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 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.Comment: 46 pages, 2 figure

Families of one-point interactions resulting from the squeezing limit of the sum of two- and three-delta-like potentials

“…Finally, for τ > 2 (at the line L 0 ) the divergent terms in (28) and (30) vanish at all at the same resonance conditions (47). Next, from expressions (27) and (29) we conclude that the limit Λ-matrix is of form (8) with the element θ defined by formulae (50).…”

“…Consider now the realization of point interactions at the limiting right-hand points of the lines L K and L S (P 1 and P 2 , respectively), and the limiting right-hand line (L 0 ) of the region Q 2 (see figure 1). Thus, at τ = 1, the total coefficients at the divergent term in the singular matrix elementsλ 21 given by (28) and (30) become zero if the resonance equations…”

“…The point interactions of this countable family may be called 'multiple-resonant-tunnelling δ ′ -potentials of the F -type'. At the point P 2 , with τ = 2 in (28) and (30), we find that these divergent terms become finite in the ε → 0 limit under conditions (41) and (42) in which c is formally set zero, i.e., if the equations J 2 (a 1 , a 2 ) := F 2 (a 1 , a 2 ; c)| c=0 = 0 and J 3 (a 1 , a 2 , a 3 ) := F 3 (a 1 , a 2 , a 3 ; c)| c=0 = 0 (47) are fulfilled. Similarly to the F 2,3 -sets, the solutions of these equations yield curves on the (a 1 , a 2 )-plane (see green curves in figure 4) and surfaces in the (a 1 , a 2 , a 3 )-space that may be referred to as J 2 -and J 3 -sets, respectively.…”

Geometric perturbation theory and Acoustic Boundary Condition Dynamics

“…We define δ(y ∈ Γ(t)) ≡ −n y ∇1 Ω(t)\(∂Ω(t)\Γ(t)) (y) for the subdomain Γ(t) ⊂ Ω(t); cf. Lange [24] for a definition in terms of the indicator function for the entire boundary.…”

“…where δ(y ∈ Γ 0,L ) denotes the surface-Dirac-delta distribution [24] for the stationary end-caps {0/L} × Γ 0/L . On the basis of the geometry of the problem [40,41,42], we take the external sound stimulus (see [4,5,6,46] Experimental methods for adjusting the angular frequency ω and the wavevector's x-component k have been discussed in the literature [4,5,6,46].…”