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

Yosra Soussi

doi:10.3934/ipi.2021022

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2021, Volume 15, Issue 5: 929-950. Doi: 10.3934/ipi.2021022

This issue Previous Article Quantum tomography and the quantum Radon transform Next Article Convergence rates of Tikhonov regularization for recovering growth rates in a Lotka-Volterra competition model with diffusion

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

Early access: March 2021

Published: October 2021

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Abstract

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.

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Mathematics Subject Classification: Primary: 35R30; Secondary: 35J15.

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References

[1]	G. Alessandrini, Stable determination of conductivity by boundary measurements, Appl. Anal., 27 (1988), 153-172. doi: 10.1080/00036818808839730.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[19]	O. Yu. Èmanuvilov, Controllability of evolution equations, Sb. Math., 186 (1995), 879-900. doi: 10.1070/SM1995v186n06ABEH000047.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[39]	M. S. Zhdanov and G. V. Keller, The Geoelectrical Methods in Geophysical Exploration, Methods in Geochemistry and Geophysics, vol 31 (Amsterdam: Elsevier), (1994).
[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.