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**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.

### MSC:

35R30 | Inverse problems for PDEs |

34L25 | Scattering theory, inverse scattering involving ordinary differential operators |

35P25 | Scattering theory for PDEs |

### Keywords:

completeness of the set of products of some solutions to Sturm-Liouville equations; uniqueness; inverse scattering problems; inverse spectral problem
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\textit{A. G. Ramm}, Z. Anal. Anwend. 18, No. 2, 331--348 (1999; Zbl 0935.35177)

### References:

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[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. |

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[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 |

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