Recursion formula for the Green's function of a Hamiltonian for several types of Dirac delta-function potentials in curved spaces

In this short article, we non-perturbatively derive a recursive formula for the Green's function associated with finitely many point Dirac delta potentials in one dimension. We also extend this formula to the case for the Dirac delta potentials supported by regular curves embedded in two dimensional manifolds and for the Dirac delta potentials supported by two dimensional compact manifolds embedded in three dimensional manifolds. Finally, this formulation allows us to find the recursive formula of the Green's function for the point Dirac delta potentials in two and three dimensional Riemannian manifolds, where the renormalization of coupling constant is required.