  This paper links the field of potential theory -- i.e. the Dirichlet and
Neumann problems for the heat and Laplace equation -- to that of the Feynman
path integral, by postulating that the potential is equal to plus/minus the
Laplacian of the indicator of the domain D. The Laplacian of the indicator is a
generalized function: it is the d-dimensional analogue of the Dirac
delta'-function. This function has -- according to the author's best knowledge
-- not formally been defined before.
  We show, first, that the path integral's perturbation series (or Born series)
matches the classical single and double boundary layer series of potential
theory, thereby connecting two hitherto unrelated fields. Second, we show that
the perturbation series is valid for all domains D that allow Green's theorem
(i.e. with a finite number of corners, edges and cusps), thereby expanding the
classical applicability of boundary layers. Third, we show that the minus
(plus) in the potential holds for the Dirichlet (Neumann) boundary condition;
showing for the first time a particularly close connection between these two
classical problems. Fourth, we demonstrate that the perturbation series of the
path integral converges in a monotone/alternating fashion, depending on the
convexity/concavity of the domain.
  We also discuss the third boundary problem (which poses Robin boundary
conditions) and discuss an extension to moving domains.
