2

G.W. JOHNSON and M.L. LAPIDUS

near a single instant x. Moreover, p could have a nontrivial singular

part.

When n has a nonzero discrete part, the form of the generalized Dyson

series changes markedly and genuinely new phenomena occur. The combi-

natorial structure of the series is much more complicated even when v

is finitely supported. For instance, additional summations appear as

well as powers of the potential 9 evaluated at fixed times. Indeed, some

of the combinatorial complications and nearly all of the analytic

difficulties are found in the simple case n « y 4 - a 6 , where 6 is the

Dirac measure at x. Accordingly, we discuss this prototypical example

in detail in Section 1 and elsewhere in the paper and use it as a con-

ceptual aid to the general development.

Another particular case of interest is obtained when n ~ v is a

purely discrete measure with finite support:

h

(0.3) v = I a ) 6 , with 0 xn ... x, t.

p-i

p

T

P

1 h

By considering the exponential functional [i.e., by letting f(z) =

exp(z)] and further specializing, we will see in Example 3.3 that the

series reduces to a single term, the familiar h-th Trotter product.

Now approximating Lebesgue measure I by discrete measures of the form

(0.3) and applying a stability theorem with respect to the measures

(Theorem 4.3), we establish connections with the Trotter product formula

[2,28,40,48,...].

One can also use the stability theorem to see the relationship

between the relatively simple perturbation series corresponding to

continuous n and more complex Dyson series. We make this explicit in

Example 4.2 where 6 is approximated by absolutely continuous measures

whose densities are given by a 6-sequence.

Our generalized Dyson series can be represented graphically by

generalized Feynman diagrams. The n-th term of the classical Dyson

series corresponds to a single connected Feynman graph. Here, however,

the n-th term of the generalized Dyson series gives rise to many dis-

connected components, one for each summand. The complex combinatorial

structure of the generalized Dyson series is accurately reflected in the