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
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