November  2014, 8(4): 1139-1150. doi: 10.3934/ipi.2014.8.1139

Increasing stability for determining the potential in the Schrödinger equation with attenuation from the Dirichlet-to-Neumann map

1. 

Department of Mathematics, Statistics, and Physics, Wichita State University, Wichita, KS 67260-0033

2. 

Institute of Applied Mathematics, NCTS (Taipei), National Taiwan University, Taipei 106, Taiwan

Received  September 2013 Revised  January 2014 Published  November 2014

We derive some bounds which can be viewed as an evidence of increasing stability in the problem of recovering the potential coefficient in the Schrödinger equation from the Dirichlet-to-Neumann map in the presence of attenuation, when energy level/frequency is growing. These bounds hold under certain a-priori regularity constraints on the unknown coefficient. Proofs use complex and bounded complex geometrical optics solutions.
Citation: 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 and Imaging, 2014, 8 (4) : 1139-1150. doi: 10.3934/ipi.2014.8.1139
References:
[1]

G. Alessandrini, Stable Determination of Conductivity by Boundary Measurements, Appl. Anal., 27 (1988), 153-172. doi: 10.1080/00036818808839730.

[2]

G. Alessandrini and M. Di Christo, Stable determination of an inclusion by boundary measurements, SIAM J. Math. Anal., 37 (2005), 200-217. doi: 10.1137/S003614100444191X.

[3]

G. Bao, S. Hou and P. Li, Inverse scattering by a continuation method with initial guesses from a direct imaging algorithm, J. Comput. Phys., 227 (2007), 755-762. doi: 10.1016/j.jcp.2007.08.020.

[4]

A. L. Bukhgeim, Recovering a potential from Cauchy data in the two-dimensional case, J. Inv. Ill-Posed Probl., 16 (2008), 19-33. doi: 10.1515/jiip.2008.002.

[5]

I. Bushuyev, Stability of recovery of the near-field wave from the scattering amplitude, Inverse Problems, 12 (1996), 859-867. doi: 10.1088/0266-5611/12/6/004.

[6]

P. Hähner, A periodic Faddeev-type solution operator, J. Diff. Equat., 128 (1996), 300-308. doi: 10.1006/jdeq.1996.0096.

[7]

L. Hörmander, Linear Partial Differential Operators, Springer-Verlag, Berlin, 1976.

[8]

T. Hrycak and V. Isakov, Increased stability in the continuation of solutions to the Helmholtz equation, Inverse Problems, 20 (2004), 697-712. doi: 10.1088/0266-5611/20/3/004.

[9]

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), 313-325. doi: 10.1515/jip-2012-0024.

[10]

V. Isakov, Inverse Problems for Partial Differential Equations, Springer-Verlag, New York, 2006.

[11]

V. Isakov, Increased stability in the continuation for the Helmholtz equation with variable coefficient, Contemp. Math., AMS, 426 (2007), 255-269. doi: 10.1090/conm/426/08192.

[12]

V. Isakov, Increasing stability for the Schrödinger potential from the Dirichlet-to-Neumann map, Discr. Cont. Dyn. Syst.-S, 4 (2011), 631-640. doi: 10.3934/dcdss.2011.4.631.

[13]

V. Isakov and S. Kindermann, Regions of stability in the Cauchy problem for the Helmholtz equation, Methods Appl. of Anal., 18 (2011), 1-29. doi: 10.4310/MAA.2011.v18.n1.a1.

[14]

V. Isakov, S. Nagayasu, G. Uhlmann and J. N. Wang, Increasing stability of the inverse boundary value problem for the Schrödinger equation, Contemp. Math.,AMS, 615 (2014), 131-143.

[15]

F. John, Continuous dependence on data for solutions of partial differential equations with a prescribed bound, Comm. Pure Appl. Math., 13 (1960), 551-585. doi: 10.1002/cpa.3160130402.

[16]

N. Mandache, Exponential instability in an inverse problem for the Schrödinger equation, Inverse Problems, 17 (2001), 1435-1444. doi: 10.1088/0266-5611/17/5/313.

[17]

S. Nagayasu, G. Uhlmann and J. N. Wang, Increasing stability of the inverse boundary value problem for the acoustic equation, Inverse Problems, 29 (2013), 025012, 11 pp. doi: 10.1088/0266-5611/29/2/025012.

[18]

F. Natterer and F. Wübbeling, Marching schemes for inverse acoustic scattering problem, Numer. Math., 100 (2005), 697-710. doi: 10.1007/s00211-004-0580-3.

[19]

V. Palamodov, Stability in diffraction tomography and a nonlinear "basic theorem", J. d' Anal. Math., 91 (2003), 247-268. doi: 10.1007/BF02788790.

[20]

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

[21]

J. Sylvester and G. Uhlmann, Inverse boundary value problems at the boundary-continuous dependence, Comm. Pure Appl. Math., 41 (1988), 197-221. doi: 10.1002/cpa.3160410205.

show all references

References:
[1]

G. Alessandrini, Stable Determination of Conductivity by Boundary Measurements, Appl. Anal., 27 (1988), 153-172. doi: 10.1080/00036818808839730.

[2]

G. Alessandrini and M. Di Christo, Stable determination of an inclusion by boundary measurements, SIAM J. Math. Anal., 37 (2005), 200-217. doi: 10.1137/S003614100444191X.

[3]

G. Bao, S. Hou and P. Li, Inverse scattering by a continuation method with initial guesses from a direct imaging algorithm, J. Comput. Phys., 227 (2007), 755-762. doi: 10.1016/j.jcp.2007.08.020.

[4]

A. L. Bukhgeim, Recovering a potential from Cauchy data in the two-dimensional case, J. Inv. Ill-Posed Probl., 16 (2008), 19-33. doi: 10.1515/jiip.2008.002.

[5]

I. Bushuyev, Stability of recovery of the near-field wave from the scattering amplitude, Inverse Problems, 12 (1996), 859-867. doi: 10.1088/0266-5611/12/6/004.

[6]

P. Hähner, A periodic Faddeev-type solution operator, J. Diff. Equat., 128 (1996), 300-308. doi: 10.1006/jdeq.1996.0096.

[7]

L. Hörmander, Linear Partial Differential Operators, Springer-Verlag, Berlin, 1976.

[8]

T. Hrycak and V. Isakov, Increased stability in the continuation of solutions to the Helmholtz equation, Inverse Problems, 20 (2004), 697-712. doi: 10.1088/0266-5611/20/3/004.

[9]

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), 313-325. doi: 10.1515/jip-2012-0024.

[10]

V. Isakov, Inverse Problems for Partial Differential Equations, Springer-Verlag, New York, 2006.

[11]

V. Isakov, Increased stability in the continuation for the Helmholtz equation with variable coefficient, Contemp. Math., AMS, 426 (2007), 255-269. doi: 10.1090/conm/426/08192.

[12]

V. Isakov, Increasing stability for the Schrödinger potential from the Dirichlet-to-Neumann map, Discr. Cont. Dyn. Syst.-S, 4 (2011), 631-640. doi: 10.3934/dcdss.2011.4.631.

[13]

V. Isakov and S. Kindermann, Regions of stability in the Cauchy problem for the Helmholtz equation, Methods Appl. of Anal., 18 (2011), 1-29. doi: 10.4310/MAA.2011.v18.n1.a1.

[14]

V. Isakov, S. Nagayasu, G. Uhlmann and J. N. Wang, Increasing stability of the inverse boundary value problem for the Schrödinger equation, Contemp. Math.,AMS, 615 (2014), 131-143.

[15]

F. John, Continuous dependence on data for solutions of partial differential equations with a prescribed bound, Comm. Pure Appl. Math., 13 (1960), 551-585. doi: 10.1002/cpa.3160130402.

[16]

N. Mandache, Exponential instability in an inverse problem for the Schrödinger equation, Inverse Problems, 17 (2001), 1435-1444. doi: 10.1088/0266-5611/17/5/313.

[17]

S. Nagayasu, G. Uhlmann and J. N. Wang, Increasing stability of the inverse boundary value problem for the acoustic equation, Inverse Problems, 29 (2013), 025012, 11 pp. doi: 10.1088/0266-5611/29/2/025012.

[18]

F. Natterer and F. Wübbeling, Marching schemes for inverse acoustic scattering problem, Numer. Math., 100 (2005), 697-710. doi: 10.1007/s00211-004-0580-3.

[19]

V. Palamodov, Stability in diffraction tomography and a nonlinear "basic theorem", J. d' Anal. Math., 91 (2003), 247-268. doi: 10.1007/BF02788790.

[20]

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

[21]

J. Sylvester and G. Uhlmann, Inverse boundary value problems at the boundary-continuous dependence, Comm. Pure Appl. Math., 41 (1988), 197-221. doi: 10.1002/cpa.3160410205.

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