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

TL;DR

This paper derives a recursive formula for the Green's function of Hamiltonians with Dirac delta potentials in various curved spaces, extending previous results to higher dimensions and curved geometries.

Contribution

It introduces a non-perturbative recursive formula for Green's functions with delta potentials in curved spaces across multiple dimensions.

Findings

Recursive formula for 1D delta potentials derived

Extension to delta potentials on curves and manifolds in 2D and 3D

Renormalization of coupling constants in higher dimensions

Abstract

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.