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VΘ‐Bound State and Uniqueness in the Three‐Particle Sector of the Lee Model
1.S. Weinberg, Phys. Rev. 102, 285 (1956).
2.N. Mugibayashi, Progr. Theoret. Phys. (Kyoto) 25, 803 (1961).
3.G. Källén and W. Pauli, Kgl. Danske Videnskab. Selskab Mat.‐Fys. Medd. 30, No. 7 (1955).
4.J. C. Houard and B. Jouvet, Nuovo Cimento 18, 466 (1960).
5.M. T. Vaughn, R. Aaron, and R. D. Amado, Phys. Rev. 124, 1258 (1961).
5.H. Ezawa, K. Kikkawa, and H. Umezawa, Nuovo Cimento 23, 751 (1962).
5.H. Chew, Phys. Rev. 132, 2756 (1963). This last author derives, but does not solve, the equations for a spin‐dependent interaction. We have extended our method of solution to this case and hope to report it in another context.
6.T. Muta, Progr. Theoret. Phys. (Kyoto) 33, 666 (1965), has independently obtained the wavefunction for the bound state. His method is very interesting because he relates the expansion coefficients to matrix elements of the V‐current for which one can, following Amado, obtain a Muskhelisvili equation. Our result, although given in quite different form, agrees with Muta’s. To derive the bound‐state condition he uses however the full amplitude instead of only its denominator. The numerator of the amplitude [Eq. (6) in I] has a zero for This explains why the value plays a special role in Muta’s work. We may emphasize that the value of the coupling constant for which and the value for which the bound‐state condition, are independent.
7.R. D. Amado, Phys. Rev. 122, 697 (1961).
8.R. D. Amado and R. P. Kenschaft, J. Math. Phys. 5, 1340 (1964).
9.A. Pagnamenta, J. Math. Phys. 6, 955 (1965), hereafter referred to as I. We are using the same notation and Hamiltonian as in I. Also: Im and where is the ordinary cutoff function normalized to
10.That a zero on the real axis below threshold in the denominator of the scattering amplitude indicates a bound state was pointed out by R. Jost, Helv. Phys. Acta. 30, 409 (1957).
11.V. Glaser and G. Källén, Nucl. Phys. 2, 706 (1956).
12.P. K. Srivastava, Phys. Rev. 131, 461 (1963).
13.P. K. Srivastava, Phys. Rev. 128, 2906 (1962).
14.Different methods to obtain solutions of this singular type of integral equations with a displacement in the denominator have been given by G. S. Litvincuk, Izv. Akad. Nauk. SSSR. Ser. Math. 25, 871 (1961); also in Refs 7 and 8. E. Kazes, Pennsylvania State University preprint. Ch. Sommerfield, Yale University preprint. Except in the first of the above references in which, however, a circular contour is considered the homogeneous equation is not discussed. In our discussion we follow to some extent Litvincuk and Kazes.
15.Equation (62) in Ref. 3 or Eq. (13) in I.
16.N. J. Muskhelisvili, Singular Integral Equations (P. Nordhoff Ltd., Groningen, The Netherlands, 1953).
17.A function satisfies a Hölder condition of degree k if for any two points in the interval L. , where C is an arbitrary constant and
18.Equation (II) in Ref. 1.
20.M. Levy, Nuovo Cimento 13, 115 (1959).
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