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U(t),v(t)> - where BO = ~ , Jo(~(t)) 34 - = (o I1 AO = -B~ Letting coupled D e : ia~ first - me8 order and and, o combining (F2) and (F4) w e h a v e the system" ~' (t) = -iDe~(t) + Je (r ( t ) ) $' (t) = -iAor + Jo (F 5) where Je (@(t)) Finally, if we : -igBu(t)~(t) set =(t) J(=(t)) then w e can w r i t e (F 6) where (F5) : as -' (t} = - i A - ( t ) A (~(t)) four + J(-(t)) operator matrix A(Oe o) o Thus apply take we h a v e rewritten the e x i s t e n c e as our H i l b e r t our theory Ao coupled equations of S e c t i o n I.

Again, this is stronger than necessary). (38), (44) } = (cos (Bt)f,v) + (B-isin(Bt)g,v) (-B-Isin B(t-s)] F (Un(S)),v)ds n Then - 60 - It (cos(Bt)f,v) + (B-isin(Bt)g,v) + (- Fn(Un(S)),B-Isin B (t-s)v) dS o Suppose that we can show that (45) Fn (Un (x,t)) then since B - I s i n ( B ( t - s ) ) v can take the limit in (46) (u(t),v) = L' (Rnx [-T,T] )) u (x,t) p is a C= function of all its variables we (44) to conclude that: (cos(Bt)f,v) + (B-isin(Bt)g,v) + ( - u (s) P , [ B - l s i n B (t-s)~v) dx o Since f and g are nice and the integrand on the right left side is absolutely continuous d (u(t),v) =- is in L* , the and (B sin(Bt)f,v) + (cos(Bt)g,v) + (-u(s) p, [cos B(t-s)]v)ds o Again, the right hand side is absolutely d~ (u(t),v} 4 = (c0s(Bt)f, ( - u(t)P,v) + -B2v) continuous, so + (B-Isin(Bt)g, ( - u(s)P,[B-lsin B(t-s)]( -B2v) - Bav))ds o = (u(t),- for almost all t.

7) e x i s t s be the c o r r e s p o n d i n g (34) the ~n (t) = e we get that of T h e o r e m of on ~ for each ~ o G D and let ~o" Let to T h e o r e m (r) so the h y p o t h e s e s D the s o l u t i o n are u n i f o r m l y M t is u n i f o r m l y ~oG~ ~(r). Suppose in the c l 6 s u r e of for - F r o m the p r o o f of the C o r o l l a r y I IMt~ol I ~ K + ~ ~ 53 implies considered continin Sec- up after a f i n i t e amount of time. The a p p l i c a t i o n to the s i n e - G o r d o n equation is also easy.

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