Zero range three-particle equations. [Karlsson-Zeiger equations] Page: 2 of 7
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difficulty In starting from the Faddecv equations is that the two-particle
on-shell amplitude n(q ) - lii L/oq.'v-- occurs in the kernels with
the energy argumen - h and must b knon in the nonphysica' region. Worse,
tinc we wish to use on-shell amplitudes with a "left-hand cut" representing
seaon oeahanges, the integrals run over the ret -td the equations become
ambiguous. I tried to avoid this by emply using the on-shell amplitude in
the physlal region, but this did not lead toeconsistent equatunn. ubse-
quent effnrtt et avoid the cute by using she n/d separation of dispersion
theory also ran into difficulties.
eanwhile one of us (Eli) in nollaboratlon with tengt Karlsson of Gute-
bhrg tas developing a consistent set of equations using sperlacor none func-
etonn an n basis (i.e. sct tering and bound state eve functions for the pair
times a plane wave for the third particle)-r tool I hod earlier used to
conntruqt the exterior-interior separation e Th-. a equattons were published
in 1975 g and indeed depend only on half-on-shell oq-e functions and t-
Yntre with physical energy values, os unti-ipatod (J . The iZ nqutions
for coo-particlo Systems that scatter oly In a finite number of partial
.nnon have the luther advantage that the herools are real and energy inde-
pendent except for the usual three-particle Green's uncton.
The KZ equations are fully equivalent to the Faddeev equations as tan be
sean by starting from the Lo equation or tenpletrness relation uhtch elbows
us 0 o potrct Fully oft-shell t-matrices from half-off-shell t-matrices,
t(q.q0;o) - t(q;4g + 10)
+ kIdk t(q;R2)t I(o;I') ki-jt - k-q 1
- t(qO;qe + 10) + ? fk-dk t(q;ke)tiq(i;k)I2 NZ- k2 --i1
ly ning tme reversal ivarane, i.e. t(qq ;z) = t(q q;z), we find that
thin puts a constraint on the half-off-sh ll-maerix tk;qol10) which may be
fqldq iq(k)t(k';4210) qOdq.i(k;42io)(k') (2)
- i(k--) 2L; i)
(k) hq + _k2- iU)
This snntraint in a smewut different fom has bee investigated by naranger,
et al. I - Ohen the fully-off-shell r-nttri n be co..treted in this nay,
it iS easy to shoe that the existence of the coruction is both a necessary
and a sufficient condition fer the is eqations and the Faddeev equations to
define the same theory.
Last spring I finally realized rhat the K equation remains well defined
In the nero ronge limit, contains anly phase shifts, binding energies and
reduced widths, and hene detines precisely the theory I had been looking
Here’s what’s next.
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Noyes, H.P. & Zeiger, E.M. Zero range three-particle equations. [Karlsson-Zeiger equations], report, April 1, 1978; California. (https://digital.library.unt.edu/ark:/67531/metadc1192674/m1/2/: accessed March 25, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.