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 & 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.  Google Scholar

[2]

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

[3]

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

[4]

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

[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.  Google Scholar

[6]

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

[7]

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

[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.  Google Scholar

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

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

[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.  Google Scholar

[6]

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

[7]

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

[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.  Google Scholar

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

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