# American Institute of Mathematical Sciences

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:
 [1] G. Alessandrini, Stable determination of conductivity by boundary measurements,, Appl. Anal., 27 (1998), 153. [2] L. D. Faddeev, Growing solutions of the Schrödinger equation,, Dokl. Akad. Nauk SSSR, 165 (1965), 514. [3] P. Hähner, A periodic Faddeev-type solution operator,, J. Diff. Equat., 128 (1996), 300. doi: 10.1006/jdeq.1996.0096. [4] L. Hörmander, Linear Partial Differential Operators,, Springer-Verlag, (1976). [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. [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. [7] V. Isakov, Inverse Problems for Partial Differential Equations,, Springer-Verlag, (2006). [8] V. Isakov, Inverse Source Problems,, AMS, (1990). doi: 10.1090/surv/034. [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. [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. [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. [12] N. Mandache, Exponential instability in an inverse problem for the Schrödinger equation,, Inverse Problems, 17 (2001), 1435. [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. [14] V. Palamodov, Stability in diffraction tomography and a nonlinear "basic theorem",, J. d' Anal. Math., 91 (2003), 247. doi: 10.1007/BF02788790. [15] J. Sylvester and G. Uhlmann, Global uniqueness theorem for an inverse boundary value problem,, Ann. of Math., 125 (1987), 153.

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##### References:
 [1] G. Alessandrini, Stable determination of conductivity by boundary measurements,, Appl. Anal., 27 (1998), 153. [2] L. D. Faddeev, Growing solutions of the Schrödinger equation,, Dokl. Akad. Nauk SSSR, 165 (1965), 514. [3] P. Hähner, A periodic Faddeev-type solution operator,, J. Diff. Equat., 128 (1996), 300. doi: 10.1006/jdeq.1996.0096. [4] L. Hörmander, Linear Partial Differential Operators,, Springer-Verlag, (1976). [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. [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. [7] V. Isakov, Inverse Problems for Partial Differential Equations,, Springer-Verlag, (2006). [8] V. Isakov, Inverse Source Problems,, AMS, (1990). doi: 10.1090/surv/034. [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. [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. [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. [12] N. Mandache, Exponential instability in an inverse problem for the Schrödinger equation,, Inverse Problems, 17 (2001), 1435. [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. [14] V. Palamodov, Stability in diffraction tomography and a nonlinear "basic theorem",, J. d' Anal. Math., 91 (2003), 247. doi: 10.1007/BF02788790. [15] J. Sylvester and G. Uhlmann, Global uniqueness theorem for an inverse boundary value problem,, Ann. of Math., 125 (1987), 153.
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