Potential theory, path integrals and the Laplacian of the indicator

TL;DR

This paper establishes a novel connection between potential theory and Feynman path integrals by linking the Laplacian of the indicator function to boundary value problems, expanding classical methods and applicability.

Contribution

It introduces a formal definition of the Laplacian of the indicator function and demonstrates its role in unifying potential theory with path integrals, including convergence and boundary condition insights.

Findings

Path integral perturbation series matches classical boundary layer series.

Series valid for domains with finite corners, edges, and cusps.

Close connection between Dirichlet and Neumann problems established.

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 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.