May  2015, 9(2): 469-478. doi: 10.3934/ipi.2015.9.469

Increasing stability for the inverse problem of the Schrödinger equation with the partial Cauchy data

1. 

Department of Mathematics, Statistics and Physics, Wichita State University, Wichita, KS 67260, United States

Received  April 2014 Revised  January 2015 Published  March 2015

To show increasing stability in the problem of recovering potential $c \in C^1(\Omega)$ in the Schrödinger equation with the given partial Cauchy data when energy frequency $k$ is growing, we will obtain some bounds for $c$ which can be viewed as an evidence of such phenomenon. The proof uses almost exponential solutions and methods of reflection.
Citation: Li Liang. Increasing stability for the inverse problem of the Schrödinger equation with the partial Cauchy data. Inverse Problems & Imaging, 2015, 9 (2) : 469-478. doi: 10.3934/ipi.2015.9.469
References:
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L. D. Faddeev, Growing solutions of the Schrödinger equation,, Dokl. Akad. Nauk SSSR, 165 (1965), 514.   Google Scholar

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V. Isakov, S. Nagayasu, G. Uhlmann and J. N. Wang, Increasing stability of the inverse boundary value problem for the Schrödinger equation,, in \emph{Inverse Problems and Applications}, (2014), 131.   Google Scholar

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V. Isakov, Inverse Problems for Partial Differential Equations,, Springer-Verlag, (2006).   Google Scholar

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V. Isakov, Inverse Source Problems,, AMS, (1990).  doi: 10.1090/surv/034.  Google Scholar

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V. Isakov, On uniqueness in the inverse conductivity problem with local data,, Inverse Problems and Imaging, 1 (2007), 95.  doi: 10.3934/ipi.2007.1.95.  Google Scholar

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V. Isakov, Increasing stability for the Schrödinger potential from the Dirichlet-to-Newmann map,, Discr. Cont. Dyn. Syst.-S, 4 (2011), 631.  doi: 10.3934/dcdss.2011.4.631.  Google Scholar

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V. Isakov and J. N. Wang, Increasing stability for determining the potential in the Schrödinger equation with attenuation from the Dirichlet-to-Neumann map,, Inverse Problems and Imaging, 8 (2014), 1139.  doi: 10.3934/ipi.2014.8.1139.  Google Scholar

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N. Mandache, Exponential instability in an inverse problem for the Schrödinger equation,, Inverse Problems, 17 (2001), 1435.   Google Scholar

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S. Nagayasu, G. Uhlmann and J. N. Wang, Increasing stability of the inverse boundary value problem for the acoustic equation,, Inverse Problems, 29 (2013).  doi: 10.1088/0266-5611/29/2/025012.  Google Scholar

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V. Palamodov, Stability in diffraction tomography and a nonlinear "basic theorem",, J. d' Anal. Math., 91 (2003), 247.  doi: 10.1007/BF02788790.  Google Scholar

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J. Sylvester and G. Uhlmann, Global uniqueness theorem for an inverse boundary value problem,, Ann. of Math., 125 (1987), 153.   Google Scholar

show all references

References:
[1]

G. Alessandrini, Stable determination of conductivity by boundary measurements,, Appl. Anal., 27 (1998), 153.   Google Scholar

[2]

L. D. Faddeev, Growing solutions of the Schrödinger equation,, Dokl. Akad. Nauk SSSR, 165 (1965), 514.   Google Scholar

[3]

P. Hähner, A periodic Faddeev-type solution operator,, J. Diff. Equat., 128 (1996), 300.  doi: 10.1006/jdeq.1996.0096.  Google Scholar

[4]

L. Hörmander, Linear Partial Differential Operators,, Springer-Verlag, (1976).   Google Scholar

[5]

M. Isaev and R. Novikov, Energy and regularity dependent stability estimates for the Gelfand's inverse problem in multi dimensions,, J. Inverse Ill-Posed Problems, 20 (2012), 313.  doi: 10.1155/2013/318154.  Google Scholar

[6]

V. Isakov, S. Nagayasu, G. Uhlmann and J. N. Wang, Increasing stability of the inverse boundary value problem for the Schrödinger equation,, in \emph{Inverse Problems and Applications}, (2014), 131.   Google Scholar

[7]

V. Isakov, Inverse Problems for Partial Differential Equations,, Springer-Verlag, (2006).   Google Scholar

[8]

V. Isakov, Inverse Source Problems,, AMS, (1990).  doi: 10.1090/surv/034.  Google Scholar

[9]

V. Isakov, On uniqueness in the inverse conductivity problem with local data,, Inverse Problems and Imaging, 1 (2007), 95.  doi: 10.3934/ipi.2007.1.95.  Google Scholar

[10]

V. Isakov, Increasing stability for the Schrödinger potential from the Dirichlet-to-Newmann map,, Discr. Cont. Dyn. Syst.-S, 4 (2011), 631.  doi: 10.3934/dcdss.2011.4.631.  Google Scholar

[11]

V. Isakov and J. N. Wang, Increasing stability for determining the potential in the Schrödinger equation with attenuation from the Dirichlet-to-Neumann map,, Inverse Problems and Imaging, 8 (2014), 1139.  doi: 10.3934/ipi.2014.8.1139.  Google Scholar

[12]

N. Mandache, Exponential instability in an inverse problem for the Schrödinger equation,, Inverse Problems, 17 (2001), 1435.   Google Scholar

[13]

S. Nagayasu, G. Uhlmann and J. N. Wang, Increasing stability of the inverse boundary value problem for the acoustic equation,, Inverse Problems, 29 (2013).  doi: 10.1088/0266-5611/29/2/025012.  Google Scholar

[14]

V. Palamodov, Stability in diffraction tomography and a nonlinear "basic theorem",, J. d' Anal. Math., 91 (2003), 247.  doi: 10.1007/BF02788790.  Google Scholar

[15]

J. Sylvester and G. Uhlmann, Global uniqueness theorem for an inverse boundary value problem,, Ann. of Math., 125 (1987), 153.   Google Scholar

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