Property C for ordinary differential equations and applications to inverse scattering. (English) Zbl 0935.35177

Summary: Property C, that is, completeness of the set of products of some solutions to Sturm-Liouville equations is proved. Several uniqueness theorems for various inverse scattering problems are obtained in a very simple way with the help of property C. Two classical uniqueness results for inverse scattering problem on the half-axis and for the inverse spectral problem are proved in a very short new way. Earlier the author introduced property C for partial differential equations and used it extensively for proving uniqueness theorems for many inverse problems.


35R30 Inverse problems for PDEs
34L25 Scattering theory, inverse scattering involving ordinary differential operators
35P25 Scattering theory for PDEs
Full Text: DOI EuDML


[1] Levin, B.: Distribution of Zeros of Entire Function. Providence (RI): Amer. Math. Soc. 1980.
[2] Levitan, B. M.: On the completeness of the products of solutions of two Sturm-Liouville equations. Duff. mt. Eq. 7 (1994), 1 - 14. · Zbl 0831.34083
[3] Marchenko, V. A.: Sturm-Liouville Operators and Applications. Boston: Birkhauser 1986. · Zbl 0592.34011
[4] Ramm, A. G.: On completeness of the products of harmonic functions. Proc. Amer. Math. Soc. 99 (1986), 253 - 256. · Zbl 0602.35015
[5] Ramm, A. G.: Completeness of the products of solutions to PDE and uniqueness theorems in inverse scattering. Inverse Problems 3 (1987), L77 - L82.
[6] Ramm, A. C.: Multidimensional Inverse Scattering Problems. New York: Longman/ Wiley 1992. Russian translation of the expanded monograph by Mir Publishers (Moscow) 1994. · Zbl 0746.35056
[7] Ramm, A. G.: Recovery of the potential from I-function. Math. Reports Acad. of Sci. Canada 9 (1987), 177 - 182. · Zbl 0623.34033
[8] Ramm, A. C.: Inversion of fixed-frequency surface data for layered medium. J. Inv. and Ill-Posed Problems 2 (1994), 263 - 268. · Zbl 0817.35128
[9] Ram in, A. C.: Recovery of compactly supported spherically symmetric potentials from the phase shift of s-wave. In: Spectral and Scattering Theory (ed.: A. G. Ramm). New York: Plenum PubI. 1998, pp.111 - 130. · Zbl 0908.34071
[10] Ramm, A. G.: Compactly supported spherically symmetric potentials are uniquely deter- mined by the phase shift of s-wave. Phys. Lett A 242 (1998)4-5, 215 - 219. · Zbl 0948.81652
[11] Ramm, A. C.: A new approach to the inverse scattering and spectral problems for the Sturm- Liouvilleequation. Ann. Phys. 7 (1998), 321 - 338. · Zbl 0919.34027
[12] Ramm, A. C. and G. Porru: Completeness and non-completeness results for the set of products of solutions to differential equations. AppI. Anal. 60(1996), 241 -249. · Zbl 0874.35132
[13] Rio, R., Gesztesy, F. and B. Simon: Inverse spectral analysis with partial information on the potential. IMRN 15 (1997), 751 - 758. · Zbl 0898.34075
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.