Potential theory, path integrals and the Laplacian of the indicator

Abstract

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 the following seemingly ill-defined potential:

$ V(x):=\mp \frac{{{\sigma^2}}}{2}\nabla_x^2{1_{{x\in D}}} $

where the volatility is the reciprocal of the mass (i.e. m = 1/σ ²) and ħ = 1. The Laplacian of the indicator can be interpreted using the theory of distributions: it is the d-dimensional analogue of the Dirac δ′-function, which can formally be defined as $\partial_x^2{1_{x>0 }}$.

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 as follows:

mode of convergence	absorbed propagator	reflected propagator
convex domain	alternating	monotone
concave domain	monotone	alternating

We also discuss the third boundary problem (which poses Robin boundary conditions) and discuss an extension to moving domains.