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Inverse scattering. I. One dimension
1.See K. Chadan and P. C. Sabatier, Inverse Problems in Quantum Scattering Theory (Springer, New York, 1977).
2.I. Kay and H. E. Moses, Nuovo Cimento 3, 276 (1956).
3.L. D. Faddeev, Dokl. Akad. Nauk SSSR 121, 63 (1958)
3.[L. D. Faddeev, Math. Rev. 20, 773 (1959)].
4.L. D. Faddeev, Trudy Mat. Inst. Steklov 73, 314 (1964)
4.[L. D. Faddeev, Am. Math. Soc. Transl. 2, 139 (1964).] We shall refer to this paper as F.
5.L. D. Faddeev, Itogi Nauk. Tekh. Sov. Probl. Mat. 3, 93 (1974)
5.[L. D. Faddeev, J. Sov. Math. 5, 334 (1976)].
6.I. Kay, Commun. Pure Appl. Math. 13, 371 (1960).
7.P. Deift and E. Trubowitz, Commun. Pure Appl. Math. 32, 121 (1979). We shall refer to this paper as D&T.
8.I. Kay and H. E. Moses, Nuovo Cimento 22, 683 (1961);
8.I. Kay and H. E. Moses, and Commun. Pure Appl. Math. 14, 435 (1961).
9.R. G. Newton, in Scattering Theory in Mathematical Physics, edited by J. A. Lavita and J.‐P. Marchand (Reidel, Dordrecht, 1974), pp. 193–235. A slightly different version was given in lectures at the 1974 Summer Seminar on Inverse Problems American Mathematical Society, U.C.L.A. (unpublished).
10.R. G. Newton, Phys. Rev. Lett. 43, 541 (1979).
11.H. E. Moses, Studies Appl. Math. 58, 187 (1978).
12.The product decompositions given by H. E. Moses, J. Math. Phys. 5, 833 (1964), and on p. 353 of Ref. 5, may look similar but they are not “canonical” in that the factors lack analyticity properties.
13.F uses the first absolute moment in place of the second. It was pointed out both in Ref. 1 and by D&T that the second moment is needed. The place in the analysis where the second moment is required is at when there is a “half‐bound” state.
14.We use * for complex conjugation, ̃ for matrix transition, and † for Hermitian conjugation:
15.See p. 156 of Ref. 4 and p. 203 of Ref. 6.
16.Reference 6, p. 210. This is the generic case.
17.Reference 6, p. 212.
18.See Sec. 6.
19.The argument to the contrary in the three‐dimensional case, given on p. 229 of Ref. 8, is in error. It requires the use of a differential equation with respect to rotation, as given by Faddeev, Ref. 5, p. 391, to obtain There is no analog of this in one dimension.
20.See, for example, R. G. Newton, Scattering Theory of Waves and Particles (McGraw Hill, New York, 1966), pp. 355 and 356.
21.In fact, it is readily shown that Eqs. (2.46) and (2.48) require that B be of the form (3.22).
22.Reference 6, p. 172.
23.See, for example, Ref. 1, p. 36.
24.This representation was first given for a similar function by A. Sh. Bloch, Dokl. Akad. Nauk. SSSR 92, 209 (1953).
25.One may use the method of Z. S. Agranovich and V. A. Marchenko, The Inverse Problem of Scattering Theory (Gordon and Breach, New York, 1963), pp. 20ff, to prove this.
26.J. Plemelj, Monatschr. Math. Phys. 19, 211 (1908);
26.see also N. I. Muskelishvili, Singular Integral Equations (Groningen, Holland, 1953), pp. 381ff.
27.This equation was given by A. Sh. Bloch, Ref. 24, but without the connection of the Jost matrix to the S matrix.
28.See, for example, Ref. 19, pp. 616 and 617.
29.Reference 6, p. 167.
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