GENERALIZED DYSON SERIES AND FEYNMAN'S OPERATIONAL CALCULUS 9

The reader will see further on that the norm (0.10) appears in our

estimates in a natural way.

The functions 8 will be interpreted physically as potentials. The

condition that 6 be in L , is rather minimal in most respects. No

smoothness is required, and 6 is allowed to be time-dependent and

C-valued. The use of C-valued functions 6 will enable us, in particular,

to treat the diffusion case (or "imaginary time" case) as well as the

quantum mechanical case (or "real time" case). (See Remark 0.3 below.)

The importance of C-valued potentials in the study of decay systems in

quantum mechanics is discussed thoroughly in the recent book of Exner

[9]. Certainly, the most serious restriction in our assumptions is that

e(s,«) be essentially bounded for n-a.e. s. However, even this condition

seems quite reasonable in light of our goal of obtaining rigorously

justified perturbation series valid in the quantum mechanical case.

If 0 € L 1 and if n = y 4 - v is decomposed into its continuous and

0 0

1 ;

ri

r

discrete parts, then it is not difficult to show that e e L ^ n L

n

00J_ j y 00J_ ; v

and

(o.ii) H9||„

1 ; n

= I M u

; u

+ llelL

1 ; v

.

H. The multiplication operators 8(s): We remind the reader that the

*

N

operator of multiplication by a function in L (E.) belongs to

- 2

cL( L

(RN))

and has operator norm equal to the essential supremum of the

function. (See, e.g., [26, Example 2.11, p. 146].) For us, the

L°°-functions that arise will be of the form e(s,-) where 0 e L

n

. It

will be convenient to let 0(s) denote the operator of multiplication by

8(s,«), acting in L (R ) . The operator norm ||e(s)|| then satisfies

(0.12) ||e(s)|| = ||8(s,OIL•

I. The operator-valued function space integrals K,(F), X e C,:

First, let Cn = Cn[0,t] be the space of E. -valued continuous functions

x on [0,t] such that x(0) =0. We consider C~ as equipped with N-dimen-

sional Wiener measure m which is just the product of N one-dimensional

Wiener measures [14,46,50]; recall that m is a probability measure on

CQ [50, Chap.7].