Insight of the Green's function as a defect state in a boundary value problem

TL;DR

This paper presents a novel interpretation of Green's functions as defect states in boundary value problems, linking them to topological edge states and providing new insights into their properties in open, non-Hermitian systems.

Contribution

It introduces a new perspective viewing Green's functions as defect states, connecting boundary value problems with topological phenomena and edge states.

Findings

Green's function can be seen as an eigenstate in an auxiliary problem.

The approach applies to open, non-Hermitian Helmholtz systems.

Analogies with topological edge states provide new insights.

Abstract

A new perspective of the Green's function in a boundary value problem as the only eigenstate in an auxiliary formulation is introduced. In this treatment, the Green's function can be perceived as a defect state in the presence of a $\delta$-function potential, the height of which depends on the Green's function itself. This approach is illustrated in one-dimensional and two-dimensional Helmholtz equation problems, with an emphasis on systems that are open and have a non-Hermitian potential. We then draw an analogy between the Green's function obtained this way and a chiral edge state circumventing a defect in a topological lattice, which shines light on the local minimum of the Green's function at the source position.