Vol. 15, No. 11 (1959)
1959, 48 (11): 588-602. doi: 10.7498/aps.15.588
The recent advances of the "cut and try" method for solving the complicate non-linear network with constant flux are reviewed in the present paper.The network-transform method suggested by P. A. lonkin(who has solved some special non-linear networks by means of this method) is extended and applied to non-linear network however complicated.Some researches for optimum solving-plan when using this method is discussed in detail.In Appendix 1, the comparison of the network-transform method and the Kirchhoff''s equations-method suggested by Wang is given.In Appendix 2, the application of superposition theorem to a " mixed network" (containing not only non-linear elements but also the portion constructed by linear elements) is considered in a general case, where the network contains any number of non-linear elements.
In this note, a simple but not rigorous proof for dispersive relations is given. The proof proceeds by expanding the causal amplitude with respect to the intermediate states and considering the analyticity of the energy denominator and the corresponding numerator separately. While the analyticity of the energy denominator is more or less obvious, the proof of the analyticity of the numerator, say N, is achieved by considering the analyticity of the corresponding numerator where the mass μ2 has been replaced by -p2 (μ = mass of meson, p = momentum of nucleon in Breit's system, the scattering of mesons by nuc-leons being considered for definiteness) and passing to the analyticity of N with the help of a suitable transformation. The idea of replacing μ2 by another quantity is due Bokolubof (Боголюбов), but here analyticity with respect to this new quantity is not considered. In the present method, p2 is allowed to be as great as M2 -μ2(M = mass of nucleon).The analyticity of phase shifts η(k) in potential scattering is also considered and it is pointed out that if the potential V→ e-αr as r→∞(α > 0), then η(k) may be extended to where |Im k| <1/2a.
This short paper investigates two aspects of Chew-Low equations. First, it is compared with the usual formal theory of scattering, for example, that developed by Moeller. In comparison, it is proved that the wave functions occuring in the two formalisms are identical, apart from a constant multiple which represents the scalar product of the wave functions of a bare nucleon and a dressed nucleon. Next, equations of Chew-Low type with two h functions both possessing discontinuities along real axis from 1 to ∞ and from -1 to -∞ are investigated. It is shown that for the solution to exist, certain conditions on the crossing symmetry must be satisfied and that in certain special cases where the above condition is satisfied, existence of solutions requires the presence in the equations for h of an infinite number of terms representing intermediate discrete states.