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Fredholm Methods in the Three‐Body Problem. I
1.L. Rosenberg, Phys. Rev. 135, B715 (1964);
1.R. G. Newton, J. Math. Phys. 8, 851 (1967).
2.L. D. Faddeev, Mathematical Aspects of the Three‐Body Problem in Quantum Scattering Theory (Davey, New York, 1965).
3.L. D. Faddeev, Zh. Eksp. Teor. Fiz. 39, 1459 (1960)
3.[L. D. Faddeev, Sov. Phys. JETP 12, 1014 (1961)];
3.L. D. Faddeev, Dokl. Akad. Nauk SSSR 138, 565 (1961)
3.[L. D. Faddeev, Sov. Phys. Doklady 6, 384 (1961)].
4.M. Rubin, R. Sugar, and G. Tiktopoulos, Phys. Rev. 159, 1348 (1967).
5.Our notation is generally to use small letters for operators on a two‐particle Hilbert space [in the center of mass system, so that it is ] and capital letters for operators on the three‐particle Hilbert space.
6.K. Meetz, J. Math. Phys. 3, 690 (1961);
6.J. Schwinger, Proc. Natl. Acad. Sci. (U.S.) 47, 122 (1961);
6.F. Coester, Phys. Rev. 133, B1516 (1964);
6.M. Scadron, S. Weinberg, and J. A. Wright, Phys. Rev. 135, B202 (1964)., Phys. Rev.
7.We multiply by by and by and then rule out isolated singularities of the resulting analytic function of three variables.
8.Note added in proof: The author has meanwhile found alternative expressions for the that do not depend on the factorizations (1.26)–(1.27″). They will be presented in a subsequent publication.
9.These are the same variables used by C. Lovelace, in Strong Interactions and High Energy Physics, edited by R. G. Moorehouse (Oliver and Boyd, London, 1964), p. 437.
10.See K. Meetz, Ref. 6.
11.R. G. Newton, Scattering Theory of Waves and Particles (McGraw‐Hill, New York, 1966), p. 281.
12.C. Zemach and A. Klein, Nuovo Cimento 10, 1078 (1958).
13.Ref. 11, p. 284.
14.An exception to this statement is the case in which and the eigenvector is spherically symmetric. Then there is no bound state even though has the eigenvalue 1. (Note that this and all our specific statements below concerning the exceptional nature of are based on the assumption that the potentials are spherically symmetric, which otherwise has not been assumed in this paper).
15.V. Bargmann, Proc. Natl. Acad. Sci. (U.S.) 38, 961 (1952).
16.There are two exceptions to this statement: If there is a “zero‐energy resonance,” i.e., if the s‐wave cross section at zero energy is infinite; or if there is a zero‐energy bound state. In both cases the spectrum of includes the point 1, but, in the first case, is not in the point spectrum of h, while in the second case it is. In both cases is not a bounded operator and, in fact, it will generally not exist. These two exceptional cases should be kept in mind in all our subsequent results.
17.If there is a zero‐energy resonance or bound state, then is a bounded operator only for The remark of Footnote 16 implies that, for is unbounded.
18.Constants such as C, etc., will be used generically, without any implication that they must have the same values when they are used in various contexts.
19.A. Erdelyi, Tables of Integral Transforms (McGraw‐Hill, New York, 1954), Vol. I, p. 75.
20.A. Erdelyi, Higher Transcendental Functions (McGraw‐Hill, New York, 1953), Vol. II, p. 85.
21.A. Erdelyi, Ref. 19, Vol. I, p. 16.
22.If the binding energy is not zero, the bound‐state wavefunction is bounded and decreases exponentially; see Ref. 11, pp. 332, 334, and 372–73. If the binding energy is zero (which implies that the angular momentum l is greater than zero) then its decrease is as see Ref. 11, p. 375, Eq. (12.152). Hence, if the potential satisfies (3.5), then so do all the bound‐state wavefunctions, except those of P‐ and D‐wave zero‐energy bound states. The D‐wave case is close enough to (3.5) to make the arguments used here still applicable. Only the P‐wave zero‐energy bound‐state case requires separate consideration. However, these had to be excluded in any case; see Footnote 16.
23.For this argument to be valid, we must assume that the (1, 3) bound state whose wavefunction is either has nonvanishing binding energy or else, if its binding energy is zero, its angular momentum is at least equal to 2; see Footnote 22. Of course, if decreases faster than (3.5), then a P‐wave zero‐energy bound state would be all right.
24.Note that the determinants are modified Fredholm determinants of kernels, so that
25.See, for example, R. T. Gunning and H. Rossi, Analytic Functions of Several Complex Variables (Prentice‐Hall, Englewood Cliffs, N.J., 1965), p. 2.
26.Ref. 25, p. 21.
27.It is assumed for simplicity that the determinant has a simple zero at If it has a multiple zero, the argument is similar.
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