Operator methods for inverse scattering on the real line. (English) Zbl 0585.35077

Inverse scattering theory for the operator \(Hu=-u''+p(x)u,\) \(-\infty <x<\infty\), Im p(x)\(=0\), is developed. This problem was extensively studied in the literature. The author cites papers by P. Deift and E. Trubowitz [Commun. Pure Appl. Math. 32, 121-251 (1979; Zbl 0388.34005)] and L. D. Faddeev [J. Math. Phys. 4, 72-104 (1963; Zbl 0112.451) and J. Sov. Math. 5, 334-396 (1976; Zbl 0373.35014)]. In Deift and Trubowitz’s paper the theory was given under the assumption \[ (*)\quad \int^{\infty}_{-\infty}(1+| x|^ 2) p(x) dx<\infty. \] In particular, some necessary and sufficient conditions were given for a \(2\times 2\) matrix to be the scattering matrix corresponding to a potential satisfying (*). On the other hand the inversion procedure described and used in many papers is valid under the weaker assumption \[ (**)\quad \int^{\infty}_{-\infty}(1+| x|) | p(x)| dx. \] In Deift and Trubowitz’s paper the characterization problem, i.e. the problem of finding necessary and sufficient conditions on the scattering data (in particular, on the scattering matrix), for these data to correspond to a potential of the class (**), was not solved. In Faddeev’s papers an attempt to solve the characterization problem was made. Some errors in his papers were pointed out in Deift and Trubowitz’s paper.
The author solves the characterization problem under the assumption (**). This presentation is new and contains a treatment of KdV equation and a study of the algebra of pseudodifferential operators related to H. The bibliography contains 8 entries.
Reviewer: A.Ramm


35P25 Scattering theory for PDEs
35J10 Schrödinger operator, Schrödinger equation
34L99 Ordinary differential operators
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[1] Agranovich Z.S., Gordon and Breach (1963)
[2] Deift P., Coram.Pure Appl. Math 32 pp 121– (1979) · Zbl 0388.34005 · doi:10.1002/cpa.3160320202
[3] Faddeev L.D., The inverse problem in quantum theory of scattering 4 (1) pp 72– (1963)
[4] Faddeev L.D., J.Sov.Math 5 (1) pp 334– (1976) · Zbl 0373.35014 · doi:10.1007/BF01083780
[5] HÖrmander L., springer-Verlag (1983)
[6] HÖrmander L., Springer-Verlag (1983)
[7] Lax P., Comm.Pure Appl.Math 21 pp 467– (1968) · Zbl 0162.41103 · doi:10.1002/cpa.3160210503
[8] McKean H.P., Inventiones math 30 pp 217– (1975) · Zbl 0319.34024 · doi:10.1007/BF01425567
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