Advanced Search
Article Contents
Article Contents

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

Abstract Related Papers Cited by
  • 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.
    Mathematics Subject Classification: Primary: 35R30; Secondary: 78A46.


    \begin{equation} \\ \end{equation}
  • [1]

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


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


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


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


    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.


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


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


    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.


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


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


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


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


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


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


    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.


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


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


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


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


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


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


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


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


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


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

  • 加载中

Article Metrics

HTML views() PDF downloads(139) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint