Recovering singularities from backscattering in two dimensions

Communications in Partial Differential Equations

Volumn 26, Issue 3-4, 2001, Pages 697-715

  Ola, Petri ^a   Päivärinta, Lassi ^a   Serov, Valeri ^b  

^a UNIVERSITY OF OULU   (Finland)

^b MOSCOW STATE UNIVERSITY   (Russian Federation)

Author keywords

Backscattering; Inverse; Singularities

Indexed keywords

EID: 21544471233     PISSN: 03605302     EISSN: None     Source Type: Journal    
DOI: 10.1081/PDE-100001768     Document Type: Article

References (27)

1. Eskin, G.; Ralston, J. Inverse Backscattering, J. d'Analyse Math. 1992, 58, 177-190.
2. Faddeev, L.D. Growing Solutions of the Schrödinger Equation. Dokl. Akad. Nauk. SSSR 1965, 165, 514-517;
3. transl. Sov. Phys. Dokl. 1966, 10, 1033-1035.
4. Nachman, A.I. Reconstructions from Boundary Measurements, Ann. Math. 1988, 128, 531-576.
5. Sylvester, J.; Uhlmann, G. A Global Uniqueness Theorem for an Inverse Boundary Value Problem, Annals, of Math 1987, 125, 92-112.
6. Nachman, A.I. Global Uniqueness for a Two Dimensional Boundary Value Problem, Ann. Math. 1996, 143, 71-96.
7. Tsai, T.Y. The Schrödinger Operator in the Plane, Thesis: Yale University, 1989.
8. Prosser, R.T. Formal Solutions of Inverse Scattering Problems I-IV. J. Math. Phys. 1969, 10, 1819-1822;
9. Prosser, R.T. J. Math. Phys. 1976, 17, 1775-1779;
10. Prosser, R.T. J. Math. Phys. 1980, 21, 2648-2653;
11. Prosser, R.T. J. Math. Phys. 1982, 23, 2127-2130.
12. Stefanov, P. A Uniqueness Result for the Inverse Backscattering Problem, Inverse Probl. 1990, 6, 1055-1064.
13. Eskin, G.; Ralston, J. The Inverse Backscattering Problem in 3 Dimensions. Comm. Math. Phys. 1989, 124, 169-215.
14. Eskin, G.; Ralston, J. Inverse Backscattering in Two Dimensions. Comm. Math. Phys. 1991, 138, 451-486.
15. Stefanov, P. Generic Uniqueness for Two Inverse Problems in Potential Scattering, Comm. P.D.E. 1992, 17, 55-68.
16. Stefanov, P.; Uhlmann, G. Inverse backscattering for the acoustic equation, SIAM. J. Math. Anal. 1997, 28, 1191-1204.
17. Ramm, A. Multidimensional Inverse Scattering Problems and Completeness of the Products of Solutions to Homogenous PDE, Z. Angew. Math. Mech. 1989, 69, T13-T22.
18. Greenleaf, A.; Uhlmann, G. Recovering Singularities of a Potential from Singularities of the Scattering Data, Comm. Math. Phys. 1993, 157, 549-572.
19. Joshi, M.S. Recovering the Total Singularity of a Conormal Potential from Backscattering Data, manuscript. Ann. Inst. Fourier (Grenable), 1998, 5, 1513-1532.
20. Päivärinta, L.; Somersalo, E. Inversion of Discontinuities for the Schrödinger Equation in Three Dimensions, SIAM J. Math. Anal. 1991, 22, 480-499.
21. Päivärinta, L.; Serov, V.; Somersalo, E. Reconstruction of Singularities of a Multidimensional Scattering Potential, Adv. Appl. Math. 1994, 15, 97-113.
22. Päivärinta, L.; Serov, V. Recovery of Singularities of a Multidimensional Scattering Potential, SIAM J. Math. Anal. 1998, 29, 697-711.
23. Hajlasz, P. Sobolev Spaces on an Arbitrary Metric Space, Potential Anal. 1996, 5, 403-415.
24. Agmon, S. Spectral Properties of Schrödinger Operators and Scattering Theory. Annali della Scuola Norm. Sup. di Pisa serie IV 1975, II (2), 151-218.
25. Reed, M.; Simon, B. Methods of Modern Mathematical Physics III, Scattering Theory; Academic Press: San Diego, CA, 1979.
26. Stein, E. Harmonic Analysis; Princeton Mathematical Series 43; Princeton University Press: Princeton, NJ, 1993.
27. Triebel, H. Theory of Function Spaces; Akademie-Verlag: Leipzig, 1983.