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S in 6 t h e o r e m, (0,t). ^m' -*• 3C. e. in A, .. K ;J Now, by ( 4 . 6 ) and t h e norm e s t i m a t e ( 1 . 1 6 ) , we have ||KA(F)|| < A E |ani(||e|| n=0 n x n ' )n < M a n | ( | | e | | o o l ) n < + ~. i,n n=0 n Hence the conclusion follows from Eqs. ( 1 . 1 ) , ( 1 . 9, p. 83]. a REMARKS 4 . 1 . 5) is preserved. We should s t r e s s t h a t , in a l l cases, for instance, in the examples of Section 3, the form of the generalized Dyson series i s preserved j u s t as in Theorem 4,2.

X ( ur ) ) ] [ n 6 9(s . Cr ) n X dy9(s ) . L u u=l u=l Z u Note t h a t w i t h i n Am, lr n < . . < r m, whereas w i t h i n An , s 1 , < . . < sn . In o r d e r t o o b t a i n t h e formula f o r K, (F ) , we would j u s t o r d e r t h e r A m ,n and s variables with respect to each other much as in some earlier examples. However, we will simply give the formula for K,(F. 39) -(t-s)(Hn/X) • 62(s) e ° dy1(r)dy2(s). 5. 41) F(x) = f( / 61(s,x(s))dn1(s),... , / e(s,x(s))dn (s)), L q (0,t) L (0,t) q where n .

Dy(sn), n;j where ? )(H0/A) 3 ° -(s -s )(H 0 /x) l 2 11 U 0 X ( T) e 9(s2) ... +1-T)(H0/X) J +i ° e(s J+1 ) ... 6(sn) -(t-s )(HQ/X) n e ° . dp(sn). X n n-0 j-0 A n; . 27) as corresponding to an external disturbance, at the fixed time x, of the quantum system determined by 0 and y. ,T, . We consider this situation in the next example where we shall also highlight some of the connections with Feynman's operational calculus. 6. Let 0 < T 1 < . . < T, < t. 0h(Th,x(Th)). GENERALIZED DYSON SERIES AND FEYNMAN'S OPERATIONAL CALCULUS 43 Then Cx) - ni / i n e(s x(s ))] [ n e (T X ( T ))] x du(s ) F k n!

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