## Potential splitting and numerical solution of the inverse scattering problem on the line.(English)Zbl 1054.34137

A numerical method is suggested for the solution of the inverse scattering problem for the Sturm-Liouville equation $$-y''+V(x)y=\lambda y$$ on the line, where the potential $$V(x)$$ is real-valued, integrable, has a finite first moment, and contains no bound states.

### MSC:

 34L25 Scattering theory, inverse scattering involving ordinary differential operators 65L09 Numerical solution of inverse problems involving ordinary differential equations 34B24 Sturm-Liouville theory 47E05 General theory of ordinary differential operators

### Keywords:

numerical method; inverse scattering problems
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### References:

 [1] Inverse Problems in Quantum Scattering Theory (2nd edn). Springer: New York, 1989. [2] Deift, Communications on Pure and Applied Mathematics 32 pp 121– (1979) · Zbl 0388.34005 [3] Faddeev, American Mathematical Society Translations. Series 65 pp 139– (1967) · Zbl 0181.56704 [4] Sturm-Liouville Operators and Applications. Birkh?user: Basel, 1986. [5] Bayliss, Mathematics of Computation 52 pp 321– (1989) [6] Sacks, Wave Motion 18 pp 21– (1993) · Zbl 0803.34075 [7] Symes, Journal of Mathematical Analysis and Applications 71 pp 379– (1979) · Zbl 0425.35092 [8] Chen, Inverse Problems 8 pp 365– (1992) · Zbl 0760.34017 [9] Rose, Inverse Problems 13 pp l1– (1997) · Zbl 0872.35129 [10] Sylvester, SIAM Journal on Applied Mathematics 59 pp 669– (1999) [11] Newton, Journal of Mathematical Physics 21 pp 493– (1980) · Zbl 0446.34029 [12] Aktosun, Journal of Mathematical Physics 33 pp 1717– (1992) · Zbl 0760.35032 [13] Aktosun, Journal of Mathematical Physics 33 pp 3865– (1992) · Zbl 0762.35075 [14] Solitons and the Inverse Scattering Transform. SIAM: Philadelphia, 1981. [15] Outline of a theory of the KdV equation. In Recent Mathematical Methods in Nonlinear Wave Propagation, (ed.), Lecture Notes in Mathematics, vol. 1640. Springer: Berlin, 1996; 70-102. [16] Bona, Philosophical Transactions of the Royal Society of London, Series A 351 pp 107– (1995) · Zbl 0824.65095 [17] Provenzale, Journal of Computational Physics 94 pp 314– (1991) · Zbl 0729.65098 [18] Taha, Journal of Computational Physics 55 pp 231– (1984) · Zbl 0541.65083 [19] Aktosun, Journal of Mathematical Physics 35 pp 6231– (1994) · Zbl 0822.34070 [20] Novikova, Computational Seismology 18 pp 164– (1987)
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