near a single instant x. Moreover, p could have a nontrivial singular
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:
(0.3) v = I a ) 6 , with 0 xn ... x, t.
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
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
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