126
Views
33
CrossRef citations to date
0
Altmetric
Original Articles

Multidimensional inverse problem with incomplete boundary spectral data

&
Pages 27-59 | Published online: 08 May 2007

  • Börg, , G. 1946 . Eine Umkehrung der Sturm-Liouwillischen Ezgenwertaufgabe. Bestimmung der Differentialgleichung durch die Eigenwerte, . Acta Math. , 78 ( 1 ) : 1 – 96 .
  • Levinson, , N. 1949 . The inverse Sturn-Lioville problem, Mat. , 25 – 30 . Tidsskr.B .
  • Gelfand, , I.M and Levitan , B.M. 1951 . On identification of the differential expression via its spectral function, Izv. Akad. . Nauk SSSR (in Russian), Sek. Matem. , 15 ( 4 ) : 309 – 360 .
  • I. S. Kac, M.G. Krein, Spectral functzon of a string, In F. Atkinson; Discrete and Continuous Boundary Value Problems (in Russian). Mir, Moskow (1968), 749pp.
  • Krein, , M.G. 1953 . The transmission function of a second-order one-dimensional boundary value problem, . Dokl. Akad. Nauk SSSR (in Russian). , 88 ( 3 ) : 405 – 408 .
  • 1957 . Some aspects of functional analysis and algebra, . Proc. Intern. Cong. Math., , 1 : 253 – 277 .
  • Berezanskii, , Y.M. 1955 . Inverse spectral problem for a Schrodinger operator, . Dokl. akad. Nauk SSSR (in Russian). , 105 ( 2 ) : 197 – 200 .
  • Berezanskii, , Y.M. 1953 . On the unique determination of a Schrödinger operator via its spectral function, . Dokl. akad. Nauk SSSR (in Russian). , 93 ( 4 ) : 591 – 594-200 .
  • Sylvester, , J. and Uhlmann, , G. 1987 . A global uniqueness theorem for an inverse boundary value problem, . Ann. Math. , 125 ( 2 ) : 153 – 169 .
  • Novikov, , R.G. and Henkin, , G.M. 1987 . A £-equation in the multidimensional inverse scattering problem, . Uspekh. Matem. Nauk (in Russian). , 42 ( 1 ) : 93 – 152 .
  • Calderon , A. 1980 . “ On an inverse boundary value problem, Semnin. Numer. Anal. and Its Appl. to Contin. Phys. ” . 65 – 73 . Rio de Janeiro .
  • Novikov, , R.G. 1988 . A multidimensional inverse spectral problem for the equation: b = 0, . Funk. Anal. i Ego Priloz. (in Russian). , 22 ( 4 ) : 11 – 22 .
  • Nachman, , A . 1988 . Reconstruction from boundary measurements, . Ann. Math. , 128 ( 4 ) : 531 – 576 .
  • I. S. Kac, M.G. Krein, Spectral functzon of a string, In F. Atkinson; Discrete and Continuous Boundary Value Problems (in Russian). Mir, Moskow (1968), 749pp.
  • A. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Dept. Math. Preprint Ser. N19, Univ. Rochester. (1993).
  • Nakamura, , G. and Uhlrmann, , G. 1994 . Global uniqueness for an inverse boundary problem arizing in elasticity, . Invent. Math. , 118 ( 3 ) : 457 – 474 .
  • Ola, , P. , Paivarinta, , L. and Somersalo, , E. 1993 . An inverse boundary value problem in electrodynamics, . Duke Math.J. , 70 : 617 – 653 .
  • Isakov, , V. 1993 . Uniqueness and stability in multidimensional inveme problems, . Inv. Probl. , 9 ( 6 ) : 579 – 622 .
  • Romanov, , V.G. 1991 . Uniqueness theorem in inverse problems for some equations of the second order, . Dokl. Akad. Nauk (in Russian) , 321 ( 3 ) : 678 – 682 .
  • Romanov , V.G. 1989 . Questions of the well-posedness of a problem of determining the speed of sound, . Sibirsk. Mat. Zhurn. (in Russian) , 29 ( 4 ) : 125 – 134 .
  • Suzuki, , T. 1998 . Ultra-hyperbolic approach to some multidimensional inverse problems, . Proc. Japan Acad., Ser. A. , 64 : 361 – 364 .
  • Lee, , J. and Uhlmann, , G. 1989 . Determining anisotropic real-analytic conductivity by boundary measurements, . Comm. Pure. Appl. Math. , 42 ( 5 ) : 1097 – 1112 .
  • Sylvester , J. 1990 . An anisotropic inverse boundary problem, . Comm. Pure. Appl. Math. , 43 ( 1 ) : 201 – 233 .
  • Belishev, , M.I. 1987 . On an approach to multidimensional inverse problems for the wave equation, . Dokl. Akad. Nauk SSSR (in Russian). , 297 ( 3 ) : 524 – 527 .
  • Belishev , M.I. and Blagovestchenskii, , A.S. 1988 . Direct method to solving a nonstationary inverse problem for the wave equation, . In: Ill-posed Problems of Math. Phys. and Anal. (in Russian), Krasnoyarsk . , : 43 – 49 .
  • Belishev, , M.I. and Kurylev, , Ya.V. 1986 . Inverse problem of the acoustic scattering in a space with inhomogeneities, . Zap.Nauchn. Semin. LOMI (in Russian). , 156 : 24 – 34 .
  • Belishev, , M.I. and Katchalov , A.P. 1989 . Boundary control in the inverse spectral problem for an inhomogeneous string, . Zap. Nauchn. Semin. LOMI (in Russian). , 179 : 14 – 22 .
  • I. S. Kac, M.G. Krein, Spectral functzon of a string, In F. Atkinson; Discrete and Continuous Boundary Value Problems (in Russian). Mir, Moskow (1968), 749pp.
  • A. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Dept. Math. Preprint Ser. N19, Univ. Rochester. (1993).
  • M.I. Belishev, Boundary control and wave field continuation, Preprint LOMI P-1-90 (in Russian). Leningrad. (1990), 41pp.
  • Belishev, , M.I. and Kurylev, , Ya.V. 1991 . Boundary control, wave field contznuation, and inverse problems for the wave equation, . Comp. Math. Appl. , 22 ( 4 ) : 27 – 52 .
  • Kurylev, , Ya.V. 1994 . To the reconstruction of two coefficients in the multidimensional isotropic inverse problem, . Zap. Nauchn. Semin. POMI (in Russian). , 210 : 158 – 163 .
  • Belishev, , M.I. and Kurylev, , Ya.V. 1992 . To the reconstruction of a Riemannian manifold via its spectral data (BC-method), . Comm. PDE. , 17 ( 5&6 ) : 767 – 804 .
  • Belishev, , M.I. and Katchalov, , A.P. 1992 . Boundary control and quasiphotons in the problem of a Riemannian manifold reconstruction via its dynamic data, . Zap. Nauchn. Semin. POMI (in Russian) , 203 : 21 – 50 .
  • Kurylev, , Ya.V. 1993 . Multidimensional inverse boundary problems by the BC-method: groups of transformations and uniqueness results, . Math. Comput. Model. , 18 ( 1 ) : 33 – 46 .
  • Kurylev, , Ya.V. 1994 . Inverse boundary problems on Riemannian manifolds, . Contemp. Math. , 173 : 181 – 192 .
  • Kurylev, , Ya.V. 1995 . An inverse boundary problem for the Schrödinger operator with magnetic field . J. Math. Phys. , 36 ( 6 ) : 2761 – 2776 .
  • Kurylev, , Ya.V. 1995 . A multidimensional Gelfand-Levitan inverse boundary problem, . Diff. Eq. and Math. Phys. (Univ. of Alabama, Birmingham, March 13-17, 1994), , : 117 – 131 .
  • Belishev, , M.I. and Katchalov, , A.P. 1994 . Operator integral in multidimensional inverse spectral problem, . Zap. Nauchn. Semin. POMI (in Russian) , 215 : 9 – 37 .
  • Kurylev, , Ya.V. 1997 . A multidimensional Gelfand inverse problem, Inverse Probl. Related with Geom., . Proc. Symp. Tokyo Metropol. Univ, 25-25 Sept. 1996 , : 1 – 15 .
  • Isozaki, , H. 1991 . Some remarks on multidimensiona1 Börg-Levinson theorem, . J. Math. Kyoto Univ. , 31 ( 5 ) : 743 – 753 .
  • Ramm, , A.G. 1993 . Property C with constraints and inverse spectral problems with incomplete data, . J. Math. Anal. Appl. , 180 ( 1 ) : 239 – 244 .
  • Katchalov, , A.P. and Kurylev, , Ya.V. 1993 . Incomplete spectral data and reconstruction of a Riemannian manifold, . J. Inv. III-Posed Probl. , 1 ( 2 ) : 141 – 153 .
  • Katchalov, , A.P. and Kurylev, , Ya.V. 1996 . The multidimensional Gelfand inverse problem with incomplete boundary spectral data, . Dokl. Akad. Nauk (in Russian). , 346 ( 5 ) : 587 – 589 .
  • Gromoll, , D. , Klingenberg, , W. and Meyer, , W. Riemannsche Geometrie im Grossen, . Lect. Notes Math. , 55 ( 1975 ) 287
  • Tataru, . 1995 . Unique continuation for solutions to PDE's; between Hörmander's theorem and Hölmgren's theorem, . Comm. Part. Diff. Eq. , 20 ( 5&6 ) : 855 – 884 .
  • Birman, , M.S. and Solomyak, , M.Z. 1986 . “ Spectral Theory of Self-Adjoint Operators in Hilbert Space, ” . In Comp., Dordrecht-Boston. , 301 D. Reidel .
  • Vainberg, , B.R. , ed. 1982 . Asymptotic Methods in Problems of Mathematical Physics, , 294 Moscow. : MGU Press .
  • Babich, , V.M. and Ulin, , V.V. 1981 . The comlex space-time ray method and "quasiphotons", . Zap. Nauch. Semin. LOMI (in Russian). , 117 : 5 – 12 .
  • Katchalov, , A.P. 1984 . System of coordinates in describing "quasiphotons", . Zap. Nauchn. Semin. LOMI (in Russian). , 140 : 73 – 76 .
  • Babich, , V.M. and Buldyrev, , V.S. 1991 . Short-Wavelength Diffraction Theory, 445 Berlin. : Springer, .
  • Hörmander, , L. 1984 . The Analysis of Linear Partial Differential Operators. 420 Berlin-Heidelberg. : III, Springer, .
  • Duff, , G.F.D. and Spencer, , D.C. 1975 . Harmonic tenzors on Riemannian manifolds with boundary, . Ann. Math. , 56 ( 1 ) : 27 – 59 .
  • Ray, , D.B. and Singer, , I.M. 1975 . R-torsion and the Laplacian on Riemannian manifolds, . Adv. Math. , 18 ( 1 ) : 27 – 59 .

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.