab or bounded domain, Comm. Math. Phys., 312 (2012), 87-126.  doi: 10.1007/s00220-012-1431-1.  Google Scholar

[32]

K. Krupchyk and G. Uhlmann, Stability estimates for partial data inverse problems for Schrödinger operators in the high frequency limit, J. Math. Pures Appl., 126 (2019), 273-291.  doi: 10.1016/j.matpur.2019.02.017.  Google Scholar

[33]

X. Li, Inverse boundary value problems with partial data in unbounded domains, Inverse Problems, 28 (2012), 085003, 23 pp. doi: 10.1088/0266-5611/28/8/085003.  Google Scholar

[34]

X. Li, Inverse problem for Schrödinger equations with Yang-Mills potentials in a slab, J. Diff. Equat., 253 (2012), 694-726.  doi: 10.1016/j.jde.2012.04.001.  Google Scholar

[35]

X. Li and G. Uhlmann, Inverse Problems with partial data in a Slab, Inverse Problems and Imaging, 4 (2010), 449-462.  doi: 10.3934/ipi.2010.4.449.  Google Scholar

[36]

J-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Die Grundlehren der mathematischen Wissenschaften, Band 181. Springer-Verlag, New York-Heidelberg, 1972. doi: 10.1007/978-3-642-65161-8.  Google Scholar

[37]

J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153-169.  doi: 10.2307/1971291.  Google Scholar

[38]

G. Uhlmann, Electrical impedance tomography and Calderón's problem, Inverse problems, 25 (2009), 123011, 39 pp. doi: 10.1088/0266-5611/25/12/123011.  Google Scholar

[39]

M. S. Zhdanov and G. V. Keller, The Geoelectrical Methods in Geophysical Exploration, Methods in Geochemistry and Geophysics, vol 31 (Amsterdam: Elsevier), (1994). Google Scholar

[40]

Y. Zou and Z. Guo, A review of electrical impedance techniques for breast cancer detection, Med. Eng. Phys., 25 (2003), 79-90.  doi: 10.1016/S1350-4533(02)00194-7.  Google Scholar

show all references

References:
[1]

G. Alessandrini, Stable determination of conductivity by boundary measurements, Appl. Anal., 27 (1988), 153-172.  doi: 10.1080/00036818808839730.  Google Scholar

[2]

H. Ammari and G. Uhlmann, Reconstruction from partial Cauchy data for the Schrödinger equation, Indiana University Math J., 53 (2004), 169-183.  doi: 10.1512/iumj.2004.53.2299.  Google Scholar

[3]

J. Behrndt and J. Rohleder, Inverse problems with partial data for elliptic operators on unbounded Lipschitz domains, Inverse Problems, 36 (2020), 035009, 18 pp. doi: 10.1088/1361-6420/ab603d.  Google Scholar

[4]

M. Bellassoued, Y. Kian and E. Soccorsi, An inverse stability result for non compactly supported potentials by one arbitrary lateral Neumann observation, J. Differential Equations, 260 (2016), 7535-7562.  doi: 10.1016/j.jde.2016.01.033.  Google Scholar

[5]

M. Bellassoued, Y. Kian and E. Soccorsi, An inverse problem for the magnetic Schrödinger equation in infinite cylindrical domains, Publ. Research Institute Math. Sci., 54 (2018), 679-728.  doi: 10.4171/PRIMS/54-4-1.  Google Scholar

[6]

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

[7]

A. L. Bukhgeim and G. Uhlmann, Recovering a potential from partial Cauchy data, Comm. PDE, 27 (2002), 653-668.  doi: 10.1081/PDE-120002868.  Google Scholar

[8]

A. P. Calderón, On an inverse boundary value problem, Seminar on Numerical Analysis and its Applications to Continuum Physics, Rio de Janeiro, Sociedade Brasileira de Matematica, (1980), 65–73. doi: 10.1590/S0101-82052006000200002.  Google Scholar

[9]

P. Caro, D. Dos Santos Ferreira and A. Ruiz, Stability estimates for the Radon transform with restricted data and applications, Advances in Math., 267 (2014), 523-564.  doi: 10.1016/j.aim.2014.08.009.  Google Scholar

[10]

P. Caro, D. Dos Santos Ferreira and A. Ruiz, Stability estimates for the Calderón problem with partial data, J. Diff. Equa., 260 (2016), 2457-2489.  doi: 10.1016/j.jde.2015.10.007.  Google Scholar

[11]

P. Caro and K. Marinov, Stability of inverse problems in an infinite slab with partial data, Commun. Partial Diff. Equa., 41 (2016), 683-704.  doi: 10.1080/03605302.2015.1127967.  Google Scholar

[12]

P. Caro and V. Pohjola, Stability estimates for an inverse problem for the magnetic Schrödinger operator, IMRN, (2015), 11083–11116. doi: 10.1093/imrn/rnv020.  Google Scholar

[13]

M. Choulli, Une Introduction Aux Problèmes Inverses Elliptiques et Paraboliques, Mathematics & Applications, Vol. 65, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-642-02460-3.  Google Scholar

[14]

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.  Google Scholar

[15]

M. Choulli, Y. Kian and E. Soccorsi, Double logarithmic stability estimate in the identification of a scalar potential by a partial elliptic Dirichlet-to-Neumann map, Bulletin of the South Ural State University, Ser. Mathematical Modelling, Programming and Computer Software (SUSUMMCS), 8 (2015), 78-94.  doi: 10.14529/mmp150305.  Google Scholar

[16]

M. Choulli, Y. Kian and E. Soccorsi, On the Calderón problem in periodic cylindrical domain with partial Dirichlet and Neumann data, Mathematical Methods in the Applied Sciences, 40 (2017), 5959-5974.  doi: 10.1002/mma.4446.  Google Scholar

[17]

M. Choulli, Y. Kian and E. Soccorsi, Stability result for elliptic inverse periodic coefficient problem by partial Dirichlet-to-Neumann map, J. Spec. Theory, 8 (2018), 733-768.  doi: 10.4171/JST/212.  Google Scholar

[18]

M. Choulli and E. Soccorsi, An inverse anisotropic conductivity problem induced by twisting a homogeneous cylindrical domain, J. Spec. Theory, 5 (2015), 295-329.  doi: 10.4171/JST/99.  Google Scholar

[19]

O. Yu. Èmanuvilov, Controllability of evolution equations, Sb. Math., 186 (1995), 879-900.  doi: 10.1070/SM1995v186n06ABEH000047.  Google Scholar

[20]

O. Yu. Imanuvilov, G. Uhlmann and M. Yamamoto, The Calderón problem with partial data in two dimensions, Journal American Math. Society, 23 (2010), 655-691.  doi: 10.1090/S0894-0347-10-00656-9.  Google Scholar

[21]

O. Yu. Imanuvilov and M. Yamamoto, Lipschitz stability in inverse parabolic problems by the Carleman estimate, Inverse Problems, 14 (1998), 1229-1245.  doi: 10.1088/0266-5611/14/5/009.  Google Scholar

[22]

O. Kavian, Y. Kian and E. Soccorsi, Uniqueness and stability results for an inverse spectral problem in a periodic waveguide, J. Math. Pures Appl., 104 (2015), 1160-1189.  doi: 10.1016/j.matpur.2015.09.002.  Google Scholar

[23]

C. Kenig and M. Salo, The Calderón problem with partial data on manifolds and applications, Analysis & PDE, 6 (2013), 2003-2048.  doi: 10.2140/apde.2013.6.2003.  Google Scholar

[24]

C. E. Kenig, J. Sjöstrand and G. Uhlmann, The Calderón problem with partial data, Ann. of Math., 165 (2007), 567-591.  doi: 10.4007/annals.2007.165.567.  Google Scholar

[25]

Y. Kian, Recovery of non-compactly supported coefficients of elliptic equations on an infinite waveguide, Journal of the Institute of Mathematics of Jussieu, 19 (2020), 1573-1600.  doi: 10.1017/S1474748018000488.  Google Scholar

[26]

Y. Kian, Determination of non-compactly supported electromagnetic potentials in an unbounded closed waveguide, Revista Matemática Iberoamericana, 36 (2020), 671-710.  doi: 10.4171/rmi/1143.  Google Scholar

[27]

Y. Kian, Stability of the determination of a coefficient for wave equations in an infinite waveguide, Inverse Probl. Imaging, 8 (2014), 713-732.  doi: 10.3934/ipi.2014.8.713.  Google Scholar

[28]

Y. Kian, D. Sambou and E. Soccorsi, Logarithmic stability inequality in an inverse source problem for the heat equation on a waveguide, Applicable Analysis, 99 (2020), 2210-2228.  doi: 10.1080/00036811.2018.1557324.  Google Scholar

[29]

Y. Kian, Q. S. Phan and E. Soccorsi, A carleman estimate for infinite cylindrical quantum domains and application to inverse problems, Inverse Problems, 30 (2014), 055016, 16 pp. doi: 10.1088/0266-5611/30/5/055016.  Google Scholar

[30]

Y. Kian, Q. S. Phan and E. Soccorsi, Hölder stable determination of a quantum scalar potential in unbounded cylindrical domains, Jour. Math. Anal. Appl., 426 (2015), 194-210.  doi: 10.1016/j.jmaa.2015.01.028.  Google Scholar

[31]

K. Krupchyk, M. Lassas and G. Uhlmann, Inverse problems with Partial data for a magnetic Schrödinger operator in an infinite slab or bounded domain, Comm. Math. Phys., 312 (2012), 87-126.  doi: 10.1007/s00220-012-1431-1.  Google Scholar

[32]

K. Krupchyk and G. Uhlmann, Stability estimates for partial data inverse problems for Schrödinger operators in the high frequency limit, J. Math. Pures Appl., 126 (2019), 273-291.  doi: 10.1016/j.matpur.2019.02.017.  Google Scholar

[33]

X. Li, Inverse boundary value problems with partial data in unbounded domains, Inverse Problems, 28 (2012), 085003, 23 pp. doi: 10.1088/0266-5611/28/8/085003.  Google Scholar

[34]

X. Li, Inverse problem for Schrödinger equations with Yang-Mills potentials in a slab, J. Diff. Equat., 253 (2012), 694-726.  doi: 10.1016/j.jde.2012.04.001.  Google Scholar

[35]

X. Li and G. Uhlmann, Inverse Problems with partial data in a Slab, Inverse Problems and Imaging, 4 (2010), 449-462.  doi: 10.3934/ipi.2010.4.449.  Google Scholar

[36]

J-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Die Grundlehren der mathematischen Wissenschaften, Band 181. Springer-Verlag, New York-Heidelberg, 1972. doi: 10.1007/978-3-642-65161-8.  Google Scholar

[37]

J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153-169.  doi: 10.2307/1971291.  Google Scholar

[38]

G. Uhlmann, Electrical impedance tomography and Calderón's problem, Inverse problems, 25 (2009), 123011, 39 pp. doi: 10.1088/0266-5611/25/12/123011.  Google Scholar

[39]

M. S. Zhdanov and G. V. Keller, The Geoelectrical Methods in Geophysical Exploration, Methods in Geochemistry and Geophysics, vol 31 (Amsterdam: Elsevier), (1994). Google Scholar

[40]

Y. Zou and Z. Guo, A review of electrical impedance techniques for breast cancer detection, Med. Eng. Phys., 25 (2003), 79-90.  doi: 10.1016/S1350-4533(02)00194-7.  Google Scholar

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