Summary
A representation for two-body multichannel amplitudes is found in terms of the solution of a nonsingular integral equation which permits the construction of systematic unitary approximations.
Riassunto
Si trova una rappresentazione per le ampiezze a molti canali di due corpi in termini della soluzione di un’equazione integrale non singolare che permette la costruzione di approssimazioni unitarie sistematiche.
Реэюме
Определяется представление для двухчастичных многоканальных амплитуд в терминах рещения несингулярного интегрального уравнения, которое допускает конструирование систематических унитарных приближений.
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References
K. L. Kowalski andD. Feldman:Journ. Math. Phys.,2, 499 (1961).
K. L. Kowalski andD. Feldman:Journ. Math. Phys.,4, 507 (1963).
This was also shown independently byH. P. Noyes:Phys. Rev. Lett.,15, 538 (1965) from a different point of view.Noyes also pointed out several advantages and applications of this formalism which had not been realized previously. Further commentary concerning possible applications can be found inH. P. Noyes: SLAC-PUB-256 (1967) (unpublished).
K. L. Kowalski:Phys. Rev. Lett.,15, 798, 908E (1965).
Essentially the same trick used in ref.(1,2) was utilized byM. Levine, J. Tjon andJ. Wright:Phys. Rev. Lett.,16, 962 (1966), to remove the dominant singularity in the kernel of the partial-wave Bethe-Salpeter equation.
The trival multichannel case, namely the elastic scattering of two spin-bearing particles was treated in ref.(1,2).
A recent detailed exposition of this model has been given byJ. Noble:Phys. Rev.,148, 1528 (1966). Our notation will follow, to a certain extent, that of this reference.
We will use this opportunity to mention that the claim made in ref. (2) that det [V(\(\bar k,\bar k\),\(\bar k,\bar k\))]=0 implies det [T(\(\bar k,\bar k\),\(\bar k,\bar k\) E)]=0, in the present notation, is incorrect. The argument from which this conclusion followed was based on the false assertion that detD=0, where we have writtenT(\(\bar k,\bar k\),\(\bar k,\bar k\) E)=D −1 V(\(\bar k,\bar k\),\(\bar k,\bar k\)), was a sufficient condition for the solutions of theT-matrix integral equations to be nonunique.
In the single-channel case one can easily show that for physical parametric energies, in general,f is infinite when the phase shift is zero or an odd integer multiple of π unless the half-off-shell amplitude at this parametric energy is zero for all values of the off-shell momentum. Specifically,f has the formF(k′, k; E)[sin δ(k)]−1 whereF is finite and, in general, not identically zero. This observation indicates that the separable approximation proposed byNoyes ref. (3) grossly misrepresents the off-shell amplitudes at those parametric energies where sin δ(k)=0.
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This work was supported, in part, by the U.S. Atomic Energy Commission.
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Kowalski, K.L., Krauss, J. Unitary respresentation of multichannel amplitudes. Nuovo Cimento B (1965-1970) 57, 30–35 (1968). https://doi.org/10.1007/BF02710311
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DOI: https://doi.org/10.1007/BF02710311