Path Integral Solution of the Dirichlet Problem

Path integral solution of linear second order partial differential equations II: elliptic, parabolic, and hyperbolic cases

“…9 Moreover, there is only one critical path since r a = 0 excludes the path that passes throught the origin and intersects the boundary transversally. This yields 7 Incidently, this example shows that, for the n-ball, τ ⊥ xa T + a = (R 2 − x 2 a )/2 . 8 For comparison purposes, this is more convenient than solving (3.58) directly (which entails finding a suitable normalization constant N after localizing the integral over T + a ).…”

“…As previously mentioned, the Fourier/Laplace transform interpretation becomes a Lagrange multiplier interpretation for phase space constructions. That is, (2.9) can be used ( [7]) to give a phase space fixed-energy Green's function.…”

“…Remark : In [7], the integrator Dτ was chosen to be Gaussian, which in hindsight was not a good choice. However, comparing [7] with the present construction shows that the theorem in [7] still holds provided Dτ is a gamma integrator.…”

“…Remark : In [7], the integrator Dτ was chosen to be Gaussian, which in hindsight was not a good choice. However, comparing [7] with the present construction shows that the theorem in [7] still holds provided Dτ is a gamma integrator. In light of Corollary 2.1, the path integral in the theorem in [7] solves the inhomogeneous elliptic PDE with vanishing boundary conditions at infinity.…”

Path integral solution of linear second order partial differential equations I: the general construction

“…Note that (2.11) and (2.13) may be quite restrictive and F R (X) may be severely limited or even empty. 6 …”