Stability estimates for an inverse problem for the Schrödinger equation at negative energy in two dimensions

References

• Alessandrini , G . 1988 . Stable determination of conductivity by boundary measurements . Appl. Anal. , 27 ( 1 ) : 153 – 172 .
   Google Scholar
• Mandache , N . 2001 . Exponential instability in an inverse problem of the Schrödinger equation . Inverse Probl. , 17 ( 5 ) : 1435 – 1444 .
   Web of Science ®Google Scholar
• Novikov , RG . 2011 . New global stability estimates for the Gel'fand-Calderon inverse problem . Inverse Probl. , 27 ( 1 ) : 015001
   Web of Science ®Google Scholar
• Santacesaria , M . New global stability estimates for the Calderón problem in two dimensions , J. Inst. Math. Jussieu, Available on CJO doi:10.1017/S147474801200076X
   Google Scholar
• Novikov , RG . 1999 . Approximate solution of the inverse problem of quantum scattering theory with fixed energy in dimension 2 . Tr. Mat. Inst. Steklova , 225 Solitony Geom. Topol. na Perekrest. pp. 301–318 (in Russian); trans. in Proc. Steklov Inst. Math. 225(2) (1999), pp. 285–302
   Google Scholar
• Novikov , RG . 2005 . The -approach to approximate inverse scattering at fixed energy in three dimensions . IMRP Int. Math. Res. Pap. , 2005 ( 6 ) : 287 – 349 .
   Google Scholar
• Novikov , RG and Santacesaria , M . Monochromatic reconstruction algorithms for two-dimensional multi-channel inverse problems . Int. Math. Res. Notices, first published online February 18, 2012 doi:10.1093/imrn/rns025
   Google Scholar
• Isakov , V . 2011 . Increasing stability for the Schrödinger potential from the Dirichlet-to-Neumann map . Discrete Contin. Dyn. Syst. Ser. S , 4 ( 3 ) : 631 – 640 .
   Google Scholar
• Gel'fand , IM . 1954 . Some Aspects of Functional Analysis and Algebra, Proceedings of the International Congress of Mathematicians, Vol. 1, Erven P. Noordhoff N.V. Groningen , 253 – 276 . Amsterdam : North-Holland Publishing Co. .
   Google Scholar
• Novikov , RG . 1988 . Multidimensional inverse spectral problem for the equation −Δψ + (v(x) − Eu(x))ψ = 0 . Funkt. Anal. i Pril. , 22 ( 4 ) : 11 – 22 . (in Russian); English Transl.: Funct. Anal. Appl. 22(4) (1988), pp. 263–272
   Google Scholar
• Calderón , AP . 1980 . On an Inverse Boundary Problem, Seminar on Numerical Analysis and its Applications to Continuum Physics , 61 – 73 . Rio de Janeiro : Sociedade Brasiliera de Matematica .
   Google Scholar
• Dubrovin , BA , Krichever , IM and Novikov , SP . 1976 . The Schrödinger equation in a periodic field and Riemann surfaces . Dokl. Akad. Nauk SSSR , 229 ( 1 ) : 15 – 18 .
   Google Scholar
• Grinevich , PG . 2000 . The scattering transform for the two-dimensional Schrödinger operator with a potential that decreases at infinity at fixed nonzero energy . Uspekhi Mat. Nauk , 55 ( 6)(336 ) : 3 – 70 . (in Russian); Transl. Russ. Math. Surveys 55(6) (2000), pp. 1015–1083
   Google Scholar
• Novikov , RG . 1992 . The inverse scattering problem on a fixed energy level for the two-dimensional Schrödinger operator . J. Funct. Anal. , 103 ( 2 ) : 409 – 463 .
   Web of Science ®Google Scholar
• Lavrent'ev , MM , Romanov , VG and Shishat·skiĭ , SP . 1986 . Ill-posed Problems of Mathematical Physics and Analysis, Transl. from the Russian by J.R. Schulenberger. Transl. edited by Lev J. Leifman, Translation of Mathematical Monographs , Vol. 64 , Providence , , RI : American Mathematical Society . vi+290pp
   Google Scholar
• Beilina , L and Klibanov , MV . 2012 . Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems , New York : Springer . 407pp
   Google Scholar
• Bukhgeim , AL . 2008 . Recovering a potential from Cauchy data in the two-dimensional case . J. Inverse Ill-Posed Probl. , 16 ( 1 ) : 19 – 33 .
   Web of Science ®Google Scholar
• Novikov , RG and Santacesaria , M . 2010 . A global stability estimate for the Gel'fand-Calderón inverse problem in two dimensions . J. Inverse Ill-Posed Probl. , 18 ( 7 ) : 765 – 785 .
   Web of Science ®Google Scholar
• Sylvester , J and Uhlmann , G . 1987 . A global uniqueness theorem for an inverse boundary value problem . Ann. Math. , 125 ( 1 ) : 153 – 169 .
   Web of Science ®Google Scholar
• Nachman , A . 1996 . Global uniqueness for a two-dimensional inverse boundary value problem . Ann. Math. , 143 : 71 – 96 .
   Web of Science ®Google Scholar
• Druskin , VL . 1982 . The unique solution of the inverse problem in electrical surveying and electrical well logging for piecewise-constant conductivity . Izvestiya, Phys. Solid Earth , 18 ( 1 ) : 51 – 53 .
   Google Scholar
• Kohn , R and Vogelius , M . 1985 . Determining conductivity by boundary measurements: II. Interior results . Comm. Pure Appl. Math. , 38 ( 5 ) : 643 – 667 .
   Web of Science ®Google Scholar
• Alessandrini , G and Vessella , S . 2005 . Lipschitz stability for the inverse conductivity problem . Adv. Appl. Math. , 35 ( 2 ) : 207 – 241 .
   Web of Science ®Google Scholar
• Rondi , L . 2006 . A remark on a paper by G. Alessandrini and S. Vessella: “Lipschitz stability for the inverse conductivity problem” . Adv. Appl. Math. , 36 ( 1 ) : 67 – 69 .
   Web of Science ®Google Scholar
• Liu , L . Stability estimates for the two-dimensional inverse conductivity problem, Ph.D. thesis, Department of Mathematics, University of Rochester, New York, 1997
   Google Scholar
• Barceló , T , Faraco , D and Ruiz , A . 2007 . Stability of Calderón inverse conductivity problem in the plane . J. Math. Pures Appl. , 88 ( 6 ) : 522 – 556 .
   Web of Science ®Google Scholar
• Isaev , MI . 2011 . Exponential instability in the Gel'fand inverse problem on the energy intervals . J. Inverse Ill-Posed Probl. , 19 ( 3 ) : 453 – 472 . e-print available at arXiv:1012.2193
   Web of Science ®Google Scholar
• Novikov , RG . 2005 . Formulae and equations for finding scattering data from the Dirichlet-to-Neumann map with nonzero background potential . Inverse Probl. , 21 ( 1 ) : 257 – 270 .
   Web of Science ®Google Scholar
• Grinevich , PG and Novikov , SP . 1988 . Two-dimensional “inverse scattering problem” for negative energies and generalized-analytic functions: I. Energies below the ground state . Funct. Anal. Appl. , 22 ( 1 ) : 19 – 27 .
   Web of Science ®Google Scholar
• Isaev , MI and Novikov , RG . Stability estimates for determination of potential from the impedance boundary map , 2011, e-print available at arXiv:1112.3728
   Google Scholar
• Faddeev , LD . 1965 . Growing solutions of the Schrödinger equation . Dokl. Akad. Nauk SSSR , 165 ( 3 ) : 514 – 517 .
   Google Scholar
• Barceló , JA , Barceló , T and Ruiz , A . 2001 . Stability of the inverse conductivity problem in the plane for less regular conductivities . J. Differ. Eqns , 173 : 231 – 270 .
   Web of Science ®Google Scholar