The resolvent kernel for PCF self-similar fractals

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For the Laplacian Δ defined on a p.c.f. self-similar fractal, we give an explicit formula for the resolvent kernel of the Laplacian with Dirichlet boundary conditions, and also with Neumann boundary conditions. That is, we construct a symmetric function G⁽λ⁾ which solves (λI - Δ)⁻¹ f(x) = ∫ G⁽λ⁾(x,y) f(y) dμ(y). The method is similar to Kigami's construction of the Green kernel in [3.5]Kig01 and is expressed as a sum of scaled and "translated" copies of a certain function ψ⁽λ⁾ which may be considered as a fundamental solution of the resolvent equation. Examples of the explicit resolvent kernel formula are given for the unit interval, standard Sierpinski gasket, and the level-3 Sierpinski gasket SG₃.