American Institute of Mathematical Sciences

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June  2011, 4(3): 631-640. doi: 10.3934/dcdss.2011.4.631

Increasing stability for the Schrödinger potential from the Dirichlet-to Neumann map

Victor Isakov 1,

1. 

Wichita State University, 1845 Fairmount, Wichita, KS, 67260-0033

Received  April 2009 Revised  September 2009 Published  November 2010

We derive some bounds which can be viewed as an evidence of increasing stability in the problem of recovery of the potential coefficient in the Schrödinger equation from the Dirichlet-to-Neumann map, when frequency (energy level) is growing. These bounds hold under certain a-priori bounds on the unknown coefficient. Proofs use complex- and real-valued geometrical optics solutions. We outline open problems and possible future developments.
Keywords: Inverse Scattering., Inverse Problems.
Mathematics Subject Classification: Primary: 35R30; Secondary: 78A4.
Citation: Victor Isakov. Increasing stability for the Schrödinger potential from the Dirichlet-to Neumann map. Discrete and Continuous Dynamical Systems - S, 2011, 4 (3) : 631-640. doi: 10.3934/dcdss.2011.4.631
References:
[1]

G. Alessandrini, Stable determination of conductivity by boundary measurements, Appl. Anal., 27 (1988), 153-172.

[2]

G. Alessandrini and M. Di Cristo, Stable determination of an inclusion by boundary measurements, SIAM J. Math. Anal., 37 (2005), 200-217.

[3]

D. Arallumallige and V. Isakov, Increased stability in the continuation of solutions to the Helmholtz equation, Inverse Problems, 23 (2007), 1689-1698.

[4]

K. Astala and L. Päivärinta, Calderon's inverse conductivity problem in the plane, Ann. Math., 163 (2006), 265-299.

[5]

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.

[6]

A. L. Bukhgeim, Recovering a potential from Cauchy data in the two-dimensional case, J. Inv. Ill-Posed Probl., 15 (2007), 19-35.

[7]

I. Bushuyev, Stability of recovery of the near-field wave from the scattering amplitude, Inverse Problems, 12 (1996), 859-869.

[8]

A. P. Calderon, On an inverse boundary value problem, in "Seminar on Numerical Analysis and its Applications to Continuum Physics," Rio de Janeiro, (1980), 65-73.

[9]

D. Colton, H. Haddar and M. Piana, The linear sampling method in inverse electromagnetic scattering theory, Inverse Problems, 19 (2003), S105-S137.

[10]

L. Faddeev, Increasing solutions of the Schrödinger equation, Soviet Phys. Dokl., 10 (1966), 1033-1035.

[11]

P. Hähner, A periodic Faddeev-type solution operator, J. Diff. Equat., 128 (1996), 300-308.

[12]

L. Hörmander, "Linear Partial Differential Operators," Springer-Verlag, Berlin, 1963.

[13]

T. Hrycak and V. Isakov, Increased stability in the continuation of solutions to the Helmholtz equation, Inverse Problems, 20 (2004), 697-712.

[14]

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

[15]

V. Isakov, "Increased Stability in the Continuation for the Helmholtz Equation with Variable Coefficient," in "Control Methods in PDE-Dynamical Systems," Contemp. Math., 426, AMS, (2007), 255-269.

[16]

V. Isakov and A. Nachman, Global uniqueness for a two-dimensional elliptic inverse problem, Trans. AMS, 347 (1995), 3375-3391.

[17]

F. John, Continuous dependence on data for solutions of partial differential equations with a prescribed bound, Comm. Pure Appl. Math., 13 (1960), 551-587.

[18]

N. Mandache, Exponential instability in an inverse problem for the Schrödinger equation, Inverse Problems, 17 (2001), 1435-1444.

[19]

A. Nachman, Global Uniqueness for a two dimensional inverse boundary value problem, Ann. Math., 142 (1996), 71-96.

[20]

F. Natterer and F. Wübbeling, Marching schemes for inverse acoustic scattering problem, Numer. Math., 100 (2005), 697-710.

[21]

R. Novikov, The $\bar{\partial}$-approach to monochromatic inverse scattering in three dimensions, J. Geom. Anal., 18 (2008), 612-631.

[22]

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

[23]

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

[24]

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

[25]

M. Taylor, "Partial Differential Equations. II," Springer-Verlag, New York, 1997.

show all references

References:
[1]

G. Alessandrini, Stable determination of conductivity by boundary measurements, Appl. Anal., 27 (1988), 153-172.

[2]

G. Alessandrini and M. Di Cristo, Stable determination of an inclusion by boundary measurements, SIAM J. Math. Anal., 37 (2005), 200-217.

[3]

D. Arallumallige and V. Isakov, Increased stability in the continuation of solutions to the Helmholtz equation, Inverse Problems, 23 (2007), 1689-1698.

[4]

K. Astala and L. Päivärinta, Calderon's inverse conductivity problem in the plane, Ann. Math., 163 (2006), 265-299.

[5]

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.

[6]

A. L. Bukhgeim, Recovering a potential from Cauchy data in the two-dimensional case, J. Inv. Ill-Posed Probl., 15 (2007), 19-35.

[7]

I. Bushuyev, Stability of recovery of the near-field wave from the scattering amplitude, Inverse Problems, 12 (1996), 859-869.

[8]

A. P. Calderon, On an inverse boundary value problem, in "Seminar on Numerical Analysis and its Applications to Continuum Physics," Rio de Janeiro, (1980), 65-73.

[9]

D. Colton, H. Haddar and M. Piana, The linear sampling method in inverse electromagnetic scattering theory, Inverse Problems, 19 (2003), S105-S137.

[10]

L. Faddeev, Increasing solutions of the Schrödinger equation, Soviet Phys. Dokl., 10 (1966), 1033-1035.

[11]

P. Hähner, A periodic Faddeev-type solution operator, J. Diff. Equat., 128 (1996), 300-308.

[12]

L. Hörmander, "Linear Partial Differential Operators," Springer-Verlag, Berlin, 1963.

[13]

T. Hrycak and V. Isakov, Increased stability in the continuation of solutions to the Helmholtz equation, Inverse Problems, 20 (2004), 697-712.

[14]

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

[15]

V. Isakov, "Increased Stability in the Continuation for the Helmholtz Equation with Variable Coefficient," in "Control Methods in PDE-Dynamical Systems," Contemp. Math., 426, AMS, (2007), 255-269.

[16]

V. Isakov and A. Nachman, Global uniqueness for a two-dimensional elliptic inverse problem, Trans. AMS, 347 (1995), 3375-3391.

[17]

F. John, Continuous dependence on data for solutions of partial differential equations with a prescribed bound, Comm. Pure Appl. Math., 13 (1960), 551-587.

[18]

N. Mandache, Exponential instability in an inverse problem for the Schrödinger equation, Inverse Problems, 17 (2001), 1435-1444.

[19]

A. Nachman, Global Uniqueness for a two dimensional inverse boundary value problem, Ann. Math., 142 (1996), 71-96.

[20]

F. Natterer and F. Wübbeling, Marching schemes for inverse acoustic scattering problem, Numer. Math., 100 (2005), 697-710.

[21]

R. Novikov, The $\bar{\partial}$-approach to monochromatic inverse scattering in three dimensions, J. Geom. Anal., 18 (2008), 612-631.

[22]

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

[23]

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

[24]

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

[25]

M. Taylor, "Partial Differential Equations. II," Springer-Verlag, New York, 1997.

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