Zero range three-particle equations. [Karlsson-Zeiger equations] Page: 3 of 7
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for - provided it also predicts physically meaningful amplitudes. This is
not so obvious. In the on-shell limit
r (k:9=) r t(q2) . e^sii (A)
the condition becoms
t t q r 0.) + a fim. fkdk z k -if0 (5)
or using the usual dispersion-theoretic representation for I
nf2k d ks2k z + -: (6)
This condition cao only be satisfied If the Integral over the left-hand sos
aihoes for any needed value of q , a condition which not in general be
satisfied when the left-hand ct arises fro meson exchanges. For the
special case q cr0 f coovr. the condition Ia mt.
I wish to stress the point, which I should have realized from the start
but have only now folly appreciared, that the limit I am about to define
takes un outside the freaok of c onrfeional theories. This is often a
hanrier for theorists who try to understand what 1 have dcvo. They realize
that the KZ and Faddecv theories .c fully equivalent before the limit Is
taken, but ane not prepared for the possibility that the KZ equation oematn
valid in the zero range limit while the Faddecv equatloys do not. Of course,
we have known ever since Thomas pointed it out in 19)5 jf21 that the Schroedinger
equation is also nct well defined in this limit.
Provided the 0000 function inside the range of forces R is non-singu-
lar when transformed into omen- space and goes smoothly to zee as R gees
io zero, we can take this limit directly ia the KZ equations simply by re-
placing t-(k;2) by no(qI) and @^(k) by
P(k) eti [k ) cos i + S (7)
Assumi. for simplicity only s.. t ncacterings between the pairs and total
three-yaticle angular roo oen, ma then Find chat the K1. amplitudes are
separable, and dirtmined by non-nacable Integral cqoatios. For the 3-3
amplitude ian this representation In
J (P p 4 I;Wlni) x nmecpB
rB(q )J B(Pg"Po)V "ZWiO)r(q )) (g)
If we have inaddition two-particle bound states with wave functions in the
ro range limit c ixp(-Ky)/v, no find the further sI-Plifiatron that all
feur amplitudes (free-irac, roil.'nerce, breakup, clrstic and reacrrangemet)
are ie in ote ef nc saa ete- variable function by
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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/m1/3/: accessed March 23, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.