Suppose that A and B are
linear operators which generate semigroups on a Hilbert space. Then A + B may be
far from being a generator. Nevertheless, a generator may sometimes be defined by
adding operators corresponding to A and B, but with values in a larger Hilbert
space, and then restricting the sum to the original Hilbert space. Here an
explicit product formula in terms of the semigroups generated by A and
B is shown to converge to a semigroup, which is that given by this sum.
This result has application to the perturbation theory of partial differential
equations. This is illustrated by the Feynman path integral representation of
the solution of the Schrödinger equation with potential term containing
singularities.
