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On the Inverse Problem for the Klein‐Gordon s‐Wave Equation
1.F. Calogero and A. Degasperis, J. Math. Phys. 9, 90 (1968). Hereafter referred to as I.
2.F. Calogero, O. D. Corbella, A. Degasperis, and B. M. De Stefano, J. Math. Phys. 9, 1002 (1968).
3.Simultaneously with this work, an analogous investigation has been devoted by O. D. Corbella to the potential scattering of Dirac particles (J. Math. Phys.) (to be published).
4.KG will be used as an abbreviation for Klein‐Gordon.
5.Throughout this paper we use units such that
6.The appearance of these unphysical solutions has been investigated for a square‐well potential by L. I. Schiff, H. Snyder, and J. Weinberg, Phys. Rev. 57, 315 (1940).
7.R. Jost and W. Kohn, Kgl. Danske Videnskab. Selskab, Mat.‐Fys. Medd. 27, No. 9 (1953).
8.E. Corinaldesi, Nuovo Cimento 11, 468 (1954).
9.M. Verde, Nucl. Phys. 9, 255 (1958).
10.I. M. Gel’fand and B. M. Levitan, Iz. Akad. Nauk SSSR 15, 309 (1951).
11.V. De Alfaro, Nuovo Cimento 10, 675 (1958).
12.We assume, of course, that the boundary conditions do not introduce any other cut in the k plane.
13.M. J. Roberts, Proc. Phys. Soc. (London) 80, 1290 (1962).
14.When Eq. (1.2) does not contain any relation between the momentum k and the energy E, so that we can formally define as energy any function of k. If we choose the nonrelativistic definition Eq. (1.2) with reduces to the Schrödinger equation, so that the results of Paper I are implicitly contained in this paper.
15.R. G. Newton, Phys. Rev. 101, 1588 (1956).
16.The relation (A2a) is not correct for because the scalar product is given by . This, however, has no effect on the considerations that follow.
17.The completeness relations can be obtained by standard methods constructing the Green’s function for the nonstationary equation .
18.Our results do not agree completely with the formulas given by Verde, Ref. 9. In addition to Eq. (A5), he obtains a second equation for the function φ(E; ρ) that we believe to be incorrect. Furthermore, we do not find the mass‐dependent term appearing in his relation between the potential and the kernels and Our corresponding formula is Eq. (3.3).
19.See the review article by R. G. Newton, J. Math. Phys. 1, 319 (1960).
20.G. Parzen, Phys. Rev. 80, 261 (1950).
21.As in the nonrelativistic case, these zeros are simple because the expression (C2) holds.
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